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arXiv:physics/0101085v1 [physics.class-ph] 24 Jan 2001Equations of Motion of Systems with
Internal Angular Momentum
Manuel Dorado∗
CENITA, S.A., Miguel Yuste, 12, 28037 Madrid, Spain
Abstract
Using the Euler’s equations and the Hamiltonian formulatio n, an attempt has been made to obtain the equations
of motion of systems with internal angular momentum that are moving with respect to a reference frame
when subjected to an interaction. This interaction involve s the application of a torque that is permanently
perpendicular to the internal angular momentum vector.
INTRODUCTION
The angular momentum of a many-particle sys-
tem with respect to its centre of mass is known as
the ”internal angular momentum” and is a prop-
erty of the system that is independent of the ob-
server. Internal angular momentum is therefore
and attribute that characterizes a system in the
same way as its mass or charge. In the case of a
rigid body and particularly in the case of an ele-
mentary particle, the internal angular momentum
is also referred to as ”spin”.
To the author’s knowledge, the dynamical be-
haviour of systems with internal angular momen-
tum has not been systematically studied within the
framework of classical mechanics (See References
[1]). This paper approaches this type of problem
through a model comprising a rotating cylinder,
with constant velocity of rotation, around its longi-
tudinal axis. The equations of motion are obtained
using the Euler’s equations and by the Hamiltonian
procedure.
Results coincide when the problem is solved us-
ing vectorial algebra or Lagrangian formalism (un-
published). However, as a result of changes in ”per-
spective”, each method uncovers new peculiarities
regarding the intimate nature of the system’s be-
haviour.
FORMULATION OF THE PROBLEM
Let us consider a rigid cylindrical solid with internal
angular momentum (with respect to its centre of
∗E-mail: mdorado@cayacea.commass)/vectorL, whose centre of mass moves at a constant
velocity/vector νwith respect to a reference frame which
can be defined as soon as the interaction in the
cylinder starts. It is aimed to obtain the equations
of motion to describe the dynamical behaviour of
the system from the instant that it undergoes an
interaction, by applying a torque /vectorMthat is perma-
nently perpendicular to the internal angular mo-
mentum (See Fig.1).
When solving the problem, the following points
are taken into account:
(a) The cylinder will continue spinning about its
longitudinal axis at a constant angular veloc-
ity/vector ωthroughout all the movement. In other
words, its internal angular momentum mod-
ule is constant. The energy that the cylinder
possesses as a result of its rotation about its
longitudinal axis is consequently considered to
be internal and does not interfere with its dy-
namical behaviour.
(b) The derivative of the internal angular momen-
tum,/vectorL, with respect to a frame of axes of iner-
tial reference ( X,Y,Z) satisfies the equation
/parenleftBigg
d/vectorL
dt/parenrightBigg
XY Z=/parenleftBigg
d/vectorL
dt/parenrightBigg
X′Y′Z′+/vectorΩ×/vectorL(1)
in which/vectorΩ is the rotation velocity of the frame
linked to the solid ( X′,Y′,Z′) about the frame
of inertial reference axes ( X,Y,Z).
(c) Only infinitesimal motions are considered that
are compatible with the system’s configura-
tional variations brought about by the applied
torque.(d) The total energy of the system is not explicitly
dependent on time.
(e) No term is included that refers to the potential
energy. According to hypothesis, the applied
torque is a null force acting on the system and
the possibility of including a potential from
which the applied torque derives is unknown.
Figure 1. Formulation of the problem
HOW TO APPROACH THESE PROB-
LEMS THROUGH THE EULER’S EQUA-
TIONS?
The figure 1 shows a cylindrical rigid body with
angular momentum, /vectorL, about its longest axis. We
should consider two referential frames: one associ-
ated with the solid ( X′,Y′,Z′), having the Z′axis
on the direction of the body’s longest axis. We will
refer this system of coordinates to a inertial frame
(X,Y,Z).
In a given instant, a torque /vectorMis applied on the
rigid body.
We call/vectorΩ to the rotational velocity of the sys-
tem of coordinates associated with the solid ( X′,
Y′,Z′), viewed from the inertial frame ( X,Y,Z).
From here, we will consider the system of coordi-
nates (X′,Y′,Z′) all along the work.
Assuming this situation, it is known that
/parenleftBigg
d/vectorL
dt/parenrightBigg
XY Z=/parenleftBigg
d/vectorL
dt/parenrightBigg
X′Y′Z′+/vectorΩ×/vectorL(2)
It is clear that we have two different velocities
acting:(a)/vector ω: the rotational velocity of the body around
its longest axis (let us call it the intrinsic ro-
tational velocity of the body).
(b)/vectorΩ: the rotational velocity of the system of co-
ordinates associated with the body ( X′,Y′,
Z′) viewed from the frame of inertia ( X,Y,
Z).
We can write /vectorLand/vectorM, refered to ( X′,Y′,Z′),
as follows:
/vectorL=I1ω1/vector e1+I2ω2/vector e2+I3ω3/vector e3 (3)
/vectorΩ = Ω 1/vector e1+ Ω2/vector e2+ Ω3/vector e3 (4)
/vectorM=M1/vector e1+M2/vector e2+M3/vector e3 (5)
So, the equation (2) takes the expression
I1˙ω1+I3ω3Ω2−I2ω2Ω3=M1
I2˙ω2+I1ω1Ω3−I3ω3Ω1=M2 (6)
I3˙ω3+I2ω2Ω1−I1ω1Ω2=M3
These equations are known as modified Euler’s
equations and can also be found in (1.e).
In the case treated above, we must substitute the
angular momentum
/vectorL=I3ω3/vector e3 (7)
in (6):
I3ω3Ω2=M1
I3ω3Ω1=M2 (8)
I3˙ω3=M3
And then we get
Ω2=M1
I3ω3
Ω1=M2
I3ω3(9)
˙ω3=M3
I3
If the torque is on the axis Y′,M1= 0,M3= 0
and:
Ω2= 0
Ω1=M2
I3ω3(10)
˙ω3= 0Where Ω 1coincides with the known velocity of
precession.
We have obtained that the body moves along a
circular trajectory with a rotational velocity Ω 1,
but this is not enough to determinate the radius of
the orbit.
The principle of conservation for the energy
should lead us to the expression for this radius.
Assuming our initial hypothesis, the forces are
applied perpendicular to the plane of movement of
the solid and so, these forces do not produce any
work while the cylinder is moving.
We have two conditions that have to be satisfied
by the body while rotating:
(a) Ω =M
L
(b) The total energy should remain constant.
Before the torque is applied the energy of the
cylinder takes the following expression:
E=1
2mν2
o+1
2Iω2
0 (11)
When the torque is acting on the cylinder about
the line defined by the unit vector /vector e2, it rotates
along a trajectory, generally defined by r(t) and
˙θ(t) that, in this case, coincides with Ω( t) and its
energy takes the following expression:
E=1
2m/parenleftBig
˙r2+r2˙θ2/parenrightBig
+1
2Iω2
0 (12)
In this particular case our axes are principal axes
of inertia, therefore we haveδLX′
δt=M1, and then,
δ
δt(mr2˙θ) = 0.
And therefore mr2˙θ=const.
In the particular case in which ˙θ= Ω is a con-
stant,rhas to be a constant (that is, ˙ rhas to be
zero) in order to satisfy this equation. We can con-
clude that the body moves in a circular orbit. The
energy takes the following expression:
E=1
2mr2Ω2+1
2Iω2
0 (13)
Both expressions (11) and (13) have to represent
the same energy. Therefore, comparing the two ex-
pressions we can obtain the radius of the orbit:
r=ν0
Ω(14)Looking at the results arising from the previous
study, we can say that the body rotates following
a circular orbit, with a radius given by the last
expression. This circular trajectory implies that
the velocity vector must precess jointly with the
intrinsic angular momentum vector. From this last
conclusion it can be proved that the force needed
to cause the particle to draw a circular trajectory
is expressed as:
/vectorF=m/vector ν×/vectorΩ (15)
When the internal angular moment of the system
does not coincide with one of the principal axes of
inertia, the treatment of the problem is much more
complex, but as it will be seen in the next section,
the results given above area completely general.
HAMILTONIAN FORMULATION.
EQUATIONS OF MOTION AND THEIR
SOLUTION
The equations of motion are obtained by the Hamil-
tonian formulation, where the independent vari-
ables are the generalized coordinates and moments.
To do this, a basis change of the frame ( q, ˙q,t) to
(p,q,t) is made using the Legendre transformation.
From the function
H(p,q,t) =/summationdisplay
i˙qipi−L(q,˙q,t) (16)
a system of 2 n+ 1 equations is obtained.
˙qi=∂H
∂t; −˙pi=∂H
∂qi;
(17)
−∂L
∂t=∂H
∂t
If the last equation in (17) is excluded, a system
of 2nfirst order equations is obtained, known as
canonical Hamilton equations.
By substituting H, the first order equations of
motion, are obtained.
It should be noted that the Hamiltonian formula-
tion is developed for holonomous systems and forces
derived from a potential that depends on the po-
sition or from generalized potentials. A torque is
applied to the present system. The result of the
forces is zero on the centre of mass and, therefore,
it is meaningless to refer to potential energy.On the other hand, the Hamiltonian function
concept does remain meaningful.
By using polar coordinate in the movement
plane,
νr= ˙r;vθ=r˙θ (18)
The Lagrangian is
L=1
2m(˙r2+r2˙θ2) (19)
The generalized moments are
Pr=m˙r;Pθ=mr2˙θ (20)
so that
˙r=Pr
m; ˙θ=Pθ
mr2(21)
The Hamiltonian His introduced using the equa-
tion
H(p,q,t) =/summationdisplay
i˙qipi−L(q,˙q,t) (22)
resulting in
H=Pr˙r+Pθ˙θ−/parenleftbigg1
2m˙r2+1
2mr2˙θ2/parenrightbigg
If the generalized velocities are substituted by
the generalized moments, the following is obtained
H=P2
r
2m+P2
θ
2mr2(23)
The first pair of Hamilton equations is
˙r=∂H
∂Pr=Pr
m; ˙θ=∂H
∂Pθ=Pθ
mr2
The second pair of equations is
−˙Pr=∂H
∂r=−P2
θ
mr3; −˙Pθ=∂H
∂θ= 0
The second of these equations shows that the an-
gular momentum Jis conserved.
Pθ=J=const. (24)
The first gives the radical equation of motion,
˙Pr=m¨r=J2
mr3(25)
asJ=Pθ=mr2˙θ, by substituting it results that
˙Pr=m2r4˙θ2
mr3=mr˙θ2(26)The term −∂V
∂rnormally appears in this radial
equation of motion and represents the force derived
from a potential. In other words,
m¨r=mr˙θ2−∂V
∂r(27)
In the present case, this term does not exist.
To complete one’s knowledge of the system’s de-
velopment, those relations derived from the con-
straints, have to be resorted to.
1.E=1
2mν2=const. ⇒ |ν|=const
2./parenleftBig
d/vectorL
dt/parenrightBig
XY Z=/parenleftBig
d/vectorL
dt/parenrightBig
X′Y′Z′+/vectorΩ×/vectorL
where the applied external torque
/vectorM=/parenleftBigg
d/vectorL
dt/parenrightBigg
XY Z(28)
by hypothesis it is known that/parenleftBig
d/vectorL
dt/parenrightBig
X′Y′Z′=/vector0 and from (1) it is concluded that
/parenleftBig
d/vectorL
dt/parenrightBig
X′Y′Z′=/vectorΩ×/vectorL
and it is obtained that Ω =M
L.
This vector represents the rotation velocity of the
frame of axes linked to the cylinder ( X′,Y′,Z′)
about the frame of inertial references axes ( X,Y,
Z).
In polar coordinates, the variable defining the
rotation about the frame of axes is ˙θand it is con-
cluded that
˙θ= Ω =M
L(29)
substituting in the radial equation of motion, re-
sults that
m¨r=mr˙θ2=mrΩ2(30)
Moreover,Pθ=const. and at the initial moment
equals
Pθ=mr2˙θ=mrν (31)
asνis constant, it is concluded that ris also
constant. Finally it is found that
r=ν
Ωandm¨r=mνΩ (32)Further considerations
The Hamiltonian is independent of θ. This is an
expression of the system’s rotation symmetry or,
in other words, there is no preferred alignment in
the plane.
The equation∂H
∂θ= 0 means that the energy of
the system remains invariable if it is turned to a
new position without changing r,ProrPθ. This
aspect is clearly proved.
Worthy of mention in this case is the fact that
the Hamiltonian equations do not provide, in this
type of problem, complete information on the ra-
dial movement. The reason for this is that the cen-
tral force is a function of θwhich is a coordinate
that does not appear in the Hamiltonian (it can
be ignored) and can only be calculated using the
constraints.
This proposal is valid for any property of the
particle that displays vectorial characteristics and
is sensitive to an external torque of the type de-
scribed, without the need to ascertain the real phys-
ical nature of that property. The proposal can be
extended to equivalent problems, despite their not
involving the presence of internal angular momen-
tum as a central characteristic.
Poisson brackets. The Poisson bracket [ Pθ,H] is
[Pθ,H] =/bracketleftbigg∂Pθ
∂θ∂H
∂Pθ−∂Pθ
∂Pθ∂H
∂θ/bracketrightbigg
=−∂H
∂θ
as∂H
∂θ= 0, it is concluded that [ Pθ,H] = 0.
CHARGED PARTICLE IN A MAGNETIC
FIELD
Let us apply this theory to the specific case of a
charged particle with spin /vector sand magnetic momen-
tum/vector µ, moving at a velocity /vector ν0in a uniform mag-
netic field of flux density /vectorB.
As it’s well known, the interaction between /vectorB
and/vector µmakes a torque to act upon the particle. This
situation verifies every condition we have defined in
our hypothesis, and so, the behaviour of the particle
must be in agreement with the theoretical model we
have developed above.
Considering the most general of the cases, let us
suppose that /vector µforms unknown angles θwith/vectorB
andαwith/vector ν0. This situation is represented in the
following figure (See Fig. 2).
Figure 2. Charged particle in a magnetic field.
It is clear that the rotation of /vector ν0causes rotation
of the/vector ν0component on the XYplane,ν0xy. Fur-
thermore, according to the theory here presented,
the particle will trace a helix and the /vector ν0compo-
nent in the direction of the Zaxis,/vector ν0z, will not be
affected.
In this case the angular momentum /vectorL=/vector s.
If the particle stops inside /vectorB, the rotational
velocity of the angular momentum would be ex-
pressed by:
Ω =|/vector µ×/vectorB|
|/vectorL|=|/vectorM|
|/vectorL|(33)
Obviously, if the angle αformed by /vector sand/vector ν0
remains constant, the angle γformed by /vector sxyand
/vector ν0xywill also remain constant. The velocity of /vector νaxy
can be calculated merely by calculating the velocity
of/vector sxy.
sxy=ssinθ (34)
and
Ων0xy= Ωs0xy=|/vector µ||/vectorB|sinθ
Lsinθ=µB
L(35)
We can conclude from this expression that the
angular velocity of the rotation of the particle de-
pends on the magnetic momentum /vector µ, the magnetic
field/vectorBin which the particle is immersed and the
spin/vector sof the particle, and it is independent of the
relative positions of /vector µ,/vectorBand/vector ν0.
In the case in question, the angular velocity of
the rotation of the particle in /vectorBis expressed by:
/vectorΩ =γe
2m/vectorB (36)According to the theory developed above, the
particle will draw a circular trajectory. We can
determine the radius of the orbit drawn by the elec-
tron under the influence of a magnetic field
r=ν0
Ω=2mν0
γeB(37)
In the specific case in which the particle is an
electron,γhas a value of 2 and the radius of the
orbit is calculated from the following equation:
r=ν0
Ω=mν0
eB(38)
From the conclusions summarized above, we
state that the particle is submitted to a central
force affecting the charged particle inside the mag-
netic field. To calculate its value, we only need to
recall the formula:
/vectorF=m/vector ν0×/vectorΩ (39)
and substitute /vectorΩ for its value previously obtained.
Then we get:
/vectorF=m/vector ν0×γe
2m/vectorB (40)
Once again, if we apply this to the case in which
the particle is an electron, the gyromagnetic factor
is 2, and hence
/vectorF=e/vector ν0×/vectorB (41)
The behaviour predicted by the theory here
shown of a charged spinning particle penetrating
into a magnetic field with velocity ν, matches the
one that can be observed in the laboratory, and
the force to which it will be submitted is the well
known Lorentz force.
EXPERIMENTS CARRIED OUT
Airmodel with spinning disc
The airmodel is provided of a strong angular mo-
mentum by mean of a high speed spinning disc at-
tached to it by exerting different torques on it, with
the built in flight systems, we achieve modifications
in its flight trajectory.
The remote control flying model was designed
specifically to verify this theory (See Fig. 3).
Figure 3. Airmodel with spinning disc
Basically it consists of a disk that can rotate,
attached on top of the flying model, which provides
the necessary support and speed.
Manoeuvring the flying system, aileron to turn
right and left, elevators to shift upwards and down-
wards and rudder, we can obtain the adequate
torques to affect this motion.
The flying model is controlled by means of a radio
equipment affecting the normal moving elements
and the rotation of the disk.
The fist flight was performed without the disk to
confirm the normal evolution of the model.
Once the normal behaviour of the model in this
conditions was confirmed, the disk was attached on
top of the model.
With the disk still the system behaves in a sim-
ilar way that it did without the disk, although the
stability was slightly affected.
With the disk rotation pointer clock way when we
actuate the ailerons to the left, instead of drawing
a horizontal circumference, it actually goes up even
to the point of performing a loop.
Actuating the elevators upwards causes the
model to turn right, when we actuate the elevators
downwards the model turns to the left.Spinning top with double suspension
assembly
In this experiment it is proved that the spinning
top cam precess with a non zero radius.
It can be observed as well its extraordinary sen-
sitivity to the torques applied on it, when these
torques have the same direction of the vertical axis,
by modifying the angle between the symmetry axis
of the spinning disc and the earths plane.
For this experiment, a top is mounted with dou-
ble suspension (See Fig. 4) whereby it can behave
according to the general theory or adopt the type
of motion predicted by the classical treatment.
Figure 4. Top mounted with double suspension.
The top is allowed four possible degrees of rota-
tional freedom:
a)AA′axis: The top rotates about its axis of
symmetry.
b)BB′axis: The top will rotate about this axis
if nutation is caused.
c)CC′axis: The top will rotate about this axis
if there is precession.
d)DD′axis: The system will rotate about this
axis if it behaves according to this theory when
we fixed the CC′axis to the support.
The gravitational torque follows the direction of
theBB′axis and it produces an increase in angular
momentum ∆ /vectorLwhich is vectorially added to the
angular momentum /vectorLof the top, which is in the
direction of the AA′axis.Logically, the top and both the AA′andBB′
axes could occupy any position with respect to the
support frame.
We provide the top with a constant slow velocity,
/vector ν0. Due the top being mounted on the support
frame, the system will rotate about the DD′axis.
As/vector ν0is slow, the centripetal acceleration due to
rotation can be considered negligible.
According to the classical explanation, the top
should precess and even achieve a nutation motion
on the support, with an independent slow rotation
motion of the system about the DD′axis.
The general theory establishes that the velocity
of the top, /vector ν0, the orbital radius and the preces-
sion velocity of the top are related by the following
expression:
R=ν
Ω.
Surprisingly this means that if Ω and Rare con-
stant (the support frame is rigid) there would be
only one velocity /vector ν0for the movement of the top
and a single rotation velocity for the top-support
system about the DD′axis, independently of the
impulse given to the support in an attempt to
achieve the desired velocity.
For our experiment, we create a rotation veloc-
ity for the top about its axis of symmetry, this axis
(AA′) being maintained initially in a horizontal po-
sition although it can really be in any position.
When the top has reached a high rotation veloc-
ity about its axis of symmetry, the support is given
an impulse so that the top reaches a velocity /vector νand
the system is then left to evolve freely. For now
on, it is also under the influence of a gravitational
torque.
If the system behaves according to classical the-
ory, precession of the top should not occur with
respect to the support.
If on the other hand its behaviour is that de-
scribed by this theory, precession should occur with
respect to the DD′axis.
The results of the experiment confirm that the
top, in agreement with the theory, move about the
DD′axis with a velocity
ν0=R×Ω0
and is independent of the impulse it may have re-
ceived.
In fact the following situations can occur:a) That the impulse received may be bigger than
that required to reach a velocity /vector ν0. The ve-
locity of movement of the top will in all cases
be/vector ν0. However, the angle formed by the axis
of symmetry ( AA′) and the vertical axis ( CC′)
decreases and there is an increase in potential
energy
It should be remembered that /vectorΩ is constant and
independent of the angle formed by the AA′and
CC′axes while/vectorLremains constant.
b) That the impulse received is equal to that re-
quired to reach a velocity /vector ν0. In this case the
position of the top does not vary with respect
to the support. The system rotates about the
DD′axis.
c) That the impulse received is less than that re-
quired to reach a velocity /vector ν0. In this case the
system continues to move with a velocity /vector ν0,
but the angle formed by the AA′and theCC′
axes increases.
Pendulum with internal angular mo-
mentum
Two experiments are performed with this device:
a) Measure the curvature of the trajectory at the
very first instants of its evolution, where we
assess that it is close to a circular trajectory.
b) Check that the final trajectory is an ellipti-
cal one in which the perihelio of the orbit ad-
vances.
This experiment is performed with a pendulum
that consist of a fibre-glass sphere. The sphere
houses a disk that can rotate around its transversal
axis (See Fig. 5).
The pendulum is suspended from a wire attached
to a point one centimeter distant of the edge of the
disk shaft.
If we observe the evolution of the pendulum when
the inner disk stands still, it is the ordinary one.
We force the disk to rotate until the maximum
speed is reach. At that moment we let the sphere
free. The system is under the influence of a torque
caused by its weight and the wire strain. If we
measure the trajectory at the very first instants of
its evolution, it happens to be a round one.Letting the sphere to move, after while it follows
an elliptical orbit, on which the perihelium shifts in
time, in good agreement with this theory (See Fig.
6).
This phenomenon is similar to the one occurring
in the shift of a planet orbit perihelium. And it
is similar, as well, to the Larmor precession of a
charge in the presence of a magnetic field.
Figure 5. Pendulum with internal angular momentum.
As in the previous experiment, if we reverse the
direction of the disk rotation the orbit perihelium
will also shift in the opposite way that it did before.
Figure 6. Perihelium shift
Theoretical explanation
d/vectorL
dt=/vectorM (42)
/vectorM=/vector r×/vectorF (43)In agreement with this theory, the pendulum is
affected by the force
/vectorF=m/vector ν×/vectorΩ (44)
The torque will be
/vectorM=/vector r×m/vector ν×/vectorΩ =/vectorL×/vectorΩ (45)
(It can be proved that the associative property of
the vectorial product is satisfied in this case)
Substituting (45) in (42) we obtain:
d/vectorL
dt=/vectorL×/vectorΩ (46)
This one is the equation of the motion of a vector
with constant module |/vectorL|, that precesses around
the axis defined by /vectorΩ, with angular velocity |/vectorΩ|.
Larmor Effect (Samples under the influence of mag-
netic fields)
In agreement with this theory, an electron em-
bedded inside an atom in the presence of a mag-
netic field/vectorB, is submitted to the central force
/vectorF=m/vector ν×/vectorΩ (47)
which adopts the expression
/vectorF=m/vector ν×/vectorΩ =m/vector ν×q/vectorB
mc=q
c/vector ν×/vectorB (48)
and is known as the Lorentz Force.
As the most general expression, we can write:
d/vectorL
dt=/vector r×m/vector ν×/vectorΩ =/vectorL×/vectorΩ (49)
(It can be proved that the associative property of
the vectorial product is satisfied in this case)
This is the equation for the motion of /vectorL, which
rotates around the vector /vectorBwith an angular veloc-
ity:
/vectorΩ =−q
mc/vectorB (50)
The general expression of the central force, which
includes the Lorentz force as a particular case, al-
lows us to explain the Larmor effect and its mean-
ing in an easy and accurate way within the range
of a wider and more important phenomenon.Magnetic spinning top
With this experiment it is proved that the direc-
tion way of the precession can be manipulated in
accordance with the torque applied.
For the this experiment we have a magnetized
rod, a aluminium cone and a magnet.
We attach the rod to the cone. We have build
up a magnetized spinning top (See Fig. 7).
Figure 7. Magnetic spinning top.
We observe the behaviour of the magnetized
spinning top in the presence of a magnetic field.
The interaction between both fields produces a
torque which affects the spinning top making him
to draw a circular trajectory.
The spinning top will not collapse on the mag-
neto as long as it has an angular momentum.
EXPERIMENT SUGGESTED TO TEST
THIS THEORY: INTERACTION OF A
HOMOGENEOUS MAGNETIC FIELD
AND A PERPENDICULAR GYRATING
MAGNETIC FIELD, WITH A PARTICLE
WITH SPIN AND MAGNETIC MOMENT
Formulation of the problem
The particle with spin, /vectorS, moves with velocity /vector ν. It
first passes into a homogeneous magnetic field /vectorB′,
inducing the magnetic moment, /vector µ, of the particle,
polarized in the direction of the field. It later enters
a region in which a homogeneous field, /vectorB, and a
gyrating field, /vectorH0, with rotation frequency, ˙φ, are
superposed. Fields /vectorB′and/vectorBare parallel and in
fact could even be the same field, although their
function is different in each region.Reference frame (See Fig. 8).
XYZ Fixed system in the particle but which
does not rotate with this.
/vectorBis on theZaxis.
/vectorH0is on theXYplane.
X′Y′Z′Fixed frame in the particle which pre-
cesses with this.
/vectorSand/vector µare on theZ′axis.
/vector νis on theY′axis.
X0Y0Z0Reference frame from which observa-
tions are made.
/vectorBis on theZ0axis.
/vectorH0is on theX0Y0plane.
Figure 8. Reference frames.
The rotation velocity, /vectorΩ, of the frame of axes
linked to the solid, X′Y′Z′, will be expressed within
this system as:
/vectorΩ = Ω 1/vectori′+ Ω2/vectorj′+ Ω3/vectork′(51)
This rotation velocity, /vectorΩ, can also be expressed
with respect to the frame of reference axes, XYZ
by using Euler angles.
Ω1=˙φsinθsinψ+˙θcosψ
Ω2=˙φsinθsinψ−˙θcosψ (52)
Ω3=˙φcosθ+˙ψ
Our particle can also rotate with respect to the
frame of axes, X′Y′Z′, and the most general ex-
pression to describe this angular velocity is:
/vector ω=ωX′/vectori′+ωY′/vectorj′+ωZ′/vectork′(53)It is proved that ˙ψ=ωZ′. The proof is easy, for if
theZ′axis is fixed, i.e. both φandθare constant,
the rotations about the Z′axis are rotations of the
X′andY′axes in theX′Y′plane. Hence, the Euler
angle that reports the rotation is ψ. It can then be
concluded that
˙ψ=ωZ′ (54)
By agreement, but without loss of generality, let
us consider that the particle has an angular mo-
mentum,/vectorS, that is constant in module.
Frame of axes linked to the solid
In the definition of the frame of axes linked to the
solid, we can either choose:
a) The frame of axes strictly accompanies the
particle in the rotation which, in classical
terms, gives its internal angular momentum.
b) The frame of axes linked to the solid is de-
fined by a parallel axis to the internal angu-
lar momentum of the particle, and the other
two are perpendicular to each other and are
in a plane which is perpendicular to the angu-
lar momentum. Internal angular momentum
is understood to be a quality of the particle
whose mathematical characteristics coincide
with those of the angular momentum, without
considering the physical nature of this quality.
In other words, no hypothesis is formed as to
whether or not the angular momentum implies
rotations of the particle under study.
For the purposes of our problem, definition (b)
has been used.
However, it should be remembered that our par-
ticle can reach a rotation velocity, /vector ω, as a result of
the interactions to which it can be subjected.
Analysis of the interactions
Given the nature of our problem, the particle pen-
etrates a homogeneous magnetic field, /vectorB, inducing
the magnetic moment, /vector µ, of the particle, polarized
in the direction of the field.
Hence, when the particle penetrates the magnetic
field,/vectorB(parallel to /vectorB′, and can even coincide with
it) and/vectorH0, the magnetic moment, /vector µ, is only (ini-
tially) sensitive to the gyrating magnetic field, /vectorH0.This, consequently, will be the first interaction that
we analyze.
Interaction with the magnetic field,
/vectorH0
The interaction of the gyrating magnetic field, /vectorH0,
with the magnetic moment, /vector µ, does not modify the
energy of the system, as both are permanently per-
pendicular.
The energy of the interaction is expressed
EH=/vectorH0·/vector µ=H0µcosπ
2= 0 (55)
The interaction between /vectorH0and/vector µ, is equivalent
to the action of a torque, /vectorΓH, on the particle, where
/vectorΓH=/vector µ×/vectorH0 (56)
/vectorΓHis perpendicular to the X′andZ′axes, and
is located on the Y′axis.
By using the previously obtained Euler equations
(9) and applying them to our problem:
˙ωZ′=M3
I3= 0 = ˙ψ
Ω1=ΓH
S=µH0
S(57)
Ω2=M1
S= 0
so that ˙ψ= 0 in all cases, given that M3= 0 at all
times. We can avoid ψby making it equal to zero
in all case (See Fig. 9).
The Euler equations depending on the Euler an-
gles (52), are simplified with the result that
Ω1=˙θ=µH0
S
Ω2=˙φsinθ= 0 (58)
Ω3=˙φcosθ
from which it is deduced that
Figure 9. Interaction with the magnetic field, /vectorH0.
/vectorθ=µH0
S
˙φ= 0 (59)
Ω3= 0
in other words, the rotation of the frame linked to
the solid on X′Y′Z′, Ω1, is constant and equal to ˙θ,
where ˙θis the rotation of the frame of axes linked
to the solid with regard to XYZ expressed in the
Euler angles.
We have obtained the rotation velocity, ˙θ, but we
have to calculate the rotation radio r(t) if we want
to know the trajectory. To do this we will use the
following equality between differential operators,
d
dt=d∗
dt+/vectorΩ×,
where
d
dtis the derivative in the inertial frame, in our case
X0Y0Z0,
d∗
dtis the derivative in the non-inertial frame, in our
caseX′Y′Z′,
/vectorΩ is the angular rotation velocity of the frame of
axes linked to the solid in relation to the inertial
frame of axes.
We will apply this equality to /vector r, a vector of posi-
tion of one frame with respect to another, and for
convenience we will express all the vectors in the
X′Y′Z′frame.
d∗/vector r
dt=d/vector r
dt−/vectorΩ×/vector r (60)If we represent the equation in components, we
have
˙rX′+rZ′Ω2−rY′Ω3=νX′
˙rY′+rX′Ω3−rZ′Ω1=νY′ (61)
˙rZ′+rY′Ω1−rX′Ω2=νZ′
Taking into account the initial conditions (See
Fig. 10), Ω 2= 0 and Ω 3= 0.
Figure 10. Initial conditions.
The interaction does not modify the energy of the
system, as explained beforehand, so the module of
/vector νhas to remain constant.
|/vector ν|=ν= constant.
and,νX′= 0,νY′=ν,νZ′= 0.
As the energy of the system cannot vary, ˙ rX′=
˙rY′= ˙rZ′= 0. If they differed from zero tangen-
tial accelerations would be involved and therefore
a variation in the total energy of the system.
These conditions simplify the equations of our
system (61) and we obtain
rZ′=−ν
Ω1(62)
This is to say, the particle and the frame of axes
linked to the particle describe a circular trajectory
with respect to the observation frame. The move-
ment plane is perpendicular to /vectorH0.Interaction with the magnetic field /vectorB
At the moment the circular trajectory is com-
menced,θis no longer null and the interaction with
the field,/vectorB, begins (See Fig. 11).
/vectorΓ =/vector µ×/vectorB (63)
Ω2=µBsinθ
S(64)
Figure 11. Interaction with the magnetic field.
M3continues to be null and, therefor, so do ˙ψ
andψ, which do not vary, but Ω 2is now different
from zero.
Ω1=˙θ
Ω2=˙φsinθ=µBsinθ
S(65)
Ω3=˙φcosθ
We finally obtain,
˙θ=µH0
S
˙φ=µB
S(66)
Ω3=µBcosθ
S
These are the expressions of the rotation of the
frame of axes linked to the solid in the base that
defines the trihedral linked to the solid.
We can also express this same rotation velocity of
the frame of axes linked to the solid with respect to
the reference frame, by means of the Euler angles,but in the base defining the trihedral of reference
frame.
ωX=˙θcosφ+˙ψsinθsinφ
ωY=˙θsinφ−˙ψsinθcosφ (67)
ωZ=ψcosθ+˙φ
As˙ψ=ψ= 0, we obtain
ωX=˙θcosφ
ωY=˙θsinφ (68)
ωZ=˙φ
These expressions clearly reflect the behaviour of
the frame of axes linked to the solid and of the solid,
with respect to the reference axes.
Visualization of the movement is assisted by con-
sidering the following specific situations ˙φ= 0
i) Forφ= 0
ωX=˙θ
ωY= 0
ωZ= 0
We would have a rotation about the Xaxis.
j) Forφ=π
2
ωX= 0
ωY=˙θ
ωZ= 0
We would have a rotation about the Yaxis.
The rotation velocity in the XYplane is
/vector ωXY=˙θ(cosθ/vectori+ sinθ/vectorj)
(69)
|/vector ωXY|=˙θ
Therefore, the total movement is a circular tra-
jectory of radius, rZ′, perpendicular to the field,
/vectorH0, which in turn spins about the field, /vectorB, with
velocity ˙φ(classically know as Larmor rotation fre-
quency). This explains why the frequency of the
oscillating field, /vectorH0, has to coincide with the Lar-
mor frequency.The behaviour of the particle indicates that it is
subjected to two central forces
/vectorF1=m/vector ν×/vectorΩ1=m/vector ν×˙θ/vectori′(70)
/vectorF2=m/vector ν×˙φ/vectork (71)
From an energy point of view, we can confirm
that the magnetic field, /vectorH0, does not modify the
energy of the system, and that the magnetic field,
/vectorB, cyclically modifies the potential energy of the
particle, since
EB=/vectorB·/vector µcosθ
(72)
withθ(t) =µH0
St
but with its kinetic energy unvaried.
Additional considerations
The reader will have already noted the similarity
between this behaviour and the Larmor effect.
It is know that this effect is produced by applying
a magnetic field, /vectorB, to a particle of charge, q, mov-
ing in an orbit around a fixed specific charge, q′.
The result is a precession of the trajectory around
the direction of the applied magnetic field, with a
precession velocity, ωL=qB
2m, know as Larmor fre-
quency, with the proviso that the cyclotronic fre-
quency is directly obtained by this procedure.
This statement can be expressed by saying that
the angular momentum vector of the particle with
respect to the rotation axis, referred to as orbital
angular momentum, precesses about the direction
of the magnetic field, /vectorB.
From our procedure, it is particularly easy to
reach the same conclusions by considering that
d/vectorL
dt=/vectorΓB=/vector r×/vectorF
(73)
/vectorL=/vector r×m/vector ν
By substituting our development, /vectorF=m/vector ν×/vectorΩ,
in (73) we obtain
d/vectorL
dt=/vector r×(m/vector ν×/vectorΩ)
(74)
= (/vector r×m/vector ν)×/vectorΩ =/vectorL×/vectorΩFigure 12. Rabi’s experiment.
Figure 13. Experiment suggested.
(the triple vector product does not generally
fulfill the associative property, but it can be de-
mostrated in our.)
Equation (74) describes the angular momentum
precession about the magnetic field with a preces-
sion rate/vectorΩ.
The Larmor effect results from subjecting the
particle to the Lorentz central force, because it is
within/vectorB. From the following, the Lorentz force is
equivalent to our force, given that
F=mνΩ =mνµB
S=mνe
mB=eνB (75)
Which demostrates that both movements are
equivalent.Experimental conditions
This experiment can be carried out by emulating
the rotating magnetic field using a radio frequency
magnetic field, just as Rabi did in his experiments.
Rabi’s experiment (see References (2)) consists
of a collimated particle beam that crosses an inho-
mogeneous magnetic field. It later passes through a
region where a homogeneous and a radio frequency
magnetic field are superposed, and finally passes
through an inhomogeneous field that refocuses the
beam towards the detector.
The inhomogeneous fields separate the beam into
different beams according to their magnetic mo-
ment (the dependence this on the spin) as in anexperiment of Stern-Gerlach. When leaving the
first field, these beams later pass into the second
inhomogeneous field, which refocuses the beams to
the detector. By adjusting the second inhomoge-
neous magnetic field we will obtain refocusing con-
ditions for a beam or group of beams.
In the central part of the experimental arrange-
ment, the homogeneous magnetic field is super-
posed on the radio frequency field. In this region,
the spin and magnetic moment are deflected with
respect to the constant field when the oscillating
field frequency approaches the Larmor precession
frequency. This process is know as nuclear mag-
netic resonance.
After deflection, the atom is on another level
which will not fulfill the refocusing condition and
a decrease in the beam intensity will be observed
in the detector. This procedure is used to study
nuclear spin, nuclear magnetic moments and hy-
perfine structures. However, the behaviour of the
particles in Rabi’s experiment is theoretically jus-
tified in our exposition and it is therefore unnec-
essary to use inhomogeneous magnetic fields (as is
required in Rabi’s experiments).
Our experiment only requires the presence of a
homogeneous magnetic field to induce the magnetic
moment,/vector µ, of the particle, polarized in the direc-
tion of the field.
In such a way that:
1oOnly those particles whose Larmor frequency,
˙φ, coincides with the radio frequency will
abandon the rectilinear trajectory.
2oThe trajectory described by these particles
should coincide with that of the theoretical so-
lution given in this article while they remain
within the region in which the two fields co-
exist. According to this solution, the parti-
cle abandons the rectilinear path and the XY
plane.
3oWhen the particle abandons the region of co-
existing fields, it will follow a rectilinear move-
ment but will not reach the detector.
4oThis procedure facilitates selection of the dif-
ferent angular moments because of their de-
pendence on the Larmor precession.CONCLUSIONS
When a system with internal angular momentum,
moving at constant velocity with respect to a ref-
erence frame, is subjected to an interaction of the
type under study, i.e., a torque perpendicular to the
internal angular momentum vector, it will begin to
trace a circular path of radius
r=ν
Ω
where Ω =M
L
Its behaviour is equivalent to that produced by
subjecting the cylinder to a central force
/vectorF=m/vector ν×/vectorΩ
As it can be seen in figure 14, the trajectory
which particle follows is the trajectory II, while the
trajectory I is the intuitive one, but as it has been
shown in this paper it is not the real behaviour of
the particle.
ACKNOWLEDGMENTS
Through the development of this paper I have had
the privilege of maintaining some discussions about
its content with Dr. Jos´ e L. S´ anchez G´ omez.
It will also like to thank Miguel Morales Furi´ o
for his support and help to the development of this
article.
In the same way, I will like to express my grat-
itude to Juan Silva Trigo, Jes´ us Abell´ an, Berto
Gonz´ alez Carrera and to all the other people who
has worked in the design and realization of all the
experiments describe in this article, whose enumer-
ation would be too extensive.
I would also like to express that the responsabil-
ity for any affirmation here introduced lies exclu-
sively with the author.Figure 14. Classical trajectory and trajectory predicted i n this article.
REFERENCES
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SANDS, M. (1964): The Feynman Lectures on
Physics. Addison-Wesley.
[1.b ] GOLDSTEIN, H. G. (1970): Classical Mechan-
ics.Addison-Wesley.
[1.c ] KIBBLE, T. W. B. (1966): Classical Mechanics.
Mc. Graw-Hill.
[1.d ] LANDAU AND LIFSHITZ (1978): Mec´ anica.
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[1.e ] RAADA, A. (1990): Din´ amica Cl´ asica. Alianza
Universidad Textos.
[2.a ] KELLOGG, J. M. B., RABI, I. I. AND
ZACHARIAS, J. R. (1936) The Gyromagnetic
Properties of the Hydrogenes. Physical Review
Vol.50, 472.
[2.b ] RABI, I. I., ZACHARIAS, J. R., MILLMAN,
S. AND KUSCH, P. (1938) A New Method ofMeasuring Nuclear Magnetic Moment. Physical
Review Vol.53, 318.
[2.c ] RABI, I. I., ZACHARIAS, J. R., MILLMAN, S.
AND KUSCH, P. (1939) The Magnetic Moments
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526.
[2.d ] RABI, I. I., ZACHARIAS, J. R., MILLMAN,
S. AND KUSCH, P. (1938) The Molecular Beam
Resonance Method for Measuring Nuclear Mag-
netic Moments.The Magnetic Moments of 3Li6,
3Li7and9F19.Physical Review Vol.53, 495.
[2.e ] G ¨UTTINGER, P. (1932) Das Verhalten von
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[2.f ] MAJORANA, E. (1932) Atomi Orientatti in
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[2.g ] RABI, I. I. (1936) On the Process of Space Quan-
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Quantization in Time Varying Magnetic Fields.
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652. |
arXiv:physics/0101086v1 [physics.atom-ph] 24 Jan 2001Alternative Fourier Expansions for Inverse Square Law Forc es
Howard S. Cohl
Logicon, Inc., Naval Oceanographic Office Major Shared Resou rce Center Programming Environment & Training,
NASA John C. Stennis Space Center, MS, 39529
A. R. P. Rau, Joel E. Tohline, Dana A. Browne, John E. Cazes, an d Eric I. Barnes
Department of Physics and Astronomy, Louisiana State Unive rsity, Baton Rouge, LA, 70803
(September 21, 2013)
Few-body problems involving Coulomb or gravitational inte ractions between pairs of particles,
whether in classical or quantum physics, are generally hand led through a standard multipole ex-
pansion of the two-body potentials. We discuss an alternati ve based on a compact, cylindrical
Green’s function expansion that should have wide applicabi lity throughout physics. Two-electron
“direct” and “exchange” integrals in many-electron quantu m systems are evaluated to illustrate
the procedure which is more compact than the standard one usi ng Wigner coefficients and Slater
integrals.
I. INTRODUCTION
For pairwise Coulomb or gravitational potentials, one ofte n expands the inverse distance between two points xand
x′in the standard multipole form [1]
1
|x−x′|=1√
rr′∞/summationdisplay
ℓ=0/parenleftbiggr<
r>/parenrightbiggℓ+1
2
Pℓ(cosγ), (1)
wherer<(r>) is the smaller (larger) of the spherical distances randr′, andPℓ(cosγ) is the Legendre polynomial [2]
with argument
cosγ≡ˆ x·ˆ x′= cosθcosθ′+ sinθsinθ′cos(φ−φ′). (2)
In the “body frame,” separation distances within the set com posed of two points and the origin are characterized
by three variables, one choice being the above triad ( r<,r>,γ). With respect to a space-fixed “laboratory frame,”
three more angles constitute the full set of six coordinates , the choice in Eq. (2) of ( θ,θ′,φ−φ′) being suited to the
spherical polar coordinates of the individual vectors; thu s,x: (rsinθcosφ,rsinθsinφ,rcosθ).
The multipole expansion in spherical polar coordinates is a lmost universal because we treat particles or charges as
points. Thus, in the three-body problem, once the motion of t he center of mass is separated, we are left with two
vectors which may be described as in the above paragraph, the potential energy and thereby dynamics in the body
frame being a function of the three dynamical variables ( r<,r>,γ). In this paper, we present an alternative expansion
to Eq. (1) based on cylindrical (azimuthal) symmetry which m ay be of wide interest in physics and astrophysics.
As an illustration, we evaluate two-electron integrals exp ressing the “direct” and “exchange” components of the
electron-electron repulsion in atoms.
The expansion in Eq. (1) disentangles the dynamics containe d in the radial variables from symmetries, particu-
larly under rotations and reflections, pertaining to the ang leγ. Whereas the three variables at this stage are joint
coordinates of xandx′, depending on both and thus characteristic of the three-bod y system as a whole, a further
disentangling in terms of the independent coordinates so as to handle permutational and rotational symmetry aspects
of the problem is often useful and achieved through the addit ion theorem for spherical harmonics [3]. Using this to
replacePℓ(cosγ) in Eq. (1), we obtain the familiar Green’s function multipo le expansion in terms of all six spherical
polar coordinate variables xandx′,
1
|x−x′|=1√
rr′∞/summationdisplay
ℓ=0/parenleftbiggr<
r>/parenrightbiggℓ+1
2ℓ/summationdisplay
m=−ℓΓ(ℓ−m+ 1)
Γ(ℓ+m+ 1)Pm
ℓ(cosθ)Pm
ℓ(cosθ′)eim(φ−φ′), (3)
wherePm
ℓ(z) is the integer-order, integer-degree, associated Legend re function of the first kind [2]. Apart from
the first factor with dimension inverse-distance formed fro m the geometric mean of the two lengths randr′, this
1expression involves only four combinations: ( r</r>,θ,θ′,φ−φ′) of the six coordinates xandx′. This is as it should
be, the separation distance being independent of the orient ation of that separation in the laboratory frame, and thus
independent of two angles serving to specify that orientati on.
This multipole expansion is very broadly utilized across th e physical sciences. With ℓandminterpreted as the
quantum numbers of orbital angular momentum and its azimuth al projection, respectively, a whole technology of
Racah-Wigner or Clebsch-Gordan algebra is available [4] fo r handling all angular (that is, geometrical or symmetry)
aspects of an N-body problem, the dynamics being confined to r adial matrix elements of the coefficients ( r</r>)ℓ+1/2
in Eq. (1). Although many other systems of coordinates have b een studied for problems with an underlying symmetry
that is different from the spherical, Eq. (1), in combination with the addition theorem for spherical harmonics, has
gained such prominence as to have become the Green’s functio n expansion of choice even for non spherically-symmetric
situations.
II. ALTERNATIVE FOURIER EXPANSION
In a recent investigation [5] of gravitational potentials i n circular cylindrical coordinates x: (R,φ,z), two of us
discovered and used an expansion
1
|x−x′|=1
π√
RR′∞/summationdisplay
m=−∞Qm−1
2(χ)eim(φ−φ′), (4)
withQm−1
2a Legendre function of the second kind of half-integer degre e [6], andχdefined as
χ≡R2+R′2+ (z−z′)2
2RR′=r2+r′2−2rr′cosθcosθ′
2rr′sinθsinθ′. (5)
We have since uncovered this expansion as an application of t he “Heine identity” in the literature [7, 8],
1√v−cosψ=√
2
π∞/summationdisplay
m=−∞Qm−1
2(v)eimψ, (6)
which has a long history but seems not to have been exploited i n mathematical physics in recent times. However, we
have found it powerful for problems with a cylindrical geome try and have used it for compact numerical evaluation of
gravitational potential fields of several axisymmetric and nonaxisymmetric mass distributions [5]. We now set Eq. (4)
in a broader context, together with new associated addition theorems and an application in quantum physics, hoping
to encourage wider use of this expansion throughout physics .
The expansion in Eq. (4), may be viewed either, in analogy wit h Eq. (1), as an expansion in Legendre functions,
now of the second kind in the joint variable χof the whole system, with coefficients ( RR′)−1/2eim(φ−φ′), or as a
Fourier expansion in the variable ( φ−φ′) with theQ’s as coefficients. In this latter view, a further step allows u s to
develop a new addition theorem for these Legendre functions . Interchanging the ℓandmsummations in Eq. (3), we
obtain
1
|x−x′|=1√
rr′∞/summationdisplay
m=−∞eim(φ−φ′)∞/summationdisplay
ℓ=|m|/parenleftbiggr<
r>/parenrightbiggℓ+1
2Γ(ℓ−m+ 1)
Γ(ℓ+m+ 1)Pm
ℓ(cosθ)Pm
ℓ(cosθ′). (7)
Comparing with Eq. (4), we obtain a new addition theorem, thi s time for the Legendre function of the second kind,
Qm−1
2(χ) =π√
sinθsinθ′∞/summationdisplay
ℓ=|m|/parenleftbiggr<
r>/parenrightbiggℓ+1
2Γ(ℓ−m+ 1)
Γ(ℓ+m+ 1)Pm
ℓ(cosθ)Pm
ℓ(cosθ′). (8)
Note thatQm−1
2=Q−m−1
2as per Eq.(8.2.2) in [2].
Similarities and contrasts between the pairs of equations, Eqs. (1) and (3) and Eqs. (4) and (8), are worth
emphasizing. Of the four variables ( r</r>,θ,θ′,φ−φ′), the first pair of equations expresses the inverse distance as a
series in powers of the first variable with coefficients Legend re polynomials of the first kind in γ, a composite of the
other three variables and decomposable in terms of them thro ugh the addition theorem as in Eq. (3). The second pair
of equations, on the other hand, expands in Eq. (4) the invers e distance in terms of the variable φ−φ′, with expansion
2coefficients Legendre functions of the second kind in χ, a composite of the other three variables ( r</r>,θ,θ′) and
decomposable in terms of them through the addition theorem i n Eq. (8). For this comparison, it is useful to recast
Eq. (1) in the more suggestive form,
1
|x−x′|=1√
rr′∞/summationdisplay
ℓ=0Pℓ(cosγ)e−(ℓ+1
2)(lnr>−lnr<). (9)
Whereas this expansion has half-integers in the exponents a nd integer degree Legendre polynomials of the first kind,
Eq. (4) has integer m’s in the exponents and half-integer degree Legendre functi ons of the second kind.
Yet another alternative to Eq. (4) follows upon casting the s quare root in the expression for the distance in terms
ofr,r′,andγin the form Eq. (6) through the definition
v≡1
2/parenleftbiggr<
r>+r>
r</parenrightbigg
=r2+r′2
2rr′. (10)
This gives the expression
1
|x−x′|=1
π√
rr′∞/summationdisplay
n=−∞Qn−1
2(v)einγ, (11)
now a Fourier expansion in γinstead of the ( φ−φ′) of Eq. (4), with Qn−1
2(v) as the coefficients. In terms of
hyperspherical coordinates, widely used in atomic and nucl ear study of three (or more) bodies [9], the variable vis
csc2α,αbeing a “hyperangle”.
We present in this paragraph a number of alternative express ions for the functions Qm−1
2which are useful in
calculations using such expansions as Eqs. (4) and (11). Set tingθ=θ′=π/2 in Eq. (8) and using Eq. (8.756.1) of
[10] gives
Qm−1
2(v) =π∞/summationdisplay
ℓ=|m|/parenleftbiggr<
r>/parenrightbiggℓ+1
2Γ(ℓ−m+ 1)
Γ(ℓ+m+ 1)π22m
[Γ(1 +ℓ−m
2)Γ(1−ℓ−m
2)]2, (12)
which can be rewritten as
Qm−1
2(v) =πe−(m+1
2)η∞/summationdisplay
ℓ=021−2m−4ℓ/parenleftbigg
2ℓ
ℓ/parenrightbigg/parenleftbigg
2ℓ+ 2m−1
ℓ+m/parenrightbigg
e−2ℓη, (13)
where we have defined r</r>≡e−η,v= coshη. Although the ℓ-th term of these series is in different form from what
one obtains through the more familiar formula for Qas a hypergeometric function [11], namely,
Qm−1
2(v= coshη) =√πΓ(m+1
2)
Γ(m+ 1)e−(m+1
2)η2F1(1
2,m+1
2;m+ 1;e−2η), (14)
their equivalence follows from straightforward algebra. A lso, another standard expansion for Qin powers of vas in
Eq. (8.1.3) of [2],
Qν−1
2(v) =√πΓ(ν+1
2)
Γ(ν+ 1)(2v)−ν−1
22F1(ν
2+3
4,ν
2+1
4;ν+ 1;1
v2), (15)
is equivalent. However, the results directly in powers of r</r>in Eqs. (12) , (13), and (14) are more convenient in
many applications. Among specific features worth noting in t hese alternative expansions are that only even powers
of (r</r>) occur in the sum in Eq. (13) and that for any m, the sum in Eq. (12) runs over all ℓvalues compatible
with it,ℓ≥ |m|, as per their interpretation as angular momentum quantum nu mbers.
In the multipole expansion in Eq. (1), γis an angle formed out of the set ( θ,θ′,φ−φ′) and, therefore, cos γin
the functions Pℓhas range of variation from -1 to 1. On the other hand, in the ex pansions in Eqs. (4) and (6), the
argumentsvandχof the Legendre functions of the second kind range from 1 to in finity and, therefore, can be written
in terms of hyperbolic functions as cosh ηand coshξ, respectively. From Eq. (5) we have the link between them,
coshη= cosθcosθ′+ sinθsinθ′coshξ. (16)
3This disentanglement of v(orη) in terms of a triad is the counterpart of Eq. (2) and may be use d with addition
theorems given in the literature such as [7, 8]
Qm−1
2(coshη) =∞/summationdisplay
n=−∞(−1)nΓ(m−n−1
2)
Γ(n+m−1
2)Qn
m−1
2(cosθ)Pn
m−1
2(cosθ′)enξ. (17)
III. TWO-ELECTRON INTEGRALS
We contrast usage of the alternative expansions in Eqs. (1) a nd (4) for calculating the electrostatic interaction
as it appears in atomic, molecular and condensed matter phys ics. Thus, consider the so-called “direct” part of this
interaction between two electrons in the 3 d2configuration,
VD
ee=/integraldisplay /integraldisplay
dxdx′ψ∗
3d(x)ψ∗
3d(x′)|x−x′|−1ψ3d(x)ψ3d(x′). (18)
The standard treatment [12] uses Eqs. (1) and (3), carries ou t all the angular integrals through Racah-Wigner algebra,
leaving behind radial “Slater integrals” Fk(dd),k= 0,2,4,and yielding (for illustrative purposes, all mvalues have
been set equal to zero)
VD
ee=F0(dd) + (4/49)F2(dd) + (36/441)F4(dd), (19)
where the coefficients are evaluated in terms of Wigner 3 j-symbols or are available in tables [12]. The Slater integra ls,
Fk(dd) =/integraldisplay /integraldisplay
r2drr′2dr′(rk
</rk+1
>)R2
3d(r)R2
3d(r′), (20)
remain for numerical evaluation. In this example, upon eval uation with hydrogenic radial functions, we obtain
V= 0.092172 in atomic units. The alternative calculation throug h Eq. (4) involves only the m= 0 term and thereby
the integral
VD
ee=1
π/integraldisplay /integraldisplay /integraldisplay /integraldisplay
R1/2dRR′1/2dR′dzdz′Q−1
2(χ)|ψ3d(x)|2|ψ3d(x′)|2. (21)
The integrand is a function of zandRvariables alone and our numerical evaluation of this integr al reproduces the
value cited above.
As a second example, we computed an exchange integral for the 3d4fconfiguration again setting, for simplicity, all
mequal to zero:
VE
ee=/integraldisplay /integraldisplay
dxdx′ψ∗
3d(x)ψ∗
4f(x′)|x−x′|−1ψ3d(x′)ψ4f(x). (22)
The standard method through exchange Slater integrals Gand Wigner coefficients gives [12]
VE
ee= (9/35)G1(df) + (16/315)G3(df) + (500/7623)G5(df) (23)
and, again through hydrogenic radial functions, gives the v alueVE
ee= 0.0082862. We reproduce the same result upon
directly computing Eq. (22) with Eq. (4), again involving a s ingle four-dimensional integral as in Eq. (21) with Q−1
2.
As the orbital angular momenta involved of the two electrons increase, the number of terms in expressions such
as Eqs. (19) and (23) also grows, necessitating the computin g of more Wigner coefficients and Slater integrals. By
contrast, only a single term of the expansion in Eq. (4) and a s ingle integral is necessary in our suggested alternative,
theφintegrations setting m= 0 for direct terms and mequal to the difference in the mvalues of the two orbitals for
exchange terms. Of course, the price paid is that four-dimen sional integrations are needed unlike the two-dimensional
ones in the Slater integrals. This same selection rule impos ed by theφintegrations means that even in a calculation
with several configurations and the imposition of antisymme trization, such as in a multi-configuration Hartree-Fock
scheme, matrix elements of |x−x′|−1between each term in the bra and in the ket gets a contribution from only one
mvalue in the expansion in Eq. (4).
4IV. SUMMARY
The inverse distance between two points x′andxis intimately involved in Coulomb and gravitational proble ms. Its
expansion in terms of Legendre polynomials Pℓof the angle between the vector pair or a further double-summ ation
expansion involving the individual polar angles of the vect ors are well known and widely used in physics and astronomy.
We have discussed an alternative in terms of cylindrical coo rdinates, a single summation in terms of Legendre functions
Qm−1
2of the second kind in a pair variable χor double summations involving the individual coordinates . These
expansions are better suited to problems involving cylindr ical (azimuthal) symmetry as shown by applications in [5]
and by an illustration here for very common electron-electr on calculations throughout many-electron physics. Furthe r
variants are possible for other coordinates such as ring or t oroidal, parabolic, bispherical, cyclidic and spheroidal [13],
and we plan to return to them in future publications. Connect ions to the theory of Lie groups will also be of interest
[14].
ACKNOWLEDGMENTS
ARPR thanks the Alexander von Humboldt Stiftung and Profs. J . Hinze and F. H. M. Faisal of the University of
Bielefeld for their hospitality during the course of this wo rk. This work has been supported, in part, by NSF grant
AST-9987344 and NASA grant NAG5-8497.
[1] See, for instance, J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, New York, 1975 ), Second edition, Sec.
4.1; H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, 1980), Second edition, Sec. 5.8 ; J. J. Sakurai, Modern
Quantum Mechanics (Addison-Wesley, Reading, 1994), Sec. 6.4.
[2] See, for instance, M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions , National Bureau of Standards,
Applied Mathematics Series 55, Tenth Printing (U. S. Govern ment Printing Office, Washington, D. C., 1972), Chapters 8
and 22.
[3] Secs. 3.5 and 3.6 of Jackson in [1]; Sec. 3.6 of Sakurai in [ 1].
[4] See, for instance, A. R. Edmonds, The Quantum Theory of Angular Momentum (Princeton, 1957); U. Fano and A. R. P.
Rau,Symmetries in Quantum Physics (Academic, New York, 1996), Sec. 5.1; Sec. 3.7 of Sakurai in [ 1].
[5] H. S. Cohl and J. E. Tohline, Astrophys. J., 527, 86 (1999).
[6] See, for instance, A. Erd´ elyi, Higher Transcendental Functions , Volume I (McGraw-Hill Book Company, Inc., New York,
1953), Chapter 3; Chapter 8 of Abramowitz and Stegun in [2];
[7] See, for instance, H. Bateman, Partial Differential Equations of Mathematical Physics (Cambridge University Press, New
York, 1959), Sec. 10.2; E. Heine, Handbuch der Kugelfunctionen (Physica-Verlag, Wuerzburg, 1961), Vol.2: Anwendungen,
Sec. 74.
[8] E. W. Hobson, The Theory of Spherical and Ellipsoidal Harmonics (Chelsea, New York, 1965), p. 443.
[9] See, for instance, U. Fano and A. R. P. Rau, Atomic Collisions and Spectra (Academic, Orlando, 1986), Sec. 10.3.1; Sec.
10.2.2 of Fano and Rau in [4]; C. D. Lin, Phys. Reports, 257, 1 (1995).
[10] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products , Fifth edition (Academic, New York, 1994)
[11] Ref. 8, p. 438.
[12] E. U. Condon and G. H. Shortley, The Theory of Atomic Spectra , (Cambridge University Press, Cambridge, 1963), Sec.
86. Entries in Table 16ford f, k= 5 are in error and have been corrected in Eq. (23).
[13] P. Moon and D. E. Spencer, Field Theory Handbook: Including Coordinate Systems, Diffe rential Equations and Their
Solutions (Springer-Verlag, Berlin, 1961), Chapter IV; H. S. Cohl, J. E. Tohline, A. R. P. Rau, and H. M. Srivastava,
Astron. Nachr. 321, 363 (2000).
[14] See, for instance, W. Miller, Jr. Symmetry and Separation of Variables (Addison-Wesley, London, 1977), Sec. 3.6; W.
Miller, Jr., Symmetry Groups and Their Applications (Academic, New York, 1972); W. Miller, Jr., Lie Theory and Special
Functions (Academic Press, New York, 1968).
5 |
arXiv:physics/0101087v1 [physics.bio-ph] 25 Jan 2001Kinetic model of DNA replication in eukaryotic organisms
John Herrick1, John Bechhoefer2∗, Aaron Bensimon1∗
1Laboratoire de Biophysique de l’ADN, D´ epartement des Biot echnologies, Institut Pasteur, 25-28, rue du Dr. Roux, 7572 4
Paris Cedex 15, France
2Department of Physics, Simon Fraser University, Burnaby, B ritish Columbia, V5A 1S6, Canada
∗To whom correspondence should be addressed. E-mail: johnb@ sfu.ca (J.B.) and abensim@pasteur.fr (A.B.)
We formulate a kinetic model of DNA replication that quantit atively describes recent results
on DNA replication in the in vitro system of Xenopus laevis prior to the mid-blastula transition.
The model describes well a large amount of different data with in a simple theoretical framework.
This allows one, for the first time, to determine the paramete rs governing the DNA replication
program in a eukaryote on a genome-wide basis. In particular , we have determined the frequency
of origin activation in time and space during the cell cycle. Although we focus on a specific stage
of development, this model can easily be adapted to describe replication in many other organisms,
including budding yeast.
Although the organization of the genome for DNA
replication varies considerably from species to species,
the duplication of most eukaryotic genomes shares a num-
ber of common features:
1) DNA is organized into a sequential series of replication
units, or replicons, each of which contains a single origin
of replication (Hand, 1978; Friedman et al., 1997).
2) Each origin is activated not more than once during the
cell-division cycle.
3) DNA synthesis propagates at replication forks bidirec-
tionally from each origin (Cairns, 1963).
4) DNA synthesis stops when two newly replicated re-
gions of DNA meet.
Understanding how these parameters are coordinated
during the replication of the genome is essential for elu-
cidating the mechanism by which S-phase is regulated in
eukaryotic cells. In this article, we formulate a stochasti c
model based on these observations that yields a mathe-
matical description of the process of DNA replication and
provides a convenient way to use the full statistics gath-
ered in any particular replication experiment. It allows
one to deduce accurate values for the parameters that
regulate DNA replication in the Xenopus laevis replica-
tion system, and it can be generalized to describe replica-
tion in any other eukaryotic system. This type of model
has also been shown to apply for the case of RecA poly-
merizing on a single molecule of DNA (Shivashankar et
al, 1999). The model turns out to be formally equivalent
to a well-known stochastic description of the kinetics of
crystal growth, which allows us to draw on a number
of previously derived results and, perhaps equally im-
portant, suggests a vocabulary that we find useful and
intuitive for understanding the process of replication.KINETIC MODEL OF DNA REPLICATION
In the 1930s, several scientists independently de-
rived a stochastic model that described the kinetics of
crystal growth (Kolmogorov, 1937; Johnson and Mehl,
1939; Avrami, 1939). The “Kolmogorov-Johnson-Mehl-
Avrami” (KJMA) model has since been widely used by
metallurgists and other scientists to analyze thermody-
namic phase transformations (Christian, 1981).
In the KJMA model, freezing kinetics result from three
simultaneous processes:
1) nucleation, which leads to discrete solid domains.
2) growth of the domain.
3) coalescence, which occurs when two expanding do-
mains merge.
Each of these processes has an analog in DNA replica-
tion in higher eukaryotes, and more specifically embryos:
1) The activation of an origin of replication is analo-
gous to the nucleation of the solid domains during crystal
growth.
2) Symmetric bidirectional DNA synthesis initiated
(nucleated) at the origin corresponds to solid-domain
growth.
3) Coalescence in crystal growth is analogous to multiple
dispersed sites of replicating DNA (replication fork) that
advance from opposite directions until they merge.
In the simplest form of the KJMA model, solids nu-
cleate anywhere in the liquid, with equal probability
for all spatial locations (“homogeneous nucleation”), al-
though it is straightforward to describe nucleation at pre-
specified sites (“heterogeneous nucleation”), which would
1correspond to a case where replication origins are speci-
fied by fixed genetic sites along the genome. Once a solid
domain has been nucleated, it grows out as a sphere at
constant velocity v. When two solid domains impinge,
growth ceases at the point of contact, while continuing
elsewhere. KJMA used elementary methods to calculate
quantities such as f(τ), the fraction of the volume that
has crystallized by time ( τ). Much later, more sophis-
ticated methods were developed to describe the detailed
statistics of domain sizes and spacings (Sekimoto, 1991;
Ben-Naim and Krapivsky, 1996).
DNA replication, of course, corresponds to one-
dimensional crystal growth; the shape in three dimen-
sions of the one-dimensional DNA strand does not di-
rectly affect the kinetics modeling. (In the model, repli-
cation is one dimensional along the DNA. The configu-
ration of DNA in three dimensions is not directly rele-
vant to the model but can enter indirectly via the nucle-
ation function I(x, τ). For example, if, for steric reasons,
certain regions of the DNA are inaccessible to replica-
tion factories, those regions would have a lower (or even
zero) value of I.) The one-dimensional version of the
KJMA model assumes that domains grow out at veloc-
ityv, assumed to remain constant. The nucleation rate
I(x, τ) =I0is defined to be the probability of domain
formation per unit length of unreplicated DNA per unit
time, at the position xand time τ. Following the analogy
to the one-dimensional KJMA model, we can calculate
the kinetics of DNA replication during S-phase. This re-
quires determining the fraction of the genome f(τ) that
has already been replicated at any given moment during
S-phase. One finds
f(τ) = 1−e−I0vτ2, (1)
which defines a sigmoidal curve. (Eq. 1 assumes an infi-
nite genome length. The relative importance of the finite
size of chromosomes is set by the ratio (fork velocity * du-
ration of S-phase) / chromosome length (Cahn, 1996). In
the case of the experiment analyzed in this paper, this ra-
tio is≈10 bases/sec * 1000 sec / 107bases/chromosome
≈10−3, which we neglect.)
A more complete description of replication kinetics
requires detailed analysis of different statistical quanti -
ties, including measurements made on replicated regions
(eyes), unreplicated regions (holes), and eye-to-eye size s
(the eye-to-eye size is defined as the length between the
center of one eye and the center of a neighboring eye.)
The probability distributions may be expressed as func-
tions either of time τor replicated fraction f. For exam-
ple, the distribution of holes of size ℓat time τ,ρh(ℓ, τ)
can be derived by a simple extension of the argument
leading to Eq. 1:
ρh(ℓ, τ) =I0τ·e−I0τℓ. (2)
From Eq. 2, the mean size of holes at time τisℓh(τ) =1
I0τ. (3)
Determining the probability distributions of replicated
lengths (eye sizes) is complicated because a given repli-
cated length may come from a single origin or it may
result from the merger of two or more replicated regions.
Thus, one must calculate in effect an infinite number of
probabilities; by contrast, holes of a given length arise
in only one way (Ben-Naim and Krapivsky, 1996). One
can nonetheless derive a simple expression for ℓi(τ), the
mean replicated length at time τ, from a “mean-field”
hypothesis (Plischke and Bergersen, 1994): the probabil-
ity distribution of a given replicated length is assumed
to be independent of the actual size of its neighbor. One
can show that this mean-field hypothesis must always be
true in one-dimensional growth problems, but not neces-
sarily in the ordinary three-dimensional setting of crys-
tal growth (Suckjoon Jun, private communication). In
particular, if I(τ) depends on space, one expects correla-
tions to be important. Using the mean-field hypothesis,
we find
ℓi(τ) =ℓh(τ)f
1−f=eI0vτ2−1
I0τ(4)
and
ℓi2i(τ) =ℓi(τ) +ℓh(τ) =ℓh(τ)
1−f=eI0vτ2
I0τ. (5)
These expressions for ℓi(τ) and ℓi2i(τ) allow one to col-
lapse the experimental observations of ℓh,ℓi, andℓi2i(the
mean eye-to-eye separation) onto a single curve. (See Fig.
3D, below.)
Finally, we can calculate the average distance between
origins of replication that were activated at different
times during the replication process, which is just the
inverse of Itot, the time-integrated nucleation probability
per unit length:
ℓ0≡I−1
tot=2√π·/radicalbiggv
I0(6)
The last expression shows that, as might have been
guessed by dimensional analysis of the model parame-
ters (I0andv), the basic length scale in the model is set
byℓ∗≡/radicalbig
v/I0.
Since the kinetics of DNA replication in any cell sys-
tem depends on two fundamental parameters, replication
fork velocity and initiation frequency, one of the princi-
pal goals of this kind of analysis is to derive accurate
values for these parameters, along with inferences about
any variation during the course of S-phase. As repli-
con size and the duration of S-phase depend on the val-
ues of these parameters, this information is indispensable
for understanding the mechanisms regulating S-phase in
2any given cell system (Pierron and Benard, 1996; Walter
and Newport, 1997; Hyrien and Mechali, 1993; Coverly
and Laskey, 1994; Blow and Chong, 1996; Shinomiya and
Ina, 1991; Brewer and Fangman, 1993; Gomez and An-
tequera, 1999).
APPLICATION OF THE KJMA MODEL TO DNA
REPLICATION IN X. LAEVIS
Recent experimental results obtained on the kinetics of
DNA replication in the well-characterized Xenopus lae-
viscell-free system were used here to derive parameter
values for that particular system. In those experiments,
fragments of DNA that have completed one cycle of repli-
cation are stretched out on a glass surface using molecu-
lar combing (Bensimon et al., 1994; Michalet et al., 1997;
Herrick et al., 1999). The DNA that has replicated prior
to some chosen time tis labeled with a single fluorescent
dye, while DNA that replicated after that time is labeled
with two dyes. The result is a series of samples, each of
which corresponds to a different time tduring S-phase.
Using an optical microscope, one can directly measure
eye, hole, and eye-to-eye lengths at that time. We can
thus monitor the evolution of genome duplication from
time point to time point, as DNA synthesis advances.
(See Fig. 1.)
Cell-free extracts of eggs from Xenopus laevis support
the major transitions of the eukaryotic cell cycle, includ-
ing complete chromosome replication under normal cell-
cycle control and offers the opportunity to study the way
that DNA replication is coordinated within the cell cy-
cle. In the experiment, cell extract was added at t=
2’, and S-phase began 15 to 20’ later. DNA replication
was monitored by incorporating two different fluorescent
dyes into the newly synthesized DNA. The first dye was
added before the cell enters S-phase in order to label the
entire genome. The second dye was added at successive
time points t= 25, 29, 32, 35, 39, and 45’, in order to
label the later replicating DNA. DNA taken from each
time point was combed, and measurements were made
on replicated and unreplicated regions. The experimental
details are described elsewhere (Herrick et al., 2000), but
the approach is similar to DNA fiber autoradiography, a
method that has been in use for the last 30 years (Huber-
man and Riggs, 1966; Jasny and Tamm, 1979). Indeed
the same approach has recently been adapted to study
the regulatory parameters of DNA replication in HeLa
cells (Jackson and Pombo, 1998). Molecular combing,
however, has the advantage that a large amount of DNA
may be extended and aligned on a glass slide which en-
sures significantly better statistics (over several thousa nd
measurements corresponding to several hundred genomes
per coverslip). Indeed, the molecular combing experi-
ments provide, for the first time, easy access to the quan-
tities of data necessary for testing models such as the oneadvanced in this paper.
LtotE1E2 E3
f(t=39) = (E1 + E2 + E3) / LtotS phase
t = 25'
t = 29'
t = 32'
t = 39'
t = 45't = 35'
FIG. 1. Schematic representation of labeled and combed
DNA molecules. Since replication initiates at multiple dis -
persed sites throughout the genome, the DNA can be dif-
ferentially labeled, so that each linearized molecule cont ains
alternating subregions stained with either one or both dyes .
The thick segments correspond to sequences synthesized in
the presence of a single dye (eyes). The thin segments cor-
respond to those sequences that were synthesized after the
second dye was added (holes). The result is an unambiguous
distinction between eyes and holes (earlier and later repli -
cating sequences) along the linearized molecules. Replica tion
is assumed to have begun at the midpoints of the thick se-
quences (dotted lines) and to have proceded bidirectionall y
from the site where DNA synthesis was initiated (arrows).
Measurements between the centers of adjacent eyes provide
information about replicon sizes (eye-to-eye distances). The
fraction of the molecule already replicated by a given time,
f(τ), is determined by summing the lengths of the thick seg-
ments and dividing that by the total length of the respective
molecule.
Generalization of the simple version of the KJMA
model
Analyzing the experimental results obtained on the ki-
netics of DNA replication in the in vitro cell-free sys-
tem of Xenopus laevis (Herrick et al., 2000; Lucas et al.,
2000), we found that the simple version of the crystal-
growth model needed to be generalized in a number of
ways:
1) Instead of assuming that the nucleation function I(τ)
has the form I(τ) =I0forτ≥0, we allowed for an
arbitrary form I(τ). Nucleation is believed to occur syn-
chronously during the first half of S-phase in Drosophila
melanogaster early embryos (Shinomiya and Ina, 1991;
Blumenthal et al., 1974). Nucleation in the myxomycete
Physarum polycephalum, on the other hand, occurs in
3a very broad temporal window, suggesting that nucle-
ation occurs continuously throughout S-phase (Pierron
and Benard, 1996). Finally, recent observations suggest
that in Xenopus laevis , early embryos nucleation may oc-
cur with increasing frequency as DNA synthesis advances
(Herrick et al., 2000; Lucas et al., 2000). By choosing an
appropriate form for I(τ), one can account for any of
these scenarios. Below, we show how measured quan-
tities may, using the model, be inverted to provide an
estimate for I(τ).
2) The model assumes implicitly that the DNA analyzed
began replication at τ= 0, but this may not be so, for
two reasons:
i) In the experimental protocols, the DNA analyzed
comes from approximately 20,000 independently replicat-
ing nuclei. Before each genome can replicate, its nuclear
membrane must form, along with, presumably, the repli-
cation factories. This process takes 15-20 minutes (Blow
and Laskey, 1986; Blow and Watson, 1987; Wu et al.,
1997). Because the exact amount of time can vary from
cell to cell, the DNA analyzed at time tin the laboratory
may have started replicating over a relatively wide range
of times.
ii) In eukaryotic organisms, origin activation may be
distributed in a programmed manner throughout the
length of S-phase, and, as a consequence, each origin
is turned on at a specific time (early and late) (Simon et
al., 1999).
In the current experiment, the lack of information about
the locations of the measured DNA segments along the
genome means that we cannot distinguish between asyn-
chrony due to reasons (i) or (ii). We can however account
for their combined effects by introducing a starting-time
distribution φ(t′), which is the probability—for whatever
reason—that a given piece of analyzed DNA began repli-
cating at time t′in the lab. We assume that the distri-
bution is Gaussian, with unknown mean and standard
deviation, an assumption that will be justified by the fits
to the data.
3) The models described above assumed that statistics
could be calculated on infinitely long segments of DNA.
In the experimental approach, the combed DNA is bro-
ken down into relatively short segments (200 kb, typi-
cally). Although it is difficult to account for this effect
analytically, we wrote a Monte Carlo simulation that can
mimic such “finite-size” effects. As we show below (Fig.
3D), we find evidence that there is no spatial variation
in nucleation rates on scales lessthan 200 kb.
4) The experiments are all analyzed using an epifluores-
cence microscope to visualize the fluorescent tracks ofcombed DNA on glass slides. The spatial resolution ( ≈
0.3µm) means that smaller signals will not be detectable.
Thus, two replicated segments separated by an unrepli-
cated region of size <0.3µm will be falsely assumed to
be one longer replicated segment. We accounted for this
in the simulations by calculating statistics on a coarse
lattice whose size equalled the optical resolution, while
the simulation itself takes place on a finer lattice.
We can redo the analysis of the DNA kinetics for gen-
eralI(τ). Eq. 1 then generalizes to
f(τ) = 1−e−g(τ),with g(τ) = 2v/integraldisplayτ
0I(τ′)(τ−τ′)dτ′,
(7)
and, similarly, Eq. 3 becomes
ℓh(τ) =/bracketleftbigg/integraldisplayτ
0I(τ′)dτ′/bracketrightbigg−1
. (8)
The other mean lengths, ℓi(τ) and ℓi2i(τ), continue to be
related to ℓh(τ) by the general expressions given in Eqs.
4 and 5. In the experiment, one measures ℓh,ℓi, and ℓi2i
as functions of both τandf. (Because of the start-time
ambiguity, the fdata are easier to interpret.) The goal
is to invert this data to find I(τ). Using Eqs. 7 and 8,
we find
τ(f) =1
2v/integraldisplayf
0ℓi2i(f′)d f′=1
2v/integraldisplayf
0ℓh(f′)
1−f′d f′.(9)
Because τ(f) increases monotonically, one can numeri-
cally invert it to find f(τ). From f(τ), one can derive all
quantities of interest, including I(τ).
Using the generalizations discussed above, we analyzed
recent results obtained on DNA replication in the Xeno-
pus laevis cell-free system. DNA taken from each time
point was combed, and measurements were made on
replicated and unreplicated regions. Statistics from each
time point were then compiled into four histograms (24
histograms for the 6 time points): ρ(f, t),ρh(ℓ, t),ρi(ℓ, t),
andρi2i(ℓ, t), where ρis the distribution of replicated
fractions fat time t,ρhis the hole-length ℓdistribu-
tions at time t, etc. For reasons of space, only the ρ(f, t)
distributions are shown (Fig. 2).
44
3
2
1
0
1.0 0.5 0.0294
3
2
1
0
1.0 0.5 0.032
20
10
0
1.0 0.5 0.04515
10
5
0
1.0 0.5 0.0394
3
2
1
0
1.0 0.5 0.03515
10
5
0Probability Density1.0 0.5 0.0
Fraction Replicated (f)25
data
simulation
theoryA B C
D E F
FIG. 2. ρ(f, t) distributions for the 6 time points. The
curves show the probability that a molecule at a given time
point (A-F) has undergone a certain amount of replication be -
fore the second dye was added. The red points represent the
experimental data. The results of the Monte Carlo simulatio n
are shown in blue, analytical curves in green.
40
20
0<Eye-to-Eye Len> (µm)1.00.50.040
20
0<Replicated Len> (µm)1.00.50.040
20
0<Hole Length> (µm)1.00.50.0
fraction replicated (f) data
simulation
theoryA B
C
0.1110100Length (µm)
1.00.50.0 holes
eyes
eye-to-eye
lengthsD
FIG. 3. Mean quantities vs. replication fraction. (A)
ℓh(f);(B)ℓi(f);(C)ℓi2i(f). Red points are data; blue points
are from the Monte-Carlo simulation; the green curve is a
least-squares fit, based on a two-segment I(τ) and excluding
data points larger than 10 µm (because of finite-size effects);
(D)curves in (A)-(C) collapsed onto a single plot, confirm-
ing mean-field hypothesis. (The discrepancies near f= 0 and
1 reflect the added errors in measuring very small eyes or
holes, because of optical-resolution limitations, or very large
eyes or holes, because of finite-segment limitations.)0.4
0.3
0.2
0.1
0.0Cum. initiation density (µm-1)
25 20 15 10 5 0
Elapsed time τ (min)0.4
0.3
0.2
0.1
0.0Nucleation Rate (µm-1/min)
25 20 15 10 5 0
Elapsed time τ (min)1.0
0.8
0.6
0.4
0.2
0.0Fraction replicated (f)
25 20 15 10 5 0
Elapsed time τ (min)break point
data
theory
S phaseA
B
C
FIG. 4. (A)Fraction of replication completed, f(τ). Red
points are derived from the measurements of mean hole, eye,
and eye-to-eye lengths. Green curve is an analytic fit (see
below). Shaded area runs from 10% to 90% replicated (10.5
min.) The time from the first origin initiation to the last co-
alescence is approximately 25 min. (B)Initiation rate I(τ).
The large statistical scatter arises because the data point s
are obtained by taking two numerical derivatives of the f(τ)
points in A. (C)Integrated origin separation, Itot(τ), which
gives the average distance between all origins activated up to
timeτ. In A-C, the green curves are from fits that assume
thatI(τ) has two linear regimes of different slopes. The form
we chose for I(τ) was the simplest analytic form consistent
with the data in B. The parameters for the least-squares fits
(slopes I1andI2, break point τ1) are obtained from a global
fit to the three data sets in Fig. 3A-C, i.e.,ℓh(f),ℓi(f), and
ℓi2i(f).
One can immediately see from the distribution of repli-
cated fractions ρ(f, t) the need to account for the spread
in starting times. If all the segments of DNA that were
analyzed had started replicating at the same time, then
the distributions would have been concentrated over a
very small range of f. But, as one can see in Fig. 2C,
5some segments of DNA (within the same time point) have
already finished replicating ( f= 1) before others have
even started ( f= 0). This spread is far larger than
would be expected on account of the finite length of the
segments analyzed.
Because of the need to account for the spread in start-
ing times, it is simpler to begin by sorting data by the
replicated fraction fof the measured segment. We thus
assume that all segments with a similar fraction fare at
roughly the same point in S-phase, an assumption that
we can check by partitioning the data into subsets and
redoing our measurements on the subsets. In Fig. 3A-C,
we plot the mean values ℓh,ℓi, and ℓi2iagainst f. We
then find f(τ),I(τ), and the cumulative distribution of
lengths between activated origins of replication, Itot(τ).
(See Fig. 4.)
The direct inversion for I(τ) (Fig. 4b) shows sev-
eral surprising features: First, origin activation takes
place throughout S-phase and with increasing proba-
bility (measured relative to the amount of unreplicated
DNA), as recently inferred by a cruder analysis of data
from the same system using plasmid DNA (Lucas et al.,
2000). Second, about halfway through S-phase, there is
a marked increase in initiation rate, an observation that,
if confirmed, would have biological significance. It is not
known what might cause a sudden increase (break point)
in initiation frequency halfway through S-phase. The in-
crease could reflect a change in chromatin structure that
may occur after a given fraction of the genome has under-
gone replication. This in turn may increase the number
of potential origins as DNA synthesis advances (Pasero
and Schwob, 2000).
The smooth curves in Fig 3A-C are fits based on the
model, using an I(τ) that has two linearly increasing re-
gions, with arbitrary slopes and “break point” (three free
parameters). The fits are quite good, except where the fi-
nite size of the combed DNA fragments becomes relevant.
For example, when mean hole, eye, and eye-to-eye lengths
exceed about 10% of the mean fragment size, larger seg-
ments in the distribution for ℓh(f), etc., are excluded and
the averages are biased down. We confirmed this with the
Monte-Carlo simulations, the results of which are over-
laid on the experimental data. The finite fragment size in
the simulation matches that of the experiment, leading
to the same downward bias. In Fig. 4, we overlay the fits
on the experimental data. We emphasize that we obtain
I(τ) directly from the data, with no fit parameters. The
analytical form is just a model that summarizes the main
features of the origin-initiation rate we determine via our
model, from the experimental data. The important result
isI(τ).
From the maximum of Itot(τ), we find a mean spac-
ing between activated origins of 4.8 ±0.5 kb, which is
much smaller than the minimum mean eye-to-eye sepa-
ration 14 ±1 kb. Interestingly, the former value agrees
well with the calculated distribution of chromatin-boundORC molecules (Walter and Newport, 1997), while the
mean eye-to-eye size coincides with the average chro-
matin loop size (Buongiorno et al., 1982, Blow and
Chong, 1996). In our model, the two quantities differ if
initiation takes place throughout S-phase, as coalescence
of replicated regions leads to fewer domains (and hence
fewer inferred origins). The mean eye-to-eye separation
is of particular interest because its inverse is just the
domain density (number of active domains per length),
which can be used to estimate the number of active repli-
cation forks at each moment during S-phase. For ex-
ample, the saturation value of Itotcorresponds to the
maximum number (about 600,000/genome) of active ori-
gins of replication. Since there are about 400 replication
foci/cell nucleus, this would indicate a partitioning of
approximately 1,500 origins (or, equivalently, about 7.5
Mb) per replication focus (Blow and Laskey, 1986; Mills
et al., 1989).
Because the distribution of fvalues in the ρ(f, t)
plots depends on the unknown starting-time distribution
(φ(t′)), we used the parameters for I(τ) in order to de-
rive the fork velocity vand the mean and width of the
Gaussian form assumed for φ(t′). As with the fdata, we
did a global fit to data from all six time points. We find
v= 640 ±40 bases/min., in excellent agreement with
previous estimates (Mahbubani et al., 1992; Lu et al.,
1998).
One can test whether adding higher moments to the
assumed Gaussian form significantly improves the fits,
and, in our case, they do not. (Specifically, we added
skewness and kurtosis, i.e., third and fourth moments.)
This implies that the actual shape of the starting-time
distribution does not differ greatly from a Gaussian form.
In a future experiment, it would be very desirable to
obtain independent information about the form of φ(t).
One could then constrain other parameters more tightly.
For example, there is a high correlation between the mean
starting time of molecules (here, 20.4 min.) and the ve-
locity. (The width of the distribution, 2.6 min., is much
less coupled in the fits.) An effect that might then be
included in the model is a variable fork velocity. For ex-
ample, vmight be decrease as forks coalesce or as replica-
tion factor becomes limiting toward the end of Sphase
(Blow and Laskey, 1986; Blow and Watson, 1987; Wu
et al., 1997; Pierron and Benard, 1996). Such effects, if
present, are small enough that they are difficult to see in
the present case.
Another important question is to separate the effects of
any intrinsic distribution due to early and late-replicati ng
regions of the genome of a single cell from the extrinsic
distribution caused by having many cells in the experi-
ment. One approach would be to isolate and comb the
DNA from a single cell. Although difficult, such an exper-
iment is technically feasible. The latter problem could be
resolved by in situ fluorescence observations of the chosen
cell.
6Finally, an implicit assumption of our analysis has been
that there is no spatial organization in the nucleation
origins—i.e., that I(τ) does not depend on the position
x. Directly testing the mean-field hypothesis by the data
collapse shown in Fig. 3D justifies this assumption on
length scales up to 200 kb.
DISCUSSION
The view that we are led to here, of random initi-
ation events occurring continuously during the replica-
tion of Xenopus sperm chromatin in egg extracts, is in
striking contrast to what has until recently been the ac-
cepted view of a regular periodic organization of replica-
tion origins throughout the genome (Buongiorno et al.,
1982; Laskey, 1985; Coverly and Laskey, 1994; Blow and
Chong, 1996). For a discussion of experiments that raise
doubts on such a view, see (Berezney, 2000). The applica-
tion of our model to the results of Herrick et al. indicate
that the kinetics of DNA replication in the X. laevis in
vitrosystem closely resembles that of genome duplication
in early embryos. Specifically, we find that the time re-
quired to duplicate the genome in vitro agrees well with
what is observed in vivo . In addition, the model yields
accurate values for replicon sizes and replication fork ve-
locities that confirm previous observations (Mahbubani
et al., 1992; Hyrien and Mechali, 1993). Though replica-
tionin vitro may differ biologically from what occurs in
vivo, the results nevertheless demonstrate that the kinet-
ics remains essentially the same. Of course, the specific
finding of an increasing rate of initiation invites a bio-
logical interpretation involving a kind of autocatalysis,
whereby the replication process itself leads to the release
of a factor whose concentration determines the rate of
initiation. This will be explored in future work.
One can entertain many further applications of the ba-
sic model discussed above, which can be generalized, if
need be. For example, Blumenthal et al. interpreted
their results on replication in Drosophila melanogaster
forρi2i(ℓ, f) to imply periodically spaced origins in the
genome (Blumenthal et al., 1974). (See their Fig. 7.) It
is difficult to judge whether their peaks are real or statis-
tical happenstance, but one could check the mean-field
hypothesis independently for that data. If the conclusion
is indeed that the origins in that system are arranged pe-
riodically, the kinetics model could be generalized in a
straightforward way (introducing an I(x, τ) that was pe-
riodic in x). Similar generalizations can be easily incor-
porated into the model to account for DNA replication
in other organisms, including yeast and Physarum .CONCLUSION
In this article, we have introduced a class of theoret-
ical models for describing replication kinetics that is in-
spired by well-known models of crystal-growth kinetics.
The model allows us to extract the rate of initiation of
new origins, a quantity whose time dependence is has
not previously been measured. With remarkably few pa-
rameters, the model fits quantitatively the most detailed
existing experiment on replication in Xenopus . It repro-
duces known results (for example, the fork velocity) and
provides the first reliable description of the temporal or-
ganization of replication initiation in a higher eukaryote .
Perhaps most important, the model can be generalized in
a straightforward way to describe replication and extract
relevant parameters in essentially any organism.
ACKNOWLEDGMENTS
We thank M. Wortis and S. Jun for helpful comments
and insights. This work was supported by grants from
the Fondation de France, NSERCC, and NIH.
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9 |
arXiv:physics/0101088v1 [physics.atom-ph] 25 Jan 2001A rainbow of cold atoms caused by a stochastic process
B.T. Wolschrijn, D. Voigt, R. Jansen, R.A. Cornelussen, N. B hattacharya, R.J.C. Spreeuw and H.B. van Linden van
den Heuvell
Van der Waals-Zeeman Institute, University of Amsterdam,
Valckenierstraat 65, 1018 XE Amsterdam, the Netherlands
e-mail: spreeuw@wins.uva.nl
(September 28, 2013)
We report direct observation of a rainbow caustic in the
velocity distribution of87Rb atoms, bouncing inelastically on
an evanescent-wave atom mirror. In contrast to known ex-
amples, this caustic is caused by a stochastic process, name ly
a spontaneous Raman transition during the bounce. The re-
sults are in good agreement with a classical calculation. We
observed that although energy is extracted from the atoms,
the phase-space density is in most cases not increased.
32.80.Lg, 42.50.Vk, 03.75.-b
Caustics are ubiquitous phenomena in nature. Exam-
ples are the cusp-shaped patterns of light reflection on the
inside of a coffee-cup and the patterns of bright lines ob-
served on the bottom of a swimming pool [1]. The prime
example of a caustic is the common rainbow, which can
be understood in a ray-optics picture by considering how
the scattering angle of a light ray depends on its impact
parameter on a water droplet [2] . Whereas the incident
rays have smoothly distributed impact parameters, the
outgoing rays pile up where the scattering angle has a
local extremum. Such a divergence of the ray density,
the caustic, appears at the rainbow angle. In atomic [3]
and nuclear [4] scattering experiments analogous rainbow
phenomena have also been observed.
The examples of caustics that have been known so far
have in common that the outgoing parameter (scattering
angle) is a deterministic function of the incoming param-
eter (impact parameter). In this Letter we report on our
observation of a new type of rainbow caustic existing by
virtue of a stochastic process, which distributes a single-
valued “impact parameter” over a range of “scattering
angles”. To our knowledge, this type of caustic has not
been observed before.
We have observed this caustic in the velocity distri-
bution of cold atoms, after bouncing inelastically off an
evanescent-wave mirror [5–7]. The incident atoms are
monochromatic, i.e. they all have the same velocity. Dur-
ing the bounce the atoms are optically pumped to a dif-
ferent hyperfine ground-state, by a spontaneous Raman
transition. They leave the surface with less kinetic en-
ergy, due to the difference in optical potential. Our mea-
surements and analysis show that the resulting velocity
distribution is highly asymmetric, containing a caustic at
the minimum velocity. This divergence demonstrates the
strong preference for making the Raman transition in theturning point.
We briefly discuss the divergence in the velocity dis-
tribution in the context of evanescent-wave cooling [5,7].
Calculations show that although energy is extracted from
the atoms, a single inelastic bounce generally leads to a
decrease of phase space density.
The experiment is performed in a rubidium vapor cell.
We trap about 107atoms of87Rb in a magneto-optical
trap (MOT) and subsequently cool them in optical mo-
lasses to 15 µK. The MOT has a 1 /e2radius of 0 .6 mm
and is centered 5.8 mm above the horizontal surface of
a right-angle BK7 prism. At this surface, we create an
evanescent wave (EW) by a Gaussian shaped laser beam
of 0.8 mm 1 /e2radius, which undergoes total internal
reflection.
When blue detuned, the EW induces a repulsive op-
tical dipole potential UF(z) =UF(0)exp( −2κz), where
the subscript F= 1,2 denotes the hyperfine ground
state, κ=k0/radicalbig
n2sin2θ−1, with k0= 2π/λthe free
space wave vector of the light, n= 1.51 the refractive
index and θthe angle of incidence [8–10]. The maxi-
mum potential UF(0)∝I0/δF, where I0is the incident
intensity at the glass surface, and δFis the detuning.
Here we define the detuning δ1,2relative to the transi-
tion 5 S1/2(F= 1,2)→5P3/2(F′=F+ 1) of the D2line
(780 nm), see Fig.1.
An atom entering the EW in the F= 1 state is slowed
down by the potential U1(z). Spontaneous Raman-
scattering can transfer the atom to the higher hyperfine
ground state ( F= 2). When transferred into this state,
the potential acting on the atom is U2(z), which is weaker
due to the increase of the detuning by approximately the
hyperfine splitting δ2≈δ1+δhfs(Fig.1). The ratio of the
two potentials, β≡U2(0)/U1(0), quantifies the reduction
in potential energy. As a result the atoms will bounce in-
elastically, i.e. to a lower height than their initial relea se
position.
Experimental data of bouncing atoms are taken by ab-
sorption imaging. After a variable time delay the atomic
cloud is exposed to 20 µs of probe light, resonant with
the 5S1/2(F= 2)→5P3/2(F′= 3) transition. Thus only
atoms which have been transferred to F= 2 contribute
to the signal. The atomic cloud is imaged on a digital
frame-transfer CCD camera. The presented images show
an area of 2 .2×4.5 mm2with a pixel resolution of 15 µm
(Fig.2) . The initial position of the MOT is outside the
field of view. The horizontal line at the bottom shows
1the prism surface.
A typical series with 2 ms time increments is shown in
Fig.2. Each density profile has been converted in a hori-
zontal line sum. The solid curve is the result of a calcula-
tion described below. The amplitude of the experimental
curves is rescaled such that the maximum optical density
of the experimental curve coincides with the theoretical
maximum value. This is the only fit parameter.
As expected, the atoms bounce up less high than
their initial height. Furthermore, the spatial distributi on
shows another striking feature: it displays a high den-
sity peak at low z, and a long low-density tail extending
upward. Note also that there is a time-focus: the den-
sity peak is sharpest when the slowest atoms reach their
upper turning point. This density peak is an immedi-
ate consequence of a caustic appearing at the minimum
possible velocity.
The caustic can be understood by considering the
atoms as point particles moving in the EW potential.
This corresponds to the ray-optics limit for the optical
rainbow. We consider an atom arriving at the surface in
itsF= 1 hyperfine ground state, with an initial down-
ward velocity vi<0. Its trajectory through phase space
is determined by energy conservation: U1(0)exp( −2κz)+
1
2mv2=1
2mv2
i, and is depicted in Fig.4 by the thick line.
While it is slowed down by the EW it may scatter a pho-
ton at a velocity vp, and be transferred to the F= 2 state.
The atom continuous on a new trajectory determined by
U2(z). To illustrate the formation of the caustic, possible
trajectories starting at various positions in phase space
are depicted as thin black ( vp<0) and gray ( vp>0)
curves. For asymptotically large zthe density of curves
represents the outgoing velocity distribution, showing th e
caustic where the trajectories pile up. This distribution
is similar to a rainbow. The density of outgoing tra-
jectories diverges at the ”rainbow velocity” . Below this
velocity the intensity is zero, similar to Alexander’s dark
band in the optical rainbow. Above the rainbow velocity
the intensity distribution decreases smoothly.
Despite the similarities, there is a crucial difference
between a rainbow created by sunlight refracted by wa-
terdroplets and our ’velocity caustic’. The appearance of
a rainbow is usually a deterministic process, where the
scattering angle is uniquely determined by the impact
parameter. In our experiment the incoming velocity vi
is single-valued, and is distributed over a range of out-
going velocities by the stochastic process of spontaneous
Raman scattering.
The position of our caustic is independent of κ, like in
the optical rainbow, where the rainbow angle does not de-
pend on the droplet size. To check this, we measured for
three different values of the decay length κ−1the atomic
density profile at the upper turning point (Fig.3c). This
is defined as the highest position reached by the peak
density. The shape of the cloud did not change, but the
number of atoms decreased with decreasing κsince the
total number of scattered photons is lower.
By varying the detuning, βis changed, resulting in achange of the rainbow velocity. Note that this amounts
to varying the “degree of dissipation.” This is analogous
to the dependence of the rainbow angle on the refractive
index. For various values of the detuning we measured
the upper turning point, as a fraction of the initial MOT
height (Fig.4a,b).
In order to quantitatively analyze our experimental
data, we write the optical hyperfine-pumping rate dur-
ing the bounce as Γ′(z) = (1 −q)ΓU1(z)/¯hδ1, where q
is the branching ratio to F= 1. We define η(v) as the
survival probability for the atom to reach the velocity v
without undergoing optical pumping. This function de-
creases monotonically as ˙ η=−Γ′η, with η(vi) = 1. For
|v| ≤ |vi|, the solution is η(v) = exp( −(v− |vi|)/vc) ,
where vc≡2κ¯hδ1/(1−q)mΓ, with mthe atomic mass.
When the pumping process takes place at a certain veloc-
ityv=vp, the atom leaves the surface with velocity vf=/radicalBig
v2p(1−β) +βv2
i(Fig.1). This results in a distribution
of bouncing velocities, resulting from atoms which were
pumped while moving toward ( v−
p) or away from ( v+
p) the
surface : n(vf) =η(v−
p(vf))×|∂v−
p
∂vf|+η(v+
p(vf))×|∂v+
p
∂vf|for√β|vi| ≤vf<|vi|, which diverges at vf=√β|vi|. This
velocity caustic originates from atoms which are pumped
near the turning point. There are two reasons for scat-
tering preferentially at this position. An atom spends a
relatively long time in the turning point since its velocity
is lowest there. In addition, the intensity of the EW, and
thus the photon scattering rate is highest in the turn-
ing point. The divergence is an artefact of the ray-optics
description. It disappears due to diffraction when the
atoms are treated as matter waves.
To compare the experimentally obtained spatial dis-
tributions with the model, we first calculated the one-
dimensional phase-space density ρ(z, v). The spatial
distribution is obtained by projecting ρon the z-axis.
Initially the MOT is described by a normalized gaus-
sian,ρ(z, v)∝exp(−(z−z0)2/2σ2
z)exp(−v2/2σ2
v) with
z0= 5.8 mm, σz= 0.3 mm and σv= 3.8 cm/s. This dis-
tribution falls due to gravity and expands due to thermal
motion. When it arrives at the EW it is reflected, and
the velocity distribution is changed in the way described
above. We measure the spatial distribution of inelasti-
cally bounced atoms after a time of flight t. To illustrate
the agreement with the experimental data, the result for
a time t= 7 ms is drawn as the thick line in Fig.2.
Considering the divergence in the distribution of out-
going velocities, it is interesting to calculate the max-
imum phase space density ρmax, in the context of
evanescent-wave cooling. Due to the broad outgoing ve-
locity distribution, an unexpected result followed: al-
though energy is removed from the atoms, the pump-
ing process leads to a decrease of the maximum value
of the phase-space distribution. After the bounce, ρmax
is roughly five times lower than the initial 105sm−2
per atom of the MOT for our experimental conditions.
Note that this is a one-dimensional calculation; trans-
2verse spreading is ignored.
When we compare ρmaxof the bounced atoms with
that of MOT: ρmax=α×ρMOT, we find that the con-
dition for α >1 is mainly determined by βandvc(both
depending on the detuning), and the falling height h.
Note that the increase of ρmaxdoes not violate Liou-
ville’s theorem. This is possible due to the spontaneous
character of the Raman transition. In Fig. 5, αis plotted
as a function of δ1andh. For a fixed falling height, α
reaches a maximum for a detuning δ1such that 2 vc=|vi|.
This condition corresponds to a maximum pumping rate,
η(0)Γ′in the turning point. In the limit of low vc, the
scattering rate at the turning point is low. When vcis too
high the F= 1 state is too quickly depleted which also
results in a low scattering rate. Decreasing the fall height
also yields a higher αsince this determines the velocity
of the atoms arriving at the EW, hence the duration of
the bounce. This means that in EW-cooling experiments,
the first bounces reduce the falling height but lead to a
decrease of ρmax. Only during later bounces can ρmax
increase, i.e. can the cloud be cooled. It should be noted
that this calculation is only valid for a single inelastic
bounce with an initial narrow distribution. After many
bounces the distribution will become barometric [11].
A very intriguing aspect of our experiment is the pos-
sibility to observe supernumeraries: interference of two
trajectories with the same outgoing velocity, but with
different pumping velocity vp. In the velocity distribu-
tion after the inelastic bounce, the high velocity tail is
the sum of two contributions: from atoms which move
towards and atoms moving away from the surface when
they are pumped to F= 2. Interference between these
two paths should result in oscillations in the velocity dis-
tribution. This is a nontrivial effect because it involves
the spontaneous emission of a photon.
Given our experimental parameters, we expect a typi-
cal oscillation period of 1 cm/s. This spacing depends on
κ, just as in a rainbow the supernumeraries depend on
the droplet size. These oscillations are not yet resolved in
our measurements, because in the EW-field the magnetic
substates are not degenerate, each producing a different
velocity distribution.
In conclusion, we observed a new type of caustic due
to the stochastic distribution of a monochromatic input.
This distribution is similar to the common rainbow. The
caustic appears in the velocity distribution of inelastica lly
bouncing rubidium atoms on an optical evanescent-wave
atom mirror. Calculations describing the phase space
evolution of the atomic cloud show that an increase as
well as a decrease of the phase-space density is possible.
This work is part of the research program of the
“Stichting voor Fundamenteel Onderzoek van de Ma-
terie” (FOM) which is financially supported by the “Ned-
erlandse Organisatie voor Wetenschappelijk Onderzoek”
(NWO). R.S. has been financially supported by the Royal
Netherlands Academy of Arts and Sciences.[1] M.V. Berry, in Les Houches, Session XXXV (1980),
Physics of defects , R. Balian et al., eds., (North Holland,
1981) p. 455-543.
[2] H.M. Nussenzveig, Sc. Am. 236, (1977).
[3] K.W. Ford and J.A. Wheeler, Ann. Phys. 7, 287 (1959);
D. Beck J. Chem. Phys. 37, 2884 (1962).
[4] M.E. Brandan and G.R. Satchler, Phys. Rep. 285, 143
(1997).
[5] J. S¨ oding, R. Grimm and Yu.B. Ovchinnikov, Opt.
Comm. 119, 652 (1995)
[6] P. Desbiolles, M. Arndt, P. Szriftgiser, and J. Dalibard ,
Phys. Rev. A 54, 4292 (1996);
[7] D.V. Laryushin, Yu.B. Ovchinnikov. V.I. Balykin, and
V.S. Letokhov, Opt. Comm 135, 138 (1997);
[8] R.J. Cook and R.K. Hill, Opt. Comm. 43, 258 (1982).
[9] V.I. Balykin, V.S. Letokhov,Yu.B. Ovchinnikov, and A.I
Sidorov, JETP Lett. 45, 353 (1987)
[10] D. Voigt, B.T. Wolschrijn, R. Jansen, N. Bhattacharya,
R.J.C. Spreeuw, and H.B. van Linden van den Heuvell,
Phys. Rev. A 61, 063412 (2000)
[11] Yu.B. Ovchinnikov, I. Manek, and R. Grimm, Phys. Rev.
Lett.79, 2225 (1997)
v/c1001/c1002
F’=3
6.8□GHzU□(z)
Height□above□prism,□zF’=0
F=2
F=1
a) b)F=1F=2
vvf
p
i2
U□(z)1
FIG. 1. (a) Involved atomic levels of87Rb (b) An atom
enters the evanescent wave in its F= 1 state with initial
velocity vi. It is decelerated and spontaneously scatters a
photon at a velocity vp. After being pumped to F= 2 it
accelerates and leaves the potential with asymptotic veloc ity
vf.
3Height□above□prism□(mm)Optical□Density3□ms 5□ms 7□ms 9□ms 11□ms 13□ms
0 04.4
1.42.97□ms
FIG. 2. Absorption images (2 .2×4.5mm2) at different
moments after bouncing on an EW, with detuning 92Γ, and
κ−1= 1µm. A line sum shows the atomic density distribution
above the prism. The dotted line indicates the prism surface .
The initial MOT was located at 5.8 mm above the surface.
The thick curve in the middle is the result of our calculation
at 7 ms.
0 400 800 1200 16000.000.040.080.120.160.20
detuning /c1001/2/c112(MHz)/c98
Height□(mm)b)a)
c)Height□(mm)2 3 1 0 2 3 1 0Optical□density
FIG. 3. a) Measured height of the upper turning point
as a fraction of the initial (MOT) height. This represents
β, the relative optical potential strength of the F= 1 and
F= 2 ground states. The curve results from a calculation
including the full level structure. b) Density profiles for v ar-
ious detunings: δEW= 67,150 and 233Γ. The curves are
measured at the moment that the density peak reaches its
highest point and have the same scale. c) Density profiles
att= 14 ms after the bounce for κ= 1.96µm, 0.80µm and
0.62µm,δEW= 150Γ. The results are rescaled such that the
density peaks are equally high. Variation of κin this range,
has no influence on the position and shape of the spatial den-
sity profile..n(v□)
0.1 0.3 0.2 -0.1 -0.2 -0.3 0vz(nm) 400
300
200
1000.1 0.2 0.3
i -vib
v□( )m/sbv□( )m/sb
FIG. 4. Construction of the velocity caustic in terms of
phase space trajectories. The velocity of the incident atom s
isvi=−0.34 m/s. The thick solid line shows the trajectory
of the lower hyperfine state bouncing elastically. If the ato m
is pumped to the other hyperfine state it continues on a dif-
ferent trajectory. The thin lines represent possible outgo ing
paths, each starting at a different pumping velocity, depend -
ing on the position of Raman-transfer. The density of out-
going trajectories diverges, yielding a caustic in the velo city
distribution. Shown in the upper curve are the total distri-
bution (solid line), and the the contribution of atoms movin g
towards (dashed) and away from (dotted) the surface while
being transferred.
02
1
0 /c49/c48/c48/c71 /c50/c48/c48/c711
1.30.70.4
fall□height□(mm)
detuning /c1001
FIG. 5. Contourplot of the ratio ( α) of phase space density
after and before the inelastic bounce. The thick line corre-
sponds to α= 1. Other parameters were fixed: κ= 1.3×106
m−1, MOT radius 0.3 mm, T= 15 µK.
4 |
Observation of transformation of chemical elements during electric discharge
Urutskoev L.I., Liksonov V.I.
"RECOM" RRC “Kurchatov Institute”
Moscow, Shchukinskaya st. 12-1
tel. 196-90-90, fax 196-1635
e-mail: sergeysmr@mail.ru
Tsinoev V.G.
RRC “Kurchatov Institute”
123182 Moscow, Kurchatov square, 1, tel. 196-73-65
Abstract.
Results of experimental studies of electric explosion, in water, of foils made of extremely pure materials
are presented. New chemical elements detected both by spectroscopic measurements during the electric
discharge and by a mass-spectrometer analysis of sediments after the discharge have been found to appear. A
"strange" radiation associated with the transformation of chemical elements has been registered. A hypothesis
has been put forward that particles of the "strange" radiation have magnetic charge.
Introduction
The physics of electric explosion of wires in water has been discussed in many papers, reviews, and
monographs [1-3], which is mainly due to the large interest to this phenomenon for numerous practical
applications. One of such tasks is shattering of concrete foundations. As a rule, devices for this purpose use
relatively low-voltage high-capacity batteries (U~ 5 kV) to obtain the required energy storage of order of
several tens of kJ. The characteristic discharge time for such batteries is about several hundreds microseconds.
Ordinarily, the electric discharges are produced in a narrow (d~20 mm) pit filled with a liquid and the discharge
is initiated by exploding wires.
A feature of such an electric explosion scheme is that reflected waves act on the plasma channel
produced in a closed volume. They rapidly brake the motion of the channel boundaries, the channel stops
expanding, and the pressure at the surface significantly increases. In this processes pressure in the channel can
exceed the one attainable at a shock front.
An additional increase of the energy input into the channel can be attained by initiating the discharge by
wires made of materials which has a larger thermal effect in reactions with water. These are titanium,
zirconium, and beryllium. This possibility has been studied in both the very early [1-4] and later papers [5].
The present work was initially dedicated to study of the efficiency of electric explosion of titanium foils
in water to shatter concrete. The experiments revealed that the concrete has been smashed by electric explosion
and its fragments have flown away with substantial velocities. A rough estimate of kinetic energy of the
fragments based on a visual rapid taking pictures (with a rate of 300 frames per second) was ~8 kJ. It is the
willing to study such an effective mechanism of the capacitor battery energy transformation into the kinetic
energy of concrete splinters that initiated the experiments results of which are presented in this paper.The experiment scheme, diagnostics, and results.
The scheme of the experiment is shown in Fig. 1. The capacitor battery has been discharged into a foil
in water. The energy storage of the battery at a charging voltage of U ~ 4.8 kV was W~50 kJ. The capacitor
spark-gaps [6] provided the commutation of the capacitor bank. The energy was supplied to the load by cables
3 with an induction of L=0.4 mcH. The load was a Ti foil that was welded to Ti electrodes 5 by pressure
contact welding. The electrodes were mounted on a polyethylene cover 6 which in turn was attached through
seals 7 to the explosion chamber 8 also made of polyethylene. The explosion chamber represented a torus
which was filled with a liquid through eight holes 9 uniformly drilled around a circle. Distilled water was used
as a working liquid in most experiments. The number of load varied from one to eight in different experiments.
Analog oscillographs and rapid analog-cipher transformers attached to computers were used to register electric
signals. The typical oscillograms of the current and voltage are shown in Fig. 2. Since various diagnostics were
used in the experiments, we believe worthwhile describing diagnostics and methods in line with experimental
material presentation.
Fig. 1.
1 - capacitor battery, 2 - spark-gap, 3 - cable,
4 -foil, 5 - electrode, 6 - polyethylene cover, 7 - seal,
8 - explosion chamber, 9 - distilled water .
During experimental studies of electric explosion of foils in water, an intensive glowing was found to
appear above the dielectric cover. In Fig. 2 we present oscillograms of signals from the photodiode (PD) and
the photo-multiplying tube (PMT-35) mounted above the dielectric cover. As seen from the oscillograms, at the
moment of the current disruption, which is noted by many authors [3], a glowing emerges above the explosion
chamber, which persists over the time period exceeding the current pulse duration by more than 10 times. Since
the beginning of the glowing coincides with the voltage drop (see Fig. 2), it is tempting to explain the
appearance of the glowing by the ordinary electric break-down in the supplying high-voltage inputs. However,
the experimental results described below can not be explained by electric break-down only.
The first argument is that by supplying a static voltage of U~10 kV (just such a tension amplitude
appears at the moment of the current disruption), we do not observe electric break-down on the power high-
voltage inputs.Fig. 2.
a. Tension oscillogram
b. Load current oscillogram
c. Signal from photodiode
d. Signal from PMT-35 attached to interference filter λ=432 nm
The second argument relates to a significant difference between the duration of the current pulse ~0.15
ms and that of the glowing ~5 ms. However, the recombination time of plasma, which appeared in the air, ~0.1ms, is much smaller than the observed glowing duration, which does not allow us to explain the observed
glowing duration by electric break-down during the current pulse [7]. In the experiments the spectrum of the
glowing and the dynamics of the ball-like plasma formation (BPF) were studied.
To study the spectral composition of the radiation, three types of spectrographs were used STE-1 (4300
A-2700 A), ISP-51 (6500 A-4500 A), and DFS-452 (4350 A-2950 A), which allowed us to obtain the time-
integrated spectrum (during one shot). The temporal behavior of the narrow spectral fragment was studied with
two PMT-35 located 1 meter above the setup with two different interferometric filters ( λ1=432 nm, λ2 =457
nm) (Fig. 3).
Fig. 3.
Scheme of diagnostics set up
1. Torus
2. High-voltage lead-ins
3. Ball-like plasma formation
4. Mirror
5. Dielectric cover
To register the image of the glowing, three methods with different time resolution were used. The most
"quick" method used electron-optical transformers (EOT)[8]. Six EOTs in the frame regime with the exposure
time T~130 mcs and the frame delay time T~1 ms allowed to get 6 frames in one shot. The EOTs were set at a
distance of 2.5 m from the axis of the setup as shown in Fig. 3. A mirror was mounted ~1 m above the setup by
the angle 45 degrees to the vertical line, which permitted us to simultaneously register two projection of the
glowing.
To register the glowing an industrial high-speed camera "IMAGE-300" was also used, which enabled us
to register 300 frames per second with an exposure time of ~2 ms. For the camera synchronization, a special
quartz clocks was designed. The time-integrated wide-field pictures were taken with a standard TV-camera.
Fig. 4b presents an EOT-gram which clearly shows that the glowing appears in the middle between the
electrodes above the dielectric cover and has a spherical shape. Using signals from calorimeters, photodiodes,
and with account of the results of spectral measurements, the light energy emitted by BPF was estimated to be
W~1kJ.
Based on the results of more than 100 tests, the typical dynamics of the spherical-like glowing can be
described as follows. At the moment of the current disruption a very bright diffuse glowing emerges in the
channel above the setup (Fig. 4a), as if the total space is glowing. Then the glowing fades and in the next spot aspherical-like glowing is clearly seen. No dynamics is observed during the subsequent 3-4 ms (Fig. 4c,d,e), and
then the glowing sphere starts dividing into many small "balls". In some experiments the "ball" was noted to
rise by 15-30 cm above the dielectric cover and then dissociate (Fig. 4f).
a b
c d
e f
Fig. 4.
Pictures seen on the EOT screens.
Exposure time 130 mcs.
The moment of exposure in Fig, 4a coincides with
the time of current pulse.
Time delay between the frames 1 ms
It should be noted that the characteristic feature of the spherical plasma formation (BPF) is its selectivity
with respect to the earth coating of the power and diagnostic cables. In experiments where "earths" of the high-
voltage cables were not thoroughly insulated, the BPF often "shorted" on the cable coating, as is seen from
EOT-grams. This fact was also supported by measurements of currents, I, on a shunt built in the power cable
coating. As seen from Fig. 5, at the moment when BPF touches the cable coating, a current emerges in the
circuit, the so-called "echo".7ms
Fig. 5.
The signal form the shunt built in the high-voltage
cable coating
Long-lived plasma formations in vacuum were observed in some experiments in various laboratories
[9,10]. A distinctive feature of the experiment under discussion is spectral measurements. It is the results of the
spectral measurements that became a key to understand the physics of BPF and largely determined the direction
of further studies. Fig. 6 demonstrates the fragments of the optical spectra obtained with spectrographs located
as shown in Fig. 2. Fig. 6 reveals a spectral line structure in the entire spectral band. In addition, a continuum is
also seen, especially in the red part of the optical spectrum.
a)
b)
Fig. 6
a). Fragment of the plasma emission spectrum obtained with ISP-51
spectrometer (the upper and bottom spectra are from Cu and Zn standards)
b). Fragment of the plasma emission spectrum (upper) and the iron
standard spectrum (bottom) in the wavelength range 3800-4100A obtained
with STE-1 spectrometer.
The identification of spectral lines led to two unexpected results. First, no oxygen and nitrogen lines
were found (only very weak traces of them were present in separate "shots"), whereas just these lines arealways seen during electric discharge in the air. Second, a lot of lines (more than 1000 lines in individual
"shots") and, accordingly, a lot of the corresponding chemical elements were discovered. The spectral analysis
revealed that most abundant elements in plasma were Ti, Fe (even very weak lines were detected), Cu, Zn, Cr,
Ni, Ca, Na. If the presence of Cu and Zn lines in the spectrum can be explained by a sliding discharge along the
setup units and power-supply cables, the presence of other lines can not be interpreted. Variation of the
experiment conditions, in particular change in mass of the exploding foil, led only to the redistribution in the
lines intensity, with the element composition changing insignificantly.
Since titanium foils were exploded, the presence of Ti lines suggested that some fraction of the foil
material penetrates through the seals to occur above the setup. In order to check this assumption, the mixture of
water and the foil (the "sample" below) was extracted from the channels and subjected to a mass-spectrometer
analysis. The results of the analysis are shown in Table 1. The table demonstrates that the original foil consists
of 99.7% of titanium. The isotope analysis of the foil shows that Ti isotopes are present in their natural
abundance.
Element Fraction of atoms ,
%
Ti 99.71643
Na 0.00067
Mg 0.00068
Al 0.00921
Si 0.00363
P 0.03078
S 0.03570
Cl 0.00337
Ka 0.00253
Ca 0.03399
V 0.00195
Cr 0.00844
Mn 0.00253
Fe 0.10613
Ni 0.04193
Co 0.00202
Table 1.
The original composition of titanium foil.
The method of studying the "samples" was as follows. The "sample" was first evaporated to yield a dry
sediment, which was then carefully mixed to a homogeneous state, and after that was subjected to mass-
spectrometer analysis. The mass-spectrometer in use measured atomic masses starting from carbon. Clearly,
gases could not be measured with mass-spectrometer. It should be noted that since the mass of the powder
under study was about 0.5 grams, it could be visually seen that after the evaporation the "sample" had a
inhomogeneous structure.
Unexpected were the results of mass-spectrometer analysis of the "samples", with the typical example
presented in Fig. 7a. The total number of all atoms detected in the "sample" is taken for 100%. Fig. 7b shows a
histogram of Ti isotope distribution discovered in the same "sample" and in the original foil for comparison (the
natural ratio). The total mass of Ti is taken for 100% in histograms in Fig. 7b. Note that the isotope ratio
strongly changes in Ti remained after the "shot". The comparison of histograms reveals that the percentage of
"shortage" of Ti48 in Fig. 7b coincides with the "shortage" of Ti in Fig. 7a. Abundance of atoms V, Ni, Ba, Pb 0.1-0.01%
00,511,52
B Na Mg Al Si Ca Cr Fe Cu Zn
a)
Ti isitope content, %
020406080
Ti46 Ti47 Ti48 Ti49 Ti50Experiment
Nature ratio
b)
Fig. 7.
Results of mass-spectrometer analysis of products in test 226(Ti load).
a) Percent composition of atoms of "alien" elements in the sample. The
fraction of Ti atoms in the experiment products is 92 %.
b) The ratio of Ti isotopes before and after the experiment
All measures were taken to provide the "purity" of the experiment. All electrodes were made of highly
purified titanium, new polyethylene caps were used in each "shot". All seals were also made of polyethylene.
Since the pressure in the chamber increases during the "shot" due to ohmic heating and chemical reaction of Ti
with water, nothing can penetrate the chamber from outside. So only Ti and possibly carbon is expected to be in
the "sample". However, in mass-spectra of the "samples" obtained in more than 200 experiments lines of
elements ("alien" elements) were detected which were absent in the original material of the exploding foil and
electrodes.
To avoid possible errors in measuring mass-spectra, some control "samples" were divided into three
parts and directed to three different mass-spectrometers in different laboratories.
Other methods such as electron sounding, X-ray structure, X-ray phase, and X-ray fluorescent analyses,
have also been used. The results of electron sounding of a fragment of one of the "samples" are shown in Fig. 8.Of course, the results obtained by the different methods differ numerically, but qualitatively all methods reveal
the presence of a substantial amount of "alien" elements.
Fig. 8.
The result of electron sounding of a fragment of one of the samples.
The averaged result of mass-spectrometric analyses performed for the "samples" obtained in different
shots is shown in Fig. 9. The mean fraction of Ti transformation is 4%. The comparison of histograms in Fig. 7a
and Fig. 9 discovers the same elements among the "alien" ones, although their relative contributions in the
mass-spectra are of course different. This difference in the specific weight is explained by different conditions
of the experiments. The following parameters were changed in the experiments: the energy input into the foil,
the number of channels, the mass and size of the foil, an external magnetic field (in some experiments). Thus
the experiments using Ti foil as a load revealed the presence of the same "alien" elements. The same conclusion
was obtained from spectrometric measurements.051015202530
Na Mg Al Si K Ca V Cr Fe Ni Cu Zn
Fig. 9.
The mean percent of "alien" elements atoms in 24 tests (N 169-240) for titanium load.
As noted above, a correlation has been observed between the fraction of mixtures in the "text" and the
"skewness" of the isotopic distribution of titanium remaining in the "sample". In all isotopic analyses of
sediments the relative fraction of isotopes Ti46, Ti47, Ti49, Ti50 was observed to increase and Ti48 to decrease.
This experimental fact enabled us to suppose that all the decrease of Ti is due to "disappearing" of Ti48 isotope.
The plot in Fig. 10 is drawn by assuming that the total "disappearing" (burning out) of Ti load is due to only the
burning out of Ti48. Only experiments with titanium foils as a load were taken to this plot. The plot shows that
the points either fall along a straight line y=x, or lie in the upper hemiplane. The last fact indicates that
predominantly "alien" elements fly out of the channel, which is in qualitative agreement with spectral
measurements from which follows that the fraction of the "alien" elements in plasma is quite significant.
The correlation of decrease of Ti48
fraction (percents by weight) and "alien"
elements fraction
(21 experiments )
010203040
0 5 10
"alien" elements fraction , %decrease of Ti48
fraction, %
Fig 10.
The correlation of decrease of Ti48 fraction and increase of the"alien" elements fraction (percents by weight).
Fig. 11 presents the histogram of the mean composition of the products for experiments with zirconium
Zr foil as a load. The original zirconium foil contained 1.1% of niobium, which was subtracted from the final
product composition. Comparing Fig. 9 and Fig. 11 suggests each original loads produces individual spectrumof chemical elements. This statement holds for other foils (Fe, Ni, Pb, V, Ta), which were used in other
experiments.
0102030
Al Si Ca Ti Cr Fe Ni Cu Zn
Fig. 11.
The mean percent of atoms of "alien" elements in 5 tests with zirconium load.
Since the transformation of elements must have been associated with some radioactive emission,
intensive search for gamma-radiation and neutrons have been done. To register gamma-radiation integral
dosimeters, X-ray films, and CsI scintillator detectors with PMT-30 have been used. No significant X-ray flux
has been detected in any experiment. As followed from mass-spectrometric results, the number of acts of
transformation was 1019-1020 per shot, so clearly even one gamma-ray photon per transformation act would led
to an enormous gamma-ray flux of P~1020.
In order to register neutrons we used 2 plastic scintillator detectors with PMT-30. The detectors were
mounted at a distance α1~0,4m, α2~0,8m from the setup axis. In Fig. 12 the typical signal from PMT-30 is
presented with a duration of T~100 ns. Such a short duration was a great surprise since the current pulse
duration was ~ 20 ms. In order to measure the time of arrival of particles, a special transformer was designed
which formed a standard pulse of ~10 ms from external signal with ~10 ns. Thus the studied pulse from the
detector triggered the oscillograph, then was directed to the oscillograph input through a delay line, and only
after that entered the transformer and ACP. The time delay in the turn-on of two oscillographs registering
signals from two plastic detectors allowed us to measure the radiation propagation velocity. It was found to be
V~20-40 m/s. Such a low velocity precluded signals to be neutrons, since then they must have been ultra cold
and could not reach the detector and moreover to overcome the light-shielding cover made of aluminum. To
understand the nature of the radiation and obtain its "self-portrait", a method using photo emulsions was
applied.Fig. 12.
The signal from polystyrol scintillator detector.
The following materials have been used in the experiment: a fluorographic film RF-ZMP with
sensitivity 1100 R-1 at a level 0.85 above the haze, a radiographic medical film RM-1MD with sensitivity 850R-
1 at a level 0.85 above the haze, nuclear photo plates of type R with a thickness of the emulsion layer of 100
mcm, high-resolution photo emulsions with a sensitivity ~0.1 GOST units and a resolution of up to 3000
lines/mm.
All the materials were processed after the exposure in the corresponding developers: the fluorographic
films in D-19 developer during 6 min at temperature 200C, the plates in a phenydon-hydrochinon developer
using isothermal method.
The inspection of the processed materials revealed micro and macro effects. Macro effects included
those that can be seen by naked eye or using a magnification glass with up to 5 times magnification. Micro
effects included those seen under magnification from 75 to 2025 times. Films and photo plates were set at
different distances from the center of electric explosion (from 20 cm to 4 m) in radial and normal planes
assuming cylindric symmetry of the experiment (Fig. 13).Fig. 13. The scheme of photo detectors location
1 - the site of electric explosion of foils
2 - permanent magnets
3 - the plate with nuclear emulsion
4 - films
5 - magnetic field coil
6 - films near the permanent magnet
All the materials were carefully wrapped in the black paper, which had preliminary been inspected to
have damages. After the exposure in the setup and developing photodetectors, the paper has been expected once
again.
The very first experiments revealed a wide variety of track forms, including continuous straight lines,
dumbbell-like ("caterpillar") tracks, and long tracks with a complex form similar to spirals and gratings.
Fig. 14a demonstrates a typical very long (1-3 mm) track similar to that of a caterpillar or a tire-cover
protector. These tracks are characterized by having the second parallel trace with darkening and length different
from the main one. The track presented in Fig. 14a was formed on the fluorographic film RF-ZMP with
emulsion layer thickness 10 mcm. In Fig. 14b a magnified fragment of the track is shown which clearly
demonstrates a complicated pattern. Notably that with a grain size of ~1 mcm, the track width is about ~20
mcm. The estimate of the energy of particles obtained from the darkening area is E~700 MeV assuming
Coulomb braking. Taking into account the position of the photo detector (shown in Fig. 13) and the track size,
the track can not be explained by alpha, beta, or gamma-radiation (recall that RF-film is wrapped in the black
paper and is surrounded by the air). To check the nature of the "strange" radiation, the remnants of the foil and
water were extracted from the channel after the explosion and put into a Petri cap (the "sample"), and the photo
detector was set 10 cm away as shown in Fig. 15a. The film RF pressed to fiber glass washer was used, and the
entire detector was enveloped in the black paper. The fiber glass washer was used because in the previous
experiments we noted that the "strange" radiation clearly demonstrates properties of the transition radiation.
The exposure time was T~18 hours. The result is presented in Fig. 15b. The inspection of Fig. 14 and 15
suggests that the darkening of the film in both cases was due to identical reasons. This in turn implies that the
radiation was not caused by acceleration and had a nuclear origin. It should be noted that the position of the
detector planes normal to the radius vector in both cases allows the interpretation that the source of the
registered emission moved with a non-zero angular velocity.Fig. 14.
The typical track on the film.
Fig. 15.
a) The scheme of experiment. 1 - Petri cap; 2 - the sample; 3 - film; 4 - fiber glass.
b) The track and its magnified fragment.
The detection of the same tracks using nuclear emulsions with thickness 100 mcm permits us to state
that the source caused the darkening flies strictly in the photo emulsion plane, since the depth of the beginning
and the end of the track inside the emulsion differs by less than 10-15 mcm.
Assuming the electric pulse in Fig. 12 and the track are due to one and the same reason and accounting
for the track length and the pulse duration, we arrive at the estimate of the radiation source velocity β=10-3.A series of experiments was performed to study the effect of the external magnetic field on the picture
observed. Using a magnetic coil located as shown in Fig. 13, a weak magnetic field H~20 G was imposed in the
site of explosion.
The photodetectors were set as shown in Fig. 13(3). The typical tracks registered are shown in Fig.
16(a,b), with a nuclear photo emulsion as photo detector. It is seen from the Figure that the track strongly
changes and the trace becomes "comet"-like in shape.
a
b
c
Fig. 16.
a) The "comet"-like trace
b) A magnified fragment of the "comet head"
c) The "flaky" trace500 mcm
100 mcm
3 mm 1,5 mm0,7 mmA more detailed study of the structure under microscope with 225 times magnification allowed us to
single out a round head (Fig. 16b) with the darkening density D>3 and a long tail with decreasing darkening
density similar to the "comet tail" (Fig. 16a). Six such "comets" were detected inside the area 4 cm2. Their sizes
varied from 300 mcm to 1300 mcm and the energy of particles derived from the darkening area reached
E~1GeV.
In some experiments the detector (and not the entire setup) was put in the magnetic field. Fig. 16c
shows the traces obtained by a photo detector consisting of three RF-films folded together and located near a
samarium-cobalt magnet (B ~ 1.2 kG) as shown in Fig. 13(b).
The positions of darkenings in Fig. 16c coincide geometrically, which exclude them to be artifacts. The
energy absorbed in the three films with account only the photo emulsion layer thickness (~10 mcm) is
estimated to be E~700 MeV.
Thus it is clearly seen that the magnetic field affects the "strange" radiation. In addition to the tracks
shown, we have registered some tracks quite different in shape from the "classical" ones. Part of these tracks
not presented in this paper is very similar to scratches or ink spots. Exactly the same tracks were observed in
experiments by Matsumoto [11] using other types of photo detectors and in experiments with breakdown in
water. This precludes us from unconditionally relating them to artifacts. However, already today we can say
with a significant reliability that one or two types of particles can hardly explain all tracks which appear during
the registration of the "strange" radiation.
Discussion of experimental results. A magneto-nucleon catalysis hypothesis.
Let us try to emphasize the main properties of the observed phenomena. Apparently, it is the symmetry
of location of plasma channels that was the reason for the BPF appearance, which in turn allowed the dynamics
and optical spectrum of the emergent radiation to be studied. >From Fig. 5 a purely visual association between
the BPF and ball lightning arises. There are at least two major problems in physics of ball lightning: the reason
for appearance and failure to explain the source of radiation. The experimental results reported here can suggest
a reason for the BPF appearance. Indeed, the optical spectra analysis and mass spectrometry results are in
qualitative agreement. This allows us to suppose that during the process a fraction of materials leaks through
the seals. However, even if not to analyze how plasma penetrates through the seals, two questions remains.
Why all the plasma coalesces into a ball and does not dissociate and why this occurs in such a short time since
we do observe the BPF already on the first EOT frame (in some "shots").
One can try to explain the BPF using the cluster model [9] of the fractal ball model [12], however we
have looked for a hypothesis which can explain all experimental facts. Such a hypothesis, in our opinion, could
be the formation of magnetically-charged particles (magnetic monopoles). The first attempt to explain the ball
lightning by magnetic monopoles was done in paper [13]. This paper suggested to explain the properties of ball
lightning by Rubakov's effect [14] predicted for super-heavy monopoles which should exist in the Great
Unification Theory [15,16] (the so-called GUT monopoles). In our opinion, the experimental results obtained in
our paper provides no serious support for the hypothesis [13]. However the assumption that magnetic
monopoles form in the plasma discharge in water could be one possible reason for the obtained experimental
results, how unusual this assumption might appear.
Indeed, a wide track similar to the trace of a "crawling caterpillar" [17] was expected just for classical
monopoles [18,19]. The estimate of energy dissipated in the emulsion by radiation E~1 GeV coincides with that
expected for magnetic monopoles. A visual change in the track shape observed in the presence of magnetic
field also gives support to this assumption.
To confirm directly the fact of magnetic monopole creation in plasma discharge an experiment was
performed using the idea from paper [20], in which iron foils were suggested to be taken as a monopole trap. In
our experiment we used three 57Fe foils, which has an ideal structure and a significant field near the nucleus.
Since both N and S magnetic monopoles must appear, the foils under study were located near different
poles of a strong magnet with the magnetic field strength H~1 kG in anticipation of selection of monopoles.
Thus N-monopoles were expected to attract by the S-pole and S-monopoles by the N-pole of the magnet. Themagnets were set at a distance of h~70 cm away from the site of electric explosion. The third foils was used as
a standard.
Due to a large magnetic charge the monopoles "captured" in the trap must change the magnetic field
near 57Fe nucleus, which can be measured by Moessbauer effect for a sufficiently large number of the "trapped"
monopoles.
The results of measurements showed that in the foils located near the N-pole the absolute value of the
superfine magnetic field increased by 0.24 kG. In another foil (S) it decreased by approximately the same
amount 0.29 kG. The measurement error was 0.012 kG.
Fe-standard: H = 330,42 kG
Fe-north -N : H = 330,66 kG, ΔN = 0,24kG
Fe-south -S : H = 330,13 kG, ΔS = -0,29 kG
Taking into account that the magnetic field in 57Fe has the opposite sign with respect to its
magnetization, we can state with certainty that S-particles (at the N-pole of the magnet) increase the negative
superfine field, while the particles with the opposite sign decrease it, with the relative change being ~8·10-4.
From analysis of Moessbauer spectra of ferromagnetics, the absorption line width is known to increase.
This phenomenon is related to inhomogeneity of internal magnetic fields near nuclei. An analysis of spectra of
irradiated foils revealed an additional absorption line broadening comparable with the ordinary magnetic
broadening. This possibly relates to a chaotic absorption of the monopoles in the lattice of iron.
Fe-standard: r_1=0.334/0.300/0.235 mm/s
Fe-north -N : r_1=0.363/0.328/0.250 mm/s
Fe-south -S : r_1=0.366/0.327/0.248 mm/s
The measurements error 0.003 mm/s.
No quadruple shift of levels is discovered, i.e. no change of the electric field gradient in crystal is
observed. The results of this experiment strongly support the magnetic monopoles hypothesis. Unfortunately,
these measurements do not allow to decide whether magnetic monopoles have electric charge.
Using the hypothesis of magnetic monopole formation we can suggest that the observed BPF are
magnetic clusters. In analogy with [9] we can suppose that the role of ion is played by a monopole coupled with
the foil atom nucleus, and the solvation occurs due to interaction of the monopole magnetic charge with
magnetic moment of oxygen atoms.
The main regularities experimentally observed during the transformation of chemical elements can be
summarized as follows.
1. The transformation occurs predominantly with even-even isotope, which leads to a notable distortion
of the original isotope content.
2. Experiments with foils made of different chemical elements have shown that they transform into
individual spectra of elements, and the statistical weight of each element is determined by concrete conditions.
3. For the set of chemical elements resulted from the transformation there is a minimal difference Δ Åb
between the binding energy of the original element and the mean over spectrum binding energy of the formed
elements. The difference of binding energies ΔEb = Eorig − Eform (with account of the real isotope ratios)
calculated from mass-spectrometric measurements in different tests, falls within the range ΔEb < 0.1
MeV/atom, which is clearly due to mass-spectrometric measurements errors.
4. No increase in the binding energy difference ΔEb as a function of the transformation fraction of the
original chemical element has been detected.
5. All nuclei of chemical elements resulted from the transformation are in the ground (non-excited)
state, i.e. no appreciable radioactivity has been found.
To explain the element transformation, we have put forward the working hypothesis of magneto-
nucleon catalysis (MNC). We introduced this term to designate the process which supposedly occurs in the
plasma channel. The essence of MNC is that the magnetic monopole with a large magnetic charge and an evensmall kinetic energy can overcome Coulomb barrier and become bound with atomic nucleus. The MNC must
have many common features with muon catalysis [21], in which the Coulomb barrier is substantially decreased
due to the large mass of mu-meson. Since the magnetic monopole seems to be a stable particle, the MNC may
be more effective.
The experiments established that the transformation and hence the MNC occurs only inside the plasma
channel.
In conclusion, the authors greatly acknowledge the members of the staff of "RECOM" A.G. Volkovich,
S.V. Smirnov, V.L. Shevchenko, S.B. Shcherbak, and to members of the staff of "Kurchatov Institute" V.A.
Kalenskii, R.V. Ryabova, Yu.P. Dontsov, B.V. Novoselov, and A.Yu. Shashkov for assistance in the
experiments. We also deeply thank A.I. Voikov for financial support of the present study and A.A. Rukhadze
for support and help.
References
1. Electric explosion of conductors. Moscow, Mir, 1965.
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3. Burtsev V.A., Kalinin N.V., Luchinskii A.V. Electric explosion of conductors and its application in
electro-physical units. Moscow, "Energoizdat", 1990.
4. Naugol'nykh K.A., Roi N.A., Electric discharges in water. Moscow, Nauka, 1971.
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6. Mesyats G.A. Generation of powerful nanosecond pulses. Moscow, "Sov. Radio", 1974, p.90.
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10. Ball lightning in the laboratory. Moscow, "Chemistry", 1994, p. 256.
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Fusion Tech . V. 23, , 1993, p. 103-113.
12. Smirnov B.M. Fractal ball - a new state of matter. UFN (Physics-Uspekhi), v. 161, N8, p. 141-153,
1991.
13. Korshunov V.K. Polyakov-t'Hooft magnetic monopole drift in the air and the "Ball lightning"
phenomenon. Moscow, IVT AN SSSR, 1991, vyp. 2, p. 133.
14. Rubakov V.A. Super-heavy magnetic monopoles and proton decay. Pis'ma ZhETP (JETP Lett.), v.
33, N 12, p. 658-660, 1981.
15. Polyakov A.M., Spectrum of particles in quantum field theory. Pis'ma ZhETP (JETP Lett.) , 1974,
v.20, p.430 -433.
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17. Amaldi Å., Baroni G., Braduer H. and et. Search for Dirac Magnetic Poles. - CERN Report, 63-13.
18. Dirac P. A. M. - 1931, Proc. Roy. Soc. Ser. A, v.133, p60.
19. Shwinger J. Magnetic Poles and quantum the field theory - 1966, Phys. Rev., v.144, p.1087.
20. Martemjanov V.P., Khakimov S.Kh. Dirac monopole braking in metals and ferromagnetics. ZhETP
(JETP), v. 62, N 1, p. 35-41, 1972.
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vol. LXXI, N 4, p. 581-630, 1960. |
arXiv:physics/0101090v1 [physics.acc-ph] 26 Jan 2001Some Limit Theorems for Linear Oscillators
with Noise in the Coefficients∗
V. Balandin∗and H. Mais†
∗Institute for Nuclear Research of RAS,
60th October Anniversary Pr., 7a, Moscow 117 312, Russia
†Deutsches Elektronen-Synchrotron DESY, Hamburg
November 24, 2013
Abstract
Using the tools and methods developed in [1] limit theorems a re
proven for the linear oscillator with random coefficients. Th e asymp-
totic behaviour of the moments is studied in detail. The tech nique
presented in this paper can be applied to general linear syst ems with
noise and is well suited for the investigation of stochastic beam dy-
namics in accelerators.
∗extended version including proofs of a contribution by V. B. at the Workshop ”Non-
linear and Stochastic Beam Dynamics in Accelerators - a Chal lenge to Theoretical and
Computational Physics” L¨ uneburg (1997)
1Contents
1 Linear Oscillator with Noise in the Coefficients 3
2 Special Basis in the Space of Polynomials 6
3 Stopped Process 8
4 Asymptotic Behaviour of Moments 9
5 Nonresonant Case 12
6 First and Second Order Moments 17
7 Comparison with White Noise Model 19
8 Another Noise Model 21
9 Proof of the Theorems 24
9.1 Proof of the Theorem A . . . . . . . . . . . . . . . . . . . . . 24
9.2 Proof of the Theorem B . . . . . . . . . . . . . . . . . . . . . 26
9.3 Sketch of the Proof of the Theorem D . . . . . . . . . . . . . . 40
21 Linear Oscillator with Noise in the Coeffi-
cients
Starting point of our investigation is a nondegenerate ( ω0/ne}ationslash= 0 ) damped
linear oscillator under the influence of noise
¨x+ε(γ(t) +εα) ˙x+ω2
0/parenleftbigg
1 +ε
ω0η(t)/parenrightbigg
x=εω0ξ(t) (1)
or written as a system of two first-order differential equatio ns
˙x=ω0z
˙z=−ω0x+ε(ξ−γz−ηx)−ε2αz(2)
εis small parameter |ε|<1 .
Theε2proportionality of the deterministic term in the damping pa rt is
connected with the fact that we will discuss the dynamics on t ime scales
O(1/ε2) (it is the minimum time scale where the stochastic effects co uld es-
sentially influence the dynamics of our oscillator). If the d amping will be
weaker it will not affect the dynamics and we can neglect it, an d if it will
be stronger it will completely change the picture of the dyna mics, the typ-
ical time scales become exponentially large O(exp(1/εa)),a>0 for positive
damping and it will require other methods (see, for example [ 2]) that are
beyond the scope of this paper.
Noise has been introduced in the damping part ( γ(t)), as a modulation
of the frequency ω0(η(t)) and as an external driving force ξ(t).
As a model of noise we shall take stochastic processes defined by the
following scalar products
η(t) =/vectorb(t)·/vector y(t), ξ(t) =/vectorh(t)·/vector y(t), γ(t) =/vectord(t)·/vector y(t)
with nonrandom n-dimensional vectors /vectorb,/vectorhand/vectordwhich are quasiperiodic
intand which can be expanded into Fourier series
/vectorb(t) =+∞/summationdisplay
m=−∞/vectorbmexp(iνmt),/vectorb−m=/parenleftig/vectorbm/parenrightig∗
/vectorh(t) =+∞/summationdisplay
m=−∞/vectorhmexp(iνmt),/vectorh−m=/parenleftig/vectorhm/parenrightig∗
3/vectord(t) =+∞/summationdisplay
m=−∞/vectordmexp(iνmt),/vectord−m=/parenleftig/vectordm/parenrightig∗
with real frequencies νmsatisfying the condition
νl+νm= 0⇔m+l= 0.
In the main part of this paper the vector /vector y(t)∈Rnis assumed to be a
solution of the linear system of Ito’s stochastic differenti al equations
d/vector y=A/vector y·dt+Bd/vector w(t) (3)
whereAandBare (n×n) and (n×r) real constant matrices respectively,
and/vector w(t) is anr-dimensional Brownian motion, other choices for the noise
model will be described later on.
As smoothness properties of the vector functions /vectorb,/vectorhand/vectordwe shall
require the convergence of the series1
+∞/summationdisplay
m=−∞|νm|p/parenleftig/vextendsingle/vextendsingle/vextendsingle/vectorbm/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/vectorhm/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/vectordm/vextendsingle/vextendsingle/vextendsingle/parenrightig
<∞, p = 0,1 (4)
We denote
F=/braceleftig
p∈Z:|/vectorbp|+|/vectorhp|+|/vectordp| /ne}ationslash= 0/bracerightig
and introduce
Fk={(p, q)∈F×F:|νp+νq−kω0| /ne}ationslash= 0}.
Besides the smoothness condition (4) we also require
min
k∈ {0,1,2,3,4}inf
(p, q)∈Fk|νp+νq−kω0| ≥δ2
f>0. (5)
The condition (5) does not exclude resonances but requires t hem to be iso-
lated. This can be easily changed to some kind of Diophantine conditions
1For a complex vector /vector w∈Cnwe use the usual spherical norm |/vector w|=√
/vector w·/vector w=/radicalbig
w1·w∗
1+. . .+wn·w∗nand for a ( n×n) matrices with complex coefficients we shall
use the norm |M|=√
λwhere λis the greatest eigenvalue of the matrix M∗M, which is
compatible with the spherical norm for vectors.
4with increasing smoothness properties (4). Note that (5) is always satis-
fied for periodic functions (i.e. νp=p·ν) and for finite trigonometrical
polynomials with arbitrary frequencies.
In this paper we will assume that all eigenvalues λkof the matrix Ain
(3) have negative real parts, i.e.
Reλ k≤ −δ2
s<0, k = 1,...,n (6)
From this it follows (see, for example [3]) that if the initia l random vector
/vector y0, independent of the r-dimensional Brownian motion /vector w(t)−/vector w(0) for
0< t < ∞, has a normal distribution with mean value /an}bracketle{t/vector y0/an}bracketri}ht=/vector0 and
covariance matrix
/angbracketleftig
/vector y0·/vector y⊤
0/angbracketrightig
=∞/integraldisplay
0exp(τA)BB⊤exp(τA⊤)dτdef=D
then the solution of (3) /vector y(t,/vector y0) is a stationary, zero-mean Gaussian process,
with covariance function
ρ(τ) =
exp(τA)D;τ≥0
Dexp(τA⊤) ;τ≤0(7)
Although, later on we shall not restrict the initial conditi ons for/vector y
in our noise model to be equal to the above mentioned initial c onditions
generating stationary solutions of the system (3)2, all results will nevertheless
be expressed in terms of the spectral density associated wit h the covariance
function (7) Ψ( ω) = Ψ c(ω)−iΨs(ω) where3
Ψc(ω) =∞/integraldisplay
0cos(ωτ)ρ(τ)dτ=−A
A2+ω2I·D
2For simplicity we even shall take the initial condition to be a point in n-dimensional
Euclidean space, but if one will follow the proofs of the theo rems it will be clear that
all results of this paper will be correct if we use as initial c ondition an arbitrary random
vector, independent of the r-dimensional Brownian motion /vector w(t)−/vector w(0) for 0 < t < ∞,
additionally assuming that some moments of /vector y0are finite.
3Note that if the matrices AandB−1commute we use notationA
Bfor the product
AB−1.
5Ψs(ω) =∞/integraldisplay
0sin(ωτ)ρ(τ)dτ=ωI
A2+ω2I·D
For further purposes let us note that independently from rea lωthe norm
of the matrix Ψ( ω) admits the estimate
|Ψ(ω)| ≤¯C (8)
where ¯Cis some positive constant whose exact value depends on δ2
sand/vextendsingle/vextendsingle/vextendsingleBB⊤/vextendsingle/vextendsingle/vextendsingleand is unimportant for us.
2 Special Basis in the Space of Polynomials
Often, the influence of noise in systems such as (1) is studied by considering
its influence on the unperturbed invariants of motion such as energy
r=1
2/parenleftig
x2+z2/parenrightig
or functions of the energy. For our later study of arbitrary m oments we
introduce a special time dependent (non-autonomous) basis in the space of
polynomials.
For all nonnegative integers m,kwe define
Im, k= exp (i(m−k)ω0t)/parenleftbiggx+iz
2/parenrightbiggm/parenleftbiggx−iz
2/parenrightbiggk
It is easy to check that the functions introduced above admit the following
properties
a./parenleftig
∂
∂t+ω0/parenleftig
z∂
∂x−x∂
∂z/parenrightig/parenrightig
Im, k= 0
b. Im1, k1·Im2, k2=Im1+m2, k1+k2
c. I m, k=I∗
k, m
d. Im, m=/parenleftig
r
2/parenrightigm
e.|Im, k|2=Im, k·I∗
m, k=/parenleftig
r
2/parenrightigm+k
6Representing xandzas
x=x+iz
2+x−iz
2= exp(iω0t)I0,1+ exp( −iω0t)I1,0
z=x+iz
2i−x−iz
2i=iexp(iω0t)I0,1−iexp(−iω0t)I1,0
and using property bwe can express xm−k·zk(0≤k≤m) with the help
of the binomial theorem in the form of a linear combination of the functions
Ip, q
xm−k·zk= (i)k·
·m−k/summationdisplay
p= 0k/summationdisplay
q= 0(−1)q/parenleftigg
m−k
p/parenrightigg/parenleftigg
k
q/parenrightigg
exp(i(m−2(p+q))ω0t)·Ip+q, m−(p+q).
Form/ne}ationslash=k I m, kare functions with complex values. However, we can
also use as basis real valued functions Um, kandVm, kwhich are defined by
Um, k=Im, k+Ik, m
2=Uk, m, V m, k=Im, k−Ik, m
2i=−Vk, m.
Note further that the functions Um, kandVm, kcan be easily expressed
through the real valued functions ¯Um, kand¯Vm, k
¯Um, k=/parenleftig
x+iz
2/parenrightigm/parenleftig
x−iz
2/parenrightigk+/parenleftig
x+iz
2/parenrightigk/parenleftig
x−iz
2/parenrightigm
2
¯Vm, k=/parenleftig
x+iz
2/parenrightigm/parenleftig
x−iz
2/parenrightigk−/parenleftig
x+iz
2/parenrightigk/parenleftig
x−iz
2/parenrightigm
2i
which do not depend on time twith help of the following simple formula
Um, k
Vm, k
=
cos((m−k)ω0t)−sin((m−k)ω0t)
sin((m−k)ω0t) cos((m−k)ω0t)
·
¯Um, k
¯Vm, k
73 Stopped Process
Although a suitable choice for AandBin (3) allows one to approximate
a wide range of spectral functions (with appropriate choice ofAandBone
can obtain for the y1component of the vector /vector yevery spectral function
which is the ratio of two polynomials), the solution of this e quation has
the disadvantage that it also allows with positive probabil ity arbitrary big
excursions during finite fixed time intervals. In order to rem ove this effect
and also to apply our proof technique we have to freeze and tru ncate the
process.
Letc(ε) be some positive function of εdefined on the set ε/ne}ationslash= 0 . For
every natural mand for every point /vector y0∈Rnwe introduce a random value
τε
m=τε
m(/vector y0) = inf {t≥0 : (t, /vector y(t))/ne}ationslash∈[0, m)× {/vector y:|/vector y|< c(ε)}}
where/vector y(t) is the solution of the system (3) which with probability one
satisfies the initial condition /vector y(0) =/vector y0. So with probability one for m1≤
m2
0≤τε
m1≤τε
m2.
Then with probability one there exists a limit (finite or infin ite) whenm→ ∞
of the sequence τε
mwhich we will denote as
τε(/vector y0)def= limm→∞τε
m(/vector y0).
In other words τε(/vector y0) is the exit time from an open ball |/vector y|< c(ε) for
the solution of (3) starting with probability one from initi al point/vector y0. Note
that if the matrix BB⊤is nondegenerate then this exit time is finite with
probability one.
The joint solution of the systems (2), (3) ( x(t), z(t), /vector y(t)) is a Marko-
vian diffusion process in ( n+ 2)-dimensional Euclidean space. Let sε
t=
min{t, τε}. For the noise model (3) for reasons which we explained above
we shall not study the moments of the stochastic process ( x(t), z(t), /vector y(t)) ,
but the moments of the stochastic process ( x(sε
t), z(sε
t), /vector y(sε
t)) (stopped
process). We shall use the time scale O(ε−2) and the difference between t
andsε
tfor this time scale can be estimated with the help of the follo wing
Theorem A: There exist positive constants aandbso that for any
initial point /vector y0and for any positive L
P/parenleftbigg
τε<L
ε2/parenrightbigg
≤/parenleftbigg
exp(a|/vector y0|2) +aL
ε2/parenrightbigg
exp(−bc2(ε)) (9)
8Rewriting the left hand side of the inequality (9) in the form
P/parenleftbigg
τε<L
ε2/parenrightbigg
=P/parenleftigg
max
0≤t≤L/ε2|t−sε
t|>0/parenrightigg
we see that on the time scale considered the measure of the set wheret/ne}ationslash=sε
t
will go to zero as ε→0 ifc2(ε)→ ∞ faster then b−1log (ε−2). On the
other hand to apply the technique of our proof we require that
lim
ε→0εc3(ε) = 0
so that we can not allow c(ε) go to infinity too fast.
4 Asymptotic Behaviour of Moments
Let us introduce functions ¯ cl(m, k) of integer arguments m, k≥0 with
the help of
¯c1(m, k) =m
4
/summationdisplay
νl−νp=ω0/braceleftig
(m−1) Ψ∗(ω0+νp)/vectorhp·/parenleftig/vectorbl+i/vectordl/parenrightig
−
−(k+ 1) Ψ∗(2ω0+νp)/parenleftig/vectorbp+i/vectordp/parenrightig
·/vectorhl−
−kΨ∗(νp)/parenleftig/vectorbp+i/vectordp/parenrightig
·/vectorhl−kΨ∗(ω0+νp)/vectorhp·/parenleftig/vectorbl−i/vectordl/parenrightig/bracerightig
+
+/summationdisplay
νl−νp=−ω0/braceleftig
mΨ⊤(νp)/parenleftig/vectorb∗
p−i/vectord∗
p/parenrightig
·/vectorh∗
l−kΨ⊤(ω0+νp)/vectorh∗
p·/parenleftig/vectorb∗
l−i/vectord∗
l/parenrightig/bracerightig
¯c2(m, k) =−m(m−1)
4/summationdisplay
νl−νp= 2ω0Ψ∗(ω0+νp)/vectorhp·/vectorhl
¯c3(m, k) =m(m−1)
4/summationdisplay
νl−νp= 3ω0/braceleftig
Ψ∗(ω0+νp)/vectorhp·/parenleftig/vectorbl−i/vectordl/parenrightig
+
9+ Ψ∗(2ω0+νp)/parenleftig/vectorbp+i/vectordp/parenrightig
·/vectorhl/bracerightig
¯c4(m, k) =−m(m−1)
4/summationdisplay
νl−νp= 4ω0Ψ∗(2ω0+νp)/parenleftig/vectorbp+i/vectordp/parenrightig
·/parenleftig/vectorbl−i/vectordl/parenrightig
¯c5(m, k) =m
4
/summationdisplay
νl−νp= 2ω0/braceleftig
kΨ∗(νp)/parenleftig/vectorbp+i/vectordp/parenrightig
·/parenleftig/vectorbl−i/vectordl/parenrightig
+
+ (k+ 1) Ψ∗(2ω0+νp)/parenleftig/vectorbp+i/vectordp/parenrightig
·/parenleftig/vectorbl−i/vectordl/parenrightig
−
−(m−1) Ψ∗(2ω0+νp)/parenleftig/vectorbp+i/vectordp/parenrightig
·/parenleftig/vectorbl+i/vectordl/parenrightig/bracerightig
−
−/summationdisplay
νl−νp=−2ω0mΨ⊤(νp)/parenleftig/vectorb∗
p−i/vectord∗
p/parenrightig
·/parenleftig/vectorb∗
l−i/vectord∗
l/parenrightig
¯c6(m, k) =mk
4∞/summationdisplay
p=−∞/bracketleftig
Ψ(ω0+νp) + Ψ∗(ω0+νp)/bracketrightig/vectorhp·/vectorhp
¯c7(m, k) =
−m+k
2α+∞/summationdisplay
p=−∞/braceleftiggmk
4/bracketleftig
Ψ(νp) + Ψ∗(νp)/bracketrightig /parenleftig/vectorbp+i/vectordp/parenrightig
·/parenleftig/vectorbp+i/vectordp/parenrightig
−
−m2
4Ψ(νp)/parenleftig/vectorbp−i/vectordp/parenrightig
·/parenleftig/vectorbp+i/vectordp/parenrightig
−k2
4Ψ∗(νp)/parenleftig/vectorbp+i/vectordp/parenrightig
·/parenleftig/vectorbp−i/vectordp/parenrightig
+
+/bracketleftigg(m+ 1)k
4Ψ(2ω0+νp) +m(k+ 1)
4Ψ∗(2ω0+νp)/bracketrightigg/parenleftig/vectorbp+i/vectordp/parenrightig
·/parenleftig/vectorbp+i/vectordp/parenrightig/bracerightigg
By using (4) and (8) it is not hard to show that cl(m, k) are correctly
defined because the series converge absolutely for every fixe d values of m
andk.
10Now in correspondence with an arbitrary two index array am, kand
nonnegative integer N
m, k≥0, m +k≤N
we define a vector /vectorV(am, k;N) with (N+1)(N+2)/2 components with the
help of the rule
Vl(am, k;N) =am, k, l =(m+k)(m+k+ 1)
2+k+ 1
This ordering corresponds to the following ordering of the e lements of the
arrayam, k(take by rows)
a0,0
a1,0a0,1
a2,0a1,1a0,2
...
aN,0aN−1,1aN−2,2... a 0, N
Consider now the system of ordinary differential equations w ith constant
coefficients
d
dτ/vectorV(am, k;N) = ¯KN/vectorV(am, k;N) (10)
generated with the help of the rule
d
dτam, k= ¯c2(m, k)am−2, k + ¯c∗
2(k, m)am, k−2+
¯c1(m, k)am−1, k + ¯c∗
1(k, m)am, k−1+
¯c3(m, k)am−2, k+1+ ¯c∗
3(k, m)am+1, k−2+
¯c4(m, k)am−2, k+2+ ¯c∗
4(k, m)am+2, k−2+
¯c5(m, k)am−1, k+1+ ¯c∗
5(k, m)am+1, k−1+
¯c6(m, k)am−1, k−1+ ¯c7(m, k)am, k(11)
where on the right hand side of (11) we take into account only t erms with
nonnegative indices.
11Theorem B: Let the function c(ε)satisfy the condition
lim
ε→0εc3(ε) = 0
Then for arbitrary initial points x0,z0,/vector y0, and for arbitrary nonnegative
integerNand positive L
lim
ε→0max
0≤t≤L/ε2/vextendsingle/vextendsingle/vextendsingle/angbracketleftig¯M−1
N(ε2sε
t)/vectorV(Im, k(sε
t);N)−/vectorV(Im, k(0);N)/angbracketrightig/vextendsingle/vextendsingle/vextendsingle= 0
where the matrix ¯MN(τ)is the fundamental matrix solution of the system
of linear ordinary differential equations with constant coe fficients (10).
Remark 1: For further purposes it is important to note that the state-
ment of the theorem B can also be written in the form
lim
ε→0max
0≤t≤L/ε2/vextendsingle/vextendsingle/vextendsingle/angbracketleftig
/vector aN(ε2sε
t)·/vectorV(Im, k(sε
t), N)−/vector aN(0)·/vectorV(Im, k(0), N)/angbracketrightig/vextendsingle/vextendsingle/vextendsingle= 0
where/vector aN(τ) is an arbitrary ( N+1)(N+2)/2-dimensional vector satisfying
d/vector aN
dτ=−¯K⊤
N/vector aN (12)
Remark 2: For physical applications one can neglect the small differen ce
betweentandsε
t(see theorem A)and we have
/angbracketleftig/vectorV(Im, k(t), N)/angbracketrightig
≈¯MN(ε2t)/vectorV(Im, k(0), N)
5 Nonresonant Case
Let us now define what we mean by nonresonant.
Definition: We shall say that there are no resonances of order m≥0
if for all integers p,qsuch that (p, q)∈F×F
mω0/ne}ationslash=νp+νq
Definition: We shall say that there are no resonances up to order m≥0
if for all integers p,qsuch that (p, q)∈F×F
kω0/ne}ationslash=νp+νqfork= 1,...,m
12In the nonresonant case only the values of ¯ c6(m, k) and ¯c7(m, k) will
be different from zero. Introduce for them special notations
Am, k= ¯c7(m, k), Cm, k= ¯c6(m, k)
Note that Cm, kis a symmetrical function of its arguments, i.e. Cm, k=Ck, m
and it is also a real valued function i.e. Cm, k=C∗
m, k, and the function Am, k
satisfies Am, k=A∗
k, m.
For the following let us also introduce special notations fo r the real and
imaginary parts of Am, k
¯Am, k=Am, k+Ak, m
2, ¯Bm, k=Am, k− A k, m
2i
We shall call ¯Am, kand¯Bm, kfor reasons which will become clear later diffu-
sion coefficient and tune shift respectively. Note that ¯Am, m=Am, mand
¯Bm, m= 0.
Theorem C: Let there be no resonances up to order 4and let the
functionc(ε)satisfy the condition
lim
ε→0εc3(ε) = 0
Then for any initial points x0,z0,/vector y0, for any nonnegative integers m,k
and for any positive L
lim
ε→0max
0≤t≤L/ε2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/angbracketleftiggq/summationdisplay
p=0am, k
p(ε2sε
t)Im−p, k−p(sε
t)−q/summationdisplay
p=0am, k
p(0)Im−p, k−p(0)/angbracketrightigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle= 0
whereq= min {m, k}and the functions am, k
p(τ)are an arbitrary
solution of the system of linear ordinary differential equat ions with constant
coefficients
dam, k
0
dτ=−Am, kam, k
0
dam, k
p
dτ=−Am−p, k−pam, k
p− C m−p+1, k−p+1am, k
p−1
p= 1,...,q
13The proof of this theorem can be obtained from the remark to th e theorem
B with help of some straightforward calculations.
Remark 1: We would like to note that for the study of the behaviour
of first order moments (i.e. when m+k= 1) we actually need to avoid
resonances in theorem B up to order 2 only.
Remark 2: The general solution of the system of differential equations
for the coefficients am, k
phas the form
am, k
0(τ) =am, k
0(0)·exp (−Am, kτ)
am, k
p(τ) =
am, k
p(0)− Cm−p+1, k−p+1τ/integraldisplay
0am, k
p−1(ζ)·exp (Am−p, k−pζ)dζ
·
·exp (−Am−p, k−pτ)
p= 1, ..., q
Choosing the initial conditions
am, k
0(0) = 1, am, k
p(0) = 0, p = 1, ..., q
the statement of the theorem C can be rewritten in the form
lim
ε→0max
0≤t≤L/ε2
/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/angbracketleftigg
exp(−ε2Am, ksε
t)Im, k(sε
t)−
Im, k(0)−q/summationdisplay
p=1am, k
p(ε2sε
t)Im−p, k−p(sε
t)
/angbracketrightigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle= 0
In the case when m/ne}ationslash=kwe can use the real valued functions Um, k
andVm, kinstead of the complex valued Im, k. Due to the symmetries
Um, k=Uk, mandVm, k=−Vk, mit is enough to consider only the case
whenm > k . So we have
Corollary 1: Let there be no resonances up to order 4and let the
functionc(ε)satisfy the condition
lim
ε→0εc3(ε) = 0
14Then for any initial points x0,z0,/vector y0, for any nonnegative integers m, k
satisfyingm > k , and for any positive L
lim
ε→0max
0≤t≤L/ε2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/angbracketleftiggk/summationdisplay
p=0Mm, k
p(ε2sε
t)·/vectorWm, k
p(sε
t)−k/summationdisplay
p=0Mm, k
p(0)·/vectorWm, k
p(0)/angbracketrightigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle= 0
where
/vectorWm, k
p(τ) =
Um−p, k−p(τ)
Vm−p, k−p(τ)
, Mm, k
p(τ) =
αm, k
p(τ)−βm, k
p(τ)
βm, k
p(τ)αm, k
p(τ)
and the functions αm, k
p(τ)andβm, k
p(τ)are an arbitrary real solution of
the system of linear ordinary differential equations with co nstant coefficients
d
dτ
αm, k
0
βm, k
0
=¯Rm, k
0
αm, k
0
βm, k
0
d
dτ
αm, k
p
βm, k
p
=¯Rm, k
p
αm, k
p
βm, k
p
− Cm−p+1, k−p+1
αm, k
p−1
βm, k
p−1
p= 1,...,k
¯Rm, k
p=
−¯Am−p, k−p¯Bm−p, k−p
−¯Bm−p, k−p−¯Am−p, k−p
For the important particular case when we do not have an exter nal noise
in our system, i.e. ξ(t)≡0 (that means that we can put /vectorh(t)≡/vector0 and hence
allCm, k= 0) the differential equations defining the functions am, m
p,αm, k
p
andβm, k
padmit the simple solution
am, m
0(τ) = exp( −¯Am, mτ)am, m
0(0)
αm, k
0(τ)
βm, k
0(τ)
= exp( −¯Am, kτ)
cos(¯Bm, kτ) sin( ¯Bm, kτ)
−sin(¯Bm, kτ) cos( ¯Bm, kτ)
αm, k
0(0)
βm, k
0(0)
15am, m
p(τ)≡0, αm, k
p(τ)≡0, βm, k
p(τ)≡0, p /ne}ationslash= 0
Choosing initial conditions
am, m
0(0) = 1, αm, k
0(0) = 1, βm, k
0(0) = 0
we get the following
Corollary 2: Letξ(t)≡0and let there be no resonances of orders 2
and4, and let the function c(ε)satisfy the condition
lim
ε→0εc3(ε) = 0
Then for any initial points x0,z0,/vector y0, for any nonnegative integers m, k
satisfyingm≥k, and for any positive L
lim
ε→0max
0≤t≤L/ε2/vextendsingle/vextendsingle/vextendsingle/angbracketleftig
exp/parenleftig
−ε2¯Am, msε
t/parenrightig
rm(sε
t)−rm(0)/angbracketrightig/vextendsingle/vextendsingle/vextendsingle= 0
fork=mand
lim
ε→0max
0≤t≤L/ε2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/angbracketleftigg
exp/parenleftig
−ε2¯Am,ksε
t/parenrightig¯Mk
m(sε
t)
¯Um,k(sε
t)
¯Vm,k(sε
t)
−
¯Um,k(0)
¯Vm,k(0)
/angbracketrightigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle= 0
where
¯Mk
m(τ) =
cos/parenleftig
∆ε
m,kτ/parenrightig
−sin/parenleftig
∆ε
m,kτ/parenrightig
sin/parenleftig
∆ε
m,kτ/parenrightig
cos/parenleftig
∆ε
m,kτ/parenrightig
,∆ε
m,k= (m−k)ω0−ε2¯Bm,k
otherwise.
For the important case of constant vectors /vectorb,/vectorhand/vectordthe formulae
for¯Am, k,¯Bm, kandCm, ktake the simplified form
¯Am, k=−m+k
2α−(m−k)2
4Ψc(0)/vectorb·/vectorb+(m+k)2
4Ψc(0)/vectord·/vectord+
16+m+ 2mk+k
4/bracketleftbigg
Ψc(2ω0)/vectorb·/vectorb+ Ψ c(2ω0)/vectord·/vectord+/parenleftig
Ψs(2ω0)−Ψ⊤
s(2ω0)/parenrightig/vectord·/vectorb/bracketrightbigg
¯Bm, k=m2−k2
4/parenleftig
Ψc(0) + Ψ⊤
c(0)/parenrightig/vectord·/vectorb+
+m−k
4/bracketleftbigg
Ψs(2ω0)/vectorb·/vectorb+ Ψ s(2ω0)/vectord·/vectord−/parenleftig
Ψc(2ω0)−Ψ⊤
c(2ω0)/parenrightig/vectord·/vectorb/bracketrightbigg
Cm, k=mk
2Ψc(ω0)/vectorh·/vectorh
6 First and Second Order Moments
First order moments in the nonresonant case:
Corollary C1: Let there be no resonances up to order 2and let the
functionc(ε)satisfy the condition
lim
ε→0εc3(ε) = 0
Then for any initial points x0,z0,/vector y0and for any positive L
lim
ε→0max
0≤t≤L/ε2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/angbracketleftigg
exp/parenleftig
−ε2¯A1,0sε
t/parenrightig
M(sε
t)
x(sε
t)
z(sε
t)
−
x0
z0
/angbracketrightigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle= 0
where
M(τ) =
cos/parenleftig/parenleftig
ω0−ε2¯B1,0/parenrightig
τ/parenrightig
−sin/parenleftig/parenleftig
ω0−ε2¯B1,0/parenrightig
τ/parenrightig
sin/parenleftig/parenleftig
ω0−ε2¯B1,0/parenrightig
τ/parenrightig
cos/parenleftig/parenleftig
ω0−ε2¯B1,0/parenrightig
τ/parenrightig
Second order moments in nonresonant case:
Corollary C2: Let there be no resonances up to order 4and let the
functionc(ε)satisfy the condition
lim
ε→0εc3(ε) = 0.
Then for any initial points x0,z0,/vector y0∈Rnand for any positive L
17lim
ε→0max
0≤t≤L/ε2/vextendsingle/vextendsingle/vextendsingle/angbracketleftig
r(sε
t)−r0−ε22C1,1sε
t/angbracketrightig/vextendsingle/vextendsingle/vextendsingle= 0
for¯A1,1= 0and
lim
ε→0max
0≤t≤L/ε2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/angbracketleftigg/parenleftigg
r(sε
t) +2C1,1
¯A1,1/parenrightigg
exp/parenleftig
−ε2¯A1,1sε
t/parenrightig
−/parenleftigg
r0+2C1,1
¯A1,1/parenrightigg/angbracketrightigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle= 0
otherwise.
To estimate the behaviour of the remainder of the second mome nts we
shall use the functions
¯U2,0=x2−z2
4¯V2,0=xz
2
Corollary C3: Let there be no resonances up to order 4and let the
functionc(ε)satisfy the condition
lim
ε→0εc3(ε) = 0
Then for any initial points x0,z0,/vector y0and for any positive L
lim
ε→0max
0≤t≤L/ε2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/angbracketleftigg
exp/parenleftig
−ε2¯A2,0sε
t/parenrightig
M(sε
t)
¯U2,0(sε
t)
¯V2,0(sε
t)
−
¯U2,0(0)
¯V2,0(0)
/angbracketrightigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle= 0
where
M(τ) =
cos/parenleftig/parenleftig
2ω0−ε2¯B2,0/parenrightig
τ/parenrightig
−sin/parenleftig/parenleftig
2ω0−ε2¯B2,0/parenrightig
τ/parenrightig
sin/parenleftig/parenleftig
2ω0−ε2¯B2,0/parenrightig
τ/parenrightig
cos/parenleftig/parenleftig
2ω0−ε2¯B2,0/parenrightig
τ/parenrightig
187 Comparison with White Noise Model
As a special case we consider white noise in this chapter i.e.
/vector y=C˙/vector w(t) (13)
whereCis a real constant ( n×r) matrix and /vector w(t) is anr-dimensional
Brownian motion. Substituting (13) into (2) we have
dx=ω0zdt
dz=−ω0xdt−ε2αzdt +εC⊤/parenleftig/vectorh−z/vectord−x/vectorb/parenrightig
·d/vector w(t)(14)
As usual for the case of multiplicative noise we shall treat t he system (14)
as a system of Stratonovich’s stochastic differential equat ions.
Introduce the matrix Φ =1
2CC⊤which plays the role of the spectral
density for the noise model (13) and define functions ˘ cl(m, k) with the help
of
˘c1(m, k) =m
2/summationdisplay
νl−νp=ω0/braceleftig
(m−2k−1) Φ/vectorhp·/vectorbl−i(m+ 2k) Φ/vectorhp·/vectordl/bracerightig
,
˘c2(m, k) =−m(m−1)
4/summationdisplay
νl−νp= 2ω0Φ/vectorhp·/vectorhl,
˘c3(m, k) =m(m−1)
2/summationdisplay
νl−νp= 3ω0Φ/vectorhp·/parenleftig/vectorbl−i/vectordl/parenrightig
,
˘c4(m, k) =−m(m−1)
4/summationdisplay
νl−νp= 4ω0Φ/parenleftig/vectorbp+i/vectordp/parenrightig
·/parenleftig/vectorbl−i/vectordl/parenrightig
,
˘c5(m, k) =m
2/summationdisplay
νl−νp= 2ω0/braceleftig
−(m+k) Φ/vectordp·/vectordl+
+ (k−m+ 1) Φ/vectorbp·/vectorbl+i(2k+ 1) Φ/vectorbp·/vectordl/bracerightig
,
19˘c6(m, k) =mk
2∞/summationdisplay
p=−∞Φ/vectorhp·/vectorhp,
˘c7(m, k) =−m+k
2α+4mk−m(m−1)−k(k−1)
4∞/summationdisplay
p=−∞Φ/vectorbp·/vectorbp+
+4mk+m(m+ 1) +k(k+ 1)
4∞/summationdisplay
p=−∞Φ/vectordp·/vectordp+im2−k2
2∞/summationdisplay
p=−∞Φ/vectordp·/vectorbp.
Consider now the system of ordinary differential equations w ith constant
coefficients
d
dτ/vectorV(am, k;N) = ˘KN/vectorV(am, k;N) (15)
generated with the help of the rule
d
dτam, k= ˘c2(m, k)am−2, k + ˘c∗
2(k, m)am, k−2+
˘c1(m, k)am−1, k + ˘c∗
1(k, m)am, k−1+
˘c3(m, k)am−2, k+1+ ˘c∗
3(k, m)am+1, k−2+
˘c4(m, k)am−2, k+2+ ˘c∗
4(k, m)am+2, k−2+
˘c5(m, k)am−1, k+1+ ˘c∗
5(k, m)am+1, k−1+
˘c6(m, k)am−1, k−1+ ˘c7(m, k)am, k(16)
where on the right hand side of (16) we take into account only t erms with
nonnegative indices.
Theorem D: For any initial points x0,z0,/vector y0, for any nonnegative
integerNand for any positive L
lim
ε→0max
0≤t≤L/ε2/vextendsingle/vextendsingle/vextendsingle/angbracketleftig˘M−1
N(ε2t)/vectorV(Im, k(t);N)−/vectorV(Im, k(0);N)/angbracketrightig/vextendsingle/vextendsingle/vextendsingle= 0
20where the matrix ˘MN(τ)is the fundamental matrix solution of the system
of linear ordinary differential equations with constant coe fficients (15).
Note that in this case and for the noise model introduced belo w we have
not to distinguish between sε
tandt. We also mention that if we substitute
into the expressions of ¯ cl(m, k) the matrix Φ instead of the matrix Ψ
(”spectral density” of white noise) we exactly get ˘ cl(m, k) .
8 Another Noise Model
The technique derived in this paper can be applied to a wide cl ass of noise
models. As a model of noise in this section we consider the sto chastic pro-
cesses represented by the following trigonometrical polyn omials4(cosine and
sine functions with random phases)
η(t) =q/summationdisplay
m=−qηmexp (i(νmt+/vector vm·/vector y)), η −m= (ηm)∗
ξ(t) =q/summationdisplay
m=−qξmexp (i(νmt+/vector vm·/vector y)), ξ −m= (ξm)∗
γ(t) =q/summationdisplay
m=−qγmexp (i(νmt+/vector vm·/vector y)), γ −m= (γm)∗
with realνmand/vector vm∈Rnsatisfying the conditions
|νl+νm|+|/vector vm+/vector vl|= 0⇔m+l= 0
where the integers m, lobeym, l=−q, ..., q .
The vector /vector y∈Rnis assumed to be a solution of the following Ito’s
system
d/vector y=√
2Bd/vector w(t)
4In order not to deal with conditions similar to (4) and (5) we c onsider the case of a
finite trigonometrical sum. The extension to the case of infin ite series and also the proof
of the theorem E we leave as an exercise for the interested rea der.
21whereBis a real constant ( n×r) matrix and /vector w(t) is anr-dimensional
Brownian motion. For simplicity we assume that the ( n×n) matrixBB⊤
is nondegenerate and |/vector vm| /ne}ationslash= 0 for all m=−q,...,q (i.e. we do not have
deterministic harmonics in our perturbation model).
Forp=−q, ..., q we introduce real vectors /vector up=B⊤/vector vp∈Rrwhich
satisfy |/vector up| /ne}ationslash= 0, and a function Ω ( ω,/vector u p)
Ω (ω,/vector u p) =|/vector up|2+iω
|/vector up|4+ω2
and define ˜ cl(m, k) as follow
˜c1(m, k) =m
4
/summationdisplay
|νp+νl+ω0|+
+|/vector vp+/vector vl|=0{(m−1) Ω (ω0+νp, /vector up)ξp(ηl−iγl)−
−(k+ 1) Ω (2ω0+νp, /vector up) (ηp+iγp)ξl−
−kΩ (ω0+νp,/vector up)ξp(ηl+iγl)−kΩ (νp, /vector up) (ηp+iγp)ξl}+
+/summationdisplay
|νl−νp+ω0|+
+|/vector vl−/vector vp|=0/braceleftig
mΩ∗(νp, /vector up)/parenleftig
η∗
p−iγ∗
p/parenrightig
ξl−kΩ∗(ω0+νp, /vector up)ξ∗
p(ηl+iγl)/bracerightig
˜c2(m, k) =−m(m−1)
4/summationdisplay
|νp+νl+2ω0|+
+|/vector vp+/vector vl|=0Ω (ω0+νp,/vector up)ξpξl
˜c3(m, k) =m(m−1)
4/summationdisplay
|νp+νl+3ω0|+
+|/vector vp+/vector vl|=0{Ω (ω0+νp,/vector up)ξp(ηl+iγl) +
22+ Ω (2ω0+νp,/vector up) (ηp+iγp)ξl}
˜c4(m, k) =−m(m−1)
4/summationdisplay
|νp+νl+4ω0|+
+|/vector vp+/vector vl|=0Ω (2ω0+νp, /vector up) (ηp+iγp) (ηl+iγl)
˜c5(m, k) =m
4
/summationdisplay
|νp+νl+2ω0|+
+|/vector vp+/vector vl|=0{kΩ (νp, /vector up) (ηp+iγp) (ηl+iγl) +
+ (k+ 1) Ω (2ω0+νp, /vector up) (ηp+iγp) (ηl+iγl)−
−(m−1) Ω (2ω0+νp, /vector up) (ηp+iγp) (ηl−iγl)} −
−/summationdisplay
|νl−νp+2ω0|+
+|/vector vl−/vector vp|=0mΩ∗(νp, /vector up)/parenleftig
η∗
p−iγ∗
p/parenrightig
(ηl+iγl)
˜c6(m, k) =mk
2q/summationdisplay
p=−q|/vector up|2
|/vector up|4+ (νp+ω0)2|ξp|2
˜c7(m, k) =−m+k
2α+mk
2q/summationdisplay
p=−q|/vector up|2
|/vector up|4+ν2
p|ηp+iγp|2+
+m2+k2
4q/summationdisplay
p=−q|/vector up|2
|/vector up|4+ν2p/parenleftig
|γp|2− |ηp|2/parenrightig
+
23+m+ 2mk+k
4q/summationdisplay
p=−q|/vector up|2
|/vector up|4+ (νp+ 2ω0)2|ηp+iγp|2+
+im2−k2
4q/summationdisplay
p=−q|/vector up|2
|/vector up|4+ν2p/parenleftig
ηpγ∗
p+η∗
pγp/parenrightig
+
+im−k
4q/summationdisplay
p=−qνp+ 2ω0
|/vector up|4+ (νp+ 2ω0)2|ηp+iγp|2
Theorem E: For any initial points x0,z0,/vector y0, for any nonnegative
integerNand for any positive L
lim
ε→0max
0≤t≤L/ε2/vextendsingle/vextendsingle/vextendsingle/angbracketleftig˜M−1
N(ε2t)/vectorV(Im, k(t);N)−/vectorV(Im, k(0);N)/angbracketrightig/vextendsingle/vextendsingle/vextendsingle= 0
where the matrix ˜MN(τ)is the fundamental matrix solution of the system
of linear ordinary differential equations with constant coe fficients
d
dτ/vectorV(am, k;N) = ˜KN/vectorV(am, k;N)
generated with the help of the rule (16) in which we use ˜cl(m, k)instead of
˘cl(m, k).
9 Proof of the Theorems
The purpose of this section is to give a detailed proof of the t heorems.
9.1 Proof of the Theorem A
1.From the fact that all eigenvalues of the matrix Ahave negative
real parts it follows that there exists a quadratic form v(/vector y) satisfying the
conditions
C1|/vector y|2≤v(/vector y)≤C2|/vector y|2
A/vector y·grad/vector yv(/vector y)≤ −C3|/vector y|2
24Here and below Ciare some positive constants the exact values of which are
unimportant for us.
2.LetˆLbe the generating differential operator of the n-dimensional
Markovian diffusion process /vector yi.e.
ˆL=∂
∂t+A/vector y·grad/vector y+1
2BB⊤grad/vector y·grad/vector y (17)
3.Introduce the function w(/vector y) = exp(ψv(/vector y)) for wich
grad/vector yw=ψw·grad/vector yv
∂2w
∂ym∂yk=ψ/parenleftigg∂2v
∂ym∂yk+ψ∂v
∂ym·∂v
∂yk/parenrightigg
w
and hence
ˆLw=ψ
A/vector y·grad/vector yv+1
2n/summationdisplay
m, k= 1/parenleftig
BB⊤/parenrightig
m,k/parenleftigg∂2v
∂ym∂yk+ψ∂v
∂ym·∂v
∂yk/parenrightigg
w
From this it follows that there exist constants C4andC5independent of
the value of ψsuch that
ˆLw≤ψ/parenleftbigg
−C3|/vector y|2+1
2C4+ψC5|/vector y|2/parenrightbigg
w
Taking now ψ=C3/2C5we get
ˆLw≤ψ
2/parenleftig
−C3|/vector y|2+C4/parenrightig
w (18)
4.Define the constant χas the maximum of the right side in inequality
(18) with respect to the variables /vector y
χ= max
/vector y∈Rn/parenleftiggψ
2/parenleftig
−C3|/vector y|2+C4/parenrightig
w/parenrightigg
(19)
Obviously one has
0≤χ≤ψ
2C4exp/parenleftbigg
ψC2C4
C3/parenrightbigg
25From (18) and (19) it follows, that the function
ˆw=w+χ/parenleftbiggL
ε2−t/parenrightbigg
(20)
will satisfy the inequality
ˆLˆw≤0 (21)
5.Let ˜sε
t=sε
t∧L
ε2. From (20) and (21) immediately follows that the
stochastic process ˆ w( ˜sε
t) is a nonnegative supermartingale, and hence
P/parenleftbigg
τε<L
ε2/parenrightbigg
≤P/parenleftigg
sup
t≥0|/vector y(˜sε
t)| ≥c(ε)/parenrightigg
≤
P/parenleftigg
sup
t≥0ˆw(˜sε
t)≥exp(ψC1c2(ε))/parenrightigg
≤/parenleftbigg
exp(ψv(/vector y0)) +χL
ε2/parenrightbigg
exp(−ψC1c2(ε))
The first two inequalities in the sequence shown above are alm ost obvi-
ous, and the last one follows from the property of the stochas tic process
ˆw(˜sε
t) to be a nonnegative supermartingale. For finishing the proo f take
a= max(χ, C 2ψ) andb=C1ψ.
9.2 Proof of the Theorem B
1.The joint solution of the systems (2), (3) is a Markovian diffu sion
process in the ( n+ 2)-dimensional Euclidean space. Let Lbe the gener-
ating differential operator of this stochastic process. Sep arating the orders
according to εwe can represent Lin the form
L=L0+εLε+ε2Lε2 (22)
where the differential operators L0,Lε,Lε2are defined as follows
L0=ω0/parenleftigg
z∂
∂x−x∂
∂z/parenrightigg
+ˆL, L ε= (ξ−γz−ηx)∂
∂z, L ε2=−αz∂
∂z
andˆLis the generating differential operator of the n-dimensional Markovian
diffusion process /vector ygiven by (17).
262.Now we wish to show that there exist functions uε
m, ksatisfying
L0uε
m, k=−LεIm, k (23)
Representing the operator Lεin the form
Lε=/parenleftbigg
ξ−(η+iγ) exp(iω0t)I0,1−(η−iγ) exp( −iω0t)I1,0/parenrightbigg
·∂
∂z
and calculating5
∂Im, k
∂z=im
2exp(iω0t)Im−1, k−ik
2exp(−iω0t)Im,k−1 (24)
and taking into account property bwe have
LεIm, k=i
2/parenleftbigg
mξexp(iω0t)Im−1, k−kξexp(−iω0t)Im, k−1+
[k(η+iγ)−m(η−iγ)]Im, k+
k(η−iγ) exp( −i2ω0t)Im+1, k−1−m(η+iγ) exp(i2ω0t)Im−1, k+1/parenrightbigg
Looking for the uε
m, kin analogous form
uε
m,k=i
2/parenleftbigg
−ma1exp(iω0t)Im−1, k+ka∗
1exp(−iω0t)Im, k−1+
(ma∗
2−ka2)Im, k+
ma3exp(i2ω0t)Im−1, k+1−ka∗
3exp(−i2ω0t)Im+1, k−1/parenrightbigg
5Starting from this point it is convenient to extend the defini tion of the function Ip, q
to negative indices assuming that if p <0 or q <0 then Ip, q≡0 . In general, after
this extension one has to be careful with respect to the appli cation of the property b, but
we have not to worry about it, because the only source of lower ing indices in this paper
is differentiation and hence if the function Ip, qwith negative index will appear we shall
have automatically zero multiplyer in front of it.
27we get the system defining the unknown al
ˆLa1+iω0a1=ξ
ˆLa2=η+iγ
ˆLa3+i2ω0a3=η+iγ(25)
Choosingalin the form al=/vector al(t)·/vector y, where
/vector al(t) =+∞/summationdisplay
p=−∞/vector al, pexp(iνpt) (26)
and taking into account that
ˆLal=/parenleftiggd/vector al
dt+A⊤/vector al/parenrightigg
·/vector y
we reduce the system (25) to a system of algebraic equations f or the Fourier
coefficients
Λ⊤(ω0+νp)/vector a1, p=/vectorhp
Λ⊤(νp)/vector a2, p=/vectorbp+i/vectordp
Λ⊤(2ω0+νp)/vector a3, p=/vectorbp+i/vectordp(27)
where we have used the notation
Λ(ω) =A+iωI.
So among the characteristic roots of Awe have no purely imaginary or zero
values the matrix Λ( ω) is invertiable for an arbitrary real ωand hence the
system (27) has a unique solution, which can be expressed as f ollows
/vector a1, p= Λ−⊤(ω0+νp)/vectorhp=Λ∗(ω0+νp)
A⊤A⊤+ (ω0+νp)2I/vectorhp
/vector a2, p= Λ−⊤(νp)/parenleftig/vectorbp+i/vectordp/parenrightig
=Λ∗(νp)
A⊤A⊤+ν2pI/parenleftig/vectorbp+i/vectordp/parenrightig
28/vector a3, p= Λ−⊤(2ω0+νp)/parenleftig/vectorbp+i/vectordp/parenrightig
=Λ∗(2ω0+νp)
A⊤A⊤+ (2ω0+νp)2I/parenleftig/vectorbp+i/vectordp/parenrightig
Using the estimate
/vextendsingle/vextendsingle/vextendsingleΛ−⊤(ω)/vextendsingle/vextendsingle/vextendsingle≤1
δ2
s
which is valid for an arbitrary real ωwe get for /vector al, p
|/vector al, p| ≤1
δ2s/parenleftig/vextendsingle/vextendsingle/vextendsingle/vectorhp/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/vectorbp/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/vectordp/vextendsingle/vextendsingle/vextendsingle/parenrightig
which together with (4) guarantees the absolute convergenc e and the possi-
bility of differentiating the series (26) term by term.
3.Calculating Lε2Im, kandLεuε
m, kwe get
Lε2Im, k+Lεuε
m, k=
c2(m, k)Im−2, k +c∗
2(k, m)Im, k−2+
c1(m, k)Im−1, k +c∗
1(k, m)Im, k−1+
c3(m, k)Im−2, k+1+c∗
3(k, m)Im+1, k−2+
c4(m, k)Im−2, k+2+c∗
4(k, m)Im+2, k−2+
c5(m, k)Im−1, k+1+c∗
5(k, m)Im+1, k−1+
c6(m, k)Im−1, k−1+c7(m, k)Im, k(28)
29where the functions cl(m, k) are given by the following expressions
c1(m, k) =m
4/bracketleftig
(ka2−ma∗
2+ (k+ 1)a3)ξ+
k(a1+a∗
1)(η+iγ)−(m−1)a1(η−iγ)/bracketrightig
exp(iω0t)
c2(m, k) =m
4/bracketleftig
(m−1)a1ξ/bracketrightig
exp(i2ω0t)
c3(m, k) = −m
4/bracketleftig
(m−1)a1(η+iγ) + (m−1)a3ξ/bracketrightig
exp(i3ω0t)
c4(m, k) =m
4/bracketleftig
(m−1)a3(η+iγ)/bracketrightig
exp(i4ω0t)
c5(m, k) =m
4/bracketleftig
2α−(ka2−ma∗
2+ (k+ 1)a3) (η+iγ) +
(m−1)a3(η−iγ)/bracketrightig
exp(i2ω0t)
c6(m, k) = −m
4/bracketleftig
k(a1+a∗
1)ξ/bracketrightig
c7(m, k) = −m
4/bracketleftig
2α+ (ka2−ma∗
2+ (k+ 1)a3)(η−iγ)/bracketrightig
−
k
4/bracketleftig
2α+ (ma∗
2−ka2+ (m+ 1)a∗
3)(η+iγ)/bracketrightig
Note thatc6(m, k) =c∗
6(k, m) andc7(m, k) =c∗
7(k, m) .
4.Introduce a ( n×n) matrixK(/vector y) =/vector y·/vector y⊤with the elements kij=yiyj.
It is easy to check, that this matrix satisfies the equation
ˆLK=AK+KA⊤+BB⊤
The usefulness of this matrix for the following is connected with the fact that
for arbitrary complex vectors /vector aand/vector c
(/vector a·/vector y)·(/vector c·/vector y) =K/vector a·/vector c∗=K/vector c·/vector a∗(29)
5.Define the ( n×n) matrix-function Pω=Pω(/vector y) with the help of the
integral
Pω=−∞/integraldisplay
0exp(iωτ) exp(Aτ)K(/vector y) exp(A⊤τ)dτ (30)
30This integral converges because all the characteristic roo ts ofAhave negative
real parts. Introduce the new integration variable τ′=τ+t, wheretis
some parameter. Then (30) becomes
Pω=−∞/integraldisplay
texp(iω(τ′−t)) exp(A(τ′−t))K(/vector y) exp(A⊤(τ′−t))dτ′(31)
Differentiating (31) with respect to tand using that due to (30) Pωdoes
not depend on t, we obtain
dPω
dt=K−APω−PωA⊤−iωPω= 0 (32)
Calculating ˆLPωand taking into account (32) we get
ˆLPω=APω+PωA⊤−C(ω) =−iωPω+K−C(ω) (33)
where we have introduced the notation
C(ω) =∞/integraldisplay
0exp(iωτ) exp(Aτ)BB⊤exp(A⊤τ)dτ, C (0) =D
For the following let us rewrite (33) in the form
ˆLPω+iωPω=K−C(ω) (34)
Note that for some positive constant C1the norms of the matrices Pωand
C(ω) can be estimated uniformly with respect to real ωas follows (using 6)
|Pω| ≤C1
δ2s|/vector y|2, |C(ω)| ≤C1
δ2s(35)
6.Now we wish to show that there exist functions gl(m, k)6satisfying
ˆLgl(m, k) +cl(m, k) = ¯cl(m, k), l = 1,...,7
and these functions have continuous first and continuous firs t and second
derivatives with respect to the variables tand/vector yrespectively, and these
6Of course, gl(m, k) like cl(m, k) are also functions of tand/vector y
31functions together with the above derivative are bounded wi th respect to the
variabletfor fixed values of m, k, /vector y .
We shall show this for l= 2 and the rest can be done by analogy.
Due to (29) and the reality of the vector /vectorh(that is/vectorh=/vectorh∗) we have
c2(m, k) =m(m−1)
4a1ξexp(i2ω0t) =
m(m−1)
4/parenleftig
K/vector a1·/vectorh∗/parenrightig
exp(i2ω0t) =m(m−1)
4/parenleftig
K/vector a1·/vectorh/parenrightig
exp(i2ω0t)
Substituting in the last expression the Fourier series of th e vectors/vector a1and
/vectorhwe transform c2(m, k) into the form
c2(m, k) =m(m−1)
4∞/summationdisplay
p, l=−∞/parenleftig
K/vector a1, p·/vectorhl/parenrightig
exp(i(νp−νl+ 2ω0)t) =
m(m−1)
4
/summationdisplay
νl−νp= 2ω0K/vector a1, p·/vectorhl+/summationdisplay
νl−νp/ne}ationslash= 2ω0/parenleftig
K/vector a1, p·/vectorhl/parenrightig
exp(i(νp−νl+ 2ω0)t)
Forω/ne}ationslash= 0 introduce the matrix
Q(ω) =i
ωC(ω)−Pω
and denote P=−P0. Taking into account (34) we have
ˆL(Q(ω) exp(iωt)) = −Kexp(iωt) and ˆLP=D−K
Choosing now
g2(m, k) =m(m−1)
4
/summationdisplay
νl−νp= 2ω0P/vector a1, p·/vectorhl+
/summationdisplay
νl−νp/ne}ationslash= 2ω0/parenleftig
Q(νp−νl+ 2ω0)/vector a1, p·/vectorhl/parenrightig
exp(i(νp−νl+ 2ω0)t)
32we obtain
ˆLg2(m, k) +c2(m, k) =m(m−1)
4/summationdisplay
νl−νp= 2ω0D/vector a1, p·/vectorhl
which just coincides with the expression for ¯ c2(m, k) if we take into account
that
D/vector a1, p=DΛ∗(ω0+νp)
A⊤A⊤+ (ω0+νp)2I/vectorhp=−Ψ∗(ω0+νp)/vectorhp (36)
Due to (35) and (5) we have
max/braceleftigg
|P|,max
νl−νp/ne}ationslash= 2ω0|Q(νp−νl+ 2ω0)|/bracerightigg
≤C1
δ2s/parenleftigg
|/vector y|2+1
δ2
f/parenrightigg
and hence, as it can be easily shown, the series defining the fu nctiong2(m,k)
converges absolutely with
|g2(m, k)| ≤m(m−1)
4C1
δ2
s/parenleftigg
|/vector y|2+1
δ2
f/parenrightigg
∞/summationdisplay
p=−∞|/vector a1, p|
∞/summationdisplay
p=−∞/vextendsingle/vextendsingle/vextendsingle/vectorhp/vextendsingle/vextendsingle/vextendsingle
(37)
The function g2(m, k) is a quadratic polynomial in /vector yand so we need
to worry about their partial derivative with respect to tonly. Expressing
∂g2(m, k)
∂t=m(m−1)
4/summationdisplay
µp, l/ne}ationslash= 0iµp, l/parenleftig
Q(µp, l)/vector a1, p·/vectorhl/parenrightig
exp(iµp, lt)
whereµp, l=νp−νl+ 2ω0and using the very rough estimate for |µp, l|
|µp, l| ≤C2(1 +|νp|)(1 +|νl|), C 2= max {1,2|ω0| }
we get that the series defining the partial derivative with re spect to the
variabletconverges absolutely with
/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂g2(m, k)
∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤m(m−1)
4C1
δ2s/parenleftigg
|/vector y|2+1
δ2
f/parenrightigg
C2·
·
∞/summationdisplay
p=−∞(1 +|νp|)|/vector a1, p|
∞/summationdisplay
p=−∞(1 +|νp|)/vextendsingle/vextendsingle/vextendsingle/vectorhp/vextendsingle/vextendsingle/vextendsingle
(38)
33Note that expressions similar to (36) hold also for D/vector a2, pandD/vector a3, p
D/vector a2, p=−Ψ∗(νp)/parenleftig/vectorbp+i/vectordp/parenrightig
D/vector a3, p=−Ψ∗(2ω0+νp)/parenleftig/vectorbp+i/vectordp/parenrightig
7.Defininguε2
m, kas
uε2
m, k=g2(m, k)Im−2, k +g∗
2(k, m)Im, k−2+
g1(m, k)Im−1, k +g∗
1(k, m)Im, k−1+
g3(m, k)Im−2, k+1+g∗
3(k, m)Im+1, k−2+
g4(m, k)Im−2, k+2+g∗
4(k, m)Im+2, k−2+
g5(m, k)Im−1, k+1+g∗
5(k, m)Im+1, k−1+
g6(m, k)Im−1, k−1+g7(m, k)Im, k
and acting on the function
˜Im, k=Im, k+εuε
m, k+ε2uε2
m, k (39)
by means of the operator Lwe have
L˜Im, k=/parenleftig
L0+εLε+ε2Lε2/parenrightig˜Im, k=
L0Im, k+ε/parenleftig
L0uε
m, k+LεIm, k/parenrightig
+
ε2/parenleftig
L0uε2
m, k+Lε2Im, k+Lεuε
m, k/parenrightig
+ε3Rm, k (40)
where for the remainder Rm, kwe have the expression
Rm, k=Lε2uε
m, k+Lεuε2
m, k+εLε2uε2
m, k (41)
34Due to our construction of the functions Im, k,uε
m, kanduε2
m, kfrom (40) it
follows that
L˜Im, k=ε2(¯c2(m, k)Im−2, k + ¯c∗
2(k, m)Im, k−2+
¯c1(m, k)Im−1, k + ¯c∗
1(k, m)Im, k−1+
¯c3(m, k)Im−2, k+1+ ¯c∗
3(k, m)Im+1, k−2+
¯c4(m, k)Im−2, k+2+ ¯c∗
4(k, m)Im+2, k−2+
¯c5(m, k)Im−1, k+1+ ¯c∗
5(k, m)Im+1, k−1+
¯c6(m, k)Im−1, k−1+ ¯c7(m, k)Im, k) +
ε3Rm, k(42)
8.So
|Ip, q|=/parenleftbiggr
2/parenrightbiggp+q
2≤1 +/parenleftbiggr
2/parenrightbiggm+k
2= 1 + |Im, k|
forp+q≤m+k, then for some positive constant C3independent of m
andkthe functions uε
m, k,uε2
m, kdefined above and Rm, kcan be roughly
estimated as follows
/vextendsingle/vextendsingle/vextendsingleuε
m, k/vextendsingle/vextendsingle/vextendsingle≤C3(m+k)|/vector y|(1 +|Im, k|) (43)
/vextendsingle/vextendsingle/vextendsingleuε2
m, k/vextendsingle/vextendsingle/vextendsingle≤C3(m+k)2/parenleftig
1 +|/vector y|2/parenrightig
(1 +|Im, k|) (44)
|Rm, k| ≤C3(m+k)3/parenleftig
1 +|/vector y|3/parenrightig
(1 +|Im, k|) (45)
9.So|Im, m|=Im, mthen for some positive constant C4independent
ofmwe have from (42)
L˜Im, m≤ε2C4m2(1 +Im, m) +ε3|Rm, m| (46)
Using the estimate (45) we can rewrite (46) in the form
L˜Im, m≤ε2Hm(1 +Im, m) (47)
35and foruε
m, m+εuε2
m, mwe have from (43) and (44)
/vextendsingle/vextendsingle/vextendsingleuε
m, m+εuε2
m, m/vextendsingle/vextendsingle/vextendsingle≤ G m(1 +Im, m) (48)
Here
Hm=m2/parenleftig
C4+ε8mC3/parenleftig
1 +|/vector y|3/parenrightig/parenrightig
(49)
Gm= 2mC3/parenleftig
|/vector y|+ε2m/parenleftig
1 +|/vector y|2/parenrightig/parenrightig
(50)
Let¯Hmand¯Gmbe some positive constants. Consider the function
vm=/parenleftig
1 +˜Im, m/parenrightig
exp/parenleftigg
−ε2¯Hm
1−ε¯Gmt/parenrightigg
for which after some straightforward calculations we have
Lvm≤ε2/parenleftigg/parenleftig
Hm−¯Hm/parenrightig
+ε¯Hm
1−ε¯Gm/parenleftig
Gm−¯Cm/parenrightig/parenrightigg
·
·(1 +Im, m)exp/parenleftigg
−ε2¯Hm
1−ε¯Gmt/parenrightigg
(51)
Choosing now
¯Hm=m2/parenleftig
C4+ε8mC3/parenleftig
1 +c3(ε)/parenrightig/parenrightig
¯Gm= 2mC3/parenleftig
c(ε) +ε2m/parenleftig
1 +c2(ε)/parenrightig/parenrightig
and assuming that εis small enough to guarantee 1 −ε¯Gm>0 we get from
(51) thatLvm≤0 on the set |/vector y| ≤c(ε) and hence
/an}bracketle{tvm(sε
t)/an}bracketri}ht ≤ /an}bracketle{tvm(0)/an}bracketri}ht (52)
Using that with probability one
1 +˜Im, m(sε
t)≤/parenleftig
1 +ε¯Gm/parenrightig
(1 +Im, m(sε
t))
361 +˜Im, m(sε
t)≥/parenleftig
1−ε¯Gm/parenrightig
(1 +Im, m(sε
t))
and hence with probability one
/parenleftig
1−ε¯Gm/parenrightig
(1 +Im, m(sε
t)) exp/parenleftigg
−ε2¯Hm
1−ε¯Gmt/parenrightigg
≤vm(sε
t)
vm(0)≤/parenleftig
1 +ε¯Gm/parenrightig
(1 +Im, m(0))
we have from (52) the estimate
/an}bracketle{t1 +Im, m(sε
t)/an}bracketri}ht ≤1 +ε¯Gm
1−ε¯Gm/an}bracketle{t1 +Im, m(0)/an}bracketri}htexp/parenleftigg
ε2¯Hm
1−ε¯Gmt/parenrightigg
which for the following is rewritten in the form
max
0≤t≤L/ε2/an}bracketle{t1 +Im, m(sε
t)/an}bracketri}ht ≤ ¯Dm/an}bracketle{t1 +Im, m(0)/an}bracketri}ht (53)
where
¯Dm=1 +ε¯Gm
1−ε¯Gmexp/parenleftiggL¯Hm
1−ε¯Gm/parenrightigg
,lim
ε→0¯Dm= exp/parenleftig
m2LC4/parenrightig
10.Denotingum, k=uε
m, k+εuε2
m, kand introducing the new remainder
˜Rm, k=Rm, k−¯c2(m, k)um−2, k−¯c∗
2(k, m)um, k−2−
¯c1(m, k)um−1, k−¯c∗
1(k, m)um, k−1−
¯c3(m, k)um−2, k+1−¯c∗
3(k, m)um+1, k−2−
¯c4(m, k)um−2, k+2−¯c∗
4(k, m)um+2, k−2−
¯c5(m, k)um−1, k+1−¯c∗
5(k, m)um+1, k−1−
¯c6(m, k)um−1, k−1−¯c7(m, k)um, k
which admits the estimate (as follows from (43)-(44))
/vextendsingle/vextendsingle/vextendsingle˜Rm, k/vextendsingle/vextendsingle/vextendsingle≤C5(m+k)3(1 +ε(m+k))/parenleftig
1 +|/vector y|3/parenrightig
(1 +|Im, k|) (54)
37we get from (42)
L˜Im, k=ε2/parenleftig
¯c2(m, k)˜Im−2, k+ ¯c∗
2(k, m)˜Im, k−2+
¯c1(m, k)˜Im−1, k + ¯c∗
1(k, m)˜Im, k−1+
¯c3(m, k)˜Im−2, k+1+ ¯c∗
3(k, m)˜Im+1, k−2+
¯c4(m, k)˜Im−2, k+2+ ¯c∗
4(k, m)˜Im+2, k−2+
¯c5(m, k)˜Im−1, k+1+ ¯c∗
5(k, m)˜Im+1, k−1+
¯c6(m, k)˜Im−1, k−1+ ¯c7(m, k)˜Im, k/parenrightig
+
ε3˜Rm, k(55)
Let nowNbe as in the theorem B. Using notation /vectorV(∗;N) we can
rewrite (55) in the form of the following system
L/vectorV(˜Im, k;N) =ε2¯KN/vectorV(˜Im, k;N) +ε3/vectorV(˜Rm, k;N) (56)
where the matrix ¯KNis the same as in (10).
The matrix ¯MN(τ) is assumed to be the fundamental matrix solution
of (10). That means that the matrix ¯M−1
N(ε2t) satisfies
d
dt¯M−1
N(ε2t) =−ε2¯M−1
N(ε2t)¯KN, ¯M−1
N(0) =I (57)
Applying the operator Lto the vector ¯M−1
N(ε2t)/vectorV(˜Im, k(t);N) and taking
into account (56), (57) we obtain
L/parenleftig¯M−1
N(ε2t)/vectorV(˜Im, k(t);N)/parenrightig
=ε3¯M−1
N(ε2t)/vectorV(˜Rm, k(t);N) (58)
From (58) and Dynkin’s formula (see, for example [4]) it foll ows that
/angbracketleftig¯M−1
N(ε2sε
t)/vectorV(˜Im, k(sε
t);N)−/vectorV(˜Im, k(0);N)/angbracketrightig
=
=ε3/angbracketleftiggsε
t/integraldisplay
0¯M−1
N(ε2τ)/vectorV(˜Rm, k(τ);N)dτ/angbracketrightigg
38or, equivalently
/angbracketleftig¯M−1
N(ε2sε
t)/vectorV(Im, k(sε
t);N)−/vectorV(Im, k(0);N)/angbracketrightig
=
=ε·/angbracketleftig/vectorV(um, k(0);N)−¯M−1
N(ε2sε
t)/vectorV(um, k(sε
t);N)/angbracketrightig
+
+ε3/angbracketleftiggsε
t/integraldisplay
0¯M−1
N(ε2τ)/vectorV(˜Rm, k(τ);N)dτ/angbracketrightigg
(59)
From (59) we obtain
max
0≤t≤L/ε2/vextendsingle/vextendsingle/vextendsingle/angbracketleftig¯M−1
N(ε2sε
t)/vectorV(Im, k(sε
t);N)−/vectorV(Im, k(0);N)/angbracketrightig/vextendsingle/vextendsingle/vextendsingle≤
≤ε·max
0≤τ≤L/vextendsingle/vextendsingle/vextendsingle¯M−1
N(τ)/vextendsingle/vextendsingle/vextendsinglemax
0≤t≤L/ε2/angbracketleftig
2/vextendsingle/vextendsingle/vextendsingle/vectorV(um, k(sε
t);N)/vextendsingle/vextendsingle/vextendsingle+L/vextendsingle/vextendsingle/vextendsingle/vectorV(˜Rm, k(sε
t);N)/vextendsingle/vextendsingle/vextendsingle/angbracketrightig
(60)
Let us define now [ m] as the smallest integer which is bigger or equal to m.
Using (43), (44), (54) and simple inequalities like
|/vector y| ≤1 +|/vector y|2
1 +|Im, k| ≤2/parenleftbigg
1 +I[N
2],[N
2]/parenrightbigg
, m +k≤N
we can obtain
max/braceleftig/vextendsingle/vextendsingle/vextendsingle/vectorV(um, k;N)/vextendsingle/vextendsingle/vextendsingle,/vextendsingle/vextendsingle/vextendsingle/vectorV(˜Rm, k;N)/vextendsingle/vextendsingle/vextendsingle/bracerightig
≤
≤C6N5(1 +εN)/parenleftig
1 +|/vector y|3/parenrightig/parenleftbigg
1 +I[N
2],[N
2]/parenrightbigg
(61)
Taking into account (53) we have from (61)
max
0≤t≤L/ε2/angbracketleftig
2/vextendsingle/vextendsingle/vextendsingle/vectorV(um, k(sε
t);N)/vextendsingle/vextendsingle/vextendsingle+L/vextendsingle/vextendsingle/vextendsingle/vectorV(˜Rm, k(sε
t);N)/vextendsingle/vextendsingle/vextendsingle/angbracketrightig
≤
≤C6N5(1 +εN)/parenleftig
1 +c3(ε)/parenrightig
(2 +L)¯D[N
2]/parenleftbigg
1 +I[N
2],[N
2](0)/parenrightbigg
39which together with
max
0≤τ≤L/vextendsingle/vextendsingle/vextendsingle¯M−1
N(τ)/vextendsingle/vextendsingle/vextendsingle≤exp(C7N2)
and (60) gives the final estimate we need
max
0≤t≤L/ε2/vextendsingle/vextendsingle/vextendsingle/angbracketleftig¯M−1
N(ε2sε
t)/vectorV(Im, k(sε
t);N)−/vectorV(Im, k(0);N)/angbracketrightig/vextendsingle/vextendsingle/vextendsingle≤
≤ε/parenleftig
1 +c3(ε)/parenrightig
P/parenleftbigg
1 +I[N
2],[N
2](0)/parenrightbigg
(62)
where
P=C6N5(1 +εN) (2 +L)¯D[N
2]exp(C7N2)
Taking the limit ε→0 we have from (62) the proof of the theorem B
with speed of convergence εc3(ε).
11.To prove the remark to the theorem B we apply the operator Lto
the function /vector aN(ε2t)·/vectorV(˜Im, k(t);N) with the result that
L/parenleftig
/vector aN(ε2t)·/vectorV(˜Im, k(t);N)/parenrightig
=ε3/vector aN(ε2t)·/vectorV(˜Rm, k(t);N) (63)
The rest of the proof is just repeating all the steps from the p revious point
with the usage (63) instead of (58).
9.3 Sketch of the Proof of the Theorem D
1.The solution of the system (14) is a Markovian diffusion proce ss in the
2-dimensional Euclidean space. Let Lbe a generating differential operator
of this stochastic process. Separating the orders accordin g toεwe can
representLin the form
L=L0+ε2Lε2 (64)
where the differential operators L0andLε2are defined as follows
L0=∂
∂t+ω0/parenleftigg
z∂
∂x−x∂
∂z/parenrightigg
40Lε2=/parenleftig
z/parenleftig
Φ/vectord·/vectord−α/parenrightig
+xΦ/vectorb·/vectord−Φ/vectorh·/vectord/parenrightig∂
∂z+
+/parenleftig
Φ/vectorh·/vectorh−2xΦ/vectorh·/vectorb−2zΦ/vectorh·/vectord+
+x2Φ/vectorb·/vectorb+ 2xzΦ/vectord·/vectorb+z2Φ/vectord·/vectord/parenrightig∂2
∂z2
2.Now we wish to calculate Lε2Im, k. Representing the operator Lε2in
the form
Lε2=/parenleftbigg
−Φ/vectorh·/vectord+/bracketleftig
Φ/vectorb·/vectord−iα+iΦ/vectord·/vectord/bracketrightig
exp(iω0t)I0,1+
+/bracketleftig
Φ/vectorb·/vectord+iα−iΦ/vectord·/vectord/bracketrightig
exp(−iω0t)I1,0/parenrightbigg
·∂
∂z+
+/parenleftbigg
Φ/vectorh·/vectorh−2/bracketleftig
Φ/vectorh·/vectorb+iΦ/vectorh·/vectord/bracketrightig
exp(iω0t)I0,1−
−2/bracketleftig
Φ/vectorh·/vectorb−iΦ/vectorh·/vectord/bracketrightig
exp(−iω0t)I1,0+ 2/bracketleftig
Φ/vectorb·/vectorb+ Φ/vectorb·/vectord/bracketrightig
I1,1+
+/bracketleftig
Φ/vectorb·/vectorb−Φ/vectord·/vectord+ 2iΦ/vectord·/vectorb/bracketrightig
exp(i2ω0t)I0,2+
+/bracketleftig
Φ/vectorb·/vectorb−Φ/vectord·/vectord−2iΦ/vectord·/vectorb/bracketrightig
exp(−i2ω0t)I2,0/parenrightbigg
·∂2
∂z2
taking into account (24), property band the expression
∂2Im, k
∂z2=mk
2Im−1, k−1−
−m(m−1)
4exp(i2ω0t)Im−2, k−k(k−1)
4exp(−i2ω0t)Im,k−2
41we obtain that Lε2Im ,kis given by the right hand side of (28) with cl(m, k)
as follows
c1(m, k) =m
2/bracketleftig
(m−2k−1) Φ/vectorh·/vectorb−i(m+ 2k) Φ/vectorh·/vectord/bracketrightig
exp(iω0t)
c2(m, k) = −m(m−1)
4/bracketleftig
Φ/vectorh·/vectorh/bracketrightig
exp(i2ω0t)
c3(m, k) =m(m−1)
2/bracketleftig
Φ/vectorh·/parenleftig/vectorb−i/vectord/parenrightig/bracketrightig
exp(i3ω0t)
c4(m, k) = −m(m−1)
4/bracketleftig
Φ/parenleftig/vectorb+i/vectord/parenrightig
·/parenleftig/vectorb−i/vectord/parenrightig /bracketrightig
exp(i4ω0t)
c5(m, k) =m
2/bracketleftig
α−(m+k) Φ/vectord·/vectord+ (k−m+ 1) Φ/vectorb·/vectorb+
i(2k+ 1) Φ/vectorb·/vectord/bracketrightig
exp(i2ω0t)
c6(m, k) =mk
2Φ/vectorh·/vectorh
c7(m, k) = −m+k
2α+4mk−m(m−1)−k(k−1)
4Φ/vectorb·/vectorb+
4mk+m(m+1)+ k(k+1)
4Φ/vectord·/vectord+im2−k2
2Φ/vectord·/vectorb
3.Now, like in the proof of theorem B, we want to show that there e xist
for fixedmandkfunctionsgl(m, k) bounded in tand satisfying
∂gl(m, k)
∂t+cl(m, k) = ˘cl(m, k), l = 1,...,7
We shall show it for l= 4 and the rest can be done by analogy.
Substituting in the expression for c4(m,k) the Fourier series of the vectors
/vectorband/vectordwe obtain
c4(m, k) =−m(m−1)
2∞/summationdisplay
p, l=−∞/bracketleftig
Φ/parenleftig/vectorbp+/vectordp/parenrightig
·/parenleftig/vectorbl−/vectordl/parenrightig/bracketrightig
exp(iµp, lt) =
= ˘c4(m, k)−m(m−1)
4/summationdisplay
µp, l/ne}ationslash= 0/bracketleftig
Φ/parenleftig/vectorbp+/vectordp/parenrightig
·/parenleftig/vectorbl−/vectordl/parenrightig/bracketrightig
exp(iµp, lt)
whereµp, l=νp−νl+ 4ω0.
42So the problem will be solved if the series
c4(m, k) =im(m−1)
4/summationdisplay
µp, l/ne}ationslash= 0Φ/parenleftig/vectorbp+/vectordp/parenrightig
·/parenleftig/vectorbl−/vectordl/parenrightig
µp, lexp(iµp, lt)
converges and can be differentiated term by term. The absolut e convergence
and differentiability can be easily shown using (4) and (5) wi th the final
estimates
|g4(m, k)| ≤m(m−1)
2|Φ|
δ2
f
∞/summationdisplay
p=−∞/vextendsingle/vextendsingle/vextendsingle/vectorbp/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/vectordp/vextendsingle/vextendsingle/vextendsingle
2
/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂g4(m, k)
∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤m(m−1)
2|Φ|
∞/summationdisplay
p=−∞/vextendsingle/vextendsingle/vextendsingle/vectorbp/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/vectordp/vextendsingle/vextendsingle/vextendsingle
2
4.The rest follows the simplified version of the proof of the the orem B.
References
[1] Balandin, V.V., A Method of Perturbed Lyapunov Functions in the Study
of the Asymptotical Behaviour of the Solutions of Ordinary D ifferential
Equations with Small Stochastic Perturbations Leading to M arkovian
Diffusion Processes , Candidate Dissertation, Moscow State University,
1987.
[2] Freidlin, M.I., Wentzell, A.D. Random Perturbations of Dynamical Sys-
tem, Springer-Verlag, 1984.
[3] Karatzas, I., Shreve, S.E., Brownian Motion and Stochastic Calculus ,
Graduate Texts in Mathematics 113, Springer-Verlag, 1987.
[4] Kushner, H.J., Stochastic Stability and Control , ACADEMIC PRESS,
New-York/London, 1967.
43 |
arXiv:physics/0101091v1 [physics.gen-ph] 26 Jan 2001The ”True Transformations Relativity” Analy-
sis of the Michelson-Morley Experiment
Tomislav Ivezi´ c
Ruder Boˇ skovi´ c Institute
P.O.B. 180, 10002 Zagreb, Croatia
E-mail: ivezic@rudjer.irb.hr
Abstract
In this paper we present an invariant formulation of special relativity, i.e., the
”true transformations relativity.” It deals either with tr ue tensor quantities
(when no basis has been introduced) or equivalently with coo rdinate-based
geometric quantities comprising both components and a basi s (when some ba-
sis has been introduced). It is shown that this invariant for mulation, in which
special relativity is understood as the theory of a four-dim ensional spacetime
with the pseudo-Euclidean geometry, completely explains t he results of the
Michelson-Morley experiment. Two noncovariant approache s to the analy-
sis of the Michelson-Morley experiment are discussed; the c oventional one
in which only the path lengths (optical or geometrical) are c onsidered, and
Driscoll’s approach (R.B. Driscoll, Phys. Essays 10,394 (1997)), in which
the increment of phase is determined not only by the segment o f geometric
path length, but also by the wavelength in that segment. Beca use these anal-
yses belong to the ”apparent transformations relativity,” they do not agree
with the results of the Michelson-Morley experiment.
Key words : Michelson-Morley experiment, true transformations rela tivity,
apparent transformations relativity
11. INTRODUCTION
Recently Driscoll(1)analyzed the Michelson-Morley(2)experiment taking
into account, in the calculation of the fringe shift, the Dop pler effect on
wavelength in the frame in which the interferometer is movin g. In contrast
to the traditional analysis the non-null fringe shift was fo und and the author
concluded: ”that the Maxwell-Einstein electromagnetic eq uations and spe-
cial relativity jointly are disproved, not confirmed, by the Michelson-Morley
experiment.” In this paper we present an invariant approach to special rel-
ativity (SR) with tensor quantities (and tensor equations) to the analysis of
the Michelson-Morley experiment and find a null fringe shift in agreement
with the experiment. We also show why the calculation from(1)leads to the
non-null result and why the traditional analysis gives an ap parent (not true)
agreement with the experiment.
In Sec. 2. we briefly discuss different approaches to SR. In the first ap-
proach SR is formulated in terms of true tensor quantities and true t ensor
equations , which we call the ”true transformations (TT) relativity.” That
approach is compared with the usual covariant approach, which mainly deals
with the basis components of tensors in a specific, i.e., Eins tein’s coordinatiza-
tion(3)of the chosen inertial frame of reference (IFR). The general discussion
is illustrated in Sec. 2.1 by two examples: the spacetime len gth for a moving
rod and the spacetime length for a moving clock. The usual, i. e., Einstein’s
formulation of SR, which is based on his two postulates, and w hich deals
with the Lorentz contraction and the dilatation of time, is a lso considered
in Sec. 2. It is shown that the Lorentz contraction and the dilatation of
time are the apparent transformations (AT). (The notions of the TT and the
AT are introduced in Ref. 4.) Any approach to SR which uses the AT we
call the ”apparent transformations (AT) relativity.” Einstein’s formulation
of SR obviously belongs to the ”AT relativity.” The same two e xamples as
mentioned above are considered in the ”AT relativity” in Sec . 2.2.
Then in Sec. 3.1 we discuss the nonrelativistic analysis and in Sec. 3.2
the traditional analysis of the Michelson-Morley experime nt. In Sec. 3.3
we repeat in short Driscoll’s calculation of the fringe shif t in the Michelson-
Morley experiment. Both, the traditional analysis and Dris coll’s analysis of
the Michelson-Morley experiment are shown to belong to the ” AT relativ-
ity.” In Sec. 4. we present the analysis of the Michelson-Mor ley experiment
in the ”TT relativity” explicitly using two very different sy nchronizations of
distant clocks (they are explained in Sec. 2.) and we find the n ull result in
agreement with the experiment. It is important to note that t his result holds
2in all permissible coordinatizations since the whole phase of a light wave, the
true tensor, φ=kagablb(see Eqs. (21) and (22)) is used in the calculation.
In Sec. 4.1 we explain Driscoll’s non-null fringe shift as a c onsequence of an
”AT relativity” calculation of the increment of phase. Dris coll’s calculation
takes into account only a part k0l0of the above mentioned whole phase φ
and considers that part in two relatively moving IFRs S(k0l0) and S′(k0′l0′)
but only in the Einstein coordinatization (the Sframe is the interferometer
rest frame). Finally in Sec. 4.2 we explicitly show that the a greement with
experiment obtained in the traditional ”AT relativity” cal culation is actu-
ally an apparent agreement. This calculation also deals wit h the part k0l0
of the whole phase φand considers that part in IFRs SandS′but again
only in the Einstein coordinatization. In contrast to Drisc oll’s calculation
the traditional analysis considers the contribution k0l0in the interferometer
rest frame S,but in the S′frame, in which the interferometer is moving, it
considers the contribution k0l0′; thek0factor is taken to be the same in Sand
S′frames. This fact caused an apparent agreement of the tradit ional analysis
with the results of the Michelson-Morley experiment. Since only a part of
the whole phase φis considered both results, Driscoll’s and the traditional
one, are synchronization, i.e., coordinatization, depend ent results. Thus the
agreement between the traditional analysis and the experim ent exists only
when Einstein’s synchronization of distant clocks is used a nd not for another
synchronization. This is also proved in Sec. 4.2.
2. THE COMPARISON OF THE ”TT RELATIVITY” WITH
THE USUAL COVARIANT APPROACH AND WITH THE
”AT RELATIVITY”
The above mentioned approaches to SR are partly discussed in Refs. 5-8.
Rohrlich,(4)and also Gamba,(9)emphasized the role of the concept of same-
ness of a physical quantity for different observers. This concept determines
the difference between the mentioned approaches and also it d etermines what
is to be understood as a relativistic theory. Our invariant a pproach to SR,
i.e., the ”TT relativity,” and the concept of sameness of a ph ysical quantity
for different observers in that approach, differs not only fro m the ”AT rela-
tivity” approach but also from the usual covariant approach (including Refs.
4 and 9).
We first explain the difference between the ”TT relativity” an d the usual
covariant approach to SR. In the ”TT relativity” SR is unders tood as the
theory of a four-dimensional (4D) spacetime with pseudo-Eu clidean geome-
3try. All physical quantities (in the case when no basis has be en introduced)
are described by true tensor fields , that are defined on the 4D spacetime,
and that satisfy true tensor equations representing physical laws. When the
coordinate system has been introduced the physical quantit ies are mathe-
matically represented by the coordinate-based geometric quantities (CBGQs)
that satisfy the coordinate-based geometric equations (CBGEs). The CBGQs
contain both the components and the basis one-forms and vect orsof the cho-
sen IFR. (Speaking in mathematical language a tensor of type (k,l) is defined
as a linear function of k one-forms and l vectors (in old names , k covariant
vectors and l contravariant vectors) into the real numbers, see, e.g., Refs. 10-
12. If a coordinate system is chosen in some IFR then, in gener al, any tensor
quantity can be reconstructed from its components and from t he basis vectors
and basis 1-forms of that frame, i.e., it can be written in a co ordinate-based
geometric language, see, e.g., Ref. 12.) The symmetry trans formations for
the metric gab, i.e., the isometries(10), leave the pseudo-Euclidean geometry of
4D spacetime of SR unchanged. At the same time they do not chan ge the true
tensor quantities, or equivalently the CBGQs, in physical e quations. Thus
isometries are what Rohrlich(4)callsthe TT . In the ”TT relativity” different
coordinatizations of an IFR are allowed and they are all equi valent in the
description of physical phenomena. Particularly two very d ifferent coordina-
tizations, the Einstein (”e”)(3)and ”radio” (”r”)(13)coordinatization, will be
briefly exposed and exploited in the paper. In the ”e” coordin atization the
Einstein synchronization(3)of distant clocks and cartesian space coordinates
xiare used in the chosen IFR. The main features of the ”r” coordi natization
will be given below, see also.Refs. 13, 6 and 7. The CBGQs representing
some 4D physical quantity in different relatively moving IFR s, or in different
coordinatizations of the chosen IFR, are all mathematicall y equal since they
are connected by the TT (i.e., the isometries). Thus they are really the same
quantity for different observers, or in different coordinati zations. Hence it
is appropriate to call the ”TT relativity” approach (which d eals with the
true tensors or with the CBGQs) as an invariant approach in co ntrast to
the usual covariant approach (which deals with the componen ts of tensors
taken in the ”e” coordinatization). We suppose that in the ”T T relativity”
such 4D tensor quantities are well-defined not only mathemat ically but also
experimentally , as measurable quantities with real physical meaning. The
complete and well-defined measurement from the ”TT relativi ty” viewpoint is
such measurement in which all parts of some 4D quantity are me asured.
However in the usual covariant approach one does not deal with the tru e
4tensors, or equivalently with CBGQs, but with the basis comp onents of ten-
sors (mainly in the ”e” coordinatization) and with the equat ions of physics
written out in the component form. Mathematically speaking the concept of
a tensor in the usual covariant approach is defined entirely i n terms of the
transformation properties of its components relative to some coordinate sys-
tem. The definitions of the same quantity in Refs. 4 and 9 also r efer to such
component form in the ”e” coordinatization of tensor quanti ties and tensor
equations. It is true that the components of some tensor refe r to the same
tensor quantity considered in two relatively moving IFRs SandS′and in
the ”e” coordinatization, but they cannot be equal, since th e bases are not
included.
The third approach to SR uses the AT of some quantities. In con trast
to the TT the AT are not the transformations of spacetime tens ors and they
do not refer to the same quantity. Thus they are not isometrie s and they
refer exclusively to the component form of tensor quantitie s and in that form
they transform only some components of the whole tensor quan tity.In fact,
depending on the used AT, only a part of a 4D tensor quantity is transformed
by the AT. Such a part of a 4D quantity, when considered in diffe rent IFRs (or
in different coordinatizations of some IFR) corresponds to d ifferent quantities
in 4D spacetime. Some examples of the AT are: the AT of the synchronously
defined spatial length,(3)i.e., the Lorentz contraction(4−9)and the AT of the
temporal distance, i.e., the conventional dilatation of ti methat is introduced
in Ref. 3 and considered in Refs. 7 and 8. The formulation of SR which
uses the AT we call the ”AT relativity.” An example of such for mulation is
Einstein’s formulation of SR which is based on his two postul ates and which
deals with all the mentioned AT.
The differences between the ”TT relativity,” the usual covar iant approach
and the ”AT relativity” will be examined considering some sp ecific examples.
First the spacetime lengths, corresponding in ”3+1” pictur e to a moving rod
and to a moving clock, will be considered in the ”TT relativit y.” Furthermore
the spatial and temporal distances for the same examples wil l be examined
in the ”AT relativity.” The comparison with the experiments on the Lorentz
contraction and the time dilatation is performed in Ref. 8 an d it shows that
all experiments can be qualitatively and quantitatively ex plained by the ”TT
relativity,” while some experiments cannot be adequately e xplained by the
”AT relativity.” This will be also shown below considering M ichelson-Morley
experiment.
Before doing this exploration we discuss the notation, diffe rent coordina-
5tizations and the connections between them. In this paper I u se the following
convention with regard to indices. Repeated indices imply s ummation. Latin
indices a, b, c, d, ... are to be read according to the abstract index notation,
see Ref. 10 Sec. 2.4; they ”...should be viewed as reminders o f the number
and type of variables the tensor acts on, notas basis components.” They
designate geometric objects in 4D spacetime. Thus, e.g., la
ABandxa
A(a dis-
tance 4-vector la
AB=xa
B−xa
Abetween two events AandBwhose position
4-vectors are xa
Aandxa
B) are (1,0) tensors and they are defined independently
of any coordinate system. Greek indices run from 0 to 3, while latin indices
i, j, k, l, ... run from 1 to 3, and they both designate the components of some
geometric object in some coordinate system, e.g., xµ(x0, xi) and xµ′(x0′, xi′)
are two coordinate representations of the position 4-vecto rxain two different
inertial coordinate systems SandS′.Similarly the metric tensor gabdenotes
a tensor of type (0,2) (whose Riemann curvature tensor Ra
bcdis everywhere
vanishing; the spacetime of SR is a flat spacetime, and this de finition in-
cludes not only the IFRs but also the accelerated frames of re ference). This
geometric object gabis represented in the component form in some IFR S,
and in the ”e” coordinatization, i.e., in the {eµ}basis, by the 4 ×4 diago-
nal matrix of components of gab,gµν,e=diag(−1,1,1,1),and this is usually
called the Minkowski metric tensor (the subscript′e′stands for the Einstein
coordinatization).
Different coordinatizations of some reference frame can be o btained using,
e.g., different synchronizations. On the other hand differen t synchronizations
are determined by the parameter εin the relation t2=t1+ε(t3−t1), where t1
andt3are the times of departure and arrival, respectively, of the light signal,
read by the clock at A, and t2is the time of reflection at B, read by the
clock at B, that has to be synchronized with the clock at A. In Einstein’s
synchronization convention ε= 1/2.We can also choose another coordinati-
zation, the ”everyday” or ”radio” (”r”) cordinatization,(13)which differs from
the ”e” coordinatization by the different procedure for the s ynchronization
of distant clocks. In the ”r” synchronization ε= 0 and thus, in contrast
to the ”e” synchronization, there is an absolute simultanei ty. As explained
in Ref. 13: ”For if we turn on the radio and set our clock by the s tandard
announcement ”...at the sound of the last tone, it will be 12 o ’clock”, then
we have synchronized our clock with the studio clock in a mann er that corre-
sponds to taking ε= 0 in t2=t1+ε(t3−t1).” The ”r” synchronization is an
assymetric synchronization which leads to an assymetry in t he coordinate,
one-way, speed of light.(13)However from the physical point of view the ”r”
6coordinatization is completely equivalent to the ”e” coord inatization. This
also holds for all other permissible coordinatizations. Su ch situation really
happens in the ”TT relativity.” As explained above the ”TT re lativity” deals
with true tensors and the true tensor equations (when no basi s has been cho-
sen),or equivalently (when the coordinate basis has been introdu ced) with
the CBGQs and CBGEs. Thus the ”TT relativity” deals on the sam e foot-
ing with all possible coordinatizations of the chosen refer ence frame. As a
consequence the second Einstein postulate referred to the constancy of t he
coordinate velocity of light, in general, does not hold in th e ”TT relativity.”
Namely, only in Einstein’s coordinatization the coordinat e, one-way, speed
of light is isotropic and constant.
In the following we shall also need the expression for the cov ariant 4D
Lorentz transformations Lab, which is independent of the chosen synchro-
nization, i.e., coordinatization of reference frames, see the works.(14,6,7)It
is
La
b≡La
b(v) =ga
b−((2uavb)/c2) + (ua+va)(ub+vb)/c2(1 +γ),(1)
where uais the proper velocity 4-vector of a frame Swith respect to itself
(only u0/ne}ationslash= 0, see also Ref. 13) ua=cna, nais the unit 4-vector along
thex0axis of the frame S,andvais the proper velocity 4-vector of S′
relative to S.Further u·v=uavaandγ=−u·v/c2.When we use the
”e” coordinatization then Labis represented by Lµν,e,the usual expression
for pure Lorentz transformation, but with vi
e(the proper velocity 4-vector
vµ
eisvµ
e≡dxµ
e/dτ= (γec, γevi
e), dτ≡dte/γeis the scalar proper-time, and
γe≡(1−v2
e/c2)1/2) replacing the components of the ordinary velocity 3-vecto r
V.Obviously, in the usual form, the Lorentz transformations c onnect two
coordinate representations, basis components (in the ”e” c oordinatization)
xµ
e, xµ′
eof a given event; xµ
e, xµ′
erefer to two relatively moving IFRs (with the
Minkowski metric tensors) SandS′,
xµ′
e=Lµ′
ν,exν
e, L0′
0,e=γe, L0′
i,e=Li′
0,e=−γevi
e/c,
Li′
j,e=δi
j+ (γe−1)vi
evje/v2
e. (2)
Since gµν,eis a diagonal matrix the space xi
eand time te(x0
e≡cte) parts of
xµ
edo have their usual meaning.
The invariant spacetime length (the Lorentz scalar) betwee n two points
(events) in 4D spacetime is defined as
l= (gablalb)1/2, (3)
7where la(lb) is the distance 4-vector between two events AandB,la=
la
AB=xa
B−xa
A. In the ”e” coordinatization the geometrical quantity l2can
be written in terms of its representation l2
e,with the separated spatial and
temporal parts, l2=l2
e= (li
elie)−(l0
e)2. Such separation remains valid in
other inertial coordinate systems with the Minkowski metri c tensor, and in
S′one finds l2=l′2
e= (li′
eli′e)−(l0′
e)2,where lµ′
einS′is connected with lµ
ein
Sby the Lorentz transformation Lµ′
ν,e(2).
This is not so in the ”r” cordinatization. In order to explain this statement
we now expose the ”r” cordinatization in more detail. The bas is vectors
in the ”r” cordinatization are constructed as in Refs. 13, 6 a nd 7. The
temporal basis vector e0is the unit vector directed along the world line
of the clock at the origin. The spatial basis vectors by defini tion connect
simultaneous events, the event ”clock at rest at the origin reads 0 time” wi th
the event ”clock at rest at unit distance from the origin read s 0 time,” and
thus they are synchronization-dependent. The spatial basi s vector eiconnects
two above mentioned simultaneous events when Einstein’s sy nchronization
(ε= 1/2) of distant clocks is used. The temporal basis vector r0is the same
ase0.The spatial basis vector riconnects two above mentioned simultaneous
events when ”radio” clock synchronization ( ε= 0) of distant clocks is used.
The spatial basis vectors, e.g., r1, r1′, r1′′..are parallel and directed along
an (observer-independent) light line. Hence, two events th at are ”everyday”
(”r”) simultaneous in Sare also ”r” simultaneous for all other IFRs.
The connection between the basis vectors in the ”r” and ”e” co ordinati-
zations is given(13,6,7)as
r0=e0, ri=e0+ei.
The geometry of the spacetime is generally defined by the metr ic tensor
gab,which can be expand in a coordinate basis in terms of its compo nents
asgab=gµνdxµ⊗dxν,and where dxµ⊗dxνis an outer product of the
basis 1-forms. The metric tensor gabbecomes gab=gµν,rdxµ
r⊗dxν
rin the
coordinate-based geometric language and in the ”r” coordin atization, where
the basis components of the metric tensor are
g00,r=g0i,r=gi0,r=gij,r(i/ne}ationslash=j) =−1, gii,r= 0.
dxµ
r, dxν
rare the basis 1-forms in the ”r” coordinatization and in S,and
dxµ
r⊗dxν
ris an outer product of the basis 1-forms, i.e., it is the basis for
(0,2) tensors (the subscript′r′stands for the ”r” coordinatization).
8The transformation matrix(6,7)Tµ
ν,rtransforms the ”e” coordinatization
to the ”r” coordinatization. The elements that are different from zero are
Tµ
µ,r=−T0
i,r= 1.
For the sake of completeness we also quote the Lorentz transf ormation Lµ′
ν,r
in the ”r” coordinatization. It can be easily found from Lab(1) and the
known gµν,r,and the elements that are different from zero are
x′µ
r=Lµ′
ν,rxν
r, L0′
0,r=K, L0′
2,r=L0′
3,r=K−1,
L1′
0,r=L1′
2,r=L1′
3,r= (−βr/K), L1′
1,r= 1/K,
L2′
2,r=L3′
3,r= 1, (4)
L1′
0,r=L1′
2,r=L1′
3,r= (−βr/K), L1′
1,r= 1/K, L2′
2,r=L3′
3,r= 1,where
K= (1 + 2 βr)1/2,andβr=dx1
r/dx0
ris the velocity of the frame S′as mea-
sured by the frame S,βr=βe/(1−βe) and it ranges as −1/2≺βr≺ ∞.
Sincegµν,r,in contrast to gµν,e,is not a diagonal matrix, then in the spacetime
length l,i.e.,l2,the spatial and temporal parts are not separated. Expressin g
lµ
rin terms of lµ
eone finds that l2=l2
r=l2
e,as it must be. It can be eas-
ily proved(13)that the ”r” synchronization is an assymetric synchronizat ion
which leads to an assymetry in the measured ”one-way” veloci ty of light (for
one direction c+
r=∞whereas in the opposite direction c−
r=−c/2). The
round trip velocity, however, does not depend on the chosen s ynchronization
procedure, and it is ≡c.Although in the ”e” coordinatization the space and
time components of the position 4-vector do have their usual meaning, i.e.,
as in the prerelativistic physics, and in l2
ethe spatial and temporal parts are
separated, it does not mean that the ”e” coordinatization do es have some
advantage relative to other coordinatizations and that the quantities in the
”e” coordinatization are more physical.
A symmetry transformation for the metric gabis called an isometry and it
does not change gab; if we denote an isometry as Φ∗then (Φ∗g)ab=gab.Thus
an isometry leaves the pseudo-Euclidean geometry of 4D spac etime of SR
unchanged. An example of isometry is the covariant 4D Lorent z transforma-
tionLab(1). In our terminology the TT are nothing else but - the isometries.
When the coordinate basis is introduced then, for example, t he isometry
Lab(1) will be expressed as the isometries, the coordinate Lore ntz trans-
formation Lµ′
ν,e(2) in the ”e” coordinatization, or as Lµ′
ν,r(4) in the ”r”
coordinatization. In our treatment mainly the coordinate- based geometric
9form will be used for tensors representing physical quantit ies and for tensor
equations representing physical laws. The basis component s of the CBGQs
will be transformed, e.g., by Lµ′
ν,ewhile the basis vectors eµby the inverse
transformation ( Lµ′
ν,e)−1=Lµν′,e.
The above consideration enable us to better explain the diffe rence in the
concept of sameness of a physical quantity for the ”TT relativity” approach
and the usual covariant approach. We consider a simple examp le the dis-
tance 4-vector (the (1 ,0) tensor) la
AB=xa
B−xa
Abetween two events Aand
B(with the position 4-vectors xa
Aandxa
B). It can be equivalently repre-
sented in the coordinate-based geometric language in differ ent bases, {eµ}
and{rµ}in an IFR S,and{eµ′}and{rµ′}in a relatively moving IFR S′,as
la
AB=lµ
eeµ=lµ
rrµ=lµ′
eeµ′=lµ′
rrµ′,where, e.g., eµare the basis 4-vectors,
e0= (1,0,0,0) and so on, and lµ
eare the basis components when the ”e”
coordinatization is chosen in some IFR S.The decompositions lµ
eeµandlµ
rrµ
(in an IFR S,and in the ”e” and ”r” coordinatizations respectively) and
lµ′
eeµ′andlµ′
rrµ′(in a relatively moving IFR S′, and in the ”e” and ”r” co-
ordinatizations respectively) of the true tensor la
ABare all mathematically
equal quantities. Thus they are really the same quantity consider ed in dif-
ferent relatively moving IFRs and in different coordinatiza tions. This is the
treatment of the distance 4-vector in the ”TT relativity.” O n the other hand
the usual covariant approach does not consider the whole ten sor quantity,
the distance 4-vector la
AB,but only the basis components, mainly lµ
eandlµ′
e
in the ”e” coordinatization (or(6−8)lµ
randlµ′
rin the ”r” coordinatization).
Note that, in contrast to the above equalities for the CBGQs, the sets of
components, e.g., lµ
eandlµ′
e,taken alone, are not equal, lµ
e/ne}ationslash=lµ′
e,and thus
they are not the same quantity from the ”TT relativity” viewp oint. From
the mathematical point of view the components of, e.g., a (1 ,0) tensor are its
values (real numbers) when the basis one-form, for example, eα,is its argu-
ment (see, e.g., Ref. 11). Thus, for example, la
AB(eα) =lµ
eeµ(eα) =lα
e(where
eαis the basis one-form in an IFR Sand in the ”e” coordinatization), while
la
AB(eα′) =lµ′
eeµ′(eα′) =lα′
e(where eα′is the basis one-form in S′and in the
”e” coordinatization). Obviously lα
eandlα′
eare not the same real numbers
since the basis one-forms eαandeα′are different bases.
2.1 The TT of the Spacetime Length
In order to explore the difference between the ”TT relativity ” and the
”AT relativity” we consider the spacetime length for a movin g rod and then
for a moving clock. The same examples will be also examined in the ”AT
10relativity.”. Let us take, for simplicity, to work in 2D spac etime. Then we
also take a particular choice for the 4-vector la
AB.In the usual ”3+1” picture
it corresponds to an object, a rod, that is at rest in an IFR Sand situated
along the common x1
e, x1′
e−axes; L0is its rest length. The decomposition
of the geometric quantity la
ABin the ”e” coordinatization and in Sisla
AB=
l0
ee0+l1
ee1= 0e0+L0e1,while in S′,where the rod is moving, it becomes
la
AB=−βeγeL0e0′+γeL0e1′,and, as explained above, it holds that
la
AB= 0e0+L0e1=−βeγeL0e0′+γeL0e1′. (5)
la
ABis a tensor of type (1,0) and in (5) it is written in the coordin ate-based
geometric language in terms of basis vectors e0, e1,(e0′, e1′) and the basis
components lµ
e(lµ′
e) of a specific IFR.
We note once again that in the ”TT relativity” the basis compo nents lµ
e
inSandlµ′
einS′,when taken alone, do not represent the same 4D quantity.
Only the geometric quantity la
AB,i.e., the CBGQs lµ
eeµ=lµ′
eeµ′comprising
both, components and a basis, is the same 4D quantity for diffe rent relatively
moving IFRs; Ref. 11: ”....the components tell only part of t he story. The
basis contains the rest of information.” Of course we could e quivalently work
in another coordinatization, e.g., the ”r” coordinatizati on, as shown in Refs.
6 and 7. Then we would find that la
AB=lµ
eeµ=lµ
rrµ=lµ′
eeµ′=lµ′
rrµ′,where
rµandrµ′are the basis 4-vectors, and lµ
randlµ′
rare the basis components in
the ”r” coordinatization, and in SandS′respectively. (The expressions for
lµ
randlµ′
rcan be easily found from the known transformation matrix Tµ
ν,r.)
We see from (5) that in the ”e” coordinatization, which is com monly used in
the ”AT relativity,” there is a dilatation of the spatial par tl1′
e=γeL0with
respect to l1
e=L0and not the Lorentz contraction as predicted in the ”AT
relativity.” Hovewer it is clear from the above discussion t hat comparison of
only spatial parts of the components of the distance 4-vecto rla
ABinSandS′is
physically meaningless in the ”TT relativity.” When only so me components
of the whole tensor quantity are taken alone, then, in the ”TT relativity,”
they do not represent some definite physical quantity in the 4 D spacetime.
Also we remark that always the whole tensor quantity la
ABcomprising com-
ponents and a basis is transformed by the Lorentz transforma tion from S
toS′.Note that if l0
e= 0 then lµ′
ein any other IFR S′will contain the time
component l0′
e/ne}ationslash= 0.The spacetime length for the considered case is frame and
coordinatization independent quantity l= (lµ
e,rlµe,r)1/2= (lµ′
e,rlµ′e,r)1/2=L0.
In the ”e” coordinatization and in S,the rest frame of the rod, where the
temporal part of lµ
eisl0
e= 0,the spacetime length lis a measure of the spatial
11distance, i.e., of the rest spatial length of the rod, as in th e prerelativistic
physics.
In a similar manner we can choose another particular choice f or the dis-
tance 4-vector la
AB,which will correspond to the well-known ”muon experi-
ment,” and which is interpreted in the ”AT relativity” in ter ms of the time
dilatation. First we consider this example in the ”TT relati vity.” The dis-
tance 4-vector la
ABwill be examined in two relatively moving IFRs SandS′,
i.e., in the/braceleftBig
eµ/bracerightBig
and{eµ′}bases. The Sframe is chosen to be the rest frame
of the muon. Two events are considered; the event Arepresents the creation
of the muon and the event Brepresents its decay after the lifetime τ0inS.
The position 4-vectors of the events AandBinSare taken to be on the
world line of a standard clock that is at rest in the origin of S.The distance
4-vector la
AB=xa
B−xa
Athat connects the events AandBis directed along
thee0basis vector from the event Atoward the event B.This geometric
quantity can be written in the coordinate-based geometric l anguage. Thus
it can be decomposed in two bases {eµ}and{eµ′}as
la
AB=cτ0e0+ 0e1=γcτ0e0′+βeγecτ0e1′. (6)
Similarly we can easily find(6,7)the decompositions of la
ABin the ”r” coordina-
tization. We again see that these decompositions, containi ng both the basis
components and the basis vectors, are the same geometric qua ntityla
AB. la
AB
does have only temporal parts in S, while in the {eµ′}basisla
ABcontains not
only the temporal part but also the spatial part. It is visibl e from (6) that the
comparison of only temporal parts of the representations of the distance 4-
vector is physically meaningless in the ”TT relativity.” Th e spacetime length
lis always a well-defined quantity in the ”TT relativity” and f or this exam-
ple it is l= (lµ
elµe)1/2= (lµ′
elµ′e)1/2= (lµ
rlµr)1/2= (lµ′
rlµ′r)1/2= (−c2τ2
0)1/2.
Since in Sthe spatial parts l1
e,roflµ
e,rare zero the spacetime length linS
is a measure of the temporal distance, as in the prerelativis tic physics; one
defines that c2τ2
0=−lµ
elµe=−lµ
rlµr.
2.2 The AT of the Spatial and Temporal Distances
In order to better explain the difference between the TT and th e AT we
now consider the same two examples as above but from the point of view of
the conventional, i.e., Einstein’s(3)interpretations of the spatial length of the
moving rod and the temporal distance for the moving clock.
The synchronous definition of the spatial length , introduced by Einstein,(3)
defines length as the spatial distance between two spatial points on the (mov-
12ing) object measured by simultaneity in the rest frame of the observer. One
can see that the concept of sameness of a physical quantity is quite different
in the ”AT relativity” but in the ”TT relativity.” Thus for th e Einstein def-
inition of the spatial length one considers only the component l1
e=L0oflµ
eeµ
(when l0
eis taken = 0 ,i.e., the spatial ends of the rod at rest in Sare taken
simultaneously at t= 0) and performs the Lorentz transformation Lµν,eof
the basis components lµ
e(but not of the basis itself) from S′toS,which
yields
l0
e=γel0′
e+γeβel1′
e
l1
e=γel1′
e+γeβel0′
e. (7)
Then one retains only the transformation of the spatial comp onent l1
e(the
second equation in (7)) neglecting completely the transformation of the tem-
poral part l0
e(the first equation in (7)). Furthermore in the transformati on
forl1
eone takes that the temporal part in S′l0′
e= 0,( i.e., the spatial ends of
the rod moving in S′are taken simultaneously at some arbitrary t′=b). The
quantity obtained in such a way will be denoted as L1′
e(it is not equal to l1′
e
appearing in the transformation equations (7)) This quanti tyL1′
edefines in
the ”AT relativity” the synchronously determined spatial length of the mov-
ing rod in S′. The mentioned procedure gives l1
e=γeL1′
e,that is, the famous
formula for the Lorentz contraction,
L1′
e=l1
e/γe=L0/γe, (8)
This quantity, L1′
e=L0/γe,is the usual Lorentz contracted spatial length ,
and the quantities L0andL1′
e=L0/γeare considered in the ”AT relativ-
ity” to be the same quantity for observers in SandS′. The comparison
with the relation (5) clearly shows that the quantities L0andL1′
e=L0/γe
are two different and independent quantities in 4D spacetime .Thus, in the
”TT relativity” the same quantity for different observers is the tensor quan-
tity, the 4-vector la
AB=lµ
eeµ=lµ′
eeµ′=lµ
rrµ=lµ′
rrµ′;only one quantity
in 4D spacetime. However in the ”AT relativity” different quantities in 4D
spacetime, the spatiall distances l1
e=L0andL1′
e(or similarly in the ”r”
coordinatization) are considered as the same quantity for d ifferent observers.
The relation for the Lorentz ”contraction” of the moving rod in the ”r” co-
ordinatization can be easily obtained performing the same p rocedure as in
the ”e” coordinatization, and it is
L1′
r=L0/K= (1 + 2 βr)−1/2L0, (9)
13see also Refs. 6 and 7. We see from (9) that there is a length ”di latation” ∞ ≻
L1′
r≻L0for−1/2≺βr≺0 and the standard length ”contraction” L0≻
L1′
r≻0 for positive βr,which clearly shows that the Lorentz contraction is
not physically correctly defined transformation. Thus the Lorentz contraction
is the transformation that connects different quantities (i n 4D spacetime) in
SandS′,or in different coordinatizations, which implies that it is - an AT.
The same example of the ”muon decay” will be now considered in the
”AT relativity” (see also(7)). In the ”e” coordinatization the events AandB
are again on the world line of a muon that is at rest in S.We shall see once
again that the concept of sameness of a physical quantity is q uite different in
the ”AT relativity.” Thus for this example one compares the basis component
l0
e=cτ0oflµ
eeµwith the quantity, which is obtained from the basis component
l0′
ein the following manner; first one performs the Lorentz trans formation of
the basis components lµ
e(but not of the basis itself) from the muon rest frame
Sto the frame S′in which the muon is moving. This procedure yields
l0′
e=γel0
e−γeβel1
e
l1′
e=γel1
e−γeβel0
e. (10)
Similarly as in the Lorentz contraction one now forgets the transformation
of the spatial part l1′
e(the second equation in (10)) and considers only the
transformation of the temporal part l0′
e(the first equation in (10)). This is,
of course, an incorrect step from the ”TT relativity” viewpo int. Then taking
thatl1
e= 0 (i.e., that x1
Be=x1
Ae) in the equation for l0′
e(the first equation in
(10)) one finds the new quantity which will be denoted as L0′
e(it is not the
same as l0′
eappearing in the transformation equations (10)). The tempo ral
distance l0
edefines in the ”AT relativity,” and in the ”e” basis, the muon
lifetime at rest, while L0′
eis considered in the ”AT relativity,” and in the
”e” coordinatization, to define the lifetime of the moving mu on in S′.The
relation connecting L0′
ewithl0
e,which is obtained by the above procedure, is
then the well-known relation for the time dilatation,
L0′
e/c=t′
e=γel0
e/c=τ0(1−β2
e)−1/2. (11)
By the same procedure we can find(7)the relation for the time ”dilatation”
in the ”r” coordinatization
L0′
r=Kl0
r= (1 + 2 βr)1/2cτ0. (12)
14This relation shows that the new quantity L0′
r,which defines in the ”AT rel-
ativity” the temporal separation in S′,where the clock is moving, is smaller
- time ”contraction” - but the temporal separation l0
r=cτ0inS,where the
clock is at rest, for −1/2≺βr≺0,and it is larger - time ”dilatation” - for
0≺βr≺ ∞. From this consideration we conclude that in the ”TT relativity”
the same quantity for different observers is the tensor quant ity, the 4-vector
la
AB=lµ
eeµ=lµ′
eeµ′=lµ
rrµ=lµ′
rrµ′;only one quantity in 4D spacetime. How-
everin the ”AT relativity” different quantities in 4D spacetime, the temporal
distances l0
e, L0′
e, l0
r, L0′
rare considered as the same quantity for different
observers. This shows that the time ”dilatation” is the tran sformation con-
necting different quantities (in 4D spacetime) in SandS′,or in different
coordinatizations, which implies that it is - an AT.
We can compare the obtained results for the determination of the space-
time length in the ”TT relativity” and the determination of t he spatial and
temporal distances in the ”AT relativity” with the existing experiments. This
comparison is presented in Ref. 8. It is shown there that the ” TT relativity”
results agree with all experiments that are complete from th e ”TT relativ-
ity” viewpoint, i.e., in which all parts of the considered te nsor quantity are
measured in the experiment. However the ”AT relativity” res ults agree only
with some of examined experiments.
The difference between the ”AT relativity” and the ”TT relati vity” will
be now examined considering the famous Michelson-Morley(2)experiment.
3. THE MICHELSON-MORLEY EXPERIMENT
In the Michelson-Morley(2)experiment two light beams emitted by one
source are sent, by half-silvered mirror O, in orthogonal directions. These
partial beams of light traverse the two equal (of the length L) and perpen-
dicular arms OM1(perpendicular to the motion) and OM2(in the line of
motion) of Michelson’s interferometer. The behaviour of th e interference
fringes produced on bringing together these two beams after reflection on
the mirrors M1andM2is examined. The experiment consists of looking for
a shift of the intereference fringes as the apparatus is rota ted. The expected
maximum shift in the number of fringes (the measured quantit y) on a 900
rotation is
△N=△(φ2−φ1)/2π, (13)
where△(φ2−φ1) is the change in the phase difference when the interferomete r
is rotated through 900. φ1andφ2are the phases of waves moving along the
15paths OM1OandOM2O,respectively.
3.1 The Nonrelativistic Approach
In the nonrelativistic approach the speed of light in the pre ferred frame is
c.Then, on the ether hypothesis, one can determine the speed of light, in the
Earth frame, i.e., in the rest frame of the interferometer (t heSframe). This
speed is ( c2−v2)1/2for the path along an arm of the Michelson interferometer
oriented perpendicular to its motion. (The motion of the int erferometer is at
velocity vrelative to the preferred frame (the ether); the Earth toget her with
the interferometer moving with velocity vthrough the ether is equivalent to
the interferometer at rest with the ether streaming through it with velocity
−v.) Since in Sboth waves are brought together to the same spatial point
the phase difference φ2−φ1is determined only by the time difference t2−t1;
φ2−φ1= 2π(t2−t1)/T,where t1andt2are the times required for the
complete trips OM1OandOM2O,respectively, and T(=λ/c) is the period
of vibration of the light. From the known speed of light one fin ds that t1is
t1=tOM1+tM1O, where tOM1=L/c(1−v2/c2)1/2=tM1O, whence
t1= 2L/c(1−v2/c2)1/2.
(In the following we shall denote that tOM1=t11andtM1O=t12.) Similarly,
the speed of light on the path OM2isc−v,and on the return path is c+v,
giving that t2=tOM2+tM2O=t21+t22=L/(c−v) +L/(c+v). Thence
t2= 2L/c(1−v2/c2).
We see that according to the nonrelativistic approach the ti met1is a little
less than the time t2,even though the mirrors M1andM2are equidistant
fromO.The time difference t2−t1is
t2−t1= (2L/c)γ(γ−1),
(γ= (1−v2/c2)−1/2) and to order v2/c2it ist2−t1≃(L/c)(v2/c2).The phase
difference φ2−φ1is determined as φ2−φ1=ω(t21+t22)−ω(t11+t12) =ω(t2−
t1) and the change in the phase difference when the interferomet er is rotated
through 900is△(φ2−φ1) = 2ω(t2−t1).Inserting it into △N(13) (ω= 2πc/λ,
and the measured quantity △Nis in this case △N= 2(t2−t1)c/λ) we find
that△N= (4L/λ)γ(γ−1).To the same order v2/c2this△Nis
△N≃(2L/λ)(v2/c2).
16This result is obtained by the classical analysis in the Eart h frame (the
interferometer rest frame).
Let us now consider the same experiment in the preferred fram e (the
S′frame). In the nonrelativistic theory the two frames are con nected by
the Galilean transformations. Consequently the correspon ding times in both
frames are equal, t1=t′
1andt2=t′
2,whence t2−t1=t′
2−t′
1and, supposing
that again the phase difference is determined only by the time difference,
△N′=△N.However, for the further purposes, it is worth to find explici tly
t′
1andt′
2considering the experiment directly in the preferred frame . Since
the speed of light in the preferred frame is c,the preferred-frame observer
considers that the light travels a distance ct′
OM′
1along the hypotenuse of a
triangle; in the same time t′
OM′
1=t′
11the mirror M1moves to M′
1, i.e., to
the right a distance vt′
11.From the right triangle this observer finds t′
11=
L/c(1−v2/c2)1/2.The return trip is again along the hypotenuse of a triangle
and the return time t′
M′
1O′=t′
12is =t′
11(the half-silvered mirror Omoved
toO′int′
1). The total time for such a zigzag path is, as it must be, t′
1=
t′
11+t′
12=t1. For the arm oriented parallel to its motion the preferred-
frame observer considers that the light, when going from OtoM′
2, must
traverse a distance L+vt′
OM′
2(we denote t′
OM′
2ast′
21) at the speed c,whence
L+vt′
21=ct′
21andt′
21=L/(c−v). The time t′
M′
2O′′is denoted as t′
22(the
half-silvered mirror Omoved to O′′int′
2). Then, in a like manner, the time
t′
22for the return trip is t′
22=L/(c+v). The total time t′
2=t′
21+t′
22is, as
it must be, equal to t2, t′
2=t2. The phase difference in S′is determined as
φ′
2−φ′
1=ω(t′
21+t′
22)−ω(t′
11+t′
12) =ω(t′
2−t′
1),and it is φ′
2−φ′
1=ω(t2−t1).
Consequently the expected maximum shift in the number of fri nges on a 900
rotation in the S′frame is
△N′=△N≃(2L/λ)(v2/c2). (14)
It has to be noted here that the same ω,i.e.,T,orλ, is used for all paths,
both in SandS′.This discussion shows that the nonrelativistic theory is a
consistent theory giving the same △Nin both frames. However it does not
agree with the experiment. Namely Michelson and Morley found from their
experiment that was no observable fringe shift.
3.2 The Traditional ”AT Relativity” Approach
Next we examine the same experiment in the traditional ”AT re lativity”
approach. We remark that the experiment is usually discusse d only in the
17”e” coordinatization, and again, as in the nonrelativistic theory, the phase
difference φ2−φ1is considered to be determined only by the time difference
t2−t1.In the ”AT relativity” and in the ”e” coordinatization it is p ostulated
(Einstein’s second postulate), in contrast to the nonrelat ivistic theory, that
lightalways travels with speed c.
Hence in the Sframe (the rest frame of the interferometer), and with the
same notation as in the preceding section, we find t1=t11+t12=L/c+L/c=
2L/cand also t2=t21+t22= 2L/c=t1.With the assumption that only the
time difference t2−t1matters, it follows that φ2−φ1=ω(t21+t22)−ω(t11+
t12) =ω(t2−t1) = 0, whence △N= 0,in agreement with the experiment.
In the S′frame (the preferred frame) the time t′
1is determined in the same
way as in the nonrelativistic theory, i.e., supposing that a zigzag path is taken
by the light beam in a moving ”light clock”. Thus, the light-t ravel time t′
1
is exactly equal to that one in the nonrelativistic theory, t′
1=t′
11+t′
12=
2L/c(1−v2/c2)1/2.Comparing with t1= 2L/cwe see that, in contrast to
the nonrelativistic theory, it takes a longer time for light to go from end to
end in the moving clock but in the stationary clock,
t′
1=t1/(1−v2/c2)1/2=γt1. (15)
This relation is Eq. (11) for the dilatation of time in the ”e” coordinatiza-
tion that is considered in Sec. 2.2. The presented derivatio n is the usual
way in which it is shown how, in the ”AT relativity,” the time d ilatation is
forced upon us by the constancy of the speed of light, see also , e.g., Ref. 15
p.15-6 and Ref. 16 p.359, or an often cited paper on modern tes ts of special
relativity.(17)However, in the ”AT relativity,” the light-travel time t′
2is de-
termined by invoking the Lorentz contraction. It is argued t hat a preferred
frame observer measures the length of the arm oriented paral lel to its motion
to be contracted to a length L′=L(1−v2/c2)1/2, Eq. (8) in Sec. 2.2. Then
t′
2is determined in the same way as in the nonrelativistic theor y but with L′
replacing the rest length L, t′
2=t′
21+t′
22= (L′/(c−v)) + (L′/(c+v)), i.e.,
t′
2= 2L/c(1−v2/c2)1/2=γt2=t′
1. (16)
Thence t′
2−t′
1= 0 and
φ′
2−φ′
1=ω(t′
21+t′
22)−ω(t′
11+t′
12) =ω(t′
2−t′
1) = 0,
where again, in the same way as in the nonrelativistic theory , the same ω,
i.e.,T,orλ, is used for all paths, both in SandS′.As a consequence it is
18found in the ”AT relativity” that △N′in the S′frame is the same as △Nin
theSframe
△N′=△N= 0. (17)
We quoted such usual derivation in order to illustrate how th e time dilatation
and the Lorentz contraction are used in the ”AT relativity” t o show the
agreement between the theory and the famous Michelson-Morl ey experiment.
Although this procedure is generally accepted by the majori ty of physicists
as a correct one and quoted in all textbooks on the subject, we note that such
an explanation of the null result of the experiment is very aw kward and does
not use at all the 4D symmetry of the spacetime. The derivatio n deals with
the temporal and spatial distances as well defined quantitie s, i.e., in a similar
way as in the prerelativistic physics, and then in an artifici al way introduces
the changes in these distances due to the motion. Our results from Secs. 2.,
2.1, and 2.2 reveal why the Lorentz contraction and the time d ilatation are
not physically correctly defined transformations. Consequ ently the approach
which uses them for the explanation of the experimental resu lts cannot be in
agreement with the 4D symmetry of the 4D spacetime.
3.3 Driscoll’s ”AT Relativity” Approach
In Ref. 1 the above discussed ”AT relativity” calculation in the ”e” coor-
dinatization (Sec. 3.2) of the fring shift in the Michelson- Morley experiment
is repeated, and, of course, the observed null fringe shift i s obtained. This
result is independent of changes of v, the relative velocity of S(the rest frame
of the interferometer) and S′(inS′the interferometer is moving) ,and/or θ,
the angle that the undivided ray from the source to the beam di vider makes
withv. However, it is noticed(1)that in such a traditional calculation of
△(φ2−φ1) only path lengths (optical or geometrical), i.e., the temp oral dis-
tances (for example, the times t1andt2required for the complete trips OM1O
andOM2O,respectively), are considered. The Doppler effect on wavele ngth
in the S′frame, in which the interferometer is moving, is not taken in to
account.
Then the same calculation of △(φ′
2−φ′
1) as the traditional one is performed,(1)
but determing the increment of phase along some path, e.g. OM′
1inS′, not
only by the segment of geometric path length (i.e., the tempo ral distance for
that path) but also by the wavelength in that segment (i.e., t he frequency
of the wave in that segment). Accordingly, the phase differen ce (in our no-
tation) φ′
1−φ′
2,in the S′frame, between the ray along the vertical path
19OM′
1O′and that one along the longitudinal path OM′
2O′′respectively, is
found (see(1)) to be
(φ′
1−φ′
2)(b)/2π= 2(Lν/c)(1 +ε+β2)−2(Lν/c)(1 + 2 β2)
= 2(Lν/c)(ε−β2), (18)
Eqs.(23-25) in Ref. 1. Lis the length of the segment OM2andL=L(1 +ε)
(ε≪1) is taken in Ref. 1 to be the length of the arm OM1.As explained:(1)
”The difference L−L=εLis usually a few wavelengths ( ≺25) and is essen-
tial for obtaining useful interference fringes.” L,Landνare determined in
S, the rest frame of the interferometer. In this expression th e Doppler effect
ofvon the frequencies, and the Lorentz contraction of the longitudinal arm ,
are taken into account. In a like manner Driscoll finds the pha se difference
in the case when the interferometer is rotated through 900
(φ′
1−φ′
2)(a)/2π= 2(Lν/c)(1 +ε+ 2β2)−2(Lν/c)(1 +β2)
= 2(Lν/c)(ε+β2), (19)
Eqs.(19-21) in Ref. 1. Hence it is found(1)a ”surprising” non-null fringe shift
△N′=△(φ′
2−φ′
1)/2π= 4(Lν/c)β2, (20)
where △(φ′
2−φ′
1) = (φ′
1−φ′
2)(b)−(φ′
1−φ′
2)(a),and we see that the entire
fringe shift is due to the Doppler shift. From the non-null re sult (20) the au-
thor of(1)concluded: ”that the Maxwell-Einstein electromagnetic eq uations
and special relativity jointly are disproved, not confirmed , by the Michelson-
Morley experiment.” However such a conclusion cannot be dra wn from the
result (20). The origin of the appearance of △N′/ne}ationslash= 0 (20) is quite different
than that considered in Ref. 1, and it will be explained below . While(1)
investigates those changes which are caused by the Doppler e ffect another
work(18)considers the changes in the usual derivation of △N′,which are
caused by the aberration of light. Both changes are examined only in the
”e” coordinatization, and both would be different in, e.g., t he ”r” coordi-
natization. This means that △N′inS′will be dependent on the chosen
synchronization. Also, both works(1,18)deal with the Lorentz contraction in
the same way as in the traditional analysis. But the Lorentz c ontraction is
an AT, as shown in Secs. 2., 2.1, and 2.2. Consequently, the tr aditional
analysis and the works(1,18)belong to the ”AT relativity,” which, as found in
Ref. 8, and as follows from the dependence of the theoretical results on the
20chosen synchronization, is not capable to explain in a satis factory manner
the results of the Michelson-Morley experiment.
4. THE ”TT RELATIVITY” APPROACH
Next we examine the Michelson-Morley experiment from the ”T T rela-
tivity” viewpoint. The relevant quantity is the phase of a li ght wave, and it
is (when written in the abstract index notation)
φ=kagablb, (21)
where kais the propagation 4-vector, gabis the metric tensor and lbis the
distance 4-vector. All quantities in (21) are true tensor qu antities. As dis-
cussed in Sec. 2. these quantities can be written in the coord inate-based
geometric language and, e.g., the decompositions of kainSandS′and in
the ”e” and ”r” coordinatizations are
ka=kµ
eeµ=kµ′
eeµ′=kµ
rrµ=kµ′
rrµ′,
where the basis components kµof the CBGQ in the ”e” coordinatization
are transformed by Lµ′
ν,e(2), while the basis vectors eµare transformed by
the inverse transformation ( Lµ′
ν,e)−1=Lµν′,e.Similarly holds for the ”r”
coordinatization where the Lorentz transformation Lµ′
ν,r(4) has to be used.
By the same reasoning the phase φ(21) is given in the coordinate-based
geometric language as
φ=kµ
egµν,elν
e=kµ′
egµν,elν′
e=kµ
rgµν,rlν
r=kµ′
rgµν,rlν′
r, (22)
(Note that the Lorentz transformation Lµ′
ν,e(2) and also Lµ′
ν,r(4) are the
TT, i.e., the isometries, and hence gµν,e=gµ′ν′,e,gµν,r=gµ′ν′,r, what is al-
ready taken into account in (22).) The traditional derivati on of△N(Sec.
3.2) deals, as already said, only with the calculation of t1andt2inSandt′
1
andt′
2inS′,but does not take into account either the changes in frequenc ies
due to the Doppler effect or the aberration of light. The ”AT re lativity”
calculations(1,18)improve the traditional procedure taking into account the
changes in frequencies,(1)and the aberration of light.(18)But all these ap-
proaches explain the experiments using the AT, the Lorentz c ontraction and
the time dilatation, and furthermore they always work only i n the ”e” co-
ordinatization. None of the ”AT relativity” calculations d eals with the true
tensors or with the CBGQs (comprising both components and a b asis). In
21this case such 4D tensor quantity is the phase (21) or (22). It will be shown
here that the non-null theoretical result obtained in Ref. 1 is a consequence
of the fact that Driscoll’s calculation also belongs to the ” AT relativity” ap-
proach. It considers only a part of the 4D tensor quantity φ(21), or (22), uses
the AT and works only in the ”e” coordinatization. In the ”TT relativity”
approach to special relativity neither the Doppler effect no r the aberration of
light exist separately as well defined physical phenomena. T he separate con-
tributions to φ(21), or (22), of the ωt(i.e.,k0l0) factor(1)andkl(i.e.,kili)
factor(18)are, in general case, meaningless in the ”TT relativity.” Fr om the
”TT relativity” viewpoint only their indivisible unity, th e phase φ(21), or
(22), is a correctly defined 4D quantity. All quantities in (21), i.e., ka,gab,
lbandφ,are the true tensor quantities, which means that in all relat ively
moving IFRs and in all permissible coordinatizations alway sthe same 4D
quantity , e.g., ka,orlb,orφ,is considered. (Eq. (22) shows it for φ.) This is
not the case in the ”AT relativity” where, for example, the re lation t′
1=γt1
is not the Lorentz transformation of some 4D quantity, and t′
1andt1do
not correspond to the same 4D quantity considered in S′andSrespectively
but to different 4D quantities, as can be clearly seen from Sec . 2.2 (see Eq.
(11)). Only in the ”e” coordinatization the ωtandklfactors can be con-
sidered separately. Therefore, and in order to retain the si milarity with the
prerelativistic and the ”AT relativity” considerations, w e first determine φ
(21), (22), in the ”e” coordinatization and in the Sframe (the rest frame of
the interferometer). This means that φwill be calculated from (22) as the
CBGQ φ=kµ
egµν,elν
e.
Let now A, B andA1denote the events; the departure of the transverse
ray from the half-silvered mirror O,the reflection of this ray on the mirror M1
and the arrival of this beam of light after the round trip on th e half-silvered
mirror O,respectively. In the same way we have, for the longitudinal a rm
of the inteferometer, the corresponding events A, CandA2.To simplify the
notation we omit the subscript ’e’ in all quantities. Then kµ
ABandlµ
AB(the
basis components of ka
ABandla
ABin the ”e” coordinatization and in S) for
the wave on the trip OM1(the events AandB) arekµ
AB= (ω/c,0,2π/λ,0),
lµ
AB= (ctM1,0,L,0). For the wave on the return trip M1O,(the events B
andA1)kµ
BA1= (ω/c,0,−2π/λ,0) and lµ
BA1= (ctM1,0,−L,0). Hence the
increment of phase φ1for the the round trip OM1O,is
φ1=kµ
ABlµAB+kµ
BA1lµBA1= 2(−ωtM1+ (2π/λ)L), (23)
where ωis the angular frequency and, for the sake of comparison with ,(1)the
22length of the arm OM1is taken to be L=L(1 +ε),andLis the length
of the segment OM2.Using the Lorentz transformation Lµ′
ν,e(2) one can
findkµ′andlµ′in the ”e” coordinatization and in S′for the same trips as in
S. Then it can be easily shown that φ′
1inS′is the same as in S, φ′
1=φ1.
Also using the transformation matrix Tµ
ν,r(Sec. 2), which transforms the ”e”
coordinatization to the ”r” coordinatization, one can get a ll quantities in the
”r” coordinatization and in S, and then by the Lorentz transformation Lµ′
ν,r
(4) these quantities can be determined in the ”r” coordinati zation and in S′.
φ1will be always the same in accordance with (22). Note that gµν,rfrom
Sec. 2 has to be used in the calculation of φin the ”r” coordinatization. As
an example we quote kµ
AB,randlµ
AB,r:kµ
AB,r= ((ω/c)−2π/λ,0,2π/λ,0) and
lµ
AB,r= (ctM1−L,0,L,0).Hence, using gµν,rone easily finds that
φAB,r=kµ
rgµν,rlν
r= (−ωtM1+ (2π/λ)L) =φAB,e.
For further purposes we shall also need kµ′
AB,randlµ′
AB,r.They are kµ′
AB,r=
((γω/c)(1+β)−2π/λ,−βγω/c, 2π/λ,0) and lµ′
AB,r= (γctM1(1+β)−L,−βγct M1,L,0)
which yields
φ′
AB,r=φAB,r=φ′
AB,e=φAB,e.
In a like manner we find kµ
ACandlµ
ACfor the wave on the trip OM2,(the
corresponding events are AandC) askµ
AC= (ω/c,2π/λ,0,0) and lµ
AC=
(ctM2, L,0,0).For the wave on the return trip M2O(the corresponding events
areCandA2)kµ
CA2= (ω/c,−2π/λ,0,0) and lµ
CA2= (ctM2,−L,0,0)), whence
φ2=kµ
AClµAC+kµ
CA2lµCA2= 2(−ωtM2+ (2π/λ)L). (24)
Of course one finds the same φ2inSandS′and in the ”e” and ”r” coordi-
natizations. Hence
φ1−φ2=−2ω(tM1−tM2) + 2(2 π/λ)(L−L). (25)
Particularly for L=L,and consequently tM1=tM2,one finds φ1−φ2= 0.
It can be easily shown that the same difference of phase (25) is obtained in
the case when the interferometer is rotated through 900,whence we find that
△(φ1−φ2) = 0,and△N= 0.According to the construction φ(21), or (22),
is a frame independent quantity and it also does not depend on the chosen
coordinatization in a considered IFR. Thus we conclude that
△Ne=△N′
e=△Nr=△N′
r= 0. (26)
23This result is in a complete agreement with the Michelson-Mo rley(2)experi-
ment.
4.1 Explanation of Driscoll’s Non-Null Fringe Shift
Driscoll’s improvement of the traditional ”AT relativity” derivation of the
fringe shift can be easily obtained from our ”TT relativity” approach tak ing
only the product k0′
el0′ein the calculation of the increment of phase φ′
einS′
in which the apparatus is moving. All quantities in the ”e” co ordinatization
and in S′are obtained by the Lorentz transformation Lµ′
ν,e(2) from the
corresponding ones in S.We remark once again that Driscoll’s ”AT relativity”
approach refers only to the ”e” coordinatization (of course it holds for the
traditional ”AT relativity” approach from Sec. 3.2 as well) . Therefore we
again omit the subscript′e′in all quantities. Then we find that in the S′
frame kµ′
AB= (γω/c,−βγω/c, 2π/λ,0) and lµ′
AB= (γctM1,−βγct M1,L,0),and
alsokµ′
BA1= (γω/c,−βγω/c, −2π/λ,0) and lµ′
BA1= (γctM1,−βγct M1,−L,0),
giving that
(−1/2π)(k0′
ABl0′AB+k0′
BA1l0′BA1) = 2γ2νtM1≃2(Lν/c)(1 +ε+β2),(27)
which is exactly Driscoll’s result △PHb; for our notation see (18). Similarly
one finds that
(−1/2π)(k0′
ACl0′AC+k0′
CA2l0′CA2) = 2 γ2(νtM2+β2L/λ)
≃2(Lν/c)(1 + 2 β2), (28)
which is Driscoll’s result △PΞb,see (18). In the same way we can find in S′
Driscoll’s result (19) and finally the non-null fringe shift (20).
We remark that the non-null fringe shift (20) would be quite d ifferent in
another coordinatization, e.g., in the ”r” coordinatizati on, since only a part
k0′
el0′eof the whole 4D tensor quantity φ(21) or (22) is considered. The basis
components of the metric tensor in the ”r” coordinatization , i.e., gµν,r,do
not form a diagonal matrix and therefore the temporal and spa tial parts of
φ=kµ
rgµν,rlν
rcannot be separated. From the above expressions we can easil y
show that the part k0′
rg00,rl0′
r(i.e.,k0′
rl0′r)is quite different but the Driscoll’s
expression k0′
eg00,el0′
e(i.e.,k0′
el0′e). However the physics must not depend on
the chosen coordinatization. Thus when only a part of the who le phase φ
(21) or (22) is taken into account then it leads to an unphysic al result.
The same calculation of ki′li′,the contribution of the spatial parts of kµ′
andlµ′to△N′
e,shows that this term exactly cancel the k0′l0′contribution
24(Driscoll’s non-null fringe shift (20)), yielding that △N′
e=△Ne= 0.Thus
the ”TT relativity”approach to SR naturally explaines the r eason for the
existence of Driscoll’s non-null fringe shift (20).
4.2 Explanation of the ”Apparent” Agreement Between the Tra di-
tional Analysis and the Experiment
The results of the usual ”AT relativity” calculation, which are presented
in Sec. 3.2, can be easily explained from our true tensor formulation of S R
taking only the part k0
el0′eof the whole phase φ(21) or (22) in the calculation
of the increment of phase φ′
einS′.In contrast to Driscoll’s treatment the
traditional analysis considers the part k0
el0e(of the whole phase φ(21), (22))
inS,the rest frame of the interferometer, and k0
el0′einS′, in which the
apparatus is moving. k0
eis not changed in transition from StoS′. Thus the
increment of phase φ1for the round trip OM1OinS, is
φ1=k0
ABg00,el0
AB+k0
BA1g00,el0
BA1=−2(ω/c)(ctM1) =−2ωtM1.(29)
In the S′frame we find for the same trip that
φ′
1=k0
ABl0′AB+k0
BA1l0′BA1=−2(ω/c)(γctM1) =−2ω(γtM1). (30)
This is exactly the result obtained in the traditional analy sis in Sec. 3.2,
which is inerpreted as that there is a ”time dilatation” t′
1=γt1. In the same
way we find that the increment of phase φ2for the round trip OM2OinS, is
φ2=k0
ACl0AC+k0
CA2l0CA2=−2ωtM2, (31)
andφ′
2inS′is
φ′
2=k0
ACl0′AC+k0
CA2l0′CA2=−2(ω/c)(γctM2) =−2ω(γtM2). (32)
This is again the result of the traditional analysis, the ”ti me dilatation,”
t′
2=γt2. Fort1=t2, i.e., for L=L,one finally finds the null fringe shift that
is obtained in the traditional analysis △N′
e=△Ne= 0.We see that such a
null fringe shift is obtained taking into account only a part of the whole phase
φ(21) or (22), and additionally, in that part, k0
eis not changed in transition
fromStoS′. Obviously this correct result follows from a physically in correct
treatment the phase φ(21) or (22). Furthermore it has to be noted that the
usual calculation is always done only in the ”e” coordinatiz ation.
25Since only the part k0
el0eof the whole phase φ(21) or (22) is taken into
account (and also k0′
e=k0
e) the results of the usual ”AT relativity” calcu-
lation are coordinatization dependent. We explicitly show it using the ”r”
coordinatization.
In the ”r” coordinatization the increment of phase φris calculated from
φr=k0
rg00,rl0
rinSand from φ′
r=k0
rg00,rl0′
rinS′.Hence we find that φ1rfor
the round trip OM1OinSis
φ1r=−2(ωtM1+ (2π/λ)L), (33)
andφ2rfor the round trip OM2OinSis
φ2r=−2(ωtM2+ (2π/λ)L). (34)
ForL=L,and consequently tM1=tM2,we find that φ1r−φ2r= 0, whence
△Nr= 0.Remark that the phases φ1randφ2rdiffer from the corresponding
phases φ1eandφ2ein the ”e” coordinatization. As shown above this is not
the case when the whole phase φ(21) or (22) is taken into account.
However, in S′,we find for the same trips that
φ′
1r=−2(γωtM1(1 +β) + (2π/λ)L), (35)
φ′
2r=−2γ2(1 +β2)(ωtM2+ (2π/λ)L). (36)
Obviously φ′
1r−φ′
2r/ne}ationslash= 0 and consequently it leads to the non-null fringe shift
△N′
r/ne}ationslash= 0, (37)
which holds even in the case when tM1=tM2.This result clearly shows
that the agreement between the usual ”AT relativity” calcul ation and the
Michelson-Morley experiment is only an ”apparent” agreeme nt. It is achieved
by an incorrect procedure and it holds only in the ”e” coordin atization.
We also remark that the traditional analysis, i.e., the ”AT r elativity,” gives
different values for the phases, e.g., φ1e, φ′
1e, φ1randφ′
1r,since only a part
of the whole phase φ(21) or (22) is considered. These phases are frame
and coordinatization dependent quantities. When the whole phase φ(21) or
(22) is taken into account, i.e., in ”TT relativity,” all the mentioned phases
are exactly equal quantities; they are the same, frame and co ordinatization
independent, quantity.
265. CONCLUSIONS
In(1)the usual ”relativistic” calculation of the fringe shift in the Michelson-
Morley experiment is objected on the grounds that it does not take into ac-
count the changes in frequencies due to the Doppler effect. Ou r discussion
shows that Driscoll’s calculation is not free from ambiguit ies either. It also
does not work with the complete expression for the phase of th e light waves
((21) or (22)) travelling along the arms of the Michelson-Mo rley interferome-
ter. Both calculations are shown to belong to the ”AT relativ ity,” which does
not deal with the whole 4D tensor quantities and their true tr ansformations.
In this paper we have exposed the approach to SR that deals wit h true tensors
and the true tensor equations (when no basis is chosen) or equ ivalently with
the CBGQs and equations (when the coordinate basis is introd uced), i.e.,
the ”TT relativity.” This approach uses the whole phase φ(21) or (22) and
yields in all IFRs and in all permissible coordinatizations the observed null
fringe shift. At the same time it successfully explains an ap parent agreement
(it holds only in the ”e” coordinatization) of the tradition al ”AT relativity”
approach and disagreement of Driscoll’s ”AT relativity” ap proach with the
experimental results. They are simply consequences of the u se of only some
parts of the 4D tensor quantity φ(21) or (22) and the use of the AT, the
Lorentz contraction and the time dilatation, in the calcula tion of the incre-
ment of phase. The results of the traditional analysis are ex actly obtained
taking into account only the part k0
el0′eof the whole phase φ(21) or (22) in
the calculation of the increment of phase φ′
einS′.Similarly the results of
Driscoll’s analysis are obtained taking only the part k0′
el0′ein the calculation
ofφ′
einS′.In conclusion, the analysis performed in this paper reveals that
the Michelson-Morley experiment does not confirm either the validity of the
traditional Einstein approach or the validity of Driscoll’ s approach. In other
words, the experiment does not confirm the ”AT relativity” ap proach, but
rather an invariant ”TT relavitity” approach to SR.
27References
1. R.B. Driscoll, Phys. Essays 10, 394 (1997).
2. A.A. Michelson and E.W. Morley, Philos. Mag. 24, 449 (1887); Am. J.
Sci.34, 333 (1887).
3. A. Einstein, Ann. Physik 17,891 (1905), tr. by W. Perrett and G.B.
Jeffery, in The principle of relativity (Dover, New York).
4. F. Rohrlich, Nuovo Cimento B 45, 76 (1966).
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6. T.Ivezi´ c, Found. Phys. Lett. 12, 507 (1999).
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in Found. Phys.
8. T. Ivezi´ c, preprint Lanl Archives: physics/0007031.
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10. R.M. Wald, General relativity (The University of Chicago Press, Chicago,
1984).
11. B.F. Schutz, A first course in general relativity (Cambridge University
Press, Cambridge, 1985).
12. C.W. Misner, K.S. Thorne and J.A. Wheeler, Gravitation , (Freeman,
San Francisco, 1970).
13. C. Leubner, K. Aufinger and P. Krumm, Eur. J. Phys. 13,170 (1992).
14. D.E. Fahnline, Am. J. Phys. 50, 818 (1982).
15. R.P. Feynman, R.B. Leightonn and M. Sands, The Feynman lectures on
physics, Vol.1 (Addison-Wesley, Reading, 1964).
16. C. Kittel, W.D. Knight and M.A. Ruderman, Mechanics (McGraw-Hill,
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28 |
arXiv:physics/0101092v1 [physics.atm-clus] 26 Jan 2001AnAb Initio Study of the Structures and Relative Stabilities of Doubly C harged
[(NaCl) m(Na) 2]2+Cluster Ions
Andr´ es Aguado∗
Physical and Theoretical Chemistry Laboratory, Oxford Uni versity, South Parks Road, Oxford OX1 3QZ, UK.
We present ab initio perturbed ion calculations on the structures and relative s tabilities of doubly
charged ((NaCl) m(Na) 2)2+ions. The obtained stabilities show excellent agreement wi th exper-
imental abundances obtained from mass spectra. Those enhan ced stabilities are found to be a
consequence of highly compact structures that can be built o nly for certain values of m. Nearly all
magic number clusters can be shown to be constructed in one of the two following ways: (a) by
adding tri- or penta-atomic chains to two edges of a perfect n eutral (NaCl) ncuboid, with n=m-2 or
n=m-4, respectively; (b) by removing a chloride anion from a per fect singly charged (NaCl) nNa+
cuboid, with n=m+1.
PACS numbers: 36.40.Wa; 36.40.Mr; 36.40.Qv; 61.46.+w
I. INTRODUCTION
Small alkali halide clusters have attracted in the last
years the interest of both experimentalists and theo-
reticians because their simple ioniclike bonding char-
acteristics make them very easy to produce in pure
form and also easily amenable to theoretical modelling.
They are ideal candidates, for example, to study the
behavior of solvated excess electrons and the insula-
tor to metal transition upon alkali enrichment in fi-
nite systems.1–12Structural isomerizations induced by
a temperature increase,13,14as well as the finite sys-
tem analogues of the bulk melting,15–17freezing,18,19and
glass20transitions have also been studied. Apart from
their inherent interest, it has been shown that natu-
ral alkali halide clusters at the marine atmosphere can
be partially responsible for catalytic ozone depletion.21
All of these interesting properties of alkali halide clus-
ters largely depend on the specific structures adopted by
them. Thus, a precise knowledge of the cluster struc-
tures is of paramount importance. Very recently, exper-
imental techniques like electron diffraction from trapped
clusters22or measurements of cluster mobilities23–25have
been succesfully applied to study the structures of ionic
clusters, and photoelectron spectroscopy has also been
applied to study isomerization transitions in small alkali
halide clusters.26At the moment, however, these tech-
niques need parallel theoretical calculations to make a
definite assignment of the observed diffraction pattern,
mobility or ionization potential to a specific isomer ge-
ometry.
The work on alkali halide clusters has been centered
mostly in the singly charged clusters (AX) nA+and in
the neutral clusters (AX) n, where A is an alkali cation
and X a halide anion. The abundance patterns obtained
from the mass spectra of singly charged alkali halide clus-
ter ions27,28point towards a prompt establishment of
bulk rock-salt symmetry. Theoretical calculations have
been able to rationalize the structures adopted by neu-tral stoichiometric clusters in terms of cation/anion size
ratios.29,30These studies show that small sodium iodide
and lithium halide neutral clusters adopt ground state
structures based on the stacking of hexagonal rings, while
the rest of the materials adopt rocksalt-like ground state
structures. Theoretical calculations on the singly charge d
cluster ions are also available,31and they conform to
the experimental expectations of bulk rocksalt symmetry
even for those elements that crystallize in the CsCl-type
structure, namely CsCl, CsBr, and CsI. The emergence
with increasing size of bulk CsCl structure in (CsCl) nCs+
cluster ions has been recently considered.32
The structural problem in neutral and singly charged
alkali halide clusters can thus be considered well under-
stood. Work on doubly charged [(AX) m(A2)]2+cluster
ions has been much more scarce, probably due to the in-
herent instability produced by the two excess charges.
Sattler et al.33published mass spectra of sodium io-
dide clusters, and deduced a critical size for stability of
the doubly charged series of m=18. In the next year,
Martin34reported pair potential model calculations on
the structures adopted by doubly charged sodium chlo-
ride clusters, in an attempt to explain the experimen-
tal findings. He found that all the cluster sizes had
at least one metastable bound state, even though for
m <8 the removal of an Na+cation was an exother-
mic reaction. Li and Whetten35showed that there are
important kinetic effects influencing the critical size de-
duced by Sattler et al, and found that it is possible to
populate the metastable minima for sizes smaller than
that critical size by using a less aggresive ionization
technique. In that way they were able to find a “sta-
bility island” in the size region m=11–12. More re-
cently, Kolmakov et al.36have been able to observe such
small clusters as Cs 5I2+
3by embedding the alkali halide
clusters inside a rare gas coating that serves to dissi-
pate the vibrational energy excess adquired after ion-
ization. Finally, Zhang and Cooks37have published
very recently mass spectra of free [(NaCl) m(Na) 2]2+clus-
1ter ions in the size range m=11–62 by using electro-
spray ionization, and found magic numbers for the sizes
m=11,12,17,20,21,26,30,34,36,44,54, and 61. They also
report collision induced fragmentation spectra for those
magic sizes, from which a specific structural assignment
is suggested. We would also like to mention here the
closely related case of metal oxide clusters, where only
one excess cation is needed to produce doubly charged
isomers. Magnesium and calcium oxide doubly charged
cluster ions have been studied both experimentally38–40
and theoretically.41,42
To reach the same understanding level for the dou-
bly charged clusters as that already achieved for the
singly charged ones, we present in this work the re-
sults of an extensive and systematic theoretical study of
[(NaCl) m(Na) 2]2+cluster ions with mranging from 6 to
28. The experimental results by Zhang and Cooks37con-
cerning enhanced stabilities will serve as an ideal check
of the theoretical calculations. We will show that we can
reproduce all of their magic numbers, although we also
obtain some magic numbers not found in the experimen-
tal mass spectra. The structural assignment suggested by
these authors is also examined and shown to be partially
correct. The rest of the paper is organized as follows: in
Section II we give just a brief resume of the theoretical
model employed, as full an exposition has been reported
already in previous works29and does not deserve in our
opinion the use of more journal space. The results are
presented in Section III, and the main conclusions to be
extracted from our study in Section IV.
II. THE AIPI MODEL. BRIEF LOCATION
RESUME
Theab initio Perturbed Ion (aiPI) model provides a
computational framework ideally suited to deal with ionic
systems, and its performance has been well tested both in
the crystal43–46and cluster29–32,41,42,47limits. The the-
oretical foundation of the aiPI model48lies in the theory
of electronic separability.49,50Very briefly, the HF equa-
tions of the cluster are solved stepwise, by breaking the
cluster wave function into local group functions (ionic in
nature in our case). In each iteration, the total energy is
minimized with respect to variations of the electron den-
sity localized in a given ion, with the electron densities
of the other ions kept frozen. In the subsequent itera-
tions each frozen ion assumes the role of nonfrozen ion.
When the self-consistent process finishes,29the outputs
are the total cluster energy and a set of localized wave
functions, one for each geometrically nonequivalent ion of
the cluster. These localized cluster-consistent ionic wav e
functions are then used to estimate the intraatomic cor-
relation energy correction through Clementi’s Coulomb-
Hartree-Fock method.51,52The large multi-zeta basis sets
of Clementi and Roetti53are used for the description of
the ions. At this respect, our optimizations have been
performed using basis sets (5s4p) for Na+and (7s6p)for Cl−, respectively. Inclusion of diffuse basis functions
has been checked and shown unnecessary. One impor-
tant advantage coming from the localized nature of the
model is the linear scaling of the computational effort
with the number of atoms in the cluster. This has al-
lowed us to perform full structural relaxations of clusters
with as many as 58 ions at a reasonable computational
cost. Moreover, for each cluster size, a large number of
isomers (between 10 and 15) has been investigated. The
generation of the initial cluster geometries was accom-
plished by using a pair potential, as explained in previous
publications.41,42The optimization of the geometries has
been performed by using a downhill simplex algorithm.54
III. RESULTS AND DISCUSSION
A. Lowest Energy Structures of [(NaCl) m(Na) 2]2+
Cluster Ions
In Fig.1 we present the optimized aiPI structures
of the ground state (GS) and lowest lying isomers or
[(NaCl) m(Na) 2]2+(m=8–28) cluster ions. Below each
isomer we show the energy difference (in eV) with re-
spect to the ground state. The GS structures for m=6
and 7 are not shown in the figure because they were cal-
culated just to show the special stability of the m=8 size
(see next section). Nevertheless, they can be obtained by
simply removing one and two NaCl molecules from the
GS structure for m=8. All the low-lying isomers of m=8
are based on the 3 ×2×2 structure of the (NaCl) 6neutral
cluster (where the notation indicates the number of ions
along each of the three perpendicular cartesian axes), and
just differ in the way the six extra ions are added to it.
The most favorable location for these extra ions is along
two opposite edges of the neutral, so that the two tri-
atomic NaClNa+units minimice their mutual repulsion
without distorting too much the structure of the neutral
cluster. Zhang and Cook have visualized this structure
as the combination of two 3 ×3×1 planar sheets.37Given
the bending of these sheets observed in the ab initio cal-
culations, we prefer to use the notation 3 ×2×2+3+3 for
this cluster, and will do it for the similar structures along
this paper. Irrespective of the notation used, however,
we must point out that our calculations essentially agree
with their suggestion. The GS structures of m=9 and 10
are much more distorted and difficult to visualize, but
simply result from the addition of one and two NaCl
molecules, respectively, to the GS isomer of m=8. For
m=11, it is possible again to construct a quite compact
GS isomer by forming a 3 ×3×2+3+3 structure, again
in good agreement with the suggestions of Zhang and
Cook.37This time the two added triatomics are on the
same face of the neutral structure, due to the specific dis-
position of the ionic charges, and thus the screening of
the excess charge is not as complete as for the m=8 case.
Form=12 a specially compact structure of a different
2kind appears. The GS for this size can be obtained from
that of the singly charged (NaCl) 13Na+cluster ion by
removing the inner six-coordinated chloride anion. This
structure had been suggested by the pair potential model
calculations of Martin34and by the experiments of Li and
Whetten35and Zhang and Cooks.37These last authors
use the term “defect structure” to refer to this kind of
structure, and in this case we will use the same notation.
These two kinds of structures seem to have a very high
stability in the whole size range considered in this study.
For example, a ×b×c+3+3 fragments are observed for
m=8,11,14,20 and 26. Defect structures are adopted as
GS structures for m=12 and 21. Note that for this last
size the anion vacancy is not located in the center of the
cluster but on an edge position. Anions are more stable
the larger their coordination number (the opposite holds
for cations)29,30, so the removal of an anion with six co-
ordination will be in general not favored energetically.
m=12 is an exceptional case in the sense that a highly
compact and symmetrical structure can be obtained by
removing the central anion from a 3 ×3×3 singly charged
cluster ion. A symmetrical structure tends to be favored
by the Madelung energy component, which is the most
important contribution to binding in ionic systems, and
this compensates for the loss of the most stable anion in
the cluster. The same will not be true for most of the
other values of mwhere a defect structure can be formed.
Another specially compact cluster that could fit into the
defect structure category is m=24, which can be obtained
from the 4 ×4×3+3 structure of (NaCl) 25Na+by remov-
ing a corner anion. The only compact cluster that does
not fit into any of these two categories is m=17, that
can be viewed as the combination of two singly charged
blocks, namely 3 ×3×3 and 3 ×3×1. Although this last
structure coincides also with that advanced by Zhang
and Cooks,37a detailed comparison with their sugges-
tions shows that the agreement is not completely good
for other sizes. For example, the GS structure of m=20
is predicted to result from the merging of two 3 ×3×3
blocks. We obtain indeed this structure as a low lying
isomer (see Fig. 1), so that those authors were not too
far from the real answer. Similarly, the GS structure for
m=26 was predicted to be a combination of 5 ×3×3 and
3×3×1 blocks instead of the structure shown in Fig. 1.
The GS structures for the rest of the sizes are mainly
obtained by adding or removing NaCl molecules from the
compact clusters of one of the two families mentioned in
the last paragraph. One exception could be m=23, which
is formed by adding a bent NaClNa+triatomic unit to a
compact 5 ×3×3 structure.
Comparing to the results of our previous papers on
neutral and singly charged alkali halide clusters,29–31
we appreciate that the structures found in those cases
can serve as “seeds” for the generation of those of the
doubly charged clusters. Specifically, the magic num-
ber structures of the neutrals (AX) n(n=6, 9, 12, 15,
etc) serve to generate specially stable [(NaCl) m(Na) 2]2+
cluster ions with m=8, 11, 14, 17, 20, etc, by edge at-taching of NaClNa+triatomic units. Specially compact
doubly charged isomers can also be obtained by removing
a chloride anion from one of the singly charged (AX) nA+
cluster ions, being this the case for m=12 and 21, or by
adding a triatomic to the singly charged clusters, for ex-
ample m=17.
B. Relative stabilities and connection to
experimental mass spectra
In the experimental mass spectra,37the populations
observed for some cluster sizes are enhanced over those
of the neighboring sizes. These “magic numbers” are
a consequence of the evaporation/fragmentation events
that occur in the cluster beam, mostly after ionization.55
A magic cluster of size mhas a stability that is large com-
pared to that of the neighboring sizes ( m-1) and ( m+1).
Thus, on the average, clusters of size mundergo a smaller
number of evaporation/fragmentation events, and this
leads to the maxima in the mass spectra. A most conve-
nient quantity to compare with experiment is the second
energy difference
∆2(m) = [E(m+ 1) + E(m−1)]−2E(m), (1)
where E( m) is the total energy of the [(NaCl) m(Na) 2]2+
cluster ion. A positive value of ∆ 2(m) indicates that the
m-stability is larger than the average of the ( m+1)- and
(m-1)-stabilities.
We show in figure 2 our results concerning the
stabilities of the doubly charged cluster ions. The
magic numbers can be divided into two subsets: sizes
m=8,11,14,17,20 and 26 show large maxima in the ∆ 2(m)
curve; sizes m=9,12,21 and 24 show smaller but positive
values of ∆ 2(m). All the enhanced stabilities found in the
experiments in this size range, namely m=11,12,17,20,21
and 26,37are reproduced by our calculations. Sizes
m=8 and 9 are too small to be observed in the exper-
iments by Zhang and Cooks, who found a critical size
for the stability of the doubly charged cluster ions of
m=11. After the stability island found at sizes m=11–
12, no doubly charged cluster ion was observed in the
experiments37until reaching a value of m=17, so that
the magic number m=14 is not observed either. Fi-
nally, although m=24 is not considered a magic num-
ber in their paper37due to some scatter in the experi-
mental data, it is concluded that it might exhibit some
enhanced stability. Thus, the agreement with experi-
ment can be considered excellent. It is a very interest-
ing question that deserves further investigation, however ,
why the metastable potential energy minima of cluster
ions in the size range m=13–16 can not be populated
in the experiments. Li and Whetten35produced the
doubly charged cluster series by soft anion photoejec-
tion from the singly charged (AX) nA+series, and found
that the stability island observed for m=11–12 is a con-
sequence of the high efficiency of that process for the
3parent cluster with n=13. We note that halogen pho-
toejection from the GS structure of (AX) 14A+found in
previous publications31would lead directly to the GS iso-
mer of [(NaCl) 13(Na) 2]2+shown in Fig. 1, but in this
case the process is not so efficient as for n=13. One
could speculate that the one-coordinated cation left in
the structure is very prone to dissociate even for very
modest excess vibrational energies. On the other hand,
photoejection of a halide anion from the GS structure
of (AX) 15A+, which is also based on attaching ions to
a 3×3×3 compact cube, would not directly populate the
GS structure of [(NaCl) 14(Na) 2]2+, which is an elongated
structure. Nevertheless, calculations on the evaporation
kinetics processes would be needed in order to draw def-
inite conclusions.
Now we try a rationalization of the enhanced stabil-
ities in terms of structural properties. Hopefully, this
will allow the GS structures of clusters larger than those
explicitely included here to be predicted with some con-
fidence. A general feature of [(NaCl) m(Na) 2]2+cluster
ions in the size range considered in this paper is that
a×b×c+3+3 fragments are specially stable compared to
other isomers whenever they can be formed. The ap-
parent reason is that those structures tend to minim-
ice the repulsion between the two excess positive charges
while not distorting too much the compact a ×b×c struc-
tures of the neutrals, which are energetically favored by
purely Madelung energy considerations.29,30As the pref-
ered place to attach the NaClNa+triatomic units is along
edges of the neutral structures, for larger sizes (where
none of the three edges will contain just three ions) one
can advance a corresponding relevance of a ×b×c+5+5
structures. In all cases, at least one of the three edges
of the neutral structures has to contain an even num-
ber of ions in order to preserve charge neutrality and
expose a convenient binding site for the tri- or penta-
atomic chains. To these structural families we have to
add the defect structures obtained by removing a halide
anion from the a ×b×c compact structures that occur for
the singly charged (AX) nA+cluster ions when all three
edges contain an odd number of ions. In Table I we
show all the relevant fragments of those kinds. Each
series (except the defect one) has a typical periodicity
that could in principle be reflected in different portions
of the mass spectra, given the high stability of these
fragments. We can see that the great majority of the
magic numbers observed in the experiments by Zhang
and Cooks37can be explained in terms of the structures
shown in the table. Thus, m=12,21,(24),30,36,52 and
61 are ascribed to defect structures based on 3 ×3×3,
5×3×3, (4×4×3+3), 7 ×3×3, 5×5×3, 7×5×3 and 5 ×5×5
parent singly charged structures ( m=52 was observed to
show some enhanced stability, although less than those
of the others).37m=8,11 and 14 are p ×2×3+3+3 struc-
tures with p=2,3 and 4, respectively. m=20,26 and 32
are p×4×3+3+3 with p=3,4 and 5. Finally, m=34,44,54
and 64 are p ×4×5+5+5 structures with p=3,4,5 and 6.
Note that the values of pare as close as possible to thelengths of the other edges, as expected. With the only ex-
ception of m=17, all the experimental magic numbers fit
into any of these categories, which we consider evidence
enough for the correcteness of the structures proposed.
We note that the stability of the defect-like structures is
reduced with respect to that shown by the other magic
numbers (see Fig. 2). Note also that the inclusion of
the merged block structures of Zhang and Cooks37is not
needed to explain the enhanced stabilities, even though
they will surely be low energy isomers for those sizes
where they can be formed.
IV. CONCLUSIONS
The structures and stabilities of doubly charged
[(NaCl) m(Na) 2]2+cluster ions have been studied in the
size range m=6–28 by means of ab initio Perturbed Ion
calculations. For this size range, we have found two main
groups of specially compact structures: (a) those ob-
tained by adding triatomic chains to the edges of a ×b×3
perfect neutral cuboids (NaCl) n, with n=m-2, and (b)
those obtained by removing one chloride anion from the
perfect a ×b×c cuboids of the singly charged cluster ions
(NaCl) nNa+, with n=m+1. The way in which these
structures are constructed indicates that there is a cor-
relation with the structures found previously for neutral
and singly charged alkali halide clusters.29–31A compar-
ison with the structural assignment suggested by Zhang
and Cooks37after an interpretation of their collision in-
duced fragmentation spectra shows a good level of agree-
ment. Nevertheless, there are some minor discrepancies.
For example, the merged block structures suggested in
their work are not found to be the GS isomers for any
size, even though they are low lying structural isomers.
The calculated enhanced stabilities show an excellent
agreement with the experimental results, being this an
ideal check for the correctness of our theoretical calcula-
tions. The only calculated magic numbers that are not
present in the experimental mass spectra are those of
m=8, 9 and 14. The experiments are not able to pop-
ulate the metastable potential energy minima of these
structures, however, so that no comparison is possible
in these cases. With the only exception of m=17, we
find that all the enhanced cluster stabilities are a con-
sequence of the highly compact structures that can be
built for certain values of mand mentioned in the previ-
ous paragraph. Given the high stability of the structures
obtained by adding triatomic chains to compact neutral
structures, we have proposed that the structures result-
ing from adding pentaatomic chains to the edges of larger
neutral clusters should also be specially stable. Taking
all these structural families altogether, we have shown
that we can reproduce all the magic numbers observed
in the experimental mass spectra.
4ACKNOWLEDGEMENTS: The author is grateful
to the Ministerio de Educaci´ on y Ciencia of Spain for the
concesion of a postdoctoral grant.
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1983,50, 27–30.Captions of Figures and Tables.
Figure 1 . Lowest-energy structure and low-lying iso-
mers of [(NaCl) m(Na) 2]2+cluster ions. Dark balls are
Na+cations and light balls are Cl−anions. The energy
difference (in eV) with respect to the most stable struc-
ture is given below the corresponding isomers.
Figure 2 . Size evolution of ∆ 2(m) (eq. 1). The local
maxima in the second energy difference curve are shown
explicitely.
Table I Structural series, together with their inherent
periodicities, used to explain the experimentally observe d
magic numbers. Those cluster sizes mthat are actually
observed to show an enhanced stability in the mass spec-
tra of [(NaCl) m(Na) 2]2+clusters are written in boldface.
m=17 is the only exception (see text).
Structure Periodicity Cluster size n
p×2×3+3+3 3 8,11,14 ,17,...
p×4×3+3+3 6 14,20,26,32 ,38...
p×4×5+5+5 10 24,34,44,54 ,64...
a×b×c-1 – 12,21,(24),30,36,52,61 ,...
6This figure "fig1a.gif" is available in "gif"
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format from:
http://arXiv.org/ps/physics/0101092v15 10 15 20 25 30
Cluster Size m−2−1012∆2(m) (eV)81114172026
912 2124 |
arXiv:physics/0101093v1 [physics.acc-ph] 27 Jan 2001SLAC-AP-127
July 2000
Analytical Formula for Weak Multi-Bunch Beam
Break-Up in a Linac∗
Karl L.F. Bane and Zenghai Li
Stanford Linear Accelerator Center, Stanford University, Stanford,
CA 94309
∗Work supported by Department of Energy contract DE–AC03–76 SF00515.Analytical Formula for Weak Multi-Bunch Beam
Break-Up in a Linac
Karl L.F. Bane and Zenghai Li
In designing linac structures for multi-bunch application s we are often in-
terested in estimating the effect of relatively weak multi-b unch beam break-
up (BBU), due to the somewhat complicated wakefields of detun ed struc-
tures. This, for example, is the case for the injector linacs of the JLC/NLC
linear collider project (see Ref. [1]). Deriving an analyti cal formula for such
a problem is the subject of this report. Note that the more stu died multi-
bunch BBU problem, i.e.the effect on a bunch train of a single strong mode,
the so-called “cumulative beam break-up instability” (see ,e.g.Ref. [2]), is a
somewhat different problem, and one for which the approach pr esented here
is probably not very useful.
In Ref. [3] an analytical formula for single-bunch beam break-up in a
smooth focusing linac, for the case without energy spread in the beam, is
derived, the so-called Chao-Richter-Yao (CRY) model for be am break-up.
Suppose the beam is initially offset from the accelerator axi s. The beam
break-up downstream is characterized by a strength paramet er Υ(t,s), where
trepresents position within the bunch, and sposition along the linac. When
Υ(t,s) is small compared to 1, the growth in betatron amplitude in t he linac
is proportional to this parameter. When applied to the speci al case of a
uniform longitudinal charge distribution, and a linearly g rowing wakefield,
the result of the calculation becomes especially simple. In this case the
growth in orbit amplitude is given as an asymptotic power ser ies in Υ( t,s),
and the series can be summed to give a closed form, asymptotic solution for
single-bunch BBU. The derivation of an analytic formula for multi-bunch
BBU is almost a trivial modification of the CRY formalism. We w ill here
reproduce the important features of the single-bunch deriv ation of Ref. [3]
(with slightly modified notation), and then show how it can be modified to
obtain a result applicable to multi-bunch BBU.
Let us consider the case of single-bunch beam break-up, wher e a beam
is initially offset by distance y0in a linac with acceleration and smooth
focusing. We assume that there is no energy spread within the beam. The
2equation of motion is
1
E(s)d
ds/bracketleftbigg
E(s)dy(t,s)
ds/bracketrightbigg
+y(t,s)
β2(s)=e2Nt
E(s)/integraldisplayt
−∞dt′y(t′,s)λt(t′)W(t−t′),(1)
withy(t,s) the bunch offset, a function of position within the bunch t, and
position along the linac s; with Ethe beam energy, [1 /β(s)] the betatron
wave number, eNtthe total bunch charge, λt(t) the longitudinal charge
distribution, and W(t) the short-range dipole wakefield. Our convention is
that negative values of tare toward the front of the bunch. Let us, for the
moment, limit ourselves to the problem of no acceleration an dβa constant.
A. Chao in Ref. [3] expands the solution to the equation of mot ion for this
problem in a perturbation series
y(t,s) =∞/summationdisplay
n=0y(n)(t,s), (2)
with the first term given by free betatron oscillation [ y(0)=y0cos(s/β)]. He
then shows that the solution for the higher terms at position s=L, after
many betatron oscillations, is given by
y(n)(t,L)≈y0
n!/parenleftbiggie2NtLβ
2E/parenrightbiggn
Rn(t)eiL/β, (3)
with
Rn(t) =/integraldisplayt
−∞dt1λ(t1)W(t−t1)/integraldisplayt1
−∞dt2λ(t2)W(t1−t2)
· · ·/integraldisplaytn−1
−∞dtnλ(tn)W(tn−1−tn), (4)
andR0(z) = 1. An observable yis meant to be the real part of Eq. 2.
The effects of adiabatic acceleration, i.e.sufficiently slow acceleration so
that the energy doubling distance is large compared to the be tatron wave
length, and βnot constant, can be added by simply replacing ( β/E) in
Eq. 3 by /angbracketleftβ/E/angbracketright, where angle brackets indicate averaging along the linac
froms= 0 to s=L.1For example, if the lattice is such that β∼Eζ
1Note that the terms y0eiL/βin Eq. 3, related to free betatron oscillation, also need
to be modified in well-known ways to reflect the dependence of βonE. It is the other
terms, however, which characterize BBU, in which we are inte rested.
3then /angbracketleftβ/E/angbracketright= (β0/E0)g(Ef/E0,ζ), where subscripts “0” and “ f” signify,
respectively, initial and final parameters, and
g(x,ζ) =1
ζ/parenleftbiggxζ−1
x−1/parenrightbigg
[β∼Eζ]. (5)
Chao then shows that for certain simple combinations of bunc h shape and
wake function shape the integrals in Eq. 4 can be performed an alytically, and
the result becomes an asymptotic series in powers of a streng th parameter.
For example, for the case of a uniform charge distribution of length ℓ(with
the front of the bunch at t= 0), and a wake that varies as W=W′t, the
strength parameter is
Υ(t,L) =e2NtLW′t2β0
2E0ℓg(Ef/E0,ζ). (6)
If Υ is small compared to 1, the growth is well approximated by Υ. If Υ
is large, the sum over all terms can be performed to give a clos ed form,
asymptotic expression.
Formulti-bunch BBU we are mainly concerned with the interaction of
the different bunches in the train, and will ignore wakefield f orces within
bunches. The derivation is nearly identical to that for the s ingle-bunch
BBU. However, in the equation of motion, Eq. 1, the independe nt variable
tis no longer a continuous variable, but rather ttakes on discrete values
tm=m∆t, where mis a bunch index and ∆ tis the bunch spacing. Also, W
now represents the long-range wakefield. Let us assume that t here are M,
equally populated bunches in a train; i.e.Nt=MN, with Nthe particles
per bunch. The solution is again expanded in a perturbation s eries. In the
solution, Eq. 3, the Rn(t), which are smooth functions of t, are replaced by
R(n)
m=1
Mm−1/summationdisplay
j=1W[(m−j)∆t]R(n−1)
j , (7)
(with R0
j= 1), which is a function of a discrete parameter, the bunch in dex
m. Note that R(1)
m=Sm/M, with Smthe sum wake.
Generally the sums in Eq. 7 cannot be given in closed form, and there-
fore a closed, asymptotic expression for multi-bunch BBU ca nnot be given.
We can still, however, numerically compute the individual t erms equivalent
to Eq. 3 for the single bunch case. For example, the first order term in
amplitude growth is given by
Υm=e2NLS mβ0
2E0g(Ef/E0,ζ) [ m= 1,... ,M ]. (8)
4If this term is small compared to 1 for all m, then BBU is well characterized
by Υ. If it is not small, though not extremely large, the next h igher terms can
be computed and their contribution added. For Υ very large, t his approach
may not be very useful.
From our derivation we see that there is nothing that fundame ntally
distinguishes our BBU solution from a single-bunch BBU solu tion. If we
consider again the single-bunch calculation, for the case o f a uniform charge
distribution of length ℓ, we see that we need to perform the integrations
forRnin Eq. 4. If we do the integrations numerically, by dividing t he
integrals into discrete steps tn= (n−1)∆tand then performing quadrature
by rectangular rule, we end up with Eq. 7 with M=ℓ/∆t. The solution is
the same as our multi-bunch solution. What distinguishes th e multi-bunch
from the single-bunch problem is that the wakefield for the mu lti-bunch case
is not normally monotonic and does not vary smoothly with lon gitudinal
position. For such a case it may be more difficult to decide how m any terms
are needed for the sum to converge.
In Fig. 1 we give a numerical example: the NLC prelinac with th e op-
timized 3 π/4 S-band structure, but with 10−5systematic frequency errors,
with the nominal (2.8 ns) bunch spacing (see Ref. [1]). The di amonds give
the first order (a) and the second order (b) perturbation term s. The crosses
in (a) give the results of a smooth focusing simulation progr am (taking
β∼E1/2), where the free betatron term has been removed. We see that t he
agreement is very good; i.e.the first order term is a good approximation
to the simulation results. In (b) we note that the next order t erm is much
smaller.
Acknowledgments
The authors thanks V. Dolgashev for carefully reading this m anuscript.
References
[1] K. Bane and Z. Li, “Dipole Mode Detuning in the Injector Li nacs of the
NLC,” SLAC/LCC Note in preparation.
[2] R. Helm and G. Loew, Linear Accelerators , North Holland, Amsterdam,
1970, Chapter B.1.4.
[3] A. Chao, “Physics of Collective Instabilities in High-E nergy Accelera-
tors”, John Wiley & Sons, New York (1993).
5Figure 1: A numerical example: the NLC prelinac with the opti mized
3π/4 S-band structure, but with 10−5systematic frequency errors, with
the nominal (2.8 ns) bunch spacing (see Ref. [1]). Here N= 1.2×1010,
E0= 1.98 GeV, Ef= 10.GeV, L= 558 m; the rms of the sum wake
Srms=.005 MV/nC/m2. The diamonds give the first order (a) and the
second order (b) perturbation terms. The crosses in (a) give smooth focusing
simulation results with the free betatron term removed.
6 |
arXiv:physics/0101094v1 [physics.acc-ph] 27 Jan 2001SLAC-AP-128
July 2000
Obtaining the Wakefield Due to Cell-to-Cell
Misalignments in a Linear Accelerator Structure∗
Karl L.F. Bane and Zenghai Li
Stanford Linear Accelerator Center, Stanford University, Stanford,
CA 94309
∗Work supported by Department of Energy contract DE–AC03–76 SF00515.Obtaining the Wakefield Due to Cell-to-Cell
Misalignments in a Linear Accelerator Structure
Karl L.F. Bane and Zenghai Li
We are interested in obtaining the long-range, dipole wakefi eld of a linac
structure with internal misalignments. The NLC linac struc ture is composed
of a collection of cups that are brazed together, and such a ca lculation, for
example, is important in setting the straightness toleranc e for the composite
structure. Our derivation, presented here, is technically applicable only to
structures for which all modes are trapped. The modes will be trapped at
least at the ends of the structure, if the connecting beam tub es have suffi-
ciently small radii and the dipole modes do not couple to the f undamental
mode couplers in the end cells. For detuned structures (DS), like those in
the injector linacs of the JLC/NLC[1], most modes are trappe d internally
within a structure, and those that do extend to the ends coupl e only weakly
to the beam; for such structures the results here can also be a pplied, even if
the conditions on the beam tube radii and the fundamental mod e coupler do
not hold. We believe that even for the damped, detuned struct ures (DDS)
of the main linac of the JLC/NLC[2], which are similar, thoug h they have
manifolds to add weak damping to the wakefield, a result very s imilar to
that presented here applies.
We assume a structure is composed of many cups that are misali gned
transversely by amounts that are very small compared to the c ell dimensions.
For such a case we assume that the mode frequencies are the sam e as in the
ideal structure, and only the mode kick factors are affected. To first order
we assume that for each mode, the kick factor for the beam on-a xis in the
imperfect structure is the same as for the case with the beam f ollowing the
negative of the misalignment path in the error-free structu re. In Fig. 1 we
sketch a portion of such a misaligned structure (top) and the model used
for the kick factor calculation (bottom). Note that the rela tive size of the
misalignments is exaggerated from what is expected, in orde r to more clearly
show the principle. Given this model, the method of calculat ion of the kick
factors can be derived using the so-called “Condon Method”[ 3],[4] (see also
[5]). Note that this application to cell-to-cell misalignm ents in an accelerator
2structure is presented in Ref. [6]. The results of this pertu rbation method
have been shown to be consistent with those using a 3-dimensi onal scattering
matrix analysis[7]. We will only sketch the derivation belo w.
bunch
bunch
Figure 1: Sketches of part of a misaligned structure (top) an d the model
used for the kick factor calculation (bottom). Note that the relative size of
the misalignments here is much exaggerated.
Consider a closed cavity with perfectly conducting walls. F or such a
cavity the Condon method expands the vector and scalar poten tials, in the
Coulomb gauge, as a sum over the empty cavity modes. As functi on of
position x(x,y,z) and time tthe vector potential in the cavity is given as
A(x,t) =/summationdisplay
λqλ(t)aλ(x), (1)
3where
∇2aλ+ω2
λ
c2aλ= 0 , (2)
withωλthe frequency of mode λ, andaλ׈ n= 0 on the metallic surface ( ˆ n
is a unit vector normal to the surface). Using the Coulomb gau ge implies
that∇ ·aλ= 0. The qλare given by
¨qλ+ω2
λqλ=1
2Uλ/integraldisplay
VdVj·aλ, (3)
with the normalization
ǫ0
2/integraldisplay
VdVaλ′·aλ=Uλδλλ′, (4)
withjthe current density. Note that the integrations are perform ed over
the volume of the cavity V.
The scalar potential is given as
Φ(x,t) =/summationdisplay
λrλ(t)φλ(x), (5)
where
∇2φλ+Ω2
λ
c2φλ= 0 , (6)
with Ω λthe frequencies associated with φλ, and with φλ= 0 on the metallic
surface. The rλare given by
rλ=1
2Tλ/integraldisplay
VdVρφλ, (7)
withρthe charge distribution in the cavity. Note that one fundame ntal
difference between the behavior of A(x,t) and Φ( x,t) is that when there are
no charges in the cavity the vector potential can still oscil late whereas the
scalar potential must be identically equal to 0.
Let us consider an ultra-relativistic driving charge Qthat passes through
the cavity parallel to the zaxis, and (for simplicity) a test charge following
at a distance sbehind on the same path. Both enter the cavity at position
z= 0 and leave at position z=L. The transverse wakefield at the test
charge is then
W(s) =1
QLx 0/integraldisplayL
0dz[c∇⊥Az− ∇ ⊥Φ]t=(z+s)/c
4=1
QLx 0/summationdisplay
λ/integraldisplayL
0dz/bracketleftbigg
cqλ/parenleftbiggz+s
c/parenrightbigg
∇⊥aλz(z)
−rλ/parenleftbiggz+s
c/parenrightbigg
∇⊥φλ(z)/bracketrightbigg
, (8)
where the integrals are along the path of the particle trajec tory. The param-
eterx0is a parameter for transverse offset (the transverse wake is u sually
given in units of V/C per longitudinal meter per transverse m eter); for a
cylindrically-symmetric structure it is usually taken to b e the offset, from
the axis, of the driving bunch trajectory. For s > L we can drop the scalar
potential term (it must be zero when there is no charge in the c avity), and
the result can be written in the form[5]
W(s) =/summationdisplay
λc
2UλωλLx0ℑm/bracketleftBig
V∗
λ∇⊥Vλeiωλs/c/bracketrightBig
[s > L], (9)
with
Vλ=/integraldisplayL
0dz aλz(z)eiωλz/c. (10)
Note that the arbitrary constants associated with the param eteraλin the
numerator and the denominator of Eq. 9 cancel. Note also that —to the same
arbitrary constant— |Vλ|2is the square of the voltage lost by the driving
particle to mode λandUλis the energy stored in mode λ.
Consider now the case of a cylindrically-symmetric, multi- cell acceler-
ating cavity, and let us limit our concern to the effect of the d ipole modes
of such a structure. We will allow the charges to move on an arb itrary,
zig-zag path in the x−zplane that is close to the axis, and for which the
slope is everywhere small (so that ∇⊥∼∂/∂x). For dipole modes in a
cylindrically-symmetric, multi-cell accelerator struct ure, it can shown that
the synchronous component of aλz(the only component that, on average,
is important) can be written in the form aλz=xfλ(z) (see e.g.Ref. [8]).
Then Eq. 9 becomes
Wx(s) =/summationdisplay
λc
2UλωλLx0× (11)
×ℑm/bracketleftbigg
eiωλs/c/integraldisplayL
0dz′x(z′)fλ(z′)e−iωλz′/c/integraldisplayL
0dz fλ(z)eiωλz/c/bracketrightbigg
[s > L].
Note that this equation can be written in the form:
Wx(s) =/summationdisplay
λ2k′
λsin/parenleftBigωλs
c+θλ/parenrightBig
[s > L], (12)
5withk′
λa kind of kick factor and θλthe phase of excitation of mode λ.
Note that in the special case where the particles move parall el to the axis,
at offset a,k′
λ=kλ=c|Vλ|2/(4Uλωλa2L), the normal kick factors for the
structure, and θλ= 0. For this case it can be shown that Eq. 12 is valid for
alls >0[5]. Finally, note that, for the general case, Eq. 12 can obv iously
not be extrapolated down to s= 0, since it implies that Wx(0)/negationslash= 0, which is
nonphysical, since a point particle cannot kick itself tran sversely. To obtain
the proper equation valid down to s= 0 we would need to include the scalar
potential term that was dropped in going from Eq. 8 to Eq. 9.
To estimate the wakefield associated with very small, random cell-to-
cell misalignments in accelerator structures we assume tha t we can use the
mode eigenfrequencies and eigenvectors of the error-free s tructure. We ob-
tain these from the circuit program. Then to find the kick fact ors we replace
x(z) in the first integral in Eq. 11 by the zig-zag path representi ng the neg-
ative of the cell misalignments, a path we generate using a ra ndom number
generator. The normalization factor x0is set to the rms of the misalign-
ments.
In Ref. [1] this method is used to estimate the wake at the bunc h spacings
in the S-band injector linacs of the JLC/NLC. How can we justi fy this?
For example, for the 3 π/4 S-band structure, one possible bunch spacing is
only 42 cm whereas the whole structure length L= 4.46 m. Therefore,
in principle, Eq. 11 is not valid until the 11thbunch spacing. We believe,
however, that the scalar potential fields will not extend mor e than one or two
cells behind the driving charge (the cell length is 4.375 cm) , and therefore
this method will be a good approximation at all bunch positio ns behind the
driving charge. This belief should be tested in the future by repeating the
calculation, but now also including the contribution from s calar potential
terms.
In Fig. 2 we give a numerical example. Shown, for the optimize d 3π/4
S-band structure for the injector linacs of the NLC(see Ref. [1]), are the kick
factors and the phases of the modes as calculated by the metho d described
here. Note that θλis not necessarily small.
Acknowledgments
The authors thanks V. Dolgashev for carefully reading this m anuscript.
References
6Figure 2: The kick factors and phases of the modes for a cell-t o-cell mis-
alignment example. The structure is the optimized 3 π/4 S-band structure
for the injector linacs of the NLC (see Ref. [1]).
[1] K. Bane and Z. Li, “Dipole Mode Detuning in the Injector Li nacs of the
NLC,” SLAC/LCC Note in preparation.
[2] R.M. Jones, et al, Proc. EPAC96, Sitges, Spain, 1996, p. 1292.
[3] E. U. Condon, J. Appl. Phys. 12, 129 (1941).
[4] P. Morton and K. Neil, UCRL-18103, LBL, 1968, p. 365.
[5] K.L.F. Bane, et al, in “Physics of High Energy Accelerators,” AIP Conf.
Proc.127, 876 (1985).
[6] R. M. Jones, et al, “Emittance Dilution and Beam Breakup in the
JLC/NLC,” Proc. of PAC99, New York, NY, 1999, p. 3474.
[7] V. Dolgashev, et al, “Scattering Analysis of the NLC Accelerating Struc-
ture,” Proc. of PAC99, New York, NY., 1999, p. 2822.
[8] K. Bane and B. Zotter, Proc. of the 11thInt. Conf. on High Energy
Acellerators, CERN (Birkh¨ auser Verlag, Basel, 1980), p. 5 81.
7 |
arXiv:physics/0101095v1 [physics.acc-ph] 27 Jan 2001SLAC-AP-129
July 2000
On Resonant Multi-Bunch Wakefield Effects in
Linear Accelerators with Dipole Mode Detuning∗
Karl L.F. Bane and Zenghai Li
Stanford Linear Accelerator Center, Stanford University, Stanford,
CA 94309
∗Work supported by Department of Energy contract DE–AC03–76 SF00515.On Resonant Multi-Bunch Wakefield Effects in
Linear Accelerators with Dipole Mode Detuning
Karl L.F. Bane and Zenghai Li
In this report we explore resonant multi-bunch (transverse ) wakefield
effects in a linear accelerator in which the dipole modes of th e accelerator
structures have been detuned. For examples we will use the pa rameters of
a slightly simplified version of an optimized S-band structu re described in
Ref. [1]. Note that we are also aware of a different analysis of resonant
multi-bunch wakefield effect[2].
It is easy to understand how resonances can arise in a linac wi th bunch
trains. Consider first the case of the interaction of the beam with one single
structure mode. The leading bunch enters the structure offse t from the axis
and excites the mode. If the bunch train is sitting on an integ er resonance,
i.e.iff∆t=n, with fthe mode frequency, ∆ tthe bunch spacing, and nan
integer, then when the 2nd bunch arrives it will excite the mo de at the same
phase and also obtain a kick due to the wakefield of the first bun ch. The
mth bunch will also excite the mode in the same phase and obtain (m−1)
times the kick from the wakefield that the second bunch experi enced (for
simplicity we assume the mode Qis infinity). On the half-integer resonance,
i.e.when f∆t=n+.5, the mth bunch will also receive kicks from the
wakefield left by the earlier bunches, but in this case the kic ks will alternate
in direction, and no resonance builds up. For a transverse wa kefield effect,
such as we are interested in here, however, this simple descr iption of the
resonant interaction needs to be modified slightly. For this case the wake
varies as sin(2 πft), and neither the integer nor the half-integer resonance
condition will excite any wakefield for the following bunche s. In this case
resonant growth is achieved at a slight deviation from the co ndition f∆t=n,
as is shown below.
In the following, for simplicity, we will use the “uncoupled ” model to
investigate resonant effects in the sum wake for a structure w ith modes with
a uniform frequency distribution. According to this model ( see, for example,
2Ref. [3])
W(t)≈Nc/summationdisplay
n2ksnsin(2πfsnt/c) [ tsmall] , (1)
where Ncis the number of cells in the structure, and fsnandksnare, re-
spectively, the frequency and kick factor at the synchronou s point, for a
periodic structure with dimensions of cell n. Therefore, one can predict
the short time behavior of the wake without solving for the ei genmodes of
the system. The point of using the uncoupled model is that it a llows us to
study the effect of an idealized, uniform frequency distribu tion. As is well
known, an ideal (input) frequency distribution becomes dis torted by the
cell-to-cell coupling of an accelerator structure. (For si mplicity we will drop
thesin the subscripts for frequency below.) For examples we will use the
parameters of a slightly simplified version (all kick factor s are equal, the fre-
quency distribution is uniform instead of trapezoidal) of t he optimized 3 π/4
S-band structure described in Ref. [1]: there are Nc= 102 cells (also modes),
the central frequency ¯f= 3.92 GHz, and the full-width of the distribution
∆δf= 5.8%; for bunch structure we consider the nominal configuratio n of
M= 95 bunches in a train and a bunch spacing ∆ t= 2.4 ns. The re-
sults for the real structure, with coupled modes, will be sli ghtly different yet
qualitatively the same.
Consider first the case of a structure with only one dipole mod e, with
frequency f, and a kick factor that we will normalize (for simplicity) to 1/2.
Suppose there are Mbunches in the bunch train. The sum wake at the mth
bunch is given by
S(1)
m(f∆t) =m/summationdisplay
i=1sin(2π[i−1]f∆t)
=sin (π[m−1]f∆t) sin(πmf∆t)
sin (πf∆t). (2)
As with the nominal (2.8 ns) bunch spacing in the S-band preli nacs, let us,
for an example, consider M= 95 bunches and the region near the 11th
harmonic. In Fig. 1 we plot f∆tvsthe sum wake for the Mth (the last)
bunch, S(1)
M, near the 11th integer resonance. It can be shown that, if Mis
not small, the largest resonance peaks (the extrema of the cu rve) are at
f∆t≈n±3
8M[Mnot small] , (3)
with values ±.72M. Note that at the exact integer and half-integer resonant
spacings the sum wake is zero.
3Figure 1: The sum wake at the last bunch in a train vsbunch spacing, due
to a single mode (Eq. 2); M= 95 bunches.
Now let us consider a uniform distribution of mode frequenci es. For
simplicity we will let all the kick factors be equal, and be no rmalized to 1 /2.
The sum wake, according to the uncoupled model, becomes
Sm(¯f∆t) =1
NcNc/summationdisplay
n=1S(1)
m/bracketleftbigg
¯f∆t/parenleftbigg
1 +(n−Nc/2)
Nc∆δf/parenrightbigg/bracketrightbigg
, (4)
withNcthe number of cells (also the number of modes), ¯fthe central fre-
quency, and ∆ δfthe total (relative) width of the frequency distribution.
As an example, let us consider the optimized 3 π/4 S-band structure, with
Nc= 102 and ∆ δf= 5.8%. The sum wake at the last (the Mth) bunch
position, SM, is plotted as function of ¯f∆tin Fig. 2. Note that the uni-
form frequency distribution appears to suppress the intege r resonance. The
extrema of the curve (the “horns”) that are seen at ¯f∆t= 11±.32 are res-
onances due to the edges of the frequency distribution, with the condition
¯f∆t≈11/(1±∆δf/2). Note, however, that the sizes of even these spikes
are small compared to those of the single mode case.
Suppose we add frequency errors to our model. We can do this by , in
each term in the sum of Eq. 4, multiplying the frequency by the factor
(1+δferrrn), with δferrthe rms (relative) frequency error and rna random
4Figure 2: The sum wake at the last bunch in a train vsbunch spacing, due
to a uniform distribution of mode frequencies (Eq. 4). The to tal frequency
spread ∆ δf= 5.8%, and Nc= 102.
number with rms 1. Doing this, considering a uniform distrib ution in fre-
quency errors with rms δferr= 10−4, Fig. 2 becomes Fig. 3. Note that this
perturbation is small compared to the frequency spacing 5 .7×10−4, so it
does not really change the frequency distribution significa ntly. Nevertheless,
because of resonance-like behavior we can see a large effect o nSMthrough-
out the range between the horns of Fig. 2 (10 .68≤¯f∆t≤11.32). To model
cell-to-cell misalignments, we multiply each term in the su m of Eq. 4 by the
random factor rn. The results, for a uniform distribution of errors with rms
1, are shown in Fig. 4. Again resonance-like behavior is seen throughout the
range between the horns of Fig. 2.
We can understand these results in the following manner: Onl y when
there are no errors does using a uniform frequency distribut ion suppress the
resonance in the region near the integer resonance. But othe rwise, using a
uniform frequency distribution basically only reduces the size of the reso-
nances, at the expense of extending the range in bunch spacin gs where they
can be excited. Instead of being localized in the region near the integer
resonance ( ¯f∆t≈n), resonance-like behavior can now be excited anywhere
5Figure 3: The sum wake at the last bunch in a train vsbunch spacing, due
to a uniform distribution of frequencies, including freque ncy errors. The
total frequency spread ∆ δf= 5.8%, the number of modes Nc= 102, and
rms relative frequency error is 10−4.
between the limits
(¯f∆t)±=n
1∓∆δf/2. (5)
Note that this implies that if ∆ δf>1/(¯f∆t), then the resonance-like be-
havior cannot be avoided no matter what bunch spacing (fract ional part) is
chosen. For example, for the X-band linac in the NLC, where th e total width
of the dipole frequency distribution (of the dominant first b and modes) is
10%, even for the alternate (1.4 ns) bunch spacing, where the integer part
of¯f∆tis 21, the resonance region cannot be avoided.
Acknowledgments
The authors thanks V. Dolgashev for carefully reading this m anuscript.
References
[1] K. Bane and Z. Li, “Dipole Mode Detuning in the Injector Li nacs of the
NLC,” SLAC/LCC Note in preparation.
6Figure 4: The sum wake at the last bunch in a train vsbunch spacing, due to
a uniform distribution of frequencies, including random mi salignment errors
with rms 1. The total frequency spread ∆ δf= 5.8% and then number of
modes Nc= 102.
[2] D. Schulte, presentation given in an NLC Linac meeting, s ummer 1999.
[3] K. Bane and R. Gluckstern, Part. Accel. ,42, 123 (1994).
7 |
1FRAGMENTARY ELECTRODYNAMICS OF SOLITONS EMITTED BY
ATOMS
Pavel S. Kamenov
Sofia University, Faculty of Physics
Summary
In the recent years there was published some papers in which the photons are
represented as electromagnetic solitons [1,2,3]. All particles – solitons – representsome electromagnetic field restricted in a very small volume, length, cross-sectionand propagate in vacuum with light velocity in one and the same directions at a verygreat unrestricted distances [3]. These unordinary properties of the electromagneticfield – soliton – require some more detailed investigations of the dynamics ofinteractions of charged particles and electromagnetic fields.
In this paper I make an attempt to solve in part (fragmentary) some questions
about energy density of the soliton, electrodynamics of soliton and the simplesthydrogen atom, the acceleration of a charged particle, the path of the electron wile itchanges its position, the emitted electromagnetic soliton energy and the electron in astationary state. The descriptions are restricted only to the properties of the solitonsin paper [3] and to the hydrogen atom, but I think that obtained results could beapplied to the more complicated systems.
Introduction .
As was shown in [3], the electromagnetic field of a single photon must be
concentrated in a very small volume, V. The relation between the maximal electric
field (E
0) of a soliton and the frequency ( ω) of a free photon is: E 02 = (8m 0 /G21/e2)ω3,
where ω is the frequency obtained from interference phenomena, de Broglie’s
frequency; m 0 and e are the mass and electric charge of the electron. The energy of
the photon is /G21ω = E02V; volume V = (e2/8m0 /G21ω2); the energy density of the soliton is
ρs = E02; the effective time of action is t e = 1/2ω and the effective length (l e) of the
electromagnetic field of the soliton and the wave length ( λ) of de Broglie are related:2le = λ/4π. So, the electromagnetic field occupies only a small part of λ and a very
small part of the photon wave package (wave function).
1. Consistence with Planck’s density of radiation.
In the Fig.1 schematically is presented a soliton with the above mentioned
properties. S e = e2/8m0cω is macroscopic cross-section of the soliton.
Fig.1. Schematic representation of soliton electromagnetic field. The shape of the field is not known
exactly, but the effective time is t e = 1/2ω.
As it is known, Planck’s density of radiation is
()()[]ρωω
πω
ω==−/G28
/G46/G47
/G4E/G37/G15/G16
/G15/G16/G14/G21
/G21 /G48/G5B /G53 /G12( /G14)
The unit frequency interval is dω = 1 s-1. The part which depends on temperature ( T)
is usually interpreted as the average number of photons ( /G31) in unit volume. So, (1)
can be represented as
()ρωω
π==EcN23
23/G21(2)3When /G31= /G14, equation (2) can be compared with the equation for the soliton energy
density (ρs):
3
20 2
08ω ρ
emEs/G21== (3)
Or
E
Ee
mc2
022
2
038=π≈1.2x10-25(3a)
Ee
mcE =2
2
0308π≈3.4x10-13E0
This confirms the assumption that in a larger volume in comparison with the real
volume of the electromagnetic field, the energy density (larger volume) appears to besmaller [3]. The soliton energy density is consistent with energy density of Planck.The only difference between (2) and (3) is that the soliton volume ( V) is different
from the unit volume. One can calculate the average number (
/G31) of photons for
which Planck’s energy density (2) is equal to the energy density of a soliton (3).
E
EKc
N02
20323
31==−ωπ
ω /G21(4)
For /G2E/G50
/G48/G13/G13
/G15/G1B=/G21 this number is a constant:
NKc m c
e== ≈ ×023 2
03
28ππ
/G21 8.1 1024(5)
(5) is reciprocal to (3a). These results explain the first unsuccessful attempt to
describe the photoelectric effect within the framework of the electromagnetic wavetheory. It is clear that the electric field E does not change with time t and frequency
ω, as it was thought before, but the number of photons in unit volume remains
proportional to E
2 . The soliton (particle) and the energy density is concentrated in a
very small volume. The frequency ω is the frequency of de Broglie’s field D, like for
all particles. The amplitude D of the real field of de Broglie accounts for all
interference phenomena and for all particles.
2. Soliton electrodynamics and hydrogen atom4As it is known from electrodynamics only when a charged particle is
accelerated it can emit an electromagnetic field. In the hydrogen atom of Bohr theelectron in a stationary state dose not emit the photons. The photon is emitted onlywhen the radial coordinates change. In this time the electron changes the velocitypassing from one upper excited state ( k) to some lower state ( n). If one know the time
of transition, then acceleration of electron could be found. Up to now this time oftransition is not known and something more, we accept that it is useless to think about
this time.
On the other hand, if the photon can be represented as an electromagnetic
particle – soliton – with a determined volume, length and cross-section [3], then
effective action time of the soliton depends of it de Broglie’s frequency ( ω) as t
e =
1/2ω. It must be accepted that effective action time of a soliton electric field (emitted
from hydrogen) corresponds to the effective time of spontaneous transition in
hydrogen atom. One can find the acceleration of electron while the transition occurand knowing this acceleration it is possible to obtain the energy of the soliton (theenergy of the emitted photon).
Acceleration . As was mentioned the effective time (t
e = 1/2ω) of the soliton
require a corresponding acceleration time of electron when the electron passes from
upper (k) to lower level ( n) in hydrogen atom. On the Fig.2 schematically are shown
the velocities ( Vk, Vn) and acceleration ( a).
Fig.2. A scheme of velocities (V k, Vn) and acceleration (a). The shape of acceleration curve is not
exactly known, but the effective time of velocity change is t e = 1/2ω.tea=(Vn-Vk)/te
timeI
elocity
VkVn5The effective time of acceleration and the shape of acceleration curve must
correspond to the effective time of the soliton and to the shape of electric field of thesoliton (Fig.1 and Fig.2). When the electron is in a stationary state (no acceleration)the electric field of the soliton is zero [4,5]. Knowing the time (t
e) and the velocities
(Vk and Vn) we can obtain the effective acceleration ( a) of the electron when passing
from upper level ( k) to lower level ( n). The corresponding velocities are:
Vk = e2/ /G21k; Vn = e2/ /G21n (6)
The effective time of acceleration correspond to the effective time of the
emitted electromagnetic soliton (t e) and effective acceleration must be:
a = (Vn – Vk)/te = (e2/ /G21)(1/n – 1/k)/te /G0B/G1A/G0C
Substituting ( te = 1/2ω) we obtain:
a = 2ω(Vn – Vk) = (2e2ω/ /G21)(1/n – 1/k) (8)
The effective way ( path) of electron ( Hnk) while changing it state is:
Hnk = (a/2)t2
e + Vkte (9)
Substituting here ( te), (8) and Vk from (6) , one can find the effective path:
Hnk = (e2ω/ /G21)(1/n – 1/k)(1/4ω2) + e2/ /G21k(1/2ω)
Hnk = e2/ /G21(1/2ω)((1/2)(1/n – 1/k) + 1/k) (10)
Energy of the soliton . According to the electrodynamics when the energy is
emitted the average force (taking into account the force of reaction) is F = am0 . This
force is a result of the Coulomb field of hydrogen atom and the reaction ofelectromagnetic field of the soliton. So, the energy ( E
nk) which the accelerated
electron emits can be found, substituting the necessary quantities in equation:
Enk = FHnk = am0Hnk (11)6Substituting here (8) and (10) we obtain:
Enk =
−+
−kknknme 111211
22
204
/G21
Or which is the same
Enk =
−22 20411
2 knme
/G21(12)
This is the energy which carry the electromagnetic particle – soliton – and the
frequency of de Broglie ( ω), as for all frequencies of the particles is:
Enk/ /G21 = ω =
−22 30411
2 knme
/G21(13)
As it is seen, in this way obtained energy (accelerated electron), coincide
exactly with the results of Bohr and electromagnetic energy of the soliton, when it isemitted from a hydrogen atom [4].
From classical electrodynamics we know that a free electromagnetic field is
proportional to the acceleration of charged particle, but the energy of the emitted fieldis redistributed in whole space and diminish with distance (r) as 1/r
2 . The field is
maximal at perpendicular direction to the acceleration vector and is delayed in time as(t – r/c). Up to now it was not possible to calculate the effective acceleration ofelectron in hydrogen atom because the time of velocity change was not accessible forinvestigation. Even the questions “how long is this time”, and “when the transitionoccur” are forbidden [4,5]. Now, the properties of the soliton determine this time andthe effective acceleration is calculated. The energy of the emitted soliton is exactlyequal to the energy losses in the hydrogen. Something more, all energy losses fromthe atom can be transferred (in vacuum) at a very great unrestricted distance as anelectromagnetic particle – soliton. This means that whole electromagnetic energy isemitted in some well defined direction and the momentum of soliton is equal to themomentum of atom, but in opposite direction, as predicted by Einstein [6,7].
These properties of the photon-soliton and the hydrogen atom are not trivial
and they must be examined in more details. If the solitons with these properties [3]7exist in the nature, then the transitions in hydrogen atoms must take a time te = 1/2ω
and the atom in a stationary state must be comparatively stable. In the following
paragraphs I describe the hydrogen atom as a solitary (single) quantum systembecause every soliton must be emitted by only one solitary hydrogen atom.
3. “When the transitions occur?”
We must accept that the hydrogen atom emits a soliton only wile the electron
and proton change the velocities. (Remember that the proton and electron move aboutthe center of masses and change simultaneously its states). But why the electroncannot be accelerated when moving in a stationary state?
Contemporary quantum physics deals with statistical ensembles of quantum
objects: atoms, nuclei, photons, electrons and other particles. The Bohr’s modelconcern a solitary Hydrogen atom. This paper deals (also) with the single (separate)quantum object: one particle, one electron, one soliton, one single hydrogen atom,one nucleus, the “solitary quantum system” (SQS). Some specific properties of a SQS(hydrogen) are derived on the basis of experimental facts and the theory ofcontemporary quantum physics (QP). Thus, the properties of a solitary hydrogenatom do not contradict the results of QP, but allow us to think about and search forunknown and unexpected applications of quantum physics.
Remember that all quantum laws were initially derived from the results of
experiments of statistical ensembles of quantum systems. Subsequently these lawswere applied to solitary quantum systems which are the elementary constituents of thestatistical ensemble. This is easy and trivial. Easy because it is not necessary to searchfor other properties of the solitary object and trivial because this transition does notcontradict the laws which govern an ensemble of identical objects (quantum systems,QS). For a statistical ensemble of quantum systems the introduction of probabilitiesand the statistical interpretation of results are inevitable, but it is not sure that asolitary quantum system must be governed by the same principles. To be morespecific, I can explain the above assertions with the help of an example:8The law of radioactive decay, N=N0exp(-t/τ), was at first observed experimentally
and after that derived from statistical considerations. N is the number of QS which
have not decayed for the time t; N 0 is the number of QS at the initial moment of time
(t = 0) and τ is the mean life time of all QS. This law concerns all decays of any
excited states of nuclei, atoms, molecules and so on (except some deviations in very
short and very long times). It is easy to transfer this law from a statistical ensemble toone solitary object by introducing the probability (W) that this object does not decay
for a time (t): W=N/N
0= exp(-t/τ). But this probability is a trivial application of a law
which concerns only a statistical ensemble of quantum objects . This probability is not
a proof that a solitary object does not have another cause for decay.
In the paper [8] it was shown that in the case of waves on the surface of a liquid the
floating classical particles which pass through only one of the two opened slits are
guided by the interfering surface waves in the same directions (angle θ) as predicted
by quantum laws Ψ(θ)2=max; (the directions Ψ(θ)2=0 are not allowed.) This is
an indirect confirmation of de Broglie's ideas that wave and particle exist
simultaneously and that this coexistence is real [9]. Most of the scientists think that
the field of de Broglie is not real and they accept the statistical interpretation of Born[10]. One of the most often stressed disadvantages of the model of Bohr is the
impossibility to determine (calculate) the probabilities of transition (intensities) of theemitted hydrogen lines...
4. Return to the real unitary field-particle of de Broglie and to Bohr's model of
hydrogen atom.
Why is the assumption that a wave-particle cannot exist simultaneously more real
than de Broglie's assumption that they always exist simultaneously ?. The results of
this work will show there is simultaneous existence of de Broglie's field and Bohr's
atom ... and that (for one atom) no statistical interpretation is necessary . De
Broglie's waves in the hydrogen atoms are such that in the stationary state the mass ofthe electron (m
0), its velocity Vn , and average radius r n are related with the principal
quantum number ( n= 1,2,3....) according to:9m0Vnrn= nh
2π = n /G21 (14)
and the field-particle (electron) is in a potential well which keeps the electron in orbit
n, and the electron cannot be accelerated (does not emit a photon). The length of de
Broglie's wave λn exactly satisfies the condition:
n
nn nVmnhr λ π ==
02 (15)
De Broglie's unitary "wave-particle" is in a stationary ("steady") state which never
changes. The "wave-particle" electron is bound together with the "wave-particle"proton by electromagnetic forces and de Broglie’s real field. They interfere andremain in their potential well (position) forever, like classical particles on the surfaceof a liquid [8]. The field of de Broglie is real and strong so that the electromagnetic
forces cannot destroy this interference and field-particle (electron) cannot be
accelerated. To explain the decay of a stationary state it is necessary to assume someinfinitely small "external perturbation" which would disturb the exact equalities (14)and (15) and (after some time of destructive interference) permit the transition tolower states (acceleration of electron). Only the state n=1 cannot be disturbed by an
"infinitely small perturbation" because the field-particle cannot be destroyed ( n
cannot be smaller than 1 and 2 πr1=λ
1). In this case only if the perturbation energy is
sufficiently great, can the electron make a transition from ground to upper levels by
absorption, [8]).I suppose that energetically excited the electron can randomly occur at any distance
(r
ni) around the exact radius of the stationary orbit (n ninrλπ≈2 ;). The difference
between the trajectory of the electron (2 πrni) and nλn can be very small, yet -
destructive interference leads (after some time) to a transition to lower states. Imagine
the "wave-particle" electron self-interferes as long as the minima of the wavecoincides with the maxima of the preceding waves so that the amplitude (D) of the
interfering electron-wave becomes
()02=tD . A transition (acceleration) occurs and
energy is emitted. The greater the difference rrni n−, the smaller the time necessary
for destructive interference. If rrni n−→0, the time for destructive interference would
be very long [11]. When the energy of excitation is exact (r ni = rn), a true stationary10state would be established and without external perturbation this state could not be
changed. So, it is evident that the wave-particle electron can be excited so, as to occurat all possible distances (r) from the proton.
5. Own lifetime of a single hydrogen atom.
In Fig.3 a schematic wave-particle in some excited state of the hydrogen atom is
shown. The particle-wave electron moves from left to right (for example, n = 2). In
Fig.3a) the velocity of the electron Vn is such that λ
n and rn correspond exactly to
Bohr's conditions:
nnVmh
0=λ (16)
Such a wave-particle electron returns from the left always with the same phase and
reiterates its motion for an infinitely long time. If the velocity of the electron ( V) is
slightly different, the new λ will also be slightly different (compared with λn):
Vmh
0=λ (17)
Fig.3. A scheme of the first hydrogen excited states. Wave-particle electron and its interference; a)
true stationary state; b) almost stationary state.11Such a particle-wave electron would arrive from the left (Fig. 3b) with a slightly
different phase. With time this difference increases and the moment when the sum ofthe amplitudes becomes zero (for the first time) can be calculated; electron is no more
in the potential well of the wave (like classical particles, [8], when Ψ
2=0) and
could be accelerated. When this occurs defines the time of life of this excited atom.
The sum of the amplitudes of de Broglie' field (D) can be written (like classicalparticles, [8]):
() ()
+
− = Vt rVt Dλπ
λπ 2sin2sin (18)
where r is the new radius which is only slightly different from rn. The relation
between λ, ω and velocity ( V) is:
ωπλV2= (19)
Substitute this in (18) to give:
()tVrt D ωωω sin sin +
−= (20)
In Bohr’s model, r/ V = 1/ω, therefore (20) becomes
()( ) t t D ω ωsin1sin +−= (21)
which is the sum of the de Broglie's amplitudes (D), expressed by the time and the
frequency of a not exactly stationary state. From (16) and (17) the small difference
Δλ and Δω are found:
ωω
λλλ
λλ
Δ=
−=−=Δ1
nn
VV(22)
Taking into account that in Bohr's model
3 3
nn
n nVV
ωωω
ωω Δ+== (23)
from (22) ( ω) is found:
−Δ+Δ= 1 3
nn
ωωωωω (24)12The moment (t) when D20= has to be found (the electron is not in the potential
well of its wave and it is accelerated):
()( ) 0 sin1sin2 2= +−= t t D ω ω (25)
Hence
() ( ) t t ω ω sin1sin −=− (26)
or
ωωtt−= −1 ; t=1
2ω(26a)
So, substituting ω from (24), gives the necessary "own lifetime" (t):
− Δ=
1 21
3
nt
ωωω=
−Δ+Δ 1 21
3
nn
ωωωω(27)
As it is seen, when ω=ωn (or Δω=0), the time is t →∞, as it should be for a
stationary state. For Δω<<ωn, the expression for the time (27) is symmetric (for
positive and negative Δω). It is more convenient to transform eqs.(27) in terms of the
binding energy:
−+ΔΔ=
11 2 3
nEEEt/G21(28)
where the energy can be measured in units eV and /G21 [eV.s]. In this case the energy of
the different excited states can be expressed through the Rydberg constant (R). Thus,the life time of each single excited hydrogen atom depends on the small energy
difference (
ΔE) and the principal quantum number ( n):
−Δ+Δ=
1 1232
REnEt/G21(29)
In the case when REn<<Δ2, the cubic root can be expanded in a series, and taking
only two first terms of the expansion (1+2n()RE3/Δ...) to give:
()2223
nERt
Δ=/G21(30)13Part of the results are shown in the Fig.4 (for /G21=6.59x10 -16 eV.s and R=13.595 eV).
These curves are different for different excited states ( n). They could be compared
with the normalized "own lifetimes" of nuclei (t/ τ and EΔ/Γ) [11].
Fig. 4. Time (t) versus energy ( EΔ) for n=2,3 and 4. These curves are symmetrical to the curves for
energy differences (- EΔ) (to the left of En ).
6. The natural width and mean life time of an ensemble of excited hydrogen atoms
Similar to the results in [11], the "own life time" (t) of one single excited atom
(in state ( n)) depends exactly on the energy difference ( EΔ) (30). The own life time
(t [s]) is determined by the exact energy of excitation ( EEEn−=Δ ), the Planck ( /G21)
and Rydberg ( R) constants, and the principle quantum number ( n) of the excited state.
This time cannot be measured experimentally (except in the case shown in [4,11] forresonant Mossbauer transitions in nuclei). Experiments with hydrogen measure onlythe mean lifetime of an ensemble of excited atoms.
The statistical natural width of the levels ( Γ
n) and mean life times ( τn) (for
different excited states) of an ensemble of hydrogen atoms will be found and
compared to reference data. Assume that N0 [cm-3] atoms (thin target) are irradiated
by a flux of photons with uniform energy distribution Φ()E = Φ0[cm-2s-1] = const.14(in the region of some quantum level n). If the effective cross-section of excitation is
σE, then, the activity of excited level can be obtained as:
() )/exp(1 )(0 0 n E t N tdtdNτ σ −− Φ= (31)
As it is well known, after irradiation stops, the activity changes with time in the
following way:
() )/exp( )(0 0 n E t N tdtdNτ σ − Φ= (32)
On the other hand, the differential cross-section (Edσ) is:
()220
4nn
EEdEd
Γ+ΔΓ=σσ (33)
(σ0 is the cross-section in the maximum; Γn is statistical natural width of level ( n)).
Then the integral cross-section (Eσ) will be:
20πσσ=E (34)
Substituting (34) in (32) gives the variation of activity with time after excitation:
() )/exp(2)(00
0 nt N tdtdNτπσ− Φ= (35)
Under the same conditions, but using the differential cross-section (33), shows
how activity ()EdtdN increases with irradiation time:
()()()()n
nnt
EdENEdtdNτσ/exp1
422000−−
Γ+ΔΓΦ= (36)
To derive an expression for this activity after irradiation, from (30) the variation of
the own life time ( t) with energy is:
()233
nERdEdt
Δ=/G21(37)
Because of the symmetry of (30), (Figs.5,6) with respect of energy, in the time
interval (dt) decay the atoms in the two intervals ΔE on both sides of En:
() () ()23 23 236 3 3
nERdE
nERdE
nERdEdt
Δ=
Δ+
Δ=/G21 /G21 /G21(38)
or15 ()
RdtnEdE
/G21623Δ= (39)
Substituting ( dE) in (36) gives the activity of hydrogen atoms after irradiation :
()()
()() R EdtnE NEdtdN
nn
/G216 42223
000
Γ+ΔΔΓΦ=σ(40)
Two expressions for the activities are found: (40), depends on the energy of
excitation ( EΔ), and (35), depends on time (t). In the experiments where the energy
(ΔE) cannot be measured, the two activities (35) and (40) must be equal [11]:
()
()() R EdtnE N
nn
/G216 42223
000
Γ+ΔΔΓΦσ= () )/exp(200
0 nt N τπσ− Φ (41)
In the specific case (Fig.5 and 6) when exp ()ntτ/−=1/2, then EΔ=Γn/2, and the
expression (41) becomes:
Γnnd t
R22
24 /G21=π (42)
Hence, the natural width ( Γn) of a statistical ensemble of atoms (for unit time interval,
dt=1) can be calculated as:
Rnn/G21π241=Γ (43)
For the population of a statistical ensemble, natural levels width (normalized in the
maximum) are shown in Fig. 5.
Fig. 5. The normalized natural lines of hydrogen atom (n=2,3 and 4). The energy ( EΔ=En-E) is
calculated in absolute units [eV].16From the natural width ( Γn) of level ( n) it is easy to derive the mean lifetime of
all excited atoms (at level n):
Rn
nnπτ24/G21 /G21=Γ= (44)
Thus, for calculation of the mean life time of an statistical ensemble of excited
hydrogen level ( n), only Rydberg's constant ( R) and Planck's constant ( /G21) are needed.
The corresponding decay constant (the spontaneous coefficient of Einstein) isAn=1/τ
n.
Fig. 6. Time (t/ τ) versus energy ( ΔΓEn/ ) of excitations and normalized natural width
of the first excited state. When decay moves from the wings of the level ( ΔE=∞) to
the place ΔΓEn=/2, then half of the excited atoms have decayed and
exp(−tn/τ)=1/2.17
Fig. 6. Time (t/τ) versus energy ( ΔΓEn/) of excitations and normalized natural width of the first
excited state. When decay moves from the wings of the level ( ΔE=∞) to the place ΔΓEn=/2,
then half of the excited atoms have decayed and exp( −tn/τ)=1/2.
7. Comparison with reference data.
In the numerous reference tables on hydrogen gave quite different values for τn
(especially for low binding energy of the excited states; n>2). In Table 1 below there18are data from [12] (1966) and [13] (1986) compared to the calculations (formula 44,
1997) [4,5].
Table 1. The values of τn=1/An (and natural width of the levels) from present paper (1997) are
closer to the values of data source [13] (1986). The difference between the data from [12] (1966) and
[13] (for n>2) are impermissible .
As it is seen, for the second excited state ( n=2) the calculated τn is equal to 1.603x10 -
9 s, while in [12] this time is τn=2.127x10 -9 s and in [13] τn=1.60x10-9s. So, the
result from the present calculations is in excellent agreement with reference data [13]
(for n=2). It is necessary to stress that the calculations fit better to the values in [13]
(1986). The differences between the values in [12] and [13] are greater than thedifferences between the calculations and the data in [13]. So, the Bohr’s model
(complemented with de Broglie’s ideas) continues to describe hydrogen properties(mean life time, natural width of the levels) as exactly as Bohr’s hydrogen modeldescribes the frequency of radiation .
8. Differences between the data.
As it is known, the experimental accuracy for frequency measurement is very
great in comparison with accuracy of time measurements. An attempt to explain thegreat differences [12,13] between reference data (for n>2) will be made.
Experimental results are very good only for the first excited states... The differences19between reference data (for n > 2) are caused by experimental difficulties and
incorrect application of the relation between Einstein's coefficients, which isexplained in [14,15].
In [12] the transition probability for spontaneous emission from upper state k to
lower state i, Aki, is related to the total intensity Iki of a line of frequency ν
ik by
kikki ki NhAI νπ41= (expression (1) on page ii of [12]) (45)
where h is Planck's constant and Nk the population of state k. It was shown in [14,15]
that this relation holds for transitions from any excited state k to the ground state i
only. If (i) is also an excited state, then relation (45) must be:
Iki=1
4π(Aki +g
gi
kAix)hikνNk (46)
where Aix is the full decay constant of level (i) and gi, gk are the corresponding
statistical weights. Only when Aix=0 (ground state), (46) coincides with (45). The
same applies for the transition probability of absorption Bik and the transition
probability of induced emission Bki in [12]:
Bik=6.01 ki
ikAgg3λ (expr. (6), p. vi of [12]) (47)
Bki =6.01λ3Aki (expr. (7), p. vi of [12]) (48)
(λ is the wavelength in Angstrom units). When (i) is an excited state, these relations
are also wrong. According to [14,15], these relations (in the same units as in [12])will be:
Bik=6.01
+ix ki
ikAAgg3λ (49)
Bki =6.01
+ix
ki
kiAggA3λ (50)
It is also seen that if (i) is a ground state, 0 =ixA, these relations correspond to the
relations in [12]. It is clear that even based on experimental results (when n > 2), τn
can have wrong values if processed using the inappropriate (but commonly accepted)
relations [12].20The mean lifetimes of excited levels of the simplest atom - hydrogen - obtained
herein are in surprising agreement with the known data. At the same time, thedifferences between the reference values for n>2, shows that all reference data fortransition probabilities in hydrogen must be critically examined and adjustedaccurately according the present results.
9. The shape of acceleration curve and the shape of electromagnetic field of the
soliton.
On the Fig.2 it is shown only one example of the acceleration curve. This curve
cannot be known exactly, because the shape of electromagnetic field of the soliton(Fig.1) also is not known exactly [3]. We know from electrodynamics only that theshape of two curves must exactly coincide. The two curves can be symmetrical or notsymmetrical, but independently of its exact shape we can calculate the integral valuesof necessary parameters for solitons (acceleration) emitted by hydrogen atom (8),(10).The effective time of transition (acceleration) is
+==
22043
11 21
knmete/G21
ω(51)
Effective acceleration ( a) is:
−
− =22 4061111
knknmea
/G21(52)
and the effective path Hnk is:
−+
+= 22022
111
1121
knkknmeHnk/G21(53)
As it is seen for n = 1 and k = 2 the effective path (53) is equal to the Bohr’s radius
0r:
≈==0
022
r
meHnk/G215.3x10-11m (54)
and it is smaller from the distance between the two orbits (30r). The energy of the
soliton Enk (12), as it must be, do not depend on effective time te (51), but this21effective time is necessary for comparison between the shapes of two curves: electric
field of soliton and acceleration of electron.
The possible shape of electromagnetic soliton.
The most often the shape of soliton curve (with a form like a bell) is described
from equation [16]:
E = (2/τ)sech[(t – x/v)/ τ] (55)
Where τ is connected with the width of the impulse. (55) can be only one example; it
is not sure that electromagnetic field of the soliton [3] correspond exactly to (55), but
according to definition in [3] the electric field in the maximum of the curve ( E0) and
effective time ( te) are related:
E0te = ∫∞
∞−dttE)( (56)
The electric field of the soliton in vacuum can be written:
E(t) = E0) (2
/ / e e tt ttee−+ or E(x) = E0) (2
/ / e e lx lxee−+(57)
If the soliton electric field correspond to (57), then acceleration of electron ( a(x))
must have the same shape (in the space along the unknown trajectory x, but known
effective path Hnk):
a(x) = a0)/exp()/(exp(2
nk nk Hx Hx −+(58)
Here a0 is acceleration in the maximum (a 0 ∼ E0) and
a0Hnk = ∫∞
∞−dxxa)( (59)
The shapes of the two curves (57 and (58) cannot be accepted as exact, but it is sure
that they must correspond to each other and the equations (56 and 59) are exact bydefinition. Knowing effective length of electromagnetic soliton in vacuum ( l
e) and
effective path of electron ( Hnk) one can estimate the average velocity of electron ( vnk)
when passing the distance Hnk:
Hnk = vnkte (60)22and
le = cte (61)
So, the ratio of the two velocities is
vnk/c = Hnk/le (62)
When the transition (hydrogen) occur between k=2 and n = 1 this ratio is:
vnk/c ≈ 5.6x10-3
The velocity of the electron is about 3 orders of magnitude smaller than the velocity
of the light.
10. Some inevitable conclusions.
Here we are finding the effective acceleration of electron in hydrogen but it is
evident that the acceleration vector changes in time and direction. These details Icannot find now because only integral acceleration is known. If we know the exact
shape of electromagnetic field of the solitons (Fig.1), the exact shape of accelerationcurve (Fig.2) would be known also and vise versa. Now we can make only somesupposition about the changes of acceleration vector in space and time. This vector isprobably not radial, as it is not radial the path ( H
nk) of the electron. The acceleration
vector describes some complicated curve different of acceleration in an ordinarydipole and it is sure that emission of a hydrogen atom cannot be presented asemission of a vibrating dipole. In the beginning and on the end the acceleration ofelectron must be zero and must have one positive maximum (Fig.1 and Fig.2). Theacceleration vector must describes some curve in the space for a very short effectivetime t
e. This curve probably lay on the plane of velocity vectors and determines the
direction of soliton propagation.
Using only the bases of classical electrodynamics we can conclude that
electromagnetic field in every moment is emitted perpendicularly to the accelerationvector in this moment. Electromagnetic field must be redistributed in differentdirections and cannot form a soliton (electromagnetic particle) which propagate onlyin one direction.
These simple calculations show that the most difficult question about the time
of transition in atoms can be answered combining the soliton properties and the23Bohr’s model of atom. We think some inevitable changes in classical electrodynamics
are needed and this probably will change the quantum electrodynamics also.
Now it is clear that energy of the soliton is discrete in time and space as it is
discrete acceleration of the electron. We else do not know what is the exact shape ofacceleration and the exact shape of soliton electromagnetic field (Fig.1 and Fig.2), butI hope that they can be found and consequently it can be found the direction of thesoliton propagation.
The calculations show that Bohr's model of hydrogen is as useful as the real
field of L. de Broglie is. The hydrogen atomic model of Bohr-de Broglie allow (for
the first time) to calculate exactly the mean lifetime (τ
n = 1/A n) of an ensemble of the
excited levels . If the energy of excitation is different from that corresponding to the
exact conditions for a stationary state, after some evolution of the excited state theCoulomb field can change the state of the electron (acceleration) because theamplitude of de Broglie's waves becomes zero and the electron is no more in apotential well (the electron can emit a photon-soliton [1,2,3]). The main result of thiswork is that excited states of the hydrogen atom decay after some exactly predictable
time (t) (30) and the emission of the photon-soliton takes an exact time ( ∼ t
e). Decay
is not an accidental event as it is believed by the majority of scientists (except
Einstein who wrote that a weakness of the theory of radiation is that the time ofoccurrence of an elementary process is left to "chance"). The mean life time ( τ
n) is a
characteristic only of a statistical ensemble of excited atoms (40).
If a transition occurs between two excited states (E n = R(1/n2) and E k = R(1/k2)
the frequency of the emitted soliton is calculated according to: (E n - Ek)/ /G21 = ωnk. This
frequency correspond to the maximum of the line distribution of a statistical
ensemble of hydrogen atoms. The width of the photon line is the sum of the levels
width: Γnk = Γn+Γk.
So, for a statistical ensemble of hydrogen atoms the distances of electrons
from the protons (or energies), are very different. In such an ensemble the probabilityto find an electron at some distance (or energy) from the proton is maximal at theplaces of Bohr’s stationary orbits. This probability is smaller at other places, butnever becomes zero. For the coordinate systems related with the center of mass of
every hydrogen atom , these probabilities are presented on Fig.5. For a laboratory24coordinate system the probabilities depend by motions of the center of masses and
become consistent with contemporary quantum physics.
I think that all these solitary objects do not contradict Quantum Mechanics
(QM) - especially the properties of a solitary hydrogen atom - but only reveal someunknown details of SQS. It seems to me now, that the properties of a solitary
quantum object must be different from the properties of the statistical ensemble ofsuch objects and cannot be further neglected... As it is known, Bohr’s model of the
hydrogen atom concerns a solitary hydrogen atom . However, all experimental results
which confirm this model - excited states, frequency of the emitted lines, thecalculation of Rydberg’s constant and so on, are obtained from spectroscopic dataabout statistical ensembles of hydrogen atoms. The frequencies, exactly calculated byBohr, correspond to the frequencies of the stationary states (at the maxima of thelines). Now, it is clear that the natural width of the lines (statistical ensemble) cannotbe obtained from Bohr’s conditions only. Bohr himself probably could solve thisproblem, if he had accepted de Broglie’s ideas about the coexistence of waves andparticles.
To pay honor to Luis de Broglie who wrote:
“In the spring of my life, I was obsessed with the problems of quanta and thecoexistence of waves and particles in the world of micro-physics: I made decisiveefforts, although incomplete, to discover the solution. Now, in the autumn of myexistence, the same problem still preoccupies me because, despite of the manysuccesses and the long way passed, I do not believe that the enigma is indeedresolved. The future, a future which I undoubtedly will never see, will maybe decidethe problem: it will tell whether my present point of view is an error of an alreadysufficiently old man who is still devoted to the ideas of his youth, or, on the contrary,this is a clairvoyance of a researcher who all his life has meditated on the most
important question of contemporary Physics”. (L. de Broglie, Certitudes et
incertitudes de la Science , Edition Albin Michel, Paris, 1966, p. 22; a free translation
from French).
I realize that there are not some consecutive descriptions of a new
electrodynamics of electromagnetic fields and charged particles. The knownproperties of solitons [3] and hydrogen atom [4,5] permit to think about and to25describe some fragmentary, but very convincing and essential results, concerning the
reality of solitons and its interactions with charged particles. It is evident that thisfragmentary electrodynamics is not complete, but I hope that in the future it can becompleted.26References:
1. J. P. Vigier, Foundations of Physics, 21,2, 125 (1991).
2. T. Waite, Ann. Fond. Louis de Broglie (Paris) 20, 4, (1995) 4273. P. Kamenov, B. Slavov, Foundations of Physics Letters, v.11, No4, (1998) 325.
4. P. S. Kamenov, Physics of solitary quantum systems , Paradigma, Sofia, 1999,
p.56 – 74;5. P. Kamenov, Comp. Rend. Acad. Bulg. Sci., 52, 5/6, (1999) 276. A. Einstein, Mitt. Phys. Ges. (Zuerich), 18, (1916), p. 47-627. A. Einstein, Verh. Dtsch. Phys. Ges., 18 (1916) p.3188. P. Kamenov, I. Christoskov, Phys.Lett. A, v.140, 1,2 (1989) 13-18.9. L. de Broglie, Rev. Sci. Prog. Decouvert, 3432, (April 1971), 4410. M. Born, Z. Phys. 37 (1926) 863; - M. Born, Z. Phys. 38 (1926) 803.11. P. Kamenov, Nature Phys.Sci. v.231 (1971) 105-107.12. Atomic Transition Probabilities , Volume I Hydrogen Through Neon (A critical
Data Compilation), W. L. Wiese, M. W. Smith, and B. M. Glennon, National Bureauof Standards, (May 20, 1966).13. A. A. Radzig, B. M. Smirnov, Parametry atomov i atomnih ionov (Data),
ENERGOATOMIZDAT, Moskva, 1986 (in Russian).14. P. Kamenov, A. Petrakiev, and A. Apostolov, Laser Physics, Vol.5, No 2, (1995)314-317.15. P. Kamenov, Nuovo Cimento D, 13/11, Nov. 1991, 1369-1377
/G14/G19/G11 /G35/G11 /G2E/G11 /G25/G58/G4F/G4F/G52/G58/G4A/G4B/G0F /G33/G11 /G2D/G11 /G26/G44/G58/G47/G55/G48/G5C/G0F /GB3/G36/G52/G4F/G4C/G57/G52/G51/G56 ”, Ed. Springer – Verlag Berlin
Heidelberg New York 1980 (Russian Ed. Moskwa, “Mir” 1983 p. 85). |
arXiv:physics/0101097v1 [physics.atom-ph] 29 Jan 2001A numerical ab initio study of harmonic generation from a rin g-shaped model
molecule in laser fields
D. Bauer and F. Ceccherini
Theoretical Quantum Electronics (TQE), Darmstadt Univers ity of Technology,
Hochschulstr. 4A, D-64289 Darmstadt, Germany
(July 26, 2013)
When a laser pulse impinges on a molecule which is invariant u nder certain symmetry opera-
tions selection rules for harmonic generation (HG) arise. I n other words: symmetry controls which
channels are open for the deposition and emission of laser en ergy—with the possible application
of filtering or amplification. We review the derivation of HG s election rules and study numerically
the interaction of laser pulses with an effectively one-dime nsional ring-shaped model molecule. The
harmonic yields obtained from that model and their dependen ce on laser frequency and intensity are
discussed. In a real experiment obvious candidates for such molecules are benzene, other aromatic
compounds, or even nanotubes.
I. INTRODUCTION
Harmonic generation (HG) from atoms (L’Huillier & Balcou 19 93), molecules (Liang et al. 1994), clusters (Donelly
et al. 1996), and solids (von der Linde et al. 1995) as a short- wavelength source is of great practical relevance. In
recent years a huge amount of publications were devoted to “h armonic engineering,” under which we subsume either
the study of phase matching during the propagation of the emi tted light through gaseous media (Gaarde et al. 1998),
generation of attosecond pulses using varying ellipticity (Antoine et al. 1997), multi-color studies (Milosevic et al .
2000), or HG from thin crystals (Faisal & Kaminski 1996 and 19 97). Experimental results on HG from cyclic organic
molecules were also reported in the literature (Hay et al. 20 00a and 2000b). However, the authors made no attempt
to verify theoretically predicted selection rules. In the w ell established physical picture of HG one assumes that an
electron tunnels out of the atom or ion, moves in the laser fiel d and eventually rescatters with its parent (or other) ion
where it might recombine—leading to an emission of a photon w ith several times the fundamental frequency (Becker
et al. 1997). Assuming such a viewpoint one can easily explai n prominent features of HG spectra such as the famous
cut-off atIp+ 3.17Upin the single atom case (where Ipis the ionization energy and Upis the ponderomotive energy,
i.e., the cycle-averaged quiver energy of the electron in th e laser field). For a general overview on harmonic generation
in laser fields see the recent review by Sali` eres et al. (1999 ).
Fortunately, to derive the selection rules for harmonic emi ssion only symmetry considerations are necessary (Alon
et al. 1998). We now briefly summarize this approach. Let us co nsider a Hamiltonian which is periodic in time, i.e.,
H(t) =H(t+τ). Such a Hamiltonian might describe an electron in a longlaser pulse where the pulse envelope is
sufficiently adiabatic. The time-dependent Schr¨ odinger eq uation (TDSE) reads (we use atomic units ¯ h=e=m= 1
throughout, if not noted otherwise)
/bracketleftbigg
H(t)−i∂
∂t/bracketrightbigg
ΨE(r,t) = 0. (1)
According to the Floquet theorem (see, e.g., Faisal 1987) we can write Ψ E(r,t) =ψE(r,t)exp( −iEt) withψE(r,t) =
ψE(r,t+τ) leading to the Schr¨ odinger equation
HF(t)ψE(r,t) =EψE(r,t),HF(t) =H(t)−i∂
∂t(2)
with Ethe so-called quasi energy and HFthe Floquet Hamiltonian. Since ψE(r,t) is periodic in time it might be
expanded in a Fourier series ψE(r,t) =/summationtext
nexp[−in(ωt+δ)]Φ(n)
E(r). This Floquet approach is a well-known method
in multiphoton physics (Gavrila 1992) and a widely used nume rical simulation technique also (Potvliege 1998). In
order to derive the HG selection rules we now assume (for reas ons which will become clear soon) that the system is
in a single and non-degenerate Floquet state ψE. The HG spectra (HGS) peak no. nis present only if the Fourier
transformed dipole moment µ(r) does not vanish for the frequency nω,
1/integraldisplay
dtexp(−inωt)/integraldisplay
d3rΨ∗
E(r,t)µ(r)ΨE(r,t)/negationslash= 0. (3)
Now let us suppose we worked out a symmetry operation Punder which the Floquet Hamiltonian is invariant,
PHF(t)P−1=HF(t). It follows that (in the case of non-degenerated Floquet st atesψE, see assumption above)
PψE=aψEholds. Here ais a phase factor, |a|= 1. For convenience, we rewrite (3) as
/angbracketleft/angbracketleftΨE|µ(r)exp( −inωt)|ΨE/angbracketright/angbracketright=/angbracketleft/angbracketleftψE|µ(r)exp( −inωt)|ψE/angbracketright/angbracketright /negationslash= 0 (4)
where the double brackets indicate spatial andtemporal integration [this is the so-called extended Hilbe rt space
formalism, see, e.g., Sambe (1973)]. The HG selection rule c an be derived from
/angbracketleft/angbracketleftψE|µ(r)exp( −inωt)|ψE/angbracketright/angbracketright=/angbracketleft/angbracketleftPψE|Pµ(r)exp( −inωt)P−1|PψE/angbracketright/angbracketright
=/angbracketleft/angbracketleftψE|Pµ(r)exp( −inωt)P−1|ψE/angbracketright/angbracketright (5)
leading to
µ(r)exp( −inωt) =Pµ(r)exp( −inωt)P−1. (6)
The last step in (5) is not possible if ψEis not a pure Floquet state. In the next Section we apply (6) to derive the
selection rule for HG from ring-shaped molecules. Here, for the sake of illustration, we rederive the selection rule
for a single atom with spherically symmetric potential V(r) and linearly (in x-direction) polarized laser field E(t) =
ˆEexsinωt. The Floquet Hamiltonian HF(t) =−1
2∇2+V(r) +ˆExsinωt−i∂tis invariant under the transformation
x→ −xandt→t+π/ω. If we look for harmonics polarized in the x-direction we have µ(r) =xand from (6)
xexp(−inωt) =−xexp[−inω(t+π/ω)] follows exp( −inπ) =−1. Therefore only odd harmonics are generated.
The paper is organized as follows: in Section II we introduce our model and the selection rule which holds in its
case. In Section III we present and discuss our numerical res ults. Finally, in Section IV we give a summary and an
outlook.
II. A SIMPLE ONE-DIMENSIONAL MODEL FOR RING-LIKE MOLECULES
The time-dependent Schr¨ odinger equation (TDSE) for a sing le electron in a laser field E(t) and under the influence
of an ionic potential V(r) reads in dipole approximation and length gauge
i∂
∂tΨ(r,t) =/parenleftbigg
−1
2∇2+V(r) +E(t)·r/parenrightbigg
Ψ(r,t). (7)
The dipole approximation is excellent since in all the cases studied in this paper the wavelength of the laser light is
much greater than the size of the molecule.
If we force the electron to move along a ring of radius ρin thexy-planeV(r) becomes V(ϕ) whereϕis the
usual polar angle. With an electric field of the form E(t) =ˆE(t)/bracketleftBig
ξcos(ωt)ex+/radicalbig
1−ξ2sin(ωt)ey/bracketrightBig
, where ˆE(t) is a
slowly varying envelope, ωis the laser frequency, and ξis the ellipticity parameter, the TDSE (7) becomes effective ly
one-dimensional (1D) in space and reads
i∂
∂tΨ(ϕ,t) =/parenleftbigg
−1
2ρ2∂2
∂ϕ2+V(ϕ) +ˆE(t)ρ/bracketleftBig
ξcos(ωt)cosϕ+/radicalbig
1−ξ2sin(ωt)sinϕ/bracketrightBig/parenrightbigg
Ψ(ϕ,t). (8)
In case of circularly polarized light ( ξ= 1/√
2) this simplifies to
i∂
∂tΨ(ϕ,t) =/parenleftBigg
−1
2ρ2∂2
∂ϕ2+V(ϕ) +ˆE(t)ρ√
2cos(ϕ−ωt)/parenrightBigg
Ψ(ϕ,t). (9)
We now assume that the potential V(ϕ) has anN-fold rotational symmetry, V(ϕ+2π/N) =V(ϕ). Then, with the
help of (6), we can easily derive the selection rule for HG in t he system described by the TDSE (9). The transformation
(ϕ→ϕ+ 2π/N,t→t+ 2π/Nω ) leaves the corresponding Floquet Hamiltonian invariant. For, e.g., anti-clockwise
polarized emission µ(r) =ρexp(iϕ) holds, and from (6) we have
2ρexp(iϕ)exp( −inωt) =ρexp[i(ϕ+ 2π/N)] exp[ −inω(t+ 2π/Nω )]
leading ton=Nk+ 1,k= 1,2,3.... For the clockwise emission one finds accordingly n=Nk−1. Thus we expect
pairs of HG peaks at kN±1 (Alon et al. 1998).
The TDSE (8) or (9) can be easily solved ab initio on a PC. We did this by propagating the wavefunction in
time with a Crank-Nicholson approximant to the propagator U(t+ ∆t,t) = exp[ −i∆tH(t+ ∆t/2)] where H(t) is the
explicitly time-dependent Hamiltonian corresponding to t he TDSE (8). Our algorithm is fourth order in the grid
spacing ∆ϕand second order in the time step ∆ t. The boundary condition is Ψ(0 ,t) = Ψ(2π,t) for all times t.
III. NUMERICAL RESULTS AND DISCUSSION
We now present results from single active electron (SAE)-ru ns withρ= 2.64 (bond length and radius of benzene
C6H6) and an effective model potential
V(ϕ) =−V0
2[cos(Nϕ) + 1] (10)
withN= 6 andV0= 0.6405. This leads to an electronic ground state energy E0=−0.34 which is the experimental
ionization potential for removing the first electron in benz ene (see, e.g., Talebpour et al. 1998 and 2000). Note, that
in our simple model we have no continuum but discrete states o nly. The first six excited states are located at −0.27
(two-fold degenerated), −0.07 (two-fold degenerated), 0 .16 (non-degenerated), 0 .48 (non-degenerated), 0 .85 (two-fold
degenerated), 1 .48 (two-fold degenerated). The energy levels of our model re semble, apart from an overall downshift
and the removal of degeneracies of certain states, those of t he isoperimetric model where V0= 0, the energy levels
are given by Em=m2/2ρ2,m= 0,1,2,...with the states m/negationslash= 0 two-fold degenerated. Therefore, the energy level
spacing, and thus typical electronic transitions, are diffe rent from those in real benzene. However, it is not our goal
to present quantitatively correct results for laser benzen e-interaction in this paper but we rather want to demonstrat e
some of the underlying principles of HG from ring-shaped mol ecules in general.
In Fig. 1 we present HGS for a q= 240 cycle pulse of the shape ˆE(t) =ˆEsin2(ωt/2q) with an electric field
amplitude ˆE= 0.5 a.u. and frequency ω= 0.18. In Fig. 1a the result for linear polarization ξ= 1 is shown, in Fig. 1b
the result for circular polarization ξ= 1/√
2. To obtain those plots we evaluated the Fourier transforme d dipole in
x-direction, i.e., /angbracketleft/angbracketleftΨ(t)|ρcos(ϕ)exp( −inωt)|Ψ(t)/angbracketright/angbracketright. In theξ= 1-case the dipole with respect to yis clearly zero while
in the circular case there is simply a phase shift of π/2 with respect to the dipole in x. As expected, in the linearly
polarized field all odd harmonics are emitted whereas in the c ircular case only the harmonics 6 k±1,k= 1,2,...are
visible. Other emission lines are many orders of magnitude w eaker. Those lines can form band-like structures which
are interesting in itself. However, in this paper we focus on ly on the laser harmonics. They dominate the HGS, at
least as long as no resonances are hit.
In Fig. 2 the emitted yield of the fundamental and the first fou r harmonics (5th, 7th, 11th, 13th) in a circularly
polarized laser pulse are presented as a function of the lase r frequency. The pulse length T= 8378 a.u. (corresponding
to≈200 fs) and peak field strength ˆE= 0.2 a.u. (corresponding to 1 .4×1015W/cm2) were held fixed. The frequency
is plotted in units of the smallest level spacing of the model molecule, i.e., the energy gap between first excited and
ground state, which is in our case Ω = 0 .34−0.27 = 0.07. For laser frequencies ω <Ω we do not find the 7th or
higher harmonics. The 11th and 13th harmonic show an overall decrease with increasing laser frequency whereas
the fundamental, the 5th and the 7th stay relatively constan t in intensity. For frequencies ω <2.5Ω there is a
complicated dependency of the harmonic yield on ω. All harmonics show a local maximum around 1 .3Ω. However,
at that frequency one apparently hits a resonance since the h armonic peaks become broad and show a substructure
(see left inlay in Fig. 2). In the interval 2 < ω/Ω<2.5 the fundamental drops whereas the 7th harmonic increases
in strength. Note that the 7th harmonic is anti-clockwise po larized, like the incident laser field, whereas the 5th is
polarized in the opposite direction. For frequencies ω>2.5Ω the behavior becomes more smooth apart from another
resonance near 3 .8Ω. In general, the HGS look clean for sufficiently high freque ncies and far away from resonances
(like in the right inlay of Fig. 2). A rich substructure near r esonances is visible in the HGS (cf. inlay for ω= 2Ω).
In Fig. 3 harmonic yields for the fixed frequency ω= 2.8Ω as a function of the field amplitude ˆEare shown. The
higher harmonics (11th–19th) appear only for higher field st rengths whereas fundamental, 5th and 7th are rather
weakly dependent on the field strength. The anti-clockwise p olarized harmonics (polarized like the incident laser
light, drawn thick) tend to overtake the clockwise polarize d ones (drawn thin) at higher laser intensities. However,
ˆE= 0.6 corresponds already to a laser intensity 1 .3×1016W/cm2where a real benzene molecule would probably
break.
3It is interesting to study the scaling of the TDSE (8) with res pect to the size of the molecule. If one scales the
molecule radius like ρ′=αρ, the TDSE (8) remains invariant if t′=α2t,V′=V/α2,E′=E/α3,ω′=ω/α2is chosen.
From this and our numerical result that laser frequencies >Ω are preferable for clean HGS we learn that molecules
bigger than benzene are more promising candidates for HG wit h realistic laser frequencies [from Nd ( ω= 0.04 a.u.)
to KrF (ω= 0.18 a.u.)].
One might object that electron correlation could spoil the s election rule because in reality it is not only a single
electron which participates in the dynamics. However, Alon et al. (1998) have proven that this is not the case. This is
due to the fact that (i)the electron interaction part of the Hamiltonian is still in variant under the transformation P,
and(ii)Pcommutes with the (anti-) symmetrization operator. Even ap proximate theories or numerical techniques
like time-dependent Hartree-Fock or density functional th eory do not spoil the selection rule since they involve only
functionals which depend on scalar products of single parti cle orbitals, all invariant under the transformation P
(Ceccherini & Bauer 2001).
IV. SUMMARY AND OUTLOOK
In this paper we demonstrated numerically HG from a model mol ecule with discrete rotational symmetry which is
subject to a circularly polarized laser pulse. In particula r the harmonic emission from an effectively 1D model with
N= 6 (i.e., a simple model for benzene) as a function of laser fr equency and intensity was discussed. It was found
that for frequencies below the characteristic level spacin g Ω HG is strongly affected by resonances. The situation
relaxes for higher frequencies. For the efficient generation of higher harmonics laser frequencies >Ωandrather strong
fields are necessary. In such fields real aromatic compounds p robably ionize and dissociate already.
Numerical studies for a more realistic, effectively 2D model molecule will be presented elsewhere (Ceccherini &
Bauer 2001).
In order to obtain short wavelength radiation it is desirabl e to have either korNin the selection rule kN±1 as big
as possible. From our numerical results we infer that it is pr obably hard to push efficiently towards high kwithout
destroying the target molecule. For that reason, in a real ex periment nanotubes are promising candidates because N
can be of the order of 100 or more (Dresselhaus et al. 1998), an d, moreover, HG should be even more efficient when
the laser propagates through the tube. However, it remains t he problem of the proper alignment of the laser beam
and the symmetry axis of the molecule. Crystals might be bett er candidates in that respect.
ACKNOWLEDGEMENT
This work was supported by the Deutsche Forschungsgemeinsc haft in the framework of the SPP “Wechselwirkung
intensiver Laserfelder mit Materie.”
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FIG. 1. HGS for a 240 cycle sin2-shaped pulse with an electric field amplitude ˆE= 0.5 a.u. and frequency ω= 0.18. The
polarization was (a) linear ( ξ= 1) and (b) circular ( ξ= 1/√
2). In (a) odd harmonics up to n= 23 dominate whereas in (b)
the harmonics obey the selection rule 6 k±1,k= 1,2, . . ., i.e., the 5th, 7th, 11th, 13th, 17th, 19th harmonics are vis ible.
FIG. 2. Fundamental and harmonic yield vs. laser frequency ωin units of the smallest level spacing Ω = 0 .07. Laser
field intensity and pulse length was kept fixed. The anti-cloc kwise polarized fundamental, 7th, and 13th are plotted thic k,
the clockwise polarized 5th and 11th are drawn thin. The inla ys show spectra (harmonic yield vs. harmonic order) for thre e
particular frequencies (indicated by arrows). See text for further discussion.
FIG. 3. Harmonic and fundamental yields for fixed frequency ω= 2.8Ω but different field amplitude ˆE. The inlays show
spectra (harmonic yield vs. harmonic order) for three parti cular ˆE(indicated by arrows). See text for further discussion.
5Harmonic orderHarmonic yield (arb. u.)(a) linear
(b) circular
Fig. 1: D. Bauer and F. Ceccherini, ”A numerical ab initio stu dy ...”❍❍❍❍ ❥❍❍❍❍ ❥✟✟✟✟ ✙
1
5
7
1113
ω/ΩHarmonic yield (arb. u.)
Fig. 2: D. Bauer and F. Ceccherini, ”A numerical ab initio stu dy ...”❍❍❍❍ ❥❅❅ ❘ |
arXiv:physics/0101098v1 [physics.atom-ph] 29 Jan 2001AN ATOMIC LINEAR STARK SHIFT VIOLATING P BUT NOT T
ARISING FROM THE ELECTROWEAK
NUCLEAR ANAPOLE MOMENT
M. A. BOUCHIAT
Laboratoire Kastler Brossel1,
24, rue Lhomond, F-75231 Paris Cedex 05, France
C. BOUCHIAT
Laboratoire de Physique Th´ eorique de l’Ecole Normale Sup´ erieure2
24, rue Lhomond, F-75231 Paris Cedex 05, France
We propose a direct method of detection of the nuclear anapol e
moment. It is based on the existence of a linear Stark shift fo r
alkali atoms in their ground state perturbed by a quadrupola r
interaction of uniaxial symmetry around a direction ˆ nand a
magnetic field. This shift is characterized by the T-even pse u-
doscalar (ˆ n·/vectorB)(ˆn∧/vectorE·/vectorB)/B2. It involves on the one hand the
anisotropy of the hyperfine interaction induced by the quadr upo-
lar interaction and, on the other, the static electric dipol e mo-
ment arising from electroweak interactions inside the nucl eus.
The case of ground state Cs atoms trapped in a uniaxial (hcp)
phase of solid4He is examined. From an explicit evaluation of
both the hyperfine structure anisotropy and the static dipol e de-
duced from recent empirical data about the Cs nuclear anapol e
moment, we predict the Stark shift. It is three times the expe r-
imental upper bound to be set on the T-odd Stark shift of free
Cs atoms in order to improve the present limit on the electron
EDM.
PACS. 11.30.Er - 21.90 +f - 67.80.Mg - 31.15.Ct
1Laboratoire de l’Universit´ e Pierre et Marie Curie et de l’E cole Normale Sup´ erieure, associ´ e au
CNRS (UMR 8552)
2UMR 8549: Unit´ e Mixte du Centre National de la Recherche Sci entifique et de l’ ´Ecole Normale
Sup´ erieure
1Introduction
It has been well demonstrated that parity violation in atomi c transitions can be used
to test electroweak theory [1]. In this way, the Standard Mod el has been confirmed
convincingly in the domain of low energies. At present, refin ements in experiments
and theory allow more precise measurements to look for a brea kdown of the Standard
Model predictions and hence, new physics [2, 3, 4, 5]. The ess ential parameter ex-
tracted from atomic parity violation (PV) measurements is t he weak nuclear charge
QW. This electroweak parameter appears in the definition of the dominant electron-
nucleus PV potential induced by a Z0exchange:
V(1)
pv(r) =GF/√
2·QW/2·γ5·PV(r), (1)
where theZ0couples to the nucleus as a vector particle, just as the photo n does
in the Coulomb interaction. In this Z0exchange,QWplays the same role as the
electric charge in the Coulomb interaction. γ5is the Dirac matrix which reduces
to the electron helicity, /vector σ·/vector p/m ec, in the non-relativistic limit. The distribution
PV(r) normalized to unity represents the weak charge distributi on inside the nucleus.
The physical quantity measured in atomic PV experiments is a transition electric
dipole moment, Epv
1between states with the same parity, like the nS1/2→n′S1/2
transitions. In particular the 6 S1/2→7S1/2transition in cesium has been the subject
of several experiments, the accuracy of which has been stead ily increasing with time
[6, 7, 8, 9, 10].
On top of this the PV electron-nucleus interaction involves also a nuclear spin-
dependent contribution which can provide valuable and orig inal information regarding
Nuclear Physics. It is generated by an interaction of the cur rent-current type with a
vector coupling for the electron and an axial coupling for th e nucleus. The associated
PV potential V(2)
pvis given by the following expression:
V(2)
pv=GF/√
2·AW/(2I)·/vector α·/vectorI·PA(r), (2)
where/vector αis the Dirac matrix associated with the electron velocity op erator,/vectorIthe
nuclear spin and PA(r) a nuclear spin distribution normalized to unity. The weak
axial moment of the nucleus, AW, receives several contributions. The most obvious
one comes from the weak neutral vector boson Z0with axial coupling to the nucleons.
However, in the standard electroweak model the coupling con stants involved nearly
cancel accidentally. As first pointed out by Flambaum et al. [11], a sizeable contri-
bution toAWis induced by the contamination of the atom by the PV interact ions
between the nucleons which take place inside the nucleus. The concept relevant to
describe this interaction is the nuclear anapole moment [12 ]. In fact the interac-
tion can be interpreted simply in terms of a chiral contribut ion to the nuclear spin
2Figure 1: Simplified representation of the nuclear helimagn etism (figure adapted from C. Bouchiat
[13]). The normal spin magnetization /vectorMS(/vector r) is assumed to be a constant vector parallel to the
nuclear spin, distributed uniformely inside a sphere. Unde r the influence of PV nuclear forces,
the nuclear magnetization distribution inside the nucleus acquires a chiral parity non-conserving
component /vectorMpnc(/vector r), obtained by rotating /vectorMS(/vector r) through the very small angle β(r) around /vector r. Three
chiral magnetization lines in the equatorial plane are show n. The vertical normal magnetization is
actually larger than /vectorMpnc(/vector r) by about six orders of magnitude. It can be shown [13] that th e vertical
anapole moment is given, within a constant, by the magnetic m oment obtained by identifying the
chiral magnetization lines with lines of electric currents .
magnetization [13, 14], as illustrated in Fig 1. In other wor ds, one can say that the
PV nuclear forces inside any stable nucleus are responsible for the nuclear anapole
moment or equivalently a nuclear helimagnetism. The presen t paper addresses the
problem of how to detect directly this unique static nuclear property c haracteristic of
parity violation in stable nuclei.
Up to now there has been only one experimental demonstration of the nuclear
anapole moment, namely that obtained very recently by the Bo ulder group [10]. In
their experiment which gives a high precision determinatio n of parity violation in the
atomic 6S1/2→7S1/2Cs transition, this effect appears as a small relative differe nce,
actually ∼5%, between the Epv
1transition dipole amplitudes measured on two dif-
ferent hyperfine lines belonging to that same transition. In this case the dominant
source of P without T violation comes from the electron-nucl eonZ0exchange asso-
ciated with the weak charge QWof the nucleus. This makes the extraction of the
3nuclear spin-dependent part a most delicate matter. In view of the importance of this
result for the determination of the PV pion-nucleon couplin g constant, f1
π(see [15]),
a totally independent determination is highly desirable.
It is well known that T reversal invariance forbids the manif estation of V(1)
pvin
an atomic stationary state. However, we shall show in the fol lowing sections that in
such a state T reversal invariance does not forbid the manife station ofV(2)
pv, hence
that of the nuclear helimagnetism. For a free atom, the rotat ion symmetry of the
Hamiltonian leads to an exact cancellation of the diagonal m atrix elements. This
property still holds true if the rotation symmetry is broken by the application of
static uniform electric and magnetic fields. However, if the symmetry is broken by
the application of a static potential of quadrupolar symmet ry, for instance by trapping
the atoms inside a crystal of hexagonal symmetry, then, the s tationary atomic states
are endowed with a permanent electric dipole moment which ca n give rise to a linear
Stark shift. This offers a novel possibility of detecting the nuclear helimagnetism
having a twofold advantage:
i) In a stationary state it is the sole cause of P without T violation .
ii) It manifests itself by a modification of the atomic transi tion frequencies in
an applied electric field, i.e. a linear Stark shift, providing for the first time an
opportunity for demonstrating the static character of this unusual nuclear property.
There exist in the literature other proposals for a direct detection in atoms of the
nuclear spin-dependent effect, i.e.without any participation from V(1)
pv:
i) One is based on the difference between the selection rules o f the potentials V(1)
pv
andV(2)
pv. While the former acts as a scalar in the total angular moment um space and
mixes only states of identical angular momentum (and opposi te parity), the latter acts
like a vector and mixes states of different total angular mome ntum. Consequently,
one can find atomic transitions between states of the same par ity which are allowed
for the nuclear spin dependent contribution but remain forb idden for the nuclear spin
independent one [16]. One such example is the (6 p2)3P0→(6p2)1S0lead transition at
339.4 nm, strictly forbidden for even isotopes, which acqui res a non-vanishing matrix
elementEpv
1in odd isotopes owing to the PV interaction involving the nuc lear spin,
which mixes the (6 p2)1S0state to the (6 p7s)3P1state of opposite parity [16].
ii) A second approach, invoked by several groups in the past a nd now under seri-
ous consideration [17], consists in the detection of an Epv
1amplitude via a right-left
asymmetry appearing in hfs transition probabilities for th e ground state of potas-
sium in the presence of a strong magnetic field (magnetic and h yperfine splittings of
comparable magnitude).
iii) There is also the possibility of detecting the energy di fference in the NMR
spectrum of enantiomer molecules [18].
In view of the extreme difficulty of these other projects, we be lieve that, over
4and above its intrisic scientific interest, the linear Stark shift discussed in this paper
deserves careful consideration.
The first section of this paper recalls the main angular momen tum properties of
the permanent nuclear spin-dependent PV electric dipole op erator arising from the
nuclear anapole moment. In addition we compute its magnitud e for the cesium atom
using recent empirical data relative to the Cs 6 S→7Stransition. The next section
(sec. 2) shows that this dipole can manifest itself via a line ar Stark shift only if
the free atom symmetry is broken. After this we consider the c ase where the atom
is perturbed by a crystal field of uniaxial symmetry. Here, th e crystal axis /vector n, and
the applied electric and magnetic fields create a chiral envi ronment permitting the
existence of a linear Stark shift, the explicit expression f or which is given. In the
section 3, we examine a realistic experimental situation wh ere its observation looks
reasonably feasible: this deals with Cs atoms trapped insid e a4He crystal matrix
of hexagonal symmetry. We have investigated quantitativel y how, by breaking the
atomic symmetry, the matrix induced perturbation manages t o generate a linear Stark
shift. Moreover, we evaluate both the matrix induced anisot ropy and the shift. The
details of the necessary calculation based on a semi-empiri cal method are given in the
Appendix. In the final section we suggest another experiment al approach in which
the atoms are no longer submitted to a crystal field, but are in stead perturbed by an
intense nonresonant radiation field.
1 The permanent nuclear spin-dependent PV electric dipole
1.1 Symmetry considerations
The space-time symmetry properties of the atomic electric d ipole induced by the
nuclear spin dependent PV interaction have been presented b efore in many review
papers (see for instance[19]). We recall them here for compl eteness, since they con-
stitute the starting point of the linear Stark shift calcula tion developed in the present
paper.
First, we wish to stress that the existence of the anapole mom ent interaction not
only implies the existence of a transition dipole proportio nal to the nuclear spin,
but also that of an electric dipole operator having diagonal matrix elements between
stationary atomic states. This electric dipole is found to b e proportional to the
operator/vector s∧/vectorI. Therefore it does not undergo the same transformation unde r P as
does an ordinary dipole, since it is a pseudovector instead o f a vector. We also note
that it is even under T-reversal, so that the quantity ( /vector s∧/vectorI)·/vectorE, associated with a
linear Stark shift, violates P, but does not violate T invari ance.
It is convenient to define /vectordpv(n′,n) as the effective pv electric dipole moment
operator acting in the tensor product ES/circlemultiplytextEIof the electronic and nuclear angular
5momentum spaces, which describes the transition between tw oS1/2subspaces corre-
sponding to given radial quantum numbers nandn′. This effective dipole operator
includes both contributions from potentials V(1)
pvandV(2)
pv. Rotation invariance to-
gether with the fact that V(2)
pvis linear in /vectorIimplies that /vectordpv(n′,n) can be written
under the following general form:
/vectordpv(n,n′) =−iImE(1)
1pv(n,n′)/vector σ+ia(n,n′)/vectorI+b(n,n′)/vector s∧/vectorI, (3)
where the real quantities a(n,n′) andb(n,n′) parametrize the contribution of the
nuclear spin-dependent pv potential. Time reversal invari ance ofV(1)
pvandV(2)
pvimplies
the following relations under the exchange n↔n′:
ImE(1)
1pv(n,n′) = −ImE(1)
1pv(n′,n),
a(n,n′) = −a(n′,n),
b(n,n′) =b(n′,n). (4)
The effective pv static dipole moment /vectorDpv=/vectordpv(6,6) relative to the ground state is
then given by :
/vectorDpv=b(6,6)/vector s∧/vectorI=dI/vector s∧/vectorI. (5)
If we introduce the total angular momentum /vectorF=/vector s+/vectorI, using simple relations of
angular momentum algebra, one can derive the useful identit y:
/vector s∧/vectorI≡[/vectorF2,−i
2/vector s]. (6)
It then becomes obvious that, in low magnetic fields and witho ut external pertur-
bation, the dipole operator /vectorDpvhas no diagonal matrix elements between atomic
eigenstates. In fact, as demonstrated in the next section of this paper, a manifesta-
tion of this dipole requires special conditions for breakin g the free-atom rotational
symmetry.
1.2 Magnitude of the permanent dipole.
The magnitude, dI, of the permanent dipole will play a decisive role in the asse ssment
of the feasibility of an experiment. We are now going to perfo rm the evaluation of
dIin the interesting case of cesium. We proceed in two steps : fir st we compute
directlyb(6,7) from experimental data, then we give a theoretical evalua tion of the
ratiob(6,6)/b(6,7). It is convenient to use the notations of ref [7] and [13] an d to
rewrite/vectordpv(6,7) as:
/vectordpv(6,7) =−iImEpv
1(6,7)
/vector σ+η/vectorI
I+iη′/vector σ∧/vectorI
I
. (7)
6The nuclear spin dependent potential V(2)
pvinduces a specific dependence of the pv
transition dipole on the initial and final hyperfine quantum n umbers,FandF′. In
order to isolate the V(2)
pvcontribution, we are led, following ref [7], to the introduc tion
of the reduced amplitudes dF F′:
dF F′(η,η′) =∝angbracketleft7S,F′M′|/vectordpv|6S,FM ∝angbracketright
∝angbracketleftF′M′|/vector σ|FM∝angbracketright. (8)
The amplitudes dF F′(η,η′) are tabulated in Table XXII of ref[7] and reduce to
−iImEpv
1(6,7) for vanishing ηandη′. The quantity of interest here is the ratio
rhf=d43/d34which is given, to second order in ηandη′, by :
rhf≃1−2I+ 1
Iη′. (9)
Using the empirical value for the ratio rhfgiven by the last Boulder experiment [10]:
rhf−1 = (4.9±0.7)×10−2, we obtain:
η′=−7
16(rhf−1) = (−2.1±0.3)×10−2. (10)
We deduce b(6,7) by a simple identification:
b(6,7) =ImEpv
1(6,7)2
Iη′= (1.04±0.15×)10−13|e|a0, (11)
where we have used for ImEpv
1(6,7) the empirical value obtained in ref [10]:
ImEpv
1(6,7) = (−0.837±0.003)×10−11|e|a0.
To compute the ratio b(6,6)/b(6,7), we are going to use an approximate relation,
derived in ref [13], which relates the potential V(2)
pvtoV(1)
pv:
V(2)
pv(/vector r) =KAAW
QW2/vectorj·/vectorI
IV(1)
pv(/vector r). (12)
HereKAis a constant very close to unity which depends weakly upon th e shape of
the nuclear distributions PV(r) andPA(r);/vectorjis the single electron angular momentum
and since, as we shall see, only single particle states with j= 1/2 are involved, we
can write hereafter 2 /vectorj=/vector σ.
This relation, valid for high Z atoms like cesium, hinges upo n the fact that the
matrix elements ∝angbracketleftn′p3/2|V(2)
pv|ns1/2∝angbracketrightinvolvingp3/2states are much smaller- by a factor
2×10−3- than those which involve p1/2states, ∝angbracketleftn′p1/2|V(2)
pv|ns1/2∝angbracketright. This is easily
7verified in the one-particle approximation since the radial wave functions at the sur-
face of the nucleus are very close to Dirac Coulomb wave funct ions for an unscreened
chargeZ. It is argued in ref [13] that this property remains true, to t he level of few
%, whenV(2)
pv(/vector r) is replaced by the non local potential U(2)
pv(/vector r,/vector r′), which describes the
core polarization effects within the R.P.A. approximation3.
The contributions of V(i)
pvto the effective dipole operator /vectordpv(n,n′) are given as
the sum of the two operators:
/vectorA(i)=P(n′S1/2)V(i)
pvG(En′)/vectordP(nS1/2),
/vectorB(i)=P(n′S1/2)/vectordG(En)V(i)
pvP(nS1/2), (13)
where/vectordis the electric dipole operator, G(En) = (En−Hatom)−1the Green function
operator relative to the atomic hamiltonian; P(nS1/2) andP(n′S1/2) stand for the
projectors upon the subspaces associated with the configura tionsnS1/2andn′S1/2;
EnandEn′are the corresponding binding energies. It follows immedia tly from the
Wigner-Eckart theorem that the operators /vectorA(1)and/vectorB(1)can be written as:
/vectorA(1)=ih(n,n′)/vector σ;/vectorB(1)=ik(n,n′)/vector σ. (14)
Using now the relation given in equation (12) and the commuta tion of/vector σ·/vectorIwith the
pseudoscalar V(1)
pv, one gets the following expressions for /vectorA(2)and/vectorB(2):
/vectorA(2)=iKAAW
QWh(n,n′)
/vector σ·/vectorI
I
/vector σ,
/vectorB(2)=iKAAW
QWk(n,n′)/vector σ
/vector σ·/vectorI
I
. (15)
We arrive finally at an expression for /vectordpv(n,n′) which can be used to compute the
ratiob(6,6)/b(6,7):
/vectordpv(n,n′) =i(/vector σ+KAAW
QW/vectorI
I) (h(n,n′) +k(n,n′)) +
/vector σ∧/vectorI
IKAAW
QW(h(n,n′)−k(n,n′)). (16)
3To check the validity of the relation (12) we have compared th e values for ηandη′obtained in
this way with those deduced from a direct computation [20] of /vectordpv(6,7). The two results agree to
better than 10 %.
8Time reversal invariance implies h(n,n) =−k(n,n) so that we can write the sought
for ratiob(6,6)/b(6,7) as:
b(6,6)
b(6,7)=2h(6,6)
h(6,7)−k(6,7). (17)
The amplitudes h(6,6), h(6,7) andk(6,7) can be computed from the formulas given
in Eqs. (13) and (14). We have used the explicit values of the r adial matrix elements
(parity mixing and allowed electric dipole amplitudes) for the intermediate states4
6P1/2−9P1/2and the energy differences involved, which are tabulated in r ef. [20]
(Table IV)5. We obtain in this way:
b(6,6)/b(6,7) = 4.152/1.86 = 2.27. (18)
Combining the above result with the value of b(6,7) given by equation (11) we obtain
the following estimate for dI:
dI≃2.36×10−13|e|a0, (19)
believed to be about 15 % accurate.
It is of interest to compare dIwith the P-odd T-odd EDM of the Cs atom ob-
tained from a theoretical evaluation using the latest exper imental upper bound for
the electron EDM [21] :
|de| ≤7.5×10−19|e|a0.
Using for the cesium anti-screening factor the theoretical value [22]: 120 ±10,one gets
the following upper bound for the cesium EDM, namely the expe rimental sensitivity
to be reached for improving the existing bound on |de|:
|dCsEDM | ≤9.0×10−17|e|a0. (20)
We are going to use Eqs.(5) and (19) for calculating the linea r Stark shift. It is
interesting to note here that these equations predict also t he magnitude of the pv
transition dipole involved in an eventual Cs project which w ould be based on the
observation of hyperfine transitions in the Cs ground state, analogous to the potassium
project mentioned in the introduction (see also [17]). Ther efore both a project of this
kind and the linear Stark shift discussed here aim at the dete rmination of the same
physical parameter, dI, but only the observation of a dc Stark shift would prove its
static character.
4We use here the fact that, as noted by several authors, most of the sum ( ≈98%) comes from
the four states 6 P1/2,7P1/2,8P1/2,9P1/2.
5Note that a misprint in table IV of ref [20] has caused an inter change between the contents of
columns 1 and 2 of its lower half (entitled “7S perturbed”).
92 The linear Stark shift induced by V(2)
pv
2.1 Need for breaking the rotation symmetry of the atomic Ham iltonian
The parity conserving spin Hamiltonian in presence of a stat ic magnetic field /vectorB0is:
Hspin=A/vector s·/vectorI−gsµB/vector s·/vectorB0−γI/vectorI·/vectorB0. (21)
From section 2, we have seen that, to first order in the electri c field, the effect of
V(2)
pvin presence of an applied electric field can be described by th e following Stark
Hamiltonian:
Hst
pv=dI/vector s∧/vectorI·/vectorE≡ −dIi
2[/vectorF2, /vector s·/vectorE]. (22)
We have noted that the above identity implies the vanishing o f the average value of
Hst
pvin the low magnetic field limit. We are going to show that this n ull result still
remains valid for arbitrary values and orientations of the magnetic field.
To do this we consider the transformation properties of both HspinandHst
pvunder
the symmetry Θ, defined as the product of Treversal by a rotation of πaround
the unit vector ˆ u=/vectorE∧/vectorB0/|EB0|, the rotation R(ˆu,π). It should be stressed that
the rotation R(ˆu,π) and the symmetry Θ considered here are quantum mechanical
transformations acting only on the spin states. The externa l fields are considered as
realc−numbers and are not affected. One sees immediately that Hspinis invariant
under the symmetry Θ = TR(ˆu,π), whileHst
pvchanges sign. We conclude that, in
order to suppress the linear Stark shift cancellation we hav e to break the Θ symmetry.
This symmetry breaking can be achieved, for instance, by per turbing the atomic
S1/2state with a crystal field compatible with uniaxial symmetry along the unit vector
/vector n. A practical realization looks feasible, since it has been d emonstrated that Cs atoms
can be trapped in a solid matrix of helium having an hexagonal symmetry [23] (see
also section 3). In this case the alkali S state is perturbed b y the Hamiltonian6:
Hb(/vector n) =λb·(e2
2a0)/parenleftbigg
(/vector ρ·/vector n)2−1
3ρ2/parenrightbigg
, (23)
where both λbandρ=r/a0are expressed in atomic units. The perturbed atomic
state is now a mixture of S and D states, with no component of th e orbital angular
momentum along the /vector naxis. The spin Hamiltonian is modified and an anisotropic
hyperfine interaction is induced by the D state admixture. Th e new spin Hamiltonian
reads:
/tildewiderHspin=A⊥/vector s·/vectorI+ (A/bardbl−A⊥)(/vector s·/vector n)(/vectorI·/vector n)−gsµB/vector s·/vectorB0−γI/vectorI·/vectorB0. (24)
6It has been shown [23] that in the bubble enclosing the cesium atom there is a small overlap be-
tween the cesium and the helium orbitals. As a consequence, t he axially symmetric crystal potential
inside the bubble can be well approximated by a regular solut ion of the Laplace equation.
10It is easily verified that, if /vector nlies in the ( /vectorB,ˆu) plane, with non-zero components along
both/vectorBand ˆu, this perturbed atomic Hamiltonian is no longer invariant u nder the
transformation Θ.
Another possible method for breaking the symmetry of Hspinwill be presented in
section 4.
2.2 Strong magnetic field limit ( γsB0≫A⊥,A/bardbl)
The anisotropy axis is defined as:
/vector n= cosψˆz+ sinψˆx. (25)
Let us consider the nuclear spin Hamiltonian associated wit h the restriction of
Hspinto the electronic eigenstate E( ˜ns,ms) perturbed by the quadrupolar potential
Hb(/vector n):
Heff
(ms)=A⊥msIz+ms(A/bardbl−A⊥)(sinψcosψIx+ cos2ψIz) +γsB0ms−γIB0Iz
=ms[γsB0+Iz(A⊥sin2ψ+A/bardblcos2ψ−γIB0
ms) +Ix(A/bardbl−A⊥) sinψcosψ]
Heff
(ms)is identical to the Hamiltonian seen by an isolated nucleus c oupled to an effective
magnetic field, /vectorBeff(ms), having the following components:
Beff
x=−(A/bardbl−A⊥) sinψcosψms
γI
Beff
y= 0
Beff
z=B0−(A⊥sin2ψ+A/bardblcos2ψ)ms
γI, (26)
or equivalently :
Beff
x=Beffsinα, Beff
y= 0, Beff
z=Beffcosα,
where
tanα=(A/bardbl−A⊥) sinψcosψm s
−γIB0+ (A⊥sin2ψ+A/bardblcos2ψ)ms.
In other words, the direction ˆ zeffof/vectorBeffcan be deduced from the ˆ zaxis by a
rotation R(ˆy,α) by an angle αaround the ˆ yaxis. Hence, the eigenstates of Heff
(ms)are
|ms˜mI>, where ˜mInow stands for the z-component of the spin/tildewider/vectorIresulting from /vectorI
through the rotation R(ˆy,−α).
/tildewideIz=/tildewider/vectorI·ˆz=/vectorI· R(ˆy,α)ˆz= cosαIz+ sinαIx.
11We can now compute the linear Stark shift associated with the Hamiltonian Hpv
stgiven
by Eq.(22), supposing the /vectorEfield directed along the ˆ yaxis:
∆Est=∝angbracketleftms˜mI|dIE szIx|ms˜mI∝angbracketright
=dIE m s∝angbracketleft˜mI|Ix|˜mI∝angbracketright
=dIE m s∝angbracketleftmI|I· R(ˆy,α)ˆx|mI∝angbracketright
=−dIE m smIsinα. (27)
If we suppose γIB0≪A/bardbl, A⊥≪ |γs|B0and|A/bardbl−A⊥| ≪A/bardbl+A⊥, we obtain:
tanα≈A/bardbl−A⊥
1
2(A/bardbl+A⊥)sinψcosψ≈sinα,
which yields the simplified expression:
∆Est=−dIE m smIA/bardbl−A⊥
A/bardbl+A⊥sin 2ψ. (28)
In this approximation, ∆ Estcan be considered as a modification of the hyperfine
constant linear in the applied electric field.
In order to show up the transformation properties of ∆ Est, it is useful to express
this last result in terms of the two fields, /vectorEand/vectorB, and the unit vector ˆ nwhich defines
the anisotropy axis:
∆Est=−2dImsmI(ˆn·/vectorB0)(ˆn·/vectorE∧/vectorB0)
/vectorB2
0A/bardbl−A⊥
A/bardbl+A⊥. (29)
From this expression, it is clearly apparent that the linear shift breaks space reflexion
symmetry but preserves time reversal invariance. It differs from the P and T violating
linear Stark shift arising from an electron EDM by the fact th at it cancels out when
the quadrupolar anisotropy of the ground state vanishes. It is also obvious from Eq.
(29) that, in the strong field limit, the size of the Stark shif t depends only on the
orientation of /vectorBrelative to /vectorEand/vector nand not on the strength of the magnetic field.
Figure 2 represents two mirror-image configurations of the e xperiment.
2.3 Limit of low magnetic fields and small anisotropy
We now consider the limit |A/bardbl−A⊥| ≪γsB0≪A⊥,A/bardbl.
The linear Stark shift can be computed by using second order p erturbation the-
ory.Hspinis perturbed by both Hst
pvandHb(/vector n), the latter being responsible for the
12BE
o
^nBE
o^n
Figure 2: Two mirror-image and T-reversal symmetric experi mental configurations corresponding
to opposite values of the pseudoscalar (ˆ n·/vectorB0)(ˆn·/vectorE∧/vectorB0)//vectorB2
0.
anisotropy contribution to Hspin, i.e. (A/bardbl−A⊥)(/vector s·ˆn)(/vectorI·ˆn). The fields /vectorB0and/vectorEare
still taken parallel to ˆ zand ˆyrespectively. We find:
∆Est(F,M) = 2(A/bardbl−A⊥)dIEcosψsinψ×
×/summationdisplay
F′/negationslash=F,M′∝angbracketleftF M|szIx+sxIz|F′M′∝angbracketright∝angbracketleftF′M′|szIx−sxIz|F M∝angbracketright
EFM−EF′M′.
Since the operator /vector s∧/vectorIis identical to the commutator [ /vectorF2,−i
2/vector s], we see that only
the hyperfine states F′∝negationslash=FwithM′=M±1 contribute to the sum. Therefore,
in the energy denominator we can neglect the Zeeman contribu tion which is small
compared to the hyperfine splitting and, in the sum, we can fac torize out the energy
denominator 2( F−I)A/bardbl(I+1
2). SinceF′=Fdoes not contribute, the resulting sum
can be performed using a closure relation:
∆Est(F,M) =(A/bardbl−A⊥)dIE
2(F−I)A/bardbl(I+1
2)sin 2ψ∝angbracketleftF M|(szIx+sxIz)(szIx−sxIz)|F M∝angbracketright.(30)
Using standard properties of spin 1 /2 matrices, we can transform the diagonal matrix
element above into1
4∝angbracketleftF M|(I2
x−I2
z)|F M∝angbracketright. Once taken into account the axial sym-
metry of the unperturbed atomic state, it still simplifies to1
8∝angbracketleftF M|(/vectorI2−3I2
z)|F M∝angbracketright.
We arrive at the final expression:
∆Est(F,M) =k(F,M)A/bardbl−A⊥
A/bardbl+A⊥dIEsin 2ψ, (31)
where
k(F,M) = 2(F−I)∝angbracketleftF M|1
2(/vectorI2−3I2
z)|F M∝angbracketright/(2I+ 1). (32)
The Stark shift coefficients k(F,M) for133
55Cs (I=7/2) are listed in Table 1.
We note that ∆ Estdepends on M2so the linear Stark shifts of the Zeeman
splittingsE(F,M)−E(F,M−1) have opposite signs for M > 0 andM < 0 (see
figure 3):
E(4,M)−E(4,M−1) = ¯hωsM/(2I+ 1) + ∆Est(4,|M|)−∆Est(4,|M−1|).
13-4-3-2-10123M=4
-4-3-2-10123M=4
-2
-1
0
1
2
3M=-3
-2
-1
0
1
2
3M=-3
Figure 3: Energies of the different F, M states of the ground co nfiguration of133Cs showing the
linear Stark shift largely magnified. The two figures corresp ond to situations realizing opposite signs
of the pseudoscalar Pdefined in the text (Left: P>0; right: P<0). Top: F=4; bottom: F=3.
14|M| 0 1 2 3 4
k(4,|M|)15/16 51/64 3/8 -21/64 -21/16
k(3,|M|)-15/16 - 45/64 075/64
Table 1: Linear Stark shift coefficients k(4, M) and k(3, M) of t he different F, M substates of the
natural cesium ground state
As expected, once again the pseudoscalar
P=A/bardbl−A⊥
A/bardbl+A⊥(ˆn·/vectorB0)(ˆn·/vectorE∧/vectorB0)
B2
0, (33)
plays an essential role. If P>0, there is a contraction of the Zeeman splittings
belonging to the F=4 hyperfine state for positive values of Mand a dilatation for
negative ones, as shown by Figure 3. The situation is reverse d when the sign of P
is changed. In the F=3 hyperfine state, splitting contractio n also occurs for M > 0
withP>0 and forM <0 with P<0. This behavior could help to discriminate the
linear Stark shift induced by the nuclear helimagnetism fro m spurious effects. The
largest shift between two contiguous sublevels is expected to occur for the couple of
statesF= 3,M=|3| →F= 3,M=|2|.
From Table 1 and Eqs. (19) and (31) we predict:
∆Est(3,3)−∆Est(3,2) =75
64A/bardbl−A⊥
A/bardbl+A⊥sin 2ψdI·E , (34)
with75
64dI≃2.76×10−13|e|a0.
As in the strong field limit, we note that the size of ∆ Estdepends only on the
direction of /vectorB.
2.4 Analogy between this shift and the PV energy shift search ed for in enantiomer
molecules
We would like to stress that from the point of view of symmetry considerations there
exists a close analogy between the linear Stark shift induce d by the anapole moment
and the energy shift which is searched for in enantiomer mole cules [24]. Indeed in
the present configuration the three vectors /vectorE,/vectorBand/vector nwhich are non-coplanar are
sufficient to place the atom in a chiral environment similar to that experienced by an
atomic nucleus inside a chiral molecule. Between two mirror -image environments an
energy difference is predicted exactly like between two mirr or-image molecules.
153 Experimental considerations and order of magnitude estim ate
We now consider an experimental situation which looks like a possible candidate for
the observation of the linear Stark shift discussed in the pr evious sections. It has been
demonstrated experimentally [25] that cesium atoms can be t rapped in solid matrices
of4He. At low pressures, solid helium cristallizes in an isotro pic body-centered cubic
(bcc) phase, but also in a uniaxial hexagonal close packed (h cp) phase. Optically
detected magnetic resonance has proved to be a sensitive too l for investigating the
symmetry of the trapping sites. The group of A. Weis has repor ted the observation
in the hexagonal phase of zero-field magnetic resonance spec tra and magnetic dipole-
forbidden transitions which they interpret in terms of a qua drupolar distorsion of the
atomic bubbles [26]. Particularly relevant here is their ob servation of the matrix-
induced lifting of the Zeeman degeneracies in zero field. Thi s is attributed to the
combined effect of two interactions, the quadrupolar intera ction of the form Hb(/vector n) =
λb/parenleftBig
(/vector ρ·/vector n)2−1
3ρ2/parenrightBig
between the cesium atom and the He matrix on the one hand,
and the hyperfine interaction in the Cs atom on the other.
Provided that /vectorF2is still a good quantum number, it is easily shown from genera l
symmetry considerations, that the anisotropy of the hyperfi ne interaction induced by
the (hcp) crystal potential can be represented, within a giv en hyperfine multiplet, by
the effective perturbation:
Heff=Ceff(F)·((/vectorF·ˆn)2−1
3/vectorF2).
The constants Ceff(F) can be easily related to the anisotropic hyperfine constant s
appearing in the spin hamiltonian/tildewiderHspinintroduced in Eq.(23) of the previous section:
Ceff(F=I±1/2) =±(A/bardbl−A⊥)/8.
In a uniaxial crystal, when the atoms are optically polarize d along the crystal axis in
the absence of external magnetic fields, the lifting of the de generacy between Zeeman
sublevels induced by Heffshould make it possible to drive magnetic resonance tran-
sitions between these levels. One would expect to deduce the hyperfine anisotropy
from the observed spectra. At first sight, the zero-field magn etic resonance spectra
observed by Weis et al. would seem to match this prediction. However, their ex-
periment has been performed in a polycristalline (hcp) samp le. The effects observed
in this situation result from averaging over the distributi on of the microcrystal axes.
For each microcrystal, there exists a quantization axis, ˆ z, which diagonalizes the hy-
perfine level density matrix. Immediately a question arises as to the direction of the
quantization axis ˆ zwith respect to the microcrystal symmetry axis /vector n. If the popu-
lation differences resulted, say, from the Boltzmann factor , then ˆzwould be along /vector n,
since in the zero magnetic field limit there is no other prefer red direction. In such
16a situation, there would be no difference between the spectra for a polycrystal and
a monocrystal. But in the experimental situation considere d here, the population
differences are induced by an optical pumping mechanism whic h provides a second
preferred direction: the direction of the photon angular mo mentum along /vectork. The
microcrystal density matrix is then expected to keep some me mory of the direction
of/vectork. So, two directions /vector nand/vectorkcompete in the determination of the quantization
axis ˆz. To proceed further, we consider the extreme case where ˆ zis taken along /vectork,
together with an assumed isotropic distribution of microcr ystal axes. It is then easily
seen that the lines associated with the hyperfine anisotropy Heffcollapse into a single
asymmetric line when the average is performed over the polyc rystal. Clearly, one at
least of the two preceeding assumptions is too drastic, most likely the isotropy of the
/vector ndistribution. It is indeed likely that the optical pumping p rocess is more efficient
for microcrystals having a preferred orientation with resp ect to the photon angular
momentum. Such a selection mechanism would then lead to an eff ective anisotropic
distribution of /vector n, and in this way a spectrum of separated lines can be recovere d.
From the above qualitative considerations, it clearly foll ows that the final interpreta-
tion of the the zero-field resonances requires a detailed ana lysis of the optical pumping
process for Cs atoms trapped inside deformed bubbles of arbi trary orientation. The
corresponding theoretical investigation is currently und erway in A. Weis’s group.
Meanwhile, to plan any experiment, we still need to know abou t the physical origin
and the magnitude of the ratioA/bardbl−A⊥
A/bardbl+A⊥, which governs the magnitude of the electroweak
linear Stark shift. We are going to present now the result of a n investigation which
has led us both to a physical understanding and a reasonably a ccurate estimate of
the sought after parameter. We have chosen to devote an appen dix to a detailed
description of our semi-empirical approach, which consist s in relating the hyperfine
anisotropy to another measured physical quantity. Here we s hall give a brief summary
of our procedure and present the final result.
We start from the remark that there really does exist a mechan ism able to generate
an hyperfine anisotropy to first order in the “bubble” Hamilto nianHb(/vector n). ThenD3/2
state is indeed mixed to the 6 S1/2state under the effect of Hb(/vector n), and we note then
that the hyperfine interaction has non-zero off-diagonal mat rix elements between S1/2
andD3/2states. In fact, it has been shown previously [27] that the ∝angbracketleftnS1/2|Hhf|n′D3/2∝angbracketright
matrix elements are not easy to calculate, because they are d ominated by the con-
tribution coming from many-body effects, due to the existenc e of an approximate
selection rule which suppresses the single particle matrix element. However, as we
show in the appendix, the variation of the matrix elements ∝angbracketleftn′S1/2|Hhf|n′′D3/2∝angbracketrightwith
respect to the binding energies En′S1/2andEn′′D3/2, -expressed in Rydberg- can be
reasonably well predicted in the limit |En′S1/2|,|En′′D3/2| ≪1. In this way, we are
left with a single parameter which can be deduced from the emp irical knowledge of
17another physical quantity involving the same matrix elemen ts. We have in mind the
quadrupolar amplitude Ehf
2induced by the hyperfine interaction which is present in
the cesium 6 S→7Stransition in the absence of a static electric field [28]. In o rder to
show the relation between the quantities A/bardbl−A⊥andEhf
2, we express them explicitly
in terms of the matrix elements M(n′,n) given by:
M(n′,n) =/summationdisplay
n′′∝angbracketleftn′S1/2|Hhf|n′′D3/2∝angbracketright∝angbracketleftn′′D3/2|ρ2|nS1/2∝angbracketright
En′S1/2− En′′D3/2. (35)
The basic formula used in our numerical evaluation of A/bardbl−A⊥can be cast in a very
compact form:
A/bardbl−A⊥=−4λb
∆E2M(6,6)
M(7,6) +M(6,7)a3(7,6)Ry (36)
where ∆ Eis the energy of the 6 S→7Stransition and a3∝Ehf
2/µBis the empirical
quadrupolar amplitude (see Eq. (A.2) for a precise definitio n). A second empirical
input is used to determine the coupling constant λb: this is the S−Dmixing coefficient
which is obtained from the hyperfine frequency shifts measur ed by Weis et al. for Cs
atoms trapped either in the (bcc) or the (hcp) phases [26] in t he low magnetic field
limit. The ratio involving the matrix elements M(n′,n) is evaluated in the appendix,
using the approximation scheme sketched above. Its absolut e value is found to lie
close to unity. Let us quote now the final result given by our se mi-empirical method7
described in the appendix: |A/bardbl−A⊥
A/bardbl+A⊥|= 1.07×10−3. The uncertainty is believed not to
exceed 20%.
For observing the electroweak linear Stark shift discussed in the present paper, it
is important to work with a uniaxial hexagonal crystal. Inde ed, in a polycristalline
phase, where the individual crystals are oriented totally a t random, the average value
of the pseudoscalar Ptaken over the isotropic distribution of ˆ nis expected to be
suppressed and thus is the Stark shift computed in the previo us section. Although
trapping of cesium atoms has not yet been achieved in a monocr ystalline hexagonal
phase, the prospect does not look unfeasible [29] and a deter mination of the magni-
tude of the hyperfine anisotropy appears to be the first step to be achieved. Using
|A/bardbl−A⊥
A/bardbl+A⊥|= 1.07×10−3and Eq.(34), we find that the effective P-odd T-even electric
dipole moment of the trapped cesium atoms associated with th e nuclear anapole mo-
ment reaches 2 .96×10−16|e|a0. For comparison, it is interesting to note that this is
about three times as large as the Cs EDM limit (Eq.20) to be mea sured on unper-
turbed Cs atoms for improving our present knowledge about a p ossible P-odd T-odd
EDM of the electron.
7This method can be seen as a generalization of that used in sec .1.2 to evaluate the static dipole
starting from the empirical knowledge of the transition dip ole.
184 Breaking the free atom symmetry by application of a nonreso nant
radiation field
In this last section we want just to mention another possibil ity for breaking the atomic
Hamiltonian rotation symmetry by other means than static un iform electric and mag-
netic fields. We have in mind the application of a strong nonre sonant radiation field
which generates an anisotropic electron gyromagnetic rati o. In the presence of an
external magnetic field /vectorBit has been shown [30] that the effect of the nonresonant
radiation field can be described by the introduction of an effe ctive magnetic field:
/vectorB′=/parenleftBig
g⊥/vectorB+ (g/bardbl−g⊥)ˆn·/vectorBˆn/parenrightBig
//radicalBig
(g2
/bardbl+g2
⊥),
where ˆndefines the direction of polarization of the radiation field, g⊥=gFandg/bardbl=
gFJ0(ω1/ω),J0is the zero-order Bessel function, and ω1is the Rabi angular frequency
associated with the radiation field. The above formula sugge sts the existence of a
uniaxial symmetry, but it is valid only within an atomic hype rfine multiplet. It is
clear that the “dressing” by a nonresonant radiation field off ers new possibilities for
placing the atoms in a quadrupolar environment. However, it is important to bear in
mind that at least two stringent requirements must be satisfi ed if one wants to detect
an electroweak Stark shift in the ground state. First, the un iaxial perturbation has
to mix the two hyperfine substates, otherwise the matrix elem ent ofHst
pvcancels.
Second, it is imperative to avoid a broadening of the transit ion lines for allowing
precise frequency measurements. We are currently investig ating how to achieve the
proper conditions in a realistic way.
Conclusion
This paper investigates a way to get around the well known no- go theorem:
no linear Stark shift can be observed in a stationary atomic s tate unless T reversal
invariance is broken.
The perturbation of an atom by the nuclear spin-dependent pa rity-odd potential
generated by the nuclear anople moment leads to a static elec tric dipole moment
dI/vector s∧/vectorI, which clearly is T-even. However, if one considers an atom p laced in arbitrarily
oriented electric and magnetic uniform static fields /vectorB0and/vectorE0, the quantum average
/vectorE0· ∝angbracketleft/vector s∧/vectorI∝angbracketrightis found to vanish. This can be understood by noting that /vector s∧/vectorI·/vectorE0is
odd under the quantum symmetry transformation Θ defined as th e product of the
time reflexion Tby a space rotation of πabout an axis normal to a plane parallel to
the fields/vectorB0,/vectorE0, while the atomic hamiltonian stays even. Our strategy to ob tain a
linear Stark shift is to break the Θ symmetry while keeping T i nvariance.
As a possible practical realization of such a situation, we h ave studied the case of
ground state Cs atoms trapped in a uniaxial (hcp) phase of sol id4He, which has been
19recently the subject of detailed spectroscopic studies [26 ]. The required breaking of
space rotation is provided by the uniaxal crystal field. As a r esult of the deformation
of the atomic spatial wave function the hyperfine interactio n acquires an anisotropic
part, which plays an essential role in the determination of t he size of the linear Stark
shift. We have performed a numerical estimation of the hyper fine anisotropy, believed
to be accurate to the 20% level, using a semi-empirical metho d. We use as an input
the recent experimental measurement of the E2amplitude of the 6 S1/2→7S1/2
transition induced in cesium by the hyperfine interaction. W e arrive in this way at a
numerical evaluation of the linear Stark shift induced by th e nuclear anapole moment:
the expected effect is found to be about three times the experi mental upper limit to
be set on the T-odd Stark shift of free Cs atoms for improving t he present limit on
the electron EDM.
Besides the obvious remark that the T-even Stark shift studi ed here could be
a possible source of systematic uncertainty in EDM experime nts designed to reach
unprecedented sensitivity[31, 21, 32, 33], we believe that there are strong physical
motivations for measuring the Stark shift itself. First, it would lead to a direct
measurement of the nuclear anapole moment in absence of any c ontribution coming
from the dominant PV potential due to the weak nuclear charge . It would also provide
an evidence for a truly static manifestation of the electrow eak interaction, something
which is still lacking. Second, this experiment would rely o n the measurement of
frequency shifts rather than transition amplitudes. While transition probabilities are
difficult to measure very accurately, high precision measure ments of frequency shifts
have already been achieved.
Acknowledgements
We thank Ph. Jacquier for continuous interest in the subject of this work and his
encouragements. We acknowledge many stimulating discussi ons with A. Weis and
S. Kanorsky. We are grateful to M. Plimmer and J. Gu´ ena for ca reful reading of the
manuscript.
This work has been supported by INTAS (96-334).
20APPENDIX: Semi-empirical calculation of the hyperfine anis otropy of Cs
atoms trapped inside a4He hexagonal matrix
In this appendix we present our evaluation of the hyperfine st ructure anisotropyA/bardbl−A⊥
A/bardbl+A⊥
resulting from the matrix induced bubble deformation of qua drupolar symmetry, a
quantity frequently referred to in this paper.
1. Two processes induced by hyperfine mixing
Our approach is based on the fact that hyperfine mixing plays q uite similar roles
in two different processes. The first process concerns the Cs 6 S→7Squadrupolar
transition amplitude in zero electric field while the second process deals with the
parameterA/bardbl−A⊥
A/bardbl+A⊥.
We start by rewriting the standard mixed M1−E2transition operator in atomic
units:
TM1+E2= (/vector ǫ∧/vectork)·/vectorM
µB−i1
2∆E(/vector ρ·/vector ǫ)(/vector ρ·/vectork), (1)
where/vector ρis the electron coordinate in Bohr radius unit and ∆ Eis the transition
energy expressed in Rydberg unit8. We are first going to study the perturbation
effect onTM1+E2caused by the hyperfine interaction Hhf. This phenomenon has
been observed experimentally in the forbidden 6 S1/2→7S1/2transition. It provides
a useful calibration amplitude in cesium parity violation e xperiments. To analyse the
experimental results, it was found convenient, to introduc e the effective transition
operatorThfacting upon the tensor products of the electron spin and nucl ear spin
states:
Thf=ia2(n′,n) (/vector s∧/vectorI)·(/vector ǫ∧/vectork) +ia3(n′,n) ((/vector s·/vectork)(/vectorI·/vector ǫ) + (/vector s·/vector ǫ)(/vectorI·/vectork)).(2)
The second physical process to be analysed in this section is not at first sight
closely connected but happens to be described by the same for malism. This will allow
us to establish a very useful connection between measuremen ts coming from rather
different experimental situations. Recently optical pumpi ng has been observed with
cesium atoms trapped inside an hexagonal matrix of solid hel ium [26]. Among the
new effects to be expected, we have seen earlier in this paper t hat the existence of an
anisotropic hyperfine structure opens the possiblity of obs erving a linear Stark shift
induced by the nuclear anapole moment, an effect which cannot exist for an atom in
a spherically symmetric environment. It is known that in the bubble enclosing the
8The phase difference, π/2, between the two amplitudes expresses the fact that the mag netic
moment /vectorMand the quadrupole operator behave differently under time re flexion: the first is odd,
while the second is even.
21cesium atom there is a small overlap between the cesium and th e helium orbitals [23].
As a consequence, the axially symmetric crystal potential i nside the bubble can be
well approximated by a regular solution of the Laplace equat ion:
Hb(/vector n) =λb(e2
2a0)/parenleftbigg
(/vector ρ·/vector n)2−1
3ρ2/parenrightbigg
. (3)
The perturbation of the hyperfine interaction by the bubble q uadrupole potential
Hb(/vector n) induces an anisotropic hyperfine structure for cesium nS1/2states. This is
described by the effective Hamiltonian:
Hanis
hf= (A/bardbl−A⊥)/parenleftbigg
(/vector s·/vector n)(/vectorI·/vector n)−1
3/vector s·/vectorI/parenrightbigg
. (4)
We present now the basic formulas which allow the computatio n of the parameters
relevant for the two physical problems in hand. They will be g iven in such a way as
to exhibit their close anology. We have chosen to use the Dira c equation formalism.
Besides the fact that formulas are more compact, it is well kn own that relativistic
corrections play an important role in cesium hyperfine struc ture computation. Ne-
glecting the contribution of the quadrupole nuclear moment of the Cs nucleus9, the
hyperfine hamiltonian is written as:
Hhf=/vectorI·/vectorA, (5)
/vectorA=Chf/vector α∧/vector ρ
ρ3+δ/vectorA(1)(/vector ρ,/vector ρ′) +.... (6)
The first term gives the hyperfine interaction of the valence e lectron treated as a Dirac
particle; the second represents the non-local modification of the hyperfine interaction
induced by the excitation of core electron-hole pairs to low est order and the dots
stand for higher order contributions10. It has been shown in reference [27] that the off-
diagonal matrix element ∝angbracketleftn′′D3/2|/vector α∧/vector ρ
ρ3|nS1/2∝angbracketrightis strongly suppressed by an approximate
selection rule which does not apply to the many-body non loca l operatorδ/vectorA(1)(/vector ρ,/vector ρ′).
An evaluation of the latter contribution led to a semi-quant ative agreement with
the experimental measurements of the ratio a3(7,6)/a2(7,6), while the single particle
result is too small by about two orders of magnitude.
9As shown in ref [28], the quadrupole contribution for133Cs plays a negligible role in the effects
discussed in this appendix.
10An explicit construction of δ/vectorA(1)(/vector ρ, /vector ρ′) together with a resummation of an infinite set of higher
order terms, within the many body field theory formalism, is g iven in reference [34]. See also [35]
for more advanced analysis.
22To obtain an estimate of the ratio ( A/bardbl−A⊥)/a3(7,6) it is convenient to introduce
the cartesian tensor operator Ti1i2i3(E). This object appears naturally in the lowest-
order pertubation expressions for the quantities of intere st:
Ti1i2i3(E) =Ai1G+
3/2(E) (ρi2ρi3−1
3δi2,i3ρ2), (7)
where the indices i1, i2, i3take any value between 1 and 3. The scalar operator
G+
3/2(E) is the atomic Green function operator restricted to the sub space ofD3/2
configurations (total atomic angular momentum J=3/2 and pos itive parity). We now
proceed to isolate in Ti1i2i3(E) the part transforming as a vector; this is the only part
to survive after the operator is sandwiched between the proj ectorsP(n′S1/2) and
P(nS1/2). This operation is achieved by a decomposition of Ti1i2i3(E) into a traceless
tensor ¯Ti1i2i3(E) and a remainder [36]:
Ti1i2i3(E) = ¯Ti1i2i3(E) +3
10(δi1,i2Tααi3(E) +δi1,i3Tααi2(E))
−2
10δi2,i3Tααi1(E), (8)
where we have used the fact that Tααi=TαiαandTiαα= 0. It is a simple matter
to verify from the above equation that we have indeed ¯Tααi3=¯Tαi2α=¯Ti1αα= 0.
The fully symmetric part of the traceless tensor ¯TS
i1i2i3(E) is easily identified with
an octupole spherical tensor having seven independant comp onents. By a simple
counting argument, the left over term is seen to have five comp onents; it is to be
identified with the quadrupole tensor which appears in the fu ll decomposition of
Ti1i2i3(E) into irreducible representations of the rotation group O(3). Let us have a
look at the vector operator, /vectorV, the components of which appear in the right hand
side of Eq.(A.8):
/vectorV= (/vectorAG+
3/2(E)·/vector ρ)/vector ρ−1
3/vectorAG+
3/2(E)ρ2.
The second term in the above expression does not contribute w hen it acts upon an
nS1/2state so, we are led for our purpose to introduce the vector op erator/vectorT(n′,n)
/vectorT(n′,n) =3
10P(n′S1/2)/parenleftBig/vectorAG+
3/2(Ei)·/vector ρ)/vector ρ+ (h.c,E f→Ei)/parenrightBig
P(nS1/2)
=γ(n′,n)/vector s, (9)
whereEfandEiare respectively the binding energies of the n′S1/2andnS1/2atomic
states. The second line of the above equation follows direct ly from the Wigner-Eckart
theorem applied to a vector operator.
23In order to calculate a3(n′,n) we have to perform the contraction of Ii1ǫi2ki3with
the tensor:
Fi1i2i3=P(n′S1/2) (Ti1i2i3(Ef) + (h.c.,E f→Ei))P(nS1/2).
Using Eq.(A8) and (A9) ,Fi1i2i3can be cast into the simple form :
Fi1i2i3=γ(n′,n)/parenleftbigg
δi1,i2si3+δi1,i3si2−2
3δi2,i3si1/parenrightbigg
.
The required index contraction with the tensor Ii1ǫi2ki3is now easily performed and
one obtains directly a3(n′,n), up to a prefactor whose value is found by identification
with Eq.(A.1):
a3(n′,n) =−1
2∆Eγ(n′,n). (10)
To calculate the hyperfine anisotropy A/bardbl−A⊥, we follow the same lines but this time
the contraction involves the tensor Ii1/parenleftBig
ni2ni3−1
3δi2,i3/parenrightBig
, the prefactor is fixed by com-
parison with Eq.(A.3) and the exchange i2↔i3leads to two identical contributions.
Hence,
A/bardbl−A⊥= 2λb(e2
2a0)γ(n,n) (11)
=−(e2
2a0)4λb
∆Eγ(n,n)
γ(n′,n)a3(n′,n). (12)
The expression (A.12) looks to us a good starting point for nu merical evaluation
ofA/bardbl−A⊥: besides the fact that several sources of uncertainties in t he evaluation
ofγ(n′,n) are eliminated in the ratioγ(n,n)
γ(n′,n), it lends itself to the use of empirical
information. One may note, here, a certain similarity with E q. (7) of sec 1.2 used for
the evaluation of the permanent dipole, dI.
2. Numerical evaluation
We proceed now to a numerical evaluation of A/bardbl−A⊥in three steps, starting from
formula (A.12).
The numerical value of the 6 S1/2→7S1/2quadrupole amplitude a3(7,6) is readily
obtained from measurements [28, 2] of the ratio
a3(7,6)/a2(7,6) =E2/Mhf
1= (5.3±0.3)×10−2,
24combined with a precise theoretical evalution of the magnet ic dipole amplitude11
a2(7,6) =−Mhf
1
2µB=−0.4047±4×10−4. We obtain finally:
a3(7,6) = (2.14±0.12)×10−7. (13)
The second step is the numerical estimate of the ratio γ(6,6)/γ(7,6). This is
more delicate and requires an assumption which has been show n to work in similar
situations. To begin with, we have addressed the question12of the origin and size of
the variations of the off-diagonal matrix elements ∝angbracketleftnS1/2|Hhf|n′′D3/2∝angbracketrightwith the radial
quantum numbers nandn′′. It is of interest to remind that a very precise answer to
this question has been already obtained in the case of cesium single particle matrix
elements ∝angbracketleftnLJ|Hsp
hf|n′LJ∝angbracketrightwithL= 0 or 1 and with nandn′referring to the radial
quantum numbers of any pair of valence states. For simplicit y, we are going to express
the answer within a non-relativistic formalism, but it shou ld be borne in mind that
all of what is said holds true within a relativistic framewor k. It is convenient to
introduce the notion of overline matrix elements such as tho se computed with radial
wave functions Rnlj(ρ) which have a starting coefficient at the origin equal to unity
instead of a unit norm13. More explicitly we can write:
∝angbracketleftnLJ|Hsp
hf|n′LJ∝angbracketright=∝angbracketleftnLJ|Hsp
hf|n′LJ∝angbracketright
AnljAn′lj, (14)
whereAnlj= lim ρ→0ρ−lRnlj(ρ) is the starting coefficient of the space normalized
wave function. (In the relativistic case the above conditio n is replaced by energy
independent boundary conditions imposed on the Dirac radia l wave functions at the
nuclear radius). It was found in references [27, 37] that the overlined matrix elements
are independent of the valence orbital radial quantum numbe rsnandn′to better than
10−4forS1/2states and better than 10−3forP1/2states. This result is understood by
noting that, in the domain of the ρvalues relevant for the evalutation of the matrix
elements of/vector α∧/vector ρ
ρ3forS1/2andP1/2states, the potential energy is larger than valence
binding energies by more than three orders of magnitude. Thi s implies that, in this
domain, the overlined radial wave functions have no depende nce upon the binding
energy or equivalently upon the radial quantum numbers of th e valence orbitals .
11The theoretical method used to get Mhf
1is based upon the factorization rule: ∝angbracketleft6S|Hhf|7S∝angbracketright=/radicalbig
∝angbracketleft6S|Hhf|6S∝angbracketright∝angbracketleft7S|Hhf|7S∝angbracketright. This rule was first established with an accuracy of a few part s in 103in
ref.[27]. It has been confirmed by a direct many-body relativ istic computation [38] of ∝angbracketleft6S|Hhf|7S∝angbracketright,
accurate to the 1% level. More recently the validity of the ru le has been pushed to the level of a
fraction of 10−3[39].
12Arguments similar to those given below and in references [27 , 37] are developed in [39].
13The wave function Rnlj(ρ) is known to be an analytic function of the energy. This prope rty is
the starting point of the quantum defect theory.
25The above argument has to be reconsidered for the lowest orde r many body correc-
tion involving the matrix element of the non local operator: δ/vectorA(1)(/vector ρ,/vector ρ′). The relevant
domain ofρvalues is now determined by the “radii” of the core outer orbi tals involved
in the computation, which in the case of S1/2andP1/2matrix elements are 5 s,5p,
while in the case of the off-diagonal matrix element ∝angbracketleftnS1/2|δ/vectorA(1)|n′D3/2∝angbracketrightonly 5pis
relevant. We measure the variation of the overlined matrix e lements ∝angbracketleftnLJ|Hmb
hf|n′L′
J′∝angbracketright
with the valence state binding energies EnLJby the parameters δLJdefined as their
logarithmic derivative with respect to EnLJ, (hereHmb
hfstands for the many-body mod-
ification to the hyperfine interaction). From results of refe rences [27, 37], we can infer
the relative variation of ∝angbracketleftnL1/2|δ/vectorA(1)|n′L1/2∝angbracketrightforL= 0,1 and we arrive to the values
δ(1)
S1/2=−0.12 andδ(1)
P1/2=−0.30. The fact that −δ(1)
P1/2is about three times larger
than−δ(1)
S1/2is coming from the fact that Pstate binding energies have to be com-
pared with the potential energy minus the centrifugal energ y. Let us, now, consider
the more difficult case of the S-D off-diagonal matrix elements ∝angbracketleftn′S1/2|δ/vectorA(1)|n′′D3/2∝angbracketright.
The corresponding parameter δ(1)
S1/2is expected to be somewhat larger in absolute
value, due to the fact that the relevant 5 porbital is less tightly bound than the 5 s
orbital which gives the dominant contribution to the S1/2diagonal matrix element.
The relative variation versus the D3/2energy is expected to be on the order of few
units, since the centrigugal barrier is three times higher t han in the case of Pstates.
This expectation is borne out by a preliminary estimate whic h givesδ(1)
D3/2∼ −3.
We proceed now to a numerical evaluation of the ratio ranis=γ(6,6)/γ(7,6),
leaving, for the moment, δS1/2andδD3/2as free parameters. As an intermediate step,
we compute the quantities M(n′,n), written as sums over the intermediate n′′D3/2
states :
M(n′,n) =/summationdisplay
n′′∝angbracketleftn′S1/2|Hhf|n′′D3/2∝angbracketright∝angbracketleftn′′D3/2|ρ2|nS1/2∝angbracketright
En′S1/2− En′′D3/2. (15)
The ratioranisis given in terms of M(n′,n) by the following formula :
ranis=γ(6,6)/γ(7,6) =2M(6,6)
M(7,6) +M(6,7). (16)
An explicit numerical computation of ranishas been performed according to the
following procedure. First, any binding energy independen t factor appearing in
M(n′,n) is dropped since it disappears in the ratio. This is indicat ed below by the
symbol ∝. The sum/summationtext
n′′appearing in M(n′,n) is limited to 5 ≤n′′≤8. The set of
the quadrupole matrix element ∝angbracketleftn′′D3/2|ρ2|nS1/2∝angbracketrightwere obtained by a relativistic ver-
sion of the Norcross model. In order to test the sensitivity o f the result to quadrupole
amplitudes, we have also used a set calculated by an extensio n of the Bates-Damgaard
method. The energy denominators are taken from experiment. The hyperfine matrix
26elements ∝angbracketleftn′S1/2|Hhf|n′′D3/2∝angbracketright, to second order in the binding energies are, given by
the following formulas:
∝angbracketleftn′S1/2|Hhf|n′′D3/2∝angbracketright ∝An′′D3/2An′S1/2×
/parenleftBig
1 +δS1/2En′S1/2+δD3/2En′′D3/2/parenrightBig
, (17)
AnLJ∝(−EnLJ)3
4. (18)
In formula (17), we have dropped, according to the above pres cription, the zero
energy limit of the overlined matrix element ∝angbracketleftn′S1/2|Hhf|n′′D3/2∝angbracketright. Equation (A.18)
follows from a result obtained in [16], where the Fermi-Segr ´ e formula was extended
to arbitrary orbital angular momentum states. For simplici ty we have ignored for
simplicity a factor involving the derivative of the quantum defects, which in the
present context would introduce few percent corrections.
We now have all the elements needed to calculate the sought af ter ratio:
ranis=γ(6,6)
γ(7,6)=−0.8173−0.0255δD3/2+ 0.1456δS1/2. (19)
The negative sign of raniscan be traced back to the fact that the 7 S1/2level lies just
in between 5 D3/2and 6D3/2levels. If we adopt the rough estimate given above : a
few tens of % for −δS1/2and a few units for −δD3/2, the first-order energy correction
remains well below the 10% level, due in part to a cancellatio n between the two
correcting terms. To reduce the absolute value of ranisby more than 10% would
require unrealistic values of −δD3/2so we believe the estimate of ranis=−0.82±0.10
to be reasonably safe14.
The final step in our evaluation of the hfs anisotropy is devot ed to the empirical
determination of the coupling constant λbappearing in front of the crystal electronic
potential. As experimental input we are going to use the hype rfine energy shift which
is observed for trapped cesium atoms, when one passes from th e cubic to the hexagonal
phase. This shift is attributed to the effect of the anisotrop ic bubble potential Hb(/vector n).
We shall ignore, for the moment, the possible contribution o f the anisotropic hyperfine
interaction Hanis
hfand assume that the shift is essentially due to the renormali zation
of the 6S1/2component of the atomic wave function by the admixtures αnD3/2of the
nD3/2states. The corresponding variation of the hyperfine splitt ingδWis then given
by :
δW
W=−/summationdisplay
n,J|αnDJ|2=−λ2
bJSD, (20)
14The validity of the procedure leading to this estimate has be en checked, to the 10% level, upon
a significative subset of the many-body Feynman diagrams con tributing to γ(n′, n).
27where we have isolated λ2
bby introducing the purely atomic quantity JSD. Let us write
down the explicit expression of JSD, neglecting spin-orbit coupling and assuming that
/vector nlies along the quantization axis:
JSD=/summationdisplay
n|∝angbracketleft6S|(cos2θ−1/3)ρ2|nD∝angbracketright/(E6S− EnD)|2. (21)
We limit the sum to nvalues ranging from 5 to 8. With the same radial quadrupole
matrix elements as before, we obtain the numerical value: JSD= 9512. Using the
empirical number given in ref [26],/radicalBig
−δW/W = 0.035, we arrive at the following
absolute value of the coupling constant λb(in Ry):
|λb|=/radicalBigg
−δW
JSDW= 0.000359. (22)
It should be pointed out that if the crystal axis /vector nis not aligned along the quantization
axis, one obtains values of JSDsmaller than the one quoted above, so the value of
|λb|should be considered, strictly speaking, as a lower bound. A t last, we have in
hand all the ingredients needed to perform a numerical evalu ation of |A/bardbl−A⊥|from
the formula (A.12) since ∆ E= 0.169 is taken directly from experiment:
|A/bardbl−A⊥|=4|λb|
∆E|γ(6,6)|
|γ(7,6)||a3(7,6)|Rydberg(MHz) = 4 .9 MHz (23)
= 1.07×10−3×(|A/bardbl+A⊥|). (24)
As a final topic, we should discuss the effect of the anisotropi c hyperfine interaction
itself on the the empirical splitting δW, since this could modify the value of |λb|
and so play a role in the assessment of the uncertainty affecti ng the result given by
Eq. (A.24). Due to this effect, the constant λbis no longer given by Eq. (A.22)
but rather by a second order equation where the linear term is associated with the
anisotropic hyperfine interaction. It is convenient to intr oduce the variable x=λb/λ0
b
withλ0
b=/radicalBig
−δW/(JSDW). The equation giving λbtakes then the simple form:
x2−2bx−1 = 0, where the coefficient bis given by the following formula:
b=(A/bardbl−A⊥)(0)
A/bardbl+A⊥W
δW∆F∝angbracketleftszIz−1
3/vector s·/vectorI∝angbracketright
2I+ 1.
The superscript(0)indicates that the hf anisotropy is given, up to a well defined sign,
by Eq. (A.24). The symbol ∆ Fmeans that one should take the difference between
the two hyperfine states of the quantum average over which it i s applied. To obtain
an over-estimate of bwe have assumed that optical pumping works at its maximum
28efficiency so that the microwave transition takes place betwe en the hyperfine levels
(4,4) and (3,3). In this case we obtain b= 0.20 and the two possible solutions for λb
are :
λ(±)
b=±3.6 10−4(1±0.2).
The actual experimental situation is expected to lie far fro m the extreme case consid-
ered here, so the difference between the two absolute values i s certainly smaller than
the upper limit given by the above calculation.
In conclusion, including all sources of uncertainties, we c onsider our evaluation
of Eq.(A.24), |A/bardbl−A⊥
A/bardbl+A⊥|= 1.07×10−3, as reliable within uncertainty limits of about
20%. However, if, during hyperfine shift measurements /vector nis not aligned along the
quantization axis, the central value of λb, and therefore that ofA/bardbl−A⊥
A/bardbl+A⊥, may be pushed
upwards.
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31 |
Mathematical Model of Shock Waves
Alexei Krouglov
1550 16th Ave., Bldg. F, 2nd Floor, Richmond Hill, Ontario L4B 3K9, Canada
Email: Alexei.Krouglov@siconvideo.com
This article represents the author’s personal view 2
ABSTRACT
Presented her e is the mathematical model describing the phenomenon of shock
waves. The underlying concept is based on the time -space model of wave propagation.
Keywords : Shock Waves, Wave Equation 31. Concept behind the Phenomenon
The first assertion is that a wave i s caused by the difference between energy’s
value at some point and “average” energy’s values (coined as an energy’s level) in the
“neighborhood” of this point. That difference is called energy’s discrepancy at this point,
which is wavily propagated both i n space and in time. Besides that a wave travels with a
finite speed (see [1]).
The second assertion is that additionally to wavy propagation the energy’s
discrepancy spreads directly as well. But direct spreading decelerates quickly whereas
wave may propa gate on a big distance.
The conclusion is that when speed of direct spreading of energy’s discrepancy
exceeds the speed of wave propagation, the wave acquires an additional energy while
travels in space. And we have to apply the wave’s attenuation as well.
Therefore an accumulation of energy’s discrepancy by traveling waves creates the
phenomenon, which received a name of shock waves.
2. Model Description
Here I describe the one -dimensional case.
I assume that at time 0tt= in point 0xx= we have a positive energy’s
discrepancy,
( )0 ,0 00> ΔΦxt 4According to [1], this discrepancy propagates wavily in space with a finite speed
0>c . While traveling the wave attenuates. Therefore, at time 0 1ttt >= when a
discrepancy reaches the point 1xx=, where
( ) 0 0 1 0xctt xx >⋅−+=
it will have the amplitude
()()()()0 00 0 00 1 10, , ,1
0xt dxx xt xtx
xΔΦ< − ΔΦ= ΔΦ∫r (1)
where ()0>xr represents the losses of energy ’s discrepancy at points [ ]1 0,xxx∈.
On the other hand, energy’s discrepancy spreads directly in interval [ ]2 0,xx
quickly attenuating. I assume that the energy’s discrepancy spreads with a speed
0>>cv .
Thus th e discrepancy accumulates, and the accumulated value is
()()
()
>≤
=
∫∫
22
2
00
xxfor dxxxxforxdx
x x
xx
x
mm
e (2)
where ()0>xm is a positive monotonically decreasing function.
Hence the overall energy’s di screpancy at time 1tt= in point 1xx= (where it
holds 2 1 0xx x <<) is
()()()()( )dxx x xt xtx
x∫−+ ΔΦ= ΔΦ1
00 00 1 1, , r m (3)
Therefore when () ()x xr m> for [ ]1 0,xxx∈ it holds the inequality,
( ) ( )0 00 1 1, , xt xt ΔΦ> ΔΦ
and the energy’s discrepancy grows for the traveling wave. 5The foregoing description explains the phenomenon of shock waves.
References
1. A. Krouglov, “Dual Time -Space Model of Wave Propagation,” Workin g Paper
physics/9909024, Los Alamos National Laboratory, September 1999 (available at
http://xxx.lanl.gov ). |
arXiv:physics/0101100v1 [physics.class-ph] 29 Jan 2001Parametric autoresonance
Evgeniy Khain and Baruch Meerson
Racah Institute of Physics, Hebrew University of Jerusalem , Jerusalem 91904, Israel
We investigate parametric autoresonance: a persisting pha se locking which occurs when the driving
frequency of a parametrically excited nonlinear oscillato r slowly varies with time. In this regime, the
resonant excitation is continuous and unarrested by the osc illator nonlinearity. The system has three
characteristic time scales, the fastest one corresponding to the natural frequency of the oscillator. We
perform averaging over the fastest time scale and analyze th e reduced set of equations analytically
and numerically. Analytical results are obtained by exploi ting the scale separation between the two
remaining time scales which enables one to use the adiabatic invariant of the perturbed nonlinear
motion.
PACS numbers: 05.45.-a
I. INTRODUCTION
This work addresses a combined action of two mech-
anisms of resonant excitation of (classical) nonlinear os-
cillating systems. The first is parametric resonance . The
second is autoresonance .
There are numerous oscillatory systems which interac-
tion with the external world amounts only to a periodic
time dependence of their parameters. The corresponding
resonance is called parametric [1,2]. A textbook example
is a simple pendulum with a vertically oscillating point of
suspension [1]. The main resonance occurs when the ex-
citation frequency ωis nearly twice the natural frequency
of the oscillator ω0[1,2]. Applications of this basic phe-
nomenon in physics and technology are ubiquitous.
Autoresonance occurs in nonlinear oscillators driven by
a small external force, almost periodic in time. If the
force is exactly periodic, and in resonance with the nat-
ural frequency of the oscillator, the resonance region of
the phase plane has a finite (and relatively small) width
[3,4]. If instead the driving frequency is slowly varying
in time (in the right direction determined by the nonlin-
earity sign), the oscillator can stay phase-locked despite
the nonlinearity. This leads to a continuous resonant ex-
citation. Autoresonance has found many applications. It
was extensively studied in the context of relativistic par-
ticle acceleration: in the 40-ies by McMillan [5], Veksler
[6] and Bohm and Foldy [7,8], and more recently [9–12].
Additional applications include a quasiclassical scheme o f
excitation of atoms [13] and molecules [14], excitation of
nonlinear waves [15,16], solitons [17,18], vortices [19,2 0]
and other collective modes [21] in fluids and plasmas, an
autoresonant mechanism of transition to chaos in Hamil-
tonian systems [22,23], etc.
Until now autoresonance was considered only in sys-
tems executing externally driven oscillations. In this
work we investigate autoresonance in a parametrically
driven oscillator.
Our presentation will be as follows. In Section 2 we
briefly review the parametric resonance in non-linear os-cillating systems. Section 3 deals, analytically and nu-
merically, with parametric autoresonance. The conclu-
sions are presented in Section 4. Some details of deriva-
tion are given in Appendices A and B.
II. PARAMETRIC RESONANCE WITH A
CONSTANT DRIVING FREQUENCY
The parametric resonance in a weakly nonlinear oscil-
lator with finite dissipation and detuning is describable
by the following equation of motion [2,24,25]:
¨x+ 2γ˙x+ [1 +ǫcos((2 +δ)t)]x−βx3= 0. (1)
where the units of time are chosen in such a way that the
scaled natural frequency of the oscillator in the small-
amplitude limit is equal to 1. In Eq. (1) ǫis the ampli-
tude of the driving force, which is assumed to be small:
0< ǫ≪1,δ≪1 is the detuning parameter, γis
the (scaled) damping coefficient (0 < γ≪1) andβis
the nonlinearity coefficient. For concreteness we assume
β >0 (for a pendulum β= 1/6).
Working in the limit of weak nonlinearity, dissipa-
tion and driving, we can employ the method of aver-
aging [2,3,26,27], valid for most of the initial conditions
[3,4]. The unperturbed oscillation period is the fast time.
Puttingx=a(t)cosθ(t) and ˙x=−a(t)sinθ(t) and per-
forming averaging over the fast time, we arrive at the
averaged equations
˙a=−γa+ǫa
4sin 2ψ,
˙ψ=−δ
2−3βa2
8+ǫ
4cos2ψ, (2)
where a new phase ψ=θ−[(2 +δ)/2]thas been in-
troduced. The averaged system (2) is an autonomous
dynamical system with two degree of freedom and there-
fore integrable. In the conservative case γ= 0 Eqs. (2)
become:
1˙a=ǫa
4sin 2ψ,
˙ψ=−δ
2−3βa2
8+ǫ
4cos2ψ. (3)
As sin2ψand cos2ψare periodic functions of ψwith a
periodπ, it is sufficient to consider the interval −π/2<
ψ≤π/2. For small enough detuning, δ < ǫ/ 2, there is
an elliptic fixed point with a non-zero amplitude:
a∗=±/bracketleftbigg2ǫ
3β/parenleftbigg
1−2δ
ǫ/parenrightbigg/bracketrightbigg1/2
;ψ∗= 0.
We need to calculate the period of motion in the phase
plane along a closed orbit around this fixed point (such
an orbit is shown in Fig. 1)./G01/G02/G03/G04/G01/G02/G03/G04 /G05/G05 /G02/G03/G04/G02/G03/G04
/G05/G05
/G05/G03/G06/G05/G03/G06
/G05/G03/G07/G05/G03/G07
/G05/G03/G08/G05/G03/G08
/G05/G03/G09/G05/G03/G09
/G02/G02
/G01/G02/G03/G04/G05/G03/G06/G01/G07/G08/G09/G0A/G0B/G05
FIG. 1. Parametric resonance with a constant driving fre-
quency. Shown is a typical closed orbit in the phase plane
with period T(ǫ= 0.04,δ=−0.04,β= 1/6 andγ= 0.). For
a time-dependent driving frequency ν(t), the autoresonance
will occur if the characteristic time for variation of ν(t) is
much greater than T, see criterion (11).
This calculation was performed by Struble [24]. For a
zero detuning, δ= 0, Hamilton’s function (we will call it
the Hamiltonian) of the system (3) is the following:
H(I,ψ) =ǫI
4cos2ψ−3βI2
8=H0=const., (4)
where we have introduced the action variable I=a2/2.
Solving Eq. (4) for Iand substituting the result into the
Hamilton’s equation for ˙ψwe obtain:
˙ψ=∓ǫ
4/parenleftbigg
cos22ψ−24βH0
ǫ2/parenrightbigg1/2
, (5)
where the minus (plus) sign corresponds to the upper
(lower) part of the closed orbit. The period of the am-
plitude and phase oscillations is therefore
T=8
ǫ/integraldisplayψ
−ψdψ
/parenleftBig
cos22ψ−24βH0
ǫ2/parenrightBig1/2, (6)where −ψandψare the roots of the equation cos22ψ=
24βH0/ǫ2.Calculating the integral, we obtain
T=8
ǫK(m), (7)
whereK(m) is the complete elliptic integral of the first
kind [28], and m= 1−24βH0/ǫ2. This result will be
used in Section 3 to establish a necessary condition for
the parametric autoresonance to occur.
III. PARAMETRIC RESONANCE WITH A
TIME-DEPENDENT DRIVING FREQUENCY:
PARAMETRIC AUTORESONANCE
Now let the driving frequency vary with time. This
time dependence introduces an additional (third) time
scale into the problem. The governing equation becomes
¨x+ 2γ˙x+ (1 +ǫcosφ)x−βx3= 0, (8)
where ˙φ=ν(t). We will assume ν(t) to be a slowly de-
creasing function which initial value is ν(t= 0) = 2 + δ.
Using the scale separation, we obtain the averaged equa-
tions. The averaging procedure of Section 2 can be re-
peated by replacing (2 + δ)tbyφin all equations. There
is one new point that should be treated more accurately.
The averaging procedure is applicable (again, for most
of the initial conditions) if there is a separation of time
scales. It requires, in particular, a strong inequality
2˙θ+ν(t)≫2˙θ−ν(t). This inequality can limit the time
of validity of the method of averaging. Let us assume,
for concreteness, a linear frequency “chirp”:
ν(t) = 2 +δ−2µt, (9)
whereµ≪1 is the chirp rate. In this case the averaging
procedure is valid as long as µt≪1.
Introducing a new phase ψ=θ−φ/2, we obtain a
reduced set of equations (compare to Eqs. (2)):
˙a=−γa+ǫa
4sin 2ψ,
˙ψ=−δ
2+µt−3βa2
8+ǫ
4cos2ψ. (10)
The first of Eqs. (10) is typical for parametric resonance:
to get excitation one should start from a non-zero os-
cillation amplitude. As we will see, the µtterm in the
second of Eqs. (10) (when small enough and of the right
sign) provides a continuous phase locking, similar to the
externally driven autoresonance.
Consider a numerical example. Fig. 2 shows the time
dependence a(t) found by solving Eqs. (10) numerically.
One can see that the system remains phase locked which
allows the amplitude of oscillations to increase, on the
average, with time in spite of the nonlinearity. The time-
dependence of the amplitude includes a slow trend and
2relatively fast, decaying oscillations. These are the two
time scales remaining after the averaging over the fastest
time scale./G01/G01 /G02/G01/G01/G02/G01/G01 /G03/G01/G01/G01/G03/G01/G01/G01 /G03/G02/G01/G01/G03/G02/G01/G01 /G04/G01/G01/G01/G04/G01/G01/G01 /G04/G02/G01/G01/G04/G02/G01/G01
/G01/G01
/G01/G05/G04/G01/G05/G04
/G01/G05/G06/G01/G05/G06
/G01/G05/G07/G01/G05/G07
/G01/G05/G08/G01/G05/G08
/G03/G03
/G03/G05/G04/G03/G05/G04
/G03/G05/G06/G03/G05/G06
/G03/G05/G07/G03/G05/G07
/G03/G05/G08/G03/G05/G08
/G01/G02/G03/G04/G05/G03/G06/G07/G02/G01/G08/G09/G04
FIG. 2. An example of parametric autoresonance. Shown
is the oscillation amplitude versus time, computed nu-
merically from the averaged equations (10). The system
remains phase-locked which allows the amplitude to in-
crease, on the average, with time. The parameters are
µ= 6.5·10−5,ǫ= 0.04,δ=−0.01,β= 1/6 andγ= 0.001.
Similar to the externally-driven autoresonance, a per-
sistent growth of the oscillation amplitude requires the
characteristic time of variation of ν(t) to be much greater
than the “nonlinear” period T[see Eq. (7)] of oscillations
of the amplitude:
/vextendsingle/vextendsingle/vextendsingle/vextendsingleν(t)
˙ν(t)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≫T. (11)
Like its externally-driven analog, the parametric au-
toresonance is insensitive to the exact form of ν(t). For
a given set of parameters, the optimal chirping rate can
be found: too low a chirping rate means an inefficient ex-
citation, while too high a rate leads to phase unlocking
and termination of the excitation.
In the remainder of the paper we will develop an an-
alytical theory of the parametric autoresonance. The
first objective of this theory is a description of the slow
trend in the amplitude (and phase) dynamics. When the
driving frequency νis constant, there is an elliptic fixed
pointa∗(see Section 2). When νvaries with time, the
fixed point ceases to exist. However, for a slowly -varying
ν(t) one can define a “quasi-fixed” point a∗(t) which is
a slowly varying function of time. It is this quasi-fixed
point that represents the slow trend seen in Fig. 2 and
corresponds to an “ideal” phase-locking regime. The fast,
decaying oscillations seen in Fig. 2 correspond to oscil-
lations around the quasi-fixed point in the phase plane
[this phase plane is actually projection of the extended
phase space ( a,ψ,t) on the (a,ψ)-plane].
In the main part of this Section we neglect the dissi-
pation and use a Hamiltonian formalism. First we will
consider excitation in the vicinity of the quasi-fixed point .Then excitation from arbitrary initial conditions will be
investigated. Finally, the role of dissipation will be brie fly
analyzed.
For a time-dependent ν(t), the Hamiltonian becomes
[compare to Eq. (4)]:
H(I,ψ,t) =ǫI
4(α(t) + cos2ψ)−3βI2
8, (12)
whereα(t) = (4/ǫ)(1−ν(t)/2).The Hamilton’s equations
are:
˙I=ǫI
2sin 2ψ,
˙ψ=ǫ
4(α+ cos2ψ)−3βI
4. (13)
Let us find the quasi-fixed point of (13), i.e. the spe-
cial autoresonance trajectory I∗(t), ψ∗(t) corresponding
to the “ideal” phase locking (a pure trend without oscil-
lations).
Assuming a slow time dependence, we put ˙ψ∗= 0, that
is
ǫ
4(α+ cos2ψ∗)−3βI∗
4= 0. (14)
Differentiating it with respect to time and using Eqs.
(13), we obtain an algebraic equation for ψ∗(t):
2α(t)sin 2ψ∗+ sin 4ψ∗=16µ
ǫ2. (15)
At this point we should demand that ˙ψ∗(t), evaluated on
the solution of Eq. (15), is indeed negligible compared
to the rest of terms in the equation (13) for ˙ψ(t). It is
easy to see that this requires 16 µ/ǫ2≪1. In this case
the sines in Eq. (15) can be replaced by their arguments,
and we obtain the following simple expressions for the
quasi-fixed point:
I∗≃ǫ
3β(α+ 1),
ψ∗≃k
α+ 1, (16)
wherek= 4µ/ǫ2.
A. Excitation in the vicinity of the quasi-fixed point
Let us make the canonical transformation from vari-
ablesIandψtoδI=I−I∗andδψ=ψ−ψ∗.Assuming
δIandδψto be small and keeping terms up to the second
order inδIandδψ, we obtain the new Hamiltonian:
H(δI,δψ,α (t)) =−ǫk
α+ 1δIδψ−
−3β
8(δI)2−ǫ2
6β(α+ 1)(δψ)2.(17)
3Here and in the following small terms of order of k2are
neglected. Let us start with the calculation of the local
maxima ofδI(t) andδψ(t), which will be called δImax(t)
andδψmax(t), respectively. As α(t) is a slow function of
time [so that the strong inequality (11) is satisfied], we
can exploit the approximate constancy of the adiabatic
invariant [1,29]:
J=1
2π/contintegraldisplay
δId(δψ)≃const. (18)
|J|is the area of the ellipse defined by Eq. (17) with the
time-dependencies “frozen”. Therefore,
J=2
ǫH
(α+ 1)1/2≃const. (19)
This expression can be rewritten in terms of δIandδψ:
|J|=2k
(α+ 1)3/2δIδψ+3β
4ǫ1
(α+ 1)1/2(δI)2
+ǫ
3β(α+ 1)1/2(δψ)2. (20)
Ifk= 4µ/ǫ2≪1, the term with δIδψ in (20) can be
neglected (in this approximation one has ψ∗= 0). Then
Jbecomes a sum of two non-negative terms, one of them
having the maximum value when the other one vanishes.
Therefore,
δImax(t) = 2/parenleftbiggǫJ
3β/parenrightbigg1/2
(α+ 1)1/4, (21)
and
δψmax(t) =/parenleftbigg3βJ
ǫ/parenrightbigg1/21
(α+ 1)1/4. (22)
Now we calculate the period of oscillations of the ac-
tion and phase. Using the well-known relation [1] T=
2π(∂J/∂H ), we obtain from Eq. (19):
T=4π
ǫ1
(α+ 1)1/2. (23)
The period of oscillations versus time is shown in Fig. 3.
The theoretical curve [Eq. (23)] shows an excellent agree-
ment with the numerical solution.
Now we obtain the complete solution δI(t) andδψ(t).
The Hamilton’s equations corresponding to the Hamilto-
nian (17) are:
˙δI=ǫ2
3β(α+ 1)δψ+ǫk
α+ 1δI,
˙δψ=−3β
4δI−ǫk
α+ 1δψ. (24)
Differentiating the second equation with respect to
time and substituting the first one, we obtain a linear
differential equation for δψ(t):¨δψ+ω2(t)δψ= 0, (25)
whereω(t) = (ǫ/2)(α(t) + 1)1/2. For the linear ν(t) de-
pendence (Eq. (9)) we have α(t) = 4µt/ǫ−2δ/ǫ, therefore
fork≪1 the criterion ˙ ω/ω2≪1 is satisfied, and Eq.
(25) can be solved by the WKB method (see, e.g.[4])./G01/G01 /G02/G01/G01/G02/G01/G01 /G03/G01/G01/G01/G03/G01/G01/G01 /G03/G02/G01/G01/G03/G02/G01/G01 /G04/G01/G01/G01/G04/G01/G01/G01 /G04/G02/G01/G01/G04/G02/G01/G01
/G05/G01/G05/G01
/G06/G01/G06/G01
/G03/G01/G01/G03/G01/G01
/G03/G04/G01/G03/G04/G01
/G03/G07/G01/G03/G07/G01
/G03/G05/G01/G03/G05/G01
/G03/G06/G01/G03/G06/G01
/G04/G01/G01/G04/G01/G01
/G01/G02/G03/G04/G05/G04/G06/G02/G07/G08/G09/G07/G0A/G09/G07/G0B/G0C/G02/G0D/G0D/G0E/G01/G02/G07/G0F/G0B
FIG. 3. Excitation in the vicinity of the quasi-fixed point:
the time-dependence of the period Tof the action and phase
oscillations. The solid line is the theoretical curve, Eq. ( 23),
the asterisks are points obtained numerically. The paramet ers
areµ= 6.5·10−5,ǫ= 0.04,δ=−0.01 andβ= 1/6.
The WKB solution takes the form (details are given in
Appendix A):
δψ(t) =/parenleftbigg3βJ
ǫ/parenrightbigg1/21
(α+ 1)1/4
×cos/parenleftbigg
q0+(α+ 1)3/2
3k/parenrightbigg
, (26)
where the phase q0is determined by the initial conditions.
The full solution for the phase is ψ=δψ+ψ∗and Fig. 4
compares it with a numerical solution of Eq. (13). Also
shown are the minimum and maximum phase deviations
predicted by Eqs. (22) and (16). One can see that the
agreement is excellent.
The solution for δI(t) can be obtained by substituting
Eq. (26) into the second equation of the system (24). In
the same order of accuracy (see Appendix A)
δI(t) = 2/parenleftbiggǫJ
3β/parenrightbigg1/2
(α+ 1)1/4sin/parenleftbigg
q0+(α+ 1)3/2
3k/parenrightbigg
.
(27)
Fig. 5 shows the dependence of the action variable
with the trend I∗(t) subtracted, δI(t), on time predicted
by Eq. (27), and found from the numerical solution. It
also shows the minimum and maximum action deviations
(21). Again, a very good agreement is obtained.
4/G01/G01 /G02/G01/G01/G02/G01/G01 /G03/G01/G01/G01/G03/G01/G01/G01 /G03/G02/G01/G01/G03/G02/G01/G01 /G04/G01/G01/G01/G04/G01/G01/G01 /G04/G02/G01/G01/G04/G02/G01/G01/G05/G01/G06/G03/G05/G01/G06/G03/G01/G01/G01/G06/G03/G01/G06/G03/G01/G06/G04/G01/G06/G04
/G07/G08/G09/G0A/G0B/G0C/G0D/G0E/G0A
FIG. 4. Parametric autoresonance excitation in the vicin-
ity of the quasi-fixed point. Shown is the phase ψ(t) found
analytically [Eqs. (16) and (26)] and by solving Eq. (13) nu-
merically. The analytical and numerical curves are indisti n-
guishable. Also shown are the minimum and maximum phase
deviations predicted by Eq. (22) and (16). The parameters
are the same as in Fig. 3.
/G01/G01 /G02/G01/G01/G02/G01/G01 /G03/G01/G01/G01/G03/G01/G01/G01 /G03/G02/G01/G01/G03/G02/G01/G01 /G04/G01/G01/G01/G04/G01/G01/G01 /G04/G02/G01/G01/G04/G02/G01/G01/G05/G01/G06/G01/G07/G02/G05/G01/G06/G01/G07/G02/G01/G01/G01/G06/G01/G07/G02/G01/G06/G01/G07/G02
/G08/G09/G0A/G0B/G0C/G0D/G08/G09/G0E/G0F/G10/G10/G09/G10/G10/G11/G09/G08/G12/G0E/G13/G08
/G10/G08/G14/G0B/G0F/G15/G10/G10/G09/G01
FIG. 5. Parametric autoresonance excitation in the vicinit y
of the quasi-fixed point. Shown is the action variable δI(t)
from Eq. (27) and from the numerical solution. Also shown
are the minimum and maximum action deviations predicted
by Eq. (21). The parameters are the same as in Fig. 3.
B. Excitation from arbitrary initial conditions
In this Subsection we go beyond the close vicinity of
the quasi-fixed point and calculate the maximum devi-
ations of the action Iand phase ψfor arbitrary initial
conditions. Again, these calculations are made possible
by employing the adiabatic invariant for the general case.
Correspondingly, the period of the action and phase os-
cillations will be also calculated.
Let us first express the maximum and minimum action
deviations in terms of the Hamiltonian Hand driving
frequencyν(t). Solving Eq. (12) as a quadratic equationforI, we obtain:
I1,2=ǫ
3β(α+ cos2ψ)±/bracketleftbiggǫ2
9β2(α+ cos2ψ)2−8H
3β/bracketrightbigg1/2
.
The time derivative of Ivanishes when I=Imaxor
I=Imin. Therefore, from the first equation of the sys-
tem (13)ψ= 0 so that
Imax,min =ǫ
3β(α+ 1)±/bracketleftbiggǫ2
9β2(α+ 1)2−8Hup,down
3β/bracketrightbigg1/2
,
(28)
whereHup,down =H(Imax,min,ψ= 0).
Now we express the maximum and minimum phase
deviations through the Hamiltonian Hand driving fre-
quencyν(t). The time derivative ˙ψvanishes ifψ=ψmax
orψ=ψmin, then the second equation of the system
(13) yields I= (ǫ/3β)(α+ cos2ψ). In this case the
Hamiltonian (12) becomes Hright,left = (ǫ2/24β)(α+
cos2ψmax,min )2. Finally, the expression for ψmax,min is
ψmax,min =±1
2arccos/bracketleftBigg/parenleftbigg24βHright,left
ǫ2/parenrightbigg1/2
−α/bracketrightBigg
.
(29)
Fig. 6 shows a part of a typical autoresonant orbit
in the phase plane. For ν(t) =const. this orbit is de-
termined by the equation H(I,ψ,ν ) =const. , and it is
closed. As in our case ν(t) changes with time, the tra-
jectory is not closed. To calculate the maximum and
minimum deviations of action and phase we should know
the values of the Hamiltonian at 4 points of the orbit
that we will call “up”, “down”, “left”, and “right” in the
following./G01/G02/G03/G04/G05/G01/G02/G03/G04/G05 /G01/G02/G03/G06/G05/G01/G02/G03/G06/G05 /G01/G02/G03/G07/G05/G01/G02/G03/G07/G05 /G01/G02/G03/G02/G05/G01/G02/G03/G02/G05 /G02/G03/G02/G05/G02/G03/G02/G05 /G02/G03/G07/G05/G02/G03/G07/G05 /G02/G03/G06/G05/G02/G03/G06/G05 /G02/G03/G04/G05/G02/G03/G04/G05
/G02/G03/G08/G02/G03/G08
/G02/G03/G09/G02/G03/G09
/G02/G03/G0A/G02/G03/G0A
/G02/G03/G0B/G02/G03/G0B
/G07/G07
/G01/G02/G03/G04/G05/G03/G06/G07/G08/G09/G0A
/G0C/G0D/G0E/G0F/G10/G11/G0C
/G0C/G12/G13/G14/G11/G0C/G15
/G0C/G16/G17/G18/G19/G0C/G15/G0C/G1A/G1B/G0C/G15
FIG. 6. A part of the autoresonant orbit in the phase plane.
Knowing the Hamiltonian at the 4 points, we can calculate the
maximum and minimum deviations of the action and phase.
The parameters are the same as in Fig. 3.
5Knowing the values of the Hamiltonian at these 4
points, we calculate Imax,min from Eq. (28) and ψmax,min
from Eq. (29). Figs. (7) and (8) show these deviations
for action and phase correspondingly, and the values of
Iandψ, found from numerical solution. The theoretical
and numerical results show an excellent agreement./G01/G01 /G02/G01/G01/G02/G01/G01 /G03/G01/G01/G01/G03/G01/G01/G01 /G03/G02/G01/G01/G03/G02/G01/G01 /G04/G01/G01/G01/G04/G01/G01/G01 /G04/G02/G01/G01/G04/G02/G01/G01
/G01/G01
/G01/G05/G04/G01/G05/G04
/G01/G05/G06/G01/G05/G06
/G01/G05/G07/G01/G05/G07
/G01/G05/G08/G01/G05/G08
/G03/G03
/G03/G05/G04/G03/G05/G04
/G03/G05/G06/G03/G05/G06
/G03/G05/G07/G03/G05/G07
/G01/G02/G03/G04/G05/G06/G01/G02/G07/G08
FIG. 7. The maximum and minimum deviations of the ac-
tion, calculated from Eq. (28) (thick line) and from numeric al
solution (thin line). The parameters are the same as in Fig.
3./G01/G01 /G02/G01/G01/G02/G01/G01 /G03/G01/G01/G01/G03/G01/G01/G01 /G03/G02/G01/G01/G03/G02/G01/G01 /G04/G01/G01/G01/G04/G01/G01/G01 /G04/G02/G01/G01/G04/G02/G01/G01
/G05/G01/G06/G07/G05/G01/G06/G07
/G05/G01/G06/G08/G05/G01/G06/G08
/G05/G01/G06/G04/G05/G01/G06/G04
/G05/G01/G06/G03/G05/G01/G06/G03
/G01/G01
/G01/G06/G03/G01/G06/G03
/G01/G06/G04/G01/G06/G04
/G01/G06/G08/G01/G06/G08
/G01/G06/G07/G01/G06/G07
/G01/G06/G02/G01/G06/G02
/G01/G06/G09/G01/G06/G09
/G01/G02/G03/G04/G05/G06/G07/G08/G04
FIG. 8. The maximum and minimum deviations of the
phase, calculated from Eq. (29) (thick line) and from numer-
ical solution (thin line). The parameters are the same as in
Fig. 3.
Now we are prepared to calculate the adiabatic invari-
antJ(H,ν(t)). Its (approximate) constancy in time al-
lows one, in principle, to find the Hamiltonian H(t) at
any timet, in particular at the points of the maximum
and minimum action and phase deviations (see Fig. 6).
It is convenient to rewrite the adiabatic invariant in
the following form:
J=1
2π/contintegraldisplay
ψdI. (30)
Using Eq. (12), we can find ψ=ψ(H,I,α (t)):
ψ=±1
2arccos/parenleftbigg8H+ 3βI
2ǫI−α/parenrightbigg
, (31)so that Eq. (30) becomes:
J=1
2π/integraldisplayImax
Iminarccos/parenleftbigg8H+ 3βI
2ǫI−α/parenrightbigg
dI, (32)
whereImaxandIminare given by Eq. (28). Notice that
H(t) andα(t) should be treated as constants under the
integral (32), see Refs. [1,3,29]. This integral can be ex-
pressed in terms of elliptic integrals (see Appendix B for
details). For definiteness, we used the values of H(t)
andα(t) in the “up” points, see Fig. 6. We checked
numerically that the adiabatic invariant J(H(t),α(t)) is
constant in our example within 0.12 per cent.
Now we calculate the period of action and phase os-
cillations. From the first equation of system (13) we
have:
T= 2/integraldisplayImax
ImindI
(ǫI/2)sin2ψ, (33)
whereImaxandIminare given by Eq. (28), while ψ=
ψ(I) is defined by (31).
Using Eq. (12), we obtain after some algebra:
T=8
3β/integraldisplayImax
ImindI
G(I)1/2, (34)
whereG(I) is given in Appendix B, Eq. (B2). Again, we
treatH(t) andα(t) as constants under the integral (34),
and take their values in the “right” points, see Fig. 6.
The final result is:
T=C2K(C3), (35)
whereC2= 4(2/3βHǫ2)1/4and
C3=1
2−C2
2
16/bracketleftbigg3βH
2+ǫ2
16/parenleftbig
1−α2/parenrightbig/bracketrightbigg
./G01/G01 /G02/G01/G01/G02/G01/G01 /G03/G01/G01/G01/G03/G01/G01/G01 /G03/G02/G01/G01/G03/G02/G01/G01 /G04/G01/G01/G01/G04/G01/G01/G01 /G04/G02/G01/G01/G04/G02/G01/G01 /G05/G01/G01/G01/G05/G01/G01/G01
/G06/G01/G06/G01
/G07/G01/G07/G01
/G03/G01/G01/G03/G01/G01
/G03/G04/G01/G03/G04/G01
/G03/G08/G01/G03/G08/G01
/G03/G06/G01/G03/G06/G01
/G03/G07/G01/G03/G07/G01
/G04/G01/G01/G04/G01/G01
/G01/G02/G03/G04/G05/G04/G06/G02/G07/G08
FIG. 9. The period Tof the phase (action) oscillations ob-
tained from Eq. (35) (solid line), and from numerical soluti on
(asterisks). The parameters are the same as in Fig. 3.
6Figure 9 shows the period Tof the phase and action
oscillations versus time obtained analytically and from
numerical solution. This completes our consideration of
the parametric autoresonance without dissipation.
C. Role of dissipation
Now we very briefly consider the role of dissipation
in the parametric autoresonance. Consider the averaged
equations (10) and assume that the detuning is zero. The
non-trivial quasi-fixed point exists when the dissipation
is not too strong: γ <ǫ/ 4, and it is given by
a∗=/parenleftbigg2ǫ
3β/parenrightbigg1/2/bracketleftBigg
α(t) +/parenleftbigg
1−16γ2
ǫ2/parenrightbigg1/2/bracketrightBigg1/2
,
ψ∗=1
2arcsin/parenleftbigg4γ
ǫ+2k
α(t) + (1 −16γ2/ǫ2)1/2/parenrightbigg
.(36)
Again, we assume k≪1. This quasi-fixed point de-
scribes the slow trend in the dissipative case. As we
see numerically, fast oscillations around the trend, δa=
a−a∗andδψ=ψ−ψ∗decay with time. Therefore, one
can expect that the a(t) will approach, at sufficiently
large times, the trend a∗(t). Fig. 10 shows the time de-
pendence of the amplitude, found by solving numerically
the system of averaged equations (10), and the amplitude
trend from (36). We can see that indeed the amplitude
a(t) approaches the trend a∗(t) at large times./G01/G01 /G02/G01/G01/G02/G01/G01 /G03/G01/G01/G01/G03/G01/G01/G01 /G03/G02/G01/G01/G03/G02/G01/G01 /G04/G01/G01/G01/G04/G01/G01/G01 /G04/G02/G01/G01/G04/G02/G01/G01 /G05/G01/G01/G01/G05/G01/G01/G01
/G01/G01
/G01/G06/G04/G01/G06/G04
/G01/G06/G07/G01/G06/G07
/G01/G06/G08/G01/G06/G08
/G01/G06/G09/G01/G06/G09
/G03/G03
/G03/G06/G04/G03/G06/G04
/G03/G06/G07/G03/G06/G07
/G03/G06/G08/G03/G06/G08
/G03/G06/G09/G03/G06/G09
/G04/G04
/G01/G02/G03/G04/G05/G03/G06/G07/G02/G01/G08/G09/G04
FIG. 10. Parametric autoresonance with dissipation: the
time dependence of the amplitude of oscillations, obtained
from numerical solution of Eqs. (10), and the amplitude
trenda∗(t), predicted by Eq. (36). The parameters are
µ= 6.5·10−5,ǫ= 0.04,δ= 0,γ= 0.002 andβ= 1/6.
Therefore, a small amount of dissipation enhances
the stability of the parametric autoresonance excitation
scheme. A similar result for the externally-driven au-
toresonance was previously known [30].IV. CONCLUSIONS
We have investigated, analytically and numerically, a
combined action of two mechanisms of resonant excita-
tion of nonlinear oscillating systems: parametric reso-
nance and autoresonance. We have shown that para-
metric autoresonance represents a robust and efficient
method of excitation of nonlinear oscillating systems.
Parametric autoresonance can be extended for the ex-
citation of nonlinear waves. We expect that parametric
autoresonance will find applications in different fields of
physics.
ACKNOWLEDGEMENTS
This research was supported by the Israel Science
Foundation, founded by the Israel Academy of Sciences
and Humanities.
APPENDIX A: CALCULATION OF PHASE AND
ACTION DEVIATIONS BY THE WKB-METHOD
Changing the variables from time ttoα, we can rewrite
Eq. (25) in the following form:
δψ′′+/parenleftbiggα(t) + 1
4k2/parenrightbigg
δψ= 0, (A1)
where′′denotes the second derivative with respect to α.
Solving this equation by the WKB-method [4], we obtain
forδψ:
δψ(t) =(2kC)1/2
(α+ 1)1/4cos/parenleftbigg
Ω0+(α(t) + 1)3/2−1
3k/parenrightbigg
,(A2)
where Ω 0andCare constants to be found later. Now
we obtain the solution for δI. Substituting (A2) into the
second equation of the system (24), we obtain in the same
order of accuracy:
δI(t) =2ǫ
3β(2kC)1/2(α+ 1)1/4
×sin/parenleftbigg
Ω0+(α(t) + 1)3/2−1
3k/parenrightbigg
. (A3)
The constant Ccan be expressed through the adiabatic
invariantJ, given by (20). From Eqs. (A2) and (A3) we
have:
2kC=/parenleftbigg3β
2ǫ/parenrightbigg21
(α+ 1)1/2(δI)2+ (α+ 1)1/2(δψ)2.
Comparing it with (20) we find: C≃3βJ/2kǫ.Substi-
tuting this value into Eqs. (A2) and (A3) we obtain the
final expressions (26) and (27) for δψ(t) andδI(t).
7APPENDIX B: CALCULATION OF THE
ADIABATIC INVARIANT
After integration by parts and some algebra, using
Eqs. (12) and (28), we obtain the following expression
for the adiabatic invariant:
J=1
2π/integraldisplayImax
Imin/parenleftBigg
I2−8H
3β
G(I)1/2/parenrightBigg
dI, (B1)
where
G(I) = (Imax−I)(I−Imin)/bracketleftBigg/parenleftbigg
I+ǫ(1−α)
3β/parenrightbigg2
−16D
9β2/bracketrightBigg
,
(B2)
and we assume D= (ǫ2/16)(1−α)2−3βH/2<0.Calcu-
lation of this integral employs several changes of variable
shown in the best way by Fikhtengolts [31]. Using the
reduction formulas [28], we arrive at:
J=C1/bracketleftbigg1 +mm′
(1−m)2(1 +m′)Π/parenleftbiggm
m−1\k2/parenrightbigg
−1
1−mK/parenleftbig
k2/parenrightbig
+m+m′
(1−m)(1 +m′)E/parenleftbig
k2/parenrightbig/bracketrightbigg
,(B3)
where
m=(ǫ/3β)(1 +α)−(8H/3β)1/2
(ǫ/3β)(1 +α) + (8H/3β)1/2>0,
m′=(ǫ/3β)(1−α) + (8H/3β)1/2
−(ǫ/3β)(1−α) + (8H/3β)1/2>0.
k2=m
m+m′, C 1=c·64H
3β(m+m′)1/2,
and
c=1
2π/bracketleftBigg
ǫ
3β(1 +α) +/parenleftbigg8H
3β/parenrightbigg1/2/bracketrightBigg−1/2
×/bracketleftBigg
−ǫ
3β(1−α) +/parenleftbigg8H
3β/parenrightbigg1/2/bracketrightBigg−1/2
.
HereK,Eand Π are the complete elliptic integrals of
the first, second and third kind, respectively.
[1] L.D. Landau and E.M. Lifshits, Mechanics (Pergamon
Press, Oxford, 1976).[2] N.N. Bogolubov and Y.A. Mitropolsky, Asymptotic
Methods In The Theory of Non-linear Oscillations (Gor-
don and Breach Science Publishers, New York, 1961).
[3] R.Z. Sagdeev, D.A. Usikov, and G.M. Zaslavsky, Nonlin-
ear Physics (Harwood Academic, Switzerland, 1988).
[4] A.J. Lichtenberg and M.A. Lieberman, Regular and
Chaotic Dynamics (Springer-Verlag, Oxford, 1992).
[5] E.M. McMillan, Phys. Rev. 68, 143 (1945).
[6] V. Veksler, J.Phys.(USSR) 9, 153 (1945).
[7] D. Bohm and L. Foldy, Phys. Rev. 70, 249 (1947).
[8] D. Bohm and L. Foldy, Phys. Rev. 72, 649 (1947).
[9] K.S. Golovanivsky, Phys. Scripta 22, 126 (1980).
[10] B. Meerson, Phys. Lett. A 150, 290 (1990).
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283 (1991).
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(1990).
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B3, 1773 (1991).
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82, 4444 (1999); Phys. Plasmas 6, 4497 (1999).
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8 |
arXiv:physics/0101101v1 [physics.optics] 29 Jan 2001Chirality and helicity in terms of
topological spin and topological torsion
R. M. Kiehn
University of Houston
rkiehn2352@aol.com
Abstract: In this article the concept of enantiomorphism is de-
veloped in terms of topological, rather than geometrical, c on-
cepts. Chirality is to be associated with enantiomorphic pa irs
which induce Optical Activity, while Helicity is to be assoc iated
enantiomorphic pairs which induce a Faraday effect. Experim en-
tally, the existence of enantiomorphic pairs is associated with the
lack of a center of symmetry, which is also serves as a necessa ry
condition for Optical Activity. However, Faraday effects ma y or
may not require a lack of a center of symmetry. The two species
of enantiomorphic pairs are distinct, as the rotation of the plane
of polarization by Optical Activity is a reciprocal phenome non,
while rotation of the plane of polarization by the Faraday eff ect
is a non-reciprocal phenomenon. From a topological viewpoi nt,
Maxwell’s electrodynamics indicates that the concept of Ch irality
is to be associated with a third rank tensor density of Topolo gical
Spin induced by the interaction of the 4 vector potentials {A,φ}
and the field excitations ( D,H). The distinct concept of Helicity
is to be associated with the third rank tensor field of Topolog ical
Torsion induced by the interaction of the 4 vector potential s and
field intensities ( E,B).
1 Introduction
It is a remarkable result of experimental chemistry, recogn ized by Pasteur
and others, that there can exist enantiomorphic pairs of sta tes of chemical
systems that cannot be smoothly mapped into one another, sta rting from
the identity. From a geometrical perspective, right handed quartz and left
handed quartz are systems with apparent equivalent energie s, yet with de-
cidedly different behavior when interacting with electroma gnetic fields. If
1fact, the rotation of the plane of optical polarization is of ten used as a tool to
distinguish the enantiomorphic isotopes. However, the iss ue is deeper than
that of geometry. There exist topologically equivalent iso topes that cannot
be smoothly mapped into one another starting from the identi ty. Geo-
metrical properties are those subsets of topological prope rties that depend
upon size and shape. However, herein, the category of intere st is that subset
of topological properties which do not depend upon the geome trical issues
of size and shape, and yet are to be associated with enantiomo rphic pairs.
These distinct topological isotopes (enantiomers) are of d ifferent size and
shape and are connected by homeomorphisms, but they are not s moothly
connected to one another by a map about the identity. The prop erties and
existence of such topological enantiomers is the theme of th is article. For
example, a right handed Moebius band is topologically equiv alent to a left
handed Moebius band, but possibly of different size and disto rted shape.
The homeomorphism between the two topological isotopes con sists of more
than one step: the right handed Moebius band can be transform ed into a left
handed Moebius band by first cutting the band, applying a 360 d egree twist,
and then reconnecting the cut ends such that points that were initially near
to one another remain near to one another. Such a combination of processes
is not C2 smooth but can be continuous in the topological sens e.
The objective of this article is to examine those special fea tures of electro-
magnetic systems that can exhibit topological enantiomers , and to determine
how such enantiomers can be created. As all chemical systems are special
examples of electromagnetic interactions, the methods to b e developed are
useful to the understanding of the more constrained (geomet rical) features
of chemical enantiomorphism. A remarkable, but little appr eciated, fact is
that the electromagnetic field itself has certain propertie s that exhibit the
topological enantiomorphism mentioned above. Hence a stud y of these elec-
tromagnetic properties (defined below as topological torsi on and topological
spin) delivers a necessary foundation for the existence, co ntrol and modifi-
cation of the geometric properties of enantiomorphism disp layed in modern
chemistry.
In modern steriochemistry, Optical Activity and Faraday Ro tation have
a dominant experimental role. Optical Activity and Faraday Rotation have
many similarities, yet they are distinct, different, electr omagnetic phenom-
ena. A necessary, but not sufficient, condition for Optical Ac tivity in crys-
talline structures is the lack of a center of symmetry. This l ack of a center of
symmetry is often used as the basis for defining ”chirality”, and, conversely,
2chirality is often associated with Optical Activity. Howev er, according to
Post [1], there exist 3 crystal classes without a center of sy mmetry (crys-
tal classes 26,27,29) that do not support Optical Activity, hence a lack of
a center of symmetry is not sufficient condition for Optical Ac tivity. Note
that Optically Active media have the capability of ”rotatin g” the plane of
polarization, as linearly polarized light passes through t he media, a practical
effect used by the wine grower to estimate the sugar content in his grapes.
Similar rotation of the plane of polarization occurs when li nearly polarized
light passes through Faraday media. However, Faraday effect s can exist both
in crystalline structures that have a center of symmetry and in crystalline
structures that do not have a center of symmetry. Post report s that there
are 9 crystalline structures that support both optical Acti vity and Faraday
rotation. What are the intrinsic differences between Optica l Activity and
Faraday Rotation? In his book ”Formal Structure of Electrom agnetics”,
E. J. Post [1] clearly delineates the differences between Opt ical Activity and
Faraday rotation, and demonstrates solutions to Maxwell’s equations for both
effects. The crucial result is that Optical Activity is recip rocal and Faraday
Rotation is not.
In short, the lack of a center of symmetry, the rotation of the plane of
polarization, and the existence of enantiomorphic pairs, a re necessary but not
sufficient properties to define the concept of chirality. Ther e exist two species
of phenomena that exhibit the three properties stated above , one species is
”reciprocal” and defines Chirality, and the other species is ”non-reciprocal”
and defines Helicity. These differences between Chirality an d Helicity deserve
attention, clarification, and exploitation. Such is the pur pose of this article.
1.1 Transverse Inbound and Outbound Waves
First consider a complex four vector potential solution to t he vector wave
equation which propagates as a transverse wave in the ±zdirection with a
phase θ=±kz∓ωt.There are 4 possibilities: The Efield rotates about the
zaxis in a Right Handed manner as viewed by an observer looking towards
the positive zdirection, or it rotates in a Left Handed manner. Outbound
θ=kz−ωtand Inbound θ=−kz−ωtwaves are to be distinguished as
ORH, OLH, IRH, and ILH..
3ORH =/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1
i/angbracketrightBigg
ei(kz−ωt)IRH=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1
i/angbracketrightBigg
ei(−kz−ωt)(1)
OLH =/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1
−i/angbracketrightBigg
ei(kz−ωt)I LH =/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1
−i/angbracketrightBigg
ei(−kz−ωt)(2)
For media with the symmetries of the Lorentz vacuum, the phas e velocities
v=ω/kare the same for all four modes. Addition or subtraction of OR H and
OLH produces a Linearly polarized state outbound. Addition or subtraction
of IRH and ILH produces a Linearly polarized state inbound.
Next, recall the experimental differences between Optical A ctivity and
Faraday Rotation:
1.1.1 Optical Activity
Consider an optically active fluid (sugar in water) in a cylin drical tube of
length L. For Optical Activity, there are also two distinct p hase velocities,
ω/k1andω/k2.Outbound Right Handed (ORH) circularly polarized light
propagates with a phase speed equal to the phase speed of Inbo und Left
Handed (ILH) circularly polarized light. Outbound Left Han ded (OLH)
polarized light propagates with a phase velocity different f rom the phase
velocity of Outbound Right Handed polarized light (ORH), bu t with the
same speed as that of Inbound Right Handed (IRH) polarized li ght. In
summary,
Optical Activity Phase Velocity, VORH=VILH/ne}ationslash=VOLH=VIRH (3)
The wave solutions for optical activity are of the format:
ORH =/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1
i/angbracketrightBigg
exp i(k1z−ωt)IRH=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1
i/angbracketrightBigg
exp i(−k2z−ωt) (4)
OLH =/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1
−i/angbracketrightBigg
exp i(k2z−ωt)ILH=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1
−i/angbracketrightBigg
exp i(−k1z−ωt)
The addition of |ORH /an}bracketri}ht+|OLH/an}bracketri}htproduces a Linearly Polarized state
propagating outbound, whose plane of polarization rotates . When the
4two inbound states are added, |IRH/an}bracketri}ht+|ILH/an}bracketri}ht,a linearly polarized state
is achieved, and its plane of polarization also rotates but in the opposite
direction as the outbound rotation. In other words, the round trip (out-
bound+reflection+inbound) motion causes the plane of polar ization to re-
turn to its initial value. This result defines what is meant by a reciprocal
effect. If the plane of polarization of the original linearly polarized light beam
suffers a rotation in the amount of θdegrees as it traverses the Optically Ac-
tive media, when reflected in a mirror, the plane of polarizat ion suffers an
negative rotation of θdegrees, as the light beam traverses the media in the
reverse direction. The plane of polarization returns to its original state after
the round trip. (The sense of Right Handed and Left Handed pol arization
is determined by an observer looking away from himself.)
1.2 Faraday Rotation
Consider a gas of He-Ne in a cylindrical tube of length L. Surr ound the tube
with a coil of wire that will produce a coaxial magnetic field t hat partially
aligns the spins of the gas atoms. For such Faraday media, the re are two
distinct phase velocities, ω/k1andω/k2. Outbound Right Handed (ORH)
circularly polarized light propagates with a phase speed eq ual to the phase
speed of Inbound Right Handed (IRH) circularly polarized li ght. Outbound
Left Handed (OLH) polarized light propagates with a phase ve locity different
from the phase velocity of Outbound Right Handed polarized l ight (ORH),
but with the same speed as that of Inbound Left Handed (ILH) po larized
light. In summary,
Faraday Effect Phase Velocity, VORH=VIRH/ne}ationslash=VOLH=VILH (5)
The wave solutions for the Faraday effect are of the format:
|ORH /an}bracketri}ht=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1
i/angbracketrightBigg
exp i(k1z−ωt)|IRH/an}bracketri}ht=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1
i/angbracketrightBigg
exp i(−k1z−ωt) (6)
|OLH/an}bracketri}ht=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1
−i/angbracketrightBigg
exp i(k2z−ωt)|ILH/an}bracketri}ht=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1
−i/angbracketrightBigg
exp i(−k2z−ωt)
The formulas represent circularly polarized waves. The add ition of |RHO /an}bracketri}ht+
|LHO/an}bracketri}htproduces a Linearly Polarized state propagating outbound, whose
5plane of polarization rotates. When the two inbound states a re added,
|RHI/an}bracketri}ht+|LHI/an}bracketri}ht,a linearly polarized state is achieved, and its plane of pola r-
ization also rotates in the same direction as the outbound rotation. In other
words, the round trip (outbound+reflection+inbound) motio n does not cause
the plane of polarization to return to its initial value. Thi s result defines
what is meant by a non-reciprocal effect. If the plane of polar ization of the
original linearly polarized light beam suffers a rotation in the amount of θ
degrees as it traverses the Faraday media, when reflected in a mirror, the
plane of polarization suffers an additional rotation of θdegrees, as the light
beam traverses the media in the reverse direction. The plane of polarization
does not return to its original state, but instead ratchets b y 2θdegrees upon
completing the round trip. (The sense of Right Handed and Lef t Handed
polarization is determined by an observer looking away from himself.)
1.3 Polar and Axial vectors
Following Schouten [2], Post points out that Faraday Rotati on is ”generated”
by a ”W vector”, while Optical Activity is generated by a ”vec tor”. Under
certain constraints, the W vector plays the role of an ”Axial ” vector, while
the ”vector” becomes a ”polar” vector. Upon reflection, a pol ar vector
changes its sense (determined by the arrow head). Point your finger into a
mirror. The image points back at you. The sense of the image is opposite
to the sense of the object. For polar vectors with a line of act ion parallel to
the mirror surface, the opposite result is obtained. The sen se of the image
is the same as the sense of the object. Note the differences of o rthogonal
and parallel reflections.
A reflected axial vector does not change its sense if the line o f action
is orthogonal to the mirror. Curl you fingers and align your th umb in a
direction orthogonal to the mirror. It does not matter wheth er the thumb
points into or away from the mirror. The sense of the ”axial ve ctor” is
determined by the curl of the fingers. The sense of the reflecte d image is the
same as the sense of the object. The opposite effect occurs whe n the line
of action of the axial vector is parallel to the reflection sur face. The sense,
as determined by the curl of the fingers, is opposite to that of the reflected
image.
The magnetic field Band the angular velocity Ωare examples of spatial
”W vectors”. On the other hand, the Dfield is a spatial ”polar vector”
6in the sense used by Post. The anti-symmetric spatial compon ents of the
covariant field intensity tensor 2-form, F=dA,are formed by the spatial
”W vector” field B.The anti-symmetric spatial components of the tensor
density, N-2 form, G,where J=dG,are formed by the spatial ”polar vector”
fieldD.These facts yield a clue for distinguishing Faraday Rotatio n and
Optical Activity on topological grounds. As will be shown be low, Faraday
Rotation is to be associated with the concept of Topological Torsion, and
Optical Activity is to be associated with the concept of Topo logical Spin.
2 Topological Formulation of Maxwell’s Equa-
tions.
2.1 Exterior Differential Systems
It is known that Maxwell’s system of PDE’s (without constitu tive constraints)
can be expressed as an exterior differential system [3] on a va riety of indepen-
dent variables. Exterior differential systems impose topol ogical constraints
on a differential variety. For the Maxwell electromagnetic s ystem on a do-
main {x,y,z,t }the two topological constraints have been called the Postul ate
of Potentials, and the Postulate of Conserved Currents. [4] . These two
topological constraints lead to the system of Partial Differ ential Equations,
known as Maxwell’s equations, for any coordinate system so c onstrained.
No metric, no connection, nor other restraints of a geometri cal nature are
required on the 4 dimensional differential variety of indepe ndent variables,
typically written as {x, y, z, t }.
Postulate of Potentials (an exact 2-form) F−dA= 0 (7)
Postulate of Conserved Currents (an exact 3-form) J−dG= 0 (8)
The method of exterior differential systems insures that the description
is not only diffeomorphically invariant in form (natural cov ariance of form
with respect to all invertible smooth coordinate transform ations), but also
the description is functionally well defined with respect to maps which are C2
continuous, but not necessarily invertible. This statemen t implies that those
7exterior differential forms which are defined on a final state v ariety can be
”pulled back” in a functionally well defined manner to an init ial state variety,
even though the map from initial to final state of coordinate v ariables is NOT
a diffeomorphic coordinate transformation. The inverse map ping need not
exist. This result is truly a remarkable property of Maxwell electrodynamics,
for it permits the analysis of certain irreversible electro dynamic processes
without the use of statistics. The ”push forward” process is not functionally
well defined when the inverse map does not exist, a fact that de monstrates
that topological evolution induces an ”arrow of time” [5].
2.2 Constitutive Constraints
In practical applications, it is possible to impose constra ints on the Maxwell
system in the form of constitutive relations between the the rmodynamically
conjugate variables of field intensity ( E,B) and field excitations ( D,H).
Post has demonstrated that the constitutive tensor (densit y) has many of
the properties of the Riemann tensor [6]. These constraints are NOT nec-
essarily equivalent to the Riemann tensor generated by a Rie mannian metric
imposed upon the variety {x, y, z, t }.In many circumstances the equiva-
lence classes of such constitutive constraints can be put in to correspondence
with the geometrical symmetries of the 32 crystal classes th at are used to
discriminate between the many different observed physical s tructures. As
mentioned above, a complex 6x6 constitutive constraint has been used by
Post to delineate between Optical Activity, Faraday Phenom ena, Birefrin-
gence and Fresnel-Fizeau motion induced effects in electrom agnetic signal
propagation. The complex constitutive tensor cannot be ded uced from a
real metric tensor. However it would appear that the constit utive tensor
has a constructive definition in terms of a non-symmetric con nection.
Indeed, the work of Post, who subsumed a complex constitutiv e tensor,
has been extended [7] to demonstrate the existence of irredu cible ”quater-
nion” solutions to the Maxwell system. Quaternion waves can not be rep-
resented by complex functions, which are the usual choice fo r describing
electromagnetic signals. Complex wave solutions generate a 4th order char-
acteristic polynomial for the phase speed which is doubly de generate. The
wave speeds have only two distinct magnitudes depending upo n direction
and polarization. For cases where a center of symmetry is not available,
and yet the medium supports both Optical Activity and Farada y rotation,
the wave solutions can NOT be expressed as complex functions , but can be
8written as quaternions. The resulting 4th order characteri stic polynomial
for the wave speeds is not degenerate and has four distinct ro ot magnitudes.
The results indicate that the phase propagation speed of lig ht is different for
each direction of propagation and for each mode of polarizat ion. The theory
has been used to explain the experimental results measured i n dual polarized
ring laser apparatus.
In contrast, in a medium with the Lorentz symmetries, the cha racteristic
polynomial is 4-fold degenerate; e.g., all polarizations a nd all directions have
the same propagation speed. The result leads to the ubiquito us statement
that the speed of light is the same for all observers, which is incorrect for me-
dia that do not have the Lorentz symmetries. For Birefringen t, or Optically
Active, or Faraday media, the characteristic polynomial fo r phase velocity is
doubly degenerate, implying a relationship exists between for the 4 modes of
propagation. There exist only two distinct phase velocity m agnitudes. The
correlation speeds for direction and polarization pairs ha ve been presented
above. Faraday rotation and Optical Activity have different propagation
direction-polarization handedness correlations. The Far aday rotation is not
reciprocal; the rotation induced by Optical Activity is rec iprocal.
2.3 Topological Three Forms
The classic formalism of electromagnetism is a consequence of a system of
two fundamental topological constraints as defined above on a domain of four
independent variables. The theory requires the existence o f two fundamental
exterior differential forms, {A, G},where the postulates permit the construc-
tion of the differential ideal {A, F=dA, G, J =dG}.This system of differ-
ential forms may be prolonged (by construction all possible exterior products)
to yield the Pfaff sequence of forms: {A, F, G, J, A ˆG, AˆF, AˆG, FˆF, FˆG, AˆJ, GˆG}.
On a domain of four independent variables, the complete Pfaff sequence con-
tains three 3-forms: the classic 3-form of charge current de nsity, J,and the
(apparently novel to many researchers) 3-forms of Spin Curr ent density,
AˆG,[9] and Topological Torsion-Helicity, AˆF[10].
The 3-form of Charge Current density J=dG (9)
The 3-form of Topological Spin density S=AˆG (10)
The 3-form of Topological Torsion T=AˆF (11)
9In most elementary descriptions of electromagnetic theory , the 3-forms of
Spin and Torsion are ignored. By direct evaluation of the ext erior prod-
uct, and on a domain of 4 independent variables, each 3-form w ill have 4
components that can be symbolized (in engineering format) b y the 4-vector
arrays
Spin−Current :S4= [A×H+Dφ,A◦D]≡[S,σ], (12)
and
Torsion −vector :T4= [E×A+Bφ,A◦B]≡[T,h], (13)
which are to be compared with the four construction componen ts of the
charge current 4-vector density:
Charge−Current :J4= [J, ρ]. (14)
2.4 Topological Invariants
The closed integral of each of the three 3-forms is a deformat ion invariant
(hence a topological property) if the selected 3-form is clo sed in an exterior
derivative sense ( dJ= 0, dS= 0, dT= 0 respectively). For example,
for any 3-form, J,such that dJ= 0,the Lie derivative of the closed inte-
gral relative to an arbitrary process path denoted by βVis given by the
expression,
LβV/integraltext/integraltext/integraltext
closed(J) =/integraltext/integraltext/integraltext
closed{i(βV)d(J)+d(i(βV)(J)}=/integraltext/integraltext/integraltext
closed{0+d(i(βV)(AˆG)}= 0.
(15)
The zero result is interpreted by the statement ”the closed i ntegral is a defor-
mation invariant” for the process can be deformed by any non- zero function
β(x, y, z, t ),and the integral is unchanged.
As the charge current 3-form, J,is closed by construction, ( dJ=ddG= 0)
it follows that its closed integral is always a deformation i nvariant. The result
leads to another ubiquitous statement known in electromagn etic theory as
the ”Conservation of electric charge”. It is not equivalent to the quantization
of charge. The additional topological constraints of closu re imply that the
exterior derivative of each of the three forms is empty (zero ). By direct
10computation, such a constraint of differential closure lead s to the Poincare
invariants for the electromagnetic system.
Poincare 1 = d(AˆG) =FˆG−AˆJ
={div3(A×H+Dφ) +∂(A◦D)/∂t}dxˆdyˆdzˆdt
={(B◦H−D◦E)−(A◦J−ρφ)}dxˆdyˆdzˆdt(16)
Poincare 2 = d(AˆF) =FˆF
={div3(E×A+Bφ) +∂(A◦B)/∂t}dxˆdyˆdzˆdt
={−2E◦B}dxˆdyˆdzˆdt (17)
For a (vacuum) state, with J= 0,zero values of the Poincare invari-
ants require that the magnetic energy density is equal to the electric energy
density (1 /2B◦H= 1/2D◦E), and, respectively, that the electric field is
orthogonal to the magnetic field ( E◦B= 0).Note that these constraints
often are used as elementary textbook definitions of what is m eant by elec-
tromagnetic waves. The possible values of the topological q uantities, as
deRham period integrals [11], form rational ratios and topo logical quantum
numbers. These quantum numbers should NOT be considered as t opolog-
ical ”charge”. Electromagnetic (topological) charge is re lated to the two
dimensional closed integrals of G,not the three dimensional closed integrals
described above.
2.5 Field Momentum, Propagation direction, and the
4-Vector Potential
The 4 vector potential A4= [A, φ] is different from the 4 dimensional
propagation vector k4= [k, ω]; in many cases the two vectors are not
even proportional (although they can be). In the language of differential
forms, it must be recognized that there is a difference betwee n the 1-form,
A=Axdx+Aydy+Azdz−φdt,and the 1-form, k=kxdx+kydy+kzdz−ωdt.
The integral of kdefines the phase θ=/integraltextk.The propagation 1-form kis de-
fined from the equations that generate the singular solution s [7] to Maxwell’s
equations:
kˆF= 0,and kˆG= 0. (18)
11Note that these equations for singular solutions are derive d from the 3-forms
kˆFandkˆGconstrained to be zero. The 3-form kˆFis not necessarily the
same as the 3-form, AˆF.Similarly, the 3-form kˆGis not necessarily the
same as the 3-form, AˆG.These singular solutions are always transverse in
a geometrical sense that the wave 3 vector kis in the direction of the field
momentum, D×B.Both the Dvector and the Bvector are orthogonal to
the wave vector, k, that generates singular solutions.
The 1-forms, k, that satisfy the equations
Associated 1-forms k {kˆF= 0, kˆG= 0} (19)
are defined as the ”associated” 1-forms relative to FandG. The 1-forms,
k, that satisfy the equations
Extremal 1-forms k {kˆdF= 0, kˆdG= 0} (20)
are defined as ”extremal” 1-forms relative to FandG. 1-forms such that
characteristic 1-forms k {kˆF= 0, kˆG= 0, kˆdF= 0, kˆdG= 0},(21)
are defined as the ”characteristic” 1-forms, k.These 1-forms kare dual to
the associated, extremal, and characteristic vector fields ,V, which satisfy
the equations
Associated vector fields V {i(V)A= 0, i(V)G= 0} (22)
or
Extremal vector fields V {i(V)dA= 0, i(V)dG= 0}, (23)
or
Characteristic vector fields V for A {i(V)A= 0, i (V)dA= 0,(24)
Characteristic vector fields V for G i(V)G= 0, i(V)dG= 0},(25)
respectively.
The concepts of Spin Current and the Torsion vector have been utilized
hardly at all in applications of classical electromagnetic theory. Just as the
vanishing of the 3-form of charge current, J= 0,defines the topological do-
main called the vacuum, the vanishing of the two other 3-form s will refine
the fundamental topology of the Maxwell system. Such constr aints permit
12a definition of transversality to be made on topological (rat her than geomet-
rical) grounds. If both AˆGandAˆFvanish, the vacuum state supports
topologically transverse modes only (TTEM). Examples lead to the conjec-
ture that TTEM modes do not transmit power, a conjecture that has been
verified when the concept of geometric transversality (TEM) and topologi-
cal transversality (TTEM) coincide. A topologically trans verse magnetic
(TTM) mode corresponds to the topological constraint that AˆF= 0.A
topologically transverse electric mode (TTE) corresponds to the topological
constraint that AˆG= 0.Examples, both novel and well-known, of vac-
uum solutions to the electromagnetic system which satisfy ( and which do
not satisfy) these topological constraints are given [12]. The ideas should
be of interest to those working in the field of Fiber Optics. Re call that clas-
sic waveguide solutions which are geometrically and topolo gically transverse
(TEM ≡TTEM) do not transmit power [13]. However, in [12] an example
vacuum wave solution is given which is geometrically transv erse (the fields
are orthogonal to the field momentum and the wave vector), and yet the geo-
metrically transverse wave transmits power at a constant ra te: the example
wave is not topologically transverse as AˆF/ne}ationslash= 0.
2.6 Connections for Right handed vs Left handed evo-
lution
In spaces (such as Finsler spaces) which may or may not be Riem annian, the
topological concept of differential neighborhoods that are linearly connected
implies the existence, over the domain, of a matrix of functi ons/bracketleftBig
Fk
a(qb)/bracketrightBig
that
will linearly map 1-forms (linear combinations of different ials|ωa(q, dq)/an}bracketri}ht) into
1-forms (linear combinations of differentials/vextendsingle/vextendsingle/vextendsingleσk(q, dq)/angbracketrightBig
).
/bracketleftBig
Fk
a(qb)/bracketrightBig
◦ |ωa(q, dq)/an}bracketri}ht ⇒/vextendsingle/vextendsingle/vextendsingleσk(q, dq)/angbracketrightBig
. (26)
The map between differentials is linear, but the matrix eleme nts are not
domain constants (non-linearity is built in). The columns o f the matrix
of functions forms a matrix of basis vectors over the domain w hich may be
used to express any tensorial properties. The matrix of basi s vectors defines
what Cartan called the Repere Mobile, or the moving frame, fo r its values
change as a point p moves along some curve in the domain. If the mapping
of 1-forms is integrable,
13integrable basis :/bracketleftBig
Fk
a(qb)/bracketrightBig
◦ |dqa/an}bracketri}ht ⇒/vextendsingle/vextendsingle/vextendsingledxk/angbracketrightBig
, implies xk=fk(qb) (27)
then there exist functions whose differentials are exact, an d the mapping is
said to be holonomic. Often it is presumed that the functiona l mapping
exists (and as such is called a coordinate transformation un der certain ad-
ditional constraints) and the Frame matrix is deduced by the differential
operations to produce the Jacobian matrix of the transforma tion. However,
there are other ways to impose or deduce a Frame matrix on a dom ain.
The fundamental question of a connection is related to the di fferential
neighborhood properties of the Frame matrix. As the Frame ma trix, by
definition of its domain, has a non-zero determinant, then it admits an in-
verse matrix, and by differential and algebraic processes th e (right Cartan)
connection matrix [ CR] (or the left Cartan matrix [ CL]) can be constructed:
d[F] = −[F]◦[dG]◦[F] (28)
=−[CL]◦[F]
= + [ F]◦[CR].
The matrix elements of [ CR] and [ CL] are differential 1-forms. These matrix
elements do not necessarily vanish nor are they necessarily equal. It is to
be noted that the left and the right Cartan matrices are (anti -) similarity
transforms of one another.
The reason that there are two distinct methods for construct ing the linear
mapping is based upon the fact that a matrix, whose determina nt is non
- zero, always has two representations, a left handed and a ri ght handed
representation. The two representations always consist of either a (right
handed) product of a unitary matrix times a Hermitean matrix , or the (left
handed) product of a Hermitean matrix times a unitary matrix . Only if
the original matrix is ”normal” ( such that the product of its elf times its
Hermitian conjugate is equal to the product of its Hermitian conjugate times
itself) will the right and left handed product representati ons be degenerately
the same [8]. It is this handedness property of topological n eighborhoods that
captures the features of electromagnetic charge distribut ions of molecules and
crystals that exhibit enantiomorphic states.
In the electromagnetic situation, the constitutive map is o ften considered
to be (within a factor) a linear mapping between two six dimen sional vector
14spaces. As such the constitutive map can have both a right or a left handed
representation, implying that there are two topologically equivalent states
that are not smoothly equivalent about the identity.
In the geometric situation, the matrix elements of frame mat rix,/bracketleftBig
Fk
a(qb)/bracketrightBig
,
are not constants. The two mechanisms (right and left handed ) for neigh-
borhood expressions imply that the connection is not genera ted from a sym-
metric metric. Such connections are said to admit torsion. I n short, the
concept of an affine connection is more general than the Hermit ian (sym-
metric) connection offered by the Christoffel symbols ( which are generated
from a metric). Any domain which is parallelizable will supp ort a linear
connection of differentials.
3 Acknowledgments
This work was presented at the International Symposium on Ch irality, ISCD
12 — Chirality 2000, Chamonix, France
4 References
[1] Post, E. J., (1997), ”Formal Structure of Electromagnet ics”, Dover p 166.
[2] Schouten, J. A., (1954), ”Tensor Analysis for Physicist s”, Dover p.32
[3] Bryant, R.L.,Chern, S.S., Gardner, R.B.,Goldschmidt, H.L., and Grif-
fiths, P. A., (1991), ”Exterior Differential Systems”, Sprin ger Verlag.
[4] Kiehn, R. M., (1998), http://www22.pair.com/csdc/pdf /classice.pdf
[5] Kiehn, R. M., (1976), ”Retrodictive Determinism”, Int. J. of Eng.
Sci.14, p. 749
[6] Ibid, Post, p. 188
[7] Kiehn, R. M., Kiehn, G. P., and Roberds, R. B. (1991) ”Pari ty and
Time-reversal Symmetry Breaking, Singular Solutions”, Ph ys Rev A, 43, p.
5665
[8] Turnbull, H. W. and Aitken, A. C., (1961) ”An Introductio n to the
Theory of Canonical Matrices”, Dover, p194.
[9] Kiehn, R. M. (1977) ”Periods on manfolds, quantization a nd gauge”,
J. of Math Phys 18, no. 4, p. 614
15[10] Kiehn, R. M., (1990) ”Topological Torsion, Pfaff Dimens ion and Co-
herent Structures”, in: H. K. Moffatt and T. S. Tsinober eds, T opological
Fluid Mechanics, Cambridge University Press, 449-458 .
[11] deRham,G. (1960) ”Varietes Differentiables”, Hermann , Paris
[12] Kiehn, R. M., ”Torsion and Spin as topological coherent structures
in a plasma” (1998), http://www22.pair.com/csdc/pdf/pla sma.pdf
....(1999) ” Topological evolution of classical electroma gnetic fields and
the photon” in ”The Photon and Poincare Group”,Valeri V. Dvo eglazov
(Ed.). Nova Science Publishers, NY ,ISBN 1-56072-718-7.
[13] See http://www.elec.qmw.ac.uk/staffinfo/mark/mot
16 |
arXiv:physics/0101102v1 [physics.acc-ph] 30 Jan 2001
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/CT/CS /CX/D2 /D8/D3 /CP/D2 /D3/D2/CT/B9/CS/CX/D1/CT/D2/D7/CX/D3/D2/CP/D0 /D7/CT/D8/D9/D4 /CQ /DD/CP/D2 /CT/DC/D8/CT/D6/D2/CP/D0 /CU/D3
/D9/D7/CX/D2/CV /AS/CT/D0/CS/BA /C1/D2 /CT/D5/D9/CX/D0/CX/CQ/D6/CX/D9/D1/B8 /D8/CW/CT /D4/CP/D6/D8/CX
/D0/CT/D7 /DB/CX/D0/D0 /CQ /CT /CT/D5/D9/CX/CS/CX/D7/D8/CP/D2 /D8 /D0/D3/D2/CV/CX/D8/D9/CS/CX/D2/CP/D0/D0/DD /BA/CF /CT /D8/D6/CT/CP/D8 /D8/CW/CT /D0/CX/D1/CX/D8 /D3/CU /CP/D2 /CX/D2/AS/D2/CX/D8/CT/B8 /CQ/D9/D8 /D4 /CT/D6/CX/D3 /CS/CX
/B8
/CW/CP/CX/D2/BA /CC/CW/CT /D4/D6/D3/CQ/D0/CT/D1 /DB/CX/D0/D0 /CQ /CT /D8/D6/CT/CP/D8/CT/CS /CX/D2 /D8/CW/CT /D6/CT/D7/D8/CU/D6/CP/D1/CT /D3/CU /CP/D2 /D3/D6/CQ/CX/D8/CX/D2/CV /D4/CP/D6/D8/CX
/D0/CT/B8
/D9/D6/DA /CP/D8/D9/D6/CT /CP/D2/CS /D6/CT/D8/CP/D6/CS/CP/D8/CX/D3/D2 /CT/AR/CT
/D8/D7 /DB/CX/D0/D0 /CQ /CT /D2/CT/CV/D0/CT
/D8/CT/CS/BA/CC/CW/CT /CZ/CX/D2/CT/D8/CX
/B8 /D4 /D3/D8/CT/D2 /D8/CX/CP/D0 /CP/D2/CS /BV/D3/D9/D0/D3/D1 /CQ /CX/D2 /D8/CT/D6/CP
/D8/CX/D3/D2 /C4/CP/CV/D6/CP/D2/CV/CX/CP/D2 /CP/D6/CT/B8 /D6/CT/D7/D4 /CT
/D8/CX/DA /CT/D0/DD
Lk=1
2m/summationdisplay
µ˙xµ·˙xµ
Lp=−mω2
ext,x
2/summationdisplay
µ(x1
µ)2−mω2
ext,y
2/summationdisplay
µ(x2
µ)2
Li=1
2/summationdisplay
µ/negationslash=νLµν=e2
2/summationdisplay
µ/negationslash=ν1/radicalbig
(xµ−xν+ (µ−ν)λ)2
/B8
/B4/BD/B5/DB/CW/CT/D6/CT /DB /CT /CW/CP /DA /CT /CX/D2 /D8/D6/D3 /CS/D9
/CT/CS /D0/D3
/CP/D0
/D3 /D3/D6/CS/CX/D2/CP/D8/CT/D7 /CP/D6/D3/D9/D2/CS /CT/CP
/CW /D4/CP/D6/D8/CX
/D0/CT/B3/D7 /CT/D5/D9/CX/D0/CX/CQ/D6/CX/D9/D1 /D4 /D3/D7/CX/D8/CX/D3/D2/BA /CC/CW/CT/D7/D9/D1/D7 /D6/D9/D2 /D3 /DA /CT/D6 /CP/D0/D0 /D0/CP/D8/D8/CX
/CT /D7/CX/D8/CT/D7/BA λ /CX/D7 /D8/CW/CT /D0/CP/D8/D8/CX
/CT /DA /CT
/D8/D3/D6/B8 /DB /CT /D9/D7/CT /CP
/D3 /D3/D6/CS/CX/D2/CP/D8/CT /D7/DD/D7/D8/CT/D1 /DB/CW/CT/D6/CT
λ= (0,0, λ). /CC/CW/CT /D4/CP/D6/D8/CX
/D0/CT /CW/CP/D7 /D1/CP/D7/D7 m /B8 /CP/D2/CS /D8/CW/CT /CT/DC/D8/CT/D6/D2/CP/D0 /CU/D3
/D9/D7/CX/D2/CV /D7/D8/D6/CT/D2/CV/D8/CW/D7 /CP/D6/CT /CV/CX/DA /CT/D2 /CQ /DD
ω2
ext,x, ω2
ext,y
/CP/D2/CS /CP/D6/CT /CP/D7/D7/D9/D1/CT/CS /D8/D3 /CQ /CT
/D3/D2/D7/D8/CP/D2 /D8 /CP/D0/D3/D2/CV /D8/CW/CT /D6/CX/D2/CV/BA /CF /CT /CP/D6/CT /D9/D7/CX/D2/CV /D2/CP/D8/D9/D6/CP/D0 /D9/D2/CX/D8/D7 /DB/CX/D8/CW
/planckover2pi1=c= 1 /BA/CF /CT /CT/DC/D4/CP/D2/CS /B4/BD/B5 /CX/D2xµ
/B8 /D8/CW/CP/D8 /CX/D7/B8 /DB /CT /DB/D6/CX/D8/CT
Lµν=∞
p=03/summationdisplay
i1,... ,i p=1∞/summationdisplay
µ1,...,µ p=11
p!/summationdisplay
L(p),µ1···µp
µν xi1
µ1···xip
µp
/BA /B4/BE/B5
L(0)
i
/CS/CX/DA /CT/D6/CV/CT/D7/B8 /CQ/D9/D8 /CX/D7 /CX/D6/D6/CT/D0/CT/DA /CP/D2 /D8 /CW/CT/D6/CT/BNL(1)
i= 0 /B8 /CP/D7 /D8/CW/CT
/D3/GU/D6/CS/CX/D2/CP/D8/CT/D7 /CP/D6/CT /CT/DC/D4/CP/D2/CS/CT/CS /CP/D6/D3/D9/D2/CS/D8/CW/CT/CX/D6 /CT/D5/D9/CX/D0/CX/CQ/D6/CX/D9/D1/BA /BY /D3/D6 /D8/CW/CT /AS/D6/D7/D8 /CX/D2 /D8/CT/D6/CT/D7/D8/CX/D2/CV /D3/D6/CS/CT/D6/D7/B8 /DB /CT /CV/CT/D8
L(2),µ1µ2
µν =∆1∆2
λ3|µ−ν|3/parenleftbig
3δi13δi23−δi1i2/parenrightbig
L(3),µ1µ2µ3
µν =∆1∆2∆3(µ−ν)
λ3|µ−ν|5/summationdisplay
Π(i)δi13/parenleftbigg3
2δi2i3−5
2δi1i2i3/parenrightbigg
L(4),µ1···µ4
µν =∆1∆2∆3∆4
λ5|µ−ν|5/summationdisplay
Π(i)/parenleftbigg3
8δi1i2δi3i4−15
4δi1i2δi33δi43
+35
8δi13δi23δi33δi43/parenrightbigg
/B8 /B4/BF/B5/DB/CW/CT/D6/CT /DB /CT /D9/D7/CT/CS /D8/CW/CT /D7/CW/D3/D6/D8/CW/CP/D2/CS /D2/D3/D8/CP/D8/CX/D3/D2 ∆i= (δµim−δµin) /BA/BE/BW/D3/CX/D2/CV /D8/CW/CT /D7/D9/D1/D1/CP/D8/CX/D3/D2 /D3 /DA /CT/D6m, n /B8 /DB /CT /CV/CT/D8
L(2)
µ1µ2=/parenleftBigg
2δµ1µ2∞/summationdisplay
±k=1Φ(3)
k0−2Φ(3)
µ1µ2/parenrightBigg
δi1i2(3δi13−1)
L(3)
µ1µ2µ3=/summationdisplay
Π(µ)/parenleftbig
−δµ1µ2Φ(4)
µ1µ3/parenrightbig/summationdisplay
Π(i)δi13/parenleftbigg3
2δi2i3−5
2δi1i2i3/parenrightbigg
L(4)
µ1···µ4=/summationdisplay
Π(µ)/parenleftBigg
1
12δµ1µ2µ3µ4∞/summationdisplay
±k=1Φ(5)
k0−1
3δµ1µ2µ3Φ(5)
µ1µ4
+1
4δµ1µ2δµ3µ4Φ(5)
µ1µ4/parenrightbigg
×/summationdisplay
Π(i)δi1i2δi3i4/parenleftbigg3
8+δi43/parenleftbigg
−15
4+35
8δi13/parenrightbigg/parenrightbigg
/B8 /B4/BG/B5/DB/CW/CT/D6/CT Φ(n)
µν=1
2e2λ−n(sgn(µ−ν))n−1|µ−ν|−n/vextendsingle/vextendsingle
µ/negationslash=ν
/CP/D2/CSΦ(n)
µµ= 0 /CP/D2/CSΠ /CS/CT/D2/D3/D8/CT/D7 /CP/D0/D0 /D4 /CT/D6/D1 /D9/D8/CP/B9/D8/CX/D3/D2/D7 /D3/CU /CP /D7/CT/D8 /D3/CU /CX/D2/CS/CX
/CT/D7/BA/CC/CW/CT /D7/D9/D1/D7 /D3 /DA /CT/D6Φk0
/CV/CX/DA /CT
∞/summationdisplay
k=1Φ(n)
k0=1
2e2λ−nζ(n) /B4/BH/B5/CU/D3/D6 /D3 /CS/CSn /CP/D2/CS /DA /CP/D2/CX/D7/CW /CU/D3/D6 /CT/DA /CT/D2n /BA /B4ζ(3)≈1.202, ζ(5)≈1.037 /B5/BA/BT/D7 /D8/CW/CT /CX/D2 /D8/CT/D6/CP
/D8/CX/D3/D2 /CX/D7 /D8/D6/CP/D2/D7/D0/CP/D8/CX/D3/D2/CP/D0/D0/DD /CX/D2 /DA /CP/D6/CX/CP/D2 /D8/B8 /DB /CT /D4/D6/D3
/CT/CT/CS /CQ /DD /BY /D3/D9/D6/CX/CT/D6 /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2/BM
xm
µ=1
2π/integraldisplayπ
−πe−ikµξm(k)
ξm(k) =∞/summationdisplay
µ=−∞eikµxm
µ
/B4/BI/B5/CF /CT /DB/D6/CX/D8/CT /CS/D3 /DB/D2 /D8/CW/CT /CX/D2 /D8/CT/D6/CP
/D8/CX/D3/D2 /C4/CP/CV/D6/CP/D2/CV/CX/CP/D2 /CX/D2 /D8/CW/CX/D7 /CQ/CP/D7/CX/D7/BA /BY /D3/D6
/D3/D2 /DA /CT/D2/CX/CT/D2
/CT/B8 /DB /CT /CX/D2 /D8/D6/D3 /CS/D9
/CT/DA /CT/D6/D8/CT/DC /CU/D9/D2
/D8/CX/D3/D2/D7
F(p)
i1...i2=1
(2π)p−1/integraldisplay
δ2π/parenleftBiggp/summationdisplay
i=1ki/parenrightBigg
˜Φ(p+1)(ki1+···+kin)dpk /BA /B4/BJ/B5/C6/D3/D8/CT /D8/CW/CP/D8 /D1/D3/D1/CT/D2 /D8/D9/D1
/D3/D2/D7/CT/D6/DA /CP/D8/CX/D3/D2 /CX/D7 /D3/D2/D0/DD /D9/D4 /D8/D3 /CX/D2 /D8/CT/CV/CT/D6 /D1 /D9/D0/D8/CX/D4/D0/CT/D7 /D3/CU2π /BA /BT/CU/D8/CT/D6 /D7/D3/D1/CT /BY /D3/D9/D6/CX/CT/D6/BF/CV/DD/D1/D2/CP/D7/D8/CX
/D7/B8 /DB /CT /CW/CP /DA /CT
L(2)= 2/parenleftBig
F(2)−F(2)
1/parenrightBig
δi1i2(3δi13−1)
L(3)=/summationdisplay
Π(k)F(3)
1/summationdisplay
Π(i)δi13/parenleftbigg3
2δi2i3−5
2δi1i2i3/parenrightbigg
L(4)=/summationdisplay
Π(k)/parenleftbigg1
12F(4)−1
3F(4)
123+1
4F(4)
12/parenrightbigg
/summationdisplay
Π(i)δi1i2δi3i4/parenleftbigg3
8+δi43/parenleftbigg35
8δi13−15
4/parenrightbigg/parenrightbigg
/BA /B4/BK/B5/BE /C9/D9/CP/D2 /D8/CX/DE/CP/D8/CX/D3/D2/CC/CW/CT /D5/D9/CP/CS/D6/CP/D8/CX
/D8/CT/D6/D1/D7 /D3/CU /D8/CW/CT /D8/D3/D8/CP/D0 /C4/CP/CV/D6/CP/D2/CV/CX/CP/D2 /CS/CT/D7
/D6/CX/CQ /CT /CP/D2 /CT/D2/D7/CT/D1 /CQ/D0/CT /D3/CU /CW/CP/D6/D1/D3/D2/CX
/D3/D7
/CX/D0/D0/CP/B9/D8/D3/D6/D7 /DB/CX/D8/CW
/D3 /GU/D6/CS/CX/D2/CP/D8/CT /DA /CP/D6/CX/CP/CQ/D0/CT/D7 ξi(k), ξ∗
i(k) =ξi(−k) /BA /CF /CT /CX/D2 /D8/D6/D3 /CS/D9
/CT /D1/D3/D1/CT/D2 /D8/CP /DA /CP/D6/CX/CP/CQ/D0/CT/D7
πi(k), π∗
i(k) =πi(−k) /D3/CQ /CT/DD/CX/D2/CV /D8/CW/CT /D9/D7/D9/CP/D0
/D3/D1/D1 /D9/D8/CP/D8/CX/D3/D2 /D6/CT/D0/CP/D8/CX/D3/D2/D7/BA/C9/D9/CP/D2 /D8/CX/DE/CP/D8/CX/D3/D2 /CX/D7 /D7/D8/D6/CP/CX/CV/CW /D8/CU/D3/D6/DB /CP/D6/CS/D0/DD /CS/D3/D2/CT /CQ /DD /CS/CT/AS/D2/CX/D2/CV
/D6/CT/CP/D8/CX/D3/D2 /CP/D2/CS /CP/D2/D2/CX/CW/CX/D0/CP/D8/CX/D3/D2 /D3/D4 /CT/D6/CP/D8/D3/D6/D7
ai(k), a+
i(k) /CQ /DD
/radicalbig
2Ω(k)aj(k) = Ω( k)ξj(k) +iπj(−k) /B8 /B4/BL/B5/DB/CX/D8/CW /D3/D7
/CX/D0/D0/CP/D8/D3/D6 /CU/D6/CT/D5/D9/CT/D2
/CX/CT/D7 Ω(k) /CS/CT/AS/D2/CT/CS /CQ /CT/D0/D3 /DB/BA /CC/CW/CT/D7/CT /D3/D7
/CX/D0/D0/CP/D8/D3/D6 /CT/CX/CV/CT/D2/D1/D3 /CS/CT/D7 /CS/CT/D7
/D6/CX/CQ /CT /D4/CW/D3/D2/D3/D2/CX
/B4/D4/CP/D6/D8/CX
/D0/CT /CS/CX/D7/D4/D0/CP
/CT/D1/CT/D2 /D8 /DB /CP /DA /CT/D7/B5 /CT/DC
/CX/D8/CP/D8/CX/D3/D2/D7 /D3/CU /D3/D9/D6 /D7/DD/D7/D8/CT/D1/BA/CF /CT /DB/D6/CX/D8/CT /D8/CW/CT /CU/D9/D0/D0 /C4/CP/CV/D6/CP/D2/CV/CX/CP/D2 /B4/BK/B5 /CX/D2 /D8/CT/D6/D1/D7 /D3/CU /D8/CW/CT /D3/D4 /CT/D6/CP/D8/D3/D6/D7 ai(k), a+
i(k) /BA /CC/CW/CT /D1/D3/D1/CT/D2 /D8/D9/D1/B9/CX/D2/CS/CT/D4 /CT/D2/CS/CT/D2 /D8 /D8/CT/D6/D1/D7 /CX/D2 /B4/BK/B5 /CP/D6/CT /CS/CX/D7/D4 /D3/D7/CT/CS /D3/CU /CQ /DD /CP/CQ/D7/D3/D6/CQ/CX/D2/CV /D8/CW/CT/D1 /CX/D2 /D8/D3 /D8/CW/CT /BY /D3/D9/D6/CX/CT/D6 /D8/D6/CP/D2/D7/CU/D3/D6/D1 /D3/CU/D8/CW/CT /D4 /D3/D8/CT/D2 /D8/CX/CP/D0/BM ˜Φ(k)→˜Φ(k)−˜Φ(0) /BA/C1/D2/D7/D4 /CT
/D8/CX/D2/CV /B4/BK/B5/B8 /D3/D2/CT /D2/D3/D8/CX
/CT/D7 /D8/CW/CP/D8 /D8/CW/CT /D8/CT/D6/D1/D7
/CP/D2 /CQ /CT /CX/D2 /D8/CT/D6/D4/D6/CT/D8/CT/CS /CS/CX/CP/CV/D6/CP/D1/D1/CP/D8/CX
/CP/D0/D0/DD/BM/BD/BAF(2)
1
/CV/CX/DA /CT/D7 /D8/CW/CT /D3/D2/CT/B9/D4/CP/D6/D8/CX
/D0/CT /D4/D6/D3/D4/CP/CV/CP/D8/D3/D6/B8 /CX/BA /CT/BA/B8 /CX/D8 /CV/CX/DA /CT/D7 /D8/CW/CT /CS/CX/D7/D4 /CT/D6/D7/CX/D3/D2 /D6/CT/D0/CP/D8/CX/D3/D2 Ω2(k) /CU/D3/D6/D8/CW/CT /D4/CW/D3/D2/D3/D2/D7 /B4/BY/CX/CV/BA /BD/B5/BE/BAF(3)
1
/CS/CT/D7
/D6/CX/CQ /CT/D7 /CP /CS/CT
/CP /DD /D4/D6/D3
/CT/D7/D7/BM /D3/D2/CT /CX/D2
/D3/D1/CX/D2/CV /D4/CW/D3/D2/D3/D2 /CS/CT
/CP /DD/D7 /CX/D2 /D8/D3 /D8 /DB /D3 /D3/D9/D8/CV/D3/CX/D2/CV /D3/D2/CT/D7/B4/BY/CX/CV/BA /BE/B5/BF/BAF(4)
123
/CS/CT/D7
/D6/CX/CQ /CT/D7 /CP /CS/CT
/CP /DD /D4/D6/D3
/CT/D7/D7/BM /D3/D2/CT /CX/D2
/D3/D1/CX/D2/CV /D4/CW/D3/D2/D3/D2 /CS/CT
/CP /DD/D7 /CX/D2 /D8/D3 /D8/CW/D6/CT/CT /D3/D9/D8/CV/D3/CX/D2/CV /D3/D2/CT/D7/B4/BY/CX/CV/BA /BF/B5/BG/BAF(4)
12
/CS/CT/D7
/D6/CX/CQ /CT/D7 /CP /D7
/CP/D8/D8/CT/D6/CX/D2/CV/BM /D8 /DB /D3 /CX/D2
/D3/D1/CX/D2/CV /D4/CW/D3/D2/D3/D2/D7 /CT/DC
/CW/CP/D2/CV/CT /D1/D3/D1/CT/D2 /D8/D9/D1 /B4/BY/CX/CV/BA /BG/B5/C6/D3/D8/CT /D8/CW/CP/D8 /D3/D9/D6 /CS/CX/CP/CV/D6/CP/D1/D7 /CP/D6/CT /CX/D2 /D8/CT/D6/D1/D7 /D3/CU /D8/CW/CT /D7/D4/CP/D8/CX/CP/D0
/D3/GU/D6/CS/CX/D2/CP/D8/CT/D7 ξ, ξ∗/BA /C1/CU /DB /CT /DB /CP/D2 /D8 /D8/D3 /CS/D6/CP /DB/D8/CW/CT /CS/CX/CP/CV/D6/CP/D1/D7 /CX/D2 /D8/CT/D6/D1/D7 /D3/CU /D4/CW/D3/D2/D3/D2/CX
/CT/CX/CV/CT/D2/D1/D3 /CS/CT/D7/B8 /DB /CT /CW/CP /DA /CT /D8/D3 /D9/D7/CTξ, ξ∗∝a+±a /CP/D2/CS /CS/D6/CP /DB /CP/D0/D0/BK /D4 /D3/D7/D7/CX/CQ/D0/CT /D8/CW/D6/CT/CT/B9/D4 /D3/CX/D2 /D8 /CP/D2/CS /BF/BE /D4 /D3/D7/D7/CX/CQ/D0/CT /CU/D3/D9/D6/B9/D4 /D3/CX/D2 /D8 /CS/CX/CP/CV/D6/CP/D1/D7/BM /BX/CP
/CW /D0/CT/CV /CX/D2 /CP/D2 /DD /D3/CU /D8/CW/CT /CS/CX/CP/CV/D6/CP/D1/D7
/CP/D2 /CQ /CT /AT/CX/D4/D4 /CT/CS /D3 /DA /CT/D6 /D8/D3 /D1/CP/CZ /CT /CP/D2 /D3/D9/D8/CV/D3/CX/D2/CV /D4/CP/D6/D8/CX
/D0/CT /CP/D2 /CX/D2/CV/D3/CX/D2/CV /D3/D2/CT /DB/CW/CX/D0/CT
/CW/CP/D2/CV/CX/D2/CV /D8/CW/CT /D7/CX/CV/D2 /D3/CU/CX/D8/D7 /D1/D3/D1/CT/D2 /D8/D9/D1/BA/BG/BY/CX/CV/D9/D6/CT /BD/BM /BY /D6/CT/CT /D8 /DB /D3/B9/D4 /D3/CX/D2 /D8 /CU/D9/D2
/D8/CX/D3/D2/BT/D0/D7/D3/B8 /DB /CT /CW/CP /DA /CT /D8/D3 /D1 /D9/D0/D8/CX/D4/D0/DD /CT/CP
/CW /CS/CX/CP/CV/D6/CP/D1 /CQ /DD /D8/CW/CT /D4 /D3/D0/CP/D6/CX/DE/CP/D8/CX/D3/D2 /D8/CT/D2/D7/D3/D6/D7/B8 /CX/BA /CT/BA /D8/CW/CT /D8/D3/D8/CP/D0/D0/DD /D7/DD/D1/B9/D1/CT/D8/D6/CX
ii
/CS/CT/D4 /CT/D2/CS/CT/D2 /D8 /D8/CT/D6/D1/D7 /CX/D2 /B4/BK/B5/BA /CF/CX/D8/CW /CP/D2 /D3/CQ /DA/CX/D3/D9/D7 /D2/D3/D8/CP/D8/CX/D3/D2 /CU/D3/D6 /D8/D6/CP/D2/D7/DA /CT/D6/D7/CT /CP/D2/CS /D0/D3/D2/CV/CX/D8/D9/CS/CX/D2/CP/D0/D4 /D3/D0/CP/D6/CX/DE/CP/D8/CX/D3/D2/D7/B8 /D8/CW/CT/D7/CT /CP/D6/CT /CV/CX/DA /CT/D2 /CX/D2 /CC /CP/CQ/D0/CT /BD/BN
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arXiv:physics/0101103v1 [physics.flu-dyn] 30 Jan 2001Hydrodynamic WavesinRegionswithSmooth Loss ofConvexity ofIsentropes.
General Phenomenological Theory
MichaelI.Tribelsky∗1, andSergeiI.Anisimov†2
1Department of AppliedPhysics, Faculty of Engineering, Fuk ui University,
Bunkyo 3-9-1, Fukui 910-8507, Japan
and Max-Planck-Institut f¨ ur Physik komplexer Systeme,
N¨ othnitzer Straße 38, D–01187 Dresden, Germany
2Landau Institutefor Theoretical Physics RAN,Kosygin St., Moscow 117334, Russia
(July26, 2013)
General phenomenological theoryof hydrodynamic waves inr egions withsmooth loss of convexity of isen-
tropes is developed based on the fact that for most media thes e regions in p-Vplane are anomalously small.
Accordingly the waves are usually weak and can be described i n the manner analogous to that for weak shock
waves of compression. The corresponding generalized Burge rs equation is derived and analyzed. The exact
solution of the equation for steadyshock waves of rarefacti on isobtained and discusses.
PACSnumbers: 47.40.-x, 43.25.+y, 52.35.Mw
Extremely great importance of the so-called elementary
wave patterns, such as shock and rarefaction waves, in hy-
drodynamicshas been known since the pioneering studies of
Riemann [1] carried out in the beginning of 1860s. The im-
portance is generally connected with (i) possibility to pic ture
general fluids flows as nonlinear superposition of elementar y
waves,propagatingasseparateentitiesand(ii)reflection with
the elementary waves the asymptotic behavior of general so-
lutionsofthesetofhydrodynamicequations. Forthisreaso na
greatdealofattentionhasbeenpayedtostudytheelementar y
waves, see, e.g., the review article [2] and referencesther ein.
Nonetheless, a certain type of the elementary waves, namely
shockwavesofrarefactionandsonicwaves[2],whichadmits
general analytical consideration, has not been considered in
sucha manner. ThegapisfilledupinthepresentLetter.
It is well known that the sign of variation of pressure and
density in a weak shock wave is associated with the sign of
(∂2V/∂p2)s. Usually the sign of the derivative is positive,
and accordinglythe shock wave is compressive. However, in
somespecificareasof p-Vplane,e.g.,closetoacriticalpoint
but still in a single-phase region, (∂2V/∂p2)smay become
negative. The phenomenon is known as smooth loss of con-
vexityofisentropes[2]andisquitegeneralformanytypeso f
equations of states of real materials. Even the van der Waals
equation possesses a region around the vapor dome, where
(∂2V/∂p2)s<0[3]. In 1942 Bethe showed that all fluids
having sufficiently large specific heats exhibit smooth loss of
convexity of isentropes in the neighborhood of the saturate d
vaporline[4].
The smooth loss of convexity of isentropes creates pre-
requisites for a shock wave of rarefaction (SWR) to come
into being. Moreover, other types of anomalous wave struc-
tures, such as sonic shocks, whose Mach number(s) before
the shock, after it, or both is (are) identically equal 1, may
be observed in this region and its vicinity, see Refs. [2,3] f or
moredetails. Bearinginmindallmentionedabove,itishigh ly
desirable to have a certain universal equation, which is not
connected with any particular equation of state but neverth e-less can describe hydrodynamics waves in the discussed re-
gion. Despite the apparentimportanceof the problemand its
longhistory,muchtooursurprise,nobodyhaspayedattenti on
to the following consequence of the fact that usually the re-
gionwiththesmoothlossofconvexityofisentropesoccupie s
a small domain in p-Vplane. The consequence is that the
corresponding waves are always weak (otherwise the wave
curves [5] go far away from the domain), and therefore the
problemofthewavedescriptionistractablewithinthefram e-
work of the macroscopic hydrodynamicsin the manner anal-
ogous to that for weak shock waves of compression. To be
precise we must stipulate that we do not consider any singu-
laritiesrelatedtotheveryvicinitytothecriticalpoint. Inother
words, it is implied the system is far enough from it to avoid
anyeffectofthesingularities.
In the present Letter the mentioned approach is applied to
the problem. As a result the generalized Burgers equation,
which describes hydrodynamic waves in the region, where
the derivative (∂2V/∂p2)sis small and may change sign is
derived. The equationis valid for both signs of the derivati ve
and therefore describes all the variety of waves, which may
exist in this region. As an example the equation is employed
to study steady shock waves of rarefaction (SWR). The gen-
eral exact solutions of the equation describing such shocks is
obtainedandanalysed.
To begin with, let us consider the conventional Burgers
equation, which governsa weak shock wave of compression.
Inthelaboratorycoordinateframeforawaveadvancinginth e
negativedirectionof x-axistheequationreads[6]
∂p′
∂t−c1∂p′
∂x−αp′∂p′
∂x=ac3
1∂2p′
∂x2, (1)
where p′is the pressure variation, so that the pressure in the
wave profile pequals to p1+p′(here and in what follows
subscript 1 indicates the value of the corresponding quanti ty
in theunperturbedmedium,i.e.,at x→ −∞, whilesubscript
2will stand forthevalueofthesame quantitybehinda wave,
1atx→ ∞),c(p)is thevelocityofsound,
α=c3
1
2V2/parenleftbigg∂2V
∂p2/parenrightbigg
s,
Vdenotes the specific volume ( V≡1/ρ),ais a dissipative
constant[6]
a≡1
2ρ1c3
1/bracketleftbigg/parenleftbigg4
3η+ζ/parenrightbigg
+κ/parenleftbigg1
cv−1
cp/parenrightbigg/bracketrightbigg
,
ηandζstandforthecoefficientsofviscosity, κforthethermal
conductivityand cv,pforthecorrespondingspecificheats.
Eq.(1)mayberegardedasleadingapproximationtoexpan-
sion of a certain more general nonlinear dissipative equati on
in powers of both weak nonlinearity and weak dissipation. It
is important that the nonlinear term in this approximation i s
consideredin non-dissipativelimit. Dissipative correct ionsto
it haveadditionalsmallnessandmaybeneglected.
Inourcase/vextendsingle/vextendsingle(∂2V/∂p2)s/vextendsingle/vextendsingleisanomalouslysmall(itevencan
vanish). Accordingly, the lowest nonlinearity αp′(∂p′/∂x)
may not correspond to the leading nonlinear term any more.
Itmeansthat higherorder(in p′) termsmustbetakenintoac-
count. Ontheotherhand,theregion,where (∂2V/∂p2)s<0,
is narrow, therefore the third derivative (∂3V/∂p3)salso has
a certain smallness in this region. Only the fourth derivati ve
(∂4V/∂p4)sdoesnothaveanysmallnessthere. Thus,atsmall
but finite p′terms O/parenleftbig
(p′)2/parenrightbig
, O/parenleftbig
(p′)3/parenrightbig
andO/parenleftbig
(p′)4/parenrightbig
may
yield contributions of the same order of magnitude and must
be taken intoaccountsimultaneously. It is importantthat d is-
sipative corrections to the nonlinear terms remain negligi ble
and all such terms may be considered in the non-dissipative
limit,inthesamemannerasitisfortheconventionalBurger s
equation. All the above mentioned is related exclusively to
nonlinear terms. Regarding the dissipative term on the righ t
hand side of Eq. (1), there are no specific reasons for it to be
small andnocorrectionto thistermis required.
To derive the desired nonlinear correctionsto the left hand
sideofEq.(1)letusconsideranarbitraryadiabaticflowint he
form of a traveling wave. Due to the adiabatic condition the
profileofentropyofsuchaflowshouldnotchange,andsince
the state of the medium ahead of the shock front is spatially
homogeneous with s=const, we obtain that s=const
anywhere. Then, we may employ the general solution of the
continuityandEulerequationsvalidforsimplewaves[6]an d
reducetheseequationsto thefollowingsingle equation
∂p′
∂t+ [v(p′)±c(p′)]∂p′
∂x= 0, (2)
wheretheflowvelocity visgivenbytheexpression
v=±p′/integraldisplay
0dp
c(p)ρ(p). (3)
Sign plus in Eqs. (2)–(3) corresponds to the wave advancing
in the positive xdirection, minus to negative. According toour choice of the direction of the wave propagation in what
followswetakesignminus.
It is importantto stress that underthe above-specifiedcon-
ditions the reduction of the set of hydrodynamicequations t o
Eqs. (2)–(3) is an exactresult valid for anynonlinear depen-
dence c(p)andρ(p),whichmaybeobtainedfortheisentropic
wave. However,if p′is small one can expandthese functions
aboutthe point p′= 0. It is seen straightforwardlythat if one
truncates the expansion at zero term, it reduces Eqs. (2)–(3 )
to the linear acoustic equation. The truncation at the term o f
orderO(p′)yieldsthelefthandsideofEq.(1). Increasingthe
order of the truncationone can obtain the desired correctio ns
totheconventionalBurgersequationwith anygivenaccuracy.
Thus,thegeneralizedBurgersequationmusthavetheform
∂p′
∂t+u(p′)∂p′
∂x=c3
1a∂2p′
∂x2, (4)
where ustands for the velocityof a point in the wave profile.
For our choice of the propagation direction u(p′) =v(p′)−
c(p′)[7].
To get the equation in the explicit form one must expand
u(p′)inpowersofsmall p′retainingthetermsupto O/parenleftbig
(p′)3/parenrightbig
.
However,beforedoingthatitisworthanalyzingsomegenera l
featuresofEq.(4).
First of all let us obtain the general expression for the ve-
locity vsofa steadytravelingshock,when p′(x, t)→p′(x+
vst)≡p′(ξ). In this case Eq. (4) may be easily integrated.
Then, taking into account the boundary conditions ( p′→
0, ∂p′/∂ξ→0atξ→ −∞andp′→p2−p1, ∂p′/∂ξ→0
atξ→ ∞),we arriveatthefollowingexpressionfor vs
vs=−1
p2−p1p2−p1/integraldisplay
0u(p)dp≡ −/angbracketleftu/angbracketright, (5)
where/angbracketleft. . ./angbracketrightdenotesaverageover p′[we remindthat u=v−
c <0seealso Eq.(3)].
Letusshownowthat vsgivenbyEq.(5)forSWRsatisfies
theevolutionaryconditions. Theconditionssaythatinthe co-
moving coordinate frame the velocity of the flow before the
shockshould be biggerthan c1, while the velocity behindthe
shock should be smaller than c2, i.e.,M1>1> M 2>0,
where M1,2stand for the corresponding Mach numbers. In
the laboratorycoordinateframe the mediumbeforethe shock
is in the rest state with zero velocity. Accordingly, in the c o-
moving coordinate frame the flow velocity here is vs. Thus,
the first evolutionary condition says vs> c1. Behind the
shock the flow velocity in the laboratory frame is v2. Con-
sequently,the secondconditionyields vs+v2< c2. Bearing
inmindEq.(5)boththeconditionsmaybewrittenasfollows
c1≡ −u1<−/angbracketleftu/angbracketright< c2−v2≡ −u2 (6)
Finally, taking into account that for a simple wave and the
chosendirection of propagationthe sign of derivative du/dp′
is opposite to that for (∂2V/∂p2)s[6], i.e., for negative
(∂2V/∂p2)sfunction −u(p′)monotonically increases with
2decrease of p′, and that for SWR p2is smaller than p1, we
reduceinequality(6)to evident.
LetusproceedwiththederivationofthegeneralizedBurg-
ers equation. From all the above-mentionedit is clear that i n
the discussed region the derivative (∂2V/∂p2)smay be ap-
proximatedasfollows
(∂2V/∂p2)s≈1
ρ11
(ρ1c2
1)2/bracketleftBigg
−ǫ2+µ6
4/parenleftbiggp−pm
ρ1c2
1/parenrightbigg2/bracketrightBigg
.(7)
Herepmis the value of p, corresponding to the local min-
imum of the derivative, ǫis a small dimensionless quantity
µ=O(1)and power µ6as well as the numerical coefficient
areintroducedforconvenienceoffurthernotations.
According to Eq (7), (∂2V/∂p2)s<0atpm−∆< p <
pm+ ∆, where ∆ = 2 ρ1c2
1ǫ/µ3=O(ǫ). Then, expanding
u(p′)in powers of p′, taking into account that for the prob-
leminquestion p′isoforder ǫ,orsmalleranddroppingterms
higherthan O(ǫ3),aftersomecalculationswereducethegen-
eralequation(4)totheformofEq.(1),whereterm αp′should
bereplacedbythefollowingexpression
αp′→ρ2
1c3
1
2/bracketleftbigg/parenleftbigg∂2V
∂p2/parenrightbigg
sp′+1
2/parenleftbigg∂3V
∂p3/parenrightbigg
sp′2
+1
6/parenleftbigg∂4V
∂p4/parenrightbigg
sp′3/bracketrightbigg
,
andallderivatives (∂nV/∂pn)saretakenat p=p1.
It is convenient to rewrite the equation in more universal
dimensionlessform. Let usintroducenewvariables
y≡p′
∆;τ≡ǫ6
µ6c1at;ζ≡ǫ3
µ3c2
1a(x+c1t).
In these variables involving expression (7) one can reduce
thegeneralizedBurgersequationto thefollowingfinal form
∂y
∂τ−f(y)∂y
∂ζ=∂2y
∂ζ2, (8)
where
f(y)≡(z2−1)y+zy2+y3
3;z≡p1−pm
∆.(9)
In what follows we examine solutions of this equation,
which correspond to SWR. In this case we should supple-
ment Eq. (8) with the boundary conditions ∂y/∂ζ →0at
ζ→ ±∞;y→y1= 0atζ→ −∞;y→y2=const < 0
atζ→ ∞and bear in mind that y≤0;−1≤z≤1.To
studysteadySWRitisconvenienttointroduceatravelingco -
ordinate η≡ζ+ντ, where in accordance with Eq (5) the
dimensionlessvelocity νequalsthefollowingexpression
ν=z2−1
2y2+zy2
2
3+y3
2
12
Then,integrationofEq.(8)yieldsη=−12/bracketleftbigg
−ln|y|
y2y3y4+ln|y−y2|
y2(y3−y2)(y4−y2)
+ln|y−y3|
y3(y3−y4)(y3−y2)+ln|y−y4|
y4(y4−y3)(y4−y2)/bracketrightbigg
,(10)
where y3,4aretherootsoftheequation
y2+ (4z+y2)y+y2
2+ 4zy2+ 6(z2−1) = 0 (11)
Letusshowthat y3,4arealwaysrealandoneoftheserootis
smaller than y2, while the other is bigger than y1(we remind
that by definition y1= 0). In other words the left hand side
of Eq.(11) is negativeat y2< y < 0and anypossible values
ofzandy2. To prove it note that for SWR pressure p2must
satisfy obvious conditions pm−∆≤p2≤p1, which in
thedimensionlessvariablesaretransformedintothefollo wing
inequalities
−(z+ 1)≤y2≤0.
Due to the fact that y2enters into Eq. (11) with a positive
coefficient to prove negativeness of this polynomial at y2<
y <0it suffices to examine its values at the marginal points
y=y1,2. Aty=y1= 0we obtain
y2
2+ 4zy2+ 6(z2−1). (12)
For the same reason it is sufficient to inspect the values of
polynomial(12)at the marginalvalues of y2, namelyat y2=
−(z+1), y2= 0. Itisstraightforwardtoseethatthemarginal
values of Eq. (12) are negative at |z|<1. Negativeness of
polynomial(11)at y=y2isprovedinthe same manner. The
proved relative position of points y1,2,3,4guarantees that the
derivative dy/dηis negativeat y2< y < y 1for any possible
values of zandy2, i.e., the profile of the steady SWR is a
monotonicallydecreasingfunctionof η.
To end this Letter we present several particular versionsof
general solution (10), when the dependence y(η)may be ob-
tainedexplicitly.
(i)z= 1y2=−2, which correspondsthe maximal possi-
bleamplitudeofSWR( p1=pm+ ∆, p2=pm−∆).
η=−3
2ln/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 +y
y/vextendsingle/vextendsingle/vextendsingle/vextendsingle−3
2√
5lny+ 1 +√
5√
5−1−y.
(ii)z= 1|y2| ≪1p1=pm+ ∆, p1−p2≪∆.In this
casethe leadingapproximationinsmall |y2|yields
y=y2/radicalbigg
1 + tanh( y2
2η/3)
2
(iii)z=−(1 +y2),|y2| ≪1 [p1=pm−∆−(p1−
p2), p1−p2≪∆]. Theleadingapproximationin |y2|inthis
caseresultsinthefollowingprofile
y=y2/bracketleftBigg
1−/radicalbigg
1−tanh ( y2
2η/3)
2/bracketrightBigg
Note, that while in the present Letter only steady solutions
ofEq.(8)–(9)correspondingtoSWRarediscussed,theequa-
tion itself describes much more broad spectrum of problems
3of steady and non-steady flows, including such very interest -
ing issues, as sonic waves, evolution of arbitrary initial p ro-
files, collision of shocks, etc. The corresponding study is i n
progressandresultswill bereportedelsewhere.
Oneoftheauthors(M.I.T.)isgratefultoPeterFuldeforin-
vitationtoMPI-PKS,whichprovidestheopportunitytocom-
plete the present study, and to the entire staff of the Instit ute
for kind hospitality. This work was supported by the Grant-
in-AidforScientific Research(No. 11837006)fromtheMin-
istry of Education, Culture, Sports, Science and Technolog y
(Japan).∗Electronic address: tribel@scroll.apphy.fukui-u.ac.jp
†Electronic address: anisimov@itp.ac.ru
[1] B. Riemann Collected Works of Bernhard Riemann , edited by
H.Weber (Dover,New York, 1953).
[2] R.Menikoff, and B.J.Plohr, Rev. Mod. Phys. 61, 75(1989).
[3] M. Cramer,and R.Sen, Phys.Fluids 30, 377(1987).
[4] H. Bethe, The theory of shock waves for an arbitrary equation
of state(Clearinghouse for Federal Scientific and Technical In-
formation, U. S. Department of Commerce, Washington D.C.),
Report No. PB-32189.
[5] A curve, which consists of states connecting initial and final
states bya given wave solution.
[6] L. D. Landau, and E. M. Lifshitz, Fluid Mechanics – 2nd ed.
(Butterworth-Heinemann, Oxford, 1995).
[7] For a wave traveling in the opposite direction we would ha ve
u(p′) =v(p′) +c(p′).
4 |
arXiv:physics/0101104v1 [physics.gen-ph] 30 Jan 2001On p-adic Stochastic Dynamics,
Supersymmetry and the Riemann Conjecture
Carlos Castro∗
(dedicated to the memory of Michael Conard)
Abstract
We construct ( assuming the quantum inverse scattering prob lem
has a solution ) the operator that yields the zeroes of the Rie mman
zeta function by defining explicitly the supersymmetric qua ntum me-
chanical model ( SUSY QM ) associated with the p-adic stochas-
tic dynamics of a particle undergoing a Brownian random walk .
The zig-zagging occurs after collisions with an infinite arr ay of scat-
tering centers that fluctuate randomly . Since the prime numbers
are themselves randomly distributed, this physical system can be
reformulated as the scattering of the particle about the infinite loca-
tions of the prime numbers positions. We are able then to refo rmulate
suchp-adic stochastic process, that has an underlying hidden Par isi-
Sourlas supersymmetry, as the effective motion of a particle in a
potential due to an infinite collection of p-adic harmonic oscillators
with fundamental (Wick-rotated imaginary ) frequencies ωp=ilog p
(pis a prime ) and whose harmonics are ωp,n=ilog pn. The p-adic
harmonic oscillator potential allow us to determine a one-t o-one cor-
respondence between the amplitudes of oscillations an( and phases )
with the imaginary zeroes of zeta λn, after solving the inverse scat-
tering problem.
The Riemann conjecture that the nontrivial zeroes of the zet a function lie on
the vertical line z= 1/2 +iyof the complex plane remains one of the most
∗Center for Theoretical Studies of Physical Systems,Clark A tlanta University, Atlanta,
GA. 30314; e-mail: castro@ctsps.cau.edu
1important unsolved problems in pure Mathematics. Hilbert- Polya suggested
long ago that the zeroes of zeta could have an spectral interp retation in terms
of the eigenvalues of a suitable self adjoint trace class diff erential operator.
Finding such operator, if it exists, will be tantamount of pr oving the Riemann
conjecture. There is a related analogy with the Laplace-Bel trami operator in
the Hyperbolic plane , a surface of constant negative curvat ure. The motion
of a billiard in such surfaces is a typical example of classic al chaotic motion.
The Selberg zeta function associated with such Laplace-Bel trami operator
admits zeroes which can be related to the energy eigenvalues of such opera-
tor. Since the zeroes of the Riemann zeta function is deeply c onnected with
the random distribution of primes it has been suggested by ma ny authors
that the spectral properties of the zeroes of zeta may be asso ciated with
the random statistical fluctuations of the energy levels ( qu antum chaos) of
a classical chaotic system , random matrix theory [1],...,t urbulence in the
cloud formation and distribution of vortices and eddies in t he form of the
logarithmic spiral with the golden mean winding number [2] , etc.
Montgomery has shown [3] that the two-level correlation fun ction of the
imaginary parts of the zeroes of zeta is exactly the same expr ession found
by Wigner-Dyson using Random Matrices techniques : the two- level spec-
tral density correlation function of the Brownian-like dis crete level statistical
dynamics associated with the random matrix model of a Gaussi an Unitary
Ensemble ( GUE) turned out to be :
1−[sin(πx)
πx]2. (1)
The functionsin(πx)
πxappears very natural as well in the self similarity of the
iterated symbolic dynamics of the Fibonacci (rabbit) numbe rs sequence .
The golden mean is the n=∞limit of the ratio of two successive Fibonacci
numbers. Physically the relation (1) means that there is a repulsion of the
energy levels, a signal of chaos . For signals of chaos in matr ix theory ( Yang-
Mills ) and its relation to the holographic properties in str ing and Mtheory
we refer to the important work of Volovich et al [4]. Since str ing theory
has a deep connection to the statistical properties of rando m surfaces, index
theory for fractal p-branes in Cantorian fractal spacetime was considered by
the author and Mahecha [16] in connection to the Riemman zeta function.
The spectrum of drums ( membranes) with fractal boundaries b ears deep
relations to the zeta function. The location of the zeroes of zeta was deeply
2related to the bidimensional energy flow inherent in the clou d formation and
dissipation process studied by Amselvam [2 ]. The role of the golden mean (
chaos ) and Cantorian fractal spacetime was an essential ing redient as well.
In this letter we will combine all these ideas with the fundam ental inclusion
of supersymmetry. Parisi and Sourlas [5] discovered in the l ate 70’s that there
is a hidden supersymmetry in classical stochastic different ial equations. The
existence of stationary solutions to the Fokker-Planck equ ation associated
with the stochastic Langevin equation can be reformulated i n terms of an
unbroken supersymmetric (SUSY) Quantum Mechanical Model . More pre-
cisely, with an imaginary time Schroedinger equation : a diff usion equation
involving a dual diffusion process forward and backward in ti me. Nagasawa
, Chapline [6,7] have used these ideas to reformulate QM as in formation fu-
sion. Ord and Nottale [8, 9] following the path pioneered by F eynman path’s
integral formulation of QM, have shown that the QM equations can be under-
stood from an underlying fractal dynamics of a particle zig- zagging back and
forth in spacetime spanning a fractal trajectory ; i.e. a par ticle undergoing
a Random walk or Brownian Motion. El Naschie [ 10 ] suggested t hat this
dual diffusion process could be the clue to prove the Riemann c onjecture.
We will propose here the physical dynamical model that furni shes , in prin-
ciple, once a solution of the quantum inverse scattering pro blem exists, the
sought-after Hilbert-Polya operator which yields the zero es of zeta . If a so-
lution to this quantum inverse scattering problem exists th is could be instru-
mental in proving the Riemann Conjecture. We will borrow all these ideas of
random matrix theory , Brownian motion, random walk, fracta ls, quantum
chaos, stochastic dynamics ...within the framework of the s upersymmetric
QM model associated with the Langevin dynamics and the Fokke r-Planck
equation of a particle moving in a randomly fluctuating mediu m; i.e. noise
due to the random fluctuations of the infinite array of particl es (atoms )
located along the one dimensional quasi periodic crystal.
By random fluctuations we mean those fluctuations with respec t to their
equilibrium configurations which, for example, could be ass umed to be the
locations of the integers numbers. Because the prime number s themselves
arerandomly distributed, the main idea of this work is to recast this phys ical
problem in terms of the scattering of the particle by the rand omly distributed
scattering centers ( prime numbers ) and in this fashion we ha ve effectively
a random process with an underlying Parisi-Sourlas hidden s upersymmetry
, and hence , a well defined SUSY QM problem.
Watkins [11 ] has also suggested that an infinite array of (cha rged ) particles
3located at the positions of the prime numbers could be releva nt in describing
the physical system which provides the evolution dynamics l inked to the
zeroes of zeta. Pitkannnen [12] has refined Riemann’s conjec ture within the
language of p-Adic numbers by constraining the imaginary pa rts of the zeros
of zeta to be members of complex rational Pythagorean phases and Berry
and Keat [ 13 ] have proposed that the SUSY QM Hamiltonian :
H=xp−i=Q2(2)
is relevant to generate the imaginary parts of the zeroes.
The imaginary time Schroedinger equation (diffusion equati on ) that we pro-
pose is :
−D∂
∂tK±(x, t) =H±K(x, t). (3)
where Dis the diffusion constant which can be set to unity , in the same
way that one can set ¯ h=m= 1 where mis the particle’s mass subject to
the random walk .
Theisospectral partner Hamiltonians, H+, H−are respectively :
H±=−D2
2∂2
∂x2+1
2Φ2±D
2∂
∂xΦ. (4)
the transition- probability density solution of the Fokker -Plank equation ,
m±(x, xo, t) , for the particle arriving at x, in a given time t, after having
started at xois :
m±(x, xo, t) =exp[−1
D(U±(x)−U±(xo))]K±(x, t). (5)
The Fokker-Planck equation obeyed by the transition- proba bility density is
:
∂
∂tm(x, xo, t) =D
2∂2
∂x2m(x, xo, t) +∂
∂xΦ(x)m(x, xo, t). (6a)
and the associated Langevin dynamical equation :
∂x
dt=F(x) +ξ(t). (6b)
F= Φ(x) is the drift momentum experienced by the particle . The quan tity
ξ(t) is the noise term due to the random fluctuations of the medium where the
4particle is immersed. The drift potential U(x) associated wit the stochastic
Langevin equation is defined to be :
U±(x) =−(±)/integraldisplayx
0dzΦ(z) (7)
Φ(x) is precisely the SUSY QM potential as we shall see below. The two
partners isospectral ( same eigenvalues ) Hamiltonians can be factorized :
H+==1
2(D∂
∂x+ Φ(x))(D∂
∂x−Φ(x)) =L−L+. (8)
H−=1
2(D∂
∂x−Φ(x))(D∂
∂x+ Φ(x)) =L+L−(9)
If SUSY is unbroken there is a zeroeigenvalue λo= 0 whose eigenfunction
corresponding to the H−Hamiltonian is the ground state :
Ψ−
0(x) =Ce−1
D/integraltextx
0dzΦ(z). (10)
Cis a normalization constant.
Notice that the random ” momentum” term ξ(t) appearing in Langevin’s
equation can be simply recast in terms of the other quantitie s as :
∂x
dt=F(x) +ξ(t)⇒∂x
dt−F(x) =ξ(t). (11)
which in essence means that the random potential term (in uni ts of ¯h=m=
1) is :
ξ(t) =p−F(x) =p+ Φ(x). (12)
which is just the L−operator used to factorize the Hamiltonian H−. One
has two random potential terms : ξ±(t) corresponding to the two operators
L−,L+associated with the two isospectral Hamiltonian partners H±.
An immediate question soon arises. Since the imaginary part s of the zeroes of
zeta do notstart at zero , the first zero begins at y= 14.1347....., how can we
reconcile the fact that the ground state eigenfunction has z ero for eigenvalue
( by virtue of SUSY) ? . The answer to this question was provide d by
Pitkannen [ 12]. Using p−adic numbers theoretical arguments, he was able
to construct the fermionic version of the zeta function. Bot h the bosonic
and fermionic zeta can be recast as partition function of a sy stem of p-
adic bosonic/fermionic oscillators in a thermal bath of tem perature T. The
5frequencies of those oscillators is log p, forp= 2,3,5...a prime number and
the inverse temperature 1 /Tcorresponds to the zcoordinate present in the
ζ(z) where real z > 1. The pole at z= 1 naturally corresponds to the
limiting Hagedom temperature.
By virtue of SUSY the zeroes of the fermionic zeta function co incided pre-
cisely with the zeroes of the bosonic zeta with the fundament al difference that
the fermionic zeta had an additional zero precisely at z= 1/2 = 1/2 +iO;
i.e the imaginary part of the first zero of the fermionic parti tion function is
precisely zero!.
Therefore, in this SUSY QM model we naturally should expect t o have a zero
eigenvalue associated with the supersymmetric ground stat e. Such ground
state does notbreak SUSY and ensures that the associated Fokker-Planck
equation has a stationary solution in the limit t=∞limit ( equilibrium
configuration is attained at t=∞). Such stationary solution is given
precisely by the modulo-squared of the ground state solutio n to the SUSY
QM model [5]:
limm−
t→∞(x, xo= 0, t) =P(x) =|Ψ−
0(x)|2=C2exp[−2
DU−(x)].(13)
Notice that the ground state solution is explicitly given in terms of the po-
tential function U−(x) given by the integral of the SUSY potential Φ( x) ; i.e
.∂
∂x(−U±(x)) =±Φ(x).
One should notice that the ordinary harmonic oscillator corresponds roughly
speaking to the case : Φ ∼xso the operators L+,L−match the raising and
lowering operators in this restricted case. However this is a very special
case and notthe SUSY QM model studied here. The physical model we
are studying is notan ordinary realharmonic oscillator but instead it is a
p-Adic one related to Pitkannen’s original p-adic bosonic/fermionic harmonic
oscillator formulation of the zeta function as p-adic partition functions.
In principle, if one has the list of the imaginary parts of allthe zeroes of zeta
one can equate them to the infinite number of eigenvalues:
λo= 0, λn=λ−
n=λ+
nn= 1,2,3..... (14)
The main problem then is to find the SUSY potential Φ( x) associated with
the zeroes of the bosonic/fermionic zeta. This would requir e solving the in-
verse quantum scattering method (that gave rise to quantum g roups). To do
this is a formidable task since it requires to have the list of theinfinite num-
ber of zeroes to begin with and then to solve the inverse scatt ering method
6problem. Wavelet analysis is very suitable to solve inverse scattering meth-
ods. Not surprisingly, Kozyrev has given convincing argume nts that Wavelet
analysis is nothing but p-adic harmonic analysis [17].
What type of SUSY potential do we expect to get ? Is it related t o the
scattering of the particle by an infinite array of atoms locat ed at the prime
numbers ? Is it related to the Coulomb potential felt by the pa rticle due to
the infinite array of charges located at the prime numbers ? Is it related to a
chaotic one-dimensional billiard ball where the bouncing ( scattering ) back
and forth from an infinite array of obstacles located at the pr ime numbers ?
Is it just the p-adic stochastic Brownian motion modeled by Pitkannen’s p-
adic bosonic/fermionic oscillators ? Since the properties of the zeta function
are associated with the distribution of primes numbers it is sensible to pose
the latter questions. We will try to answer these questions s hortly.
The ordinary QM potential associated with the SUSY Φ( x) potential is de-
fined as
V±(x) =1
2Φ2(x)±D
2∂
∂xΦ(x). (15)
The potential V(x) should be symmetric (to preserve supersymmetry) under
the exchange x→ −xand this entails that the SUSY potential Φ( x) has to
be an odd function : Φ( −x) =−Φ(x).
Clearly since we do not have at hand the algorithm to generate all the zeros of
zeta ( nor the prime numbers ) one cannot write explicitly the ansatz for the
potential nor solve the inverse scattering problem (which w ould yield the form
of the potential). Nevertheless , Euler was confronted with a similar problem
of finding out all the prime numbers when he wrote the adelic pr oduct formula
of the Riemman zeta which relates an infinite summation over t he integers
to an infinite product over the primes :
For Real z > 1ζ(z)≡∞/summationdisplay1
nz=/productdisplay
p(1−p−z)−1. (16)
Euler’s formula can be derived simply by writing any integer as a product of
powers of primes and using the summation formula for a geomet ric series with
growth parameter p−z. Euler’s formula is a simple proof of why the prime
numbers are infinite. The product is an infinite product over allprimes.
Despite the fact that we do not have the list of allthe prime numbers nor
an algorithm to generate them , due to their intrinsic random ness, this does
nor prevent us from evaluating such infinite product; i.e in c omputing the
7value of the zeta function by performing the sum of the Dirich let series over
all integers!
This will provide us with the fundamental clue for writing an ansatz for the
SUSY potential Φ( x) and its associated potentials V±(x) giving us the SUSY
QM model which yields all the imaginary zeroes of the zeta fun ction as the
eigenvalues of such SUSY QM model. The reader could ask why go through
all this trouble and take a tortuous route of writing down the SUSY QM
model instead of solving directly the ordinary QM inverse sc attering problem
; i.e . finding the ordinary potential V±(x) of an ordinary QM problem ?
If one followed such procedure one would loose the deep under lying stochas-
tic dynamics of the problem. One would have not discovered th e underlying
hidden supersymmetry associated with the stochastic Lange vin dynamics;
nor its associated Fokker-Planck equation; nor be able to no tice that the
ground state is supersymmetric and its eigenvalue is precis ezero; nor to
construct the ground state solution explicitly in terms of t he SUSY poten-
tial Φ( x) as shown in Eq.(13 ). Also one would fail to notice the crucia l
factorization properties of the Hamiltonian, and that the potential V±(x)
must be symmetric with respect to the origin while the SUSY po tential Φ( x)
is antisymmetric,etc. The SUSY QM is very restricted that na rrows down
the inverse scattering problem. The simplest analogy one ca n give is that of
a person who fails to recognize the sine function because ins tead there is the
infinite Taylor expansion of the sine function.
The crux of this work is to write down the ansatz for the potent ial associated
with the SUSY QM model. The ansatz is based in writing an infini te summa-
tion in terms of an infinite product, similar to the Euler Adel ic product form
of the zeta and to well known relation between the sums of gamma functions
in terms of products of zeta functions present in the scattering formulae of
p-adic open strings [4, 14 ]. We proceed as follows.
Φ(x)≡/summationdisplay
nV(|x−xn|) =/productdisplay
pW(xp) (17)
where xp≡p−xandW=W(xp) is a function to be determined. This is our
ansatz for the SUSY potential. We shall call this ansatz for t he potential the
adelic condition since the product is taken over all the prim es.
The zeta function has a similar form ( although it is not symme tric with
respect to the origin ): one has an infinite summation over all the integers
of the series n−z( playing the role of the potential) expressed as an infinite
product over all the primes of functions of p−z. We are just recasting the
8potential felt by the particle due to the infinite interactio ns with the objects
situated at xn, an infinite sum of terms, in terms of the infinite product over
all the primes of functions of xp≡p−x. We emphasize that what is equal to
the imaginary parts of the zeroes of zeta are the eigenvalues λ nof the SUSY
QM model. We have just recast the infinite numerical input par ameters xn
of the potential in terms of the location of the infinite numbe r of primes .
For example, based on the Euler adelic formula for the zeta fu nction, one
could have chosen the ansatz for the potential to be precisel y of the zeta
function form :
For x > 0.Φ(x) =/productdisplay
pW(xp) =/productdisplay
p(1−xp)−1. (18)
xp≡p−x,Φ(x) =−Φ(−x)
where xis the location of the particle executing the p-adic stochastic Brown-
ian motion . For applications of p-adic numbers in physics we refer to [4,14].
Due to the antisymmetry requirement of Φ( x) , the SUSY potential for x <0
must be taken to be an exact mirror copy of the x >0 region to ensure that
supersymmetry is unbroken. As a result, the partner potenti alsV±are1:
V±(x) =1
2Φ2±D
2∂
∂xΦ(x). (19)
But how can we be so sure that the eigenvalues will be precisel y equal to the
imaginary zeroes of zeta ? This would have been an amazing coincidence
! For this reason we must have an unknown function W=W(xp) to be
determined by solving the quantum inverse scattering method. Assuming
that the eigenvalues are precisely the imaginary parts of th e zeroes of zeta
, in principle , we have defined the quantum inverse scatterin g problem .
Using wavelet analysis or p-adic harmonic analysis one could find the SUSY
potential Φ( x) =/producttext
pW(xp) where xp≡p−x, where xis the location of the
particle. How does on achieve such a numerical feat ? One coul d expand the
function W(xp) in Taylor series assuming that the potential is analytic, e x-
cept at some points representing the location of the infinite array of particles,
obstacles of the chaotic one-dimensional billiard, or the a toms ( scattering
centers ) of the one dimensional quasi periodic crystal . The adelic condition
1To ensure that Φ( −x) =−Φ(x) one must add a constant such that Φ(0) = 0
9for the SUSY potential becomes then :
For x > 0 Φ(x) =/producttext
pW(xp) =/producttext
p/summationtext
nan(xp)n
=/producttext
p/summationtext
nan[p−x]n=/producttext
p/summationtext
nanp−nx.(20a)
where Φ( x) =−Φ(−x).Recasting the Taylor series as a Dirichelt series by
simply rewriting :
p−nx=e−xnlogp=ei2xnlog p→
Φ(x) =/producttext
p/summationtext
nancos[ix(log pn)] +iansin[ix(log pn)].(20b)
(where once again Φ( x) =−Φ(−x))allows us to recast the adelic potential
as an infinite collection of p-adic harmonic oscillators with fundamental
imaginary frequencies ωp=ilog p and whose harmonics are suitable powers
of the fundamental frequencies : ωp,n=ilog pn=in log p .
We have then recast the quantum inverse scattering problem a s the problem
of solving the amplitudes an(and phases ) of the (imaginary frequencies )
p-adic harmonic oscillators by simply writing the adelic pot ential in terms
ofp-adic Fourier expansion ( p-adic harmonic analysis ) . This is attained
by means of performing the usual Wick rotation in Euclidean Q FT :ω→
iω. One could Wick rotate the imaginary Schroedinger equation (a diffusion
equation ) to an ordinary Schroedinger equation by the usual Wick rotation
trickt→it. Our adelic potential ansatz coincides with Pitkannen’s id eas
on the zeta function being the partition function of the adel ic ensemble of an
infinite system of p-adic oscillators with fundamental frequencies ωp=log p,
with the only difference that in our case we are performing the Wick rotation
of those frequencies.
The infinite unknown amplitude coefficients anwill be determined numer-
ically by solving the inverse quantum scattering problem in terms of the
eigenvalues of the SUSY QM model = imaginary parts of the zero es of zeta.
A. Odlyzko [15] has computed the first 1020(or more) zeroes of zeta. Hav-
ing a list of 1020zeroes should be enough data points to find the first 1020
numerical coefficients anappearing in the Taylor expansion of the potential
function Φ( x) =/producttext
pW(xp) that is being determined via inverse quantum
scattering methods associated with this SUSY QM model that l inks stochas-
tic dynamics , supersymmetry,chaos,etc. to the zeroes of th e Riemann zeta
function.
To summarize, if one can find a solution of the inverse scatter ing problem
that determines the (symmetric) potential V±(x) then one can
101) propose the Hilbert-Polya operator in the following form :
H= 1/2(H−H++H+H−) + 1/4. (21)
and
2) postulate that the eigenfunctions Ψ nof the Hamiltonian containing quartic
derivative His the fusion ( or convolution) of two eigenfuncrions Ψ+
n−1and
Ψ−
n. The ordinary product will not be suitable, as can be verified by simple
inspection. The fusion rules of this type have been widely us ed in conformal
field theories and in string theory.
The fused Hamiltonian operator is automatically self-adjo int as a result of
thefusion of the self−adjoint isospectral Hamiltonians H+, H−which
characterize the two ” dual ” Nagasawa’s diffusion equations [10] and its
realeigenvalues are the product of the nontrivial zeroes of the R iemann zeta
function and their complex conjugates :
HΨn= (1/4 +λ2
n)Ψn= (1/2 +iλn)(1/2−iλn)Ψn. (22)
. Notice that the value n= 0 is not included since Ψ+
−1isnotdefined and the
operator Hisquartic in derivatives. Could this candidate for Hilbert-Polya
operator be instrumental in proving the Riemann conjecture ?
The fusion or convolution product of the two eigenfunctions ofH±can be
found by referring to the Fourier transform: the Fourier tra nsform of an
ordinary product equals the convolution product of their Fo urier transforms
. Hence the eigenfunctions of Hcan be written, by denoting F, F−1the
Fourier transform and its inverse :
Ψn(x) =F−1[F(Ψ+
n−1∗F(Ψ−
n)]. n= 1,2,3, ... (23)
As expected, the eigenfunction of the fused Hamiltonian is notthe naive
product of the eigenfunctions of their constituents.
The 1 /4 coefficient present in the eigenvalues of the fused-Hamilto nian op-
erator is intrinsically related to the real part of the zeroe s of zeta (1 /2 +
iλn)(1/2 +iλn).The interpretation of the 1 /4 coefficient appearing in the
fused-Hamiltonan is as an additive constant,like a zero point energy of the
ordinary Harmonic oscillator. From the conformat field theo ry and string
theory point of view one constructs unitary irreducible hig hest-weight repre-
sentations of the Virasoro algebra for suitable values of th e central charges
and weights associated with the ground sates , candhrespectively. It is very
11plausible that supersymmetry and representation theory ma y select and fix
uniquely the value of 1 /4 which then would be an elegant proof of the Riem-
man conjecture. Pitkannen [12] has used conformal field theo ry arguments to
refine the Riemman conjecture : the imaginary parts of the zer oes correspond
to complex rational Pytagorean phases : piy.
Fractal p-branes in Cantorian-fractal spacetime and its relation to the zeta
function were considered by the author and Mahecha [ 16], and most recently
the role of fractal strings and the zeroes of zeta has appeare d in the book
by Lapidus and van Frankenhuysen [ 18] . Combining p-adic numbers and
fractals we arrive at the notion of p-adic fractal strings. The fundamental
question to ask would be how to establish a one-to-one corres pondence be-
tween the zeroes of zeta and the spectrum of p-adic fractal strings. This
would mean an establishment of a relation between the exact l ocation of the
poles of the scattering amplitues of p-adic fractal strings (a generalization
of the Veneziano formula in terms of Euler gamma functions) t o the exact
location of the zeroes of zeta, i.e. a one -to-one correspond ence between the
Regge trajectories in the complex angular momentum plane an d the spec-
trum of the p-adic fractal strings with the zeroes of zeta .
Since p-adic topology is the topology of Cantonan-fractal spaceti me it is not
surprising that the Golden Mean will play a fundamental role [2, 10, 16].
p-adic fractals have been discussed in full detail by Pitkann en [12]. p-Adic
Fractals are roughly speaking just fuzzy fractals [19], as they should be,
since Cantorian-fractal spacetime involves a von Neumann’ s noncommutative
pointless geometry . Wavelet analysis = p-adic Harmonic analysis must play
a fundamental role [17] in the classification of such spectru m. After all, the
scattering of a particle off a p-adic fractal string should be another way to
look at the p-adic stochastic motion discussed in this work.
Conclusion
We have been able to construct (assuming the quantum inverse scattering
problem has a solution) the operator that yields the zeroes o f the Riemman
zeta function by defining explicitly the SUSY QM model associ ated with
thep-adic stochastic dynamics of a particle undergoing a Browni an random
walk. We have been able to construct (assuming the quantum in verse scat-
tering problem has a solution ) the operator that yields the z eroes of the
Riemman zeta function by defining explicitly the SUSY QM mode l associ-
ated with the p-adic stochastic dynamics of a particle undergoing a Browni an
12random walk . The zig-zagging occurs after collisions with a n infinite array
of scattering centers that fluctuate randomly . Since the prime numbers are
themselves randomly distributed, this physical system can be reformulated
as the scatterings of the particle about the infinite locatio ns of the prime
numbers positions. Therefore, we have reformulated such p-adic stochastic
process which has an underlying hidden Parisi-Sourlas supe rsymmetry sim-
ilar to the effective motion of a particle subjected to an infin ite collection
ofp-adic harmonic oscillators with fundamental imaginary fre quencies given
byωp=ilog p and whose harmonics are ωp,n=ilog pn. This adelic ansatz
for the SUSY potential allow us to determine a one-to-one cor respondence
between the amplitudes of oscillations an( and phases ) with the imaginary
zeroes of zeta λn, after solving the inverse scattering problem. p-adic frac-
tal strings and their spectrum may establish a one-to-one co rrespondence
between the poles of their scattering amplitudes and the zer oes of zeta.
Acknowledgements
We are very grateful to A. Granik for his help in preparing thi s manuscript
and for many discussions. We also wish to thank M. Pitkannen, M. S. El
Naschie, J. Mahecha, M. Watkins for their correspondence an d insights.
References
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14 |
arXiv:physics/0101105v1 [physics.data-an] 30 Jan 2001Deconvolution problems in x-ray absorption fine structure
K V Klementev
Moscow State Engineering Physics Institute, 115409 Kashir skoe sh. 31, Moscow, Russia
(January 15, 2001)
A Bayesian method application to the deconvolution of EXAFS spectra is considered. It is shown
that for purposes of EXAFS spectroscopy, from the infinitely large number of Bayesian solutions it
is possible to determine an optimal range of solutions, any o ne from which is appropriate. Since
this removes the requirement for the uniqueness of solution , it becomes possible to exclude the
instrumental broadening and the lifetime broadening from E XAFS spectra. In addition, we propose
several approaches to the determination of optimal Bayesia n regularization parameter. The Bayesian
deconvolution is compared with the deconvolution which use s the Fourier transform and optimal
Wiener filtering. It is shown that XPS spectra could be in prin ciple used for extraction of a one-
electron absorptance. The amplitude correction factors ob tained after deconvolution are considered
and discussed.
61.10.Ht
I. INTRODUCTION
The chief task of the extended x-ray absorption fine-
structure (EXAFS) spectroscopy, determination of inter-
atomic distances, rms fluctuations in bond lengths etc.,
is solved mainly by means of the fitting of parameterized
theoretical curves to experimental ones. However, there
exist obstacles for such a direct comparison: theory lim-
itations and systematic errors. Among latter are various
broadening effects. Fist of all, (i) this is the broadening
arising from the finite energy selectivity of monochroma-
tor and the finite angular size of the x-ray beam. (ii)
The absorption even of strictly monochromatic x-ray ir-
radiation by the electrons of a deep atomic level gives
rise to photoelectrons with the finite energy dispersion
due to the finite natural width of this level and the fi-
nite lifetime of the core-hole. (iii) For x-ray energies far
above the absorption edge the process of photoelectron
creation (the outgoing from an absorbing atom) and the
process of its propagation occur for essentially different
time intervals. In other words, just created, the photo-
electron ‘does not know’ where and how it will decay.
Therefore the photoionization from the chosen atomic
level and excitation of the remaining system can be con-
sidered as independent processes, and hence the total ab-
sorption cross-section, as a probability density of two in-
dependent random processes, is given by the convolution
of a one-electron cross-section and excitation spectrum
W(∆E). The latter is the probability density of the en-
ergy ∆ Ecapture at the electron-hole pair creation and
is the quantity measured in x-ray photoemission spec-
troscopy (XPS). For light elements there are examples of
such deep and lengthy enough XPS spectra (see figure 1,
taken from [1]).
For all three cases the measured absorption coefficient
µmis given by the convolution:
µm(E) =W∗µ≡/integraldisplay
W(E−E′)µ(E′)dE′, (1)
/G10/G15 /G18 /G13 /G15 /G18 /G18 /G13 /G1A /G18 /G14/G13 /G13
/G25 /G4C /G51 /G47 /G4C /G51 /G4A /G03 /G48 /G51 /G48 /G55 /G4A /G5C /G0F /G03 /G48 /G39/G26 /G52 /G58/G51 /G57/G56 /G03 /G52 /G49 /G03 /G48 /G50 /G4C/G57/G57 /G48 /G47/G03 /G48 /G4F/G48 /G46 /G57/G55 /G52 /G51 /G56/G2E /G03 /G48 /G50 /G4C /G56 /G56 /G4C /G52 /G51
/G53 /G4F /G44 /G56 /G50 /G52 /G51
/G4F /G52 /G56 /G56 /G48 /G56
/G2E /G2F /G03 /G48 /G50 /G4C /G56 /G56 /G4C /G52 /G51/G30 /G4A /G32
/G30 /G4A /G29 /G15/G14 /G13 /G13 /G13
FIG. 1. From [1]: XPS spectra of MgO and MgF 2in the
vicinity of the Mg 1 speak. Two secondary structures, due to
plasmon losses and to double-electrton KL2,3excitations, are
detected. Zero of energy is placed at Mg Klevel ( ∼1300 eV).
where the broadening profile W(∆E) and the meaning
of the function µdepend on the considered problem.
These can be, correspondingly: x-ray spectral density
after the monochromator and the cross-section of ide-
ally monochromatic irradiation; the Lorentzian function
and the cross-section with a stationary initial level (of
zero width); the excitation spectrum and a one-electron
cross-section.
It is the common practice in modern EXAFS spec-
troscopy to account for the broadening processes (i)–(iii)
at the stage of theoretical calculations by introducing
into the one-electron scattering potential the imaginary
correlation part which represents the average interaction
between photoelectron and hole and their own polariza-
tion cloud; in doing so, the choice of the correlation part
is dictated by empiric considerations and can be different
for different systems.
1Another approach to the account for the broadening
processes is to solve the integral equation (1) for the un-
known µ. To find some solution of this equation is quite
not difficult to do, the simplest way is to use the theorem
about the Fourier transform (FT) of convolution. How-
ever, it is known that the problem of deconvolution is
an ill-posed one: it has an unstable solution or, in other
words, the infinitely large number of solutions specified
by different realizations of the noise. Thus, there is evi-
dent necessity for determination of an optimal, in some
sense, solution. Yet a less evident approach exists: to find
an appropriate functional of the solution which itself be
stable.
A number of works have been addressed the problem
of deconvolution, among them those concerning the x-
ray absorption spectra. Loeffen et al. [2] applied decon-
volution with the Lorentzian function partly eliminating
the core-hole life-time broadening. They used fast FT
and the Wiener filter which is determined from the noise
level which, in turn, is specified by the choice of the lim-
iting FT frequency above which the signal is supposed
to be less than noise. The arbitrariness of such a choice
gives rise to rather different deconvolved spectra, which
although remained obscured in [2].
Recently, for the deconvolution problem with a
Lorentzian function Filipponi [3] used the FT and pro-
posed the idea of the decomposition of an experimental
spectrum into the sum of linear contribution, a special
analytic function representing the edge and oscillating
part. For the Lorentzian response, the deconvolution for
the first two contribution is found analytically, for the
latter one, numerically. The advance of such a decom-
position is in that fact that now the FT of the oscillat-
ing part is not dominated by the very strong signal of
low frequency, therefore the combination of forward and
backward FT gives less numerical errors. Notice that
this method is solely suitable for the analytically given
response. In addition, in [3] the choice of the filter func-
tion (Gaussian curve) and its parameterization remained
vague. Therefore the issue on the uniqueness or optimal-
ity of the found solution was left open.
In early work [4], for the solution of ill-posed problems
the statistical approach was proposed. Following that
work, in the present paper we shall consider the deconvo-
lution problem in the framework of Bayesian method, de-
tailed formalism of which was described in [5]. Since the
parameterization is naturally involved in the Bayesian
method, there exists a principle possibility to choose an
optimal, in some sense, parameter. Here we shall scru-
tinize the problem of such a choice which is relevant to
any spectroscopy. We shall show that this problem is ab-
sent in EXAFS spectroscopy because though the EXAFS
spectrum itself does depend on the regularization param-
eter, its FT does not in the range of real space used for the
analysis. In Sec. II we discuss the choice of the optimal
deconvolution for a model Gaussian response and com-
pare the results of Bayesian approach with the results of
FT combined with Wiener filtering. In Sec. III we utilizethe deconvolution to an experimental spectrum in order
to eliminate the aforementioned broadening processes.
II. THE CHOICE OF OPTIMAL
DECONVOLUTION
First of all, we shall show that the deconvolution prob-
lem really has the infinitely large number of solutions.
In the present paper we use for examples the absorp-
tion spectrum of Nd 1.85Ce0.15CuO 4−δabove Cu Kedge
collected at 8K in transmission mode at LURE (beam-
line D-21) using Si(111) monochromator and harmonics
rejecting mirror; energy step ∼2 eV, total amount of
points 826 (from 8850eV to 10500eV), each one recorded
with integration time of 10s. Let us take for a while for
the response function a simple model form: W(E) =
Cexp(−E2/2Γ2), where Cnormalizes Wto unity, Γ is
chosen to be equal to 4 eV.
In [5] we showed howto construct a regularized solu-
tion of the convolution equation in the framework of the
Bayesian approach. For that, one needs to find eigenval-
ues and eigenvectors of a special symmetric N×Nmatrix
determined by the experimental spectrum, Nis the num-
ber of experimental points. Using that approach, find a
solution µ(E) for an arbitrary regularization parameter
αand perform its convolution with W. The ˆ µmobtained,
ideally, must coincide with µm. Introduce the character-
istics of the solution quality, the normalized difference of
these curves:
/G55 /G48 /G4A /G58 /G4F /G44 /G55 /G4C /G5D /G48 /G55 /G03 /G44/G35/G15/G0F/G03 /G14/G13/G10 /G18
/G47 /G48 /G57 /G0B /G4A /G0C /G36 /G12 /G31 /G0F /G03 /G14/G13/G16
/G14 /G13/G10 /G15 /G14 /G13/G13 /G14 /G13/G15/G17/G19
/G15
/G13
/G44 /G16
/G14 /G13/G10 /G16/G14 /G13/G10 /G15 /G13 /G13/G14 /G13/G15 /G13 /G13/G14/G53/G52 /G56/G57 /G48 /G55 /G4C /G52 /G55 /G03 /G53 /G0B /G44 /G0C/G0F/G03 /G44/G11 /G58/G11
/G44 /G14/G15
/G14/G13
/G44 /G14 /G56/G44 /G15
/G44 /G56 /G03 /G4C /G56
/G14 /G13/G10 /G14 /G14 /G13/G14
FIG. 2. The quality of the deconvolution R2vs. regulariza-
tion parameter α(solid). Dashed lines — signal-to-noise ratio
before and after deconvolution. Two peaks — the posterior
density functions p(α|d) (left) and p(α|d, σ) (right) for the
regularization parameter α. Straight line — the determinant
of Bayesian matrix as a function of α.
2R2=/summationdisplay
i(µmi−ˆµmi)2/slashBig/summationdisplay
iµ2
mi, (2)
where the summation is done over all experimental
points. Figure 2 shows the dependence R2onα. That
fact that the quality of the found solutions is practically
the same for all α<∼1 is a clear manifestation of ill-
posedness of the problem: there is no a unique solution.
How to chose an optimal one? It turns out, that for pur-
poses of EXAFS spectroscopy there is no need of that and
any solution from the optimal range (here, α<∼1) is suit-
able. At arbitrary αfrom the optimal range found the de-
convolution µ(E), extract the EXAFS function χ(k)·kw
in a conventional way, where kis the photoelectron wave
number, and find its FT. In figure 3 we show the EXAFS-
functions obtained after the Bayesian deconvolution with
α= 1 and α= 0.01, and their FT’s. As seen, although
the EXAFS-function itself does depend on α, its FT prac-
tically does not. Thus, if one uses for fitting a range of
r-space (in our example, up to rmax= 8˚A) or filtered k-
space, the problem of search for the optimal αis not rele-
vant. Nevertheless, below we propose several approaches
to the solution of this problem, for instance for XANES
spectroscopy needs.
(1) For the regularization parameter αitself one can
introduce the posterior probability density function [4,5 ]
and choose αwith a maximum probability density. It can
be done either using the most probable value of noise
or assuming the standard deviation σof the noise of
the absorption coefficient to be known (for our spectrum
σ= 9·10−4, as determined from the FT following [6]).
In figure 2 these probability densities are drawn as, cor-
respondingly, p(α|d) and p(α|d, σ), and their most prob-
able values are α1= 0.021 and α1σ= 0.044.
(2) The optimal regularization can be determined
from the consideration of the signal-to-noise ratio S/N.
The Shannon-Hartley theorem states that Imax =
Bln (1 + S/N), where Imaxis the maximum informa-
tion rate, Bis the bandwidth. The authors of [2] are of
opinion that deconvolution is a mathematical operation
that conserves information. Therefore from the theorem
follows that one pays for an increase in bandwidth, re-
sulted from deconvolution, via a reduction in S/Nratio.
The thesis on Imaxconservation is quite questionable,
since for deconvolution one should introduce additional
independent information about the profile of broaden-
ing. What quantity is conserved in deconvolution is hard
to tell. Here to the contrary, for the optimal αwe de-
mand to conserve S/N. Define S/Nas the ratio of mean
values of the EXAFS power spectrum over two regions,
r <15˚A and 15 < r < 25˚A. The regularization param-
eter at which the S/Nis conserved is denoted in figure
2 asα2= 5.54. The signal-to-noise ratio can be defined
in a different way. Since the Bayesian methods work
in terms of posterior density functions, for each experi-
mental point one can find not only the mean deconvolved
value but also the standard deviation δµdeconv from which
one finds δχ=δµdeconv/µ0, where µ0is the atomic-like/G13 /G14 /G15 /G16 /G17 /G18 /G19 /G1A /G1B
/G55 /G0F /G03 /G63/G13/G15/G17/G19/G5F /G29 /G37 /G3E /G46 /G0B /G4E /G0C /G11 /G4E/G15/G40 /G5F/G44/G56/G03 /G4C /G56
/G44 /G20 /G14
/G44 /G20 /G14 /G13/G10 /G15/G13/G15/G13 /G17 /G1B /G14 /G15 /G14 /G19 /G15 /G13/G4E /G0F /G03 /G63/G10 /G14
/G13/G15/G46 /G11 /G4E/G15/G13/G15
/G13/G13/G11/G13 /G14/G13/G11/G13 /G15/G13/G11/G13 /G16/G47 /G46 /G47 /G48/G46 /G52 /G51/G59/G44 /G56 /G03 /G4C /G56/G44 /G20 /G14 /G13/G10 /G15
/G44 /G20 /G14
/G44 /G20 /G14/G44 /G20 /G14 /G13/G10 /G15
FIG. 3. χ·k2obtained without deconvolution and after
that with α= 1 and α= 0.01. In the middle — the envelope
of the initial χand rms deviations of the deconvolved values
(dots). Below — the absolute values of the Fourier transform
(the dashed and the dotted curves practically merge).
absorption coefficient constructed at the stage of EXAFS
function extraction. It is reasonable to compare δχvalues
with the envelope of EXAFS spectrum (figure 3, middle).
As seen, at small αthe noise dominates over the signal
in the extended part of the spectrum. The regularization
parameter at which they match is the optimal one, α2.
(3) For the Bayesian deconvolution it is necessary to
find eigenvalues and eigenvectors of a special symmetric
matrix g. It turns out that the determinant of this matrix
varies with αover hundreds orders of magnitude. At
small α’s the matrix is poorly defined, large α’s yield very
3/G13 /G13/G11 /G15 /G13/G11 /G17 /G13/G11 /G19
/G57 /G0F /G03 /G48 /G39/G10 /G14/G13/G11 /G14/G14/G14 /G13/G49 /G4C /G4F/G57 /G48 /G55/G13/G15/G17/G19/G1B/G5F /G29/G37 /G3E /G50 /G0B /G28 /G0C/G40 /G5F
/G3A/G0B /G57 /G0C /G20 /G48 /G5B /G53 /G0B /G2A/G15 /G57/G15 /G12 /G15 /G0C/G44 /G56 /G03 /G4C /G56/G44 /G20 /G14/G44 /G20 /G14 /G13/G10 /G14/G44 /G20 /G14 /G13/G10 /G15
/G3A /G11 /G29
/G44 /G20 /G14/G44 /G20 /G14 /G13/G10 /G14/G44 /G20 /G14 /G13/G10 /G15
FIG. 4. Module of the FT of initial µmand deconvolved µ
at different α. Below — filters transforming µm(τ) toµ(τ)
obtained after Bayesian deconvolution (thin solid lines), after
deconvolution based on the FT (dashed line), and after de-
convolution based on the combination of the FT and Wiener
filtering (thick solid line).
large det( g) (figure 2). Both cases give large numerical
errors because of ratios of very small or very large values
in calculations. As an optimal parameter we choose α3=
1.41 at which det( g)∼1.
The cases (1) and (2) require to determine the noise
level, which demands additional variables (for instance,
the limiting frequencies of FT). The case (3) does not
explicitly concern the noise. Due to the dependence of
lg[det( g)] onαappears to be linear, which readily allows
one to find the optimal parameter, the case (3) is more
preferable from the practical point of view. Below, for
deconvolution of the real broadening processes we use
the optimal parameter α3.
It is of certain interest to compare the Bayesian method
of deconvolution with the widely known method combin-
ing optimal Wiener filtering and the convolution theo-
rem, where the conjugate variables of the FT are not k
and 2 radopted in EXAFS but Eandτ. According to
the theorem, µm(τ) =W(τ)·µ(τ), where for our model
Gaussian response W(τ) = exp(Γ2τ2/2). A simple back
FT of the ratio µm(τ)/W(τ) will give the thought solu-
tionµ(E) but very noisy. Therefore µm(τ) at large τhas
to be smoothed. Figure 4 shows module of the FT of
the measured spectrum and of the Bayesian deconvolved
spectra. The latter are merged at τ<∼0.25eV−1. In the
bottom part of the figure the ratios |FTµdeconv|/|FTµm|are shown for different α. As seen, the Bayesian decon-
volution performs the effective filtration of spectra with
the limiting frequency τmaxdepending on α.
The optimal, in the least-square sense, Wiener filter
is expressed as [7]: Φ( τ) = (1 + |n(τ)|2/|FTµm|2)−1,
where |n(τ)|2is the power spectrum of the noise re-
placed here by 0.01, the mean value of |µ(τ)|2over the
range τ >0.4eV−1. As seen in figure 4, the effective
Wiener filter W(τ)Φ transforming µm(τ) toµ(τ) is close
to the effective filter of the Bayesian deconvolution with
α= 1≈α3. Notice, however, that the limiting fre-
quency for the estimation of noise power spectrum was
chosen rather arbitrarily.
In closing this section, it should be noticed that apart
from the possibility of determination of the deconvolu-
tion errors and the possibility of the optimal regulariza-
tion parameter choice, the Bayesian deconvolution has
the advantage of the capability to take into account a pri-
oriinformation about the smoothness and shape of the
solution (see details in [5]). In addition, in the Bayesian
method the response function W(E−E′) could be of
more general form W(E−E′, E), which will be useful
for deconvolution of the instrumental broadening because
monochromator energy resolution depends noticeably on
the angular position and, hence, on the energy of the
output x-ray beam.
III. APPLICATIONS OF DECONVOLUTION
We have seen that the Bayesian method proves to be
effective for deconvolution of EXAFS spectra, and the
choice of the regularization parameter appears to be ir-
relevant. Now we perform the deconvolution of various
types of broadening, for which purpose specify the corre-
sponding response functions W(E−E′, E).
A. Instrumental broadening
The monochromator resolution is determined by the
rocking curve width δθBand by the vertical beam diver-
gence δθ⊥. For the monochromator Si(111) at E= 9keV
the rocking curve width is δθB= 32.4µrad (FWHM)
[8], the beam divergence (LURE, D-21) δθ⊥= 150 µrad.
Strictly speaking, the resulting spectral distribution is
given by the convolution of rocking curve and the an-
gular beam profile. But since δθB≪δθ⊥, the energy
selectivity is determined by δθ⊥, namely: δE⊥/E=
δθ⊥cotθB=δθ⊥/radicalbig
(2Ed/ch )2−1, where θBis Bragg
angle, dis Bragg plane spacing. Modelling the spectral
distribution by a Gaussian function, one obtains:
W(E−E′, E)∝exp/bracketleftbigg
−(E−E′)2
2σ2
⊥(E)/bracketrightbigg
, σ⊥(E) =δE⊥(E)
2√
2 ln2,
where the normalization constant must be calculated at
eachEvalue. For our sample spectrum, σ⊥(8850 eV) =
42.46eV and σ⊥(10500 eV) = 3 .49eV.
B. Lifetime broadening
For deconvolution of the lifetime broadening described
by a Lorentzian function W(∆E)∝[(∆E/ΓK)2+ 1]−1,
we take as the initial spectrum µmthe spectrum µinstrob-
tained after the instrumental deconvolution. According
to [9], the width of Cu Klevel (FWHM) equals 1.55eV,
from where Γ K= 0.775eV.
C. Multielectron broadening
There are certain difficulties in measuring XPS spec-
tra near (and deeper) the deepest atomic levels: the
monochromatic x-ray sources of high energy are required;
for long enough spectra ( ∼100eV) a photoelectron an-
alyzer with a broad energy window and long integration
time are necessary. Unfortunately, for lack of experi-
mental XPS spectra in cuprates near Cu 1 slevel, we
can use a model representation of the response W(∆E).
For that we take the estimations of position, intensity
and width of the secondary KM23excitation from [10]:
EKM23−EK= 85eV; IKM23/IK= 0.03; Γ KM23= 3eV.
finally, for the broadening function we have:
W(∆E)∝IKΓK
∆E2+ Γ2
K+IKM23ΓKM23
(∆E−EKM23)2+ Γ2
KM23.
Again, as the initial spectrum we take the spectrum
µinstr.
IV. DISCUSSION
With the specified response functions W, perform the
Bayesian deconvolution of the absorption coefficient at
the optimal regularization parameter, then construct the
EXAFS function for which calculate FT and the ampli-
tudes and phases (see figure 5). Just as for the model re-
sponse in Sec. II, the deconvolution leads to the increase
of EXAFS oscillations. As appeared, the deconvolution
has practically no influence on the first FT peak origi-
nating from the shortest scattering path. It is clear why:
the oscillations corresponding to this peak are essentiall y
wider then the response W(for these, Wis almost a δ-
function), and this is more true for the extended part of
a spectrum, due to the period of the oscillations there
is even longer (in E-space). Thus, it is in the extended
part where µ, the solution of equation (1), less differs
fromµm.
In modern EXAFS spectroscopy the difference between
amplitudes of experimental and calculated spectra are
taken into account by the reduction factor S2
0which is
either treated as a fitting parameter or estimated from/G13 /G14 /G15 /G16 /G17 /G18 /G19 /G1A /G1B/G55 /G0F /G03 /G63
/G13/G15/G17/G19/G5F /G29 /G37 /G3E /G46 /G0B /G4E /G0C /G11 /G4E/G15/G40 /G5F/G50 /G44 /G51 /G5C /G10 /G48 /G4F /G48 /G46 /G57 /G55 /G52 /G51
/G4F /G4C /G49 /G48 /G57 /G4C /G50 /G48
/G4C /G51 /G56 /G57 /G55 /G58 /G50 /G48 /G51 /G57 /G44 /G4F
/G50 /G48 /G44 /G56 /G58 /G55 /G48 /G47
/G13 /G17 /G1B /G14 /G15 /G14 /G19 /G15 /G13
/G4E /G0F /G03 /G63/G10 /G14/G13/G11 /G1B/G13/G11 /G1C/G14/G24 /G0B /G4E /G0C /G20 /G44 /G50/G0B /G46 /G4C /G51 /G4C/G57 /G0C /G12 /G44 /G50/G0B /G46 /G47 /G48/G46/G52 /G51 /G59 /G0C
/G36/G13/G15
/G4C /G51 /G56 /G57 /G55 /G58 /G50 /G11 /G03 /G12 /G03 /G4F /G4C /G49 /G48 /G57 /G4C /G50 /G48
/G4C /G51 /G56 /G57 /G55 /G58 /G50 /G11 /G03 /G12 /G03 /G50 /G44 /G51 /G5C /G10 /G48 /G4F /G48 /G46 /G57 /G55 /G52 /G51
/G50 /G48 /G44 /G56 /G58 /G55 /G48 /G47 /G03 /G12 /G03 /G4F /G4C /G49 /G48 /G57 /G4C /G50 /G48
/G50 /G48 /G44 /G56 /G58 /G55 /G48 /G47 /G03 /G12 /G03 /G50 /G44 /G51 /G5C /G10 /G48 /G4F /G48 /G46 /G57 /G55 /G52 /G51
FIG. 5. Module of the FT of various EXAFS functions:
initial; obtained from the instrumentally deconvolved abs orp-
tance; the latter was deconvolved with the Lorentzian re-
sponse and with the total multielectron response, these two
give the EXAFS FT’s practically merged. Bottom: the ratio
of amplitudes of initial χand deconvolved χ. The S2
0value
was calculated from atomic overlap integrals.
the relaxation of the core-hole as the many-electron over-
lap integral. In many works this factor is considered to
be independent from energy, however, as noted in review
by Rehr and Albers [11], it must be path-dependent and
energy-dependent. At the bottom of figure 5 we draw
the ratios A(k) of the amplitude of initial EXAFS spec-
trum to that of the deconvolved one. Here, they were
calculated relatively both χmandχinstr. For comparison
we show the factor S2
0as computed by feff code [12].
At large k, where the noise become comparable with the
EXAFS signal, the ratios A(k) have significant errors.
However, the general trend of the curves corresponds to
the expected one [13]: A(k) is minimal at intermediate
EXAFS energies, while at both low and high energy A(k)
reduces to unity. In addition, there are some phase shifts
between the initial spectrum and the deconvolved ones.
But these shifts are found to be quite small: less than
0.2rad at k <4˚A and less than 0.1rad at k >4˚A.
The Lorentzian broadening of the EXAFS spectrum
with a half-width Γ is similar to the effect of the imagi-
nary part of the self-energy with ImΣ = Γ. The resulting
reduction factors, i.e. the ratios the of amplitudes calcu-
lated with and without the imaginary part, are analogous
5to the reduction factors obtained by us relatively χinstr:
A(k) = am( χinstr)/am(χdeconv). However, up to now the
reduction factors relatively measured EXAFS spectrum
were considered, A(k) = am( χm)/am(χdeconv). As seen
(figure 5), these are the noticeably different quantities.
That is why for the correct analysis and comparison of
spectra taken at different experimental conditions, the
instrumental deconvolution must be the first step.
For our example spectrum and the chosen response
functions, deconvolution of the lifetime broadening and
deconvolution of the multielectron broadening are practi-
cally undistinguishable (figure 5, top), i.e. the secondary
weak peak in the excitation spectrum W(∆E) has very
little effect. The main effect of using real excitation
spectra is expected from the presence and taking into
account the plasmon losses which have a considerable
integral weight (figure 1). Because of their very broad
spectral distribution, their effect consists in the change
of the EXAFS spectrum as a whole. In the present paper
this contribution was not taken into account for lack of
appropriate experimental information.
Near the absorption edge, where the photoelectron ki-
netic energy is low, the core-hole relaxation processes are
of certain importance for the photoelectron propagation.
Here we do not consider the validity of the neglect of this
effect, but refer to the review [11].
V. CONCLUSION
To take into account the many-electron effects, there
exist, in principle, two approaches: (a) to include into
a one-electron theory relevant amendments or (b) to ex-
tract a one-electron absorptance from the total one and
to use then a pure one-electron theory. The first, tra-
ditional, approach invokes semi-empirical rules, but not
ab initio calculations, to construct the exchange corre-
lation part of the scattering potential, with the empiri-
cism being based on the comparison with experimental
spectra already broadened . In the present paper we have
shown the principle way for the second approach, using
the solution of integral convolution equation, the kernel
in which is the excitation spectrum measured in XPS
spectroscopy. Notice, that owing to the specific way of
the structural information extraction from the EXAFS
spectra, in which an isolated signal in r-space or a fil-
tered signal in k-space is used, we have not committed
a sin against the fact that the integral convolution equa-
tion is an ill-posed problem, because from the infinitely
large number of solutions it is possible to determine an
optimal range, also infinitely large, of solutions any one
from which is appropriate.
Because of some technical difficulties, it is impossible
so far to measure XPS spectra near deep core levels.
Therefore, the desirable pure one-electron absorptance
is unavailable. Nevertheless, as we have shown, it is pos-
sible to perform an accurate instrumental deconvolutionand deconvolution of the lifetime broadening. These pro-
cedures make the comparison between calculated and ex-
perimental spectra more immediate and the final results
of EXAFS spectroscopy more reliable.
All the stages of EXAFS spectra processing including
those described here are realized in the freeware program
viper [14].
ACKNOWLEDGMENTS
The example spectrum was measured by Prof. A. P.
Menushenkov. The author wishes to thank Dr. A. V.
Kuznetsov for many valuable comments and advices. The
work was supported in part by RFBR grant No. 99-02-
17343.
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chrotron Rad. 6, 264 (1999), (Proc. of Int. Conf. XAFS
X).
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Flannery, Numerical Recipes in Fortran 77: The Art of
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6 |
arXiv:physics/0101106v1 [physics.plasm-ph] 30 Jan 2001Intermittency and turbulence in a magnetically confined fus ion plasma
V. Carbone1, L. Sorriso–Valvo1, E. Martines2, V. Antoni2,3, and P. Veltri1
1Dipartimento di Fisica, Universit` a degli studi della Cala bria, 87036 Rende (CS), Italy,
and Istituto Nazionale di Fisica della Materia Unit` a di Cos enza
2Consorzio RFX, corso Stati Uniti 4, 35127 Padova, Italy
3Istituto Nazionale di Fisica della Materia, Unit` a di Padov a
(January 10, 2014)
We investigate the intermittency of magnetic turbulence
as measured in Reversed Field Pinch plasmas. We show that
the Probability Distribution Functions of magnetic field di f-
ferences are not scale invariant, that is the wings of these
functions are more important at the smallest scales, a clas-
sical signature of intermittency. We show that scaling laws
appear also in a region very close to the external wall of the
confinement device, and we present evidences that the ob-
served intermittency increases moving towards the wall.
PACS Number(s): 52.35.Ra; 52.55.Ez; 52.55.Hc
The issue of self–similarity is of paramount importance
in turbulence studies. Indeed, self–similarity is one of th e
key hypotheses of Kolmogorov theory [1,2], which leads
for fluid turbulence to the famous −5/3 exponent for the
power spectrum decay in the inertial range (the interme-
diate range of scales between the large scales where en-
ergy is injected and the small ones where it is dissipated).
Notwithstanding the success of the Kolmogorov’s the-
ory, the study of the Probability Distribution Functions
(PDF) of velocity fluctuations at a given scale has shown
a departure from gaussianity in the PDF tails. The same
phenomenon is usually evidenced by looking at the scal-
ing exponents of higher order moments of fluctuations,
which appear to be nonlinear functions of the order in-
dex. Intermittency described by multifractal models [2]
is usually invoked to be the cause of the observed break of
pure self–similarity. Even if Kolmogorov theory was orig-
inally developed for homogeneous and isotropic turbu-
lence, the evidence for self–affine fields has been studied
also inside the boundary layer turbulence in fluid labo-
ratory experiments [3] which is neither homogeneous nor
isotropic. The only hypothesis required to perform these
studies from an experimental point of view, in order to
apply the usual Taylor’s hypothesis, is the statistical sta -
tionarity of turbulence. Recently it has been shown [4]
that only after a suitable decomposition in terms of ir-
reducible representations of the SO(3) groups one can
hope to properly disentangle isotropic from anisotropic
effects in Navier–Stokes equations. Of course this should
be true also in MHD flows, even if it is not clear how
to recover anisotropic and non homogeneous effects from
real data.
While in ordinary fluids the statistical properties of
turbulence have been well characterized, both theoreti-cally and experimentally, in magnetized fluids only re-
cently this has been undertaken, mostly in relation with
velocity and magnetic field fluctuations measured in the
solar wind [5]. In this paper we report evidences for
the presence of intermittency in another type of mag-
netized fluid, namely a plasma of interest for controlled
thermonuclear fusion research, confined in reversed field
pinch (RFP) configuration.
The RFP is a configuration of magnetic fields [6] char-
acterized by toroidal and poloidal components of com-
parable magnitude (in a tokamak the field is mainly
toroidal). The configuration represents a near-minimum
energy state to which a plasma relaxes under proper con-
straints [7]. The toroidal field changes sign in the outer
part of the plasma, a feature which gives the name to
the configuration. Such field reversal, which improves
the MHD stability of the configuration, is spontaneously
generated by the plasma, and is maintained against resis-
tive diffusion by the dynamo process [8]. This is achieved
through the action and nonlinear coupling of several re-
sistive magnetohydrodynamic (MHD) modes, which give
rise to a high level of magnetic turbulence (of the order
of 1% of the average field in present day experiments,
i.e. two orders of magnitude larger than in tokamaks).
This high fluctuation level makes the RFP very suited
for the study of MHD turbulence properties, mainly for
their magnetic part. The magnetic turbulence has been
demonstrated to be the main cause of energy and par-
ticle transport in the RFP core, whereas at the edge its
contribution is still under investigation. In this region
the electrostatic turbulence has been proved to give an
important contribution to the particle transport [9]. It
is worth mentioning that a recent investigation of the
edge electrostatic turbulence in different fusion devices
(including RFP and tokamaks) has shown the existence
of long range time and space correlations [10].
The measurements described in this paper have been
obtained in the RFX experiment, which is the largest
RFP presently in operation (major radius 2 m, minor ra-
dius 0 .457 m) [11]. RFX is designed to reach a plasma
current of 2 MA, and currents up to 1 MA have been ob-
tained up to now. The measurements were performed in
low currents discharges (300 kA) using a magnetic probe
inserted in the edge plasma. The probe consists of a coil
housed in a boron nitride protecting head. The coil mea-
sures the time derivative ∂tBof the radial component
B(t) of the magnetic field. The radial direction in this
1case goes from the core plasma to the edge. The sampling
frequency of the measurements is 2 MHz. Measurements
have been collected at different values of the normalized
radius r/a(r/a= 1 identifies the location of the last
magnetic flux surface) In RFX two different components
of the magnetic fluctuations can be identified: a localised
and stationary magnetic perturbation, originated by the
tearing modes responsible for the dynamo which tend to
be phase–locked and locked to the wall [12], and an high
frequency broadband activity, which is investigated here.
All measurements presented were made away from the
stationary perturbation.
We start by looking at the statistical properties of the
normalized variables s(t) =∂tB//radicalbig
<(∂tB)2>(brackets
being time averages). In figure 1 we show the flatness of
these stochastic variables f=< s4> /[< s2>]2for some
positions r/a. As it can be seen f(r/a) is higher than
the gaussian value and tends to decrease as r/aincreases.
This is a first rough evidence that the observed magnetic
field is intermittent, that is the time evolution of ∂tBis
dominated by strong magnetic fluctuations. The inter-
mittency (say the departure from a gaussian statistics)
is more visible near the external wall.
0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96
r/a246810
f(r/a)
FIG. 1. We show the values of the flatness f(r/a) for the
derivative of the radial magnetic field, as a function of the
insertion r/a. The gaussian value ( f= 3) is shown as a
dotted line.
To get some insight into the nature of intermittency
actually present in the fusion device, following the usual
analysis currently made in fluid flows [2], we investigate
the scaling behavior of the stochastic variables δB(τ) =
B(t+τ)−B(t), which represents characteristic fluctua-
tions across turbulent structures at the scale τ. For each
position within the device, we can study the statistical
behavior of fluctuations at different scales τ. The interest
of this resides in the fact that, if we introduce a scaling
law for magnetic fluctuations δB(τ)∼τhas MHD equa-
tions seem to indicate [13], a scale variation τ→λτ(λis the parameter which defines the change of scale) leads
to
δB(λτ) =λ−hδB(τ)
This is interpreted as an “equality in law” [2], that is
the right–hand–side of the equation has the same sta-
tistical properties of left–hand–side. If his constant we
can easily show that the PDFs of the normalized stochas-
tic variables δbτ=δB(τ)//radicalbig
<[δB(τ)]2>collapse to a
unique PDF, independent on the scale τ. This is true in a
pure self–similar (fractal) case. On the contrary we must
invoke the multifractal model to describe intermittency
[2] which is introduced by defining a range of values of h.
In figure 2 we report the PDFs of δbτat different scales
for a given value of r/a. As it can be seen the PDFs do
not collapse to a single curve, but follow a characteristic
scaling behavior which is visible for all values of r/a. At
large scales the PDF are almost gaussian, and the wings
of the distributions grow up as the scale becomes smaller.
Stronger events at small scales have a probability of oc-
currence greater than that they would have if they were
distributed according to a gaussian function. This be-
havior, that is the presence of self–affine fields, is at the
heart of the phenomenon of intermittency as currently
observed in fluid flows [2,14].
0.0010.010.11.0pdf(b)
=0.5s =2s
-4 -2 0 2 4
b0.0010.010.1pdf(b)
=32s
-2 0 2 4
b=512s
FIG. 2. We show the PDFs of the normalized magnetic
fluctuations for four different scales, at a given position
r/a= 0.95. The full line represents the fit made with the
convolution function.
The behavior of PDFs against the scale can be de-
scribed by introducing a given shape for the distribution.
At each scale τthe PDF of δbτcan be represented as a
convolution of gaussian functions of widths σwhose dis-
tribution is given by a function Gλ(σ)
P(δbτ) =1√
2π/integraldisplay∞
0Gλ(σ)exp/parenleftbig
−δb2
τ/2σ2/parenrightbigdσ
σ(1)
2which can be interpreted in the framework of a cascade
model as the signature of an underlying multiplicative
process [14–16]. We use a log–normal ansatz
Gλ(σ) =1√
2πλexp/parenleftbigg
−ln2σ/σ0
2λ2/parenrightbigg
(2)
even if other functions does not give really different re-
sults [14]. The free parameter λ2represents the width of
the distribution Gλ, while σ0is the most probable value
ofσ. The scaling behavior of P(δbτ) is translated in
the scaling variation of the parameter λ2[14,16]. In fact
when the PDF is gaussian λ2= 0 (Gλbecomes a delta
function centered around σ0), while the departure from
a gaussian function increases as λ2increases. In figure 2
we report as full line a fit of the data with equation (1).
A satisfactory agreement at all scales is evident.
Looking at the scaling laws for λ2, at different insertion
points r/a, it can be seen (figure 3) that λ2displays a
power law behavior
λ2(τ, r/a) = Λ2(r/a)τβ
all over the observed time scales, for insertion points near
the external wall. On the contrary measurements more
inside the device show a saturation of intermittency at
scales τS≃10µs. The values of βwe find are of the or-
der of β≃0.42±0.03, close to that found for the velocity
field both in fluid flows and in the solar wind turbulence
[14,16], but higher than the value found for the magnetic
field intensity in the solar wind [16]. Finally the absolute
values for λ2are decreasing going from the wall inside
the device. Namely we found λ2
max= 0.21±0.01 for
r/a= 0.98, and λ2
max= 0.086±0.006 for r/a= 0.84.
This is a further confirmation of the stronger intermit-
tency near the external wall. All the error–bars has been
estimated starting from a poisson statistical uncertainit y
on the PDFs value.
1 10 100
(s)0.010.1
2()FIG. 3. Scaling behavior of the exponent λ2(τ, r/a) for
three different insertion r/a, namely: r/a= 0.97 (black cir-
cles), r/a= 0.95 (white circles), and r/a= 0.91 (stars).
A complementary analysis of intermittency can be per-
formed by calculating the scaling exponents of the struc-
ture functions, say of the p–th moments of fluctuations
S(p)
τ=< δbp
τ>(brackets are defined as time averages).
In figure 4 we report the structure functions S(p)
τ, for two
values of p, and for two different position r/a. The dif-
ferences for different position is evident, and represent a
signature of the absence of universality.
1 10 100
(s)10-610-510-410-310-210-1
S(p)
FIG. 4. The structure functions S(p)
τare shown for p= 2
(circles) and p= 3 (squares). Open symbols refer to the
position r/a= 0.96, full symbols refer to r/a= 0.86.
To calculate the scaling exponents, we use the gener-
alized scaling introduced by Benzi and coworkers [17],
which has been found to be useful also in magnetohydro-
dynamic turbulence [13,18], thus obtaining the normal-
ized scaling exponents ζpdefined through S(p)
τ∼[S(3)
τ]ζp.
31 2 3 4 5 6
p0.00.51.01.52.0
p p = p/3
r/a = 0.96
r/a = 0.93
r/a = 0.90
r/a = 0.86
FIG. 5. The normalized scaling exponents ζpof the struc-
ture functions are shown as a function of p, for different inser-
tion points r/a. Errorbars, about 5% of the exponent values,
are not displayed for clarity. The K41 scaling ζp∼p/3 is also
reported for comparison.
In figure 5 we report the scaling exponents obtained
for some insertions r/a. The behavior of ζpagainst p
shows that scaling exponents are anomalous, say they
are different from the usual p/3 Kolmogorov’s law. Note
once more that the strength of intermittency, measured
through the difference between ζpandp/3, is greater near
the wall. In conclusion scaling laws for PDFs of magnetic
fluctuations, and anomalous scalings for structure func-
tions, are found everywhere in the outer plasma region
of the RFX thermonuclear fusion experiment. We find
that the anomaly of scaling exponents, as well as scaling
laws for PDFs, strongly depend on the position inside the
plasma, so that magnetic turbulence inside the device is
not universal, as far as scaling laws are concerned. Pos-
sible reasons for this are the presence of the first wall,
the presence of the toroidal field reversal (which takes
place at r/a≃0.9) or the strongly sheared plasma flow
measured in the RFX edge [19]. Concerning this latter
option, it is worth to mention that in principle differ-
ent plasma velocities in different points would only affect
the relationship between time and spatial scales obtained
through Taylor hypothesis, and not the PDF scaling
properties. However, the eddy breaking effect induced
by a velocity shear is well known to affect electrostatic
turbulence in fusion plasmas [20,9], and an influence on
MHD turbulence can also be envisaged, either directly
or thorugh nonlinear coupling to electrostatic modes. If
this is not the case, the reason for the observed differ-
ences could be perhaps found in the conjecture of Farge
[21]. She proposed that turbulence could be described
by interwoven sets of both intermittent structures and
background gaussian flow on each characteristic scale.
The nature of the intermittent structures can evidently
be influenced by walls [3], and/or current sheets asso-ciated with field reversal [22]. We are actually review-
ing and testing this idea on the RFX device in order to
identify structures which generate intermittency. Since a
reduction of magnetic fluctuations has been linked to im-
provements in the energy confinement [23], a better un-
derstanding of the generation of intermittency through
structures could improve the confinement physics under-
standing.
ACKNOWLEDGMENTS
We are grateful to Francesco Pegoraro for some discus-
sions and for its interest in this work.
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434, 9 (1995).
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(1999)
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R29 (1993).
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5 |
1The cosmic origin of supersymmetry and internal symmetry
Ding-Yu Chung
Supersymmetry involves a symmetrical change in spacetime labels in terms of
spacetime coordinates, while internal symmetry involves a symmetrical changein non-spacetime labels in terms of charged fields, isotopic spin, and colors. Inthis paper, the cosmic origin of supersymmetry and internal symmetry isexplained by the cyclic dual universe, the coexistence of the unobservable bigbag universe and the observable big bang universe. The cosmic cycle starts withthe eleven dimensional Planck supermembrane with the Planck mass. Thesupersymmetry of the Planck supermembrane is broken into a Kaluza-Kleinstructure where the lower dimensional spacetime has lower energy than thehigher dimensional spacetime. In the big bag universe, the Plancksupermembranes are slowly broken and decay into concerted particles that thenare slowly recombined. It undergoes a gigantic slow cosmic oscillation betweenthe high-energy eleven dimensional spacetime and the low-energy fourdimensional spacetime. It is like a gigantic cosmic particle-wave. The big banguniverse is the collapsed cosmic oscillation due to the disruption by the non-spacetime labels, independent of the spacetime of the cosmic oscillation. It islike the collapse of microscopic particle-wave by a system (such asmeasurement) independent of microscopic particle-wave. The collapse of thecosmic oscillation results in the immediate decay into the four dimensionalspacetime and the seven dimensional internal space (non-spacetime). Theremnants of the cosmic oscillation are cosmic radiation, gravity, and microscopicparticle-wave in the four dimensional spacetime. Internal symmetries for gaugebosons, leptons, quarks, and black holes are in the seven dimensional internalspace. There is no curled-up space dimension and compacitification. Except atthe very beginning of the cosmic cycle and black hole, the big bang universe haspure four-dimensional spacetime. The unobservable big bag universe is thesupersymmetry four-to-eleven dimensional spacetime universe, while theobservable big bang universe is the internal symmetry four-plus-sevendimensional universe. The two universes are separated by the dematerializedzone everywhere all the time. The big bang universe absorbs space from andemits space to the dematerialized zone, when the spacetime dimensions in bothuniverses are compatible. The results are the inflation during the big bang, thelate-time cosmic accelerated expansion, the cosmic accelerated contraction, andthe final cosmic accelerated contraction. The cosmic cycle starts with all Plancksupermembranes for both universe, and ends with all black hole universe in thebig bang universe. The inflation is not an add-on but a necessity to start thecosmic cycle. According to the calculation, we are now at the peak of the late-time cosmic accelerated expansion.2 1. Introduction
Supersymmetry involves a symmetrical change in spacetime labels in
terms of spacetime coordinates, while internal symmetry involves a symmetricalchange in non-spacetime labels in terms of charged fields, isotopic spin, andcolors. In this paper, the cosmic origin of supersymmetry and internal symmetryis explained by the cyclic dual universe, the coexistence of the unobservable bigbag universe and the observable big bang universe. The cosmic cycle for bothuniverses starts with the eleven dimensional Planck supermembrane with thePlanck mass. The supersymmetry of the Planck supermembrane is broken into aKaluza-Klein structure where the lower dimensional spacetime has lower energythan the higher dimensional spacetime. In the big bag universe, the Plancksupermembranes are slowly broken and decay into concerted particles that thenare slowly recombined. It undergoes a gigantic slow cosmic oscillation betweenthe high-energy eleven dimensional spacetime and the low-energy fourdimensional spacetime. It is like a gigantic cosmic particle-wave.
The big bang universe is the collapsed cosmic oscillation due to the
disruption by the non-spacetime labels, independent of the spacetime of thecosmic oscillation. It is like the collapse of microscopic particle-wave by asystem (such as measurement) independent of microscopic particle-wave. Thecollapse of the cosmic oscillation results in the immediate decay into the fourdimensional spacetime and the seven dimensional internal space (non-spacetime).
The two universes are separated by the dematerialized zone everywhere
all the time. The big bang universe absorbs space from and emits space to thedematerialized zone, when the spacetime dimensions in both universes arecompatible. The results are the inflation during the big bang, the late-timecosmic accelerated expansion, the cosmic accelerated contraction, and the finalcosmic accelerated contraction. The cosmic cycle starts with all Plancksupermembrane for both universe, and ends with all black hole universe for thebig bang universe.
In the Section 2, the unobservable big bag universe involving the
supersymmetry breaking of the eleven dimensional Planck supermembrane andthe cosmic oscillation will be discussed. The Section 3 covers the cosmicoscillation breaking. The Section 4 briefly reviews the observable big banguniverse. In the Section 5, the four periods of the active cosmic dematerializedzone will be discussed.
2. The Unobservable Big Bag Universe
A supermembrane can be described as two dimensional object that moves
in an eleven dimensional space-time [1]. This supermembrane can be convertedinto the ten dimensional superstring with the extra dimension curled into a circle tobecome a closed superstring. Based on this eleven-dimensional3supermembrane, the extra space dimension assumes a Kaluza-Klein substructure
for the broken Planck supermembrane.
In the Kaluza-Klein substructure, each space dimension can be described
by a fermion and a boson. The masses of fermion and its boson partner are notthe same. This supersymmetry breaking is in the form of a mass hierarchy withincreasing masses from the dimension four to the dimension eleven as
F
4 B4 F5 B5 F6 B6 F7 B7 F8 B8 F9 B9 F10 B10 F11 B11
where B and F are boson and fermion in each spacetime dimension. The
probability to transforming a fermion into its boson partner in the adjacent
dimension is same as the fine structure constant, α, the probability of a fermion
emitting or absorbing a boson. The probability to transforming a boson into its
fermion partner in the same dimension is also the fine structure constant, α. This
hierarchy can be expressed in term of the dimension number, D,
M D-1, B = M D, F αD, F, (1)
M D, F = M D, B αD, B, (2)
where M D, B and M D,F are the masses for a boson and a fermion, respectively, and
αD, B or αD,F is the fine structure constant, which is the ratio between the energies
of a boson and its fermionic partner. All fermions and bosons are related by the
order 1/α. (In some mechanism for the dynamical supersymmetry breaking, the
effects of order exp (-4 π2/ g2) where g is some small coupling, give rise to large
mass hierarchies [2].) This Kaluza-Klein substructure for the big bag universe isnot like the conventional structure where the energy in the extra space dimensionis very high, because the observed universe is the big bang universe, not the bigbag universe.
This supersymmetry breaking is through decay. The big bag universe starts
from the Planck supermembrane that decays into concerted particles in the decaymode, and undergoes a gigantic slow cosmic quantum oscillation between thehigh-energy small eleven dimensional spacetime and the low-energy large twodimensional spacetime. The expansion depends only on the decay mode, and thecontraction depends only on the combination mode that combine the concertedparticles together. There are no force field and cosmic radiation in the big baguniverse.
As in a microscopic quantum system before the measurement process,
there are no independent permanent components within the big bag during thecosmic quantum oscillation. The average dimension of the particles in the big bagchanges continuously and concertedly. All particles are in a concerted change.
The decay mode in the big bag universe is different from the decay mode
in the big bang universe. In the decay mode in the big bag universe, the decayparticles are in a concerted state as in the microscopic quantum system before4the measurement. All decay particles “remember” their parent Planck
supermembranes, as the in the microscopic system before the measurement, allindividual spin states remember their original composite spin state regardless oftime and space. Instead of explosive exponential radiation decay in the big banguniverse, the decay in the big bag universe is extremely orderly, one at the timeper parent Planck supermembrane. As soon as one particle decay, all otherdecay particles adjust their masses, so average mass of the particles in the bigbag universe decreases with time. All of them are under the concerted change.
Only basic physical constant about time is the Planck time (5.39 x10
-44
second). The decay rates for unstable hadrons in the big bang universe are closeto the Planck time. The decay rate in the big bag universe is much slower than thePlanck time, because it relates to the time span of the big bag universe in oneoscillation. It can only be a fraction of power of the Planck time. It appears thatthe decay rate is one particle per one-fourth power of the Planck time (1.52 x10
-11
second) per parent Planck supermembrane. With this decay rate, the time spanof the big bag universe in one oscillation is very long. For example, it takes 27.6billion years to decay from the Planck supermembrane to the four dimensionalbosons.
3. The Collapse of the Cosmic Oscillation
The cosmic oscillation, the cosmic particle-wave, is collapsed by non-
spacetime label, independent of spacetime of the cosmic oscillation. The non-spacetime label system is derived from the Planck supermembrane crossing insmall portion of the Planck supermembranes.
The Planck supermembrane crossing is the crossing of two Planck
supermembranes. It involves the crossing of seven space dimensions from twoPlanck supermembrane, resulting in the lepton-quark composite with fourdimensional spacetime. The lepton-quark composite consists of sevendimensions mostly for leptons and seven auxiliary dimensions mostly for quarks.The result of this crossing is shown in Fig. 1.
To make the lepton-quark composite permanent requires non-spacetime
labels, such as lepton number and quark (baryon) number, which areindependent of spacetime. All space labels other than the four dimensionalspacetime are replaced by non-spacetime labels. The spacetime labels areretained only in the four dimensional spacetime. Therefore, two elevendimensional Planck supermembranes are merged into the lepton-quarkcomposite consisting of two sets of four spacetime dimensions and seveninternal space dimensions. There is no curled-up space dimension. Thecompactification process to curl up space dimension is unnecessary. Except atthe very beginning of the cosmic cycle and black hole, the big bang universe haspure four dimensional spacetime. The big bag universe is the supersymmetryfour-to-eleven dimensional universe, while the big bang universe is the internalsymmetry four-plus-seven dimensional universe.5Different particles with non-spacetime labels behave differently in
spacetime. Without concerted behavior for all particles with non-spacetimelabels, the cosmic oscillation collapses, as the collapse of a microscopic systemby a system, such as the measurement process, that disrupts the concertedchange in the microscopic system.
The permanent non-spacetime labels require the violations of symmetries
(CP and P) in spacetime and the internal symmetry acting on the non-spacetimelabels. CP nonconservation is required to distinguish the lepton-quarkcomposite from the CP symmetrical cosmic radiation that is absence of thelepton-quark composite. P nonconservation is required to achieve chiralsymmetry for massless leptons (neutrinos), so some of the dimensional fermionscan become leptons (neutrinos). Various internal symmetry groups are requireto organize leptons, quarks, black hole particles, and force fields. The forcefields include the long-range massless force to bind leptons and quarks, theshort-range force to bind quarks, the short-range interactions to change quarksand leptons, and the short-range interaction for the particles inside black hole.Finally, the gauge bosons, leptons, and quarks absorb the common scalar fieldof four dimensional spacetime to acquire the common label of four dimensionalspacetime through Higgs mechanism. The results are the gauge bosons (Table1) and the periodic table of elementary particles (Table 2).
After the collapse, the remnants of the cosmic oscillation are in the four
dimensional spacetime. They are cosmic radiation and gravity, corresponding tothe expansion and the contraction in the cosmic oscillation. Cosmic radiationmaximizes the distance between any two objects as in the expansion, while gravityminimizes the distance between any two objects as in the contraction. Theinteractions of cosmic radiation and gravity toward all materials are indiscriminate.Everything can be involved in the same expansion and the contraction. Bothcosmic radiation and gravity follow the property of spacetime. Ubiquitous cosmicradiation represents spacetime, and gravity is the curvature of four dimensionalspacetime. Cosmic radiation and gravity do not need internal space.
The remnant of the cosmic oscillation is also shown in microscopic particle-
wave in quantum mechanics. In the wavefunciton, the spread out momentumcorresponds to the continuously changing momentum in the cosmic oscillation,and the spread out location corresponds to the continuously changing size of thecosmic oscillation. The cosmic uncertainty principle, therefore, is the uncertaintyto identify a particle with precise position and momentum in the cosmic oscillation.Microscopic particle-wave is the remnant of ever changing spacetime in the cosmicoscillation. Microscopic particle-wave does not need internal space. While thetime scale of the cosmic oscillation is the large one-fourth power of the Plancktime, the time scale of microscopic particle-wave is the small Planck time.
3. The Observable Big Bang Universe
The collapsed cosmic oscillation in the big bang universe results in four6dimensional spacetime and seven dimensional internal space. The seven
dimensional internal space is the base for the gauge bosons and the periodic tableof elementary particles as described in details in Reference 3. It is briefly reviewedhere.
For the big bang universe, the seven internal space dimensions are
arranged in the same way as the spacetime dimensions in the big bag universe.
F
5 B5 F6 B6 F7 B7 F8 B8 F9 B9 F10 B10 F11 B11
where B and F are boson and fermion in each spacetime dimension. The gauge
bosons in the big bang universe can be derived from Eqs. (1) and (2). Assuming
αD,B = αD,F , the relation between the bosons in the adjacent dimensions, then, can
be expressed in term of the dimension number, D,
M D-1, B = M D, B α2 D,( 3 )
where D= 6 to 11, and E 5,B and E 11,B are the energies for the dimension five and
the dimension eleven, respectively.
The lowest energy is the Coulombic field, E 5,B
E 5, B = α M6,F
= α Me,( 4 )
where M e is the rest energy of electron, and α = αe , the fine structure constant for
the magnetic field. The bosons generated are called "dimensional bosons" or
"BD". Using only αe, the mass of electron, the mass of Z 0, and the number of extra
dimensions (seven), the masses of B D as the gauge boson can be calculated as
shown in Table 1.7Table 1. The Energies of the Dimensional Bosons
BD = dimensional boson, α = αe
BD MD GeV Gauge Boson Interaction
B5 Me α 3.7x10-6A electromagnetic, U(1)
B6 Me/α 7x10-2π1/2 strong, SU(3)
B7 M6/αw2 cos θw91.177 Z L0weak (left), SU(2) L
B8 M7/α21.7x106XR CP (right) nonconservation,
U(1) R
B9 M8/α23.2x1010XL CP (left) nonconservation, U(1) L
B10 M9/α26.0x1014ZR0weak (right), SU(2) R
B11 M10/α21.1x1019G b black hole, large N color field
In Table 1, αw is not same as α of the rest, because there is symmetry
group mixing between B 5 and B 7 as the symmetry mixing in the standard theory of
the electroweak interaction, and sin θw is not equal to 1. As shown in Reference 3,
B5, B6, B7, B8, B9, and B 10 are A (massless photon), π1/2, ZL0, XR, XL, and Z R0,
respectively, responsible for the electromagnetic field, the strong interaction, theweak (left handed) interaction, the CP (right handed) nonconservation, the CP (lefthanded) nonconservation, and the P (right handed) nonconservation, respectively.
The calculated value for θ
w is 29.690 in good agreement with 28.70 for the
observed value of θw [4].
The calculated energy for B 11 (G b, black hole gluon, equivalent to the
Planck supermembrane) is 1.1x1019 GeV in good agreement with the Planck
mass, 1.2x1019 GeV. G b is the eleven dimensional gauge boson for the
interaction inside the event horizon of the black hole. The confinement of allparticles inside black hole is similar to the confinement of all quarks inside hardon,so the internal symmetry is based on large N color [5]. Black hole gluoncorresponds to gluon, and black hole gluino, the supersymmetrical partner,corresponds to quark. Therefore, inside the horizon of black hole, the gravitationalforce goes zero, and the large N color field appears. The eleven dimensionalblack hole gluon and gluino are surrounded by the four dimensional gravity.
These eleven dimensional black hole gluons and gluinos are still in internal
space (color) whose internal symmetry generates large N color force field. Theyare not eleven dimensional Planck supermembranes that hold together by thetangle of D-branes and strings. The event horizon is the dividing line between thefour dimensional spacetime labeled force and the internal space labeled force. Asdescribed later, all black hole universe is the end of the cosmic cycle, and allPlanck supermembrane is the beginning of the cosmic cycle.8The calculated masses of all gauge bosons are also in good agreement
with the observed values. Most importantly, the calculation shows that exactlyseven extra dimensions are needed for all fundamental interactions.
The model for leptons and quarks is shown in Fig. 1. The periodic table for
elementary particles is shown in Table 2.
Lepton
ν
e e νµ ντ l 9 l10
µ7 τ7 µ8
D = 5 6 7 8 9 10 11
a = 0 1 2 3 4 5 0 1 2
d 7 s7 c7 b 7 t 7 b 8 t8
u 7
u d 3 µ µ′ q 9 q 10
Quark
Fig. 1. Leptons and quarks in the dimensional orbits
D = dimensional number, a = auxiliary dimensional number
Table 2. The Periodic Table of Elementary Particles
D = dimensional number, a = auxiliary dimensional number
D A = 0 12a = 0 12345
Lepton Quark Boson
5 l5 = νe q5 = u = 3νe B5 = A
6l 6 = e q 6 = d = 3e B6 = π1/2
7 l7 = νµµ7τ7 q7 = 3µ u7/d7s7c7b7t7 B7 = Z L0
8 l8 = ντµ8 q8 = µ' b8t8 B8 = X R
9l 9 q9 B9 = X L
10 F 10 B10 = Z R0
11 F 11 B11 = G b
D is the dimensional orbital number for the seven extra space dimensions.
The auxiliary dimensional orbital number, a, is for the seven extra auxiliary spacedimensions, mostly for quarks. All gauge bosons, leptons, and quarks are locatedon the seven dimensional orbits and seven auxiliary orbits. Most leptons aredimensional fermions, while all quarks are the sums of subquarks.
The fermion mass formula for massive leptons and quarks is derived from
Reference 3 as follows.9∑ ∑∑ ∑∑
==
+ =+ =+ =
−
a
aD F Fa
aB FAF F F
a M Ma M MM M M
D DD DaD D aD
0404
0, 0,0,1 0,, 0, ,
2323
α (5)
Each fermion can be defined by dimensional numbers (D's) and auxiliary
dimensional numbers (a's). The compositions and calculated masses of leptonsand quarks are listed in Table 3.
Table 3. The Compositions and the Constituent Masses of Leptons and Quarks
D = dimensional number and a = auxiliary dimensional number
D
a Composition Calc. Mass
Leptons Da for leptons
νe 50 νe 0
e6 0 e 0.51 MeV (given)
νµ 70 νµ 0
ντ 80 ντ 0
µ 60 + 7 0 + 7 1 e + νµ + µ7 105.6 MeV
τ 60 + 7 0 + 7 2 e + νµ + τ7 1786 MeV
µ' 60 + 7 0 + 7 2 + 8 0 + 8 1e + νµ + µ7 + ντ + µ8 136.9 GeV
Quarks Da for quarks
u5 0 + 7 0 + 7 1 u5 + q 7 + u 7 330.8 MeV
d6 0 + 70 + 7 1 d6 + q 7 + d 7 332.3 MeV
s6 0 + 70 + 7 2 d6 + q 7 + s 7 558 MeV
c5 0 + 7 0 + 7 3 u5 + q 7 + c 7 1701 MeV
b6 0 + 70 + 7 4 d6 + q 7 + b 7 5318 MeV
t5 0 + 7 0 + 7 5 + 8 0 + 8 2u5 + q 7 + t7 + q 8 + t8 176.5 GeV
The calculated masses are in good agreement with the observed
constituent masses of leptons and quarks [5,6]. The mass of the top quark foundby Collider Detector Facility is 176 ± 13 GeV [6] in a good agreement with thecalculated value, 176.5 GeV. The masses of elementary particles can becalculated using the periodic table with only four known constants: the number ofthe extra spatial dimensions in the supermembrane, the mass of electron, the
mass of Z°, and α
e. The calculated masses are in good agreement with the
observed values.
5. The Cosmic Dematerialized zone10The two universes are together yet separate everywhere all the time. They
are separated by the cosmic dematerialized zone everywhere all the time. Nothingcan pass through the dematerialized zone. When the spacetime dimensions inthe two universes are different, the cosmic dematerialized zone remains dormant.When the spacetime dimensions in the two universes are compatible, the cosmicdematerialized zone becomes active. During this period of compatibility, the bigbang universe can either absorb space from or emit space to the cosmicdematerialized zone. During one cycle of the universe, there are four periods of active cosmicdematerialized zone. The first period of active cosmic dematerialized zone is theinflation at the start of the expansion of the universe. The Plancksupermembranes that decay in a short time into internal symmetry four-plus-sevendimensional universe. As soon as the decay takes place, the cosmicdematerialized zone becomes dormant due to the difference in the spacetimedimensions between the two universes.
The dormant cosmic dematerialized zone becomes active when the
energy level of the decay mode in the big bag is equal to the energy levelbetween the five-dimensional spacetime and the four dimensional spacetime asin the big bang universe. The big bang universe again starts to absorb spacefrom the cosmic dematerialized zone. The space absorption in the big banguniverse causes accelerated expansion in the big bang universe. It is thesecond period of active cosmic dematerialized zone, which is happening at thepresent as late-time cosmic accelerated expansion. The evidence for late-time cosmic accelerated expansion is from therecent observations of large-scale structure that suggests that the universe isundergoing cosmic accelerated expansion, and it is assumed that the universe isdominated by a dark energy with negative pressure recently [8]. The darkenergy can be provided by a non-vanishing cosmological constant orquintessence [9], a scalar field with negative pressure. However, a cosmologicalconstant requires extremely fine-tuned [10]. Quintessence requires anexplanation for the late- time cosmic accelerated expansion [11]. Why doesquintessence dominate the universe only recently? One of the explanations isthe k-essence model where the pressure of quintessence switched to a negative
value at the onset of matter-domination in the universe [11].
According to the cyclic dual universe model, this late-time cosmic
accelerated expansion is caused by the space absorption during the secondperiod of active cosmic dematerialized zone. The compatible energy level for thebig bag universe and the big bang universe is between the energy levels of thefive dimensional fermion and the four dimensional boson. From Eqs (1) and (2)and Table 1, the energies of the five dimensional fermion and the fourdimensional boson are calculated to be 2.72 x10
-8 GeV and 1.99 x10-10 GeV,
respectively. The energy of the Planck supermembrane is 1.1x1019 GeV from
Table 1. The energy ratio between the Planck supermembrane and the fivedimensional fermion and the four dimensional boson are 4.17 x10
26 and 5.71
x1028, respectively. In other words, there are 4.17 x1026 five dimensional fermions11and 5.71 x1028 four dimensional bosons per Planck supermembrane. The decay
rate is one particle per one-fourth power of the Planck time (1.52 x10-11 second)
per parent Planck supermembrane. The total time for the decay is 0.2 billionyears for the five dimensional fermion, and 27.6 billion years for the fourdimensional boson. Therefore, the space absorption starts in 0.2 billion yearsafter the Big Bang, and ends in 27.6 billions years, and the space absorptioncauses accelerated expansion in the big bang universe.
If the relative rate of the rise and the fall of the space absorption in the big
bang universe during this period follows a smooth curve with respect to time, themaximum relative rate (100) of space absorption occurs in about 13.9 billionyears after the big bang. It is coincidentally about the age of the universe. Therelative rate of space absorption versus time is shown in Fig. 2.
Fig. 2. relative rate of the space absorption in the big bang
universe versus time
In other words, the greatest acceleration occurs at the peak of the space
absorption in 13.9 billion years after the big bang.
The oscillation of the big bag now is in the expansion mode. The big bag
universe may expand all the way down to the beginning of the two dimensionalspacetime, or may expand only down to the beginning of the four dimensionalspacetime, that is about 14 billion years from now. When the oscillation is in thecontraction mode in the future, and the big bag universe again passes the fourdimensional spacetime, the big bang universe will emit space to the cosmicdematerialized zone to pay back the space absorbed during the second period ofthe active cosmic dematerialized zone. The result is the accelerated contractionof the big bang universe. This third period of the active cosmic dematerializedzone prevents the big bang universe to expand forever with an infinite largespace. Therefore, the big bang universe and the big bag universe can contracttogether.
Toward the end of the contraction mode, black holes become the
dominating feature in the big bang universe. Black hole contains elevendimensional black hole gluon and gluino surrounded by gravity. When bothuniverses are near eleven dimensional spacetime, they again becomecompatible. In this last period of active cosmic dematerialized zone, the bigbang universe again emits space to the cosmic dematerialized zone. The resultis to gather all material in the big bang universe for the final contraction into the0.050.0100.0
0.2 13.9 27.6
billions of yearsrelative rate of
space absorption12eleven dimensional black hole gluons and gluinos surrounded by four
dimensional gravity in one area.
Moment after the final accelerated contraction, the space absorption
starts to start the inflation. The inflation is anti-gravity that annihilates gravitysurrounding the black hole gluons and gluinos. Without gravity, black holegluons and gluinos disappear as quarks disappear outside of hadrons. Whatremain are the eleven dimensional Planck supermembranes that hold togetherby the tangle of D-branes and strings. Immediately, the eleven dimensionalPlanck supermembranes decay, and another cosmic cycle starts. The inflation,therefore, is not an add-on but a necessity to start the cosmic cycle.
6. Conclusion
Supersymmetry involves a symmetrical change in spacetime labels in
terms of spacetime coordinates, while internal symmetry involves a symmetricalchange in non-spacetime labels in terms of charged fields, isotopic spin, andcolors. In this paper, the cosmic origin of supersymmetry and internal symmetryis explained by the cyclic dual universe, the coexistence of the unobservable bigbag universe and the observable big bang universe.
The cosmic cycle for both universes starts with the eleven dimensional
Planck supermembrane with the Planck mass. The supersymmetry of the Plancksupermembrane is broken into a Kaluza-Klein structure where the lowerdimensional spacetime has lower energy than the higher dimensional spacetime.In the big bag universe, the Planck supermembranes are slowly broken anddecay into concerted particles that then are slowly recombined. It undergoes agigantic slow cosmic oscillation between the high-energy eleven dimensionalspacetime and the low-energy four dimensional spacetime. It is like a giganticcosmic particle-wave.
The collapse of the cosmic oscillation is caused by the lepton-quark
composite derived from the crossing of two sets of the seven dimensions fromtwo Planck supermembranes. To maintain the lepton-quark composite requiresnon-spacetime labels independent of spacetime. The spacetime labels for thetwo sets of seven dimensions are replaced by non-spacetime labels.Consequently, the cosmic oscillation collapse due to the disruption by the non-spacetime labels, independent of the spacetime of the cosmic oscillation. It islike the collapse of microscopic particle-wave by a system (such asmeasurement) independent of microscopic particle-wave. The collapse of thecosmic oscillation results in the immediate decay into the four dimensionalspacetime and the seven dimensional internal space. There is no curled-upspace dimension and compacitification. Except at the very beginning of thecosmic cycle and black hole, the big bang universe has pure four dimensionalspacetime. The unobservable big bag universe is the supersymmetry four-to-eleven dimensional spacetime universe, while the observable big bang universeis the internal symmetry four-plus-seven dimensional universe.13In the four dimensional spacetime, there are cosmic radiation, gravity, and
microscopic particle-wave. In the seven dimensional internal space, there areinternal symmetries for gauge bosons, leptons, quarks, and black hole. Thegauge bosons and the periodic table of elementary particles can be derived fromthe collapsed cosmic oscillation. There is black hole gluon specifically for theinteraction inside black hole. The masses of gauge bosons and elementaryparticles can be calculated using the periodic table with only four known constants:the number of the extra spatial dimensions in the supermembrane, the mass of
electron, the mass of Z°, and α
e. The calculated masses are in good agreement
with the observed values.
The two universes are separated by the dematerialized zone everywhere
all the time. The big bang universe absorbs space from and emits space to thedematerialized zone, when the spacetime dimensions in both universes arecompatible. The results are the inflation during the big bang, the late-timecosmic accelerated expansion, the cosmic accelerated contraction, and the finalcosmic accelerated contraction. The cosmic cycle starts with all Plancksupermembrane for both universe, and ends with all black hole universe for thebig bang universe. The inflation is not an add-on but a necessity to start thecosmic cycle. According to the calculation, we are now at the peak of the late-time cosmic accelerated expansion.14 References
[1] C.M.Hull and P.K. Townsend, Nucl. Phys. B 438 (1995) 109; E. Witten,
Nucl. Phys. B 443 (1995) 85.
[2] E. Witten 1982 Nucl. Phys. B 202 (1981) 253; T. Bank, D. Kaplan, and A.
Nelson A 1994 Phys. Rev. D 49 (1994) 779.
[3] D. Chung, Speculations in Science and Technology 20 (1997) 259;
Speculations in Science and Technology 21(1999) 277; hep-ph/0003237;
Concise Encyclopedia of on Supersymmetry to be published by KluwerAcademic Publishers.
[4] P. Langacher, M. Luo , and A. Mann, Rev. Mod. Phys. 64 (1992) 87.
[5] J. Maldacena, Adv. Theor. Math Phys. 2, 231 (1998); hep-ph/0002092.
[6] CDF Collaboration, 1995 Phys. Rev. Lett 74 (1995) 2626.
[7] C.P. Singh, Phys. Rev. D 24 (1981) 2481; D. B. Lichtenberg Phys. Rev.
D40 (1989) 3675.
[8] N. Balcall, J.P. Ostriker, S. Perlmutter, and P.J. Steinhardt, Science 284,
1481-1488, (1999).
[9] R.R. Caldwell, R. Dave, and P.J. Steinhardt, Phys.Rev.Lett. 80, 1582
(1998).
[10] J.D. Barrow & F.J. Tipler, The Antropic Cosmological Principle, Oxford UP
(1986) p.668.
[11] C. Armedariz-Picon, V. Mukhanov, and P.J. Steinhardt, Phys.Rev.Lett. 85,
4438 (2000). |
arXiv:physics/0101108v1 [physics.pop-ph] 31 Jan 2001Superluminal motions?
A bird-eye view of the experimental situation(†)
Erasmo Recami
Facolt` a di Ingegneria, Universit` a Statale di Bergamo, Da lmine (BG), Italy;
INFN—Sezione di Milano, Milan, Italy; and
CCS, State University of Campinas, Campinas, S.P., Brazil.
1. - Introduction.
The question of Super-luminal ( V2> c2) objects or waves has a long story, starting
perhaps in 50 b.C. with Lucretius’ De Rerum Natura (cf., e.g., book 4, line 201: [ <<Quone
videscitius debere et longius ire/ Multiplexque loci spatium transcurr ere eodem/ Tempore
quo Solis pervolgant lumina coelum? >>]). Still in pre-relativistic times, one meets various
related works, from those by J.J.Thomson to the papers by the great A.Sommerfeld.
With Special Relativity, however, since 1905 the convictio n spread over that the speed c
of light in vacuum was the upper limit of any possible speed. For instance, R.C.Tolman
in 1917 believed to have shown by his “paradox” that the exist ence of particles endowed
with speeds larger than cwould have allowed sending information into the past. Such a
conviction blocked for more than half a century —aside from a n isolated paper (1922) by
the Italian mathematician G.Somigliana— any research abou t Superluminal speeds. Our
problem started to be tackled again essentially in the fiftie s and sixties, in particular after
the papers[1] by E.C.George Sudarshan et al., and later on[2 ] by E.Recami, R.Mignani,
et al. [who rendered the expressions subluminal and Superlu minal of popular use by their
works at the beginning of the Seventies], as well as by H.C.Co rben and others (to confine
ourselves to the theoretical researches). The first experiments looking for tachyons wer e
performed by T.Alv¨ ager et al.
Superluminal objects were called tachyons, T, by G.Feinber g, from the Greek word
ταχ´υς, quick, and this induced us in 1970 to coin the term bradyon, B , for ordinary
subluminal ( v2< c2) objects, from the Greek word βραδ´υς, slow). At last, objects
travelling exactly at the speed of light are called “luxons” .
In recent years, terms as “tachyon” and “superluminal” fell unhappily into the
(cunning, rather than crazy) hands of pranotherapists and m ere cheats, who started
squeezing money out of simple-minded people; for instance b y selling plasters (!) that
should cure various illnesses by “emitting tachyons”... We are dealing with them here,
0 (†)E-mail for contacts: recami@mi.infn.it
1however, since at least four different experimental sectors of physics seem to indicate the
actual existence of Superluminal motions [it is an old use of ours to write Superluminal
with a capital S], thus confirming some long-standing theore tical predictions[3]. So much
so that even the N.Y.Times commented on May 30, 2000, upon two of such experiments,
imitated the next day (and again at the end of the next July) by nearly all the world
press. In this rapid informative paper, after a sketchy theo retical introduction, we are
setting forth a reasoned outline of the experimental state- of-art: brief, but accompanied
by a bibliography sufficient in some cases to provide the inter ested readers with coherent,
adequate information; and without forgetting to call atten tion —at least in the two sectors
more after fashion today— to some other worthy experiments.
2. Special and Extended Relativity .
Let us premise that special relativity (SR), abundantly ver ified by experience, can
be built on two simple, natural Postulates: 1) that the laws ( of electromagnetism and
mechanics) be valid not only for a particular observer, but f or the whole class of the
“inertial” observers: 2) that space and time be homogeneous and space be moreover
isotropic. From these Postulates one can theoretically infer that one, and only one,
invariant speed: and experience tells us such a speed to be that, c, of light in vacuum; in
fact, light possesses the peculiar feature of presenting al ways the same speed in vacuum,
even when we run towards or away from it. It is just that featur e, of being invariant,
that makes quite exceptional the speed c: no bradyons, and no tachyons, can enjoy the
same property!
Another (known) consequence of our Postulates is that the to tal energy of an or-
dinary particle increases when its speed vincreases, tending to infinity when vtends to
c. Therefore, infinite forces would be needed for a bradyon to r each the speed c. This
fact generated the popular opinion that speed ccan be neither achieved nor overcome.
However, as speed cphotons exist which are born live and die always at the speed o f
light (without any need of accelerating from rest to the ligh t speed), so particles can
exist —tachyons[4]— always endowed with speeds Vlarger than c(see Fig.1). This cir-
cumstance has been picturesquely illustrated by George Sud arshan (1972) with reference
to an imaginary demographer studying the population patter ns of the Indian subconti-
nent: <<Suppose a demographer calmly asserts that there are no peopl e North of the
Himalayas, since none could climb over the mountain ranges! That would be an absurd
conclusion. People of central Asia are born there and live th ere: they did not have to be
born in India and cross the mountain range. So with faster-th an-light particles >>. Let
us add that, still starting from the above two Postulates (be sides a third one, even more
obvious), the theory of relativity can be generalized[3,4] in such a way to accommodate
also Superluminal objects; such an extension is largely due to the Italian school, by a
series of works performed mainly in the Sixties–Seventies. Also within the “Extended
Relativity”[3] the speed c, besides being invariant, is a limiting velocity: but every limit-
2ing value has two sides, and one can a priori approach it both f rom the left and from the
right.
Actually, the ordinary formulation of SR is restricted too m uch. For instance, even
leaving tachyons aside , it can be easily so widened as to include antimatter [5]. Then,
one finds space-time to be a priori populated by normal partic les P (which travel forward
in time carrying positive energy), andby dual particles Q “which travel backwards in
time carrying negative energy”. The latter shall appear to u s asantiparticles , i.e., as
particles —regularly travelling forward in time with posit ive energy, but— with all their
“additive” charges (e.g., the electric charge) reversed in sign!: see Fig.2. To clarify this
point, let us recall that we, macroscopic observers, have to move in time along a single,
well-defined direction, to such an extent that we cannot even see a motion backwards in
time...; and every object like Q, travelling backwards in ti me (with negative energy), will
benecessarily reinterpreted by us as an anti-object, with opposite charge s but travelling
forward in time (with positive energy).[3-5]
But let us forget about antimatter and go back to tachyons. A s trong objection
against their existence is based on the opinion that by tachy ons it be possible to send
signals into the past, owing to the fact that a tachyon T which —say— appears to a first
observer Oas emitted by A and absorbed by B, can appear to a second observ erO′as a
tachyon T’ which travels backwards in time with negative ene rgy. However, by applying
(as it is obligatory to do) the same “reinterpretation rule” or switching procedure seen
above, T’ will appear to the new observer O′just as an antitachyon T emitted by B and
absorbed by A, and therefore travelling forward in time, eve n if in the contrary space
direction. In such a way, every travel towards the past, and e very negative energy, do
disappear...
Starting from this observation, it is possible to solve[5] t he so-called causal paradoxes
associated with Superluminal motions: paradoxes which res ult to be the more instructive
and amusing, the more sophisticated they are; but that canno t be re-examined here
(some of them having been proposed by R.C.Tolman, J.Bell, F. A.E.Pirani, J.D.Edmonds
and others).[6,3] Let us only mention here the following. Th e reinterpretation principle
—according to which, in simple words, signals are carried on ly by objects which appear
to be endowed with positive energy— does eliminate any infor mation transfer backwards
in time, but this has a price: That of abandoning the ingraine d conviction that the
judgement about what is cause and what is effect be independen t of the observer. In fact,
in the case examined above, the first observer Oconsiders the event at A t be the cause
of the event at B. By contrast, the second observer O′will consider the event at B as
causing the event at A. All the observers will however see the cause to happen before its
effect.
Taking new objects or entities into consideration always fo rces us to a criticism
of our prejudices. If we require the phenomena to obey the lawof (retarded) causality
with respect to all the observers, then we cannot demand also the phenomena description
“details” to be invariant: namely, we cannot demand in that c ase also the invariance of
3the “cause” and “effect” labels.[6,2] To illustrate the natu re of our difficulties in accepting
that e.g. the parts of cause and effect depend on the observer, let us cite an analogous
situation that does not imply present-day prejudices: <<For ancient Egyptians, who knew
only the Nile and its tributaries, which all flow South to Nort h, the meaning of the word
“south” coincided with the one of “upstream”, and the meanin g of the word “north”
coincided with the one of “downstream”. When Egyptians disc overed the Euphrates,
which unfortunately happens to flow North to South, they pass ed through such a crisis
that it is mentioned in the stele of Tuthmosis I, which tells u s about that inverted water
that goes downstream (i.e. towards the North) in going upstr eam>>(Csonka, 1970).
The last century theoretical physics led us in a natural way t o suppose the exis-
tence of various types of objects: magnetic monopoles, quar ks, strings, tachyons, besides
black-holes: and various sectors of physics could not go on w ithout them, even if the
existence of none of them is certain (also because attention has not yet been paid to some
links existing among them: e.g., a Superluminal electric ch arge is expected to behave as
a magnetic monopole; and a black-hole a priori can be the sour ce of tachyonic matter).
According to Democritus of Abdera, everything that was thin kable without meeting con-
tradictions had to exist somewhere in the unlimited univers e. This point of view —which
was given by M.Gell-Mann the name of “totalitarian principl e”— was later on expressed
(T.H.White) in the humorous form “Anything not forbidden is compulsory”. Applying it
to tachyons, Sudarshan was led to claim that if tachyons exis t, they must to be found; if
they do not exist, we must be able to say clearly why...
3. The experimental state-of-the-art .
Extended Relativity can allow a better understanding of man y aspects also of ordi-
naryrelativistic physics, even if tachyons would not exist in ou r cosmos as asymptotically
free objects. As already said, we are dealing with them —howe ver— since their topic is
presently returning after fashion, especially because of t he fact that at least three or four
different experimental sectors of physics seem to suggest th e possible existence of faster-
than-light motions. We wish to put forth in the following som e information (mainly
bibliographical) about the experimental results obtained in each one of those different
physics sectors.
A) Neutrinos – First: A long series of experiments, started in 1971, seems to show
that the square m02of the mass m0of muonic neutrinos, and more recently of electronic
neutrinos too, is negative; which, if confirmed, would mean t hat (when using a na ¨ive lan-
guage, commonly adopted) such neutrinos possess an “imagin ary mass” and are therefore
tachyonic, or mainly tachyonic.[7,3] [In Extended Relativ ity, the dispersion relation for
a free tachyon becomes E2−p2=−m2
o, and there is noneed therefore of imaginary
4masses...].
B) Galactic Micro-quasars – Second: As to the apparent Superluminal expansions ob-
served in the core of quasars[8] and, recently, in the so-cal led galactic microquasars[9], we
shall not deal here with that problem, too far from the other t opics of this paper: without
mentioning that for those astronomical observations there exist orthodox interpretations,
based on ref. [10],that are accepted by the astrophysicists’ majority. For a theoretical
discussion, see ref.[11]. Here, let us mention only that sim ple geometrical considerations
in Minkowski space show that a single Superluminal light source would look[11,3]: (i)
initially, in the “optical boom” phase (analogous to the aco ustic “boom” produced by
a plane travelling with constant supersonic speed), as an in tense source which appears
suddenly; and that (ii) afterwards seem to split into TWO obj ects receding one from the
other with speed V >2c.
C) Evanescent waves and “tunnelling photons” – Third: Within quantum mechan-
ics (and precisely in the tunnelling processes), it had been shown that the tunnelling time
—firstly evaluated as a simple “phase time” and later on calcu lated through the analysis
of the wavepacket behaviour— does not depend on the barrier w idth in the case of opaque
barriers (“Hartman effect”)[12]: which implies Superlumin al and arbitrarily large (group)
velocities Vinside long enough barriers: see Fig.3. Experiments that ma y verify this pre-
diction by, say, electrons are difficult. Luckily enough, how ever, the Schroedinger equation
in the presence of a potential barrier is mathematically ide ntical to the Helmholtz equa-
tion for an electromagnetic wave propagating e.g. down a met allic waveguide along the
x-axis: and a barrier height Ubigger than the electron energy Ecorresponds (for a given
wave frequency) to a waveguide transverse size lower than a c ut-off value. A segment of
undersized guide does therefore behave as a barrier for the w ave (photonic barrier)[13]:
So that the wave assumes therein —like an electron inside a qu antum barrier— an imagi-
nary momentum or wave-number and gets, as a consequence, exp onentially damped along
x. In other words, it becomes an evanescent wave (going back to normal propagation,
even if with reduced amplitude, when the narrowing ends and t he guide returns to its
initial transverse size). Thus, a tunnelling experiment ca n be simulated[13] by having
recourse to evanescent waves (for which the concept of group velocity can be properly
extended[14]). And the fact that evanescent waves travel wi th Superluminal speeds has
been actually verified in a series of famous experiments (cf. Fig.4).
Namely, various experiments —performed since 1992 onwards by G.Nimtz at
Cologne[15], by R.Chiao’s and A.Steinberg’s group at Berke ley[16], by A.Ranfagni and
colleagues at Florence[17], and by others at Vienna, Orsay, Rennes[17]— verified that
“tunnelling photons” travel with Superluminal group veloc ities. Such experiments roused
a great deal of interest[18], also within the non-specializ ed press, and were reported by
Scientific American, Nature, New Scientist, and even Newswe ek, etc. Let us add that also
Extended Relativity had predicted[19] evanescent waves to be endowed with faster-than- c
5speeds; the whole matter appears to be therefore theoretica lly selfconsistent. The debate
in the current literature does not refer to the experimental results (which can be correctly
reproduced by numerical elaborations[20,21] based on Maxw ell equations only), but rather
to the question whether they allow, or do not allow, sending s ignals or information with
Superluminal speed[21,14].
Let emphasize that the most interesting experiment of this series is the one with
two “barriers” (e.g., with two segments of udersized wavegu ide separated by a piece of
normal-sized waveguide: Fig.5). For suitable frequency ba nds —i.e., for “tunnelling” far
from resonances—, it was found that the total crossing time d oes not depend on the length
of the intermediate (normal) guide: namely, that the beam sp eed along it is infinite[22].
This agrees with what predicted by Quantum Mechanics for the non-resonant tunnelling
trough two successive opaque barriers (the tunnelling phas e time, which depends on the
entering energy, has been shown by us to be independent of the distance between the
two barriers[23]). Such an important experiment could and s hould be repeated, taking
advantage also of the circumstance that quite interesting e vanescence regions can be easily
constructed in the most varied manners, like by several “pho tonic band-gap materials” or
gratings (it being possible tu use from multilayer dielectr ic mirrors, to semiconductors, to
photonic crystals...)
We cannot skip a further topic —which, being delicate, shoul d not appear in a
brief review like this one— since the last experimental cont ribution to it (performed at
Princeton by J.Wang et al. and published in Nature on 7.20.00 ) is one of the two arti-
cles mentioned by the N.Y.Times and commented at the end of Ju ly, 2000, by the whole
world press. Even if in Extended Relativity all the ordinary causal paradoxes seem to be
solvable[3,6], nevertheless one has to bear in mind that (wh enever it is met an object, O,
travelling with Superluminal speed) one may have to deal wit h negative contributions to
the tunnelling times[24]: and this ought not to be regarded a s unphysical. In fact, when-
ever an “object” (particle, electromagnetic pulse,,...) Oovercomes the infinite speed[3,6]
with respect to a certain observer, it will afterwards appea r to the same observer as the
“anti-object” Otravelling in the opposite space direction[3,6]. For instance, when going
on from the lab to a frame Fmoving in the same direction as the particles or waves
entering the barrier region, the object Openetrating through the final part of the barrier
(with almost infinite speed[12,21,23], like in Figs.3) will appear in the frame Fas an
anti-object Ocrossing that portion of the barrier in the opposite space–direction [3,6]. In
the new frame F, therefore, such anti-object Owould yield a negative contribution to the
tunnelling time: which could even result, in total, to be neg ative. For any clarifications,
see refs.[18]. Ci` o che vogliamo qui What we want to stress he re is that the appearance of
such negative times is predicted by Relativity itself, on th e basis of the ordinary postu-
lates[3,6,24,12,21]. (In the case of a non-polarized beam, , the wave anti-packet coincides
with the initial wave packet; if a photon is however endowed w ith helicity λ= +1, the
anti-photon will bear the opposite helicity λ=−1). From the theoretical point of view,
besides refs.[24,12,21,6,3], see refs.[25]. On the (quite interesting!) experimental side, see
6papers [26], the last one having already been mentioned abov e.
Let us addhere that, via quantum interference effects in three-levels atomic systems,
it is possible to obtain dielectrics with refraction indice s very rapidly varying as a function
of frequency, with almost complete absence of light absorpt ion (i.e., with quantum induced
transparency) [27]. The group velocity of a light pulse prop agating in such a medium can
decrease to very low values, either positive or negatives, w ithnopulse distortion. It is
known that experiments were performed both in atomic sample s at room temperature, and
in Bose-Einstein condensates, which showed the possibilit y of reducing the speed of light
to few meters per second. Similar, but negative group veloci ties —implying a propagation
with Superluminal speeds thousands of time higher than the p reviously mentioned ones—
have been recently predicted, in the presence of such an “ele ctromagnetically induced
transparency”, for light moving in a rubidium condensate[2 8], while the corresponding
experiments are being dome at the Florence European laborat ory “LENS”.
Finally, let us emphasize that faster-than- cpropagation of light pulses can be (and
was, in same cases) observed also by taking advantage of anom alous dispersion near an
absorbing line, or nonlinear and linear gain lines, or nondi spersive dielectric media, or
inverted two-level media, as well as of some parametric proc esses in nonlinear optics (cf.
G.Kurizki et al.)
D) Superluminal Localized Solutions (SLS) to the wave equat ions. The “X-
shaped waves” – The fourth sector (to leave aside the others) is not less imp ortant. It
returned after fashion when some groups of capable scholars in engineering (for sociological
reasons, most physicists had abandoned the field) rediscove red by a series of clever works
that any wave equation —to fix the ideas, let us think of the ele ctromagnetic case—
admit also solutions so much sub-luminal as Super-luminal ( besides the ordinary plane
waves endowed with speed c/n). Let us recall that, starting with the pioneering work
by H.Bateman, it had slowly become known that all homogeneou s wave equations (in
a general sense: scalar, electromagnetic, spinorial,...) admit wavelet-type solutions with
sub-luminal group velocities[29]. Subsequently, also Sup erluminal solutions started to be
written down, in refs.[30] and, independently, in refs.[31 ] (in one case just by the mere
application of a Superluminal Lorentz “transformation”[3 ,32]).
A quite important feature of some new solutions of these (whi ch attracted much
of the attention of the engineering colleagues) is that they propagate as localized, non-
dispersive pulses: namely, according to the Courant and Hil bert’s[29] terminology, as
“undistorted progressive waves”. It is easy to realize the p ractical importance, for in-
stance, of a radio transmission carried out by localized bea ms, independently of their
being sub- or Super-luminal. But non-dispersive wave packe ts can be of use also in
theoretical physics for a reasonable representation of ele mentary particles[33].
Within Extended Relativity since 1980 it had been found[34] that —whilst the
simplest subluminal object conceivable is a small sphere, o r a point as its limit— the
simplest Superluminal objects results by contrast to be (se e refs.[34], and Figs.6 and
77) an “X-shaped” wave, or a double cone as its limit, which mor eover travels without
deforming —i.e., rigidly— in a homogeneous medium[3]. It is worth noticing that the
most interesting localized solutions happened to be just th e Superluminal ones, and with a
shape of that kind. Even more, since from Maxwell equations u nder simple hypotheses one
goes on to the usual scalar wave equation for each electric or magnetic field component,
one can expect the same solutions to exist also in the field of a coustic waves, and of
seismic waves (and perhaps of gravitational waves too). Act ually, such beams (as suitable
superpositions of Bessel beams) were mathematically const ructed for the first time, by
Lu et al.[35], in acoustics : and were then called “X-waves” or rather X-shaped waves.
It is more important for us that the X-shaped waves have been i n effect produced in
experiments both with acoustic and with electromagnetic wa ves; that is, X-beams were
produced that, in their medium, travel undistorted with a sp eed larger than sound, in
the first case, and than light, in the second case. In Acoustic s, the first experiment
was performed by Lu et al. themselves[36] in 1992, at the Mayo Clinic (and their pa-
pers received the first 1992 IEEE award). In the electromagne tic case, certainly more
“intriguing”, Superluminal localized X-shaped solutions were first mathematically con-
structed (cf., e.g., Fig.8) in refs.[37], and later on exper imentally produced by Saari et
al.[38] in 1997 at Tartu by visible light (see fig.2 in ref.[38 ]!), and recently by Mugnai,
Ranfagni and Ruggeri at Florence by microwaves[39] (this be ing the paper appeared in
the Phys. Rev. Lett. of May 22, 2000, which the national and in ternational press refer-
eed to). Further experimental activity is in progress, for i nstance, at Pirelli Cables, in
Milan (by adopting as a source a pulsed laser) and at the FEEC o f Unicamp, Campinas,
S.P.; while in the theoretical sector the activity is even mo re intense, in order to build
up —for example— new analogous solutions with finite total en ergy or more suitable for
high frequencies, on one hand, and localized solutions Supe rluminally propagating even
along a normal waveguide[40], on the other hand.
Let us eventually touch the problem of producing an X-shaped Superluminal wave
like the one in Fig.7, but truncated —of course– in space and i n time (by the use of a
finite, dynamic antenna, radiating for a finite time): in such a situation, the wave will
keep its localization and Superluminality only along a cert ain “depth of field”, decaying
abruptly afterwards[35,37]. We can become convinced about the possibility of realizing
it, by imaging the simple ideal case of a negligibly sized Sup erluminal source Sendowed
with speed V > c in vacuum and emitting electromagnetic waves W(each one travelling
with the invariant speed c). The electromagnetic waves will result to be internally ta ngent
to an enveloping cone Chaving Sas its vertex, and as its axis the propagation line x
of the source[3]. This is analogous to what happens for a plan e that moves in the air
with constant supersonic speed. The waves Winterfere negatively inside the cone C,
and constructively only on its surface. We can place a plane d etector orthogonally to
x, and record magnitude and direction of the Wwaves that hit on it, as (cylindrically
symmetric) functions of position and of time. It will be enou gh, then, to replace the
plane detector with a plane antenna which emits —instead of recording— exactly the
8same (axially symmetric) space-time pattern of waves W, for constructing a cone-shaped
electromagnetic wave Cthat will propagate with the Superluminal speed V(of course,
without a source any longer at its vertex): even if each wave Wtravels with the invariant
speed c. For further details, see the first one of refs.[37]. Here let us only add that
such localized Superluminal waves appear to keep their good properties only as long as
they are fed by the waves arriving (with speed c) from the dynamic antenna: taking
the time needed for their generation into account, these wav es seem therefore unable to
transmit information faster than c; however, they have nothing to do with the illusory
“scissors effect”, since they certainly carry energy-momen tum Superluminally along their
field depth (for instance, they can get two detectors at a dist anceLto click after a time
smaller thanL/c).
As we mentioned above, the existence of all these X-shaped Su perluminal (or “Super-
sonic”) seem to constitute at the moment—together, e.g., wi th the Superluminality of
evanescent waves— one of the best confirmation of Extended Re lativity. It is curious
than one of the first applications of such X-waves (that takes advantage of their prop-
agation without deformation) is in progress in the field of me dicine, and precisely of
ultrasound scanners[41]. A few years ago only, the hypothes is that “tachyons” could be
used to obtain directly 3-dimensional ultrasound scans wou ld have arisen the scepticism
of any physicist, this author included.
Acknowledgments
The author is deeply indebted to all the Organizers of this Co nference, and particularly to
Larry Horwitz, J.D.Bekenstein and J.R.Fanchi, for their ki nd invitation and warm, gen-
erous hospitality. For stimulating and friendly discussio ns he is grateful to all the partici-
pants, and in particular to J.D.Bekenstein, N.Ben-Amots, J .R.Fanchi, L.Horwitz, R.Lieu,
M.Pavˇ siˇ c. For further discussions or kind collaboration thanks are due also to F.Bassani,
A.Bertin, R.Chiao, A.Degli Antoni, F.Fontana, A.Gigli, H. E.Hern´ andez, G.Kurizki, J.-
y.Lu, A.van der Merwe, D.Mugnai, G.Nimtz, V.S.Olkhovsky, M .R.Zamboni, A.Ranfagni,
R.A.Ricci, A.Shaarawi, D.Stauffer, A.Steinberg, C.Vasini , M.T.Vasconselos and A.Vitale.
Figure captions
Fig.1 – Andamento dell’energia di un oggetto libero al varia re della sua velocit` a.[2-4]
Fig.2 – Illustrazione della “regola di switching” (o princi pio di reinterpretazione) di
Stueckelberg-Feynman-Sudarshan[3-5]: Qapparir` a essere l’antiparticella di P. Vedere
il testo.
Fig.3 – Andamento del “tempo di penetrazione” di un pacchett o d’onde al variare dello
spazio percorso all’interno di una barriera di potenziale ( da Olkhovski, Recami, Rac-
9iti & Zaichenko, ref.[12]). Secondo le predizioni della mec canica quantistica, la velocit` a
all’interno della barriera cresce illimitatamente per bar riere opache; e il tempo di tun-
nelling non dipende dalla larghezza della barriera[12].
Fig.4 – Simulazione di tunnelling mediante esperimenti con onde evanescenti (vedere il
testo), le quali pure era previsto fossero Superluminali in base alla Relativit` a Estesa[3,4].
La figura mostra uno dei risultati delle misure in refs.[15], ovvero la velocit` a media di
attraversamento della regione di evanescenza (tratto di gu ida sottodimensionata, o “bar-
riera”) al variare della sua lunghezza. Come previsto[19,1 2], la velocit` a media supera c
per “barriere” abbastanza lunghe.
Fig.5 – L’interessante esperimento in guida d’onda metalli ca con due barriere (tratti di
guida sottodimensionata), cio` e con due regioni di evanesc enza[22]. Vedere il testo.
Fig.6 – Un oggetto intrinsecamente sferico (o al limite punt iforme) appare come un elis-
soide contratto nella direzione del moto quando ` e dotato ne l vuoto di velocit` a v < c .
Qualora fosse dotato di velocit` a V > c (anche se la barriera della velocit` a cnon pu` o
essere attraversata n´ e da sinistra n´ e da destra) apparire bbe[34] non pi´ u come una parti-
cella, ma come un’onda “a forma di X” che si disloca rigidamen te (ovvero, come la regione
compresa tra un doppio cono e un iperboloide di rotazione a du e falde, o al limite come
un doppio cono, che viaggia nel vuoto —o in un mezzo omogeneo— Superluminalmente
e senza deformazione).
Fig.7 – Intersezioni con piani ortogonali alla direzione de l moto di una “X-shaped
wave”[34], secondo la Relativit` a Estesa[2-4]. L’esame de lla figura suggerisce come costru-
ire una semplice antenna dinamica atta a generare tali onde S uperluminali localizzate
(una tale antenna fu in effetti, indipendentemente, adottat a da Lu et al.[36] per la prima
produzione di questi beams non-dispersivi).
Fig.8 – Previsione teorica di onde Superluminali localizza te “a forma di X” per il caso
elettromagnetico (da Lu, Greenleaf & Recami[37], e Recami[ 37]).
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1Independence of events and quantum structure of the light
in Einsteins special relativity
Yves Pierseaux, Bruxelles and Oxford
(ypiersea@ulb.ac.be , yves.pierseaux@philosophy.ox.ac.uk ).
1- Einsteins principles of relativistic kinematics
2- Einsteins 1905 proclamation of independence of events 3- Events, points of light and quanta of light in January paper
4-Events, points of light and complex of light in June paper
5- Poincarés principles of implicit relativistic kinematics of deformable rods 6- Independent true time, material points and wave of light in Poincarés SR with ether Conclusion
1- Einsteins principles of relativistic kinematics
Everything seems have been said about the famous miraculous year 1905 of
the young Einstein. Many authors, physicists, philosophers and historians, have analysed the multiples connections between the three mains papers: quanta of light (January) brownian motion (March) and special relativity (June)[1].
The most important is perhaps this one of de Broglie whose the source of
inspiration for wavy mechanics is the contrast between the transformation of the frequency of light (Einstein January) and the transformation of the frequency of clock (Einstein June). Anyway it is well known that Einstein himself has developed a synthesis between quanta of light and brownian motion in 1909 on duality of the structure of the radiation.
However according also to Einstein himself, the three papers have not the
same statute: the first and the second are constructive theories while the last special relativity (SR) - is essentially a theory of principle [1].
January and March articles consist essentially in works of statistical
thermodynamics that show the reality of atomic structure of matter (March) and the hypothesis of atomic structure of radiation (January). Einsteins SR (June) look
deeply different because it is based on principles (more precisely two principles) that
are independent of any consideration on the structure of the matter or radiation.
But, in Einsteins own words, thermodynamics and relativity are the best
examples of theory of principles. Einstein insist often on the analogy between the principles of special relativity and the principles of thermodynamics. In his response to Ehrenfest about the question of rigidity Einstein underlines even more precisely that the principle of relativity together with the principle of constancy of velocity of 2light is to conceived as a heuristic principle and also as being similar to the
second principle of thermodynamics.
We want to analyse Einsteins analogy in depth not to contest that Einsteins
kinematics are based on principles (on the contrary!) but because there exist others principles in physics for example principle of reaction, principle of least action etc
(see for instance, Poincaré 1904 conference in Saint Louis on principles of mathematical physics).
2- Einsteins 1905 proclamation of independence of events
We suggest here a new way of research because we can find in each of the
three Einsteins articles the concept of independent events (unabhängig Ereignis): Quanta of light
(January, paragraph 5)
In calculating entropy by molecular-theoretical methods, the word probability is often used in a sense
differing from the way the word is defined in probability theory. In particular, case of equal probability are often hypothetically stipulated when the theoretical model employed are definite enough to permit a
deduction rather than a stipulation. I will show in a separate paper that in dealing with thermal
processes, it suffices completely to use the so-called statistical probability and hope thereby to
eliminate a logical difficulty that still obstructs the application of Boltzmanns principle . Here, I shall
give only its general formulation and its application to very special cases. It is meaningful to talk about the probability of a state of a system, and if furthermore, each increase of
entropy can be conceived as a transition to a state of higher probability, then the entropy S
1 of a system
is a function of the probability of its instantaneous state . Therefore if we have two systems S 1 and S 2
that do not interact which each other, we can set S 1 = ϕ (W1) S 2 = ϕ (W2)
It these two systems are viewed as a single system of entropy S and probability W, then:
S = S 1 + S 2 = ϕ (W) and W = W 1 W2
The last equation tells us that the states of the two systems are events that are independent of each
other (voneinander unabhängig Ereignis). From these equations it follows that:
ϕ (W 1 . W 2) = ϕ (W 1) + ϕ (W 2)
If S 0 denotes the entropy of a certain initial state and W is the relative probability of a state of entropy
S, we obtain [for Einstein always reference 1, always my italics and my bold ]:
Brownian motion (March, paragraph 4)
Obviously we must assume that each individual particle executes a motion that is independent of the
motions of all the other particles; the motions of the same particle in different time intervals must also be considered as mutually independent processes , so long as we think of these time intervals as chosen
not to be too small.
We now introduce a time interval τ, which is very small compared with observable time intervals but
still large enough that the motions performed by a particle during two consecutive time intervals can be
considered as mutually independent events (unabhängig Ereignis).[1]
Special relativity
(June, paragraph 1)
We have to bear in mind that our judgements involving time are always judgments about simultaneous
events (gleichzeitige Ereignisse). If, for example, I say that the train arrives here at 7 oclock that
means the pointing of the small hand of my watch to 7 and the arrival of the train are simultaneous
events .
SS kW−=0 ln 3We notice that in the three case the concept of independent event is in the core
of the article.
Quanta of light
Annus Mirabilis begins with Einsteins inversion of Boltzmanns principle:
independent states or independent events are directly connected with Einsteins conception of statistical probability in one of the two most important paragraphs of the article (see3).
I have show in my book [2] that Einsteins definition of probability with the
inversion of Boltzmanns principle is remained essentially the same from 1903 to 1925 (gas of photons). Brownian motion
Einstein treats fundamentally the Brownian motion as a random walk and the
independent events take place in the most important paragraph of the paper where Einstein deduces his famous relation of fluctuation-diffusion of Brownian particle from which Perrin will show experimentally the reality of the atoms.
Special relativity
Einsteins simultaneous events at the same place are of course independent
events. There is indeed no causal connection between the motion of the train that arrives in a given place (for example the station of Schaerbeek) and the motion of the small hand that arrives in a given place (for example the figure 7). We showed [2] that the crucial importance of independence of events is not only true for the events at the same place (in Minkowkis cone of light) but also for distance events (on and out Minkowkis cone of light).
The reader could be have the impression that we play on the word
unabhangig Ereignisse because in the two first cases the sense of word would be purely mathematical while only in the last case it would have a physical meaning connected with space and time.
We shall show in the details (this is the reason of the length of the quotations)
that it is in the three case the same physical concept.
It is obvious for the Brownian motion where each displacement D of the
particle in a time t is an event. We insist on the fact it is the first time in the history of physics that the movement of a particle is described as a serie of independent events.
In the article on quanta of light the deep physical meaning of Einsteins
concept of event is directly visible in its identification with the concept of state (see the quotation). We shall show that Einsteins definition of independent states or events is directly connected with the deduction of independent quanta of light (see 3).
The two first articles establish clearly a discontinuous structure of matter and
light. The standard look of Einsteins SR is, on the contrary, essentially based on the continuous conception of the field. The young Einstein however never uses in 1905 the concept of field while that he uses all the time the concept of event. 4 The advent of an event means deeply that suddenly something happens
somewhere et therefore implies a concept of discontinuity . The collision is of course
the best example of an event.
The classical continuous trajectory of a point material can be represented by a
series of events but it is in this case wrong events or events in a weak sense. Indeed each event is completely determined by the precedent because the initial state x, v determines completely the next positions and velocities of the material point.
The true event (in a strong sense of Prigogine) is thus a coincidence of events.
This is precisely the definition given by Einstein in his famous paragraph on simultaneity in SR.
We want to concentrate here the attention on the synthesis between quanta of
light and SR. We notice only about the Brownian motion that (see Einsteins quotation) the mutually independent process and mutually independent event are
deeply connected in Einsteins logic. So Einstein defines in his second synthesis on SR an event as the limit of a process (To determine the position of a process of infinitesimally short duration that occurs in a space element (point event) we need a Cartesian system ).
It is interesting to notice about Einsteins logic that Einsteins himself
underlines the necessity to eliminate a logical difficulty that still obstructs the application of Boltzmanns principle.
3- Events, points of light and quanta of light in January paper
We insist on the fact that Einstein characterises the state of a system according
to Boltzmanns principle by the entropy S or the probability W. After his definition of probability of independent state or event, he considers a set (or a gas) of points. I use inverted comas because these points are not the usual material points of classical mechanics.
We now treat the following special case. Let a volume V 0 contain a number n of moving points (e.g.
molecules) to which our discussion will apply. The volume may also contain any arbitrary number of
other moving points of any kind . No assumption is made about the law governing the motion of the
points under discussion in the volume except that, as concerns this motion, no part of the space (and no direction within it) is preferred over the others. Further, let the number of (aforementioned) moving points under discussion be small enough that we can disregard interactions between them.
This system, which, for example, can be an ideal gas or a dilute solution, possesses a certain entropy
S
0. Let us imagine that this state has a different value of entropy (S), and we now wish to determine the
difference in entropy with the help of Boltzmanns principle .
This space of points is homogenous and isotropic. Einstein applies therefore -
in order to find the temporal evolution of the system ( each increase of entropy can be
conceived as a transition to a state of higher probability) - not the second principle of
Newtons mechanics but the second principle of Boltzmanns thermodynamics.
In the point of view of mechanics, Newtons law is the law of change of state
defined by the position x or q and velocity v or p- of the material point. In the point of view of the young Einstein, Boltzmann law is the law of change of state
1 - defined
by V and S - of the n independently moving points:
1 I showed in my book that the mysterious expression “law of change of states” Einstein’s principle of relativity
could be the Boltzmann law. We notice that k is at the level of a principle theory while h is only a the level of a
constructive theory (see conclusion). This is the reason for which I have notice N/R by k. 5We ask: how great is the probability of the last-mentioned state relative to the original one? Or: how
great is the probability that a randomly chosen instant of time, all n independently moving points in a
given volume V 0 will be found (by chance) in the volume V? Obviously, this probability, which is a
statistical probability has the value:
From this, by applying Boltzmanns principle, one obtains
Lorentz insisted in Solvay Conference in 1911 on the fact that Einsteins
statistical probability has nothing to do with Gibbss definition of probability based on the element dpdq in the space of phase. Anyway if the state of Einsteins points were defined by classical mechanics, it is impossible that the motions of the points were fundamentally independently. This is the deep reason for which Einstein rejects the definition of equiprobability (see quotation) based on a mechanical priori model. According to Einstein, this a priori model is precisely this one of standard Gibbss statistical mechanics.
Einsteins independently moving points suppose another definition of the
state of this points. Einsteins thermodynamical definition (V, S) of state implies the definition of independent state not from a element of the phase space but from an element of volume and an element of duration and therefore - to the limit - a point event (x, y, z, t).
By an elementary event we will understand an event that is supposed to be concentrated in one point
and is of infinitely short duration.[Einstein, SR, 1910] To determine the position of a process of infinitesimally short duration that occurs in a space element (point event) [Einstein, SR, 1907]
We have so shown that concept of independent events is the same that the
concept of independently moving points whose each independent state is not characterised by ( x, v) - as in Gibbs mechanical statistics- but by (x, y, z, t). We have
however not yet shown that this latter is the same concept that independently quanta of light. Let us go on with Einsteins quotation:
Paragraph 6 Interpretation according to Boltzmanns principle of the expression for the dependence of
the entropy of monochromatic radiation on volume
In section 4 (limiting law for the entropy of monochromatic radiation at low radiation density) for the
dependence of the entropy of monochromatic radiation on volume (
)
And compare to the general formula expressing the Boltzmanns principle
We arrive in the following conclusion. WV
Vn=()
0
SS n kV
V−=0
0ln
SS kV
VkE
−=0
01
ln( )βν
SS kW−=0ln 6If monochromatic radiation of frequency ν and energy E is enclosed (by reflecting walls) in the volume
V, the probability that at a randomly instant the total radiation energy will be found in the portion V of
the volume V 0 is:
From this we further conclude that monochromatic radiation of low density behaves
thermodynamically as if it consisted of mutually independent energy quanta of magnitude βkν.
The logic structure of this extraordinary paper is that Einstein obtains in his
paragraph 4 a relation between entropy and volume for the Wien radiation and in his paragraph 5 a similar relation thanks to the applying of Boltzmanns principle as the fundamental law of movement for his independent points (W-> S). Einsteins strategy in paragraph 6 consists logically in a comparison and an inversion (S->W).
The famous disagreements between Planck and Einstein about the structure of
light and the definition of probability are the same thing. Einsteins inversion of Boltzmann principle means not only that the relation may be mathematically in the other sense but implies physically that the probability that at a randomly instant the total radiation energy will be found in the portion V in others words, if Einsteins probability has a physical sense, Einsteins quanta has also a physical sense.
Einsteins points can be as well material point or light point whose the
singularity consists in his velocity v = c. The deepest meaning of the relation E = β k
ν in Einsteins paper is that the structure of monochromatic radiation consists in a
discontinuous serie of independent point events.
4-Events, points of light and complex of light in June paper
It is well known that Einsteins kinematics implies the deletion of the ether.
The first meaning of the existence of this medium is that the light is a wave and there is no point with velocity of light (v = c).
In the first part of his article, Einstein never speak of wave of light. Einsteins
basic concept is ray of light (in the sense of geometrical optics), i.e. a point of light. In his paragraph 3, Einstein deduces the transformation of the coordinates of an event:
To any system of values x, y, z, t, which completely defines the place and the time on an event in the
stationary system K, there belongs a system of values ξ, η, ζ, τ , determining that event relatively to the
system k, and our task is now to find the system of equations connecting this quantities (
). We obtain:
x = ϕ γ (x-v t), η = ϕ y, ζ = ϕ z,
τ = ϕ γ (t - vx/c
2)
We must show that Einsteins transformations of event supposes an extension
of the classical notion of material point. In other words: it is not only in the first part of his article that Einstein characterises the light as a point but also in the second part.
In the famous polemic of priority between Poincaré and Einstein, Whittaker,
who attributes the discovery of SR to Lorentz and Poincaré, conceded that the Doppler formula was deduced by Einstein and not by Poincaré. Is it a historical contingence? WV
VkE
=()
01
βν 7The first time that Einstein speaks of the electrodynamic (sic) wave in his
famous June paper, he applies his Lorentzs transformation at
a point v = c of the plane of a wave plane (paragraph 7):
In the system K and very far from the coordinate origin, let there be a source of electrodynamic waves
[whose the phase is] Φ = ωt k x. We want to known the character of these waves when
investigated by an observer at rest in the moving system k. Applying the transformation equations for
electric and magnetic forces found in section 6 and those for coordinates in section 3, we immediately
obtain: Φ = ωτ κκκκ ξ.ξ.ξ.ξ.
It is of course forbidden to make v = c in LT but the state (x, y, z, t) of a light
point is an event. The propagation of the light consists in Einsteins SR fundamentally in a serie of events. The concept of event implies a extension of the domain of application of LT.
The physicists credit often Einstein with the discovery of the relativistic law of
composition of the speed and Poincaré with the discovery of the structure of group of Lorentz transformations (LT). But in fact the two elements are in each approach (Einsteins § 5 and Poincarés § 4). The interesting point is not in these polemical questions of priority but the fact that Einstein applies his LT for light point unlike Poincaré whose the law of composition of velocities concerns the material systems or the material points:
It is noteworthy that the Lorentz transformations form a group. For, if we put:
x = l k (x + εt), y= l y, z = l z, t = k l (t + εx)
and
x = l k(x + ε't), y = l y, z = l z, t = l k (t+ ε'x)
we find that
x = l k (x + ε"t), y= l y, z = l z, t = l k (t + ε"x)
w i t h
[2d, §4, 1905]
Einstein introduces his kinematics theorem of addition of velocities in his
paragraph 5:
If w also has the direction of the x-axis v, we get
It follows from this equation that the composition of two velocities that are smaller than c always
results in a velocity that is smaller than c. It also follows that the velocity of light c cannot be changed by compounding it with a subliminal velocity. For this case we get: uvw
vw
c=+
+12′′=+′
+′εεε
εε1
ucw
w
cc =+
+=
1 8
Poincarés relation is formally the same than Einsteins one but it concerns
only the systems or material points. The most important difference between the two conceptions consist in the case v = c. Einstein The reader could be think that Einsteins substitution v = c is not fundamental but Einstein writes in his paragraph 4:
For v=c, all moving objects considered from the rest system, - shrink into plane structures. For
superluminal velocities our considerations become meaningless.
We return now in the second part of the article. In his paragraph 8, Einstein deduced,
after the Doppler relations, the mode of transformation of energy and frequency of what he calls a complex of light.
It is noteworthy that the energy and the frequency of a light complex vary with the observers state of
motion according to the same law.
Many authors have of course underlined the similarity between quanta of light
of January and complex of light of June. They generally said that Einsteins deduction of proportionality relation between frequency and energy confirms only that
Einsteins SR is a theory of principles and therefore not depending of a classical or
quantum model of the light. It is necessary to say that but it is not sufficient because to obtain this relation energy-frequency, Einstein has one more time applies the Lorentz transformation to a light point v = c (paragraph 8).
The surface elements of the spherical surface [in K] moving with the velocity of light are not traversed
by any energy. We may therefore say that this surface permanently encloses the same light complex. We investigate the quantity of energy from the system k, i.e. the energy of the light complex relative to the system k.
In order to obtain his quantum relation, Einstein considers a quite singular limit case
not only for the applying of LT but also for Maxwell-Lorentz equations themselves.
Lorentz dont like this approach of the young Einstein writes in the final proceedings of the Conference in Brussels (1911):
There is nothing in Maxwell equation that can encloses a quantity of energy in a limited volume
The light complex were not more appreciated that by the physicists. The young
Einstein was however not wrong because the volume and the energy changes in function of
the time. In fact, Einstein treats, in a very singular manner, the Maxwell equations not as an
wave equation but as equation of movement of a light point. Light complex is a finite volume
around the light point characterised by a frequency. We find again to the limit the concept of event and that Einsteins representation of the light is deeply discontinuous.
Penrose writes:
To me it is virtually inconceivable that he would have put forward two papers in the same year which
depended upon hypothetical views of nature that he felt were in contradiction with each other.[1.
preface]
We proved there is no contradiction: quanta of light and complex of light are
on the same side of the quantum-classic frontier the most important of the present physics. When the young When Einstein considers the Maxwell equations as an application his kinematics of events, he doesnt treat them classically. 9His treatment consist to associate to a point light wavy properties (the first
being already that the light point is a point whose the speed is independent of the source that emits it). We find again de Broglie but not for the electron but the photon, in the strong quantum sense of the concept.
However, we dont want say here that SR, in the standard sense where SR is
the group properties of Lorentz transformation, imposes a quantum conception of the light. Indeed we must now prove that there exist another SR based on a purely wavy representation of the light and on a implicit kinematics of material point.
5- Poincarés principles of implicit relativistic kinematics of deformable rods
The main mistake of Poincarés supporters in the discover of SR (1905) is to extract elements in Poincarés work (for example, the synchronisation of the clocks, the scepticism with respect to the ether, the statement of principle of relativity, the structure of group of LT
) and to insert them in Einsteins logic (or axiomatic) in order to show that Poincaré had Einstein before Einstein. This is of course unpleasant for Einstein but in my opinion also for Poincaré.
Indeed with this method of extraction-insertion it is absolutely impossible to
understand Poincarés relativistic logic. Poincarés logic is deeply different than
Einsteins one because his theory of relativity proves the existence of the ether while Einsteins one proves the absence of this latter: this is truly an antinomy in the sense of Kant.
I show in my book that the Poincarés scepticism dates back to long before his
crucial interest for Lorentz theory based on the existence of the ether and the electrons. According to Poincaré the main problem of Lorentz theory is his incompatibility with the principle of reaction (Poincaré 1900): what is the dynamical force for responsible of the real contraction of length in ether ( Lorentz hypothesis, LH)?
Poincarés guide for the theory of relativity is clearly the third principle of
mechanics in the same epistemological manner that the second principle of thermodynamics is Einsteins guide. In his fundamentally 1905 work he determines this force exerted by the ether on the electron:
But in the Lorentz hypothesis [LH], also, the agreement between the formulas does not occur just by
itself; it is obtained together with a possible explanation of the compression of the electron under the
assumption that the deformed and compressed electron is subject to constant external pressure , the
work done by which is proportional to the variation of volume of this electron. (my italics) [2d, intro.]
Poincarés aim in his 1905 work is to show dynamically ( with his pressure of
ether) the compatibility between the principle of relativity and principle of real contraction:
So Lorentz hypothesis [LH] is the only one that is compatible with the impossibility of demonstrating
the absolute motion [RP]; if we admit this impossibility, we must admit that moving electrons are contracted such a manner to become revolution ellipsoids whose two axis remain constant [2d, § 7].
It is true that there is no theory of space-time in Poincarés 1905 work. There
is however a implicit kinematics (real contraction of deformable rod) behind his 10dynamics (Poincaré deduce fundamental equations of relativistic dynamics by
introducing the ether pressure) .
Implicit Poincarés kinematics is based on two postulates (principle of
relativity and LH) exactly of the same manner that explicit Einsteins kinematics is based on two postulates (principle of relativity and light principle).
Poincarés SR with ether implies another use of LT:
In accordance with LH, moving electrons are deformed in such a manner that the real electron becomes
an ellipsoid, while the ideal electron at rest is always a sphere of radius r (
) The LT replaces thus a moving real electron by a motionless ideal electron. (2d, §6)
In order to illustrate this, we can use Tonnelats diagram [22] (fig 1, we
adopt Poincarés and Einsteins respective notations in the following, see former
respective quotations about LT):
In Einsteins SR, the contraction of the moving rod γ-1L is not real but is the
reciprocal result of a comparison of measurement made on identical rods2 L from one
system to the other with the well known use of LT (dashed lines).
In Poincarés SR the contraction of the moving rod k-1L is by principle (LH)
real in K. By the use of LT the length of the rod in K (for observers in K) seems be equal to L. Reciprocally, we can of course reverse the role of K and K(where ether is now chosen at rest) and reverse the continuous lines and dashed lines.
The calculation with the LT is also very easy. Suppose the ether is chosen by
definition at rest in K. The real length of the rod placed in the moving system K is
thus k
-1 L. The first LT x = k (x εt) replaces (in Poincarés terms) in any time t
(see below) the length moving real rod k-1L by a motionless rod L.
According to Poincaré , the real differences are compensated by a good use of LT.
According to Einstein, the identical processes seem to be different by another good use of LT. This is
of course valid for the rods but also for the clocks.
Now we must show that Poincarés LT is fundamentally a transformation of
the coordinates of a material point and not a transformation of the co-ordinate of an event.
5- Independent true time, material points and wave of light in Poincarés SR with ether
We showed that ether plays an essential role in Poincarés SR. But it is not an
absolute ether in the sense of Lorentz. Indeed, according to Poincaré only a relative
speed can be measured with respect to the ether. The group structure implies that Poincarés relativistic ether can be chosen, for all the couples of inertial systems, at
rest in one of the two frames but the other is then in movement with respect to the
ether. The relativistic ether supposes that the light is a wave.
The direct consequence of wavy representation of the light is Poincar
és
convention of synchronisation based on the duality local time and true time. Indeed the two inertial system are not in the same state of movement with respect to the
2 Max Born, who was a specialist of rigidity in Einsteins special relativity: A fixed rod that is at rest in the
system S and is of length 1 cm, will, of course, also have the length 1 cm, when it is at rest in the system S,
provided that the remaining physical conditions are the same in S as in S. Exactly the same would be postulated of
the clocks. We may call this tacit assumption of Einsteins theory the principle of the physical identity of the
units of measure. [14, my italics p252] This is not really a third hypothesis b ecause Einsteins deduces the
identity of his rods from his relativity principle. The rigidity is not important. The important thing in the spirit of
the young Einsteins text, is to postulate the existence in Nature of processes giving units of length and time. 11relativistic ether. In his talk on The principles of mathematical physics, Poincar é
considers optical signals as optical perturbations:
1- (Two observers are at rest relative to ether, System K)
“The most ingenious idea has been that of local time. Imagine two observers [A and B] who wish to
adjust their watches by optical signals; they exchange signals, (…) And in fact, they [The clocks of A and B] mark the same hour at the same physical instant, but on one condition, namely, that the stations are fixed.
2- (Two observers are moving relative to ether, System K’)
In the contrary case the duration of the transmission will not be the same in the two senses, since the
station A, for example, moves forward to meet the optical perturbation emanating from B, while the station B flies away before the perturbation emanating from A.” The watches adjusted in that manner do not mark, therefore the true time; they mark the local time , so
that one of them goes slow on the other (de telle manière que l’une retarde sur l’autre). It matters little, since we have no means of perceiving it.” (…)
(for exact compensation, we must add LH, former quotation)
Unhappily, that does not suffice, and complementary hypotheses are necessary. It is necessary to admit
that bodies in motion undergo a uniform contraction in the sense of the motion.” [2b]
Poincarés synchronisation is clearly based for his second system K on the
duality of the true time t and the local time t .
This is Poincarés tour de force to have shown that his second principle
(LH) implies for the local time not the Lorentz expression t = t + εx - but the
expression given by the fourth LT: t = k (t + εx).
We proved3 that Einsteins radical elimination of ether implies that he
prepares identically his two systems K(t) and k( τ), (light clocks or better clocks
with photons) in the same state of synchronisation.
This is the reason for which the duality true time-local time have no sense in
Einsteins logic. Poincarés relativistic ether hasnt got a particular state of movement, but his two systems are never in the same state of moving relative to it . This is the
direct consequence of the wavy nature of the light that propagates in a medium. Poincarés local time is not a internal time, given by identical clocks, in the second
frame K.
The main interest here is the incompatibility of the concept of local time and
the concept of event. The local time t
A= k (t + ε xA) (or the apparent time in the
vocabulary of Poincaré) is a time that depends of three factors: firstly of course the
independent true time (independent doesnt mean absolute), secondly of the speed (the difference of state of moving of the two systems relative to ether) and thirdly of the place of the observer placed for example in A:
Let us consider one point at true time with the true coordinates x, y, z. What will be the apparent
coordinates x, y, z at apparent time t?
x = k (x + εt), y= y, z = z, t = k (t + εx)
Let us the units by such a way that the speed of light equals to 1. What is the meaning of ε? The
translation velocity of the system in the sense of the x-axis et has the value ε.
In the same way t is the apparent time, because in two points , the apparent time differs of the true time
with a quantity proportional to the abscissa. [Cours a la Sorbonne, 3a]
3 Einsteins synchronisation of identical clocks within the second system k is exactly the same than the
synchronisation in the first system because the speed of light is of course exactly the same. It is essential to have
time defined by means of stationary clocks in stationary system (
) [ §1, June 1905]. To do this [deduce LT] we
have to express in equations that τ is nothing else than the set of data of clocks at rest in system k, which have been
synchronized according to the rule given in paragraph 1. 12
The local time (apparent time) in A means that the time of an event4 in B is
not the time given at the place B where this event occurs. This is not a time of event in Einsteins sense of the coincidence of two independent events. It is not necessary to consider the distance simultaneity (necessarily independent events) in both SR because already Einsteins definition of simultaneity (coincidence) at the same place is sufficient.
Even if Poincaré doesnt use the concept of event, let us introduce an event
in weak sense i.e. in the sense of a point on the trajectory of material point. We must then of course have two material points to consider the simultaneity of two events. We underline then that in the well known laplacian deterministic point of view of the
classical mechanics the two events considered (for example the two points are simultaneously at a given distance) cannot be independent. This is exactly what Poincaré admitted in his text on Lespace et le temps en 1911 [3a].
The choice between the independence of time (Poincaré) and the
independence of events (Einstein) is then clearly antinomic.
Conclusion
The fact that SR is basically considered as a theory of space-time implies that
historically Einstein must be considered as the author of this theory.
Indeed Poincaré has not developed in 1905 as Einstein a kinematics based on
two principles but only a relativistic dynamics or more precisely a relativistic dynamic of continuous medium. But the opposition between SR with principles (Einstein) and SR without principles (Poincaré) is superficial (historical contingence) because it is easy to see that there is a implicit kinematics of material point and wave of light behind Poincarés relativistic dynamics.
The comparison of Einsteins kinematics of events with Poincarés implicit
kinematics of material point is very interesting because it shows the deep and irreducible singularity not only about Einsteins principle of identity of units of measure but also about Einsteins complex of light or photons.
The idea of a fine structure of SR means that the borderline classical-
quantum, the main cut of the physics of XX century, passes between the two SR.
The reader could think that we contest that Einsteins SR is based on
principles because we show that it exist a quantum representation hidden behind this principles. But quantum theory is no longer a constructive or an heuristic theory as at the epoch of old quantum theory. The quantum theory is now considered as a theory of principles or a genuine picture of world.
The existence of a fine structure of SR opens a new way of research that
consists to transform Einsteins SR without any change in his historical originality -
under the enormous pressure of classical Poincarés theory, CSR) in a real relativistic quantum theory (QSR) of the light (photons), the time (quanta of time) and the space (acausal zone).
In others words, we can overtake the fierce concurrency between Einstein and
Poincaré on the market of the priorities in a genuine dialectical interaction between
4 Even if Poincaré doesn’t use concept of event let us introduce an “event” in weak sense i.e. in the sense of a point
on the trajectory of material point. We must then of course have two material points to consider the simultaneity of
two “events”. We underlines then that in the laplacian point of view of the classical mechanics the two “events”
considered (for example the two points are simultaneously at a given distance) are not independent. 13the two complete and coherent relativistic logics. The philosophical opposition
continous-discontinous becomes then a physics opposition quantum-classic.
References
Einstein
[1a] Einsteins miraculous year, Five papers that change the face of physics, edited and introduced by
John Stachel, Princeton University Press, 1998. [1b] 1907. Relativitätsprinzip und die aus demselben gezogenen. Folger ungen Jahrbuch der
Radioaktivität, 4, p 411-462 § 5, p 98-99.
[1c] 1910 Principe de relativité et ses conséquences dans la physique moderne (1ère des trois
publications francophones-suisses- dEinstein). Archives des sciences physiques et naturelles, 29, p 5-28 et 125-244.
Poincaré
[2a] 1900. La théorie de Lorentz et le principe de réaction. Archives néerlandaises des sciences
exactes et naturelles, 2ème série, 5. [2b] 1904 Les principes de la physique mathématique. Congrès international dArts et de Sciences., exposition universelle à Saint Louis.
[2c] 1905 La dynamique de lélectron. Comptes rendus de lAcadémie des sciences de Paris, 5 juin,
140, 1504-1508. [2d] 1905. Sur la dynamique de lélectron. Comptes rendus de lAcadémie des sciences de Palerme;
dans Rendiconti d. Circ. mat. de Palermo, 21, 1906.
[3a] Pierseaux Yves, La structure fine de la relativite restreinte ,Paris, LHarmattan, 1998
[3b] Pierseaux Yves, Euclidean Poincarés SR and non Euclidean Einstein-Minkowskis SR, PIRT, 2000.
[4] Born M. Einsteins theory of Relativity 1949 Dover, New York, 1965. original,
[5] Tonnelat M.A. Histoire du principe de relativité (Flammarion, Paris, 1931).
|
arXiv:physics/0102001v1 [physics.plasm-ph] 1 Feb 2001Topological-Torsion and Topological-Spin
as coherent structures in plasmas
R. M. Kiehn
Physics Department, University of Houston
rkiehn2352@aol.com
http://www.cartan.pair.com
Abstract The PDE’s of classical electromagnetism can be generated
from two exterior differential systems that distinguish top ologically the field
intensities and potentials, F−dA= 0,from the field excitations and the
charge current densities, J−dG= 0. The existence of potentials, A, leads
to the independent 3-forms of T-Torsion = AˆFand T-Spin = AˆG. The
exterior derivatives (divergences) of these 3-forms produ ce anomalies that de-
fine the two classic Poincare invariants. The closed integra ls of these forms,
when deformation invariants of frozen-in fields, define topo logical coherent
structures in the plasma. Solutions when T-Torsion (T-Spin ) is zero define
transverse magnetic (electric) modes on topological groun ds. When the di-
vergence of T-Torsion is not zero there exists a classical me chanism for charge
acceleration along the magnetic field lines producing sympl ectic plasma cur-
rents; large temperature gradients along the Bfield lines can act as a source
of stellar plasma jets in neutron stars. In such circumstanc es, the Torsion
vector is uniquely defined by conformal invariance of the Act ion potentials.
Plasma currents in the direction of the Torsion vector leave bothAˆFand
AˆGconformally invariant, hence these fields are frozen-in eve n though the
processes are thermodynamically irreversible. The decayi ng coherent and
deformable topological structures associated with such fr ozen-in fields are
persistent and observable artifacts, similar to wakes, tha t can appear in any
plasma domain, such as that which surrounds stars.
1 Introduction
In the language of exterior differential systems [1] it becom es evident that
classical electromagnetism is equivalent to a set of topolo gical constraints on
a variety of independent variables. Certain integral prope rties of an electro-
magnetic system are deformation invariants with respect to all continuous
1evolutionary processes that can be described by a singly par ameterized vec-
tor field. These deformation invariants lead to the fundamen tal topological
conservation laws described in the physical literature as t he conservation of
charge and the conservation of flux. Recall the definitions:
A continuous process is defined as a map from an initial state o f
topology Tinitial into a final state of perhaps different topology
Tfinalsuch that the limit points of the initial state are permuted
among the limit points of the final state. [2]
A deformation invariant is defined as an integral over a close d
manifold,/integraltext
zωsuch that the Lie derivative of the closed integral
with respect to a singly parameterized vector field, βVk,vanishes,
for any choice of parametrization, β.
L(βVk)/integraldisplay
zω= 0 any β
The idea of a deformation invariant comes from the Cartan con cept of a
tube of trajectories as applied to Hamiltonian mechanics. C onsider the flow
lines tangent to the trajectories generated by Vk,and a closed integration
chain that connects points on a tube of different trajectorie s. Under certain
conditions (when the virtual work vanishes) the integral of the exterior 1-
form of Action, A=pdq−H(p, q, t)dtevaluated along the closed integration
chain yields a value which is the same no matter how the integr ation chain
is deformed, as long as it resides on the same tube of trajecto ries. As
the points on the trajectories have relative displacements determined by a
factor β(p, q, t) then the closed chain connecting points can be deformed by
choosing a different function β.Cartan used this idea for demonstrating
that the tube of trajectories is uniquely defined on a contact manifold by a
Hamiltonian flow that conserves energy. [3] He thereby define d conservative
Hamiltonian processes in a topological manner by requiring that processes be
the subsets of singly parameterized vector fields that leave the closed integral
of the 1-form of Action a deformation invariant.
However, for physical systems that can be defined by a 1-form o f
Action, A, the derived 2-form F=dAis a deformation invariant with respect
toallcontinuous processes that can be defined by a singly paramete rized
vector field. This concept is at the basis of the Helmholtz the orems in
2hydrodynamics, and the conservation of flux in classical ele ctromagnetism.
Herein, this topological constraint will be called the post ulate of potentials.
When written as the equation, F−dA= 0,the postulate of potentials is to be
recognized as an exterior differential system constraining the topology of the
independent variables. From Stokes theorem, the (2 dimensi onal) domain of
finite support for Fcan not in general be compact without boundary, unless
the Euler characteristic vanishes. There are two exception al cases for two
dimensional domains, the (flat or twisted) torus and the Klei n-Bottle, but
these situations require the additional topological const raint that FˆF= 0.
The fields in these exceptional cases must reside on these exc eptional compact
surfaces, which form topological coherent structures in th e electromagnetic
field. For an electromagnetic action, the exceptional compa ct cases can only
exist if E◦B= 0.The resulting statement is that there do not exist compact
domains of support without boundary when E◦B/ne}ationslash= 0,a statement that will
be of interest to thermodynamics of irreversible systems, a nd of plasma jets.
The definition of an electromagnetic system of charges and cu rrents will
require a second topological constraint imposed upon the do main of indepen-
dent variables. This second postulate will be called the pos tulate of conserved
currents. The electromagnetic domain not only supports the 1-form A, but
also supports an N-1 form density, J,which is exact. The equivalent differ-
ential system, J−dG= 0,requires that the (N-1 dimensional) domain of
support for Jcannot be compact without boundary. However, the closed
integrals of Jare deformation invariants for anycontinuous evolutionary
process that can be defined in terms of a singly parameterized vector field.
In section 2, the classical Maxwell system will be displayed in terms of
the vector formalism of Sommerfeld and Stratton. The key fea ture is to note
that the fields of intensities ( EandB) are considered as separate and distinct
from the fields of excitation ( DandH), a historical distinction (championed
by Sommerfeld) that is often masked in modern exposes of elec tromagnetic
theory.
In section 3, it will be demonstrated explicitly that the cla ssic formal-
ism of electromagnetism in section 2 is a consequence of a sys tem of two
fundamental topological constraints
F−dA= 0, J −dG= 0.
defined on a domain of four independent variables. The theory requires the
existence of four fundamental exterior differential forms, {A, F, G, J },which
3can be used to construct the complete Pffaf sequence [4] of for ms by the
processes of exterior differentiation and exterior multipl ication. On a domain
of four independent variables, the complete Pfaff sequence c ontains three 3-
forms: the classic 3-form of charge current density, J,and the (apparently
novel to many researchers) 3-forms of Topological Spin Curr ent density,
AˆG,[5] and Topological Torsion-Helicity, AˆF[6]. To shorten notation,
the terms T-Torsion and T-Spin will be used.
As the charge current 3-form, J,is a deformation invariant by construc-
tion, it is of interest to determine topological refinements or constraints for
which the 3-forms of T-Spin and T-Torsion will define physica l topological
conservation laws in the form of deformation invariants. Th e additional
constraints are equivalent to the topological statement th at the closure (ex-
terior derivative) of each of the three forms is empty (zero) . It will be
demonstrated in section 4 that these closure conditions defi ne the two clas-
sic Poincare invariants (4-forms) as deformation invarian ts, and when each
of these invariants vanish the corresponding 3-form genera tes a topological
quantity (T-Spin or T-Torsion respectively) which is also a deformation in-
variant. The possible values of the topological quantities , as deRham period
integrals [7], form rational ratios.
The concepts of T-Spin Current and the T-Torsion vector have been uti-
lized hardly at all in applications of classical electromag netic theory. Just as
the vanishing of the 3-form of charge current, J= 0,defines the topological
domain called the vacuum, the vanishing of the two other 3-fo rms will refine
the fundamental topology of the Maxwell system. Such constr aints permit
a definition of transversality to be made on topological (rat her than geomet-
rical) grounds. If both AˆGandAˆFvanish, the vacuum state supports
topologically transverse modes only (TTEM). Examples lead to the conjec-
ture that TTEM modes do not transmit power, a conjecture that has been
verified when the concept of geometric transversality (TEM) and topologi-
cal transversality (TTEM) coincide. A topologically trans verse magnetic
(TTM) mode corresponds to the topological constraint that AˆF= 0.A
topologically transverse electric mode (TTE) corresponds to the topological
constraint that AˆG= 0.Examples, both novel and well-known, of vacuum
solutions to the electromagnetic system which satisfy (and which do not sat-
isfy) these topological constraints are given in the append ix. These ideas
should be of interest to those working in the field of Fiber Opt ics. Recall
that classic solutions which are geometrically and topolog ically transverse
(TEM≡TTEM) do not transmit power [8]. However, in the appendix an
4example vacuum wave solution is given which is geometricall y transverse (the
fields are orthogonal to the field momentum and the wave vector ), and yet
the geometrically transverse wave transmits power at a cons tant rate: the
example wave is not topologically transverse as AˆF/ne}ationslash= 0.
In section 4, an additional topological constraint will be u sed to define
the plasma process as a restriction on all processes which ca n be described
in terms of a singly parameterized vector field. The plasma pr ocess (which
is to be distinguished from a Hamiltonian process) will be re stricted to those
vector fields which leave the closed integrals of Ga deformation invariant.
(Compare to the Cartan definition that a Hamiltonian process is a restriction
on arbitrary processes such that the closed integrals of Aare deformation
invariants with respect to Hamiltonian processes). A plasm a process need
not conserve energy. A perfect plasma process is a plasma process which
is also a Hamiltonian process. Again, the three forms, J, AˆGandAˆF
are of particular interested for their tangent manifolds de fine ”lines” in the
4-dimensional variety of space and time. Relative to plasma processes, the
topological evolution associated with such lines, and thei r entanglements, is
of utility in understanding solar corona and plasma instabi lity. [9]
2 The Domain of Classical Electromagnetism
2.1 The classical Maxwell-Faraday and the Maxwell-
Ampere equations.
Using the notation and the language of Sommerfeld and Stratt on [10], the
classic definition of an electromagnetic system is a domain o f space-time
{x, y, z, t }which supports both the Maxwell-Faraday equations,
curlE+∂B/∂t= 0, div B= 0, (1)
and the Maxwell-Ampere equations,
curlH−∂D/∂t=J, div D=ρ. (2)
52.2 The conservation of charge current
In every case, the charge current density for the Maxwell sys tem satisfies
the conservation law,
divJ+∂ρ/∂t = 0. (3)
The charge-current densities are subsumed to be zero [ J,ρ] = 0 for the
vacuum state. For the Lorentz vacuum state, the field excitat ions,Dand
H, are linearly connected to the field intensities, EandB, by means of the
Lorentz (homogeneous and isotropic) constitutive relatio ns:
D=εE,B=µH. (4)
The two vacuum constraints imply that the solutions to the ho mogeneous
Maxwell equations also satisfy the vector wave equation, ty pically of the form
grad div B−curl curl B−εµ∂2B/∂t2= 0. (5)
The constant wave phase velocity, vp,is taken to be
v2
p= 1/εµ≡c2(6)
Similar results can be obtained for the solid state where the constitutive
constraints can be more complex [11], and for the plasma stat e where the
charge-current densities are not zero.
2.3 The existence of potentials
It is further subsumed that the classic Maxwell electromagn etic system is
constrained by the statement that the field intensities are d educible from a
system of twice differentiable potentials, [ A, φ]:
B=curlA,E=−grad φ −∂A/∂t. (7)
This constraint topologically implies that domains that su pport non-zero val-
ues for the covariant field intensities, EandB,cannotbe compact domains
without a boundary. It is this constraint that distinguishe s classical electro-
magnetism from Yang Mills theories. Two other classical 3-v ector fields are
of interest, the Poynting vector E×Hrepresenting the flux of electromag-
netic radiative energy, and the field momentum flux, D×B.
63 The Fundamental Exterior Differential Sys-
tems.
The formulation of Maxwell theory in section 2 is relative to a choice of
independent variables {x, y, z, t }using classical vector analysis developed in
euclidean 3-space. The topological features of the formali sm are not imme-
diately evident. However, electromagnetism has a formulat ion in terms of
Cartan’s exterior differential forms [12]. Exterior differe ntial forms do not
depend upon a choice of coordinates, do not depend upon the a c hoice of
metric, and are independent of the constraints imposed by ga uge groups and
connections. In such a formulation the equations of an elect romagnetic
system become recognized as consequences of topological co nstraints on a
domain of independent variables.
The use of differential forms should not be viewed as just anot her formal-
ism of fancy. The technique goes beyond the methods of tensor calculus,
and admits the study of topological evolution. Recall that i f an exterior dif-
ferential system is valid on a final variety of independent va riables {x,y,z,t},
then it is also true on any initial variety of independent var iables that can
be mapped onto {x,y,z,t}. The map need only be differentiable, such that
the Jacobian matrix elements are well defined functions . The Jacobian
matrix does not have to have an inverse, so that the exterior d ifferential
system is not restricted to the equivalence class of diffeomo rphisms. The
field intensities on the initial variety are functionally we ll defined by the pull-
back mechanism, which involves algebraic composition with components of
the Jacobian matrix transpose, and the process of functiona l substitution.
This independence from a choice of independent variables (o r coordinates)
for Maxwell’s equations was first reported by Van Dantzig [13 ]. It follows
that the Maxwell differential system is well defined in a covar iant manner
for both Galilean transformations as well as Lorentz transf ormations, or any
other diffeomorphism. (The singular solution sets to the equ ations do not
enjoy this universal property). In addition, it should be no ted that the ideas
of the exterior differential system imply that the closure eq uations of the
Maxwell-Faraday type form a nested set, with exactly the sam e format, in-
dependent of the choice of the number of independent variables. In addition,
every physical system (such as fluid) that supports a 1-form o f Action, has
its version of the Maxwell-Faraday induction equations.
73.1 The Maxwell-Faraday exterior differential system.
The Maxwell-Faraday equations are a consequence of the exte rior differential
system
F−dA= 0, (8)
where Ais a 1-form of Action, with twice differentiable coefficients ( poten-
tials proportional to momenta) which induce a 2-form, F,of electromagnetic
intensities ( EandB,related to forces and objects of intensities). The ex-
terior differential system is a topological constraint that in effect defines
field intensities in terms of the potentials. On a four dimens ional space-time
of independent variables, ( x, y, z, t ) the 1-form of Action (representing the
postulate of potentials) can be written in the form
A= Σ3
k=1Ak(x, y, z, t )dxk−φ(x, y, z, t )dt=A◦dr−φdt. (9)
Subject to the constraint of the exterior differential syste m, the 2-form of
field intensities, F,becomes:
F=dA={∂Ak/∂xj−∂Aj/∂xk}dxjˆdxk(10)
=Fjkdxjˆdxk=Bzdxˆdy...Exdxˆdt... (11)
where in usual engineering notation,
E=−∂A/∂t−gradφ, B=curlA≡∂Ak/∂xj−∂Aj/∂xk.(12)
The closure of the exterior differential system, dF= 0,
dF=ddA={curlE+∂B/∂t}xdyˆdzˆdt−..+..−divBdxˆdyˆdz} ⇒0,
(13)
generates the Maxwell-Faraday partial differential equati ons.:
{curlE+∂B/∂t= 0, div B= 0}. (14)
The component functions ( EandB) of the 2-form, F,transform as covariant
tensor of rank 2. The topological constraint that Fis exact, implies that
the domain of support for the field intensities cannot be comp act without
8boundary, unless the Euler characteristic vanishes. These facts distinguish
classical electromagnetism from Yang-Mills field theories (where the domain
of support for Fis presumed to be compact without boundary). More-
over, the fact that Fis subsumed to be exact and C1 differentiable excludes
the concept of magnetic monopoles from classical electroma gnetic theory on
topological grounds. The closed integral of the 2-form Fover any closed 2-
manifold is a deformation (topological) invariant of any ev olutionary process
that can be described by a singly parameterized vector field, for
LV(/integraldisplay
z2F) =/integraldisplay
z2{i(V)dF+d(i(V)F)}= (15)
/integraldisplay
z2{0 +d(i(V)F)}=/integraldisplay
z2d(i(V)F) = 0 (16)
The integral is then a deformation invariant, for the result is valid even if
the 4-vector field is distorted by an arbitrary function, f{x, y, z, t },such that
V⇒f(x, y, z, t )V.The notation/integraltext
z2implies that the 2D integration chain is
closed. It can be a cycle or a boundary.
3.2 The Maxwell Ampere exterior differential system
The Maxwell Ampere equations are a consequence of second ext erior
differential system,
J−dG= 0, (17)
where Gis an N-2 form density of field excitations ( DandH,related to
sources or objects of quantity), and Jis the N-1 form of charge-current densi-
ties. The partial differential equations equivalent to the e xterior differential
system are precisely the Maxwell-Ampere equations. This se cond postulate,
on a four dimensional domain of independent variables, assu mes the existence
of a N-2 form density given by the expression,
G=G34(x, y, z, t )dxˆdy...+G12(x, y, z, t )dzˆdt...=Dzdxˆdy...Hzdzˆdt...
(18)
Exterior differentiation produces an N-1 form,
J=Jz(x, y, z, t )dxˆdyˆdt...−ρ(x, y, z, t )dxˆdyˆdz. (19)
9Matching the coefficients of the exterior expression dG=Jleads to the
Maxwell-Ampere equations,
curlH−∂D/∂t=Jand div D=ρ. (20)
The fact that Jis exact leads to the charge conservation law, dJ=ddG=
0,or
∂Jx/∂x+∂Jy/∂y+∂Jz/∂z+∂ρ/∂t = 0. (21)
The exterior differential system is a topological constrain t for by Stokes
theorem the support for Gcan be compact without boundary only if the
domain is without charge-currents. The closure of the exter ior differential
system, dJ= 0,generates the charge-current conservation law. The integr al
ofJover a closed 3 dimensional domain is a relative integral inv ariant (a
deformation invariant) of any process that can be described in terms of a
singly parametrized vector field. The formal statement is gi ven by Cartan’s
magic formula [14], which describes continuous topologica l evolution in terms
of the action of the Lie derivative, with respect to a vector fi eld, acting on
the exterior differential 3-form, J:
LV(/integraldisplay
z3J) =/integraldisplay
z3{i(V)dJ+d(i(V)J)}=/integraldisplay
z3{0 +d(i(V)J)}= 0.(22)
The Lie derivative of the closed integral is equal to zero for any 4-vector
fieldV,when dJ= 0.The integral is then a deformation invariant, for the
result is valid even if the 4-vector field is distorted by an ar bitrary function,
f{x, y, z, t },such that V⇒f(x, y, z, t )V.
3.3 The T-Torsion and T-Spin 3-forms
As mentioned above, the method of exterior differential form s goes be-
yond the domain of classical tensor analysis, for it admits o f maps from ini-
tial to final state that are without inverse. (Tensor analysi s and coordinate
transformations require that the Jacobian map from initial to final state has
an inverse - the method of exterior differential forms does no t.) Hence the
theory of electromagnetism expressed in the language of ext erior differential
forms admits of topological evolution, at least with respec t to continuous pro-
cesses without Jacobian inverse. With respect to such non-i nvertible maps,
10both tensor fields and differential forms are not functionall y well defined in
a predictive sense [15]. Given the functional forms of a tens or field on an
initial state, it is impossible to predict uniquely the func tional form of the
tensor field on the final state unless the map between initial a nd final state is
invertible. However differential forms are functionally we ll defined in a retro-
dictive sense, by means of the pullback. Covariant anti-sym metric tensor
fields pull back retrodictively with respect to the transpos e of the Jacobian
matrix (of functions) and functional substitution, and con travariant tensor
densities pullback retrodictively with respect to the adjo int of the Jacobian
matrix, and functional substitution. The transpose and the adjoint of the
Jacobian exist, even if the Jacobian inverse does not.
The exterior differential forms that make up the electromagn etic system
consist of the primitive 1-form, A, and the primitive N-2 form density, G,their
exterior derivatives, and their algebraic intersections d efined by all possible
exterior products. The complete Maxwell system of exterior differential forms
(the Pfaff sequence for the Maxwell system) is given by the set :
{A, F=dA, G, J =dG, A ˆF, AˆG, AˆJ, FˆF, GˆG}. (23)
These forms and their unions may be used to form a topological base on
the domain of independent variables. The Cartan topology co nstructed
on this system of forms has the useful feature that the exteri or derivative
may be interpreted as a limit point, or closure, operator in t he sense of
Kuratowski [16]. The exterior differential systems that defi ne the Maxwell-
Ampere and the Maxwell-Faraday equations above are essenti ally topological
constraints of closure. Note that the complete Maxwell syst em of differential
forms (which assumes the existence of A) also generates two other exterior
differential systems.
d(AˆG)−(FˆG−AˆJ) = 0, (24)
and
d(AˆF)−FˆF= 0. (25)
The two objects, AˆGandAˆFare three forms, not usually found in dis-
cussions of classical electromagnetism. The closed compon ents of the first
3-form (density) were called T-Spin [17] and the second 3-fo rm, were called
T-Torsion (or helicity) [18]. By direct evaluation of the ex terior product, and
11on a domain of 4 independent variables, each 3-form will have 4 components
that can be symbolized by the 4-vector arrays
Spin−Current :S4= [A×H+Dφ,A◦D]≡[S,σ], (26)
and
Torsion −vector :T4= [E×A+Bφ,A◦B]≡[T,h], (27)
which are to be compared with the charge current 4-vector den sity:
Charge−Current :J4= [J, ρ], (28)
The 3-forms then can be defined by the equivalent contraction processes
Topological Spin 3−form.=AˆG (29)
=i(S4)dxˆdyˆdzˆdt=Sxdyˆdzˆdt.....−σdxˆdyˆdz (30)
and
Topological Torsion −helicity 3−form.=AˆF (31)
=i(T4)dxˆdyˆdzˆdt=Txdyˆdzˆdt.....−hdxˆdyˆdz. (32)
The vanishing of the first 3-form is a topological constraint on the do-
main that defines topologically transverse electric (TTE) w aves: the vector
potential, A, is orthogonal to D,in the sense that A◦D= 0.The vanishing
of the second 3-form is a topological constraint on the domai n that defines
topologically transverse magnetic (TTM) waves: the vector potential, A,
is orthogonal to B,in the sense that A◦B= 0.When both 3-forms van-
ish, the topological constraint on the domain defines topolo gically transverse
(TTEM) waves. For classic real fields this double constraint would require
that vector potential, A,is collinear with the field momentum, D×B,and
in the direction of the wave vector, k.
The geometric notion of distinct transversality modes of el ectromagnetic
waves is a well known concept experimentally, but the associ ation of transver-
sality to topological issues is novel herein. For certain ex amples that appear
in the appendix, it is apparent that the concept of geometric and topological
transversality are the same. In the classic case, often cons idered in fiber
12optic theory, it is known that the TEM modes do transmit power . However,
in the appendix, a vacuum wave solution is given which satisfi es the geomet-
ric concept of transversality ( it is both a TM and a TE solutio n) but the
mode radiates for it is not both a TTM and a TTE solution. The co njecture
obtained from examples is that a TTEM solution does not radia te.
Note that if the 2-form Fwas not exact, such topological concepts of
transversality would be without meaning, for the 3-forms of T-Spin and T-
Torsion depend upon the existence of the 1-form of Action. Th e torsion
vector T4and theT-Spin vector S4are associated vectors to the 1-form of
Action in the sense that
i(T4)A= 0and i (S4)A= 0 (33)
3.4 The Poincare Invariants
The exterior derivatives of the 3-forms of T-Spin and T-Tors ion produce
two 4-forms, FˆG−AˆJandFˆF,whose integrals over closed 4 dimensional
domains are deformation invariants for the Maxwell system. These topo-
logical objects are related to the conformal invariants of a Lorentz system as
discovered by Poincare and Bateman. In the format of indepen dent vari-
ables{x, y, z, t },the exterior derivative corresponds to the 4-divergence of
the 4-component T-Spin and T-Torsion vectors, S4andT4.The functions
so created define the Poincare conformal invariants of the Ma xwell system:
Poincare 1 = d(AˆG) =FˆG−AˆJ (34)
={div3(A×H+Dφ) +∂(A◦D)/∂t}dxˆdyˆdzˆdt(35)
={(B◦H−D◦E)−(A◦J−ρφ)}dxˆdyˆdzˆdt(36)
Poincare 2 = d(AˆF) =FˆF (37)
={div3(E×A+Bφ) +∂(A◦B)/∂t}dxˆdyˆdzˆdt(38)
={−2E◦B}dxˆdyˆdzˆdt (39)
For the vacuum state, with J= 0,zero values of the Poincare invari-
ants require that the magnetic energy density is equal to the electric energy
13density (1 /2B◦H= 1/2D◦E), and, respectively, that the electric field is or-
thogonal to the magnetic field ( E◦B= 0).Note that these constraints often
are used as elementary textbook definitions of what is meant b y electromag-
netic waves. When either Poincare invariant vanishes, the c orresponding
closed 3-dimensional integral becomes a topological quant ity in the sense of
a deRham period integral. For example, when the first Poincar e invariant
vanishes, the closed integral of the 3-form ofT-Spin become s a deformation
invariant with quantized values:
Define Topo logical Spin =/integraldisplay
z3AˆG (40)
Let d(AˆG) = 0 , then (41)
LV(Spin) =/integraldisplay
z3{i(V)d(AˆG) +d(i(V)(AˆG)}(42)
=/integraldisplay
z3{0 +d(i(V)(AˆG)}= 0. (43)
Similarly, when the second Poincare invariant vanishes, th e closed integral
of the 3-form of T-Torsion-Helicity becomes a deformation i nvariant with
quantized values:
Define Topo logical Torsion =/integraldisplay
z3AˆF (44)
Let d (AˆF) = 0 , then (45)
LV(Torsion -Helicity ) =/integraldisplay
z3{i(V)d(AˆF) +d(i(V)(AˆF)}(46)
=/integraldisplay
z3{0 +d(i(V)(AˆF)}= 0. (47)
It is important to realize that these topological conservat ion laws are valid in
a plasma as well as in the vacuum, subject to the conditions of zero values for
the Poincare invariants. On the other hand, topological tra nsitions require
that the Poincare invariants are not zero.
4 Deformation Invariants and the Plasma State.
4.1 Special evolutionary processes. The plasma pro-
14cess
As described in a previous section, the fundamental equatio n of topological
evolution is given by Cartan’s magic formula, which acts as a propagator on
the forms that make up the exterior differential system. As st ated in the
first paragraph, an evolutionary process is defined herein as a map that can
be described by a singly parameterized vector field. If the Ac tion of the
Lie derivative on the complete system of Maxwell exterior di fferential forms
vanishes for a particular choice of process, then that proce ss leaves the entire
Maxwell system absolutely invariant. As a topology can be co nstructed in
terms of an exterior differential system, and if a special pro cess leaves that
system of forms invariant, then the topology induced by the s ystem of forms
is invariant; the process must be a homeomorphism.
However, for a given Maxwell system, it is more likely that on ly some of
the exterior differential forms that make up the Maxwell syst em are invari-
ant relative to an arbitrary process; others are not. Of part icular interest
are those forms which are relative integral invariants of co ntinuous deforma-
tions. The closed integral of the form is not only invariant w ith respect
to a process represented by particular vector field, but also with respect to
longitudinal deformations of that process obtained by mult iplying the par-
ticular vector field by an arbitrary function. For vector fiel ds which are
singly parameterized, this concept of longitudinal deform ation is equivalent
to a reparameterization of the vector field.
The development that follows is guided by Cartan’s pioneeri ng work,
in which he examined those specialized processes for mechan ical systems that
leave the 1-form of Action, A,a deformation invariant. Cartan proved that
such processes always have a Hamiltonian representation. A n electromag-
netic system has not only the primitive 1-form, A,but also the N-2 form, G,
which can undergo evolutionary processes. For electromagn etic systems, a
particular interesting choice of specialized processes ar e those that leave the
N-2 form, G,of field excitations a deformation (relative) integral inva riant.
The equations that must be satisfied are of the form
LβV(/integraldisplay
z2G) =/integraldisplay
z2i(βV)dG=/integraldisplay
z2i(βV)J (48)
=/integraldisplay
z2β{(J−ρV)xdyˆdz−...+ (J×V)xdxˆdt...⇒0(49)
It follows that deformation invariance of the N-2 form Grequires that the
15admissable evolutionary processes be restricted to those t hat satisfy the def-
initions of the classical plasma:
J=ρV. (50)
(This constraint is used to define the ”Plasma state”in this a rticle). As
the closed integrals of Gare by Gauss law, the counters of net charge within
the closed domain, the classical plasma equation is to be rec ognized as the
statement that in the closed domain the net number of charges is a deforma-
tion invariant. That is, charges can be produced only in equa l and opposite
pairs by a ”plasma process”. A plasma process does not involv e net charge
production.
This invariance principle is to be compared to the Helmholtz theorem
which checks on the validity of the deformation integral inv ariance of the
2-form F.
LβV(/integraldisplay
z2F) =/integraldisplay
z2i(βV)dF= 0 (51)
The closed integral of Helmholtz is an intrinsic topologica l (deformation) in-
variant of an electromagnetic system, for the 2-form Fis exact by construc-
tion (the postulate of potentials). The Helmholtz integral is a deformation
invariant for all evolutionary processes that can be descri bed by a singly pa-
rameterized vector field. (This statement is not true for Yan g Mills fields).
Hence in a plasma, for which the evolutionary processes are c onstrained such
thatJ=ρV, both the closed integrals of FandGare deformation invari-
ants. In the sense, the plasma is a topological refinement of t he complete
Maxwell system.
In the subsections that follow, various topological catego ries of plasma
processes will be examined. The ideal and semi-ideal plasma processes will
obey the plasma master equation, and the non-ideal plasma pr ocesses will
not. The electromagnetic flux is a local (absolute) invarian t of all semi-
ideal plasma processes. This statement is similar to the cla ssification of
hydrodynamic flows. Ideal and semi-ideal hydrodynamic flows satisfy the
Helmholtz theorem, and the local conservation of vorticity .
4.2 The ideal plasma process = a Hamiltonian process.
16Next consider the evolutionary properties of the 1-form of A ction in the
plasma state by evaluating the possible deformation invari ance of the 1-form
of Action, A,with respect to motions that preserve the plasma state:
LρV(/integraldisplay
z1A) =/integraldisplay
z1i(ρV)dA=/integraldisplay
z1W=/integraldisplay
z1{(ρE+J×B)kdxk+(J◦E)dt} ⇒0.
(52)
The 1-form Wis the 1-form of virtual work defined in terms of the Lorentz
force. The resulting equation demonstrates that the concep t of a Lorentz
force, ρE+J×B,has a topological foundation. It is apparent that if the
Lorentz force vanishes,
{ρE+J×B)⇒0, (53)
and the Plasma current density is NOT ohmic,
(J◦E) =ρ(V◦E)⇒0, (54)
then the closed integral of the Action 1-form is also a deform ation topological
invariant of the Plasma process. Such a set of constraints,
W=i(ρV)dA= 0, (55)
topologically defines the ”ideal” plasma state as a plasma pr ocess for which
the 1-form of virtual Work vanishes. By Cartan’s theorem, th e 1-form of
Action then has a unique Hamiltonian representation and the ideal plasma
process is uniquely defined as a Hamiltonian process ( the Pfa ff dimension of
the 1-form, Amust processes be 3 or less for uniqueness). The ideal plasma
is thereby a restriction of arbitrary processes to that uniq ue process that
leaves invariant both the closed integrals of flux and the clo sed integrals of
charge. Ideal plasmas are electromagnetic systems for whic h the admissable
processes are the intersection of a plasma process and a uniq ue Hamiltonian
process. The ideal plasma can not exist on a domain of a 4 dimen sional
variety where the second Poincare invariant is not zero.
4.3 The Bernoulli-Casimir plasma process is a semi-
ideal plasma process.
The topological constraint that the 1-form of virtual work v anishes is suffi-
cient but not necessary for a plasma process to preserve the c losed integrals
17of the Action 1-form. Evolutionary invariance of the closed integral of Ac-
tion does not require that the plasma process be unique. The 1 -form of
virtual Work, W,need not be zero, but only closed: dW⇒0.By analogy
to hydrodynamics, if the virtual Work 1-form is exact,
W=dΘ (56)
then the Lorentz force is represented by a spatial gradient, ρE+J×B=∇Θ,
and the Power −J◦E=∂Θ/∂t. The function Θ( x, y, z, t ) is a Bernoulli-
Casimir function, and acts as the generator of a symplectic H amiltonian flow.
The (non-unique) Bernoulli-Casimir function is an evoluti onary invariant for
each process path, but is not necessarily a constant over the domain:
LρV(Θ) = i(ρV)dΘ =i(ρV)i(ρV)A= 0. (57)
The Bernoulli-Casimir function is not the same as the Hamilt onian energy
function, but is more closely related to the thermodynamic c oncept of en-
thalpy. The Bernoulli-Casimir function can be used to gener ate a ”Hamil-
tonian process”, but the process is not uniquely defined.
For such symplectic plasma processes, the gradient of the Be rnoulli-
Casimir function is transverse to the Bfield only when the second Poincare
invariant vanishes.
ρE◦B=∇Θ◦B. (58)
Similar expression were studied in conjunction with topolo gical conservation
in MHD by Hornig and Schindler [20].
ρE◦V=∇Θ◦V. (59)
If the Ohmic assumption is made for the plasma process, J=ρV=
σ(E+V×B) , then the symplectic condition leads to a thermopower form at
of the type
J= (1/ρσ)grad(kT) (60)
when it is subsumed that the Bernoulli-Casimir function is r elated to tem-
perature. It would appear that for plasma motion along the Bfield lines,
there can exist a dynamo action to produce an Efield collinear with the
magnetic field.
18It is suggested that the large temperature gradient that exi sts in a
plasma envelope about a rotating star (with a Bfield like a neutron star) can
induce a current flow and an Eacceleration field along the polar magnetic
field lines. Like in a Bernoulli process in a fluid, the mechani cal energy is not
conserved, but the enthalpy (the Bernoulli-Casimir) is a in variant along any
trajectory, and that invariant can be different from traject ory to neighboring
trajectory.
4.4 The Stokes plasma process is a semi-ideal process
that obeys the Master equation.
The constraint that the virtual work 1-form, W, generated by a plasma
process, W=i(ρV)dA, be closed, does not require that it be exact. The
constraint of closure yields two vector conditions:
dW= 0⇒curl(ρE+J×B) = 0 and ∇(J◦E) =∂(ρE+J×B)/∂t.
(61)
The first vector condition implies that
∇ρ×(E+V×B) +ρ curl(E) +curl(V×B) = 0. (62)
By using the Maxwell-Faraday equation, this topological co nstraint becomes
the plasma master equation:
−∂B/∂t+curl(V×B) =−∇lnρ×(E+V×B). (63)
All of these ideal and semi ideal plasma processes enjoy the p roperty that
the electromagnetic flux is conserved locally. That is
LρV(dA) =LρVF=d(i(ρV)F) = 0. (64)
4.5 Frozen-in lines.
It is of some interest to examine the evolution of the differen tial forms that
make up an electromagnetic system relative to Plasma proces ses. The
19method is to construct the Lie derivative with respect a plas ma process,
J=ρV,of all forms that make up the electromagnetic Pfaff sequence.
For an arbitrary vector field Z whose tangents define a line in s pace time,
the N-1 form
W=i(γZ)dxˆdyˆdzˆdt (65)
can be tested for evolutionary invariance relative to any ot her vector V.Sup-
pose the effect of the evolutionary process is conformal:
L(V)W=i(V)d W+d(i(V)W) = Γ( x, y, z, t )W (66)
This statement implies that the points that make up the tange nt line of the
vector field W remain on the tangent line. The points may be per muted
but they do not leave the line. Such is the concept of a frozen i n field.
The points that make up a line evolve into points of the same li ne. The
evolution need not be uniformly continuous, especially whe re the points are
folded. Yet even in such cases the points of a line are still po ints of the line,
even though rearranged in order. If for a given Vthe evolution of the lines
ofWis conformal, then there exists a parametrization of Vsuch that the
evolution is uniform and invariant. A parametrization func tionβ(x, y, z, t )
can be found such that
L(βV)W=βL(V)W+L(V)βˆW= (β·Γ +i(V)dβ)W⇒0.(67)
For the electromagnetic system there are three N-1 forms, wh ich may or
may not be frozen into the evolutionary process. Consider th e 3-form of
current.
L(V)J=i(V)d J+d(i(V)J) (68)
AsdJ= 0,
L(V)J=d{i(V)i(J)dxˆdyˆdzˆdt} (69)
It follows that if i(V)i(J)dxˆdyˆdzˆdt}= 0,the field lines of Jare frozen-
in (with Γ = 0). So the plasma evolutionary process evolution ary, with
J=ρV, is an example of a process that ”freezes-in” the lines of cur rent.
However, there are many other evolutionary processes for wh ich the Jlines
are frozen in.
20The formulas created by 4.16 are valid on any set of independe nt variables,
but expressions on 4 dimensions of space time for ”frozen-in ” lines are not
quite the same as those that appear in the engineering litera ture based on
euclidean 3-space [21]. Either the time-like component of t he 4-vector W
must vanish, or the process Vmust be explicitly time-independent for the
general formulas to be in precise agreement with the enginee ring expressions.
[22]
It is important to note that in space time the ”frozen-in” lin es must
be related to 3-forms, and not to the two form components EandB.These
latter objects can produce ”frozen-in” lines only on the exc eptional 2-surfaces,
the torus and the Klein bottle, (and then only when E◦B=0).The 3-form
of T-Torsion has spatial components that are dominated by th eBfield (in
the limit E→0), such that ”frozen-in” lines of T-Torsion might have the
appearance of ”frozen-in” lines of B.The 3-form of T-Spin has lines that
can be dominated by A×H.However the explicit formulas for the 3-forms
of T-Torsion and T-Spin are not dependent upon a choice of con stitutive
relations that act as geometrical constraints on the 2-form s ofFandG.See
below.
4.6 Evolution of the lines of T-Torsion with respect to
plasma currents.
Consider the evolution of the lines of T-Torsion
L(ρV)AˆF=i(ρV)d(AˆF) +d(i(ρV)AˆF) (70)
=i(ρV)d(AˆF) +d{(i(ρV)A)ˆF−Aˆi(ρV)F}(71)
First consider those systems where the second Poincare inva riant van-
ishes, FˆF= 0.The lines in space time which are tangent to the 3-form AˆF
then have zero divergence. The lines can only start and stop o n boundary
points, or they are closed on themselves. The T-Torsion line s can be either
parallel to the plasma current or they can be orthogonal to th e plasma cur-
rent. As the electromagnetic current is exact, any three dim ensional domain
of support for a finite plasma current cannot be compact witho ut a bound-
ary. If the lines of plasma current start and stop on boundary points, then
the lines of T-Torsion can form closed loops that link the cur rent lines. It
is the concept of linkages that is of interest to the theory of magnetic knots.
21Consider that plasma process such that the evolution is in th e direc-
tion of the T-Torsion lines. As in this situation,
(i(J)AˆF) = ( i(ρV)AˆF)⇒(i(γT4)AˆF) (72)
=γ(i(T4)(i(T4)dxˆdyˆdzˆdt= 0, (73)
the 3-form of T-Torsion is a local invariant whenever the sec ond Poincare
invariant vanishes; E◦B⇒0. In other words, FˆF/ne}ationslash= 0 is a local nec-
essary condition for topological change. It is also a remark able fact that
any evolution in the direction of the Torsion vector leaves t he Action 1-form
conformally invariant, in the sense that:
L(γT4)A=i(γT4)dA+di(γT4)A=γ(E◦B)A+ 0. (74)
The torsion vector on a domain of 4 variables is transverse to the 1-form of
Action, as Aˆ(AˆF) = 0.Evolution in the direction of the Torsion vector in
not Hamiltonian, unless the second Poincare invariant vani shes. In section
6 below this idea will be related to thermodynamic irreversi bility.
4.7 Evolution of the lines of T-Spin Current with re-
spect to plasma currents.
Consider the evolution of the lines ofT-Spin current
L(ρV)AˆG=i(ρV)d(AˆG) +d(i(ρV)AˆG) (75)
=i(ρV)d(AˆG) +d{(i(ρV)A)ˆG−Aˆi(ρV)G}(76)
First consider those systems where the first Poincare invari ant vanishes,
FˆG−AˆJ= 0.The lines in space time which are tangent to the 3-form
AˆGthen have zero divergence. The lines can only start and stop o n bound-
ary points, or they are closed on themselves. The T-Spin line s are either
parallel to the plasma current or they are orthogonal to the p lasma current.
As the electromagnetic current is exact, any three dimensio nal domain of
support for a finite plasma current cannot be compact without a boundary.
If the lines of plasma current do not stop or start on boundary points (cur-
rent loops), then the T-Spin lines which terminate on bounda ry points can
be linked by the current loops.
22The concept of the T-Spin vector depends on the existence of G, but not
on the concept of J=dG. That is, the T-Spin vector can be associated
with separated domains of charges, which can be compact doma ins with-
out boundary that are compliments of the domain of finite char ge current
densities, which are domains that can not be compact without boundary.
5 Thermodynamics
5.1 Topological Thermodynamics and Irreversibility
The basic tool for studying topological evolution is Cartan ’s magic formula, in
which it is presumed that a physical (hydrodynamic) system c an be described
adequately by a 1-form of Action, A, and that a physical process can be
represented by a contravariant vector field, V, which can be used to represent
a dynamical system or a flow:
L(V)/integraltextA=/integraltextL(V)A=/integraltext{i(V)dA+d(i(V)A)} (77)
=/integraltext{W+d(U)}=/integraltextQ. (78)
The basic idea behind this formalism (which is at the foundat ion of the
Cartan-Hilbert variational principle) is that postulate o f potentials is valid:
F−dA= 0. The base manifold will be the 4-dimensional variety {x, y, z, t }
of engineering practice, but no metrical features are presu med a priori. If
relative to the process, V, the RHS of equation ??is zero,/integraltextQ.⇒0, then
/integraltextAis said to be an integral invariant of the evolution generate d byV.In
thermodynamics such processes are said to be adiabatic.
From the point of view of differential topology, the key idea i s that the
Pfaff dimension, or class [23], of the 1-form of Action specifi es topological
properties of the system. Given the Action 1-form, A, the Pfaff sequence,
{A, dA, A ˆdA, dA ˆdA, ...}will terminate at an integer number of terms ≤the
number of dimensions of the domain of definition. On a 2n+2=4 d imensional
domain, the top Pfaffian, dAˆdA, will define a volume element with a density
function whose singular zero set (if it exists) reduces the s ymplectic domain
to a contact manifold of dimension 2n+1=3. This (defect) con tact manifold
supports a unique extremal field that leaves the Action integ ral ”stationary”,
and leads to the Hamiltonian conservative representation f or the Euler flow
in hydrodynamics. The irreversible regime will be on an irre ducible sym-
plectic manifold of Pfaff dimension 4, where dAˆdA/ne}ationslash= 0.Topological defects
23(or coherent structures) appear as singularities of lesser Pfaff (topological)
dimension, dAˆdA= 0.
Classical hydrodynamic processes can be represented by cer tain nested
categories of vector fields, V. Recall that in order to be Extremal, the pro-
cess,V, must satisfy the equation
Extremal − −(unique Hamiltonian ) : i(V)dA= 0; (79)
in order to be Hamiltonian the process must satisfy the equat ion
Bernouilli − −Casimir − −Hamiltonian :i(V)dA=dΘ; (80)
in order to be Symplectic, the process must satisfy the equat ion
Helmholtz − −Symplectic : di(V)dA= 0.(81)
Extremal processes cannot exist on the non-singular symple ctic domain,
because a non-degenerate anti-symmetric matrix (the coeffic ients of the 2-
formdA) does not have null eigenvectors on space of even dimensions . Al-
though unique extremal stationary states do not exist on the domain of Pffaf
dimension 4, there can exist evolutionary invariant Bernou lli-Casimir func-
tions, Θ ,that generate non-extremal, ”stationary”states. Such Ber noulli
processes can correspond to energy dissipative symplectic processes, but
they, as well as all symplectic processes, are reversible in the thermodynamic
sense described below. The mechanical energy need not be con stant, but the
Bernoulli-Casimir function(s), Θ ,are evolutionary invariant(s), and may be
used to describe non-unique stationary state(s).
The equations, above, that define several familiar categori es of processes,
are in effect constraints on the topological evolution of any physical system
represented by an Action 1-form, A.The Pfaff dimension of the 1-form of
virtual work, W=i(V)dAis 1 or less for the three categories. The extremal
constraint of equation 79 can be used to generate the Euler eq uations of hy-
drodynamics for a incompressible fluid. The Bernoulli-Casi mir constraint of
24equation 80 can be used to generate the equations for a barotr opic compress-
ible fluid. The Helmholtz constraint of equation 81 can be use d to generate
the equations for a Stokes flow. All such processes are thermo dynamically
reversible. None of these constraints above will generate t he Navier-Stokes
equations, which require that the topological dimension of the 1-form of vir-
tual work must be greater than 2.
A crucial idea is the recognition that irreversible process es must on do-
mains of Pfaff dimension which support T-Torsion, AˆdA/ne}ationslash= 0,with its
attendant properties of non-uniqueness, envelopes, regre ssions, and projec-
tivized tangent bundles. Such domains are of Pfaff dimension 3 or greater.
Moreover, as described below, it would appear that thermody namic irre-
versibility must support a non-zero Topological Parity 4-f orm, dAˆdA/ne}ationslash= 0.
Such domains are of Pfaff dimension 4 or greater.
Although there does not exist a unique gauge independent sta tionary
state on the symplectic manifold of Pfaff dimension 4, remark ably there does
exist a unique vector field on the symplectic domain, with com ponents that
are generated by the 3-form AˆdA. This unique (to within a factor) vector
field is defined as the T-Torsion vector, T4, and satisfies (on the 2n+2=4
dimensional manifold) the equation,
i(T4)dxˆdyˆdzˆdt=AˆdA (82)
This (four component) vector field, T4, has a non-zero divergence almost ev-
erywhere, for if the divergence is zero, then the 4-form dAˆdAvanishes, and
the domain is no longer a symplectic manifold! The T-Torsion vector, T4,
can be used to generate a dynamical system that will decay to t he station-
ary states ( div4(T4)⇒0) starting from arbitrary initial conditions. These
processes are irreversible in the thermodynamic sense. It i s remarkable that
this unique evolutionary vector field, T4, is completely determined (to within
a factor) by the physical system itself; e.g., the component s of the 1-form,
A, determine the components of the T-Torsion vector.
To understand what is meant by thermodynamic irreversibili ty, realize
that Cartan’s magic formula of topological evolution is equ ivalent to the first
law of thermodynamics.
L(v)A=i(V)dA+d(i(V)A) =W+dU=Q. (83)
25Ais the ”Action” 1-form that describes the hydrodynamic syst em.Vis the
vector field that defines the evolutionary process. Wis the 1-form of (virtual)
work. Qis the 1-form of heat. From classical thermodynamics, a proc ess
is irreversible when the heat 1-form Qdoes not admit an integrating factor.
From the Frobenius theorem, the lack of an integrating facto r implies that
QˆdQ/ne}ationslash= 0.Hence a simple test may be made for any process, V, relative to
a physical system described by an Action 1-form, A:
If L (v)AˆL(v)dA/ne}ationslash= 0then the process is irreversible. (84)
This topological definition implies that the three categori es (above) of
symplectic, Hamiltonian or extremal processes, ⊂S,are reversible ( as L (S)dA=
dQ= 0).However, for evolution in the direction of the T-Torsion vec tor,T4,
direct computation demonstrates that the fundamental equa tions lead to a
conformal evolutionary process, a process which is thermod ynamically irre-
versible:
L(T4)A=σA and i (T4)A= 0, (85)
such that
L(T4)AˆL(T4)dA=QˆdQ=σ2AˆdA/ne}ationslash= 0. (86)
It is remarkable for the irreversible case that the Lie deriv ative with respect
toTacting on Ais comparable to the covariant derivative. However the
Lie derivative with respect to Tacting on dAis not equivalent to a covariant
derivative.
5.2 Applications to Electromagnetism and Plasmas
All of the development of previous sections will carry over t o the electro-
magnetic system, which also subsumes the postulate of poten tials. The
T-Torsion 3-form, AˆdA, induces the T-Torsion vector,
T4={(E×A+Bφ);A◦B} ≡ {S, h}. (87)
26Ifdiv4T=−2E◦B/ne}ationslash= 0,the electromagnetic 1-form, A,defines a do-
main of Pfaff dimension 4. Such domains cannot support topolo gically
transverse magnetic waves ( as AˆF/ne}ationslash= 0). Evolutionary processes (including
plasma currents) that are proportional to the T-Torsion vec tor are thermody-
namically irreversible, if σ=E◦B/ne}ationslash= 0. However, the conformal properties
of evolution in the direction of the T-Torsion vector lead to extraordinary
properties when the plasma current is in the direction of the T-Torsion vec-
tor. From the thermodynamic arguments presented above, bas ed on the
postulate of potentials for an arbitrary system, but using t he notation of an
electromagnetic system, it follows that
L(T4)A=σA=−(E◦B)A (88)
and
L(T4)(AˆF) = 2σA=−2(E◦B)AˆF. (89)
It follows that motion along the direction of the torsion vec tor freezes-in the
lines of the torsion vector in space time, but the process is i rreversible unless
the second Poincare invariant is zero. The time evolution of the deformable
coherent structure is recognizable even though it thermody namically decays!
Recall that the definition of a plasma current, J,is equivalent to an
evolutionary process such that
Definition ofa plasma Current J:L(J)G= 0. (90)
Consider a plasma current which is also in the direction of th e Torsion vector.
Then
L(J)AˆG= (L(J)A)ˆG+AˆL(J)G (91)
= (L(γT4)A)ˆG+AˆL(γT4)G=γ·(E◦B)AˆG+ 0 (92)
For plasma motions in the direction of the (possibly dissipa tive) torsion vec-
tor, both the ”lines”of theT-Spin vector are ”frozen in” and the lines of the
Torsion vector are ”frozen in”. Such ”frozen in”objects can be used to give a
topological definition of deformable coherent structures i n a plasma. More-
over, as the evolutionary process causes the frozen in struc tures to deform
and decay, it is conceivable that evolution could proceed to form stationary
27( but not stagnant) states (where E◦B⇒0),such that the frozen in field
line structures become local deformation invariants, or to pological defects.
In conclusion, electromagnetic coherent structures are ev olutionary de-
formable (and perhaps decaying) domains of Pfaff dimension 4 , which form
stationary states of topological defects (including the nu ll state) in regions
of Pfaff dimension 3, where E◦B= 0. (Note that all semi-ideal plasma
current processes are reversible in a thermodynamic sense. )
6 Electromagnetic Waves in the Vacuum with
T-Spin and T-Torsion
As the T-Spin 4-vector and the T-Torsion 4-vector formalism may be unfamil-
iar to many readers, it is useful to compare four classes of un usual vacuum
wave solutions with the usual waveguide solutions. The ”unu sual waves”
have their vector potential, A,orthogonal to the wave vector, k, describing
the direction of the wave front. In each unusual example, the current density
is in the direction of the vector potential and therefore als o orthogonal to the
wave vector. The usual wave solutions have their vector pote ntial parallel
to the wave vector. The four unusual cases belong to equivale nce classes
defined by the constraints
(AˆF= 0, AˆG/ne}ationslash= 0)
(AˆF/ne}ationslash= 0, AˆG= 0)
(AˆF= 0, AˆG= 0)
(AˆF/ne}ationslash= 0, AˆG/ne}ationslash= 0).
In each case, each component of the potentials satisfies the w ave equation
subject to the phase velocity relation, ( ω/k)2−1/(ξµ) = 0 .The current
density, J,is proportional to the vector potential, A, (in a fashion reminiscent
to the London conjecture) multiplied by the same phase veloc ity relation.
The examples do not generate any charge current distributio ns when the
phase velocity equation is satisfied (the phase velocity equ als the speed of
light as determined by the constitutive equations).
In each example given below, the 1-form of Action is specified and the
field intensities are computed. Then the T-Spin Current and t he T-Torsion
28vector are evaluated. The functions have been chosen to sati sfy the Lorentz
vacuum conditions of zero charge current densities, subjec t to a phase velocity
”dispersion” relation. The phase function is defined by the f ormula Θ =
(±kz∓wt) representing outbound eaves. The Poynting vector is compu ted,
and the Poincare invariants are evaluated.
Examples of the four classes of these simple (but unusual) wa ve types
correspond to:
6.1 Example 1. Real Linear Polarization:
Consider the Potentials
A= [cos( kz−ωt),cos(kz−ωt),0,0] (93)
and their induced fields:
E= [−sin(kz−ωt),−sin(kz−ωt),0]ω
B= [+ sin( kz−ωt),−sin(kz−ωt),0]k
J4= [cos( kz−ωt),cos(kz−ωt),0,0](k2−εµω2)/µ
S4= [0,0,−k/µ,−εω] 2 cos( kz−ωt) sin(kz−ωt).
T4= [0,0,0,0].
E×H= [0,0,1](ωk/µ)(2 cos( kz−ωt)2−1)
(B◦H−D◦E) = −2{cos(kz−ωt)2−sin(kz−ωt)2}(k2−εµω2)/µ
(E◦B) = 0
This class of potentials generates a set of complex field inte nsities and
excitations, and a current density proportional to the vect or potential. If
29the dispersion relation ( k2−εµω2) = 0 is satisfied, then the solutions are
acceptable vacuum solutions, with a vanishing charge curre nt density. The
T-Torsion vector vanishes identically, independent from t he dispersion con-
dition, but theT-Spin vector does not. The first Poincare inv ariant vanishes
subject to the constraint of the dispersion relation. The se cond Poincare
invariant vanishes identically. The solution corresponds to a linear state of
polarization at 45◦with respect to the x-axis, with the electric and the mag-
netic fields in phase. There is a non-zero Poynting vector alo ng the z axis.,
which is orthogonal to the vector potential. Note that the ra diated power
has a time average which is zero. If the charge current densit y is not zero
(due to a fluctuation in the dispersion relation) the charge c urrent vector is
orthogonal to theT-Spin current vector.
6.2 Example 2. Real Circular Polarization:
Consider the Potentials
A= [cos( kz−ωt),sin(kz−ωt),0,0] (94)
and their induced fields:
E= [−sin(kz−ωt),+ cos( kz−ωt),0]ω
B= [−cos(kz−ωt),−sin(kz−ωt),0]k
J4= [cos( kz−ωt),sin(kz−ωt),0,0](k2−εµω2)/µ
S4= [0,0,0,0].
T4= [0,0,−ω,−k].
E×H= [0,0,1]ω k/µ
(B◦H−D◦E) = 0 ( E◦B) = 0
30This class of potentials generates a set of complex field inte nsities and
excitations, and a current density proportional to the vect or potential. If
the dispersion relation ( k2−εµω2) = 0 is satisfied, then the solutions are
acceptable vacuum solutions, with a vanishing charge curre nt density. The
T-Spin vector vanishes identically, but the T-Torsion vect or does not. In fact,
the T-Torsion vector is constant. The solution corresponds to a circular state
of polarization with the constant magnetic and electric amp litudes rotating
about the z axis. The Poynting vector is not zero and is a const ant, time
independent, vector. This wave solution is geometrically t ransverse (TEM),
yet it produces power as it is not topologically transverse ( TTEM). If the
dispersion relation is not precisely satisfied, the current vector is orthogonal
to the T-Torsion vector and parallel to the vector potential . Both Poincare
invariants vanish identically. The soliton like solution s hould be compared
to the wave guide solution of example 5 below, which is also TE M, but does
not radiate.
6.3 Example 3. Complex Linear Polarization:
Consider the Potentials
A= [cos( kz−ωt), icos(kz−ωt),0,0] (95)
and their induced fields:
E= [−sin(kz−ωt),−isin(kz−ωt),0]ω
B= [+isin(kz−ωt),−sin(kz−ωt),0]k
J4= [cos( kz−ωt), icos(kz−ωt),0,0](k2−εµω2)/µ
S4= [0,0,0,0].
T4= [0,0,0,0].
E×H= [0,0,0]
31(B◦H−D◦E) = 0 ( E◦B) = 0
This class of potentials generates a set of complex field inte nsities and
excitations, and a current density proportional to the vect or potential. The
fields are said to be complex linearly polarized because the c omplex Bfield
is a complex scalar multiple of the complex Efield. If the dispersion rela-
tion (k2−εµω2) = 0 is satisfied, then the solutions are acceptable vacuum
solutions, with a vanishing charge current density. Note th at both the T-
Torsion vector and the T-Spin vector vanish identically. Th e complex square
of both the electric and the magnetic field vectors vanish. Bo th Poincare
invariants vanish independent from the dispersion constra int. Although the
fields are propagating, there is no momentum flux and the Poynt ing vector is
zero. The EandBfields are (complex) collinear. This example is perhaps
the simplest member of the class of Bateman-Whittaker compl ex solutions
described in Example 11, below.
6.4 Example 4. Complex Circular Polarization:
Consider the Potentials
A= [cos( kz−ωt), isin(kz−ωt),0,0] (96)
and their induced fields:
E= [−sin(kz−ωt),+icos(kz−ωt),0]ω
B= [−icos(kz−ωt),−sin(kz−ωt),0]k
J4= [cos( kz−ωt), isin(kz−ωt),0,0](k2−εµω2)/µ
S4= [0,0,−k/µ,−εω] 2 cos( kz−ωt) sin(kz−ωt).
T4=i[0,0,−ω,−k].
32E×H= [0,0,−1] (ω k/µ)(2 cos( kz−ωt)2−1)
(B◦H−D◦E) = −2{cos(kz−ωt)2−sin(kz−ωt)2}(k2−εµω2)/µ
(E◦B) = 0
This class of potentials generates a set of complex field inte nsities and
excitations, and a current density proportional to the vect or potential. If
the dispersion relation ( k2−εµω2) = 0 is satisfied, then the solutions are
acceptable vacuum solutions, with a vanishing charge curre nt density. Both
the T-Torsion vector (imaginary) and the T-Spin vector (rea l) do not vanish.
The second Poincare invariant vanishes identically, and th e first Poincare
invariant vanishes subject to the dispersion constraint. T he current vector,
if non-zero due to fluctuations in the dispersion relation, i s orthogonal to
both the T-Torsion vector and the T-Spin vector.
Examples 1 through 4 above are geometrically transverse wav es in
the engineering sense that the propagation direction of the phase (along the
z axis) is in the direction of the momentum flux, D×B.However, the waves
are not ”topologically transverse” in that the sense that th eDandBfields
are not necessarily transverse to the components of the vect or potential A.
6.5 Example 5. Waveguide TEM modes
Consider the Potentials
A= [0,0, φ(x, y),(ω/k)φ(x, y)] cos( kz−ωt) (97)
and their induced fields:
E= [−(ω/k)∂φ/∂x, −(ω/k)∂φ/∂y, 0] cos( kz−ωt)
B= [∂φ/∂y, −∂φ/∂x, 0] cos( kz−ωt)
J4= [∂φ/∂x (εµ(ω/k)2−1) sin( kz−ωt),
∂φ/∂y (εµ(ω/k)2−1) sin( kz−ωt),
∇2φcos(kz−ωt),
(εµω/k )∇2φcos(kz−ωt)]/µ
33S4= [φ∂φ/∂x cos(kz−ωt)2(1−εµ(ω/k)2),
φ∂φ/∂y cos(kz−ωt)2(1−εµ(ω/k)2),
0,
0]/µ
T4= [0,0,0,0].
E×H= [φ(∂φ/∂x )kcos(kz−ωt) sin(kz−ωt)(vg−vp),
φ(∂φ/∂y )kcos(kz−ωt) sin(kz−ωt)(vg−vp),
(vg) cos(kz−ωt)2(∇2φ)]/µ
(B◦H−D◦E)/ne}ationslash= 0 ( E◦B) = 0
Note that the vector potential, A,is parallel to both the wave vector,
k,and the field momentum, D×B.The T-Torsion vector and the second
Poincare invariant are identically zero. The solution prod uces transverse
current and T-Spin densities unless a dispersion relation, εµ(ω/k)2= 1,is
satisfied. Subject to the dispersion constraints, this clas sic solution has
both a zero T-Torsion vector and a zero T-Spin vector. Both A◦D= 0
andA◦B= 0.The wave front is in the spatial direction of the potential,
by construction. The candidate solution subject to the disp ersion relation
is both topologically transverse TTEM and geometrically tr ansverse, TEM .
However, even if the dispersion relations are satisfied, the geometric
TEM solution produces finite charge current densities, unle ss the function
φ(x, y) is a solution of the two dimensional Laplace equation, ∇2φ= 0.This
further constraint implies that the TEM solution produces n o radiated power
in the charge free state, for E×H⇒0 as∇2φ⇒0.In the next example
the constraint that the system be TTEM is relaxed, and radiat ed power is
achieved. in a TTM mode.
346.6 Example 6. Waveguide TM modes
Consider the Potentials
A= [0,0, φ(x, y) cos(kz−ωt), vgφ(x, y) cos(kz−ωt) (98)
and their induced fields (note that example 6 differs from exam ple 5 in that
a ”group” velocity vgis used in the definition of the potentials, instead of
the phase velocity, vp=ω/k):
E= [−vg∂φ/∂x, −vg∂φ/∂y, φ (x, y) tan(kz−ωt)(vgk−ω)] cos( kz−ωt)
B= [∂φ(x, y)/∂ycos(kz−ωt),−∂φ(x, y)/∂xsin(kz−ωt),0]
J4= [k∂φ/∂x (εµvgvp−1) sin(( kz−ωt),
k∂φ/∂y sin((kz−ωt)(εµvgvp−1),
−(∇2φ+αφ) cos(kz−ωt),
−vgεµ(∇2φ+βφ) cos(kz−ωt)]/µ
α=k2εµvp(vp−vg), β =k2vg(vp/vg−1)
S4= [−(vg/vp−1)φ∂φ/∂x cos(kz−ωt)2,
−(vg/vp−1)φ∂φ/∂x cos(kz−ωt)2,
−k(vg/vp−1)φ2sin(kz−ωt),
−µk(vg−vp)φ2sin(kz−ωt)]/µ
T4= [0,0,0,0].
E×H= [(vp/vg−1)φ∂φ/∂x sin(kz−ωt),
(vp/vg−1)φ∂φ/∂y sin(kz−ωt),
((∂φ/∂x )2+ (∂φ/∂y )2) cos(kz−ωt)](vg/µ) cos(kz−ωt)
35(B◦H−D◦E) = −({εµ(ω/k)2−1}/µ) cos(kz−ωt)2{(∇φ)2+φ(∇2φ)}
(E◦B) = 0
Note that in this solution, the fourth component of the Actio n is scaled
by the ”group velocity”, vg,not the ”speed of light”, as determined by the
constitutive properties: c=/radicalBig
1/ξµ.This class of potentials requires that the
function φ(x, y) be a solution of the two dimensional Helmholtz equation,
∇2φ+λ2φ= 0 . The phase velocity, vp=ω/k,differs from the group veloc-
ity,vg.Again, two constraint conditions (dispersion relations) a re required
for the solution to be a vacuum solution without charge curre nts. One of
the constraint conditions demands that the product of the gr oup and the
phase velocity, vp=ω/k,to be equal to the square of the speed of light as
determined from the constitutive properties:
vp·vg= 1/εµ=c2. (99)
The second constraint required for the vacuum state ( J= 0, ρ= 0) is
determined by the Helmholtz parameter, λ,and is satisfied when
λ2=k2(vp/vg−1). (100)
Such TM modes are also TTM modes; the T-Torsion vector is iden tically
zero, but the T-Spin vector is not. Note that the solution bec omes a TEM
mode solution when the phase velocity equals the group veloc ity, and the
function φsatisfies the Laplace equation, ∇2φ= 0.Further note that the
Efield has a longitudinal component when the group velocity an d the phase
velocity are not the same. For the transverse magnetic mode, A◦B= 0,but
A◦D/ne}ationslash= 0. The second Poincare invariant vanishes, E◦B= 0,but for this
solution, the first Poincare invariant does not vanish. Not o nly is the T-
Spin vector not zero, but also its divergence is not zero. The energy flow is
in the direction of the wave vector, k, but not in the direction of the field
momentum, D×B,and the energy propagates with the group velocity vg.
6.7 Example 7. An irreversible vacuum solution
of type 1 for which E ◦B/ne}ationslash= 0
Consider the potentials
36A= [+y,−x, ct]/λ4, φ=cz/λ4, where λ2=−c2t2+x2+y2+z2.(101)
and their induced fields:
E= [−2(cty−xz),+2(ctx+yz),−(c2t2+x2+y2−z2)]2c/λ6
B= [−2(cty+xz),+2(ctx−yz),+(c2t2+x2+y2−z2)]2/λ6.
Subject to the dispersion relation, εµc2= 1.and the Lorentz constitutive
conditions, these time dependent wave functions satisfy th e homogeneous
Maxwell equations without charge currents, and are therefo re acceptable
vacuum solutions.
J+t=dG= [0,0,0,0] (102)
The extensive algebra involved in these and other computati ons in this
article were checked with a Maple symbolic mathematics prog ram [12]. It
is to be noted that when the substitution t⇒ −tis made in the functional
forms for the potentials, the modified potentials fail to sat isfy the vacuum
Lorentz conditions for zero charge-currents. The algebrai c results for the
charge current density are somewhat complicated, but the bo ttom line is
that
J−t=dG/ne}ationslash= [0,0,0,0]. (103)
It appears that the valid vacuum solution presented above is not time-
reversal invariant.
The T-Spin current density for this first non-transverse vac uum wave
example is evaluated as:
Spin :S4= [x(3λ2−4y2−4x2), y(3λ2−4y2−4x2), z(λ2−4y2−4x2),
t(λ2−4y2−4x2)](2/µ)/λ10, (104)
and has zero divergence, subject to the condition εµc2= 1. Hence the first
Poincare invariant is zero
(B◦H−D◦E) = 0 (105)
37The T-Torsion current may be evaluated as
T−Torsion :T4=−[x, y, z, t ]2c/λ8. (106)
and has a non-zero divergence equal to the second Poincare in variant
Poincare 2 =−2E◦B= +8c/λ8. (107)
The solution has magnetic helicity as A◦B/ne}ationslash=0 and is radiative in the sense
that the Poynting vector, E×H/ne}ationslash=0.
Both the T-Spin current and the T-Torsion vector are non-zer o, which
implies that this solution represents waves which are neith er TTM nor TTE.
They are not transverse waves in any sense. However, the first Poincare
invariant vanishes, implying that the T-Spin integral is a d eformation invari-
ant, and is conserved. The second Poincare invariant is not z ero, which
implies that the T-Torsion-Helicity integral is not a topol ogical invariant.
These solutions are not simple transverse waves for both A◦B/ne}ationslash= 0,and
A◦D/ne}ationslash= 0.Note that the physical units of the second Poincare invarian t are
that of an energy density multiplied by an impedance (ohms). As the sec-
ond Poincare invariant is not zero, it is impossible to find a c ompact without
boundary two surface that contains non-zero lines of magnet ic field. That
is, a closed 2-torus of magnetic field lines does not exist.
However, as the first Poincare invariant is zero it is possibl e to con-
struct a deformation invariant in terms of the deRham period integral over
a closed 3 dimensional submanifold
Spin=/integraldisplay
z3{Sxdyˆdzˆdt−Sydxˆdzˆdt+Szdxˆdyˆdt−σdxˆdyˆdz}.(108)
6.8 Example 8. An irreversible vacuum solution of
type 2 complimentary to Example 7.
Consider the potentials
A= [+ct,−z,+y]/λ4, φ=cx/λ4, where λ2=−c2t2+x2+y2+z2(109)
and their induced fields:
E= [+(−c2t2+x2−y2−z2),+2(ctz+yx),−2(cty−zx)]2c/λ6
38B= [+(−c2t2+x2−y2−z2),+2(−ctz+yx),+2(cty+zx)]2/λ6.
As in the previous example above, these fields satisfy the Max well-Faraday
equations, and the associated excitations satisfy the Maxw ell-Ampere equa-
tions without producing a charge current 4-vector. However , it follows by
direct computation that the second Poincare invariant, and the T-Torsion 4-
vector are of opposite signs to the values computed for the pr evious example:
E◦B= +4c/λ8and A◦B= +2ct/λ8.
6.9 Example 9. Superposition of the two compli-
mentary examples of type 1 and type 2.
When the potentials of examples type 1 and type 2 above are com bined by ad-
dition or subtraction, the resulting wave is topologically transverse magnetic,
but not topological transverse electric. Not only does the s econd Poincare
invariant vanish under superposition, but so also does the T -Torsion 4 vec-
tor. Conversely, the examples above show that there can exis t topologically
transverse magnetic waves which can be decomposed into two n on-transverse
waves. A notable feature of the superposed solutions is that the T-Spin 4
vector does not vanish, hence the example superposition is a wave that is not
topologically transverse electric. However, for the examp les above and their
superposition, the first Poincare invariant vanishes, whic h implies that the
T-Spin remains a conserved topological quantity for the sup erposition. The
T-Spin current density for the combined examples is given by the formula:
S4= [−2cx(y+ct)2, cy(y+ct)(x2−y2+z2−2cty−c2t2),−2cz(y+ct)2,(110)
−(y+ct)(x2+y2+z2+ 2cty+c2t2)]4c/λ10
while the T-Torsion current is a zero vector
T4= [0,0,0,0].
In addition, for the superposed example, the spatial compon ents of the
Poynting vector are equal to the T-Spin current density vect or multiplied by
γ, such that
39E×H=γS, with γ =−(x2+y2+z2+ 2cty+c2t2)/2c(y+ct)λ2.
These results seem to give classical credence to the Planck a ssumption that
vacuum state of Maxwell’s electrodynamics supports quanti zed angular mo-
mentum, (the conserved T-Spin integral) and that the energy flux must come
in multiples of the T-Spin quanta. In other words, these comb ined irre-
versible solutions of examples type 1 and type 2 have the appe arance of the
photon
6.10 Example 10. Bateman-Whittaker solutions.
In the modern language of differential forms, Bateman [19] (a nd Whit-
taker) determined that if two complex functions α(x, y, z, t ) and β(x, y, z, t )
are used to define the 1-form of Action,
A=αdβ−βdα⇒A=α∇β−β∇α, φ =−(α∂β/∂t −β∂α/∂t ) (111)
then the derived 2-form F= 2dαˆdβgenerates the complex field intensities,
E= (∂α/∂t )∇β−(∂β/∂t )∇α and B=∇α× ∇β,
which of course satisfy the Maxwell-Faraday equations. If i n addition, the
functions αandβsatisfy the complex Bateman constraints:
∇α× ∇β=±(i/c)[(∂α/∂t )∇β−(∂β/∂t )∇α],
then the complex field excitations computed from the Lorentz vacuum con-
stitutive constraints will satisfy the Maxwell-Ampere equ ations for the vac-
uum, without charge currents. It is apparent immediately th at the second
Poincare invariant is identically zero for such solutions. It is also appar-
ent immediately that the T-Torsion vector is identically ze ro. What is not
immediately apparent is that first Poincare invariant and th e T-Spin vector
vanish identically as well. In fact, the constrained comple x solutions of the
Bateman type are examples of topologically transverse (TTE M) waves. The
Bateman solutions do not radiate!
As an explicit example, consider
40α= (x±iy)/(z−r), β= (r−ct), r=/radicalBig
x2+y2+z2).
These functions satisfy the Bateman conditions (and, it sho uld be mentioned,
the Eikonal equation subject to the dispersion relation εµc2= 1).TheE
and the Bfields are complex (and complicated algebraically)
B= [yx+√
−1(z2+y2−rz),−(z2+x2−rz)−√
−1xy,
(r2+z2−2rz)/(r−z) )(y−√
−1x)]2/(r(z−r)2)
E= [−√
−1yx+ (y2+z2−rz),√
−1(x2+z2−rz)−xy,
(z−r)(x+√
−1y)]2c/(r(z−r)2)
S4= [0,0,0,0].
T4= [0,0,0,0].
E×H= [0,0,0],D×B= [0,0,0],E◦E= 0,B◦B= 0
(B◦H−D◦E) = 0 ( E◦B) = 0
The functions αandβthat satisfy the Bateman condition may be used
to construct an arbitrary function, F(α, β),and remarkably enough, the
arbitrary function F(α, β) satisfies the Eikonal equation,
(∇F)2−εµ(∂F/∂t )2= 0. (112)
From experience with Eikonal solutions and wave equations, it might
be thought that Eikonal solutions are sufficient. However, th e Bateman
conditions are necessary, for both the candidate solutions
α= (x±iy)/(z−ct), β= (r−ct), r=/radicalBig
x2+y2+z2). (113)
satisfy the Eikonal equation, but not the Bateman condition s. They do not
generate TTEM modes in the vacuum. For arbitrary functions t he algebra
can become quite complex. A Maple symbolic mathematics prog ram for
computing the various terms is available (see references be low)
416.11 Example 11. A Plasma Accretion disc from
HedgeHog B field solutions.
An interesting static solution that exhibits chiral symmet ry breaking
can be obtained from the potentials
A= Γ( x, y, z, t )[−y, x,0]/(x2+y2), (114)
with Γ = −z m//radicalBig
(x2+y2+ǫz2) (115)
and φ = 0. (116)
These potentials induce the field intensities:
E= [0,0,0] (117)
B=m[x, y, z ]/(x2+y2+ǫ z2)3/2. (118)
TheBfield is the famous Dirac Hedgehog field often associated with ”mag-
netic monopoles”. However, the radial Bfield has zero divergence ev-
erywhere except at the origin, which herein is interpreted a s a topological
obstruction. The factor ǫis to be interpreted as an oblateness factor asso-
ciated with rotation of a plasma, and is a number between zero and 1. It
is apparent that the helicity density and the second Poincar e invariant are
zero:
E◦B= 0 and A◦B= 0. (119)
In fact, the 3-form of T-Torsion vanishes identically (as φ= 0),
T4= [0,0,0,0]. (120)
In this example, there is a non-zero value for the Amperian cu rrent den-
sity, even though the potentials are static. The Current Den sity 3-form has
components,
J4= (3m/2µ) (1−ǫ)z[−y, x,0,0]/(x2+y2+ǫ z2)5/2.. (121)
which do not vanish if the system is ”oblate” (0 < ǫ < 1).This current
density has a sense of ”circulation” about the z axis, and is p roportional to
42the vector potential reminiscent of a London current, J=λA. The ”order”
parameter is (3 /2µ) (1−ǫ)/(x2+y2+ǫ z2)2.
The Lorentz force can be computed as:
J×B=(3m2/4µ) (1−ǫ)[xz2, yz2,−z]/(x2+y2+ǫ z2)2(122)
The formula demonstrates that the Lorentz force on the plasm a, for the
given system of circulating currents, is directed radially away (centrifugally)
from the rotational axis, and yet is such that the plasma is at tracted to
thez= 0, xyplane. The Lorentz force is divergent in the radial plane
and convergent in the direction of the z axis, towards the z=0 plane. This
electromagnetic field, therefor, would have the tendency to form an accretion
disk of the plasma in the presence of a central gravitational field.
Although the 3-form of T-Torsion vanishes identically, the 3-form of
T-Spin is not zero. The spatial components of the T-Spin are o pposite to the
direction field of the Lorentz force (in the sense of a radiati on reaction).
S4= (m2/4µ)[xz2, yz2,−z,0]/(x2+y2+ǫ z2)2. (123)
The components of the T-Spin 3-form are in fact proportional to the com-
ponents of the virtual work 1-form. (See section 6) with the r atio−3(1−ǫ)
depending on the oblateness factor.
It is also true that the divergence of the 3-form of T-Spin is n ot zero, for
the first Poincare invariant is
d(AˆG)⇒P1 = (m2/4µ)(x2+y2+ 4(1−ǫ)z2)/(x2+y2+ǫ z2)3(124)
6.12 Example 11. Self dual solutions
It is possible to construct a two-form G(without using the Lorentz vacuum
constitutive definitions) in terms of two arbitrary functio ns,αandβ,from
the dual relations:
G=i(∗dα)ˆi(∗dβ)Ω =i(∗dα)ˆi(∗dβ)dxˆdyˆdzˆdt.
The functions αandβused in the dual construction are not required to be
solutions of the Bateman condition. However, the resulting ”self-dual” field
43excitations are notthe same as those generated by the Bateman method, un-
less the functions also satisfy the Bateman conditions of co mplex collinearity.
In the self dual formulas the * operator is the Hodge * operato r with respect
to the Lorentz metric modified by the impedance of free space. The resulting
self-dual excitations constructed from the two arbitrary f unctions indeed sat-
isfy the Maxwell-Ampere equations, in virtue of the Maxwell -Faraday equa-
tions and the dispersion relation. The construction yields :
H=√
−1/µc(∂α/∂t )∇β−(∂β/∂t )∇α and D=−√
−1ε/c∇α× ∇β.
The self-dual construction, however, implies a chiral (non -Lorentz) con-
stitutive relation of the type D=−[γ]◦BandH= [γ†]◦E, and will not be
considered further in this article.
7 SUMMARY
T-Torsion, AˆF,and T-Spin, AˆG, have been demonstrated to be useful
theoretical concepts that give credence to the physical rea lity of potentials
in electromagnetic theory. The closed integrals of these 3 f orms and their
divergences give precise meaning to the concept of coherent structures in a
plasma, and offer possible explanations for certain astroph ysical phenomena
such as plasma jets from neutron stars, and the formation of s table rings of
material about rotating astrophysical objects. The constr uctions can be used
to define topological transverse modes, similar to the geome tric definitions
of TM and TE modes.
8 References
1. Bryant, R.L.,Chern, S.S., Gardner, R.B.,Goldschmidt, H .L., and Grif-
fiths, P. A. (1991), Exterior Differential Systems, Springer Verlag
2. K. Kuratowski, Topology (Warsaw, 1948), Vol. I.
Lipschutz, S. (1965) General Topology (Schaum, New York), 8 8.
3. Cartan, E., (1958) Lecons sur les invariants integrauxs, Hermann, Paris
.
444. Schouten, J. A. and Van der Kulk, W., (1949) Pfaff’s Problem and its
Generalizations, Oxford Clarendon Press
5. Kiehn, R. M. (1977) ”Periods on manfolds, quantization an d gauge”,
J. of Math Phys 18, no. 4, p. 614
6. Kiehn, R. M., (1990) ”Topological Torsion, Pfaff Dimensio n and Coher-
ent Structures”, in: H. K. Moffatt and T. S. Tsinober eds, Topo logical
Fluid Mechanics, Cambridge University Press, 449-458 .
7. deRham,G. (1960) Varietes Differentiables, Hermann, Par is
8. TEM modes do not exist if the wave guide is simply connected .
9. Hornig, G and Rastatter, L. (1998) ”The Magnetic Structur e of B /ne}ationslash= 0
Reconnection”, Physica Scripta, Vol T74, p.34-39
10. Sommerfeld, A., (1952) Electrodynamics, Academic Pres s, New York.
Stratton, J.A., (1941) Electromagnetic Theory McGraw Hill N.Y.
Sommerfeld carefully distingushes between intensities an d excitations
on thermodynamic grounds.
11. Kiehn, R. M., Kiehn, G. P., and Roberds, R. B. (1991) ”Pari ty and
Time-reversal Symmetry Breaking, Singular Solutions”, Ph ys Rev A,
43, p. 5665
12. Kiehn, R. M. (1991) ”Are there three kinds of superconduc tivity” Int.
J. Mod. Phys B Vol. 5 p.1779-1790
Kiehn, R. M. and Pierce, J. F. (1969) ”An Intrinsic Transport Theo-
rem” Phys. Fluids 12, p. 1971
13. D. Van Dantzig, Proc. Cambridge Philos. Soc. 30, 421 (1934). Also
see:
D. Van Dantzig, ”Electromagnetism Independent of metrical geome-
try”, Proc. Kon. Ned. Akad. v. Wet. 37 (1934).
14. Marsden, J.E. and Riatu, T. S. (1994) Introduction to Mec hanics and
Symmetry, Springer-Verlag, p.122
4515. Kiehn, R. M. (1976) ”Retrodictive Determinism”, Int. J. of Eng. Sci.
14, p. 749
16. K. Kuratowski, Topology (Warsaw, 1948), Vol. I.
17. Lipschutz, S. (1965) General Topology (Schaum, New York ), 88.
18. Ibid 5
19. Ibid 6
20. Bateman, H. (1914, 1955) Electrical and Optical Wave Mot ion, Dover
p.12
21. G. Hornig and K. Schindler, K. ”Magnetic topology and the problem
of its invariant definition” Physics of Plasmas, 3, p.646 (1996).
22. Kochin, N.E., Kibel, I.A., Roze, N. N., (1964) Theoretic al Hydrody-
namics, Interscience, NY p.157
23. Kiehn, R. M. (1975) ”Intrinsic hydrodynamics with appli cations to
space time fluids”, Int. J. Engng Sci 13, p. 941-949
46 |
arXiv:physics/0102002v1 [physics.flu-dyn] 1 Feb 2001Some closed form solutions to theNavier-Stokes equations
R. M. Kiehn
Mazan, France
rkiehn2352@aol.com
http://www.cartan.pair.com
Abstract: An algorithm for generating a class of closed form
solutions to the Navier-Stokes equations is suggested, wit h exam-
ples. Of particular interest are those exact solutions that exhibit
intermittency, tertiary Hopf bifurcations, flow reversal, and hys-
teresis.
1 INTRODUCTION
The Navier-Stokes equations are notoriously difficult to sol ve. However, from
the viewpoint of differential topology, the Navier-Stokes e quations may be
viewed as a statement of cohomology: the difference between t wo non-exact
1-forms is exact. Abstractly, the idea is similar to the coho mology statement
of the first law of thermodynamics.
Q−W=dU (1)
For the Navier-Stokes case, define the two inexact 1-forms in terms of the
dissipative forces
WD=fD•dr=ρ{ν∇2V} •dr (2)
and in terms of the advective forces of virtual work
WV=fV•dr=ρ{∂V/∂t+grad(V•V/2)−V×curlV} •dr(3)
1Then the abstract statement of cohomology, formulated as WV−WD=−dP,
when divided by the common function, ρ,is precisely equivalent to an exterior
differential system whose coefficients are the partial differe ntial equations
defined as the Navier-Stokes equations,
{∂V/∂t+grad(V•V/2)−V×curlV} − {ν∇2V}=−grad P/ρ (4)
The cohomological constraint on the velocity field, V, is such that the kine-
matically defined vector, f,
f=fV−fD (5)
is a vector field that satisfies the Frobenius integrability t heorem [1]. That
is,
f•curlf= 0 even though v•curlv/ne}ationslash=0. (6)
The meaning of the Frobenius criteria is that the vector fhas a representa-
tion in terms of only two independent functions of {x, y, z, t }. The Navier-
Stokes equations makes this statement obvious. One of these functions has
a gradient, gradP, in the direction of the tangent vector to f,and the other
function, ρ,is a renormalization, or better, a reparametrization facto r for the
dynamical system represented by f.
These observations suggest that there must exist certain co nstraint rela-
tionships on the functional forms that make up the component s of any solu-
tion vector field, V, (which usually does not satisfy the Frobenius condition
in general) such that the covariant kinematic vector, f, is decomposable in
terms of at most two functions. If such a constraint equation can be found in
terms of the component functions that represent V, then its solutions may be
easier to deduce than the direct solutions of the Navier-Sto kes equations. For
example, the constraint relation may involve only 1 partial differential equa-
tion rather than 3. In fact such a single constraint relation can be found by
imposing a type of symmetry condition on the system, a symmet ry condition
that expresses the existence of a two dimensional (function al) representation
for the vector field, f.In this article attention will be focused on the two
spatial variables, randz, such that the solution examples will have a certain
2degree of cylindrical symmetry. As these solutions involve dissipative terms
with a kinematic viscosity coefficient, ν,they are not necessarily equilibrium
solutions of an isolated thermodynamic system.
Closed form solutions are few in number [3], but it appears th at many of
the known steady-state solutions to the Navier-Stokes equa tions fall into the
following class of systems: Consider a variety {x, y, z, t }withr2=x2+y2.
Consider three arbitrary functions, Θ( r, z) and Φ( r, z, t), and Λ( r, z) which
are defined in terms of two independent variables spatial var iables, ( r, z),and
time. Define the flow field, Vin cylindrical coordinates as,
V= Λ(r, z)uz+ Θ(r, z)ur+ Φ(r, z, t)uφ/r, (7)
where uφis a unit vector in the azimuthal direction. Note that this ve c-
tor field does not necessarily satisfy the Frobenius theorem . Note that for
simplicity, the only time dependence permitted is in the azi muthal direction.
Substitution of this format for Vinto the equation for fwill yield a vector
equation of the form
f=α(r, z)uz+β(r, z)ur+γ(r, z, t)uφ/r. (8)
The Pfaffian form W=f◦drwill become an expression in two variables
if the azimuthal factor γ(r, z, t) is constrained to the value zero. In other
words, a single constraint on the functions, Θ( r, z) and Φ( r, z, t), and Λ( r, z),
defined by the equation γ(r, z, t) = 0,can be used to reduce the Pfaffian form
to the expression
α(r, z)dz+β(r, z)dr=−dP/ρ(r, z) (9)
The left hand side represents a Pfaffian form in two variables, and therefore
always admits an integrating factor. It is this idea that is u sed to find new
solutions to the Navier-Stokes equations. First a solution to constraint equa-
tion is determined. Then the Cartan 1-form of total work is co mputed. The
1-form is either exact, or can be made exact by an appropriate integrating
factor. If the 1-form is exact then the Pressure is obtained b y integration.
It the 1-form is not exact a suitable integrating factor is fo und, and that
integrating factor represents a variable fluid density, ρ.For a given choice of
integrating factor, the Pressure is again obtained by integ ration.
3It is also useful to consider a rotating frame of reference de fined by the
equation
Ω =ωuz. (10)
It is the choice of rotational axis that defines the cylindric al symmetry. For
such rotating systems the same technique will insure that th e flow field, V,
is a solution of the Navier-Stokes equations in a rotating fr ame of reference,
∂V/∂t+grad(V◦V/2)−V×curlV (11)
=−gradP/ρ +ν∇2V−2Ω×V−Ω×(Ω×r) (12)
By direct substitution, into the Navier-Stokes equation ab ove, of the pre-
sumed format for the velocity field Vyields an expression for γ(r, z) in terms
of the three functions Θ( r, z) and Φ( r, z), and Λ( r, z) :
γ(r, z) = {∂Φ/∂t+ Λ(r, z)∂Φ/∂z+ Θ(r, z)(∂Φ/∂r−2ωr) (13)
−ν{∂2Φ/∂z2+∂2Φ/∂r2−(∂Φ/∂r)/r}. (14)
Similar evaluations of the standard formulas of vector calc ulus in terms of the
assumed functional forms for the velocity field lead to the us eful expressions:
divV=∂Θ/∂r+ Θ/r+∂Λ/∂z (15)
curlV={∂Φ/∂ruz−∂Φ/∂zur}/r+{∂Θ/∂z−∂Λ/∂r}uφ
curl curl V={−∂2Λ/∂r2+∂2Θ/∂z∂r }uz+ (16)
{−∂2Θ/∂z2+∂2Λ/∂z∂r }ur+
{−∂2Φ/∂z2−∂2Φ/∂r2+ (∂Φ/∂r/r )}uφ/r
V×curlV= Θ {∂Θ/∂z−∂Λ/∂r}uz+ (17)
Λ{∂Λ/∂r−∂Θ/∂z}ur+
{(1/r2)grad(Φ2/2)− {Λ∂Φ/∂z+ Θ∂Φ/∂r}uφ/r
4grad(V•V)/2 = {Θ∂Θ/∂z+ Λ∂Λ/∂z+ Φ(∂Φ/∂z)/r2}uz+ (18)
{Θ∂Θ/∂r+ Λ∂Λ/∂r+ Φ(∂Φ/∂r)/r2−Φ2/r3}ur
grad(divV) = {∂2Θ/∂z∂r + (∂Θ/∂z)/r+∂2Λ/∂z2}uz+ (19)
{∂Θ2/∂r2+∂2Λ/∂r∂z + (∂Θ/∂r)/r−Θ/r2}ur
It is remarkable that many solutions to the Navier-Stokes eq uations then
can be found by using the following algorithm: Choose a funct ional form
Φ(r, z) of interest and then deduce functions Λ( r, z) and Θ( r, z) to satisfy
the azimuthal constraint,
γ(r, z, t) = 0. (20)
The flow field Vso obtained is therefore a candidate solution to the
compressible, viscous, three dimensional Navier-Stokes e quations for a system
with a density distribution, ρand a pressure, P. The components of flow field
so determined then permit the evaluation of the coefficients o f the Pfaffian
form
W=α(r, z)dz+β(r, z)dr (21)
If the expression is not a perfect differential, then use the s tandard methods
of ordinary differential equations to find an integrating fac tor,ρ(r, z).The
integrating factor represents the density distribution of the resulting Navier-
Stokes solution. The Pressure follows by integration.
This method is demonstrated in the next section for the known viscous
vortex examples reported in Lugt. In addition, several new c losed form exact
solutions are generated by the technique. Among these close d form solutions
are exact solutions to the Navier Stokes equations (in a rota ting frame of
reference) that exhibit the bifurcation classifications fo r N = 3 as given by
Langford [2]. In particular, exact, non-truncated solutio ns are given that
represent the trans-critical Hopf bifurcation, the saddle -node Hopf bifurca-
tion, and the hysteresis Hopf bifurcation. It has been long s uspect that many
5phenomena in hydrodynamics exhibit Hopf bifurcation; now t hese exact so-
lutions to the Navier-Stokes equation formally justify thi s position, and are
especially interesting for the understanding of slightly p erturbed Poiseuille
flow and the onset of turbulence in a pipe.
2 EXAMPLES
In the following examples, the vector field specified has been used to compute
the various terms in the Navier-Stokes equations. The algeb ra has been
simplified by use of a symbolic computation program written i n the Maple
syntax. For each example, the two vector components that mak e up the work
one form have been evaluated and are displayed with the solut ion. For the
divergence free cases, the pressure function also has been c omputed. First,
known solutions are exhibited, and are shown to be derived fr om the above
technique. Then a few new solutions are exhibited.
2.1 Old solutions
2.1.1 Example 1. The Rankine Vortex
Φ(r, z) =a+ (b+ω)r2,Θ = 0 ,Λ = 1 (22)
fV={−(a+br2)2/r3}ur+{0}uφ/r+{0}uz (23)
fD={0}ur+{0}uφ/r+{0}uz (24)
This flow is a solution independent of the kinematic viscosit y coefficient
(the velocity field is harmonic, as fD= 0) and therefore could be construed
as an equilibrium solution. This solution, for a and b equal t o piecewise
constants, will generate the Rankine vortex.
As the flow is isochoric ( divV= 0), the steady pressure can be determined
by quadrature, and is given by the expression,
P= 1/2(b2r4+ 4abr2ln(r)−a2)/r2(25)
62.1.2 Example 2. Diffusion Balancing Advection.
Φ(r, z) =a+br2+m/ν,Θ(r, z) =m/r, Λ = 1, ω= 0 (26)
fV={−m2−(a+br(2ν+m)/ν)2/r3}ur+ (27)
{br(2ν+m)/νm(2ν+m)/νr2}uφ/r+{0}uz
fD={0}ur+{br(2ν+m)/νm(2ν+m)/νr2}uφ/r+{0}uz (28)
In this case the Laplacian of the vector field is not zero, but t he dissipative
parts exactly cancel the advective parts in the coefficient of the azimuthal
field, thereby satisfying the constraint condition. As the f unctions depend
only on r, the integrability (gradient) condition is satisfi ed, and these solu-
tions obey the Navier-Stokes equations for a system of const ant density. The
Pressure function may be computed as
P= (−νb(4a(m+ν) +bmr(2ν+m)/ν))r(2ν+m)/ν)− (29)
(m(ν+m)(a2+m2))/(2m(ν+m)r2)
The solutions are cataloged in Lugt. As these solutions expl icitly involve
the kinematic viscosity, ν,they cannot be equilibrium solutions to isolated
systems. Instead they represent steady state solutions, fa r from equilibrium.
A special case exists for m/ν=−2.
2.1.3 Example 3. Burger’s Solution, but with Helicity and Ze ro
Divergence.
Φ(r, z) =k(1−e−ar2/2ν),Θ(r, z) =−ar,Λ =U+ 2az, ω = 0 (30)
fV={−(ke(−ar2/2ν)+r2a−k)(ke(−ar2/2ν)−r2a−k)/r3}ur+
{kra2/ν e(−1/2ar2/ν)}uφ/r+
{2(U+ 2az)a}uz
7fD={0}ur+{−kra2/ν e(−1/2ar2/ν)}uφ/r+{0}uz (31)
This solution corresponds to a modification of Burger’s solu tion and ex-
hibits a 3-dimensional flow (in 2-variables) in which the diff usion is balanced
by convection to give azimuthal cancellation. The Burgers s olutions has
been modified to exhibit zero divergence. This flow in a non-ro tating frame
of reference exhibits a helicity.
Helicity = (U+ 2az)(ka/ν)e(−1/2ar2/ν)(32)
2.2 New Solutions
2.2.1 Example 4. A Beltrami Type Solution
Φ(r, z) =r2cos(z/a),Θ(r, z) =rsin(z/a),Λ(r, z) = 2acos(z/a), ω= 0
(33)
fV={−r}ur+{0}uφ/r+{−4acos(z/a)sin(z/a)}uz (34)
fD=ν/a2[{−r sin(z/a)}ur+{−rcos(z/a)}uφ/r+{−2a cos(z/a)}uz]
(35)
This solution is a Beltrami-like solution, has zero diverge nce, and can
be made time harmonic by multiplying the velocity field by any function of
t. The flow exhibits Eckman pumping and has a superficial resem blance
to a hurricane. The time independent steady flow is a strictly Beltrami
(curlv=av)with the vorticity proportional to the velocity field. In al l
cases the helicity is given by the expression,
Helicity := (r2+ 4a2cos(z/a)2)/a. (36)
The kinetic energy is a/2 times the helicity, which is a times the enstrophy.
The Pressure generated from the Navier Stokes equation is gi ven by the
expression
P= 1/2(r2+ (r2(ν/a2)−4ν)sin(z/a) + 4a2sin(z/a)2) (37)
82.2.2 Example 5. A Saddle Node Hopf Solution
Φ(r, z) =ωr2,Θ(r, z) =r(a+bz),Λ(r, z) =U−dr2+Bz2(38)
The components of the advective force and dissipative force are given by
the expressions,
fV={r(a+bz)2+ (U−dr2+Bz2)rb}ur+ (39)
{0}uφ/r+
{−2r2(a+bz)d+ 2(U−dr2+Bz2)Bz}uz
and
fD={0}ur+{0}uφ/r+ν{−4d+ 2B]}uz (40)
The divergence of the velocity field is given by the expressio n:
divV:= 2{a+ (b+B)z} (41)
The helicity of the flow depends upon the rotation, ω,
Helicity :ω(+r2b+ 2U−2bz2) (42)
but remarkably changes for finite values of randz, depending on mean flow
speed, U.
Note that when b= 0, B= 0, a= 0,the solution is equivalent to the
standard incompressible Poiseuille solution for flow down a pipe. The vector
velocity field is not harmonic, but vector Laplacian of the ve locity field is a
constant.
Without these constraints, it is remarkable that the ordina ry differential
equations that represent the components of the velocity fiel d are in one to one
correspondence with the saddle node - Hopf bifurcation of La ngford. That
is, the ODE,s representing the Langford format for the SN-Ho pf are given
by the expressions:
9dz/dt = Λ( r, z) =U−dr2+Bz2(43)
dr/dt = Θ( r, z) =r(a+bz)
dθ/dt =ω
.This first order system which exhibits tertiary bifurcation is associated
with an exact solution of the Navier Stokes partial different ial system in a ro-
tating frame of reference. In principle, the method also rel axes the constraint
on incompressibility, and allows a density distribution, o r integrating factor,
to be computed for an exact solution to the compressible Navi er-Stokes equa-
tions which can be put into correspondence with saddle node- Hopf bifurcation
process.
This example exhibits isochoric ( divV= 0) flow for B+b= 0, a= 0.The
steady isochoric pressure is then determined by quadrature , and is given by
the expression,
P=b(dr4/2−(r2−2z2)U−bz4)/2−ν(4d−2b)z (44)
where the constant U can be interpreted as the mean flow down th e pipe. Part
of the pressure is due to geometry, and part is due to the kinem atic viscosity.
Note that the pressure is independent from the viscosity coe fficient when the
velocity field is harmonic; e.g. when (2 d−b) = 0.As the vector Laplacian of
the velocity field determines the dissipation in the system, intuition would
say that the harmonic solution is some form of a limit set for t he otherwise
viscous flow.
2.2.3 Example 6. A Transcritical Hopf Bifurcation
Φ(r, z) =ωr2,Θ(r, z) =r(A−a+cz),Λ(r, z) =br2+Az+Bz2(45)
fV={r(A−a+cz)2+ (br2+Az+Bz2)rc}ur+ (46)
{0}uφ/r+
{2r2(A−a+cz)b+ (br2+Az+Bz2)(A+ 2Bz)}uz
10fD={0}ur+{0}uφ/r+ν{4b+ 2B}uz (47)
This example exhibits isochoric ( divV= 0) flow for a= 3A/2 and B=−c.
The steady isochoric pressure is then determined by quadrat ure, and is given
by the expression,
P=−1/4cbr4−1/8A2r2+ 1/2A2z2−Az3c+ 1/2c2z4−ν(4b−2c)z.(48)
Again it is apparent that the pressure splits into a viscous a nd a non-viscous
component, and when the flow is harmonic (2 b−c= 0),the pressure is
independent from viscosity, and there is no dissipation in t he flow.
The transcritical Hopf bifurcation is represented by the La ngford system
dz/dt = Λ( r, z) =br2+Az+Bz2(49)
dr/dt = Θ( r, z) =r(A−a+cz)
dθ/dt =ω
2.2.4 Example 7. A Hysteritic Hopf Bifurcation
Φ(r, z) =ωr2,Θ(r, z) =r(a+bz),Λ(r, z) =U−dr2+Az+Bz3(50)
fV={r(a+bz)2+ (U−dr2+Az+Az3)rb}ur+ (51)
{0}uφ/r+
{−2r2(a+bz)d+ (U−dr2+Az+Az3)(A+ 3Az2)}uz
fD={0}ur+{0}uφ/r+ν{−4d+ 6Az}uz (52)
This system has the remarkable property that the vector Lapl acian changes
sign at a position z= 2d/3Adown stream. There is no global way of making
this solution isochoric, for the divergence is equal to
11divV= (A+ 2a) + 2bz+ 3Az2. (53)
The hysteretic Hopf bifurcation exhibits what has been call ed intermit-
tency. The Langford system is
dz/dt = Λ( r, z) =U−dr2+Az+Bz3(54)
dr/dt = Θ( r, z) =r(a+bz)
dθ/dt =ω
3 Acknowledments
This work was supported in part by the Energy Lab at the Univer sity of
Houston in 1989, and was discussed at the Permb Conference in 1990
4 References
[1] FLANDERS, H. (1963) ”Differential Forms”. Academic Pres s, New York.
[2] LANGFORD, W. F. (1983) in ”Non-linear Dynamics and Turbu lence”
Edited by G.I. Barrenblatt, et. al. Pitman, London.
[3] LUGT, H. J. (1983) ”Vortex flow in Nature and Technology” W iley,
New York, p.33.
12 |
arXiv:physics/0102003v1 [physics.flu-dyn] 2 Feb 2001Topology and Turbulence
R. M. Kiehn
Mazan, France
rkiehn2352@aol.com
http://www.cartan.pair.com
Abstract: Over a given regular domain of independent variables
{x,y,z,t }, every covariant vector field of flow can be constructed
in terms a differential 1-form of Action. The associated Cart an
topology permits the definition of four basic topological eq uiva-
lence classes of flows based on the Pfaff dimension of the 1-for m
of Action. Potential flows or streamline processes are gener ated
by an Action 1-form of Pfaff dimension 1 and 2, respectively.
Chaotic flows must be associated with domains of Pfaff dimen-
sion 3 or more. Turbulent flows are associated with domains
of Pfaff dimension 4. It will be demonstrated that the Navier-
Stokes equations are related to Action 1-forms of Pfaff dimen sion
4. The Cartan Topology is a disconnected topology if the Pfaff
dimension is greater than 2. This fact implies that the creat ion of
turbulence (a state of Pfaff dimension 4 and a disconnected Ca r-
tan topology) from a streamline flow (a state of Pfaff dimensio n 2
and a connected topology) can take place only by discontinuo us
processes which induce shocks and tangential discontinuit ies. On
the otherhand, the decay of turbulence can be described by co n-
tinuous, but irreversible, processes. Numerical procedur es that
force continuity of slope and value cannot in principle desc ribe
the creation of turbulence, but such techniques of forced co ntinu-
ity can be used to describe the decay of turbulence.
1 INTRODUCTION
The turbulence problem of hydrodynamics is complicated by t he fact that
there does not exist a precise definition of the turbulent sta te that is uni-
versally accepted. The visual complexity of the turbulent s tate leads to the
1assumption that the phenomenon is in some way random and stat istical. It
is however a matter of experience that a real viscous liquid w hich is isolated
from its surroundings, after being put into a turbulent stat e will decay into
a streamline state and ultimately a state of rest. This proce ss of decay is
apparently continous
Although the turbulent state is intuitively recognizable, only a small num-
ber of properties necessary for the turbulent state receive the support of a
majority of researchers:
1. A turbulent flow is three dimensional
2. A turbulent flow is time dependent.
3. A turbulent flow is dissipative.
4. A turbulent flow may be intermittent.
5. A turbulent flow is irreversible.
Still, a precise mathematical definition of the turbulent st ate has not been
established.
It sometimes is argued that a turbulent flow is random, a quali ty that
has been reinforced by successes of certain statistical the ories in describing
average properties of the turbulent state. Recent advances in non-linear
dynamics indicate a sensitivity to initial conditions can l ead to deterministic
”chaos”, a quality which visually has some of the intuitive f eatures often
associated with turbulence. Moreover, the theory of non-li near dynamics has
led to several new suggestions that describe the route to tur bulence.
However, there are fundamental differences between irrever sibility, chaos,
and randomness that suggest that the turbulent state is not s imply a chaotic
regime. Similar statements can be made about the transition to turbulence
and the formation of ”coherent structures” in turbulent flow s. Although the
eye easily perceives the antithesis to turbulence as being a steady streamline
flow, the transition from the pure streamline state to the tur bulent state
is complicated by the fact that there may be intermediate cha otic states in
between an initial state of rest (or steady integrable strea mline flow) and a
final state of turbulent flow. The objective of this article is to focus attention
on and apply topological methods, rather than the customary geometric or
2statistical methods, to the problem of defining the transiti on to turbulence
and the turbulent state itself.
In hydrodynamics, it is generally accepted that the non-tur bulent state
seems to be described adequately in terms of solutions to the Navier-Stokes
equations. The description of the turbulent state is not so c lear, and to
this author the reason may be due to a lack of definition of what is the
turbulent state. As Ian Stewart [1988] states, ”the Leray Th eory of tur-
bulence...asserts that when turbulence occurs, the Navier -Stokes equations
break down. ... turbulence is a fundamentally different prob lem from smooth
flow.” In other words, one option would be that the turbulent s tate is not
among the solutions to the Navier-Stokes equations. Counte r to this op-
tion, and more in concurrence with the spirit of this article , would be the
inclusion of discontinuous solutions to the Navier-Stokes equations into the
class of evolutionary flows under consideration. However it is possible to
show that there exist continuous solutions to the Navier-St okes equations
which are thermodynamically irreversible. Such solutions are irreducibly
four dimensional and can serve as candidates decscribing th e decay of tur-
bulence. It is possible to demonstrate that the creation of t urbulence cannot
be described by a continuous process.
The Kolmogorov theory of turbulence [Kolmogorov, 1941] is a statisti-
cal option motivated by the assumption that the turbulent st ate consists of
”vortices” of all ”scales” with random intensities, but oth erwise it is a theory
which is not based upon the Navier-Stokes equations explici tly. The wavelet
theory of Zimin [1991] is a method that does use a specific deco mposition of
the solutions to the Navier-Stokes equations, and a transfo rmation to a set
of collective variables, which mimic the Kolmogorov motiva tion of vortices of
all scales. However from a topological view, scales can not h ave any intrinsic
relevance.
The Hopf-Landau theory [Landau 1959] claims that the transi tion to tur-
bulence is a quasi-periodic phenomenon in which infinitely m any periods are
sequentially generated in order to give the appearance of ra ndomness. In
contrast, the Ruelle-Takens theory [Berge, 1984] describe s the transition to
turbulence not in terms of an infinite cascade, but in terms of a few transi-
tions leading to a chaotic state defined by a strange attracto r. In other words,
it would appear that the presence of a strange attractor defin es the turbulent
state, but again there is a degree of vagueness in that the ”st range attractor”
is ill-defined. A basic question arises: ”Is chaos the same as turbulence?”
[Kiehn, 1990b].
3In this article, topological methods will be used to formula te a definition of
the turbulent state. The method, being topological and not g eometrical, will
involve concepts of scale independence, a concept in spirit recently utilized
by Frisch [1991] in an approach based on multi-fractals. How ever, at the
time of writing of the Frisch article, a specific connection b etween fractals
and solutions to the Navier-Stokes equations was unknown. O nly recently has
the suggestion been made that those special characteristic sets of points upon
which the solutions to the Navier-Stokes equations can be di scontinuous, and
which at the same time are minimal surfaces, may be the genera tors of fractal
sets [Kiehn, 1992].
A guiding feature of this article will be the idea that the abs tract Cartan
topology of interest is refined by the constraints imposed by the Navier-
Stokes equations on the equations of topological evolution . Questions of
smoothness and continuity are always related to a specific ch oice of a topol-
ogy, and the topology chosen in this article is the topology g enerated by a
constrained Cartan exterior differential system [Bryant, 1 991]. The vector
field solutions to the constrained topology (the Navier-Sto kes equations) fall
into four equivalence classes that characterize certain to pological properties
of the evolutionary system. Most of the mathematical detail s are left to
the Appendix, with the primary discussion presented in a qua litative man-
ner. The principle key feature of the constrained topology i s that two of the
equivalence classes imply that the Cartan topology induced by the vector
flow field is disconnected, while the other two equivalence cl asses imply that
the induced topology is connected.
According to the theory presented herein, the key feature of the turbu-
lent state is that its representation as a vector field can onl y be associated
with a disconnected Cartan topology, while the globally str eamline integrable
state is associated with a connected Cartan topology. It is a mathematical
fact that a transition from a connected topology to disconne cted topology
can take place only by a discontinuous transformation. This discontinuous
transformation is typically realized as a cutting or tearin g operation of sep-
aration. However, a transition from a disconnected topolog y to a connected
topology can take place by a continuous, but not reversible, transformation
of pasting. The fundamental conclusion is that the creation of the turbulent
state is intrinsically different from the decay of the turbul ent state, but both
processes involve topological change.
42 THE CARTAN TOPOLOGY
To come to grips with the topological issues of hydrodynamic s it is necessary
to be able to define a useful topology in a way naturally suited to the problem
at hand. On the set {x,y,z,t }it is possible to define many topologies. Indeed,
for many evolutionary systems, particularly those which ar e dissipative and
irreversible, the topology of the initial state need not be t he same as the
topology of the final state. Hence, not only is it necessary to construct a
topology for a hydrodynamic system at some time, t, it is also necessary that
the topology so constructed be a dynamical system in itself. The transition
to or from turbulence will involve topological change.
As mentioned above, topological change can be induced eithe r by a pro-
cess that is discontinuous, or by a process that is continuou s, but not re-
versible [Kiehn 1991a]. In this article attention is focuse d mainly on those
processes that are continuous but irreversible. Recall tha t continuity is a
topological property (not a geometrical property) that is d efined in terms of
the limit points of the initial and final state topologies. Th e methods used to
define the topology used herein are based on those techniques that E. Cartan
developed for his studies of exterior differential systems [ Bishop, 1968]. For
the hydrodynamic application it will be assumed that the evo lutionary phys-
ical system (the fluid) can be defined in terms of a 1-form of Act ion built
on a single covariant vector field. Typical of Lagrangian fiel d theory, the
equations of evolution are determined from a system of Pfaffia n expressions
constructed from the extremals to the 1-form of Action when s ubjected to
constraints. The resultant Pfaffian system corresponds to a s ystem of partial
differential equations of evolution. The extremal process i s equivalent to the
first variation in the theory of the Calculus of Variations. T he Action 1-form,
A, may be composed from the functions that make up the evoluti onary flow
field itself, as well as other functions on the set {x,y,z,t }. A unreduced typi-
cal format for the Action 1-form on the set {x,y,z,t }would be given by the
expression,
A=Axdx+Aydy+Azdz+Atdt, (1)
where the functions Akare constructed from the components of the flow
field, and other functions. Cartans idea was to examine the ir reducible rep-
resentations of this 1-form of Action. For example, in certa in domains, the
action may be represented by the differentials of a single fun ction,A=dφ.
In other cases, the representation might require two functi ons,A=αdβ, and
5so on. Given an arbitrary action, A, how do you determine its irreducible
represention? Is a given Action reducible? The answer to the last question
is given by the Pfaff dimension or class of the 1-form A. The Pfaff dimension
is computed by one differential and several algebraic proces ses producing a
sequence of higher and higher order differential forms. The c onstruction of
these forms is detailed in the next section, but the remark to be made here is
that these objects may be used to construct a topological bas is, and thereby,
a topology.
In short, a point set equivalent to the Cartan topology can be defined in
terms of those functions that form the components of the cova riant vector
field use to define the Action 1-form, the first partial derivat ives of these
functions, and their algebraic intersections. The details of this coarse Cartan
topology over a space, {x,y,z,t }, are given in Appendix A, with the refinement
that constrains the coarse topology to yield those evolutio nary fields that are
solutions to the Navier-Stokes equations.
3 PFAFF DIMENSION
The topological property of dimension is the key feature tha t distinguishes
precisely four Cartan equivalence classes of covariant vec tor fields on the set
{x,y,z,t }. The idea of Pfaff dimension (called the class of the Pfaffian sy stem
in the older literature [Forsyth, 1953] ) is related to the ir reducible number
of functions which are required to describe an arbitrary Pfa ffian form, in this
case the 1-form of Action. An Action 1-form that can be genera ted globally
in terms of a single scalar field, and its differentials, is of P faff dimension 1.
Over space time, this single function or parameter is often c alled the ”phase”
or ”potential” function, φ(x,y,z,t ). When the Action is of Pfaff dimension
1, then it may be expressed in the reduced form, A=dφ.
For the purposes described in this article, where the Action is defined
in terms of the flow field itself, the constraint of Pfaff dimens ion 1 implies
that the vector field representing the evolutionary flow is a ” potential” flow.
Such gradient vector flow fields represent a submersion from f our dimensional
space-time to a parameter space of one dimension, the potent ial function
itself.
Actions constructed for flows that admit vorticity (but are c ompletely
integrable in the sense that through every point there exist s a unique param-
eter function whose gradient determines the line of action o f the flow) can be
6represented by a submersive map to a parameter space of two di mensions.
The reduced format for the 1-form of Action is A=ψ(x,y,z,t )dφ(x,y,z,t ).
Such integrable flows are defined as globally laminar flows (in the sense that
there exists a globally sychronizable set of unique initial conditions, or pa-
rameters). Such flows are to be distinguished from flows that m ay have,
for example, re-entrant domains, and are locally layered, b ut for which it
is impossible to define a global connected set of initial conditions. Globally
laminar flows and potential flows are of Pfaff dimension 2 or les s, and are
associated with a Cartan point set topology which is connected . This con-
cept is to be interpreted as implying that there exists an N-1 dimensional set
which intersects the flow lines in a unique set of points. (As A rnold says, the
field is of co-dimension 1.) For three dimensional space, N=3 , this set is a
surface. For four-dimensional space, N=4, this set is a volu me.
In a domain where the Cartan point set topology is connected ( the 1-
form of Action is of Pfaff dimension 2 or less) it is possible to define a single
connected parameter of evolution which plays the role of ”ph ase”. In three
dimensions, the parametric value is called ”time”. This ide a of a uniquely
defined global parameter N-1 surface is the heart of the Carat heodory theory
of equilibrium thermodynamics. For such systems, there are infinitely close
neighboring points which are not reachable by closed equili brium processes.
The equilibrium process is defined to be a process whose traje ctory is confined
to the N-1 integrable hypersurface. The two irreducible fun ctions of the
integrable Pfaff representation are called temperature, T, and entropy, S, in
the Caratheodory setting. The parameter space is connected , but a constant
value of the parameter space does not intersect all points of the domain.
The evolutionary vector fields associated with completely i ntegrable Pfaffian
systems are never chaotic [Schuster, 1984].
For a 1-form of Action that is associated with a disconnected point set
topology, such a globally unique parameterization as descr ibed above is im-
possible. Those 1-forms of Action, if not of Pfaff dimension 2 (or less) glob-
ally, do not satisfy the Frobenius complete integrability c onditions [Flan-
ders,1963] The evolutionary vector fields associated with n on-integrable 1-
forms can be chaotic. If the 1-form of Action is of Pfaff dimens ion 3,
then it has an irreducible representation as A=dβ+ψdφ. The three
independent functions form a non-zero three form otf topolo gical torsion,
H=AˆdA=dβˆdψˆdφ, and represents a covariant current of rank 3, with
a dual representation as a contravariant tensor density (th e torsion current).
The torsion current has zero divergence on the domain of spac e{x,y,z,t }
7which is of Pfaff dimension 3 relative to the Action A. This res ult implies
that the ”lines” so generated by the solenoidal torsion curr ent in {x,y,z,t }
can never stop or start within the domain interior. The lines representing
the topological torsion 4-current either close on themselv es, or start and stop
on points of the boundary of the domain. The torsion lines nev er stop in the
interior of the domain where the Pfaff dimension is 3. Such a to rsion current
does not exist in domains of Pfaff dimension 2 or less. Explici t formulas for
the torsion current will be given below.
For a four dimensional domain, the 1-form of Action may be of P faff
dimension 4, and the irreducible representation of the Acti on is given by the
expression, A=αdβ+ψdφ. Each of these functions is independent, so the
topological torsion current is of the form,
AˆdA=αdβˆdψˆdφ+ψdφˆdαˆdβ (2)
The topological torsion current is a 4 component vector field . However,
the divergence of this vector field is not necessarily zero! T he lines of the
torsion current can start or stop in the interior of the domain when the Pfaff
dimension is 4, but not when the Pfaff dimension is 3.
In four dimensions, a solenoidal vector field, if homogeneou s of degree 0,
forms a minimal surface in space time. In fact, if a four dimen sional vector
field can be represented by a complex holomorphic curve, then the field is not
only solenoidal, but also harmonic, and is always associate d with a minimal
surface. It will be shown below, that in the hydrodynamics go verned by the
Navier-Stokes equations, harmonic vector field solutions a re not dissipative,
no matter what the value of the viscosity coefficient. For diss ipative irre-
versible systems, attention is therefore focused on system s of Pfaff dimension
4, for which the torsion current is not solenoidal.
The theory presented in this article insists that irreversi ble turbulence
must be time dependent and irreducibly three dimensional. T he idea of ”two
dimensional” turbulence, for time dependent continuous flo ws, is inconsis-
tent, for such flows have a maximum Pfaff dimension of 3. Flows o f Pfaff
dimension 3 can be chaotic, but they are deterministically r eversible, hence
not turbulent. In agreement with the arguments expressed by Kida [1989],
the turbulent state is more than just chaos. A turbulent doma in must be of
Pfaff dimension 4, for in space-time domains of Pfaff dimensio n 3, it is always
possible to construct flow lines that never intersect. Hence such flow lines are
8always re-traceable, without ambiguity, and such flows are n ot irreversible. In
order to break time-reversal symmetry, and hence to be irrev ersible, the flow
lines must intersect in space-time such that they cannot be r etraced without
ambiguity. Such a result requires that the Euler characteri stic of the four
dimensional domain must be non-zero, for then it is impossib le to construct
a vector field without intersections. The Euler characteris tic of space-time is
only non-zero on domains of Pfaff dimension 4. Hence, the Pfaff dimension
of turbulent domains on {x,y,z,t }must be 4, while the chaotic domains need
be only of Pfaff dimension 3 [Kiehn, 1991b].
4 TOPOLOGICAL CONNECTEDNESS VS.
GEOMETRIC SCALES
In early studies of the turbulent state, from both the statis tical point of
view and the point of view of the Navier Stokes equations, the geometric
concepts of large and small spatial scales, or short or long t emporal scales,
have permeated the discussions. From a topological point of view, length
scales and time scales have no meaning. If things are too smal l, a topologist
stretches them out, and conversely. If turbulence is a topol ogical concept,
then the ideas should be independent from scales. It is inter esting to note
that the original Kolmogorov statistical analysis of the tu rbulent state is now
interpreted in terms of the multi-fractal concept of scale i nvariance [Frisch,
1991].
A key feature of the disconnected Cartan topology is that the domain
supports non-null Topological Torsion, and is of Pfaff dimen sion 3 for chaotic
flows, and of Pfaff dimension 4 for turbulent flows. Suppose the initial state
is a turbulent state in which there exist disconnected stria ted or tubular
domains that are of Pfaff dimension 4 and are embedded in domai ns of Pfaff
dimension 2 or less. Then if the hydrodynamic system is left t o decay, these
striated domains can decay by continuous collapse into stri ations or filaments
of measure zero. The Topological Torsion of the striated dom ains cannot be
zero. The size of the striated domains is not of issue, but the existence of
such domains with non-zero measure is of interest, for if the se domains do
not exist, the flow is not chaotic and not turbulent.
The geometric idea of small domains versus large domains of s pace and/or
time is transformed to a topological idea of connected domai ns versus dis-
9connected domains. Points in disconnected components are n ot reachable
[Hermann, 1968] in the sense of Caratheodory, hence are sepa rated by ”large
scales”, while points in the same component are reachable, a nd hence are sep-
arated by ”small scales”, compared to points in disconnecte d components.
It is the view of this article that the geometric concept of sc ales is not ger-
mane to the problem of turbulence, but instead the basic issu e is one of
connectedness or disconnectedness.
5 TOPOLOGICAL TORSION
The difference between chaotic flows and turbulent flows is tha t chaotic flows
preserve time reversal symmetry and turbulent flows do not. C haotic flows
can be reversible, while turbulent flows are not. Both chaoti c and turbulent
flows support a non-zero value of Topological Torsion tensor . As constructed
in the Appendix for the Navier-Stokes system, the Topologic al Torsion 3-form
on space-time, H, has 4 components, {T,h}that transform as the compo-
nents of a third rank completely anti-symmetric covariant t ensor field, Hijk.
If the vector field used to construct the Topological Torsion tensor is com-
pletely integrable in the sense of Frobenius, then all compo nents, {T,h},
vanish, and the Pfaff dimension of the domain is 2 or less. For t he Navier-
Stokes fluid, the torsion current is given by the engineering expression given
in the appendix as equation (20). As a third rank tensor field, the Topo-
logical Torsion tensor is intrinsically covariant with res pect to all coordinate
transformations, including the Galilean translation.
For engineers, the closest analog to the Topological Torsio n tensor is the
charge-current, 4-vector density, J, of electromagnetism. The fundamental
difference is that where the electromagnetic 4-current alwa ys satisfies the
conservation law, divj+∂ρ/∂t = 0, the Topological Torsion 4-current does
not, unless the vector field use to construct the Topological Torsion tensor
is an element of an equivalence class with Pfaff dimension les s than 4. In
other words, for the turbulent state, the Topological Torsi on tensor does not
satisfy a local conservation law, where for the chaotic stat e it does:
•divT+∂h/∂t = 0 ... Pfaff Dimension 3,
(a necessary condition for chaos),
•divT+∂h/∂t /ne}ationslash= 0 ... Pfaff Dimension 4,
10(a necessary condition for turbulence).
The lines of torsion current, given by solutions to the syste m of first order
differential equations,
dx/Tx=dy/Ty=dz/Tz, (3)
can start or stop internally if the Pfaff dimension is 4, but on ly on boundary
points or limit points of the domain, if the Pfaff dimension is 3 . If the vector
field is of Pfaff dimension less than 4, then the integral over a boundary of the
Topological Torsion tensor is an evolutionary invariant, b ut some care must
be taken to insure that the integration domain is a boundary, and not just a
closed cycle. This result, which corresponds to a global hel icity conservation
theorem, is independent of any statement about viscosity. H owever, the
invariance of the Topological Torsion integral over a bound ary for a Navier-
Stokes fluid implies that integral of the 4-form of Topologic al Parity must
vanish, which in turn implies that the Euler index of the Cart an topology
for this situation is zero. The compliment of this idea leads to a variable
Topological Torsion integral, and the requirement that the Euler index is
not zero for the irreversible decay of the turbulent state. F or if the Euler
index is not zero, then every vector field has at least one flow l ine with
a singular point. If an evolutionary parameter carries the p rocess through
the singular point, a reversal of the process parameter will retrace the path
only back to the singular point uniquely. Subsequently, the return path
then becomes ambiguous, and the evolution is not reversible . Recall that
a necessary condition for a vector field to exist on a manifold without self
intersection singularities is that the Euler index of the ma nifold must vanish.
Hence a necessary condition for turbulence is that the Carta n topology must
be of Pfaff dimension 4, and the topological torsion is not sol enoidal.
It is remarkable to this author that experimentalists and th eorists (includ-
ing the present author) have been so brain-washed by the dogm a of unique
predictability in the physical sciences that they have comp letely ignored the
measurement and implications of the Topological Torsion te nsor. Although
the solutions to a Pfaffian system of equations is a problem tha t has found
use in the older literature of differential equations (where it is known as the
”subsidiary” system) [Forsyth 1959 p.95], its utilization in applied dynami-
cal systems, especially hydrodynamics, is extremely limit ed. Of course, for
11uniquely integrable systems, the equations of topological torsion are evanes-
cent, and not useful. The very existence of the Topological T orsion tensor is
an indicator of when unique predictability is impossible [K iehn, 1976], and
attention should be paid to the Pfaff dimension of a physical s ystem described
by (1).
6 TORSION WAVES
One of the predictions of the Cartan topological approach is the fact that for
systems of Pfaff dimension 4 it is possible to excite torsion w aves. Torsion
waves are essentially transverse waves but with enough long itudinal compo-
nent to give them a helical or spiral signature. In the electr omagnetic case,
where such waves have been measured, they are represented by four com-
ponent quaternionic solutions to Maxwell’s equations [Sch ultz, 1979; Kiehn,
1991]. Such electromagnetic waves represent different stat es of left or right
polarization (parity) traveling in opposite directions. T he wave speeds in
different directions can be distinct. In fluids, transverse t orsion waves can
be made visible by first constructing a Falaco soliton state [ Kiehn 1991c]
and then dropping dye near the rotating surface defect. The d ye drop will
execute transverse polarized helical motions about a guidi ng filamentary vor-
tex that connects the pair of contra-rotating surface defec ts. There is some
evidence that torsion waves can appear as traveling waves on Rayleigh cells
[Croquette, 1989].
7 THE PRODUCTION VS. THE DECAY
OF TURBULENCE
The Cartan topological theory predicts that the transition to turbulence from
a globally laminar state involves a transition from a connec ted Cartan topol-
ogy to a disconnected Cartan topology. From this fact it may b e proved
that such transitions can NOT be continuous, but they may be r eversible!
However, the theory also predicts that the decay of turbulen ce can be de-
scribed by a continuous transformation, but the transforma tion can NOT be
reversible.
The Cartan topology when combined with the Lie derivative ma y be
used to define partial differential equations of evolution [K iehn, 1990], which
12include the Navier-Stokes equations as a subset of a more refi ned topology.
However, if the Cartan topology is constructed from p-forms and vector fields
that are restricted to be C2 differentiable, then it may be sho wn that all such
solutions to the Navier-Stokes equations are continuous re lative to the Car-
tan topology. The creation of the turbulent state must involve discontinuous
solutions to the Navier-Stokes equations, which are genera ted only by shocks
or tangential discontinuities, and therefore are not descr ibable by C2 fields.
On the other hand, the decay of turbulence can be described by C2 differen-
tiable, hence continuous, solutions which are not homeomor phisms, and are
therefore not reversible. In this article, the decay of turb ulence by C0 and
C1 functions is left open.
Domains of finite Topological Torsion are topologically dis connected from
domains that have zero Topological Torsion. The anomaly tha t permits the
local creation or destruction of Topological Torsion is exa ctly the 4-form of
Topological Parity (see Appendix). If the Topological Pari ty is zero, then
the Topological Torsion obeys a pointwise conservation law . For a barotropic
Navier-Stokes fluid, the anomaly, or source term for the Tors ion current can
be evaluated explicitly, and appears as the right hand side i n the following
equation:
divT+∂h/∂t =−2ν curl v•curlcurl v. (4)
It is remarkable that for flows of any viscosity, the Topologi cal Torsion
tensor satisfies a pointwise conversation law, and the integ ral over a bounded
domain is a flow invariant, if the vorticity vector field satis fies the Frobenious
integrability conditions, curlv•curlcurl v= 0. It would seem natural
that the decay of turbulence would be attracted to such inter esting limiting
configurations of topological coherence in a viscous fluid. T hese limit sets
can be related to minimal surfaces of tangential discontinu ities which can act
as fractal boundaries of chaotic domains [Kiehn 1992b, 1992 c, 1993].
8 SUMMARY
The topological perspective of Cartan indicates that:
1. A necessary condition for the turbulent state is that the fl ow field must
generate a domain of support which is of Pfaff dimension 4, and is to
13be distinguished from the chaotic state, which is necessari ly of Pfaff
dimension 3. Moreover, the irreversible property of turbul ence decay
implies that the Euler index of the induced Cartan topology m ust be
non-zero.
2. The geometric concept of length scales and time scales sho uld be re-
placed by the topological concept of connectedness vs. disc onnected-
ness.
3. The transition to turbulence can take place only by discon tinuous so-
lutions to the Navier-Stokes equations.
4. The decay of turbulence can be described by continuous but irreversible
solutions to the Navier-Stokes equations. The decay of doma ins for
which there exists finite Topological Torsion is dependent u pon a finite
viscosity, and the lack of integrability for the lines of vor ticity. One
model for the decay of turbulence might be described as the co llapse
of tubular domains of torsion current, becoming ever finer fil amentary
domains without helicity until ultimately they have a measu re zero.
5. A coherent structure in a turbulent flow may be defined as a co nnected
deformable component of a disconnected Cartan topology. Fo r the
Cartan topology, these coherent structures are of two speci es. One
component will have null Topological Torsion and the other c omponent
will have non-null Topological Torsion. Bounded domains fo r which the
integral of Topological Torsion is a flow invariant can form i n a viscous
turbulent fluid. In particular, domains for which curlv•curlcurl v=
0, but v•curlv/ne}ationslash= 0, can have an evolutionary persistence in a viscous
fluid. The existence of helicity density is a sufficient but not necessary
signature that the Cartan topology is disconnected
9 ACKNOWLEDGMENTS
This work was supported in part by the Energy Laboratory at th e University
of Houston. The author owes the late A. Hildebrand, the direc tor of the
Energy Lab, a debt of gratitude for first, his skepticisms, th en his conversion,
and finally his insistence that the Cartan ideas of differenti al forms would
have practical importance in physical systems.
14The concepts actually began in 1975-1976 when due to the supp ort of
NASA- Ames, the idea that the concept of Frobenius integrabi lity delineated
streamline flows from turbulent flows was formulated [Kiehn 1 976]. When
the Action 1-form satisfies AˆdA= 0, the Frobenius condition is valid and
the flow is streamline. When AˆdA/ne}ationslash= 0 the Frobenius condition for uniquew
integrability fails. The 3-form AˆdAdefined as topological torsion is now
more popularly known in field theories as the Chern-Simons te rm. Most of
the results were presented at an APS meeting in 1991.
10 APPENDIX A : THE CARTAN TOPOL-
OGY
Starting in 1899, Cartan [1899,1937,1945] developed his th eory of exterior
differential systems built on the Grassmann algebraic conce pt of exterior mul-
tiplication, and the novel calculus concept of exterior diff erentiation. These
operations are applied to sets called exterior p-forms, whi ch are often de-
scribed as the objects that form an integrand under the integ ral sign. The
Cartan concepts may still seem unconventional to the engine er, and only dur-
ing the past few years have they slowly crept into the mainstr eam of physics.
There are several texts at an introductory level that the uni nitiated will find
useful [Flanders, 1963; Bamberg, 1989; Bishop,1968; Lovel ock, 1989; Nash,
1989; Greub, 1973]. A reading of Cartan’s many works in the or iginal French
will yield a wealth of ideas that have yet to be exploited in th e physical
sciences. It is not the purpose of this article to provide suc h a tutorial of
Cartan’s methods, but suffice it to say the ”raison d’ˆ etre” fo r these, perhaps
unfamiliar, but simple and useful methods is that they permi t topological
properties of physical systems and processes to be sifted ou t from the chaff
of geometric ideas that, at present, seem to dominate the eng ineering and
physical sciences.
For hydrodynamics, the combination of the exterior derivat ive and the
interior product to form the Lie derivative acting on p-form s should be in-
terpreted as the fundamental way of expressing an evolution operator with
properties that are independent of geometric concepts such as metric and
connection [Kiehn, 1975a]. Cartan’s Lie derivative yields the equivalent of
a convective derivative that may be used to demonstrate that the laws of
hydrodynamic evolution are topological laws, not geometri c laws. This phi-
15losophy is similar to that championed by Van Dantzig [1934] w ith regard to
the topological content of Maxwell’s equations of electrod ynamics.
The details of the Cartan Topology may be found at [Kiehn 2001 ], but
for purposes herein recall that there the Pfaff sequence buil t on a 1-form has
up to four terms defined as:
Topological ACTION :A=Aµdxµ(5)
Topological VORTICITY :F=dA=Fµνdxµˆdxν
Topological TORSION :H=AˆdA=Hµνσdxµˆdxνˆdxσ
Topological PARITY :K=dAˆdA=Kµνστdxµˆdxνˆdxσˆdxτ.
These elements of the Pfaff sequence can be used to produce a ba sis collection
of open sets that consists of the subsets,
B={A, Ac, H, Hc}={A, A∪F, H, H ∪K}
The collection of all possible unions of these base elements , and the null set,
∅,generate the Cartan topology of open sets:
T(open) ={X,∅, A, H,A ∪F, H∪K, A∪H, A∪H∪K, A∪F∪H}.
These nine subsets form the open sets of the Cartan topology c onstructed
from the domains of support of the Pfaff sequence constructed from a single
1-form,A. The compliments of the open sets are the closed sets of the
Cartan topology.
T(closed ) ={∅, X, F ∪H∪K, A∪F∪K,A∪F, H ∪K,F∪K,F, A }.
16From the set of 4 ”points” {A,F,H,K }that make up the Pfaff sequence
it is possible to construct 16 subset collections by the proc ess of union. It
is possible to compute the limit points for every subset rela tive to the Car-
tan topology. The classical definition of a limit point is tha t a point p is a
limit point of the subset Y relative to the topology T if and on ly if for every
open set which contains p there exists another point of Y othe r than p [Lip-
schutz, 1965]. The results appear in the given reference htt p://arXiv/math-
ph/0101033
By examining the set of limit points so constructed for every subset of the
Cartan system, and presuming that the functions that make up the forms are
C2 differentiable (such that the Poincare lemma is true, ddω= 0,any ω ), it
is easy to show that for all subsets of the Cartan topology eve ry limit set is
composed of the exterior derivative of the subset, thereby p roving the claim
that the exterior derivative is a limit point operator relat ive to the Cartan
topology. For example, the open subset, A∪H, has the limit points that
consist ofFandK.
It is apparent that the Cartan topology is composed of the uni on of two
subsets (other than ∅andX) which are both open and closed, for X=
Ac∪Hc, a result that implies that the Cartan topology is not connec ted,
unless the Topological Torsion, H, and hence its closure, vanishes. This
extraordinary result has a number of physical consequences , some of which
are described in [Kiehn, 1975, 1975a].
To prove that a turbulent flow must be a consequence of a Cartan topol-
ogy that is not connected, consider the following argument: First consider
a fluid at rest and from a global set of unique, synchronous, in itial condi-
tions generate a vector field of flow. Such flows must satisfy th e Frobenius
complete integrability theorem, which requires that AˆdA= 0. The Tor-
sion current is zero for such systems. The Cartan topology fo r such systems
(H=AˆdA= 0) is connected, and the Pfaff dimension of the domain is
2 or less. Such domains do not support Topological Torsion. S uch glob-
ally laminar flows are to be distinguished from flows that resi de on surfaces,
but do not admit a unique set of connected sychronizable init ial conditions.
Next consider turbulent flows which, as the antithesis of lam inar flows, can
not be integrable in the sense of Frobenius; such turbulent d omains support
Topological Torsion ( H=AˆdA/ne}ationslash= 0), and therefore a disconnected Cartan
topology. The connected components of the disconnected Car tan topology
can be defined as the coherent structures of the turbulent flow . The transi-
17tion from an initial laminar state ( H= 0) to the turbulent state ( H/ne}ationslash= 0) is a
transition from a connected topology to a disconnected topo logy. Therefore
the transition to turbulence can NOT be continuous. However , the decay
of turbulence can be described by a continuous transformati on from a dis-
connected topology to a connected topology. Condensation i s continuous,
gasification is not.
A topological structure is defined to be enough information t o decide
whether a transformation is continuous or not [Gellert, 197 7]. The classical
definition of continuity depends upon the idea that every ope n set in the
range must have an inverse image in the domain. This means tha t topologies
must be defined on both the initial and final state, and that som ehow an
inverse image must be defined. Note that the open sets of the fin al state may
be different from the open sets of the initial state, because t he topologies of
the two states can be different.
There is another definition of continuity that is more useful for it depends
only on the transformation and not its inverse explicitly. A transformation is
continuous if and only if the image of the closure of every sub set is included
in the closure of the image. This means that the concept of clo sure and the
concept of transformation must commute for continuous proc esses. Suppose
the forward image of a 1-form AisQ,and the forward image of the set F=
dAisZ. Then if the closure, Ac=A∪Fis included in the closure of Qc=
Q∪dQ, for all sub-sets, the transformation is defined to be contin uous. The
idea of continuity becomes equivalent to the concept that th e forward image
Zof the limit points, dA, is an element of the closure of Q[Hocking, 1961]:
A function fthat produces an image f[A] =Qis continuous iff
for every subset Aof the Cartan topology, Z=f[dA]⊂Qc=
(Q∪dQ).
The Cartan theory of exterior differential systems can now be interpreted
as a topological structure, for every subset of the topology can be tested to
see if the process of closure commutes with the process of tra nsformation. For
the Cartan topology, this emphasis on limit points rather th an on open sets is
a more convenient method for determining continuity. A simp le evolutionary
process,X⇒Y, is defined by a map Φ. The map, Φ, may be viewed as
a propagator that takes the initial state, X, into the final state, Y. For
more general physical situations the evolutionary process es are generated by
18vector fields of flow, V. The trajectories defined by the vector fields may
be viewed as propagators that carry domains into ranges in th e manner of a
convective fluid flow. The evolutionary propagator of intere st to this article
is the Lie derivative with respect to a vector field, V, acting on differential
forms, Σ [Bishop, 1968].
The Lie derivative has a number of interesting and useful pro perties.
1. The Lie derivative does not depend upon a metric or a connec tion.
2. The Lie derivative has a simple action on differential form s producing
a resultant form that is decomposed into a transversal and an exact
part:
L(V)ω=i(V)dω+di(V)ω. (6)
This formula is known as ”Cartan’s magic formula”. For those vector
fieldsVwhich are ”associated” with the form ω,such thati(V)ω= 0,
the Lie derivative becomes equivalent to the covariant deri vative of
tensor analysis. Otherwise the two derivative concepts are distinct.
3. The Lie derivative may be used to describe deformations an d topolog-
ical evolution. Note that the action of the Lie derivative on a 0-form
(scalar function) is the same as the directional or convecti ve derivative
of ordinary calculus,
L(V)Φ =i(V)dΦ +di(V)Φ =i(V)dΦ + 0 = V·gradΦ.(7)
It may be demonstrated that the action of the Lie derivative o n a 1-form
will generate equations of motion of the hydrodynamic type.
4. With respect to vector fields and forms constructed over C2 functions,
the Lie derivative commutes with the closure operator. Henc e, the
Lie derivative (restricted to C2 functions) generates tran sformations
on differential forms which are continuous with respect to th e Cartan
topology. To prove this claim:
First construct the closure, {Σ∪dΣ}. Next propagate Σ and dΣ by
means of the Lie derivative to produce the decremental forms , sayQ
andZ,
L(V)Σ =Q and L (V)dΣ =Z. (8)
19Now compute the contributions to the closure of the final stat e as
given by {Q∪dQ}. IfZ=dQ, then the closure of the initial state
is propagated into the closure of the final state, and the evol utionary
process defined by Vis continuous. However,
dQ=dL(V)Σ =di(V)dΣ +dd(i(V)Σ) =di(V)dΣ
asdd(i(V)Σ) = 0 for C2 functions. But,
Z=L(V)dΣ =d(i(V)dΣ) +i(V)ddΣ =di(V)dΣ
asi(V)ddΣ = 0 for C2 p-forms. It follows that Z=dQ, and therefore
Vgenerates a continuous evolutionary process relative to th e Cartan
topology.QED
Certain special cases arise for those subsets of vector field s that satisfy
the equations, d(i(V)Σ) = 0. In these cases, only the functions that
make up the p-form, Σ, need be C2 differentiable, and the vecto r
field need only be C1. Such processes will be of interest to sym plectic
processes, with Bernoulli-Casimir invariants.
By suitable choice of the 1-form of action it is possible to sh ow that
the action of the Lie derivative on the 1-form of action can ge nerate the
Navier Stokes partial differential equations [Kiehn, 1978] . The analysis above
indicates that C2 differentiable solutions to the Navier-St okes equations can
not be used to describe the transition to turbulence. The C2 s olutions can,
however, describe the irreversible decay of turbulence to t he globally laminar
state.
For a given Lagrange Action, A, Cartan has demonstrated that the first
variation of the Action integral (with fixed endpoints) is eq uivalent to the
search for those vector fields, V, that for any renormalization factor, ρ, satisfy
the equations
i(ρV)dA= 0 (9)
Such vector fields are called extremal vector fields [Klein, 1 962, Kiehn, 1975a].
The Cartan theorem [Cartan, 1958] states that the extremal c onstraint fur-
nishes both necessary and sufficient conditions that there ex ists a Hamilto-
nian representation for V, on a space of odd Pfaff dimension (2n+1 state
20space). The resultant equations are a set of partial differen tial equations
that represent extremal evolution. The renormalization co ndition is common
place in the projective geometry of lines, and does not requi re the Rieman-
nian or euclidean concept of an inner product or a metric [Mes erve, 1983].
Hamiltonian systems are not considered to be dissipative. T he very strong
topological constraint can be relaxed on a space of even Pfaff dimension and
still yield a Hamiltonian representation if i(ρV)dA=dθ.Moreover, it can be
shown that evolutionary processes Vthat satisfy the Helmholtz-Symplectic
constraint, (which includes all Hamiltonian processes)
d(i(ρV)dA) = 0, (10)
are thermodynamically reversible, relative to the Cartan t opology.
A more general expression for the Cartan condition is given b y the transver-
sal condition,
i(ρV)dA=−f•(dx−Vdt)−d(kT). (11)
This extension of the Cartan Hamiltonian constraint is tran sversal for the
first term is orthogonal to the vector field, ( V). The Lagrange multipliers,
f, are arbitrary for such a transversal constraint, but if cho sen to be of the
form,f=νcurlcurl V, then the constrained Cartan topology will generate
the Navier-Stokes equations.
As an example of this Cartan technique, substitute the 1-for m of action
given by the expression,
A=3/summationdisplay
1vkdxk− Hdt, (12)
where the ”Hamiltonian” function, H, is defined as,
H=v•v/2 +/integraldisplay
dP/ρ−λdivv+kT (13)
into the constraint equation given by 11. Carry out the indic ated operations
of exterior differentiation and exterior multiplication to yield a system of
necessary partial differential equations yields of the form ,
21∂v/∂t+grad(v•v/2)−v×curlv=−gradP/ρ +λgraddiv v−ν curlcurl v.
(14)
These equations are exactly the Navier-Stokes partial diffe rential equations
for the evolution of a compressible viscous irreversible flo wing fluid.
By direct computation, the 2-form F=dAhas components,
F=dA=ωzdxˆdy+ωxdyˆdz+ωydzˆdx (15)
+axdxˆdt+aydyˆdt+azdzˆdt, (16)
where by definition
ω=curlv,a=−∂v/∂t−gradH (17)
The 3-form of Helicity or Topological Torsion, H, is constru cted from the
exterior product of A and dA as,
H=AˆdA=Hijkdxiˆdxjˆdxk (18)
=−Txdyˆdzˆdt−Tydzˆdxˆdt−Tzdxˆdyˆdt+hdxˆdyˆdz,(19)
where Tis the fluidic Torsion axial vector current, and h is the torsi on
(helicity) density:
T=a×v+Hω, h =v•ω (20)
The Torsion current, T, consists of two parts. The first term represents
the shear of translational accelerations, and the second pa rt represents the
shear of rotational accelerations. The topological torsio n tensor,Hijk, is a
third rank completely anti-symmetric covariant tensor fiel d, with four com-
ponents on the variety {x,y,z,t }.
The Topological Parity becomes
K=dH=dAˆdA=−2(a•ω)dxˆdyˆdzˆdt. (21)
22This equation is in the form of a divergence when expressed on {x,y,z,t },
divT+∂h/∂t =−2(a•ω), (22)
and yields the helicity-torsion current conservation law i f the anomaly, −2(a•ω),
on the RHS vanishes. It is to be observed that when K= 0, the integral of
Kvanishes, which implies that the Euler index, χ, is zero. It follows that
the integral of Hover a boundary of support vanishes by Stokes theorem.
This idea is the generalization of the conservation of the in tegral of helicity
density in an Eulerian flow. Note the result is independent fr om viscosity,
subject to the constraint of zero Euler index, χ= 0.
The Navier-Stokes equations of topological constraint may be used to
express the acceleration term, a, kinematically; i.e.,
a=−∂v/∂t−gradH=−v×ω+ν curl ω. (23)
Substition of this expression into the definition of the Tors ion current yields a
formula in terms of the helicity density, h, the viscosity, ν, and the Lagrangian
function,
L=v•v− H, (24)
that may be written as:
T={hv−Lω}−ν{v×curlω} (25)
={hv−(v•v/2−/integraldisplay
dP/ρ−λdivv)ω}−ν{v×curlω}(26)
Note that the torsion axial vector current persists even for Euler flows, where
νandλvanish. The measurement of the components of the Torsion cur rent
have been completely ignored by experimentalists in hydrod ynamics.
Similarly, the Topological Parity pseudo-scalar for the Na vier-Stokes fluid
becomes expressible in terms of engineering quantities as,
K= 2ν(ω•curlω)dxˆdyˆdzˆdt. (27)
23The Euler index for the Navier-Stokes fluid is proportional t o the integral of
the Topological Parity 4-form, which is the ”top Pfaffian” in C hern’s analysis
[Chern, 1944, 1988]. When
(ω•curlω) (28)
the Euler index of the induced Cartan topology must vanish. T his result is
to be compared to the classic hydrodynamic principle of mini mum rate of
energy dissipation [Lamb, 1932]. For a barotropic Navier-S tokes fluid of Pfaff
dimension 4, the viscosity cannot be zero, and the lines of vo rticity must be
non-integrable in the sense of Frobenius.
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27 |
Instability patterns, wakes and topological limit sets.
R. M. Kiehn
Mazan,France
Rkiehn2352@aol.com
http://www.cartan.pair.com
Published in J.P.Bonnet and M.N. Glauser, (eds) “Eddy Structure Identification in Free
Turbulent Shear Flows”, Kluwer Academic Publishers, (1993) p. 363
Abstract
Many hydrodynamic instability patterns can be put into correspondence with a
subset of characteristic surfaces of tangential discontinuities. These topological limits sets
to systems of hyperbolic PDE's are locally unstable, but a certain subset associated with
minimal surfaces are globally stabilized, persistent and non-dissipative. Sections of these
surfaces are the spiral scrolls so often observed in hydrodynamic wakes. This method of
wake production does not depend explicitly upon viscosity.
Introduction
From the topological point of view it is remarkable how often flow instabilities and
wakes take on one or another of two basic scroll patterns. The first scroll pattern is
epitomized by the Kelvin-Helmholtz instability (Figure 1a) and the second scroll pattern is
epitomized by the Raleigh-Taylor instability (Figure 1b).
The repeated occurrence of these two patterns (one similar to a C ornu Spiral and
the other similar to a Mushroom Spiral) often deformed but still recognizable and
persistent, even in dissipative media, suggests that a basic simple underlying topological
principle is responsible for their creation. The mushroom pattern is of particular interest
to this author, who long ago was fascinated by the long lived ionized ring that persists in
Figure 1b. The pattern of the
Rayleigh-Taylor Instability.
Figure 1a. The pattern of the
Kelvin-Helmholtz instability.the mushroom cloud of an atomic explosion. Although the mushroom pattern appears in
many diverse physical systems (in the Frank-Reed source of crystal growth, in the scroll
patterns generated in excitable systems, in the generation of the wake behind an aircraft,
...), no simple functional description of the mushroom pattern is given in the literature
known to this author. Classical geometric analysis applied to the equations of
hydrodynamics has failed to give a satisfactory description of these persistent structures,
so often observed in many different situations.
Viscosity vs. Compressibility
Conventional analysis of wake production and boundary layer formation subsumes
that the dominant physical effect is governed by viscous creation (and destruction) of
vorticity. To quote Batchelor, "The term wake is commonly applied to the whole region
of non-zero vorticity on the downstream side of a body in an otherwise uniform stream of
fluid." The conventional perspective presumes that the wake is approximated as a
tangential discontinuity, defined as a vortex sheet. The viscous creation of the vortex
sheet is not precisely defined, but once the sheet is formed, its evolution is presumed to be
governed by an integral form of the Biot-Savart law, known as the Rott-Birchoff equation.
These assumptions have been shown to lead to the production of discontinuities in finite
time. Asymptotic spiral type solutions in the vicinity of the singularity have been
investigated both analytically and numerically [Kaden, Rott, Kambe, Pullin, Cayflisch,
Kransky].
The viscous process essentially is one of diffusion, and is governed by a non-
hyperbolic system of PDE's. It is most certainly applicable to the decay of wake
phenomena. However, this point of view, which has its domain of applicability in the
infinitely far field behind a body, is often at odds with experiments of the near field, which
indicate that wake features persist with sharp definition for long periods of time, over
many characteristic lengths, without diffusive blurring, and at high Reynolds numbers
where viscosity effects are minimal. To quote Browand [Browand, 1986] "At the present
time, there is no theoretical framework to describe the structural features of high Reynolds
number shear flows."
In this article an alternate approach to the creation of wakes is proposed. The
basic physical mechanism for wake production is assumed to be associated with the fact
that all real fluids have a finite speed of sound, hence a finite compressibility. Therefore,
there can exist domains in every (perhaps slightly) compressible fluid where the system of
PDE's describing the fluid evolution is hyperbolic, and not diffusively parabolic or elliptic.
Hyperbolic domains have the feature that they can be associated with topological limit sets
upon which the solutions to the PDE's are not unique. Therefore topological
discontinuities are admissible in such dynamical systems. Such discontinuities are of two
types: shock waves, where C 0 differentiability is lost for the component of velocity normal
to the discontinuity surface (but the tangential components of velocity are continuous)
and tangential discontinuities, where C 0 differentiability is lost for the tangential
components of the velocity, but the normal component is continuous. It is this set of limit
points associated with the tangential discontinuities that can be put into correspondencewith the two basic instability patterns described above. Recall that propagating
discontinuities are associated with singular solutions to wave equations. In fact, the very
definition of a wave, according to Hadamard, is a propagating discontinuity. To recap,
the point of view taken in this article is that the creation of a wake is to be associated with
a discontinuous process in a hyperbolic domain, while the decay of a wake is associated
with a diffusive process in an elliptic domain.
This alternate topological approach to the problem of wake creation is independent
from viscosity, and gains credence from the fact that not only is a mechanism offered for
the creation of the tangential discontinuity, but also closed form solutions to the vector
fields that describe the aforementioned instability patterns can be obtained. An equivalent
result is not known in the literature familiar to this author. Not only does the topological
point of view give new insight into how wakes may be generated, but also points out how
such wakes possibly may be controlled. As described below, the resultant wake creation
phenomena is closely related to the problems of diffraction and interference in
electromagnetism. In fact, it may be said that from the topological point of view the near
field hydrodynamic wake is a diffraction pattern caused by the physical obstruction.
Hence, it is plausible that phase interference mechanisms may be used to control wakes.
At high Reynolds numbers the effect of viscous diffusion is to smear out or thicken the
tangential discontinuity.
Spiral Space Curves
The clue that wave fronts, representing propagating tangential discontinuities, are
the basis for the two basic spiral instability patterns in fluid dynamics came to this author
during a study of Cartan's methods of differential topology as applied to the production of
defects and topological torsion in dynamical systems [ Kiehn 1990, 1991, 1992]. Cartan's
ideas start from a generalization of the Frenet-Serret concept of the "moving frame". The
Frenet analysis indicates that a space curve, conventionally defined by a position vector,
R(t), relative to a specific coordinate system, is completely characterized by its arc length,
s, its Frenet curvature, κ(s), and its Frenet torsion, τ(s). [Struik 1956] All curves having
the same κ(s) and τ(s) are congruent, hence are independent from the choice of
coordinates used to define the original position vector. The variables {s, κ,τ} are the
intrinsic variables of a space curve.
The null set of a function f(s, κ,τ) defines a surface in the space of the intrinsic
coordinates, and the intersection of two such surfaces defines a curve in the space of
intrinsic coordinates that has a preimage in the space of initial variables, {x,y,z}. Consider
"plane" curves given by the relations
f1(s,τ,κ) = τ = 0 and f 2(s,τ,κ) = κ - g(s) = 0 . (1) (2)
For g(s) = 0, the intrinsic curve has zero curvature and zero torsion. It is a straight line in
{x,y,z} space, too. For g(s) = +1, the curve in {s, κ,τ} space is a straight line displaced
from the origin to the right, parallel to the s axis and in the τ = 0 plane. The preimage of
this intrinsic curve with constant curvature and zero torsion in {x,y,z} space is a circle.For g(s) = 1/s in the right half plane, the resulting space curve is the logarithmic spiral in
the space {x,y,z). For g(s) = s, a straight line in the intrinsic coordinate space, {s, κ,τ}, the
resulting {x,y,z} image is the Cornu spiral. These facts have been known for more than
100 years to differential geometers.
However, a simple sequence is to be recognized:
... κ = g(s) = s -1, κ = s0 , κ = s1...
The question arises: what are the characteristic shapes of intrinsic space curves for which
the curvature is proportional to an arbitrary power of the arc length, κ = g(s) = s n, for all
positive and negative integers (or rational fractions)? Through the power of the PC these
questions may be answered quickly by integrating the equations,
dx/ds = u = sin(Q(s)) and dy/ds = v = cos(Q(s)), (3) (4)
where Q is some "phase" function of s. The vector with components, u and v, is a unit
tangent vector to a curve with arc length, s. The Frenet curvature is given by κ = dQ/ds.
Hence if Q is s n+1/n+1, then the Frenet curvature is κ = sn. The results of the numerical
integrations are presented in Figure 2 for n = 1 and n = 2. A most remarkable result is that
the Cornu spiral of Figure 2a is the deformable equivalent for all odd-integer n > 0, and
the Mushroom spiral of Figure 2b is the deformable equivalent for all even-integer n > 0.
Figure 2a. The Cornu spiral is a
harmonic function for which the
phase function, Q = s 2.
Figure 2b. The Mushroom spiral is
the harmonic function for which the
phase function, Q = s 3.
Not only has the missing analytic description of the mushroom spiral been found,
but also a raison d'être has been established for the universality of the two spiral patterns.
They belong to the even and odd classes of arc length exponents describing plane curves
in terms of the formula, κ = sn. The Cornu spiral is the first odd harmonic function that
maps the infinite interval into a bounded region of the plane, and the mushroom spiral is
the first even function that maps the infinite interval into a bounded region of the plane.
Note that the zeroth harmonic function maps the infinite interval into the bounded region
of the plane, but the trajectory is unique in that it is not only bounded but it is closed; i.e.,
a circle. A similar sequence can be generated for the half-integer exponents with n = +3/2,
7/2, 11/2... giving the Mushroom spirals and 5/2,9/2,13/2... giving the Cornu spirals. In
numeric and experimental studies of certain shear flows, the Cornu spiral appears to
dominate the motion in the longitudinal direction, while the Mushroom spiral
appears in the transverse direction. Periodic patterns can be obtained by examining phase
functions of various forms. For example, the Kelvin Helmholtz instability of Figure 1a is
homeomorphic to the choice Q(s) = 1/cos 2(s) . Similarly, the Rayleigh-Taylor instability
of Figure 1b is homeomorphic to the case Q(s) = tan(s)/cos(s).
The Kelvin-Helmoltz instability function is highly singular at points where cos(s) = 0. To
study the effect of this singularity, consider a modification of the phase function, such that
Q(s) = 1/( a 2 + cos2(s)). The constant term a represents the radius of a finite "core" or
hole. The solutions to (3-4) for such a phase function are extraordinary. Above a certain
minimum core size the wave pattern exhibits no spiral shapes or self-intersections. At a
value of a 2 = .167, the wave pattern describes a curve which exhibits the first double
tangency, in the shape of a Ying-Yang diagram. As the core radius is decreased further,
Figure 1a'. The pattern of the
Kelvin-Helmholtz instability.
Q(s) = 1/cos 2(s).
Figure 1b'. The pattern of the
Rayleigh-Taylor Instability.
Q(s) = tan(s)/cos(s).
a2= 0.067a2a2= 0.167 = 0.050
Figure 3. Ying Yang patterns Q(s) = 1/ (a 2 + cos2(s))another size is reached when the double tangency again occurs, but now the pattern
contains a dipole, or circulation pair. The process can be continued to generate a
sequence of such double tangencies, ultimately generating the spiral patterns of Figure 1a.
In the interior of each of the double tangencies is a circulation pair, about which the curve
describing the pattern winds, first in one direction, then in the other direction. At each
new double tangency the winding number about each doublet center increases by one. A
description of this effect is given in Figure 3, where the first few double tangencies are
displayed. The interpretation and application of this spin-pair production mechanism has
not been developed.
Torsion vs. Curvature
The full Frenet equations in three dimensions are given by the expressions,
d t /ds = + κ n
d n /ds = - κ t + τ b
d b /ds = - τ n .
The vectors { t, n, b} form an orthogonal moving frame at the point p along the space
curve. The variables s, κ, τ, are the arc length, curvature and torsion, respectively.
There are two extreme situations. In the first extremal case the torsion is
negligible, and the space curve is confined to the plane described by the vectors t and n.
In classical mechanics this result is interpreted as the conservation of angular momentum.
The unit binormal maintains a constant direction (d b/ds ≈ 0) along the angular momentum
vector. Space curves are generated by a parametric displacement in the direction of the
tangent vector. The tangent and the normal vectors form a harmonically conjugate pair if
the curvature is a constant, but if the curvature is exactly zero, then the directions of the
vectors n and b are indeterminate.
However, supp ose that the curvature, κ, is small but finite at some initial condition,
such that the normal and the binormal field can be specified. Now, however, assert that
the torsion of the curve is not negligible. Then the normal and binormal form a
harmonically conjugate pair. To first order, the unit tangent vector is an invariant (d t/ds ≈
0). A curve can be generated in the ( n, b) plane, parametrized by a displacement, ν,
along b. If the torsion, τ(ν), is one of the functions described above, then spiral shapes for
the curve in the ( n, b) plane are to be expected. Hence there are two extremes, the first
extreme where the torsion tends to zero, and the space curves are dominated by curvature,
and can be embedded in scrolled surfaces which locally approximate a cylinder with
generators along the binormal. The second extreme is where the space curves are
dominated by torsion, and are sections of a scrolled surface with local "cylindrical"
generators along the tangent field. In both cases, the tangent plane to the surface is
constructed (only approximately) in terms of the unit tangent and the binormal. In the first
extreme, angular momentum is conserved, d b/ds ≈ 0. In the second extreme, d t/ds ≈ 0.
As derived below, the curvature, κ, for an Eulerian flow is proportional to the componentof the pressure gradient along the normal field for a single surface. In certain situations, it
would appear that the free wake consists of a thin double layer.
From experimental photographs, it
would appear that the Kelvin-Helmholtz
instability is related to characteristic spirals
along the tangent line (and is curvature
dominated) and the Rayleigh-Taylor instability
is related to characteristic spirals along the
binormal line (and is torsion dominated). The
latter case is that associated with the rollup of
the wake behind and aircraft in flight,
schematically described in Figure 4. (The
curvature dominated case is described in Figure
8 .)
Wakes as Characteristic Minimal Surfaces
A fundamental observation is that these spiral space curves are universally among
the "characteristic" solutions to hyperbolic systems of PDE's, of which the Euler equation
of two dimensional compressible flow,
Aφηη + 2Bφηξ + C φξξ = D ,
is a prime example. Following Landau [Landau, 1959, p.380], the characteristic surfaces
contain embedded curves which are given by solutions to ordinary differential equations,
dη/dξ = [Β ± ( B2 - AC)1/2 ]/C .
Recall that characteristic surfaces represent point sets upon which the solutions to
the PDE's are not unique; i.e., the characteristic surfaces include surfaces of tangential
discontinuities (on which the normal components of the vector field are continuous, but
the tangential components are not continuous), and shocks (on which the normal
components are discontinuous, but the tangential components are continuous). Shocks
permit the flow of mass across the discontinuity and permit pressure discontinuities,
tangential discontinuities do not. Shocks can be dissipative and involve entropy change;
tangential discontinuities are not dissipative. Shocks are generally stable, tangential
discontinuities are generally unstable. Landau states that tangential discontinuities are to
be ignored, for they will lead to turbulence. However, a certain subset of characteristic
surfaces, which always are locally unstable, may be globally stabilized. It is these special
characteristic surfaces, those associated with minimal surfaces, that are of that interest to
this article.
Define a position vector {u,v,w} to a point on a surface in terms of the
characteristic coordinates, with parametrization, σ, asbtn
Figure 4. Torsion dominated wake.
Characteristic Spirals along the
binormal. u = d η/dσ = A(σ) sin(Q( σ)),
v = d ξ/dσ = A(σ) cos(Q( σ)),
and w = F(u,v) = f( σ). (5)
If F(u,v) satisfies the equation
(1 + Fv2)Fuu - 2 FuFvFuv + (1+ F u2)Fvv = 0 (6)
then F(u,v) defines a minimal surface. There is one solution of (6) which is unique in that
it is the only harmonic minimal surface; it is the solution F(u,v) = tan -1(u/v) = Q( σ), or
the right helicoid [Osserman, 1983]. It is this special subset of characteristic surfaces that
is to be associated with the spiral solutions generated by Equations (3-4). Working
backwards, assert that the surface of characteristics is a minimal surface; then to generate
the ordinary differential equations (3-4), the minimal surface must be the right helicoid.
More precisely, the fundamental result is that of the infinite number of surfaces of
tangential discontinuities, there is a special subset whose sections yield spiral space curves
that are solutions of Equations (3-4). It turns out that this special subset can be related to
the unique harmonically generated minimal surface, the right helicoid.
As mentioned above, all such surfaces of tang ential discontinuities are locally
unstable [Landau, 1959 p.114], for they are associated with a hyperbolic domains. In fact
Landau claims that these surfaces are the precursors of turbulence. However, the special
subset of locally unstable surfaces of tangential discontinuities (in fact those which contain
the spiral characteristic lines described above, and which can be associated with minimal
surfaces) can, like soap films, exhibit domains of global stability [Barbossa, 1976]. In the
domain of global stability, the locally unstable surfaces create the persistent wake patterns
observed at moderate Reynolds numbers. When the Reynolds number exceeds a certain
value such that the global stability of the minimal surface is lost, then these special surfaces
of tangential discontinuity lose their global stability and the flow becomes turbulent. It is
the thesis of this article that it is these globally stabilized surfaces of tangential
discontinuities, which are not diffusive, but which are persist and observable, which are
those surfaces that generate the distorted but persistent mushroom patterns in the Von
Karman wake, or the Lanchester tip vortices in the wake of an aircraft in flight, or the
Kelvin-Helmholtz instability pattern of a shear layer.
Such patterns are reproduced in a qualitative way by appropriate adaptations of the
phase function Q(s). For example, in Figure 5a, the phase function used to generate the
Kelvin-Helmoltz instability of Figure 1a has been modified to include an amplitude factor
A(s) = .1 s 3/2. The exponent of the amplitude factor is related to the ratio of the relative
scales between successive spiral singularities. The physical interpretation of this
mathematical observation has not been resolved, but it points to an experimental property
that could be measured. If the phase function used to describe the Rayleigh-Taylorinstability of Figure 1b is modified to include a translation term proportional to cos(s),
then the spiral wave pattern of Figure 5b is produced. It is apparent that this
modification of the phase function is related to the staggered Von Karman wake.
Further modifications of the Von Karman wake of Figure 5b to include a decaying
minimal core radius produces the complex spiral pattern of Figure 6. If the phase factor
Figure 6. Von Karman Wake past a cylinder.
includes a linear and a quadratic term in arc length s, as well as the Kelvin-Helmholtz
Figure 7. Wake past a sharp edge
term, then the generated pattern imitates the observed flow pattern generated around a
sharp edge. See Figure 7.
Figure 5b . Von Karman wake Figure 5a Q(s) = 0.1(s) 3/2 / cos2(s)The fact that the spiral characteristic lines are contained in the only harmonic
minimal surface (generated by the function w = arctan Q(s)), and their obvious connection
with the Cornu Spiral, demonstrates that the production of the Kelvin-Helmholtz
instability pattern is a first order (odd) diffraction problem. It seems natural to suggest
that the Rayleigh-Taylor instability pattern is a second order (even) diffraction problem.
Being topological limit sets, these sets of tangential discontinuities can create extensions
of material boundaries into the bulk fluid. Typically, in the region of a "sharp" edge, a
material boundary will be extended by a topological limit set of tangential discontinuities
in the form of a scroll pattern, independent from any viscous, diffusive effect. During the
transient accelerations from rest to a state of constant velocity, the flow about sharp edges
will become hyperbolic, and (in section) lines of tangential discontinuity will grow into the
bulk fluid. These surfaces (in section, lines) of topological limit sets form extensions of
the material boundary, and will evolve into the spiral shapes described above. As
mentioned above, tangential discontinuities also can be generated at points where the
pressure thermodynamically permits a change of phase, or cavitation, to take place. For
example, in potential flow around a cylinder, the kinetic energy density at the top of the
cylinder is four times the kinetic energy density at infinity. At points along the streamline
in the neighborhood of the cylindrical surface, between the leading stagnation point and
the top of the cylinder, and where the kinetic energy density exceeds one third the pressure
energy density at infinity, Bernoulli's law would indicate a domain of "zero" or "negative"
pressure. Physically, it would be expected that a surface of tangential discontinuity
(separation) would be created when the Bernoulli pressure goes to zero.
The characteristic surface is a domain where more than one solution to a set of
PDE's can exist. In the hydrodynamic case, such a surface can be used to divide a domain
where the flow in the nearby neighborhood is represented by a non-harmonic potential
function (div V = ∇2φ ≠ 0, curl V = 0) from a nearby neighborhood where the flow is
represented by a non-harmonic stream function ( div V = 0, curl V = ∇2ψ ≠ 0). In the
first case, the fluid is compressible, but without vorticity, and in the second case the fluid
is incompressible, but with vorticity. A classical wake could be modeled as a thin layer
bounded on either side by such a surface of tangential discontinuity. On the surfaces of
boundary limit points, both functions are harmonic, but admit discontinuities. The interior
of the thin layer would be diffusive, and would exhibit a evolutionary thickening. The
tangential discontinuity would not permit mass flow between the two regions. In the
classic analysis of wakes, the viscous-vorticity point of view (div V = 0, curl V = ∇2ψ ≠
0) has dominated the literature. In this article, the alternate point of view based upon
compressibility (div V = ∇2φ ≠ 0, curl V = 0) is assumed to be dominant, for it gives not
only a plausible physical theory for describing the production of the tangential
discontinuity, but also leads the way to finding close form generators for the wake patterns
created by such tangential discontinuities.
Integrability and Topological DefectsThe two ext remes of wake patterns mentioned above indicate that the vector field
tangent to the spiral characteristics is (to first order) two dimensional. More precisely,
the observation is that these characteristic lines (not necessarily the flow field) are
integrable in the sense of Frobenious. Either the tangent field is integrable, or the
binormal field is integrable. It is of some interest to see how these two extremes for the
characteristics are related to hydrodynamic flows. To this end, a useful procedure is
obtained by making a transformation to a space-time coordinate system involving the
intrinsic properties of the particle paths (in this case streamlines). That is, consider the
velocity field, V, which may be composed of a unit tangent field, t, and a normalization
factor U.
V = U t.
Assume that the unit tangent field is not explicitly time dependent (such that the particle
paths are along streamlines), but the normalization factor might be time dependent, U =
U(x,y,z,t).
This representation fo r a vector field is very convenient, for if a unit tangent field
can be found that satisfies boundary conditions at infinity and has zero normal component
along a material boundary, U(x,y,z,t) = 0, then the zero set of the product function, V =
U(x,y,z,t) t(x,y,z), can be used to define no-slip boundary conditions. The unit tangent
field, t, is permitted to have its own singularities (zeros), which are different from the zero
sets of the normalization function, U. On the surface U(x,y,z,t) = 0, the velocity field
vanishes identically.
The equations for the divergence and curl of the vector field become
V = U t, V•V/2 = U²/2,
div V = gradU • t + U div t, curl V = gradU x t + U curl t,
V • curl V = U² t • curl t,
V x curl V = UgradU - U ( t • gradU) t + U² t x curl t,
The divergence in the nearby vicinity of the set U(x,y,z,t) = 0 is related to gradU • t, and
the curl is equal to gradU x t. Note that curl V always resides in the surface U(x,y,z,t) =
0, (i.e., the vorticity is in the tangent plane of the surface whose normal is given by grad
U). The divergence, div V, is not zero unless t resides in the surface represented by
equation, U(x,y,z,t) = 0.
The non-exact differential, ds, of arc length is defined such that
dR = t ds = V dt .
Assuming locally a euclidean metric, it follows that ds = U dt. The differential dt is exact;
the differential ds is not. Closed loop integrals for dt vanish, but closed loop integrals for
ds do not. From the Frenet theory,dt/ds = κ n = ( ∂t/∂t + ∂t/∂xυ dxυ/dt )(1/U) = (1/U) ∂t/∂t + curl t x t . (7)
where κ is the curvature, n is the normal vector, orthogonal to the unit tangent field. Curl
t is the Darboux vector ,
curl t = τ t + κ b, (8)
given in terms of the binormal, b, of the Frenet frame, and the torsion, τ, of the space
curve, as well as the curvature, κ, and the unit tangent field, t.
The Frenet theory becomes useful in a hydrodynamic representation when it is
realized the convective derivative may be expressed in terms of the Frenet derivative:
d V/ds = (d V/dt )dt/ds
= ( ∂V/∂t + ∂V/∂xν dxν/dt ) (1/U)
= ( ∂V/∂t + V • grad V ) (1/U)
= ( ∂V/∂t + grad(U²/2)- V x curl V) (1/U)
= (dU/ds) t + U curl t x t + ∂t/∂t
= (dU/ds) t + U κ n + ∂t/∂t .
The Euler equations of hydrodynamics therefore have an intrinsic representation as
(dU/ds) t + U d t/ds = (dU/ds) t + Uκ n + ∂t/∂t = - grad P/ ( ρU) + f/(ρU), ( 9)
or,
[ (ρU)dU/ds + t • grad P ] t + [(ρU²κ + n • gradP] n = f - [b • gradP] b. (10)
If the external forces were identically zero, it would follow that the square bracket factors
must vanish everywhere, and the curvature or bending becomes
κ = - n • gradP / ρU² . (11)
Pressure gradients in the direction of the binormal cannot be sustained without a
component of external force in he same direction. In other words, the Euler equations
(with no external forces) constrain the flow such that the torsion is null and the curvature
is given by (11)
Equation (10) is predicated upon the assumption that the velocity vector field can
be represented by a single parameter group of evolution, such that d R - Vdt = 0.
Suppose, however that this is only true on an "equilibrium singular" surface, in the sense
that off this surface, d R - Vdt = e , where e is a small vector of fluctuations. The external
forces, f, then could be attributed to fluctuations. In any case rewrite (10) as:
A(s) t + B(s) n = (A² + B²) ½ dP/ds = f - C(s) b. (12)The zero set of the norm of this vector defines the eq uilibrium singular surface.
Construct the derivative of this constraint equation with respect to s to yield,
(dA/ds - κ B) t + (dB/ds + κ A - τ C) n + (dC/ds + τΒ ) b = df/ds. (13)
First hypothecate the vector f in the direction of the binormal is vanishingly small. Then,
subject to the assumptions imposed above, C = 0, and dC/ds = 0, and the torsion, τ, must
vanish:
τ = t • curl t = V • curl V/U2 = 0. (14)
This condition is equivalent to the Frobenius integrability condition. Note that the
integrability condition is equivalent to the vanishing of the helicity density, V • curl V = 0,
for the flow. The flow field is of Pfaff dimension 2 or less. The coefficients A and B are
harmonic in the sense that,
dA/ds - κ B) = 0,
(dB/ds + κ A) = 0 . (15)
In fact, divide (15) by (A² + B²) ½ to form a
unit tangent vector with the same format as
equations (3-4); spiral solutions in the
direction of the tangent field are to be
expected for the curvature dominated
characteristics of such a surface. Figure 8 is
a schematic of the curvature dominated
case.
Suppose that f is not identically zero, but d f/ds is approximated by zero. Then the
"power" equation given by (13) can be examined in the limit of κ ≈ 0. This would be the
case of torsion dominated evolution as discussed above. For torsion dominated flows, the
pressure gradient is either perpendicular to the normal, or zero. The spiral characteristic
lines are in the direction of the binormal. A schematic representation of the torsion
dominated case was presented in Figure 4. Note that the velocity field is NOT integrable
in the Frobenius sense, for a torsion dominated wake. The flow field is of Pfaff dimension
3 or higher.
The topological defects (characteristic limit sets) described above are associated
with flow fields of Pfaff dimension 2 (vector fields that satisfy the Frobenius integrability
theorem) and Pfaff dimension 3. Other topological defects for flow fields of Pfaff
dimension higher than 2 yield more complicated wakes. For example, in a rotating frame
of reference it is possible to find closed form solutions to the Navier-Stokes equations
which are of Pfaff dimension 3. Such vector fields have finite torsion, τ, and exhibit re-
entrant flow patterns (contained within a swirling bubble) above a critical flow parameter. t
n
b
Figure 8. Curvature Dominated Wake
Characteristic spirals along the
tangent vectorThe topological defect associated with the transition to a re-entrant flow is not a
dislocation hole, but can be put into correspondence with when the surface of zero torsion
changes from a hyperboloid of one sheet into a hyperboloid of two sheets. The surface of
zero torsion becomes disconnected . The analytic results, which mimic the "vortex"
bursting problem, are presented in [Kiehn, 1991]. Note that in this problem the sheet of
zero torsion is a point set (a surface, not flat) upon which the Frobenius integrability
conditions are true.
Lagrangian Wakes and Dislocation Defects
When a harmonic potential flow or a harmonic stream function is u sed to describe
a fluid flow, the symmetry of the Eulerian streamlines created about obstacles gives no
indication of a wake. However, if it is assumed possible to synchronize globally and
transversely a set of "points" in a fluid, then the kinematic evolution of such a set will
give an indication of a wake. In this section these ideas will be examined. Potential flows
(irrotational vector fields) are always sychronizable over the domain of uniqueness.
Stream function flows (as solenoidal vector fields constructed from two independent
variables at most, in 3 dimensions) are also globally synchronizable over the domain of
uniqueness. However, the combination of a potential flow and a solenoidal flow in 3D is
not necessarily integrable
The Frobenius integrability theorem can be used to determine if a given vector field
admits globally a set of points that can be synchronized. That is, if a set of initial points
lies on a smooth surface everywhere transversal to the flow field, then after an interval of
evolution (time increment) the initial set of points can still be connected by a smooth
surface in the final state if and only if the vector field satisfies the Frobenius condition.
This fact implies that the Pfaff dimension of the vector field never exceeds 2 over the
domain of interest, and that there exists a global submersion of this domain to a space of
two dimensions! The smooth surface of synchronous points may be grossly distorted, and
this distortion is indicative of different topological features of the flow and its wake.
For example, consider the two dimensional potential flow about a cylindrical rod.
Start a set of synchronized particles at very far distances upstream from the obstacle.
Assert that at the initial position the synchronized set of points resides on a surface that is
orthogonal to the streamlines, much like a plane wave front in optics. Now forward
integrate the kinematic equations for these points using the velocity potential for the
streamline flow. As the "wave front" advances to the obstacle, the front retains its shape
as a plane which is transversal and almost orthogonal to the streamlines. However as the
"wave front" reaches the obstacle, the forward stagnation point greatly deforms the shape
of the "synchronized" set of points that make up the "wave front". After the "wave front"
of synchronous points has passed over the obstacle, a noticeable near field wake is
observed downstream, a wake pattern that is not at all apparent from the format of the
streamlines, which do form a pattern which is symmetric with respect to time inversion.
The Eulerian description of the flow pattern is symmetric, but the Lagrangian description,
with a choice of initial conditions, is not.At very far distances to the rear of the
obstacle, the "wave front" for synchronous
points again looks as if were a plane surface
almost perpendicular to the flow lines.
Different times correspond to uniformly spaced
"almost plane waves". Only very close
inspection (on scales smaller that the obstacle
size) reveals that in the neighborhood of the
positive x-axis the set of equal time surfaces
has "sharp" bend on the downstream side of the
obstacle. The equal time surface does not cross
the x-axis downstream from the obstacle, but it
does cross the x axis on the upstream side. See
Figure 9a. On measurement scales much
larger than the radius of the cylindrical
obstruction, the wave fronts form a lattice with
only a "local" defect in the downstream side of
the obstacle. This defect (on these very large
scales) probably would be ignored
observationally.
On the other hand, if the potential flow
that generated the wake of Figure 9a is
modified to include some circulation, but no
vorticity, then the "wave front" pattern exhibits
an apparent "global" defect as it passes over the
obstacle. On scales much larger than the radius of the cylindrical obstacle, the "plane
waves" that approximate the synchronous particles have a "dislocation" defect in the
downstream wake. This topological defect, associated with the circulation of the flow
about the obstacle, has nothing to do with vorticity! It would not be ignored
observationally. See Figure 9b. An extra line, or defect, appears to be inserted into the
large scale pattern. The circulation integral around the contour that is a section of the
closed surface composed of two equal time faces is zero, but the contribution from the
loop around the circular cylinder is balanced by the contribution along the dislocation
defect.
This work was supported in part by the ISSO at the University of Houston.
Figure 9a Lagrangian wake,
no circulation
d
Figure 9b. Lagrangian wake
with circulation. Note the
zebra stripe dislocation defect.REFERENCES
J. L. Barbossa and M. do Carmo (1976), Amer. Journ. Math. 98, 515
G. K. Batchelor (1967), Fluid Dynamics , (Cambridge University Press) p.354.
H. K. Browand (1986), Physica D 118 173
R. E. Cayflish (1991), in "Vortex Dynamics and Vortex Methods", C.R. Anderson and
C. Greengard, editors, (Am. Math. Soc., Providence, RI) p. 67
L.D. Landau & E.M. Lifschitz (1959), Fluid Mechanics , Addison-Wesley, Reading Mass
J. Hadamard (1952) Lectures on Cauch's Problem in Linear Partial Differential
Equations (Dover, N.Y.) p.21
H.Kaden (1931), Ing. Arch. 2, p.140
T. Kambe (1989), Physica D 37 p.403
R.M.Kiehn (1990), "Topological Torsion, Pfaff Dimension and Coherent Structures" in
Topological Fluid Mechanics , H. K. Moffatt and A. Tsinober, editors,
(Cambridge University Press), p. 225.
R. M. Kiehn (1991), "Compact Dissipative Flow Structures with Topological Coherence
Embedded in Eulerian Environ ments" in The Generation of Large Scale
Structures in Continuous Media , (Singapore World Press).
R. M. Kiehn (1992), "Topological Defects, Coherent Structures, and Turbulence in
Terms of Cartan's theory of Differential Topology" in Developments in
Theoretical and Applied Mechanics , SECTAM XVI Conference Proceedings,
B.N.Antar,R. Engels, A.A.Prinaris, T.H.Molden. editors (University of Tennessee
Space Institute, TN)
R. Krasny (1991), in Vortex Dynamics and Vortex Methods , C.R. Anderson and
C. Greengard, editors, (Am. Math. Soc., Providence, RI) p. 385
R. Osserman (1986) A Survey of Minimal Surfaces (Dover, N.Y.) p.18
S. D. I. Pullin (1989), in Mathematical Aspects of Vortex Dynamics, R. E. Cayflisch,
editors (SIAM)
N. Rott (1956), J. Fluid Mech. 1 p.111
D. Struik (1961), Differential Geometry , (Addison Wesley, Reading, Mass) |
1The Principle of Synergy and Isomorphic Units, a revised
version
Edgar Paternina
Electrical engineer. With experience in the area of real time control centers with
State Estimator included, an algorithm that validates the complex equations of a
power system.
Contact: epaterni@epm.net.co
__________________________________________________________________
Abstract: A solution to the part and whole problem is presented in this
paper by using a complex mathematical representation that permits to define
the Holon concept as a unit that remains itself in spite of complex operationssuch as integration and derivation. This can be done because of theremarkable isomorphic property of Euler Relation. We can then define adomain independent of the observer and the object where the object isembedded. We will then be able to have a Quantum Mechanics solution
without the "observer drawback", as Karl R. Popper tried to find all his life
but from the philosophical point of view and which was Einstein mainconcern about QM. A unit that has always similar or identical structure orform, despite even complex operations such as integration and derivation, isthe ideal unit for the new sciences of complexity or just the systems scienceswhere structure or form, wholeness, organization, and complexity are mainrequirements. A table for validating the results obtained is presented in case
of the pendulum formula.
Keywords: whole, part, Holon, bus, energy, synergy, complex numbers,
isomorphic properties
__________________________________________________________________
1. Introduction: Three basic attributes of reality
In his General System Theory , Ludwig von Bertalanffy wrote:
Reality, in the modern conception, appears as a tremendous hierarchical order of
organized entities, leading, in a superposition of many levels, from the physical
and chemical to biological and sociological systems. Unity of Science is granted,not by a utopian reduction of all sciences to physics and chemistry, but by thestructural uniformity of the different levels of reality.2That structural uniformity or isomorphism of the different levels of reality is
the main concern of this paper, and its main aim will be to present a new way of"seeing" reality by means of some isomorphic units, or co-variant units , so to
speak, units in which the form is one important attribute as well. A basic recurrentdesign or pattern that can be used to interpret and explain those problems wheredynamic interactions or an organized complexity appear.
The problem of form appeared in classical physics, but precisely in those
fields where the field concept was unavoidable. The magnetic field problem,
whose existence can even be felt by putting two permanent magnets near by, isreally one of those problems of nature that after all were hiding a great mystery.With the magnet we can not only present theoretical examples of those three basicfundamental attributes that are the basic to an isomorphic unit, but the magneticfield to be well represented, from the mathematical point of view, we must also use
a complex number mathematical symbolism. Those three basic attributes are:
wholeness, oneness and openness.
The wholeness attribute can be seen easily in the case of a magnet, where
each magnet split from another magnet is precisely a whole new magnet. This new
"part" that came from an old whole is a whole too . To obtain a whole from another
whole is like to obtain a son from a father, or an object from the instantiation of a
class and in this same sense Ken Wilber[8] wrote To be a part of a larger whole
means that the whole supplies a principle(or some sort of glue) not found in theisolated parts alone, and this principle allows the parts to join, to link together, tohave something in common, to be connected, in ways that they simply could not beon their own...When it is said that "the whole is greater than the sum of its parts,"the "greater" means "hierarchy"...This is why "hierarchy" and "wholeness" areoften uttered in the same sentence
Associated with this wholeness attribute is that binary or dual aspect of
reality, where we have always two "opposites" or more appropriately,complementary or polar entities within a comprehensive whole, or just a unit thattranscends duality, the oneness attribute, that can be found physically in a magnet.The within and the without [4 ] some sort of dynamic structure embedded in a unitthat cannot be split in its two components. Structures that perform well in a
changing environment must be capable to reflect the without, they must have, as it
were, a storing capacity to reflect that without.
The magnet has also another remarkable attribute associated with form, with
the environment or interface or some sort of a medium to separate an internalmilieu from an external environment[13], as it were, a field concept, or theopenness attribute. At this point is important to recall that these three attributescannot be considered on their own. They are linked together by some sort of glue.
This is the main characteristic of this new way of "seeing" reality in which we
must always have in mind that gluing principle, we will name the Principle of3Synergy. At this point of reasoning we are taken to think in open systems, as those
systems capable of exchanging with the environment
2. Thirdness as an alternative way of codification
For representing mathematically such kind of problems, it is necessary to use
a language that permits to define a unit embedded in a comprehensive whole,
environment or dynamic structure, and which can include also a radical duality orpolarity, which can be generated, precisely because that inherent coincidentiaoppositorum or that tension, that manifests as a field that makes by definition thatunit an open system: the self-preservation attribute .
The main aim of this semiotic codification[16] is to integrate in a
comprehensive whole the radical duality of the universe that can be codified as
- (the original, (the potential/the actual))
- (Being or reality, (mind or consciousness/form)- (Form, (time/space))
- (wholeness, (Oneness/Openness))
where:
- with openness we associate the field concept,
- with oneness the dual nature properly speaking not in the monadic sense ,
- and with wholeness the nondual or qualitative nature of reality, in the thirdness
sense.
It is important to notice that we have two fundamental entities:
- one that stands on its own and
- another one dual embedded in another parenthesis and separated by a symbol
“/” of the or type in the sense that the one excludes the other in its manifestation
The symbol “,” is for differentiating two different and fundamental orders of
reality, and the symbol ( ), is for integrating them both in a comprehensivewhole. It is the relation between these two components that permits us toredefine the uncertainty principle in a new context as we will see later.
A Holon has embedded a within and a without, as it were, an analytical anda synthetic capacity, or a partness and a wholeness attribute.4 The openness attribute is then related with a capacity to generate a field that
is concomitant with those entities we can named holonic -to use a term coined byKen Wilber- such as magnetic entities, electrons, linguistic signs, cells, life, mindand beings in general.
3. Complex number and thirdness
Complex numbers have the capacity not only to represent that binary or
duality aspect of reality or the chance to have two polarities included in one unit,but also the nondual or wholeness attribute, we have related with that capacity togenerate a new whole, the self-replication attribute .
Complex numbers were born when trying to solve the simple algebraicequation
x² + 1 = 0
x² = -1
x = Sqr(-1) = J
where as a solution we have the square root of a negative one or the radical unit J .
From the point of view of "real" number perspective it does not exist a
solution for this simple problem and as so it was necessary to make a paradigmextension or paradigm shift, and to define a new type of more general numbers tosolve the problem. The solution J was named "imaginary" by Descartes for the first
time and since then, complex numbers remained as some sort of a strangemathematical tool. It was Leonard Euler in 1745, the one, that finally found amathematical symbol for representing that new entity that included both kinds of
numbers:
- those named real and which we will term nondual for reasons we will see later,
and
- those named "imaginary", we will name dual on the other hand for apparent
reasons too.Why was it written by David Berlinski in “Tour of the Calculus”(1196,
Heinemann) that “the area in thought that the calculus made possible is coming to
an end” and that “it is a style that has shaped the physical but not the biologicalscience”? A reason is that area in thought of classical mathematics was definitelyrestricted to the without of things, to the surface, to the reduced part. And as so itcould only be codified from a monadic world perspective and in a way that thewithin necessarily had to be reduced to the without of things[16 ], the quantitativeto the qualitative. In such a monadic world we have:
- (within/without)... reduced to... (without)
- (qualitative/quantitative)... reduced to... (quantitative)- (whole/part)... reduced to... (part)5- (potential/actual)... reduced to... (actual)
Being the main advantage of this mathematical methodology or language
that with it, we can work with close systems, reversible, universal, deterministicand atomistic [16 ]
Problems of growth has been associated from the point of view of a
mathematical representation with the number epsilon or Euler number since a long
time ago. And it was by studying infinite series, that Euler found that entity, that
not only could represent those two kinds of numbers, nondual and dual, but also itincluded those cyclical waveforms, sine and cosine, that occur so frequently innature wherever we have cyclical phenomena. But the most important, the mostcogent argument to use this kind of new numbers, that have been used by ElectricalEngineering since Oliver Heaviside and Steinmetz introduced them at the end ofthe 19th century to solve alternating electrical current circuits, is precisely, its
inherent isomorphic property, that permits them to make, as it were, co-variant
representations, or to make simpler complex operations.
Evidently such a useful mathematical entity, was adequate to represent
dynamic realities, and it was used for the first time for representingelectromagnetic fields at the end of the 19th century, and after that, vectors wereborn in physics, but then they are taught, in general, without making any references
to this complex number origin. Up to that moment complex numbers had not been
used in practical cases, and as so it justifies why the complex plane was delayed100 years from its real birth at the end of the 18th century.
A unit that has always similar or identical structure or form, despite even
complex operations such as integration and derivation, is the ideal unit for the newsciences of complexity or just the systems sciences where structure or form,wholeness, organization, and complexity are main requirements. Another
important point is that it can also be used to define then a Basic Unit System
concept in which uncertainty is included, and as so open systems.
Classical physics had as it main aim to resolve natural phenomena into a
play of elementary units, as it were, to resolve those phenomena into their parts,isolated parts, I mean, so the concept of particle was always the starting point ofthe whole framework. But that part or particle needed to be considered as an
isolated entity, that is, as a closed system, in which there were no interactions at
all with the environment[2]. This ideal model to represent reality was so restrictivethat it definitely failed, the way we all know.
An adequate framework for representing reality, the whole reality, must be
complex, in the sense, that it must not only include, the dual-logical-nature ofreality, but also it must include, that another aspect related with form, wholenessand oneness.
The form, the structure must be co-variant[5], that is, it must remain the
same in spite of a progressive modification of that same structure. In this wayadaptation or changeness as a fundamental process can give us persistent properties6for that structure or just despite complex operations such as integration and
derivation done upon that structure. There is another more important aspect torecall, and it is the need not to reduce uncertainty, in certain cases, to be able justto manage it. And here we come across with the fundamental problem of open andclosed systems. When we have an open system, in general, that uncertainty is anessential part of the problem, of its openness attribute, and in this sense that
uncertainty cannot be reduced unless we close the system, so that we determine its
state completely defining then ideal objects of study, that can fail in real cases. Theinteractions of a closed system are reduced almost to zero and then the systembecomes a static system and not precisely a steady state system. Open systems andthat second law are in some sense "incompatible" if we establish a hierarchicalframework in which, that second law is just a special case of the behavior of anopen system.
4. Euler Relation and Its Isomorphic Properties
Up to this moment we have been seeing the emergence of a new concept of
unit that includes in its mathematical representation the dual and nondual nature of
reality, i.e., a relationship between the part and the whole or just a unit that is awhole and a part at the same time or a Holon as Ken Wilber named it [8],represented succinctly as:
(whole, part)
What we aim at this moment is to show that Euler Relation, is precisely the
mathematical symbol necessary to make an adequate mathematical representationof the Holon concept by taking J as the symbol for differentiation. The whole/partentity we have named a Basic Unit System can be codified then as:
e J( Ø ) = Cos (Ø) + J Sin (Ø)
By assigning values to Ø, from Ø = 0 to Ø = 90 degrees, we obtain both a
horizontal and a vertical line, respectively, i.e., the complex plane, which can be"seen" as a fifth sphere of reality or a totality that can be used to represent or
contain the four dimensional space-time continuum.
It can also be seen as a mathematical representation of the domain of Form,
in the same line of that domain imagined by the perennial philosophy with Plato.Karl R. Popper in the intent to transcend dualism envisioned this domain as a"third world", independent from mind or the subject and the object. Karl R.Popper main concern in Objective Knowledge [10]was precisely to avoid what he
called an essentialist explanation by introducing this "third world", as a world7independent both from the object and the subject. Only through a mathematical
representation we can avoid any semantic pitfall. In the following figure we cansee two systems S and S´ in interrelation, and apart each other an angle, being the
whole domain of representation the complex plane.
In general we can then envision Ø acquiring all values, from zero to 360 degrees,
or just, some sort of a clock pointer, or a vector rotating about an origin at a given
frequency, where Ø = wt and w = 2¶f
8
A vector rotating about a point, is what we term the centerness attribute of Euler
Relation, that has to do with its natural cyclical behavior. We could say nature
loves the cyclical waveform behavior, as we find it everywhere, from the motionsof the stars and the planets, to the tides on earth, our heartbeats and ourpsychological states and even that motion that has always exerted such afascination upon human mind, I mean, the pendulum motion. In Euler relation wehave two types of cyclical waves separated by the radical J, and what this means is
that those two type of cyclical waves are very different in nature, and this is what
we are going to show in the following.
In Euler Relation we then have a:Dual, symmetric or binary component that can be represented with the sine
function:
(-) = Sin (- θ) (+) = Sin ( θ)
_________________________________o________________________________
and a Nondual one component that can be represented with the cosine function:
Cos(Ø ) = Cos(-Ø)
We must recall that the dual component or that component associated with
the sine function changes with changing the sign of the angle Ø, and that the
nondual component or cosine function remains the same with changing that signangle, so we have in this unit those two requisites we pointed out at the beginning
9of this paper necessary for representing reality. This mathematical process of
changing sign is associated with changing the rotation sense, counterclockwise orclockwise as we will see later, so it has to do with a very general sense of rotationof the whole structure.
Historically Euler Relation was associated with the problem of the infinite
series:
x
e = 1 + x + x² / 2! + x³ / 3! +...
where by replacing
x = J*Ø
we obtain finally Euler Relation by separating those terms that are affected by J
from the others, obtaining the nondual or cosine expression and the dual affected
by J, or sine expression. This infinite series is Euler Relation, where, e = 2.71828. The main point tonotice at this very moment, is the cosine and sine nature of the two componentsseparated by J.
Cosine or nondual nature of Euler Relation
From Euler relation we obtain the cosine function represented by
Cos (wt) = ( e J( wt ))/ 2+ ( e - J( wt )) / 2
The cosine function is expressed as the sum of two vectors rotating in
opposite directions[14], one of them in counterclockwise or positive direction at an
angular velocity w, and the second one in the clockwise or negative direction at anequal angular velocity w. As the vector rotate the two dual components cancel eachother, and as so the sum is a purely real or nondual vector, nondual, because wecannot obtain the opposite by changing the sign of the angle. The axis of the cosinefunction is in fact that axis not affected by J, the so called real axis, we have named
the nondual axis instead. It’s important to recall the radical duality, expressed in
that inherent polarity or tension of those two vectors rotating in opposite directionsbut in a comprehensive whole, i.e., the complex plane. 10
Sine or dual nature of Euler Relation
From that same Euler Relation we obtain the sine function represented by
Sin (wt) = [( e - J( wt )) / 2- ( e J( wt ))/ 2]* J
The sine function is expressed as the sum of two vectors rotating in opposite
directions, one of them in counterclockwise or positive direction at an angular
velocity w, and the second one in the clockwise or negative direction at an equalangular velocity w. As the vector rotate the two nondual components cancel eachother, and as so the sum is a purely, as it were, "imaginary" or a dual vector. Dualbecause by just changing the sign of the angle we can obtain the opposite.
The axis of the sine function is in fact the J axis(see how the sine function is
affected by J), the so-called-imaginary-axis, or the symmetry axis, where
symmetry is defined as similarity of forms or arrangement on either side of adividing line, so on one side we have a positive magnitude and on the other wehave a negative sign for that magnitude.
11
From this point of view it is not perfectly general to say that the sinusoidal
wave, as an adjective, is the same for both sine and cosine functions even though
we can obtain the one from the other by the addition of a phase angle of -90°; that
addition is not at all a trivial one though.
To avoid any ambiguity we must differentiate clearly the asymmetrical or
nondual nature of the one, and the symmetrical or dual nature of the other. Reality
is in fact composed of two components, but in the physical domain, i.e., the "imaginary" domain, the dual nature is easily grasped and as so we can really say
nature "loves" the sinusoid, and normally it hides as a mystery, its nondual nature.
The nondual nature must always be discovered. It is some sort of fixed or
nonchanging component to which the change of the system as a whole can be
referred, but as a point of reference it must be discovered, it must be chosen orrequires a decision. In this sense we can say that the foundation of all reality is anultimate frame of reference in which the nondual, the nonchanging is at thebackground[17] or up from the point of view of hierarchy and this implies then an
"inclusive" attribute that is essential to have always in mind when dealing with
reality. That frame of reference, we have named the domain of Form, permits us todefine a domain independent of the observer and the object, as within it the object
12is embedded. We will then be able to have a Quantum Mechanics solution without
the "observer drawback", as Karl R. Popper tried to find all his life but from the
philosophical point of view and which was Einstein main concern about QM too.
5. The resonant effect: "merry-go-round" effect
Phase angle and magnitude are the two main state variables of that unit
obtained with Euler relation and that has been named a phasor in EE. The Basic
Unit System concept will be a rotating entity in the complex plane at a given
frequency , that can be codified as:
(Energy, (angle/magnitude))In most practical problems all vectors, in the complex plane, must be riding
with the same frequency, so that a sort of "merry-go-round" effect[14] exists.Vectors are seen as stationary with respect to each other then, so the ordinary rulesof vector geometry can be used to manipulate them as with that "merry-go-round"
effect we obtain some sort of static framework within dynamism. This is a real
impossibility with the monadic “real” numbers mathematical representation.
We have then the chance to have the phenomenon of resonance. We are
acquainted with such a phenomenon specially in the production of musical soundsand certain type of vibrations but the important point to recall is that through
resonance we can explain those cases where small changes can often produce
large effects , [13]
We must recall the fact that the phase or the angle sign is associated with thepositive or -negative sense of the vector rotation:- In the cosine case we can interchange both rotation vectors, by changing their
sign and nothing changes, the form remains the same, they cannot be segregatedjust as in a magnet where we cannot differentiate the two polar components asthey are seen always as one, as it were, the oneness property.
- In the sine case that interchange, or changing of signs of the angle, affects the
final-resulting vector with respect to its position in relation with the nondual axis. Resuming we have with ER a fundamental structure with two componentsseparated by the radical J :
- one nondual in which wholeness is an essential attribute and which brings us to
mind that self-awareness capacity that permits us as humans to think about our
own thinking process, that permits us reflectivity or to know that we know, a
new way of "seeing"; a way to be conscious of our being that makes us differentfrom the animal world. This nondual component is precisely that one thatpermits us to represent the within of things[4]. With this within attribute we canenvision a storing capacity of energy as that one we find in magnetic fields oran information storing capacity in general. This within attribute as a storing
information capacity at its highest manifestation is very different from
consciousness , being more related with a fifth sphere of reality or just in plain
mathematical words with the complex plane. That pretension to reduce13everything to consciousness is not our pretension anymore. A greater within
means a greater complexity or a greater without, that can or cannot benecessarily a greater consciousness.
- one dual in which we can have two parts separated or clearly differentiated.
Parts separated, is precisely that condition necessary for the application ofanalytical procedures, being the second one, the chance to linearize.
We will see in what follows that ER most important isomorphic property is
precisely to reduce complexity to a minus one degree making linear the
representation of non-linear operators without reducing them. These two basiccomponents are united in that mathematical symbolism called Euler Relation andat the same time separated in a radical way by J, repeating again at a higher levelof representation the same basic structure we have found within that same relation,an inclusiveness attribute. We have then a fundamental and basic minimum
structural complexity represented by Euler Relation.
Is this basic minimum structural complexity the one necessary to represent
reality at its most profound structure? Does this basic minimum structuralcomplexity give us those isomorphic properties needed for representing anddeducing the most fundamental laws of physics and reality? The answer to thesetwo questions will be yes, and this is what this proposal is all about.
6. Complex Algebra and Isomorphic Properties
Vectors are ideal mathematical entities when relationships are important and
as so we have vector sums and differences, but also multiplication, division,
derivative and integrals and every one of these operations can be represented in the
complex plane as some sort of complex metrics and even the resulting geometricalfigures are simpler than those obtained in normal geometry[14]. The powerfuladvantage of complex number is seen in computations, or when using complex-algebra, where the isomorphic property of Euler Relation manifests all its co-variant power.
The main restriction we must pose from the very beginning is that of the
same frequency. All rotating vectors considered must have the same angularvelocity or frequency, so that we can have the "merry-go-round" effect[14]. Thefrequency in this sense is that variable that can in fact produce large effects in asystem that was previously chaotic before acquiring it[13]. The order of thissystem depends on that acquired same frequency for all entities conforming thesystem. This "restriction" is in fact the central point for the system acquiring a
higher ordered state [13] . That frequency is associated with one of the essential
state variables of the system, the angle, being the other the magnitude. The state isin fact that one in which having a "merry-go-round" effect gives us as a result anorganized complexity.14Rewriting ER again
e J( Ø ) = Cos (Ø) + J Sin (Ø)
we can note that it is a unit vector standing at an angle Ø from the nondual or real
or main axis
1 = Sqr [ Cos ²(Ø) + Sin ²(Ø) ]
that can be represented by the following figure
We can have then in general, entities represented as
A = Abs (A)* e J( Ø )
that can be represented in a rectangular form as
15or
A = a-nondual + J * a-dual
whereA-nondual = Abs (A)*Cos(Ø)
And
A-dual = Abs (A)*Sin(Ø)
With these two expressions the complex algebra was established having in mind
that we must not mix oranges and apples. Must we abandon the reduction tendency
with this complex representation?. Must we make a radical distinction between the
two basic components that are used to represent reality?.
The nondual components must be used with nonduals, and the dual components
with duals. In this way all laws of normal algebra and arithmetic are preserved.
This radical separation preserves homogeneity on the one hand and heterogeneity
on the other.
If we call qualitative those aspects related with the phase angle it is shown
they are summed instead of multiplied within complex algebra. This capacity ofcomplex numbers to reduce a nonlinear operation to a linear one, is in fact the onethat makes them such a powerful simplifier tool, and adequate to represent the
complex nature of reality. Linearization does not affect at all the quantitative part,
just the qualitative one. This gives us a new capacity to think the whole correctly
16and the chance to introduce boldly in our intellectual frameworks new categories.
A new world outlook is obtained that includes in itself an explanation anddevelopment of new things and even solves once and for all the paradoxes of waveand particle as we will see when deducing the complex Schrödinger WaveEquation.
Differentiation and integration. We have already pointed out that remarkable
property of ER we have qualified as isomorphic and that has to with the fact that
the integration and differentiation of ER, are themselves ER of the same frequency.
By the rules of elementary Calculus, the derivative of an ER with a
constant magnitude has the same magnitude as the original vector multiplied by thefrequency w, but the angle has been advanced by 90°.
And the integration is the dual operation as that of differentiation and as so the
integration of an ER with constant magnitude, has the same magnitude as theoriginal vector divided by the frequency w, the angle has been retarded by 90°, orjust the expression multiplied by -J , as were, an opposite sense of rotation .
F -1 (INTEGRATION)
F ' (DIFFERENTIATION)
17This way of reasoning is exactly the application of what can be named the
duality principle, which is a way of applying a binary logic or symmetry to obtaina dual reality from the other components.
The complex vector algebra simplifies mathematical operations by one
degree or it reduces complexity by one degree. With it a real isomorphic tool isobtained as even complex operations such as differentiation and integration are
reduced to simple algebraic operators.
This reduction of complexity by minus one degree permits us to comply with
those two conditions of analytical procedures:- linearization and
- to have weak interaction between the parts in those cases in which the generalopen system is closed, which was what Ludwig Von Bertalanffy pointed out as thecentral and methodological problem of systems theory[12].
Euler Relation is the ideal isomorphic unit, as the sums, differences,
integrals, and derivatives of Euler Relation functions of a given frequency arethemselves Euler Relation functions of the same frequency, they do not changetheir form with these operations, they remain invariant or just co-variant. No other
mathematical function is preserved in this fashion , as so it is the ideal one for
filling that requirement Einstein put in his The foundation of the General Theory of
Relativity , [5], when he wrote "The general laws of nature are to be expressed by
equations which hold good for all systems of co-ordinates, that is, are co-variant
with respect to any substitutions whatever(generally co-variant)".
7. A complex Geometry and Synergy
Our next step is to define a differential complex geometry in which we have
a differential of reality defined as a Basic Unit System or a Holon such that
DS = Abs (DS)* e J( Ø )
The late Abraham Maslow was the one who coined the term "synergy" , an
obscure term from anthropology, but he used it for the first time in business todescribe how wealth can be created from cooperation. Creation of wealth,emergence of new things and structures that modify themselves to give betterperformance are main issues of the so-called-sciences-of-complexity or systemssciences[13]. Structure and environment or thirdness(Oneness)[16] and openness they
both claim for a whole that is greater than the sum of the parts, for a basic and
fundamental framework in which three fundamental and interdependent entities areput in mutual interaction. They pose the need to have as starting point not only aminimum structure, but also a minimum system of elements in interaction.18In the physical domain synergy can be found whenever we have a real
transformation of energy to another useful way or level which presupposesinterchange or transformation of energy between systems. This "useful way" is infact the real source of new applications and in this sense synergy is there, wherewe have emergence of new realities or just open systems that have exchanging-energy-capacity with the environment.
That whole greater than the sum of its parts has nothing to do with
metaphysic, as that "greater" comes from the very nature of an open system.
Electrical energy, or the alternating current we use at home, is in fact the
result of the application of the Principle of Synergy where energy is transformedfrom a primary source, hydraulic to electrical energy, by moving a threefoldmagnetic structure, which gives at the output the AC energy we can utilize in manyways.
If we consider three-double-complex-vectors or just Buses, one in space and
one in time, codified as:
J( θθθθ) J(wt)
A = Abs(A)* e + e
J( θθθθ+120) J(wt)
A = Abs(A)* e + e
J( θθθθ+240) J(wt)
A = Abs(A)* e + e
where they correspond to a three physical disposition displaced 120° from one
another, on the one hand, and on the other to a dynamic entity or field. It is shownthat the physical disposition cancels or the cause-effect relationship is subtle. Itdepends on the interchanging of energy or information with the environmentthrough field concept.
At last we have
A' =Abs ( A')* e J( wt )
a rotating entity or Bus at a given frequency.
A given frequency is precisely the one attribute that makes it possible, the
"merry-go-around" effect, i.e., that new order or reality that comes from theapplication of the Principle of Synergy. If the frequencies are not the same for thethree independent entities we won’t have the desired "merry-go-around" effect. We19have some sort of emergence state in which one small effect -a small change in
frequency- gives us a large effect. Those three, or six independent entities, in interrelationship conform one entity, awhole picture, a form represented by a circle.
The Principle of Synergy, or the gluing principle can be represented by a
fundamental sevenfold structure
20
The main point to recall is the structure represented by
- three inner or nondual relations, or just the 1-1, 2-2 and 3-3 relations
- three outer or dual relations, or just the 1-2, 2-3 and 1-3 relations
The wholeness attribute is associated with an emergent property represented by
the 1-2-3 relation.
Even though we have a holistic property in this codification, it is embedded
in a Basic Unit System or Holon, not in a general abstraction where the hierarchy-relationship between the part and the whole is not clear at all.
The Bus(Holon) concept is embedded in a formal complex mathematical
framework, whose generality is associated with those isomorphic properties wehave already presented and that gives us a real internal consistency which allows totreat steady-state systems by the same general techniques or methodology.
Furthermore those six relations are some sort of detailed complexity as
opposed to the 1-2-3 relation that is the complexity of the whole. The traditional
problem between the part or detail and the whole is not a problem anymore with
this sevenfold structure that can be used at different levels of reality withoutreducing the one to the other, as its “actualization”[16] depends on a frequency, notnecessarily in a law or relation between the two basic components.
The fact we use a complex mathematical symbolism, as a tool, for
representing this sevenfold structure avoids that syncretic whole reasoning Galileoused when he wrote There are seven windows given to animals in the domicile of
the head, through which the air is admitted to the tabernacle of the body, to
enlighten, to warm and to nourish it. What are these parts of the microcosmos?
21Two nostrils, two eyes, two ears, and a mouth. So in the heavens, as in
macrocosmos, there are two favorable starts, two unpropitious, two luminaries,and Mercury undecided and indifferent. From this and many other similarities innature, such as seven metals, etc., which is were tedious to enumerate, we gatherthat the numbers of planets is necessarily seven. But it also prevents using as a key
guiding principle the primacy of the whole which results in arrogant role of
dominance when applied to organizations. But let’s go to the applications.
8. The Pendulum
The pendulum movement is so remarkable not just for its role as a
cornerstone in the birth of modern science that permitted Galileo a paradigm shift,
but also because, it is the most natural example of an open system. When lookingat the swinging body he saw a body that almost succeeded in repeating the samemotion over and over ad infinitum. Its success in repeating the same motion overan over lies in its interrelationship with the environment or on its opennessattribute.
By looking at the pendulum Galileo reported that the pendulum's period was
independent of amplitude for amplitudes as great as 90°, his view of the pendulumled him to see far more regularity than we can now discover there.[18] Is this view
of seeing more regularities than there really existed part of the incapacity of normalscience to explain pendulum-like regularities in a most documented way, andfollowing a mathematical methodology with no leaps? Or How else are we to
account for Galileo's discovery that the bob's period is entirely independent of
amplitude, a discovery that the normal science stemming from Galileo had to
eradicate and that we are quite unable to document today. [18]
The exact simple pendulum solution implies the solution of a first order
differential equation which implies too an integration whose solution is an ellipticintegral. This means the introduction of an approximation factor that could only befound by observations of the pendulum real behavior, some sort of trial and error
procedure. This normal mathematical symbolism, I mean, not-complex-
mathematical symbolism, cannot give reason of that approximation factor withoutusing some sort of methodological leaps, to explain deviations.
Normal science works with closed systems, that is, systems that do not
exchange with their environment. The pendulum movement seems to violate eventhat infamous second law of thermodynamics. It gives us a natural sense of eternityjust as the poet Jacques Bridaine wrote
"Eternity is a pendulum whose balance wheel says unceasingly only the two words,
in the silence of a tomb, 'always! never! always!...'"22After exploiting the cyclical wave nature of Euler Relation in EE, it is obvious to
expect we will be able to explain all those natural phenomena such as that of thependulum, in which we have cyclical or wave movements too.
For achieving this it is necessary to realize a real paradigm shift. With the
complex plane we have introduced in fact a new sphere of reality in which we haveembedded both the nondual and dual nature of reality, and as so a complex metrics,
whose main characteristics are:
- on the one hand, its isomorphic property that gives us a powerful methodologicaltool to explain those cases in which real dynamism is involved- on the other that property we have associated with a "merry-go-round" effect thatpermits us to make a mathematical representation of real dynamic entities or elsegeneration of forms. In the pendulum its form is continually generating itself byjust a mere impulse.
A new sphere of reality or the sphere of form, in which the four-dimensional
space-time continuum is embedded. A new category or totality that contains thephysiosphere, but also that sphere in which we can have animated forms, as itwere, the biosphere, or the Bergson-Theilhard de Chardin DURATION[1,4]concept which is quite different from the space time continuum.
A complex metrics is then a top-down metrics. From the notion of the Basic
Unit System and those normal regularities already known from physics, but also
from those properties obtained from geometry, we can obtain the state of our Bussystem, through a methodology in which instead of starting with the postulation ofa differential equation, we start with the Bus concept as a tool to integrate, orobtain that state.
From the point of view of the BUS, the pendulum movement is a rotational
motion. The pendulum as a Bus is then an open steady state system. The earth
gravitational field is its context.
By observing that cyclical movement we can observe a maximum angle θ, or
θmax, for a corresponding maximum displacement Smax.
Solving this problem means, integrating , some sort of constitutive
characteristic, not a summative one.23
The complex trajectory DS between S = o to S = Smax will be
Smax Smax Smax
∫ DS = Cos ( θmax )* ∫ DS + J*Sin ( θmax)* ∫ DS
0 0 0
so we have
J( θmax)
S = Smax* e
we note that this is a static and potential expression related with space, and as so
we must introduce its dynamic counterpart by multiplying both parts of theequation by the basic unit of time given by:
J(wt)
e
where θ = wt
The reality S of the pendulum is generated and represented by:
24 J(wt + θmax)
S(t) = Smax* e
Intuitively, from geometry we know that in a circle we have
Smax = L* θmax
Where L is the radius or the cord length of our bob system and θmax, that angle
subtended by the arch, so we have
J(wt +θmax)
S(t) = L* e
as a general dynamic expression in which the principle of synergy is included.
Whenever we have rotation synergy is at hand.
9. Balancing Equality and the Pendulum
To obtain the pendulum harmonic motion or just its steady state, we must apply
a balancing or a compensating equality, between two driving forces or polarities:
- an inner one related with the weighting mass of the bob and
- an outer one related with the so-called-inertial mass.
This equality played a very important role in Einstein’s works about
gravitational fields[5]. Up to that time that equality had been considered a mysteryin the Newtonian framework. It generated those famous thought experiments with
elevators, in which bodies are subjected on the one hand to an inertial force, and on
the other to the gravitational field.
When these two forces or polarities within a comprehensive whole become
equal then we have a harmonic motion or a motion that almost succeeded inrepeating itself over and over ad infinitum. A balancing loop which implies arotational motion or the Synergistic Principle means that the law of opposites mustalways be considered in a comprehensive whole, or dynamic context, producing a
cyclical movement. But if we consider inertial forces then we must apply Newton
Second Law, but this time in the fivefold continuum, or in the complex planecontext.
The first derivative of S(t)
S(t) = L* e J(wt + Ømax)
is
dS/dt = L *Ømax*e J(wt + Ømax)25and the second derivative:
d 2 S/ d t 2 = - (L*Ømax*w*w)*e J(wt + Ømax)
being
amax = ( L * Ømax * w *w *)
the maximum absolute value of acceleration.
According to that second law we have at the maximum point Smax the
inertia "vector", which tends to maintain the motion as:
Fmax = m *( L * Ømax * w * w)
if we apply, on the other hand, the vector composition of forces and using
tangential components at that point we have the mass "vector" as:
Fmax = m * g * Sen(Ømax)
so equating both the inertial and the mass "vectors" we've got:
L*Ømax*w*w = g*Sen(Ømax)the angular velocity is:
w = sqr( g * Sen(Ømax) /( Ømax * L))
but ifw = 2*Pi *f
and
T = 1/f, where f is the frequency and T the period thenT = 2*Pi *sqr((L/g) *( Ømax /Sen(Ømax))which is the exact pendulum formula.
The factor
sqr( Ømax /Sen( Ømax) )26tends to one when Ømax is small and it can be omitted as long as the amplitude
does not exceed 10°.
In Table 1, in the second row we can see the values K of the elliptic integral,
in the third row the corresponding factor according that integral, and in the fourthrow the corresponding factor obtained with the Bus concept. In the fifth row we
can see the difference in error between the elliptic factor and the Bus Factor. This
difference in errors as the amplitude increases can be taken as a fallibility criterionfor those angles greater then 30 degrees where the error between the two isdefinitely unacceptable.
Table 1. Approximation Factor for the Period of a simple Pendulum
Ø(angle) 0° 10° 20° 30° 60° 90° 120° 150° 180°
K 1.571 1.574 1.583 1.598 1.686 1.854 2.157 2.768 Infinite
2K/Pi 1.000 1.002 1.008 1.017 1.073 1.180 1.373 1.762 Infinite
Ø /Sen(Ø) 1.000 1.005 1.002 1.047 1.209 1.570 2.418 5.235 Infinite
Error(%) 0.000 0.298 -0.595 2.865 11.248 24.84 43.217 66.34 N/A
The exact solution introduces a nonlinear character in the pendulum motion which
gives reason of its real behavior defined by the well known observed laws.Normally this problem was solved by resolving a differential equation, that gavefinally an elliptic integral, or some kind of solution that must be validated with thereal observed behavior of the pendulum, as an elliptic integral cannot be expressedin terms of the usual algebraic or trigonometric functions, see Vector Mechanics
for Engineers. [3]
Galileo did not know this factor, and so he extended the application of the
observed pendulum laws far beyond its real range, even to 90 degrees. But the factthat the time of any amplitude were independent of the mass of the body made hethink in the falling of bodies and specially in the well-known Pisa Towerexperiment, in which both an iron and a wood sphere fall with the same time,which means in all cases, the trajectory followed by those bodies is an invariant.
The pendulum is essentially a device for measuring time. It is in fact the
contrivance of time. Its form unfolds itself in time, it is the dynamic device forexcellence. Differently from those movements of classical Newtonian mechanics,like those of the planets, in which time does not play really any role and in whichwe could talk about the chance to invert time, the pendulum movement in thissense is not a classical movement, and it answers the question posed by Thomas S.Kuhn about that movement.
The Principle of Synergy applied to the pendulum explains why this
principle seems opposed to that infamous second law, as with it forms can begenerated and we can obtain in natural way a steady state open system.2710. Quantum Mechanics
The Quantum Mechanics problem before being a scientific problem is a
philosophical problem as from the beginning it touches the same nature of reality
and the relation of the subject with that reality. The relation between object andsubject, the active factor of the subject in the process of cognition is in fact anepistemological problem that has to do with the way the subject defines what heunderstands by objects[10, 15]. But this philosophical problem can be found in theidea of the active role of language in the shaping of our worldviews or images ofthe world, and as so it is primitive and was the main concern not only of
philosophers but also of historians as we can see in Adam Shaff’s book History
and Truth. And we know also the efforts done by Karl Popper during all his lifeand specially in the introduction of his Quantum theory and the Schism in physics.
From the Postscript to the Logic of Scientific Discovery [11] to exorcise
consciousness or the observer from physics.
Reality is independent of human mind, of the subject perceiving it, and we
can distinguish in it three great realms:
- the subject that perceives that reality
- the object perceived by the subject as reality, the intersubjective domain- reality per se
that we can codify in a holonic way as
(The subject, (the field of consciousness /reality)or just as the big three
(“I”, (“We”/”It”))
The structure of reality perceived by the subject is crucial in the
determination of the objects of study, those objects normally studied by science.But science is a product of human activity and as so language is crucial in thedetermination of those objects of study. This problem took Karl R. Popper todefine a third world, as an intermediate world between consciousness and objective
reality[10].
As we have seen we can have two kinds of mathematical languages, a partial
one that not necessarily denies complex numbers as it uses them in "convenientways", and an integral one in which complex number are used for building a unit inwhich the radical J is some sort of operator to distinguish between two different28orders of reality, as it were, the nondual and the dual nature of reality embedded in
that same unit. In this sense this is not a new theory of everything but a way of"seeing" and interpreting reality in which uncertainty is always concomitant.
When we talk about Reality per se we are not meaning a third world as that
of Plato, divine, superhuman and eternal, but of reality as defined by Mortimer J.
Adler in his Adler's Philosophical Dictionary [9] where he wrote:
BEING The word "being" is an understanding of that which in the twentieth
century is identified with reality.
What does the word "real" mean? The sphere of the real is defined as the sphere of
existence that is totally independent of the human mind...Another distinction withwhich we must deal is that between being and becoming, between the mutable
being of all things subject to change, and the immutable being of that which is
timeless and unchangeable. That is eternal which is beyond time and change. Inthe real of change and time, past events exist only as objects remembered, andfuture events exits only as objects imagined.
Historically the prevailing structure of reality has been a dualistic one in
which reality as a whole is divided in just two domains, being each one, as it were,
parallel to the other, with no possibility of integration in case of gross dualism.
Normally there has been some sort of partialness depending on the accent puteither in philosophy as the queens of all science, or in science as the real-objectivescience. Philosophy and Ontology are not considered normally as independentdomains, but for our purpose and aim, philosophy must look for an integratedimage of reality and in the searching of methods of generalization so thatphilosophical theses are kept from contradicting science, in other words it must
contribute to oneness and not to partialness. On the other hand Ontology must look
for the more-being, and as such realism must be concomitant as a main issue.
An open system can have a variable center and as such its external
manifestation or its form cannot be determined, as its field or circle of influence isvariable, and this will be the case for the electron. Other open systems in whichboth its center and its radius are not variable will have a more determined state and
will be near to an ideal closed system. But this issue has to do directly with the
Uncertainty Principle, in which we have as in philosophy no univocal conditions asthose we can obtained with closed systems. But the important point is thecorrelation we have between closed systems, measuring instruments and opensystem and the Uncertainty Principle. The incapacity to measure the momentumand the position of an electron at the same time, has been associated with aphilosophical problem that has to do with an emphasis on the subject doing the
experiment, pointing out the fact that the structures of the subject are crucial in the
way he defines reality or its object of study, which is similar to that philosophicalproblem in which language is considered as a factor that creates reality.29We can talk about metaphysics when we have assertions for which we don’t
have procedures on the one hand, and the laws on the other, that can be used toverify those assertions. In the latter case when we have laws we can builtmeasurement instruments to verify the Data. On the other hand we have knownprocedures that when used they take us to a series of consistent results, that can bedescribed qualitatively, but not necessarily quantitatively. In both cases our main
task as scientists and philosophers must be the searching of truth.
In our case we will consider this problem of the Uncertainty Principle, in
case of an electron, as the impossibility to determine the state of the correspondingBasic Unit System because there is not a law with which we can interrelate its twocomponents, as it were, its dynamic counterpart and its static one. In this way wecan separate the scientific problem of Quantum Mechanics from a philosophicalproblem which I think was the aim of Karl Popper when defining his "third world
theory". In our case there are clear differentiation between precision and
uncertainty, as there are cases such as the electron case in which there is no way toreduce the qualitative to the quantitative, which was Karl R. Popper requisite toincrease the degree of contrastability of certain theories.
11. The complex Schrödinger’s wave Equation
The trajectory or more appropriately the reality of an electromagnetic entity
as that of the electron, which as a matter of fact behaves as a wave, can be
represented by the bus concept as
DS = Abs (DS)* e J( Ø )
we can codify it too as
(energy, (time/space)) or (energy, (t/s))
or
(Energy, (wave/particle))A general conceptualization of the Bus concept can be seen as a mathematical
codification of energy in its most primary definition, but energy "is like afrequency multiplied by Plank's constant h"
E = h*f
and that angle ø can be replaced by30ø = 2*p / h * (p* x - E *t)
a well-known expression taken from Quantum Mechanicsand
1/λ = p/h
where λ is de Broglie’s wave length, h is Plank’ s constant, and p is the momentum
of the particle –a BUS rotating at a very slow unknown rotational speed in the
complex plane- that can be written too as
p² = 2*E* m
The two state variables of the BUS in this case are x and t, so to able to
determine its state, we must find or know a relationship between them. The
geometric-like behavior of the pendulum gave us the clue to find that relationship.
In case of the planet movements it is the second Kepler’s Law associated with acentral force movement the one that permits to determine its state and in case ofthe Lorentz’s transformation group, the fundamental equation of electromagnetismit was the constancy of the velocity of light[6].
The electron does not behaves like a classical particle or a planet subjected
to a central force, we don’t know, like in the pendulum a way to link its two
components, as it were, its energy and its particle nature. Its state cannot bedetermined.
For the electron we cannot find a relationship between its two state
variables, being this another way to present the Uncertainty Principle which meansfrom the geometrical point of view we will not have a known trajectory followedby the electron, as was the case for the pendulum and the planets. In both of these
cases we have a "real" differential equation associated with. In the electron case we
have the well-known complex Schrödinger wave Equation that was presented byhim in 1926 as a postulate, i.e., as a way of saying that equation does not have justone solution starting from initial conditions, so the need to measure was replacedby qualitative methods, where one must focus in a behavioral area and not infinding laws that permit us to measure in the classical way. In case of the
pendulum its form changes with time, but at the same time, its centerness attribute
is localized, which seems not to be the case for the electron case.
Our aim is to deduce that complex wave Equation in the context of the Bus
concept. Let us suppose an unknown general solution, as a function of space andtime, as it were, of its two state variables
J(2 λ/h*(p *x – E *t))
S(x, t) = Abs(S) * e31Please note the variable Ψ(x, t) usually called “the wave function” in QM has been
replaced by S(x, t) which is a complex function that contains by definition a wave
character[7]. S(x, t) has to do with the real trajectory of the entity in question in
the complex plane so in a certain real sense it is a complex amplitude too.
Let us rename
Abs(S) by S
to distinguish the magnitude from the complex quantity. Expressing E as function
of momentum and replacing
E = p²/2mwe have
J(2p/h*(p *x – p²/2m *t))
S(x, t) = S *e
J(2p/h*(p x) – J2p/h * p²/2m *t
S(x, t) = S *e * e (1)
the point here is to follow the well-known-wave procedure by making two partial
derivatives of this expression with respect to space and one partial derivative with
respect to time. By equating them we obtain the complex Schrödinger Wave
Equation. At this point it is important to recall the isomorphic property of EulerRelation as the one that makes it possible this result, as it were, the permanence ofthe unstable.
The first partial derivative with respect to space gives
2ππππJ* p /h* x - 2πJ/h*( p²/2m)*t
∂∂∂∂S(x, t)/ ∂∂∂∂x = J(2ππππ/h)*p*S*e * e
where the internal derivative of
2ππππJ* p /h* x
e32is J(2ππππ/h)*p
and the rest of the expression can be represented by S´´ so
J( 2ππππ/h)*p - J(2ππππ/h)* ( p²/2m)*t
S´´ = S* e * e
the second partial derivative gives us
∂∂∂∂²S(x, t)/ ∂∂∂∂²x =( J(2ππππ/h)*p) ²* S´´
or else S´´
S´´ = ∂∂∂∂²S(x, t)/ ∂∂∂∂²x/( J(2ππππ/h)*p) ²
S´´ = -∂∂∂∂²S(x, t)/ ∂∂∂∂²x* h² / 4ππππ²* p ² (2)
If we now take the partial derivative of (1) with respect to time, the internal
derivative is
-2ππππJ/h*( p²/2m)
so we have
∂∂∂∂S(x, t)/ ∂∂∂∂t = -2ππππJ/h*( p²/2m) *S´´
or else
S´´ = ∂∂∂∂S(x, t)/ ∂∂∂∂t /-2ππππJ/h*( p²/2m)
S´´ = - ∂∂∂∂S(x, t)/ ∂∂∂∂t* 2hm / 2ππππJ *p² (3)33equating both Ψ" expressions 2 and 3
∂∂ ∂∂²S(x, t)/ ∂∂ ∂∂²x*h²/4ππ ππ²* p² = ∂∂ ∂∂S(x, t)/ ∂∂ ∂∂t*2hm / 2ππ ππJ *p²
and by eliminating and organizing terms on both sides we obtain finally:
∂²S(x, t) h² h ∂S(x, t)
--------- ---- * --- ---------- = -------- * -------------- --------
∂²x 8 π² * m 2 πJ ∂t
we obtain the well-known Schrödinger‘s Wave Equation, introduced by him in1926, for a free particle moving in x' s direction. This equation was presented then,as a postulate as there was no means to deduce it, from more basic principles, nonethe less, we have just applied the wave procedure to the BUS concept based on theprinciple of synergy. Richard P. Feynman wrote[7] in his famous lectures onPhysics:
In principle, Schrödinger ‘s equation is capable of explaining all atomic
phenomena except those involving magnetism and relativity.
We must take into account though that the Bus concept as a powerful tool
has been used for finding the Lorentz’s group of equations that give reason both ofrelativity and electromagnetism, but also of gravitational fields.The symbolic representation(Energy, (t/x ))establishes a hierarchy in the sense that wholeness or energy can stand on its own,and the other two state variables, t and x, cannot. When the state of the system isdetermined t and x are related by means of some law or relation, the wholeness orenergy is linked to (oneness, openness). When they cannot be linked as the case ofthe electron we have uncertainty and energy prevails on its own.There is an emergent conclusion in all this, and it is that the fundamental of
physical reality is energy , and not a particle( or the mass concept).
A particle is a Bus or Holon rotating at an unknown low frequency but in the
complex plane or in reality represented in that plane, so that its state is completely
determined, or else the qualitative aspect is reduced to the quantitative one in sucha way no possibility of a field or just a storing capacity is feasible different from itswell determined state. It is important to remember at this point that the Bus conceptas a mathematical symbolism also has that wholeness attribute that is at the base ofgrowth, each whole has the capacity to generate another whole, but always having34in mind that openness attribute that also permits us to define the Bus in general as a
symbol for representing a steady state open system. It is not a metaphysicalconcept that came from nothing. It is a concept in which a gluing principle as thatof the Principle of Synergy permits us to define that oneness attribute.
12. Conclusions and Suggested applications
From all this exposition emerges a conceptual framework that not
necessarily reduces everything to the most elementary levels of reality, but as agood engineering conceptual tool opens new avenues for future research anddevelopment. The important point to recall at the outset is the need to abandon the
old dualistic framework that has the natural tendency to put the whole or the part as
a "primacy" not as a fundamental component of the Basic Unit System. The Busconcept is a part/whole complex mathematical concept that has embedded, as wehave seen, the nondual and the dual nature of reality, but also the Principle ofSynergy in which the whole is greater than the sum of its parts that gives us amedium to interpret reality, as it were, in organic ways. We have a minimum
threshold of complexity in that structure.
In general, reality can be analyzed as buses within buses within buses, but
the Principle of Synergy implies a same frequency to obtain that "merry-go-
around" effect or that emergent steady state or new order or that higher complexstate where we have an organized complexity. One of the most important point torecall of this symbolism is that we do not need to make any reference toanthropomorphic concepts such as psyche or consciousness to explain those
emergent states where we find a whole greater than the sum of the parts. But from
the point of view of the whole reality we obtain a framework in which by theintroduction of a fifth sphere of reality, that of form, we have then a sevenfoldstructure where the space-time continuum is embedded in that fifth sphere, wherelife can be defined as an animated form, but then we have mind or the noosphere asthe sixth sphere of reality, but also the Being as that one that has embedded them
all, in an all encompassing way.
Having found such a powerful explaining tool, it is obvious to feel an
imperious need to share with the scientific community such a framework that cannot only illuminates its actual practice by defining the objects of study in certaincases but also establishing clearly the impossibility to reduce those objects toisolated units in other cases. The whole concern of Ludwig von Bertalanffy in itsGST[2] when he wrote
We may state as characteristic of modern science that this scheme of isolated units
acting in one-way causality has proved to be insufficient. Hence the appearance, inall fields of science, of notions like wholeness, holistic, organismic, gestalt, etc.,35which all signify that, in the last resort, we must think in terms of systems of
elements in mutual interaction.
is then the same concern of all this paper, but our main point is the integration of
those three big three, Being, Mind and Form or as Ken Wilber wrote in its Sex,
Ecology and Spirituality the Spirit of Evolution [8]
With Kant, each of these spheres is differentiated and set free to develop its own
potentials without violence...These three spheres, we have seen, refer in general tothe dimensions of "it", of "we", and of "I"...In the realm of "itness" or empiric-scientific truths, we want to know if propositions more or less accurately match thefacts as disclosed...In the realm of "I-ness", the criterion is sincerity ...And in the
realm of "we-ness" the criterion is goodness, or justness or relational care and
concern...What is required, of course, is not a retreat to a predifferentiated
state...what is required is the integration of the Big Three. And that, indeed, is
what might be called the central problem of postmodernity...how does one
integrate them?
Science, philosophy and ontology and its integration implies a science that
looks for truth but with an openness criterion, a philosophy that looks for oneness
without the reductionism tendency but also an ontology that looks for wholeness
as a fundamental principle.
To manage complexity properly it is very essential to have a basic structure
and among the possible conceptual structures we can have- a binary or dual one and
- a thirdness one that by mathematical inevitability becomes a sevenfold structure
This sevenfold structure has been used successfully in many fields in a
natural way but also recently we can find its application in many other fields too.
For example the three-"tiers" architectural approach to client/server solutions andwhich looks for separating the various components of a client/server system intothree "tiers" of services that must come together to create an application isprecisely a solution whose main aim is to manage the changing complexity andwhich requires a basic hierarchy that starts with the service to the client. A tool
must be adapted in every case so that we can avoid what can be named the "Galileo
syncretic whole reasoning" about that sevenfold structure. Maybe Galileo powerfulmind "saw" the powerful explaining capacity of this structure but his time was justthe beginning of a science in which the binary or dual structure would prevail.
Structure and environment are the main starting point of the new sciences of
complexity or in the studies of complex adaptive systems in which adaptation tothe environment implies always a minimum conceptual structural framework.
User services or interface or environment, business services or an adaptive plan,
and data services or representation of changing structures in a general way, are just36some of the new applications we can found in the General Systems Sciences or
Sciences of complexity of this sevenfold framework with which we definitelytranscend the dualistic worldview and which is really very different from theholistic paradigm, and not just from the conceptual point of view but even moreimportant from the practical point of view.
1. Bergson Henry.Creative Evolution. University Press of America.1983.
2. Bertalanffy von Ludwig. General System Theory George Braziller.1969.3. Beer and Johnston. Vector Mechanics for Engineers. McGraw-Hill. 19624. Chardin Teilhard de. The Phenomenon of Man. Perennial Library.19755. Einstein et All. The Principle Of relativity. Dover.19526. Epsilon Pi. Physics and The Principle of Synergy Amazon.com.1999
7. Feynman. Fìsica. Vol III. Fondo Educativo Interamericano S. A.
8. Ken Wilber. Sex, Ecology, Spirituality. Shambhala. 1995
9. Mortimer J. Adler. Adler’s Philosophical Dictionary. Touchstone Books, 199510. Popper K.R. Conocimiento Objetivo. Tecnos. 1998.11. Popper K.R. Teoría Cuántica y el cisma en Física. Tecnos. 198512. Saussure Ferdinand de. Course in General Linguistic. 199713. Stuart Kauffman. At Home in the Universe. Oxford Paperbacks.1995.
14. Scott R.E., Essigmann M.W. Linear Circuits. Addison-Wesley. PC.1967
15. Shaff Adam. Language and Cognition. McGraw-Hill.197316. Taborsky, Edwina. The Complex Information Process. Entropy. 2000, 2, 81-9717. The I Ching. Richard Wilhelm Translation. Princeton .University Press.199018. Thomas S. Kuhn. The Structure of Scientific Revolutions. The University ofChicago Press.1996
© 2000 by Epsilon Pi. All rights reserved.
|
Thermal and Nonthermal Mechanisms of the
Biological Interaction of Microwaves *
by John Michael Williams
jwill@AstraGate.net
P. O. Box 2697
Redwood City, CA 94064
2001-02-04
Copyright (c) 2001, John Michael Williams
All Rights Reserved
* This paper originally was submitted to the US Federal Communications Commission (the NTIA) as
a Comment response on the question of unlicensed operation of ultrawide-band (UWB) devices.
The author, and not the UWB Working Group itself, bears full responsibility for everything in this
paper.J. M. Williams Thermal and Nonthermal Interactions of Microwaves2
Abstract
Research in the past on the biological effects of microwaves often has been based
on faulty assumptions. The major flaw has been the premise that microwaves only
produce thermal effects in tissue. This premise easily may be proven illogical and
physically incorrect. Furthermore, assuming only thermal effects leads one to an
optimist's error of quantification in which calories are counted instead of joules.
Past investigations have been mislead both by these assumptions and by
stereotyped experiments using only narrow-band radiation sources. Recent studies
show that wide-band microwaves bring out biological effects which are unrelated to
those caused by heat flow. A review by Kenneth Foster provides a basis for
criticism and improved understanding.
PACS Codes:
87.10.+e General, theoretical, and mathematical biophysics
87.22.-q Physics of cellular and physiological processes
87.50.-a Biological effects of electromagnetic and particle radiation
87.50.Jk Radiofrequency and microwave radiation (power lines)
87.54.-n Non-ionizing radiation therapy physics
Introduction
Kenneth Foster's seminal review, "Thermal and Nonthermal Mechanisms of
Interaction of Radio-Frequency Energy with Biological Systems" will be referred to
as KF2000 in the following.
I would like to spend considerable discussion on the overall method of analysis of
the problem, as shown in KF2000 Figure 1. I think the approach, even for
expository or didactic purposes, is wrong and may be seriously misleading. As will
be shown, the underlying problem is the use of engineering technique to quantify or
formulate new understanding: Understanding, as opposed to proper application,
represents a function of science, not of engineering.
I also have comments on numerous specific points in the paper. These points
accurately exemplify methods which have become accepted in the older literature in
this field. Although many of my comments express disagreement with findings or
opinions cited in KF2000, I do in general think the paper provides a good basis of
review of the literature, with a balanced openness to future improvements in
knowledge of the subject.
The Model of Analysis Should be Changed
To begin, I do not think a paper addressed to an audience of engineers need be
cast in a familiar engineering stereotype. Engineers in general may be assumed toJ. M. Williams Thermal and Nonthermal Interactions of Microwaves3
be intelligent and well-educated and therefore capable of understanding new
approaches. All electrical or electronic engineers will have a very firm basis in
electricity and magnetism. They won't need filters or antennas to understand the
body. However, the KF2000 method of analysis seems to model a biological
response as somewhat like a message from a radio receiver; so, it looks for an
antenna isolated or decoupled from a set of RF amplifiers perhaps themselves
decoupled from audio amplifiers finally triggering a bio-servo response.
This becomes an error when it leads the author to an emphasis on the electronics
of the electromagnetic radiation (EMR), and not on the fundamental reasons for
those electronics. Filters are for radios or servo systems. There are no filters in the
body, so analysis by filter stages necessarily will be wrong. For example, on p. 18
the author says,
"The electric field induced inside the body by an external
electric or magnetic field [or EMR--jmw] depends strongly on
frequency and the geometry of the system. Because of the
conductivity of body tissues, it is difficult to couple low
frequency fields into the body through air, and DC fields are
excluded entirely."
This is wrong. The penetration of a field is attenuated at least exponentially
with depth by any uniform body tissue, as shown in KF2000 Fig. 4 and documented
in papers cited throughout Section II of KF2000. For example, a typical equation
relating the field intensity at depth d in tissue to the applied intensity in air would
take the form, for some tissue- and unit-dependent constant k, and frequency f in
Hz,
() ( ) I df I kdf tissue air, exp =⋅−⋅⋅12. (1)
The tissue depth for a given intensity becomes less, the higher the frequency, not
greater. Penetration increases with decreasing frequency. In the quote above, the
author apparently is generalizing from metallic reflection to a biological response: A
DC voltage applied across the body, whether by EMR, a static field in air, or wires
from a battery, will cause current to flow by ionic conduction in the body fluids. The
more conductive the body, the greater the current. When the coupling is through
air, the current flow will stop because of the resistivity of air, but displaced charges
will remain displaced to cancel the field applied.
In the quote above, the author is thinking of reflection of EMR from a (very
conductive) metal: There, the penetration is small; and, for DC, the internal field is
close to zero. However, the reason the penetration, or the DC field, in a metal is
small is that the charge carriers in the metal, electrons, are responding so actively
that they cancel the field by their motion (in the case of microwaves) or by their
static relocation (in the case of DC).
When oscillating electrical (or magnetic) fields are applied to the body in air, the
reason the penetration is limited is because the biological effect, an oscillating flow
of ions, is large near the surface, declines exponentially with depth, and isJ. M. Williams Thermal and Nonthermal Interactions of Microwaves4
absorptive. Deeper, the field is weaker, so motion will be less whether the deep
tissue be conductive or not. So the quote above is misdirective if not completely
wrong.
Returning specifically to the issue of describing serial filter stages as in KF2000
Figure 1, to accept the KF2000 model, one has to abandon the idea that the EMR
applied to the organism might interact anywhere it can penetrate. KF2000 Figure
1, and nearby text, presumes that EMR has sequential effects at stages separate
enough to be considered isolated (or, shielded) from one another except for
connecting filters. Such nice sequencing can be achieved by good engineering
design, but one should not insist on discovering it in a nonengineered organism.
The problem implied by KF2000 Figure 1 may be seen by considering an analogy
of two containers of 250 ml of water, one a 6-cm diameter paper cup, and the other a
50-cm diameter paper plate: Which one would warm more rapidly in a microwave
oven? Both are "high-pass" to the ~2 GHz microwave oven excitation because of the
~30 GHz (~1 cm wavelength) first E-dipole harmonic of liquid water.
The problem also may be seen by trying to diagram not the signal flow, but the
geometry of an organism that would allow such an analysis to be valid. A two-
dimensional representation of this geometry might be as follows:
Figure 1. Organism geometry implied by the KF2000, Fig. 1, filter
system.
The problem now becomes obvious: There is a contradiction in separating
KF2000's "Target Structure" from "Inside of Body". Merely because the human
body might have a certain geometry does not mean that interaction of EMR with the
target structure depends upon that geometry. Counterexamples to KF2000 Figure
1 are easy to construct:
Suppose, for example, that the target structure were a set of metal fillings in the
teeth and the EMR were in the broadcast TV or GHz range. Clearly, a fraction of
the current induced in the fillings by EMR beamed at the body would be
independent of the height of the person.J. M. Williams Thermal and Nonthermal Interactions of Microwaves5
Now, true, the body can act as an "antenna", but any small component of the
EMR "high-passed" by the body merely would be superposed on the EMR directly
interacting with the fillings. The EMR directly acting would depend on shielding,
reflection, and refraction by the teeth and lips, not by any generality representable
by "Inside of Body". It would be incorrect in this example to describe the body
(actually, the mouth) as a high-pass filter: In fact, if anything, the circumference of
the open mouth would define a bandpass for EMR capable of exciting the fillings.
The same argument might be applied anywhere in the body--for any "target
structure". The skull, for example, bandpasses EMR finally interacting in the scalp
and the flesh of the face. Both geometrical resonance and the microwave "skin
effect" (no pun intended) would channel microwave energy from the skull into the
flesh outside the skull. KF2000 Figure 1 would imply that the interaction with the
skull could be filtered only to the brain and other organs inside the skull.
We suggest that analysis of the effect of EMR on human biological response
should focus on tissue and organ boundaries as they exist in the body, not on "the
body" as a radio chassis in any general sense. This idea will be expanded
immediately below.
We also point out that the belief that microwaves do not produce tissue effects
other than heat seems to be a prejudice based on history more than science, as may
be seen by the review in Chapter 7 of Durney, et al (1999). The history seems based
on attempts to discover how the body "was engineered". KF2000 and others rightly
have reserved final judgement on this belief.
The belief that heat might exclude other responses seems connected to inaccurate
usage of the term, "biological mechanisms". "Biological mechanisms" has no
meaning in terms of the living body; actually, "biological mechanisms" merely refers
to measurements or theory typically made by people studying biology. The real,
physical mechanisms driving life involve the same atomic coulomb fields--valence
and conduction electrons--and the same molecular dipoles that change in the
picosecond scale of the fastest optical instruments. It is an error unknowingly to
avoid this level of analysis, or to claim that life at this level is as random as an ideal
gas, in trying to understand the interactions of microwaves with living tissue. If the
microwaves are well-ordered at this level, and if the tissue is well-ordered (alive) at
this level, then well-ordered interactions must be taking place before the end-result
of heat.
The difficult question is how to apply the microwaves so they are amplified by the
living organism at a level easily observable by persons studying the biology. Thus,
the question is not how to postulate a "design" for the organism but rather how to
seek empirically to use what is there, whatever science reveals of it, as an effective
interface to our instruments.
EMR Interaction Can Not All Be Heat
Consider a tissue boundary such as the mucous membrane of the stomach wall or
the endothelium of a large blood vessel. Such a boundary will be enclosed in cellJ. M. Williams Thermal and Nonthermal Interactions of Microwaves6
membranes with a regular coulomb structure, the structure being organ dependent
when relaxed in the absence of EMR.
For example, without much loss of generality, let us take a structure consisting of
electric dipoles which happened to be aligned with positive pole locally pointing into
some polar fluid such as water. This is shown in Figure 2:
Figure 2. Unaligned electric dipoles in a fluid (left) bounded by solid
tissue with a membrane layer of aligned dipoles.
Now, how would microwave EMR raise the temperature of the fluid? More
generally, how can one understand the effect of microwave EMR on the structures in
Figure 2? Especially, how could a coherent, regularly structured beam create
increased disorder (heat) in the fluid?
The effect within the bulk of the fluid is easy to understand: The dipoles initially
may be assumed arranged at random with respect to one another. When they are
rotated (torqued) together, all magnitudes in phase with the EMR, they interact at
random distances and phases with respect to one another. Thus, the EMR-induced
motions contribute random momentum and therefore random energy. This is
shown in Figure 3. In that figure, the Poynting vector r
J is shown for EMR
propagating as a plane wave from left to right, with a few illustrative EMR torque
angles shown. The Poynting vector is given to define the direction of the incident
EMR as the direction of EH×, were the medium removed and replaced by vacuum.J. M. Williams Thermal and Nonthermal Interactions of Microwaves7
Figure 3. Sketch of the effect of a plane EMR beam propagating in a
bulk fluid of polar elements, with a net result of increasing the
temperature. The E-vector magnitude and polarity (+ up) is shown on
an arbitrary vertical scale. The EMR torque on a few illustrative polar
elements is shown as a vectorial angle.
So, the net effect of the EMR is to deposit energy in the fluid. This energy
macroscopically is completely disordered and therefore is heat energy.
Now consider the same EMR near the fluid boundary: The boundary is more
ordered than the bulk of the fluid, so some EMR energy will be delivered to the
membrane as free energy, not heat. It is not physically possible that no free energy
would be delivered. As it happens, in the absence of EMR, the fluid dipoles will
tend to be somewhat ordered (correlated) close to the membrane anyway, so as to
tend to line up with the membrane dipoles. This means that random motion
because of heat is screened from the membrane dipoles to some extent by the dipoles
in the immediately adjacent fluid. The heat in the fluid immediately adjacent to
the membrane is represented by lower-amplitude, less random velocities of larger,
more correlated ionic elements. All this is sketched in Figure 4, where, for variety, a
gaussian-tapered beam envelope is assumed.J. M. Williams Thermal and Nonthermal Interactions of Microwaves8
Figure 4. Effect of a gaussian EMR beam propagating from a polar fluid
across an organ boundary with a polarized membrane. Necessarily,
some EMR energy will be transferred to the boundary as free energy,
not as heat.
If we assume in general that the Poynting vector r
J will not be perfectly
perpendicular to the boundary, then the interaction in Figure 4 will deliver some
linear momentum transverse to r
J; the EMR beam will be refracted. The angle of
refraction will be proportional to the amount of free energy delivered to the
boundary dipoles. The momentum transferred to the membrane will correspond
entirely to free energy; no heat will be involved. We ignore here the dielectric and
other properties quantifying the direction and amount of refraction. For now, we
also ignore diffraction and reflections (which would set up standing wave
modulations in the incident beam).
So, we may see that an ordering effect will occur in the direction of propagation,
and also in general transverse to it. Because all body tissue is ordered at organ
boundaries, and also is ordered in bulk (except maybe fat, and fluid such as blood,
lymph, or a digestive juice), all body tissue will receive free energy from EMR.
We therefore assert that the burden of proof should be on the claim that
microwaves act like microwave ovens and do nothing but cause heat to be
transferred to living tissue. Based on the prima facie argument of this section, EMR
should be assumed have nonthermal effects on living tissue unless explicitly
provable otherwise.
We move to one, final point in this initial, overall discussion of KF2000 with the
observation that the body cells, especially the individual nerves and sensory
receptors, function as ordered systems of boundaries in fluid. The temperature of
the body always causes disorder in the ubiquitous body fluids; nevertheless, nerves
and muscles do perform orderly functions, which of course require free energy.J. M. Williams Thermal and Nonthermal Interactions of Microwaves9
The mechanism of action of nerves, muscles, and systems of them, has been
described in terms of membrane depolarization. This depolarization occurs because
membranes are not only populated by structure-related dipoles as in the figures
above, but also are more sparsely populated by specialized channels which can open
or close to regulate exchange of ions, primarily sodium (Na+) and potassium (K+),
but also calcium and perhaps other ions.
Living cells generally internally are more negative than their local environment,
thereby storing some free energy in an electric potential across their bounding
membranes. This potential difference is electrochemical and depends on continuous
expenditure of energy to maintain higher intracellular K+ concentrations and lower
intracellular Na+ concentrations than would be so during chemical (osmotic)
equilibrium. The ion gradients are controlled by enzyme action which does its work
using the adenosine triphosphate (ATP) energy store available to cells throughout
the body.
Specifically, resting neurons typically expend energy to operate a Na-K "pump"
which keeps the potential inside the cell about 70 mV more negative than the
outside (Regan, 1972, p. 7). The membrane can liberate energy very locally by
allowing some extracellular Na+ to flow in and some intracellular K+ to flow out.
An additional osmotic potential also exists. Dendritic depolarization (e. g., "C"
nerve fiber sensory response) may cause the -70 mV potential to drop close to 0 V
immediately adjacent to the depolarized areas of the cell membrane.
When the originally resting electric field across a nerve membrane develops a
steep enough spatial gradient, the specialized closed channels are forced to open,
causing a spread of depolarization, which may become a nerve action potential
under proper circumstances. The sudden depolarization during an axonal action
potential may cause the resting potential of -70 mV to swing past 0 to perhaps +10
mV. This action probably is what KF2000 describes as having the relatively slow
time constant on the order of milliseconds. For example, see the review of
"membrane excitation" in KF2000, p. 20.
We note, aside, that diffusion permits the approximate equation of locally steep
spatial gradients with locally sudden temporal ones and is essential in the Hodgkin-
Huxley and similar approaches: For ion species j in concentration ()uxtj,,
()()()Duxt
xuxtuxt
t¶
¶r¶
¶,,,− = ; so, () ( )¶
¶¶
¶u
xu
t− ∝− loss system inp ut. (2)
Here, the hypothetical "Na-K pump" would function to keep (loss) = (system input)
when averaged over long times.
The membrane channels in a neuron originally are held closed by polar molecules
(PM) which open for depolarization because of smaller-scale, faster coulombic
(electrical) response. At some level, the time-constants of processes seen as a
nerve action potential will be on microsecond and nanosecond scales. It may be
convenient to describe the response in terms of a single characteristic time constant,
but arbitrary time constants control the response at arbitrary levels of analysis.J. M. Williams Thermal and Nonthermal Interactions of Microwaves10
Clarification or Refutation of Common
Misconceptions
These are in the form of comments on specific approaches or theoretical
assumptions reviewed in KF2000.
p. 16, Figure 4 . The field penetration vs. frequency curves tend to fall on straight
lines, except between about 108 and 1010 Hz (wavelength between a couple of cm and
a few m), where they seem to level off somewhat before starting a steeper descent.
The attenuation in muscle is greater than in fat at all frequencies, probably
reflecting the greater water (polar) fraction of muscle, but perhaps also related to
the more regular organization (larger membrane fraction in the hierarchical fascicle
structures) of muscle tissue than fat.
Some data have shown a loss of frequency dependence of electrical properties in
all soft tissue for very high frequencies around 20 GHz (see Durney et al, 1999).
Presumably, at high enough frequencies significantly water-distinct tissues would
converge to somewhat distinct levels of absorbance.
p. 17, Eq. 4 : The terminology of the interpretation seems ambiguous. What is
being referred to as "dipole"? The magnetic permeability even of blood is
insignificantly different from that of vacuum. However, in nearby text, the result
for typical human flesh in the quasistatic limit is said to be that the magnetic dipole
DM contributes about 15 x the electric dipole DE to the induced electric field Ei.
Also, the Ei (same as r
D?) is the sum of the electrically and magnetically induced
contributions; so, is it correct to write, for some constant k, ( ) E kD D i M E≅ +?
Presumably DM and DE are the (orthogonal) components of the incident EMR, if
plane wave? Would we have, D B M i=r
, and D PE i=r
?
p. 17, IIIB, the Resonance Region . It would not be unreasonable consider this
somewhere between wavelengths of about 5x the height of a human to 1/5 the
length of the smallest bone, organ, or opening in the body--say, 10 m to 0.1 mm; this
would imply free space microwave frequencies of perhaps 30 MHz to 1000 GHz.
Again, just because the math gets complicated here doesn't mean that organ, tissue,
or cellular response has to be determined here. However, in this region, the local
geometry of body structures can concentrate fields by factors of ten or more.
pp. 17-18, cites Chen & Gandhi , in 1991, with simulation results for a half-cycle
of 40 MHz at 1 kV/m peak found about 1 to 4 A, depending on target organ
geometry; the conclusion was that the internal field was about 1/3 the incident field
in air. Both in this and the quasi-static regime, an approximation is maybe 10 mV
induced for every incident V/m, assuming a grounded human in a vertically
polarized E field and pulses lasting some tens of ns. This is an interesting and
useful generality.
p. 18, IV: The Laplace representation , using a "cell" of two concentric spheres,
seems completely divorced from reality and merely a choice based on available
computer programs. Applied to a cell (rather than to the nucleus of a cell), theJ. M. Williams Thermal and Nonthermal Interactions of Microwaves11
reasoning thus resembles the assumption that microwaves induce only heat because
heat can be calculated and measured easily.
Spherical symmetry implies field cancellations unwarranted in living,
multicellular organisms. In fact, even unicellular organisms almost never are
spherical in geometry: Why? Because they need asymmetry to locomote or to sense
chemical or other field differences necessary to survival. Otherwise, the cell would
have to wait at random for life to come to it--a survival strategy good enough for the
nucleus (or a virus) to survive inside a cell, but a strategy as random as heat for the
cell as such!
The only spherical cells in the human body are fat cells or perhaps some kinds of
blood cell. Modelled as ellipsoids, sensory or motor nerve cells would have
eccentricities on the order of 105 or more.
The spherical model in this section must be assumed irrelevant to the question of
nonthermal interaction. However, perhaps it might be corrected for geometry, and
organized to match the observed data on actual living cells:
For example, allow a cell diameter of 105− m and length of 101− m. Many human
nerve cells would fit this. So, the eccentricity would be something like 104.
Therefore, allowing for living organization, one might approximate the living cell by
a chain of 104 of the objectionable spherical cells, each of twice the spherical radius
to correct for surface vector cancellation because of spherical symmetry. Because of
series addition of resistance, the capacitivity to conductivity ratio in KF2000 Eq.
(12) would be increased lengthwise by a factor of about 104; but, radially it would be
decreased by a factor of 22122≅. Using KF2000 Eq. (12), the result would be a
time-constant of maybe 0.1 ms locally (in a plane perpendicular to the long
direction), but one of about 0110 14.⋅≅msms globally, in a direction exactly on the
longest dimension of the cell. The projection of an incident EMR Poynting vector
thus would interact by the time-constant of the living cell, but depending on the
angle of incidence, over a range of something like 1014:. The membrane potentials in
KF2000 Figure 5 then would have to be corrected by some factor between 1 and 104,
depending on angle of incidence of the EMR on the elongated, living cell.
To continue the example of the living cell just described, the preceding reasoning
would lead one to expect a membrane potential somewhere between 106− and 102− V
for a 1 V/m incident E field. Because pulsed UWB fields may be expected to reach
several kV/m, EMR-induced membrane potentials on the order of several times the
normal neuronal resting levels of -70 mV should be anticipated. Thus, in this case,
the assumption of heat was based clearly on an invalid spherical simulation.
p. 19, IV discussion of time-domain response . One must recognize that standing
wave components will be found everywhere near organ boundaries, especially where
muscle and either bone or skin are close. The medium always being absorptive, the
reflected wave amplitude will be smaller than the incident, creating an oscillating,
standing field with a net DC component, the polarity depending on local phases
(Durney, et al, 1999). The geometry of the resultant DC field essentially will be
unrelated to incident frequency, except for being multiple half-wavelengths in size,J. M. Williams Thermal and Nonthermal Interactions of Microwaves12
but it will be fairly closely related to the geometry and tissue properties of the
organism. If the incident field or organism is moving, the standing wave patterns
will be transitory, persisting perhaps for milliseconds or seconds. The time-change
of the DC component then should be expected to impose cm-scale, changing, DC
spatial patterns on nerve cell receptive processes, which should cause sensory
responses.
It seems reasonable that this would be the theoretical basis for use of UWB or
other pulsed radar transmitters to harass living humans and animals.
p. 20, VA 2, Rate of temperature increase . Rapid thermal expansion of tissue
water caused by 5 ms, 1-10 GHz, ~ 104Wm/ pulses is said to explain "microwave
hearing" of radar. At 1 GHz, the wavelength would be about 30 cm; at 10 GHz, 3
cm. Presumably, different tissues would cause a temporal transient proportional in
intensity to their differential absorbance. However, a more efficient mechanism
would be differential spatial amplitude over the skull, because of refraction,
diffraction, and reflection of the EMR pulse, and possibly because of occurrence of
briefly existing standing modes.
p. 21, VC, Field-charge interactions . The conclusion here was that random
thermal motion of ions in solution would be some 8 or 9 orders of magnitude greater
than the velocity to an ion added by a (steady) field of 1kV/m; therefore, field-charge
interactions would be drowned out in thermal noise. However, there are three
serious flaws in this conclusion:
First, the extracellular fluids are not filled with small ions, but rather with
macromolecular assemblages (Regan, 1972, p. 1) with far greater organization
(correlation) than the "ideal gas" on which the use of Boltzmann's formula is based.
At a given temperature, then, the thermal velocities will be far less than the ones
presented in KF2000. Much of the fluid thermal energy will be internal to the
assemblages and will not be available to add noise to EMR-membrane interactions.
See also Schulten (2000).
Second, the conclusion refers to the distant fluid extracellular medium, not to the
partially screening region near the membrane of a living cell.
Third, as is well known, small drift displacements in general may correspond to
very large measured values of electrical current. These displacements are
superposed on the average random thermal motion and are independent of it
because they are not random. See also Huang, et al (2000).
For example, a coaxial-plate capacitor with properties similar to a 1-cm long
unmyelinated "C" nerve fiber, 0.5 mm in diameter, and with 100 Ao
-thick
membrane, at 1 GHz (KF2000, Fig. 3) would have capacitance,
( )Cd
r router inner=2pe
ln( )
( ) ( )=⋅⋅
⋅+⋅ ⋅− −
− − −22088510 10
0510 100 10 051012 2
6 10 6p .
ln. .≅ ≅−10
00250011
.pF. (3)J. M. Williams Thermal and Nonthermal Interactions of Microwaves13
Now, charging a 500 pF capacitor to 1 kV means that 5107⋅− coulomb of charge
must be moved between the plates. That amounts to over 1012 charge carriers (ions
of unit charge); so, doing 9 orders of magnitude less still amounts to a displacement
across the membrane of maybe 103 ions of the kind described in KF2000. Thermal
activity on these scales averages out to zero and should be ignored when describing
the drift movement of several, if not many hundreds or thousands of, ions. KF2000's
comments about random thermal motion must be rethought.
p. 21, VC, Field-permanent-dipole interactions . KF2000 provides an expression
for torque on a membrane dipole from an electric field E, as torque E=m qcos , with
m a dipole moment and q the angle between the field and dipole. Then, Eq. (20) is
supposed to give the response time constant t for dipole alignment by Stokes's law
as,
tph=43a
kT, (4)
in which a is the dipole element radius (membrane half-width), kT is Boltzmann's
constant times the absolute temperature, and h is a fluid-dynamic viscosity causing
the interaction to be inelastic.
The reasoning is that if the time-constant in (4) is too low, it implies that the
dipole moment must be small, so very little torque can be applied, and the
interaction of the field with the membrane will be small. KF2000 Table II gives
data for various cell elements, showing that very large field strengths are required
for significant energy transfer to the membrane.
However, the dependence on one time parameter to interface all elements of the
implied system again seems to be in error: It assumes that the only membrane
interaction, above some thermal threshold, must be thermal! This is an incredible
assumption: If any significant heat could be stored in the membrane, it would lose
structural integrity. What is the reason that energy transferred to the membrane
would not be transferred to the hypothetical "Na-K pump", to a subset of the PM
gating system, or simply to geometric displacement (local compression or dilation) of
the membrane in some phase-related way? And why should not each conceivable
channel for momentum or energy be associated with its own, arbitrary time
constant?
p. 22, final paragraph of V . KF2000 states that "The mechanisms described
above are well-established and noncontroversial". Yes, heat is noncontroversial as
the end-product of any interaction. It is definitely a safe haven. However, the
errors of concept explained in detail at the start of these comments suggest that the
use of heat by KF2000, as well as by many others in the long history cited in
KF2000, must be in error. Errors of agreement can add nonlinearly with errors of
understanding to produce unwarranted noncontroversy.
The present list of comments, as well as the final "paradox" discussion in KF2000,
seems to show that the problem of microwave-tissue interaction, especially for UWB
microwaves, is not well understood by the majority of researchers or writers whoJ. M. Williams Thermal and Nonthermal Interactions of Microwaves14
have been publishing papers in this field. Therefore, the lack of controversy
claimed by KF2000 should be replaced by serious questioning of the assumptions
underlying the reportedly widespread agreement.
In closing, it is interesting that KF2000 does not cite Lu et al (1999), who have
found that UWB microwaves can cause a drop in blood pressure in rats.
Speculation as to the apparently nonthermal mechanism of the Lu et al findings
might have been interesting.
References
Durney, C. H., Massoudi, H, and Iskander, M. F. Radiofrequency Radiation
Dosimetry Handbook (4th ed., online; updated 1999). Brooks Air Force Base
(USAFSAM-TR-85-73), Armstrong Research Laboratories (AL/OE-TR-1996-
0037). Available at
http://www.brooks.af.mil/AFRL/HED/hedr/reports/handbook/home.html .
Kenneth Foster, K. Thermal and Nonthermal Mechanisms of Interaction of Radio-
Frequency Energy with Biological Systems, IEEE Transactions on Plasma
Science, 2000, vol. 28 (1), pp. 15 - 23 (February issue).
Huang, Y., Rettner, C. T., Auerbach, D. J., and Wodtke, A. M. "Vibrational
Promotion of Electron Transfer". Science , 2000, 290, 111 - 114. For example,
p. 112: "[others have found that] the vibrational relaxation of small molecules
on insulating surfaces [occurs] on a millisecond time scale, indicating that
vibration couples very weakly to phonons of the solid."
Lu, S. T., Mathur, S. P., Akyel, Y, and Lee, J. C. "Ultrawide-Band Electromagnetic
Pulses Induced Hypotension in Rats". Physiology and Behavior , 1999, 65(4/5 -
January), 753 - 761.
Regan, D. Evoked Potentials . London: Chapman and Hall, 1972.
Schulten, K. "Exploiting Thermal Motion". Science , 2000, 290, 61 - 62. Science
Perspective on Balabin, I. A. and Onuchic, J. N., "Dynamically Controlled
Protein Tunneling Paths in Photosynthetic Reaction Centers", 114 - 117. For
example, p. 61: "[proteins and RNA] work at physiological temperatures and
thus experience thermal motion, yet their function, which requires correct
alignment of parts and steering of reactions, is executed with precision." |
arXiv:physics/0102008 5 Feb 2001Symposium E1.5 - H0
Manuscript Number E1.5 - H0 - 0018
LARGE - SCALE HOMOGENEITY OR
PRINCIPLE HIERARCHY OF THE UNIVERSE?
Chechelnitsky A.M. Laboratory of Theoretical Physics,
Joint Institute for Nuclear Research,
141980 Dubna, Moscow Region, Russia
E'mail: ach@thsun1.jinr.ru
ABSTRACT
Invariablly justified representations of the Wave Universe Concept - WU Concept (See
monography - Chechelnitsky /G3A. /G46. Extremum, Stability and Resonance in Astrodynamics..., etc.
and other publications) indicate a principle incorrectness of expectations of Standard (Model)
cosmology about homogeneity and isotropy of the Universe.
It also is connected with observational data about apparent hierarchy of giant astronomical
systems (stellar systems, galaxies, clusters of galaxies, superclusters of galaxies, etc.), their
megawave structure, quantization "in the Large", non-homogeneity of microwave background
Space radiation, adequately interpreted (in frameworks WU Concept) effects redshifts quantization
of quasars, etc.
The principle absence of a Limit of Hierarchy of Matter Levels asserts: "The Staircase of a
Matter" - is endless.
For orientation of the explorers, working with the observational data, in frameworks of WU
Concept the concrete characteristics of following (behind superclusters of galaxies) potentially
possible extremely large astronomical systems are calculated with using the Fundamental
parameter of Hierarchy – Chechelnitsky Number χ = 3.66(6).
The astronomical systems, belonging to the nearest hierarchy Levels of Solar-Like systems, are
characterized by external radiuses [a(k) = χk a(0), a(0) = 39.373 AU]
a(20) = 36.83, a(21) = 135, a(22) = 495, a(23) = 1815 Mpc.
It is possible to expect that in the Universe also exist and should show itself in observations (the
Solar-Like objects) – extremely large astronomical systems (ELAS), characterized by the external
radiuses (of peripherals)
a(26) = 89503, a(27) = 328177, a(28) = 1203318 Mpc.
COSMOLOGY - STANDARD MODEL:
At Last There Will Set of Long-Awaited Homogeneity...
Almost whole century the cosmology of new time is in painful expectation: It demands from
observational astrophysics to confirm a postulate, similar to dream, of the refined theory: The
Universe - is homogeneitic and is isotropic.
Unfortunately, for the supporters of Standard model the triumph of the prescriptions of the
prevailing theory permanently is sidetracked. It - lengthy and (would be desirable to believe)
instructive history.
TOWARDS TO (MEGA) WAVE UNIVERSE
What - Beyond the Horizon of (Visible) Universe?
Today, being grounded on competitive representations of Standard Model of a Cosmology and
of the Wave Universe Concept (WU Concept) it is interesting to attempt to answer on following
(probably, anticipatory and impatient) problem:
What will meet tomorrow (in XXI century - in III millenary) grown-up, more technically equipped
and, probably, conceptually more perfect astronomy, astrophysics, cosmology in the Universe -
Monotonic Desert of homogeneous "gas" (clusters, superclusters) galaxies?
Or, "having tightened" for some Levels of Matter, it will find out new, more extended
consolidated, close to stationary, astronomical objects of the extremely large sizes and masses?
Chechelnitsky A.M. Large-Scale Homogeneity or Pricipled Hierarchy of the Universe? 2
What - Behind a World of Superclusters of Galaxies?
In other words, whether there will be a cosmology hereafter again opened Le vels and Layers of
a Matter, physically isolated astronomical systems superior the sizes of observed at present
superclusters of galaxies?
Not sidetracking, we at once, here and now, attempt to answer this problem, being grounded on
representations of the Wave Universe Concept (WU Concept).
So,
# It is necessary to expect, that purposeful researches and the future successes of an
observational astrophysics really will lead to detection of astronomical objects more high rank, than
superclusters of galaxies.
#Potentially possible, to the greatest degree probable characteristics of these extremely large
astronomical objects can be indicated as outcome of the analysis in frameworks of WU Concept.
THE WAVE UNIVERSE CONCEPT (WU CONCEPT).
WAVE ASTRODYNAMICS
The Wave Universe Concept (WU Concept) and fundamental ideas of Wave astrodynamics
[Chechelnitsky, 1990 -1999] are connected with representations that the large astronomical
systems in the theoretical plan are not only multiparticle dynamic systems in sense Poincare-
Birkgof, but are considered as essentially Wave Dynamic Systems (WDS), systems, being
somewhat analogues of atom.
The Fundamental Wave Equations.
Stability, Quantization of Megasystems
The theoretical aspects of these problems (in particular, eigenproblem of the Fundamental wave
equations) both appropriate astronomical and astrophysical questions are discussed in the
monography [Chechelnitsky, 1980] and subsequent publications.
SHELL STRUCTURE OF ASTRONOMICAL SYSTEMS
The any astronomical systems of the Universe considered as wave dynamic systems (WDS)
have shell structure, in many respects similar with shell structure of Solar - planetary system
[Chechelnitsky, 1980, 1983-1986].
The exceptions in this sense and numerous satellite systems of planets do not constitute, it is
good verified by experience, observations and space experiments.
Shell Hierarchy
In that case, the astronomical system considered as WDS, is characterized by hierarchy
enclosed each other spatially and structurally (radially) of the divided areas - G[s] Shells (s = ..., -2,
-1, 0, 1, 2, 3...).
Inducing experience in research of wave shell structure of any astronomical systems are the
results of an experimental research of Solar system - most in details and authentically known
astronomical system.
In Solar - Planetary system some spatially divided shells - can be clearly identified, at least -
G [0] - Intra - Mercurian;
G [1] - taken by space of planets I (Earth) group;
G [2] - taken by space of planets II (Jupiter) group;
G [3] - Trans - Pluto etc.
Sound Velocities Hierarchy.
Fundamental Parameter of Hierarchy
The hierarchy of the C∗[s] sound velocities - phase velocities of the (multicomponent cosmic
medium) cosmic plasma small perturbations (megawaves) [Chechelnitsky, 1984,1986] is closely
connected with the hierarchy of G[s] Shells
/G4B∗[s] = (1/χs-1)⋅ /G4B∗[1], (s = ..., -2, -1, 0, 1, 2, 3,...),
where /G4B∗[1] = 154.3864 /G64/G66⋅s−1 is the calculated value of sound velocity in the G[1] Shell, that was
made valid by observation, and
χ=11/3=3.66(6) – is the Fundamental parameter of hierarchy (Chechelnitsky Number)
[Chechelnitsky, (1978) 1980 -1988].
Chechelnitsky A.M. Large-Scale Homogeneity or Pricipled Hierarchy of the Universe? 3
"Magic Number" (Chechelnitsky Number, FPH) χ χ=3,66(6).
Role and Status of Fundamental Parameter of Hierarchy in Universe.
Previous after primary publications [Chechelnitsky, 1980-1985] time and new investigations to
the full extent convince the theory expectations, in particular, connected with the G[s] Shells
hierarchy in each of such WDS, with the hierarchy of Levels of matter (and WDS) in Universe, with
the exceptional role of the introduced in the theory χ FPH [Chechelnitsky, (1978) 1980-1986].
The very brief resume of some aspects of these investigations may be formulated in frame of
following short suggestion.
Proposition (Role and Status of χ FPH in Universe) [Chechelnitsky, (1978) 1980-1986]
# Τhe central parameter, which organizes and orders the dynamical and physical structure,
geometry, hierarchy of Universe
∗ "Wave Universe (WU) Staircase" of matter Levels,
∗ Internal structure each of real systems - wave dynamic systems (WDS) at any Levels of
matter, is (manifested oneself) χ - the Fundamental Parameter Hierarchy (FPH) - nondimensional
number χ =3,66(6).
# It may be expected, that investigations, can show in the full scale, that χ - FPH, generally
speeking, presents and appea res everywhere - in any case, - in an extremely wide circle of
dynamical relations, which reflect the geometry, dynamical structure, hierarchy of real systems of
Universe.
We aren't be able now and at once to appear all well-known to us relations and multiple links, in
which oneself the [Chechelnitsky] χ=3.66(6) "Magic Number" manifests.
We hope that all this stands (becomes) possible in due time and with new opening opportunities
for the publications and communications.
Chechelnitsky A.M. Large-Scale Homogeneity or Pricipled Hierarchy of the Universe? 4
HIERARCHY STRUCTURE OF WAVE UNIVERSE.
STEPS OF HIERARCHY. A STAIRCASE OF MATTER
Hierarchy of Solar - Similar Systems.
According to representations of the Wave Universe Concept (WU Concept) the Hierarchy of
Solar - similar systems (Solar-like Systems - SL Systems) can be shown, first of all, by Homology -
by Homologous series of Main dynamic parameters - parameters of the Kepler
K(k) = χk K (0) k= …, -2, -1, 0, 1, 2, …
where
K(0) = K/G7E = 1.32712438 ⋅1011 /G64/G663⋅s-2 - Main dynamic parameter - parameter of the Kepler -
Gravity parameter of the Solar System (Sun);
χ = 3.66(6) - Fundamental parameters of hierarchy (Chechelnitsky Number) [Chechelnitsky,
(1978) 1980, 1980-1986];
k = 1,2,3, ... - countable parameter.
The Sun and Solar system are most well investigated objects of a population of stars and
planetary systems, in many respects are typical, steady enough, well observed representatives of
a Layer (and Levels) of Matter - stars - one of the brightest component of a Staircase of Matter. It is
necessary to expect, that been transposed in other Levels of Matter, the representatives of a
Homologous series K(k) also will appear by reference, quite typical, steady enough, it is good and
widely observed objects.
In other words, it is necessary to expect, that K(k) and at other Levels of a Matter U(k) also will
appear quite representative, widespread objects, such as the Sun and Solar system.
These expectations can be affirmed by the observational data.
The Basis for Selection of Solar-Like Systems.
We are reverted to the analysis of Hierarchy of Solar - like systems, at least, by virtue of following
circumstances:
# Determinancy.
Dynamic and physical properties of the Sun and the Solar systems are known with
extraordinary accuracy (as contrasted to by set of diverse astronomical objects). It means, as all
Homologous series of Solar-similar systems is representable quite definitely with reasonable
accuracy.
# Representativity.
The Sun - as an star is the quite typical representative of a Layer of stars in a Staircase of a
Matter. About it speaks, for example, a median position of the Sun on the Hertzsprung-Russel
diagramm. It is necessary to expect, that other generated components of a Homologous series of
Solar–like systems (for example, in a Layer of galaxies) also will be quite representative objects at
the conforming Levels of a Matter.
Hierarchy of External Sizes of Systems (Plut /G68-Like Orbits).
As a subject for the analysis it is interesting to consider also Homologous series of Pluto-Like
Orbits - PL Orbits, External PL Sizes, connected with Hierarchy of Solar - like systems.
If to accept as a generating component (Eponym) the semi-major axis of orbit of Pluto ( /G4A) in a
Solar System (SS)
/G5A/G0B /G68 /G0C = aSS,P = 39.37364 AU = 0.0001900889 pc,
than a Homologous series of external (PL) sizes of Solar-Like systems will look like this:
/G5A/G0B /G64 /G0C = χ/G64/G5A/G0B /G68 /G0C = χ/G64aSS,P = χ/G6439.37364 AU = χ/G64/G13/G11/G13/G13/G13/G14/G1C/G13/G1B/G1B/G1C/G03 /G6A/G6B /G0F
where k = 1, 2, 3, ... - countable parameter,
χ = 3.66(6) - Fundamental parameter of hierarchy (Chechelnitsky Number).
/G2C/G57/G03 /G4C/G56/G03 /G51/G48/G46/G48/G56/G56/G44/G55/G5C/G03 /G57/G52/G03 /G48/G5B/G53/G48/G46/G57/G0F/G03 /G57/G4B/G44/G57/G0F/G03 /G56/G4C/G50/G4C/G4F/G44/G55/G4F/G5C/G03 /G57/G52/G03 /G57/G4B/G48/G03 /G56/G48/G50/G4C/G50/G44/G4D/G52/G55/G03 /G44/G5B/G4C/G56/G03 /G0B/G55/G44/G47/G4C/G58/G56/G0C/G03 /G5A/G0B /G68 /G0C = aSS,P of Pluto orbit,
reflecting the peripheral size of a Solar system, the radiuses a(k) of a Homologous series of
external (PL) sizes will describe (maximal) peripheral - external sizes - radiuses of astronomical
systems of corresponding U(k) Levels of Matter. Its are the most simplly and directly observed in an
astrophysics values – linear dimensional characteristics of astronomical systems - clusters of stars,
galaxies [see the Table].
Extremely - Large Astronomical Systems. First Steps.
Analysing a Homologous series of external (PL) sizes, it is possible at once directly to indicate
potentially possible existence in the Universe of extremely - large astronomical systems (objects). Chechelnitsky A.M. Large-Scale Homogeneity or Pricipled Hierarchy of the Universe? 5
It is possible to expect, for example, that in the Universe exist and should show itself in
observations (Solar-Like objects) - extremely large astronomical systems (ELAS), characterized by
the sizes (external) radiuses of peripherals
/G5A(26) = χ26 aSS,P = 89503 M /G6A/G6B, a(27) = 328177 M /G6A/G6B, a(28) = 1203318 M /G6A/G6B.
It - apparently, steady enough objects of new, nearest, potentially existing (ELAS) Layer of
Matter. The distinguished nature of these pointed Levels of Matter in new (ELAS) Layer of Matter
can be realized also from following simple heuristic consideration.
If to consider a Level of Matter - Solar system (Layer of stars) (external size) in any dynamic
sense conforming to a Level of Matter - of our Galaxy (or of galaxy /G4631 Andromeda) (Layer of
galaxies) (external size /G5AG), than for following higher (ELAS) Layer of Matter (the external size
/G5AELAS) is possible to record a following ratio (of similarity) of external sizes of astronomical systems
aG / aSS,P ∼ aELAS / aG
From here we have aELAS ∼ (aG)2/aSS,P
In more detail such conclusion follows from consideration, connected with definite isomorphism
of Layers of Matter.
Stability - Observability.
The general reasons about a problem "Stability - Observability" can be found out in the
monography (Chechelnitsky, 1980).
But the problem of stability, so, and actual observability of objects of high Levels of hierarchy
represents the special, non trivial problem. It merits the special discussion.
Isomorphism of Layers of Matter - Stars and Galaxies.
From the point of view of WU Concept representations the hierarchical structure of themselves
Layers of Matter in definite dynamic sense - is look-alike.
The definite similarity - isomorphism of Layers of a Matter - stars and galaxies can be observed
if to compare, for example, Levels of Matter U(0) of a Layer of stars and U(13) (or U(14)) of Layer of
galaxies and further, accordingly, subsequent Levels of Matter (U(1) and U(14), U(2) and U(15), etc.).
In that case it appears, that the aggregates - clusters of galaxies (clusters, clusters of clusters,
etc) correspond to aggregates - clusters of stars (clusters of stars, spherical clusters, etc). The
capability also opens to present dynamic structure nowadays of unknowns at high Levels of a
Layer of Matter - galaxies on the basis of comparison with dynamic stucture of known high Levels
of Matter of a Layer of Matter - of stars.
The Nearest Levels of Matter.
But today, apparently, the problem of existence of grandiose astronomical systems located on
the nearest steps of Hierarchy of the Universe is most actual. It is dictated by capabilities and
technical limitations of a modern observational astrophysics.
If to consider, that at present clearly identifiable astronomical systems (the superclusters of
galaxies) are characterized by external sizes - radiuses of the order
a ∼ 30 h-1 /G46/G6A/G6B (Peebles, 1980 (1983)),
a ∼ 35 /G46/G6A/G6B (LSS: Rudnicki, Zieba),
a ∼ 40(50) /G46/G6A/G6B (LSS: Kalinkov et al)
that its, most likely, belong to (or are close to) a Level of Matter characterized by an external radius
of a Solar–Like system /G5A(20) = χ20aSS,P = 36.83 /G46/G6A/G6B.
The astronomical systems, belonging to the nearest hierarchy Levels of Solar-Like systems, in
that case are introduced by external radiuses
/G5A(21)/G03/G20/G03/G14/G16/G18/G03 /G46/G6A/G6B /G0F
/G5A(22)/G03/G20/G03/G17/G1C/G18/G03 /G46/G6A/G6B /G0F
/G5A(23) = 1815 /G46/G6A/G6B
It is interesting to mark, that some explorers working with the observational data (LSS: Rudnicki,
Zieba), indicate distinguished nature of the following sizes (radiuses)
/G5A ∼ 35, 128, 421 /G46/G6A/G6B.
(See also Einasto, 2000 with a ∼ 130 Mpc).
Whether is contingency the close conformity of these data obtained from processing of the
cataloques (Abell, Zwicky) of galaxies clusters with the analysis and expectations of the Wave
Universe Concept?
Chechelnitsky A.M. Large-Scale Homogeneity or Pricipled Hierarchy of the Universe? 6
The Law of Generalized Dichotomy and External Sizes of Astronomical Systems.
At statistical processing of the cataloques of galaxies clusters it find out a series of maxima in
distribution of the characteristics of the sizes of galaxies aggregates (clusters). Thus at processing
the isolated nature of these aggregates implicitly is meant.
Agrees [LSS: Rudnicki, Zieba], this series of maxima - preferential sizes looks like
a /G68 /G45/G56 /G03/G20/G03/G16/G18/G0F/G03/G19/G16/G0F/G03/G19/G17/G0F/G03/G19/G1C/G0F/G03/G1A/G15/G0F/G03/G1B/G15/G0F/G03/G1C/G16/G0F/G03/G14/G15/G1B/G0F/G03/G14/G17/G15/G0F/G03/G14/G1C/G13/G0F/G03/G15/G13/G19/G0F/G03/G15/G1B/G1C/G03/G30 /G6A/G6B /G11/G03
In frameworks WU Concept the latent sense of such distribution can be realized. The
preferential sizes of aggregates (clusters) of galaxies appear by connected with the Law of
generalized dichotomy
/G5Ap /G20/G03 /G5A /G68⋅2p/2 , p = 0, 1, 2, 3, …,
characteristic for the dominant sizes (and other parameters) of astronomical systems
(Chechelnitsky, 1984, 1992, 1999). This Law is some generalization (in frameworks of WU
Concept) of known Law a Titius-Bode for planetary orbits.
In this connection it is interesting to mark following conformity of the observational data with
se /G55/G4C/G48/G56/G03/G52/G49/G03/G2A/G48/G51/G48/G55/G44/G4F/G4C/G5D/G48/G47/G03/G27/G4C/G46/G4B/G52/G57/G52/G50/G5C/G03/G44/G57/G03/G0B /G5A /G68/G03/G20/G03/G16/G19/G03/G30 /G6A/G6B ∼ /G03 /G5A(20)/G03/G20/G03/G16/G19/G11/G1B/G16/G03/G30 /G6A/G6B /G0C
/G5Ap = 36⋅2p/2/G03/G30 /G6A/G6B /G0F p = 0, 1, 2, 3, …,
/G5Ap/G03/G03/G03/G03/G03/G16/G19/G03/G03/G03/G03/G18/G13/G11/G1C/G14/G03/G03/G03/G03/G1A/G15/G03/G03/G03/G03/G14/G13/G14/G11/G1B/G15/G03/G03/G03/G03/G14/G17/G17/G03/G03/G03/G03/G15/G13/G16/G11/G19/G03/G03/G03/G03/G15/G1B/G1B/G03/G30 /G6A/G6B
aobs/G03/G03/G16/G18/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G1A/G15/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G03/G14/G17/G17/G03/G03/G03/G03/G15/G13/G19/G03/G03/G03/G03/G03/G03/G03/G15/G1B/G1C/G03/G30 /G6A/G6B
Besides it is necessary to mark individual ratio of dichotomy for the falling out observational
data
a /G68 /G45/G56 ∼ /G03/G19/G17/G0F/G03/G14/G15/G15/G03/G30 /G6A/G6B /G03→ 2× /G19/G17/G03/G30 /G6A/G6B /G03/G20/G03/G14/G15/G1B/G03/G30 /G6A/G6B
a /G68 /G45/G56 ∼ /G03/G1C/G16/G0F/G03/G14/G1C/G13/G03/G30 /G6A/G6B → 2× /G1C/G16/G03/G30 /G6A/G6B /G03/G20/G03/G14/G1B/G19/G03/G30 /G6A/G6B ∼ /G03/G14/G1C/G13/G03/G30 /G6A/G6B
Causing for Hope for Homogeneity?
The researches of last time, generally speaking, have not changed the information on
availability of heterogeneities - distinguished scales (maxima) of /G6Blustering of galaxies.
But the volume of the processed data considerably has increased, as well as quantity of a
usedprocessing techniques. It gives the basis to some explorers [ /G44/G5Alinkov et al. 1998] to suppose,
for example, following:
" It seams that there are no structures of superclasters of galaxies ".
It is understandable, that such conclusion lies quite in a channel of Standard model and owes,
in next time, with gladness to be hailed by the representatives of Standard cosmology Mainstream.
Unfortunately, this brief conclusion not absolutely corresponds even to the contents of this large
and valuable work. At desire in it is possible to see, for example, availability of distinguished scales
(see Kalinkov t al., 1998, Fig. 1,2,3)
aobs ∼ 90h-1 Mpc , 325h-1 Mpc etc.
At a Hubble constant /G47=65 /G64/G66⋅s-1⋅Mpc-1 (h = 0.65) it gives distinguished scales,
aobs ∼ 138, 500 Mpc,
quite comparable with external radiuses of giant astronomical systems
/G5A(21) = 135 Mpc /G0F/G03/G03 /G5A(22) = 495 Mpc
Generally speaking, it is follows with fear to approach to results of statistical processing of
rather vast, but heterogeneous stuff. Occasionally, the desired signal (availability of maxima) is lost
("is washed") in a massif of such rich data set. Be it – case history to a maxim "best - enemy of
good".
In any case, detail critical analysis of such works merits separate, special discussion.
View Ad Infinitum:
About a Capability of Existence of Extreme - Large Astronomical Systems.
If not to limit by consideration only of nearest Levels and Layers of Matter, the Wave Universe
Concept gives a capability to investigate (for the present - theoretically) an hierarchical stucture of
the Universe, in principle, at any Levels and Layers of Matter. All Hierarchy of the Wave Universe,
all Staircase of Matter - potentially is opened for free and unb iassed researches object.
This polygon for possible researches extends (even at present familiar - with verifiable modern
experiments – physical world) many tens orders: downwards - deep into Matter in subatomic world
and hill up - in world of extreme – large astronomical systems.
In this connection the fundamental (in definite sense - epistemological or metaphysical)
statement of WU Concept is imply following: Chechelnitsky A.M. Large-Scale Homogeneity or Pricipled Hierarchy of the Universe? 7
The Assertion (Hierarchy Ad Infinitum)
There is No limit of Hierarchy (of Staircase of Matter).
This extreme brief statement, at least, does not limit by dogmas the horizon of a cosmology of
the Future, does not bar creative, search tendencies of the explorers.
DISCUSSION
The Standard Model of a modern cosmology does not test the special doubts that the Universe
"As a whole" ("All Universe" - in the issue) is homoheneitic and is isotropic. But the impartial and
critical analysis demonstrates, that such reliance, mainly, reposes on formerly none-critically
adopted by the fathers-founders (of modern cosmology) postulates, and later - it is already simple
– on tradition of following to authorities, habit, mode.
The scandalous gap with observed properties of the actual Universe is eliminated by the ad hoc
confirmation that nevertheless at a definite stage in time and space the Universe (why?) ceases to
be homogeneous and therefore appear reference observed features an actual hierarchic world -
the world of atoms, planetary, star galactic systems.
The Wave Universe Concept tumbles this model and puts it from a head on legs, first of all, - on
the apparent, fundamental observational basis. The observed Hierarchy of Levels of Matter
extends many tens orders and there are no visible causes, by virtue of which one this hierarchy
should interrupt at any Level of Matter. You see then it will be special, really physically
distinguished Level of Matter. Here again not only it is desirable, but it is necessary to result
extremely severe, nontrivial, convincing physical arguments, by virtue of which one the so series
universal property of hierarchy of Universe is for some reason upset.
Moreover, the Wave Universe Concept indicates the causes, circumstances, arguments, by
virtue of which one the phenomenon of Hierarchy is an indispensable, natural consequent of some
more fundamental laws of nature. Briefly speaking, the observed Hierarchy of the Universe is to
straight lines a consequent of a Wave (Megawave) constitution of the actual Universe, consequent
that it is grandiose composition of the enclosed Wave dynamic systems (WDS) at all Levels of
Staircase of Matter.
Thus the immanent wave (megawave) properties spontaneously and directly are connected to
properties of quantization (including, and quantization "in the Large"), commensurability (as inside
WDS, and between them - at miscellaneous Levels of Matter) and, thus - with properties and laws
of Hierarchy.
In order to prevent barren loss of time and efforts, cosmologists, which one with the tight
attention prolong to expect approach of epoch of total homogeneity and isotropy of the Universe, it
is necessary to advise - more often to recall astern wisdom: "If you very wait for the friend, do not
accept knocking the heart for stamp of hoofs of his horse".
Chechelnitsky A.M. Large-Scale Homogeneity or Pricipled Hierarchy of the Universe? 8
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HIERARCHY OF UNIVERSE. STAIRCASE OF MATTER .
HIERARCHY OF SOLAR – LIKE SYSTEMS: EXTERNAL RADIUSES
Levels
of Matter U(k)
OBSERVATIONS
Layers
of
Matter
a(k)=χ χ kass,p;
χ χ=3.66(6) External
Radiuses (Ref.: LSS, Allen, Peebles, Rudnicki, Kalinkov, Holopov, Sharov, etc.)
a(32)=χ 32ass,p 217.5⋅106 Mpc
a(31)=χ 31ass,p 59.319⋅106 Mpc
a(30)=χ 30ass,p 16.177⋅106 Mpc
a(29)=χ 29ass,p 4.412⋅106 Mpc
a(28)=χ 28ass,p 1203318 Mpc
a(27)=χ 27ass,p 328177 Mpc
a(26)=χ 26ass,p 89503 Mpc
a(25)=χ 25ass,p 24409.9 Mpc
a(24)=χ 24ass,p 6657.24 Mpc
a(23)=χ 23ass,p 1815.612 Mpc
a(22)=χ 22ass,p 495.167 Mpc 421 Mpc [LSS:Rudnicki,Zieba]
a(21)=χ 21ass,p 135.045 Mpc 128 Mpc [LSS:Rudnicki,Zieba]; 120÷150 Mpc [LSS:Kalinkov et al.]
E
L
A
S
a(20)=χ 20ass,p 36.830 Mpc 35 Mpc [LSS:Rudnicki,Zieba]; 40÷50 Mpc [LSS:Kalinkov et al.]
a(19)=χ 19ass,p 10.044 Mpc r0∼5h-1Mpc∼7÷10 Mpc [LSS:Peebles]; 5.7 Mpc (Pis) [Allen]
a(18)=χ 18ass,p 2.73946 Mpc 2÷3.5 Mpc [LSS:Peebles];
a(17)=χ 17ass,p 0.7471 Mpc 0.57 Mpc (Peg) [Allen]
a(16)=χ 16ass,p 0.2037 Mpc 0.15 Mpc (Her) [Allen]
a(15)=χ 15ass,p 55.571 kpc 46÷68 kpc (MW) [(Kukarkin; Holopov]; 30÷36 kpc (MW) [Zinn];
35÷50 kpc (M31 And) [Sharov];
a(14)=χ 14ass,p 15.155 kpc 11.5 kpc (M1O1) [Allen]
a(13)=χ 13ass,p 4.133 kpc 3.5 kpc (M82), 4 kpc (M104), 4.5 kpc (M51) [Allen]
a(12)=χ 12ass,p 1.12729 kpc 1 kpc (NGS 6822,205) [Allen]
a(11)=χ 11ass,p 0.3074 kpc 0.5 kpc (M32,NGC 147, 185) [Allen]
G
A
L
X
I
E
S
a(10)=χ 10ass,p 83.848 pc 80 pc (χ,h Per); 64pc(M3)[Holopov]; 70pc(Gum);
85pc(NGC 2070) Diffuse Nebula;
a(9) =χ 9ass,p 22.867 pc 19pc(NGC 7243) [Holopov];
16pc (NGC 2419)Globular Cluster [Allen, p.400];
50pc Sco-Cen) –Open Cluster [Allen, p.396];
a(8)=χ 8ass,p 6.236 pc 6.9pc(M44) [Holopov]; 6pc(Per);
7pc(χ Per) Open Cluster [Allen, p.396];
a(7)=χ 7ass,p 1.7009 pc 1.5pc (NGC 2632, I2602);
2pc (NGC2632,2682,4755,6531) Open clusters
a(6)=χ 6ass,p 0.4638 pc 0.3pc (M78); 0.5pc (NGC 7023) – Diffuse Nebula;
a(5)=χ52ass,p 0.1265 pc 0.1pc(NGC 3132,6720); 0.15pc (NGC6853) – Planetary Nebula;
0.15pc (NGC 2261) – Diffuse Nebula;
a(4)=χ 4ass,p 0.0345 pc 0.03pc (NGC 7662); 0.035pc (NGC 7027);
0.04pc (NGC 3918, 6210, 6543, 7009); [ Allen]
a(3)=χ 3ass,p 1940.9 AU
a(2)=χ 2ass,p 529.35 AU
a(1)=χ 1ass,p 144.37 AU
S
T
A
R
S
a(0)=χ 0ass,p 39.37364 AU Solar System: Radius of Pluto Orbit
|
arXiv:physics/0102009v1 [physics.bio-ph] 5 Feb 2001Self-adaptive exploration in evolutionary
search
Marc Toussaint
Institut f¨ ur Neuroinformatik
Ruhr-Universit¨ at Bochum, ND 04
44780 Bochum - Germany
www.neuroinformatik.ruhr-uni-bochum.de/PEOPLE/mt/
Abstract
We address a primary question of computational as well as bio logical research on evolution: How can
an exploration strategy adapt in such a way as to exploit the i nformation gained about the problem at
hand? We first introduce an integrated formalism of evolutionary s earch which provides a unified view
on different specific approaches. On this basis we discuss the implications of indirect modeling (via a
“genotype-phenotype mapping”) on the exploration strateg y. Notions such as modularity, pleiotropy
and functional phenotypic complex are discussed as implica tions.
Then, rigorously reflecting the notion of self-adaptabilit y, we introduce a new definition that captures
self-adaptability of exploration: different genotypes tha t map to the same phenotype may represent
(also topologically) different exploration strategies; se lf-adaptability requires a variation of exploration
strategies along such a “neutral space”. By this definition, the concept of neutrality becomes a central
concern of this paper.
Finally, we present examples of these concepts: For a specifi c grammar-type encoding, we observe a
large variability of exploration strategies for a fixed phen otype, and a self-adaptive drift towards short
representations with highly structured exploration strat egy that matches the “problem’s structure”.
Keywords
Exploration, self-adaptability, evolvability, neutrali ty, modularity, pleiotropy, functional phenotypic
complex.
1 Introduction
Typically, when a problem is given, the space of all potentia l solutions is too large to try all
of them in reasonable time. If not making anyfurther assumptions on the problem, there
neither exists a preferable strategy to search for solution s. Usually though, one assumes
that the problem is not notoriously arbitrary, that it has so me “structure” and that there
might exist some smart strategies to explore the space. More specifically, one hopes that
one can draw information from the quality of previously expl ored solutions on how to choose1 INTRODUCTION 2
new explorations. For example, when assuming some “continu ity”1of the problem, one may
search further in regions of previously explored good solut ions.
A more elaborated strategy is the following: Analyze the sta tistics of previously found
solutions, find correlations between certain characters (p arameters) of the solution and the
solution’s quality, find mutual information between the cha racters of good solutions, etc., and
exploit all this information to choose further exploration s — in the hope that these findings
really characterize the problem, that the problem is charac terizable by such information. In
essence, the latter approach will explore only a tiny part of P, strongly dependent on early
explorations that have been successful. Found solutions ma y lay no claim to be globally
optimal; they are a further development of early successful concepts.
The central questions become: How can we analyze the statistics of previously explored
and evaluated solutions? How can we represent this gained in formation? How can we model
an exploration strategy depending on this information?
One direct approach to these questions leads to statistical models of exploration. For ex-
ample, a Bayesian network can encode the probability of futu re explorations (the exploration
density ) and is trained with previously successful solution parame ters (as done by Pelikan,
Goldberg, and Cant´ u-Paz (2000), see appendix A). In contra st, we will argue that the explo-
ration strategy can be modeled by a mapping onto the solution space, a genotype-phenotype
mapping. This means that a (simple) density on a base space (g enotype) is lifted to the
exploration density on the search space (phenotype). The im plications of such an ansatz
are far-reaching: An exploration density now exists on both , the base space and the search
space. In both spaces notions as neighborhood or topology sh ould be constituted only by the
exploration density. In this respect, the genotype-phenot ype mapping is a liftof (topological)
structure from the base space to the search space.
To investigate the implications, we assume that the explora tion density on the base space
is one of independent random variables. Then, for a given mapping, we investigate the ex-
ploration density on the search space; in particular the cor relations and mutual information
between phenotypic variables. This structuredness of phen otypic exploration coherently im-
plies notions as “modularity” and “functional phenotypic c omplex”. Concerning the adaption
of this structure, we will argue for a self-adaptive mechani sm, in place of a statistical analysis
of characters of good solutions (as with the Bayesian ansatz ). A major goal of this paper is
formal and notational clarity of such issues.
The paper is organized as follows: The next section starts by introducing a general notation
of evolutionary search. This notation emphasizes the role o f the exploration density in the
search space and, even more, the way of parameterization of t his density. We call the latter
“exploration model”. An important point of this section is t hat most evolutionary algorithms
differ just by this exploration model. Since it distracts fro m the major line of this paper we
moved this reinvestigation of existing evolutionary algor ithms to appendix A.
In section 3 we introduce and formalize the idea of indirect m odeling. Instead of param-
1which requires to identify a topology on the search space2 THE CENTRAL ROLE OF THE EXPLORATION MODEL 3
eterizing the exploration density directly on the search sp ace, we introduce the additional
base space, parameterize a simple density thereon, and lift this density on the search space.
We compare this to the lift of a topological or metrical struc ture onto a manifold from a
simple-structured base space. Notions as pleiotropy and fu nctional phenotypic complex are
discussed as implications of such a lift. We also relate the b iological view on indirect modeling
(here, via a genotype-phenotype mapping) and adaptive expl oration to our formalism.
Section 4 begins by reflecting and criticizing the usual defin ition of self-adaptability. We
introduce a new definition which is based on the notion of neut rality: Different genotypes
that map to the same phenotype may represent (also topologic ally) different exploration
densities. Thus, such genotypes may represent very differen t information and neutrality is
not necessarily a form of redundancy as is often claimed. By t his definition, neutrality becomes
a central concern and we briefly review other research on this subject in order to argue for
the plausibility of our interpretation.
Finally, in section 5 we exemplify all these concepts with a r unning system. Simulations
show that the exploration density adapts to the problem stru cture by (self-adaptive) walks
on neutral sets. In particular, the pair-wise mutual inform ation between phenotypic variables
resembles the modularity of the fitness function. We also obs erve and explain a drift towards
short representations. The experiments are based on a gramm ar-type recursive encoding
which is thoroughly motivated by the previously developed c oncepts.
2 The central role of the exploration model
The goal of this section is to show that the central concern of evolutionary search, esp.
evolutionary algorithms, is the modeling of exploration. W e will show that the main difference
between specific evolutionary algorithms is their ansatz to model exploration.
Perhaps the most general idea of stochastic search, global r andom search, is described by
Zhigljavsky (1991). The formal scheme of global random sear ch reads:
(i)Choose a probability distribution on the search space P.
(ii)Obtain points s(t)
1,... ,s(t)
λby sampling λtimes from this distribution. Evaluate the
quality of these points.
(iii)According to a fixed (algorithm dependent) rule construct a n ew probability distribution
onP.
(iv)Check some appropriate stopping condition; if the algorith m is not terminated, then
substitute t←t+ 1 and return to step (ii).
This concept is general enough to include also evolutionary algorithms. However, the for-
mulation lacks to stress that the exploration density needs to be parameterized (and instead2 THE CENTRAL ROLE OF THE EXPLORATION MODEL 4
stresses the choice of update rule in step (iii)). We will str ess the parameterization of explo-
ration densities and call it the exploration model . It is this model that we focus on. We now
formalize evolutionary search in analogy to global random s earch, but with different focus:
In general we assume that the task is to find an element pin a search space Pwhich is
“superior” to all other points in P. Here, superiority is defined in terms of a quality measure
for the search problem at hand (usually a fitness function). I fPis too large to evaluate the
quality of all p∈P, the strategy is to explore only a few points ( p1,..,p λ), evaluate their
quality, and then try to extract information on where to perf orm further explorations. We
capture this view on evolutionary search in an abstract form alism that is capable to unify the
different specific approaches. Below, we exemplify each step of the scheme by embedding the
Simple Genetic Algorithm (SGA) (Vose 1999) in the formalism . See also figure 1.
Definition 1 (Evolutionary exploration)
(i)The only information maintained for evolutionary search is a finite set of parameters
q(t)∈Qthat uniquely define an exploration density Mq(t)onP. Here, we call M
theexploration model , actually a map from Qto the space Λof densities over P. In
general, the variety MQ={Mq|q∈Q}of representable densities is limited.
(ii)Given some parameters q(t), exploration starts by choosing λsamples s(t)
i=1..λof the
exploration density. We use brackets to indicate this sampl ing:
s(t)= [Mq(t)]λ∈Pλ. (1)
Here and in the following, we disregard the possibility of el itists. Taking them into
account would require to append selected points (p1,..,p µ)ofPtos(t),
s(t)= [Mq(t)]λ⊕(p1,..,p µ)∈Pµ+λ. (2)
(iii)We require the existence of an evaluation E:Pλ→Λwhich maps the exploration
sample (s1,..,s λ)to a density over Pwith support{s1,..,s λ}. This evaluation is applied
to our exploration points:
Es(t)=E/parenleftbig
[Mq(t)]λ/parenrightbig
∈Λ. (3)
One should interpret Eas “density of quality” rather than a probability density.
(iv)Finally, there exists an update operator
A:q(t)×Es(t)∝mapsto→q(t+1). (4)
In general, this operator is supposed to exploit the informa tion in Es(t).
Example (The Simple Genetic Algorithm)
(i)The SGA (without crossover) is a typical example of population-based modeling :
q(t)= (p1,..,p µ)∈Pµis a discrete population and ˜Mpispecifies the offspring density2 THE CENTRAL ROLE OF THE EXPLORATION MODEL 5
for each single individual. We call ˜Mpiexploration kernels . The total exploration
density reads
Mq=1
µµ/summationdisplay
i=1˜Mpi. (5)
We note that the key feature of population-based modeling is its capacity to represent
multi-modal exploration densities.
(ii)In the SGA, s(t)are new offsprings. The algorithms does not explicitly const ruct the
complete exploration density Mq(t); rather, the drawing of mutations for each individual
resembles a sampling of the exploration kernels.
(iii)For the SGA, evaluation is proportional to a given fitness fun ction.
(iv)The update rule of the SGA can be written as
q(t+1)=/bracketleftBig
E/parenleftbig
[Mq(t)]n/parenrightbig/bracketrightBig
n. (6)
In words: From the parent population q(t)generate noffsprings [ Mq(t)]n, evaluate their
fitness and select nnew individuals by sampling their evaluation.
One might assume that evolutionary algorithms mostly differ with respect to the update
rule. However, we claim that the choice of the exploration mo del is crucial and that, given
such a model, two generic update operators are canonical and widely in use:
Definition 2 (Adopting and approaching updates)
It is the adopting update to choose the update operator such that Mq(t+1)is a best possible
approximation of Es(t)within the model class MQ(with respect to some chosen metric D
onΛ):
q(t+1)= argmin
qD(Mq:Es(t)). (7)
We will abbreviate this formula by using the simplified notat ionA=M−1:
q(t+1)=M−1(Es(t)). (8)
Second, many algorithms realize not an adopting but rather a napproaching update by
slowly adapting q(t). Here, the parameters must be continuous. The generic updat e rule
reads
q(t+1)= (1−α)q(t)+α M−1(Es(t)), (9)
for some constant α∈[0,1].3 AN INDIRECT MODEL OF EXPLORATION 6
samplingdensityexploration
Mq(t)model M
evaluationpointsexplored
s(t)=/bracketleftbig
Mq(t)/bracketrightbig
λ(e.g., A=M−1)
new
informationq(t+1)
update A
Es(t)q(t)parameters parameters
Figure 1: The general scheme of evolutionary search.
Example (Update operator of the SGA)
The update operator of the SGA is strongly related to the adop ting update: The sampling
[Es(t)]nof the evaluation density can be interpreted as “finding new p arameters q(t+1)that
approximate Es(t)in the population-based model”. The quality of this approxi mation is
reflected by the sampling error.
Both of these canonical update operators are derived from M−1. Thus, when we show
that most existing evolutionary algorithms realize these o perators, then we stress the impor-
tance of the choice of exploration model. Note that any algor ithm, when embedded in the
upper formalism, is uniquely characterized by the choice of model M, the update operator A
(eventually derived form M), the evaluation E(given at hand) and the sampling size λ.
It is, of course, possible to think of exceptions that cannot be embedded in this formalism.
However, in appendix A we show how the formalism allows an emb edding of – and a unified
view on – very different state-of-the-art evolutionary algo rithms. Indeed, those evolutionary
algorithms mainly differ with respect to their exploration m odel.
3 An indirect model of exploration
After we stressed the importance of exploration modeling we concentrate on the specific case
of modeling defined as follows:
Definition 3 (Indirect exploration modeling)
To model an exploration density over P, introduce a base space G=Xnand abase
density MG
qoverGsuch that the variables x∈Xare independent with respect to MG
q.
Then, introduce a GP-map h:G→Pthat induces the exploration density Mq=MG
q◦h−1
overP. Here, h−1(p)⊂Gis a subspace of Gcalledneutral space ofp∈P; and
MG
q/bracketleftbig
h−1(p)/bracketrightbig
is evaluated via integration. The class of allowed GP-maps a nd base densities
limits this model M. The triplet (G,h,MG)is also referred to as coding .
In the following, in order to refer to their biological inter pretations, we will also use the
names phenotype space for the search space P,genotype space for the base space G, and
phenotype-genotype mapping forh.3 AN INDIRECT MODEL OF EXPLORATION 7
Also, we call the independent variables x∈Xgenes and say we introduce genes on
Pwhen introducing such a GP-map and stressing the introducin g of a representation via
independent variables. This can be seen in analogy to the introduction of local coordinates
on a manifold by a local map from a base space of (Cartesian) va riables. There is, however,
a crucial difference: The map hdoes not need to be one-to-one. If his non-injective, there
exist different genotypes githat map to the same phenotype. Then there exist different
neighborhoods Ugithat map to eventually different neighborhoods of the same phenotype.
This change of neighborhood is of major interest. It allows a variability of exploration. The
next section will address this important issue in detail.
As an example for indirect modeling, note that the CMA (see ap pendix A) may be inter-
preted as indirect modeling: it restricts the class of GP-ma ps to affine transformations; the
translational part is encoded in the population’s center of mass and the linear part is encoded
in the covariance matrix; the base space is G=Rnwith normal density N(0,1).
3.1 Characters of indirect exploration: Pleiotropy, mutua l information, lift
of topology, neutrality
The introduction of a GP-map leads to some straightforward d efinitions and notions. We use
this section to briefly introduce some.
Pleiotropy. In a biological context one may define pleiotropy as “the phen omenon of one
gene being responsible for or affecting more than one phenoty pic characteristic”. Our previous
definitions allow to translate this notion into our formalis m: Genes are independent (with
respect to the base density) variables of G. One gene affecting more than one variable of P
means that the change of one variable in Gleads to the change of many variables in P. Thus
pleiotropy means that the base density of independent varia bles is mapped on an exploration
density of non-independent variables; pleiotropy may be me asured by the correlatedness of
variables of Pwith respect to the exploration density. We refer to this als o as structure of
the exploration density. In particular, we will measure ple iotropy as the mutual information
contained in the exploration density.
Population-based indirect modeling. Population-based modeling was defined in section
2. We briefly clarify notations in the indirect modeling case : The parameters q∈Qare a
population ( g1,..,g µ)∈Gµon the base space and the exploration kernels ˜MG
giare such that
the total exploration density reads:
Mq=MG
q◦h−1=/bracketleftBig1
µµ/summationdisplay
i=1˜MG
gi/bracketrightBig
◦h−1=1
µµ/summationdisplay
i=1/bracketleftBig
˜MG
gi◦h−1/bracketrightBig
=:1
µµ/summationdisplay
i=1˜Mgi. (10)
Lift of topology. For population-based modeling, the exploration kernels as sociate a den-
sity of offsprings to each individual. Form a topological poi nt of view, this defines a neigh-
borhood (of most probable offsprings) for each individual, r eferred to as variational topology.3 AN INDIRECT MODEL OF EXPLORATION 8
In the case of indirect modeling, the kernels ˜MG
gon the base space are lifted to kernels
˜Mg=˜MG
g◦h−1on the search space. This means a lift of topology.
Neutrality. The possibility of a non-injective GP-map hautomatically leads to the defini-
tion of neutrality.2In particular we define h−1(p) as the neutral set of p∈P. Further, the
neutral degree of g∈Gis defined as the probability
˜Mg[h(g)] =˜MG
g[h−1◦h(g)]. (11)
This reads: Take some individual g∈Gand let N=h−1◦h(g) be the neutral space “around”
g. Now measure the probability ˜MG
g[N] for landing in this neutral set when exploring from
g.
Such measures are thoroughly discussed by Schuster (1996) a nd Fontana and Schuster
(1998) (see also section 4.1). However, in these publicatio ns, the variational topology rather
than the probability is emphasized. For completeness we app end: Let neighborhoods be
defined in Gand let Br(g) be the r-ball around ginG(those points linked to Gby at
least one chain of no more than rneighbors). We call the maximal connected component
Ng⊂h−1◦h(g) with g∈Ngneutral network of g∈Gand define:
|h−1(h(g))∩B1(g)| neutral degree of g∈G (12)
3.2 Indirect exploration modeling in biology
One may argue that algorithms as discussed in appendix A are h ardly plausible in nature and
thus without relevance for biology. What mechanisms should keep track of dependencies in
nature, model distributions by storing a Bayesian network o r a covariance matrix, and how
should such knowledge be taken into account when creating ne w offsprings?
Nevertheless, a biologist may in principle ask the same ques tions; we refer to Wagner and
Altenberg (1996): How comes that some phenotypic character s are obviously correlated and
others are not? How comes that a single gene in Drosophila can trigger the expression of
many others and thereby the growth of a whole eye at different p laces on the body? The
existence of pleiotropy is obvious; are its specific mechani sms an accident, an unavoidability,
or the result of evolutionary optimization? What is optimiz ed when adapting pleiotropy?
The idea of Wagner and Altenberg is that in nature the genotyp e-phenotype mapping is
adaptable and does adapt in such a way that pleiotropy betwee n independent phenotypic
characters is decreased (in order to allow for an unbiased, p arallel search) while pleiotropy
between correlated phenotypic characters may increase (in order to stabilize the optimal
relative value of these characters). For example, pleiotropy betwee n the existence of the
eye’s cornea and its photoreceptors is high because one alon e won’t contribute to selection
2More precisely, if also considering a fitness function f:P→R, we denote non-injectiveness of hby
phenotypic neutrality and non-injectiveness of fwithfitness neutrality . In this paper, only phenotypic
neutrality will be addressed to.3 AN INDIRECT MODEL OF EXPLORATION 9
probability without the other. In contrast, pleiotropy bet ween characters of the immune
system is low in order to allow a fast, parallel optimization of different protection mechanisms
which each separately contribute to selection probability . We mimic a discussion by picking
some quotations of Wagner and Altenberg (1996) and adding a c omment:
Concerning evolvability
“Evolvability is the genome’s ability to produce adaptive v ariants when acted upon by the
genetic system.” [ sec 5, par 2 ]
In our words: Evolvability denotes the capability of a syste m to model a desired exploration
distribution.
“The thesis of this essay is, that the genotype-phenotype ma p is under genetic control and
therefore evolvable.” [ sec 2, par 9 ]
In the case of indirect modeling, the GP-map induces the expl oration density on P. Conclud-
ing, though, that evolvability requires a GP-map being “und er genetic control” is questionable
from our point of view. We reflect this circumstance in detail in the section 4.
Concerning modularity
“Modularity is one example of variational property.” [ sec 1, par 3 ]
Modularity is a property of the exploration density. It deno tes correlations, i.e. mutual
information, between variables of P. We discussed such correlations in section 3.1 in the
context of pleiotropy and structure of exploration.
Concerning functional phenotypic complexes
“The key feature is that, on average, further improvements i n one part of the system must
not compromise past achievements.” [ sec 5, par 10 ]
“By modularity we mean a genotype-phenotype map in which the re are few pleiotropic
effects among characters serving different functions, with p leiotropic effects falling mainly
among characters that are part of a single functional comple x.” [abstract ]
“Independent genetic representation of functionally dist inct character complexes can be
described as modularity of the genotype-phenotype map.” [ sec 6, par 1 ]
“Evolution of complex adaptation requires a match between t he functional relationships
of the phenotypic characters and their genetic representat ion.” [ sec 6, par 6 ]
In essence, the exploration density should have the charact er that some variables in Pare
mutually independent while others are dependent. Reflectin g that adaptation can only oc-
cur by extracting information from the evaluation density Eswe claim that the notion of a
“functional complex” or a “functionally distinct [phenoty pic] character complex” may only
be constituted via this evaluation density Es. More precisely, we define a functional phe-
notypic complex as a set of variables of Pthat are highly dependent on each other (with
high mutual information) but only weakly dependent on other phenotypic characters — all
with respect to the evaluation density Es. The “required match” between these properties4 NEUTRALITY AS BASIS OF SELF-ADAPTABILITY OF EXPLORATION 10
of the exploration distribution and the evaluation distrib ution motivates the adopting or
approaching update as introduced above.
4 Neutrality as basis of self-adaptability of exploration
So far, we stressed the importance of exploration modeling a nd focused on the special case
of indirect modeling. We did not yet address the problem of ho w the exploration density can
be adapted in the indirect modeling case. This section gives an answer by providing a strict
definition of self-adaptability, which considers neutrali ty as a key feature. We will also review
other interpretations of neutrality and argue in favor of ou r interpretation.
Obviously, if exploration is described by means of fixed kern els around the positions of
individuals, the exploration density varies when individu als move on. But this does not quite
capture what we actually meant by requiring variable explor ation. Rather it is intuitive to
call for “adaptive codings”. The review (Eiben, Hinterding , and Michalewicz 1999) (and also
(Smith and Fogarty 1997)) summarizes and classifies such app roaches. Their discussion is
based on the assumption that the coding ( G,h,MG) depends on some parameters x∈X
calledstrategy parameters ; we write ( Gx,hx,MG
x). They classify different approaches by
distinguishing between different choices of X:
(i)Xare parameters altered by some deterministic rule (e.g., fu nction in time) independent
of any feedback from the evolutionary process. ( deterministic )
(ii)Xare parameters depending on feedback from the evolutionary process. ( adaptive )
(iii)Xis part of the genotype. ( self-adaptive )
Option (i) is of no interest here. It is very important to dist inguish between (ii) and (iii).
Option (ii) means to analyze the evolutionary process, name ly the evaluation density and
the exploration density itself, and deterministically ded uce an adaptation. Good examples
are the algorithms presented in appendix A. Option (iii) mea ns that adaptation becomes a
stochastic search itself — the search for a good exploration density is itself determined by
just this exploration.
However, as formulated above, following option (iii) is qui te irritating since, after adding
some strategy parameters XtoG, the GP-map hstill maps G→Pand it is formally incorrect
to think of has being parameterized by variables of G. One might want to escape this circle
by splitting Ginto two parts, the strategy part Xand the objective part ˜G,G=˜G×X.
Then, for some strategy parameters x∈X, one may define h:˜G×X→P, (g,x)∝mapsto→hx(g)
and call hxan adaptive GP-map. However, in general it is unclear which p art of Gis to
be considered as strategy part and which as objective. Only i n some cases, e.g. if simply
adding control parameters that have no direct effect on the ph enotype (neutral parameters!),
this splitting seems to be straightforward. Also, one could argue that the mutation rate of
the strategy part is kept very low. Formally and conceptuall y, though, these arguments are4 NEUTRALITY AS BASIS OF SELF-ADAPTABILITY OF EXPLORATION 11
G Ph−→g1
g2
˜MG
g2˜MG
g1h(g1) =h(g2)
˜Mg1=˜MG
g1◦h−1
˜Mg2=˜MG
g2◦h−1
Figure 2: Two different points g1,g2inGare mapped onto the same point in P. The elliptic
ranges around the points illustrate the exploration kernel s by suggesting the range of probable
mutants. Thus, the two points g1,g2belong to one neutral set but represent two different
exploration strategies.
unsatisfactory and thus we reject the definition of self-ada ptability as given by option (iii).
Instead, we circumvent such problems by defining:
Definition 4 (Self-adaptive exploration)
Given an indirect, population-based model Mwith GP-map h, exploration at x∈Pis
defined self-adaptable if the exploration kernel ˜Mg=˜MG
g◦h−1varies for different g∈
h−1(x)in the neutral set of x. The variety{˜Mg|g∈h−1(x)}of different exploration
kernels represents the scope of self-adaptability.
What does this definition mean? Assume that one individual g∈Gis drifting in a
neutral set h−1(x). Meanwhile, although its image h(g) is not changing at all, the probability
distribution of offsprings inP(i.e. the exploration kernel ˜Mgassociated to it) may change
very well. This is how the definition captures the ability of e xploration to adapt. See figure
2 for an illustration.
As a simple example we note that adding (neutral) mutation ra te parameters aligns with
this definition: Changing such strategy parameters actuall y is a neutral walk but varies the
exploration kernels (e.g. by resizing them). Such and simil ar methods, may be understood
as “local rescalings of neighborhood in P”; distances (probabilities to reach neighbors within
one generation) are rescaled. However, such methods do not a im at varying the variational
topology within P: the probabilities for mutations into the neighborhood cha nge, the neigh-
borhood itself though is not varied. The generality of our de finition also captures the latter
kind of variability and it will be a major goal of this paper to exemplify it by introducing
neutral variations that do vary the variational topology on P.
In the following we will exclusively focus on self-adaptabi lity of exploration as defined
above.4 NEUTRALITY AS BASIS OF SELF-ADAPTABILITY OF EXPLORATION 12
Note: Focusing only on self-adaptability (neglecting option (ii )), we want to emphasize that
we always consider the GP-map hto be fix, i.e. non-varying during evolution — and that
this is nota restriction, nota loss of generality. If one would protest and claim that hshould
be variable by depending on genes in G, we veto by stating that the formalism requires to
collect allgenetic parameters in the space G, that by definition the GP-map his the map
which maps allGonP, and thus it is formally incorrect to speak of has depending on genes
inG.
Of course, others may have another point of view and this does not diminish the profound
meaning of, e.g., Wagner and Altenberg’s statement that “th e genotype-phenotype map is
under genetic control and therefore evolvable.” [ sec 2, par 9 ] — though from our point of
view a questionable formulation.
4.1 Interpretations of neutrality
It is intuitive to believe that every little detail in nature fulfills “some purpose”; evolution
would abandon all useless mechanisms and redundancies. The existence of something like
neutrality in nature offends this intuition: A typical examp le is the fact that different codons
are transcribed into the same amino acid, suggesting that ce rtain nucleotide substitutions
have no effect whatever on the phenotype or its fitness — they ar e neutral. Such issues
initiated many investigations, pioneered by Motoo Kimura’ s Neutral Theory (Kimura 1983).
In a later paper (Kimura 1986), he defends his theory against the selectionists’ criticism, who
argued that neutral genes would be functionless, mere noise , and thus biologically implausible:
“Sometimes, it is remarked that neutral alleles are by defini tion not relevant to adaptation,
and therefore not biologically very important. I think that this is too short-sighted a
view. Even if the so-called neutral alleles are selectively equivalent under a prevailing
set of environmental conditions of a species, it is possible that some of them, when a
new environmental condition is imposed, will become select ed. Experiments suggesting
this possibility have been reported by Dykhuizen & Hartl (19 80) who called attention to
the possibility that neutral alleles have a ‘latent potenti al for selection’. I concur with
them and believe that ‘neutral mutations’ can be the raw mate rial for adaptive evolution.”
[Kimura (1986), page 345 ]
The last section gave a clear statement of how neutrality can be understood as “raw material
for adaptive evolution”.
The interplay between neutrality and evolvability is a cent ral topic also in other works.
Fontana and Schuster (1998), when investigating neutralit y inherent in protein folding, claim
that neutrality enables discontinuous transitions in the p rotein’s shape space (the space P):
“[Transitions] can be triggered by a single point mutation o nly if the rest of the sequence [point
inG] provides the appropriate context [neighborhood in G]; they are preceded by extended
periods of neutral drift.” [ last but one paragraph ] Their arguments focus on the connectivity
of neutral sets which can be analyzed theoretically by perco lation theory. We agree on these
generic ideas. A precondition is however that neutral sets e xist and, most important, that
exploration varies along these neutral sets — as we captured in the above definition.5 PARADIGMS OF SELF-ADAPTIVE EXPLORATION 13
A very intriguing study of such phenomena in nature is the one by Stephens and Waelbroeck
(1999). They empirically analyze the codon bias and its effec t in HIV sequences. Codon bias
means that, although there exist several codons that code fo r the same amino acid (which
form a neutral set), HIV sequences exhibit a preference of wh ich codon is used to code for a
specific amino acid. More precisely, at some places of the seq uence codons are preferred that
are “in the center of this neutral set” (with high neutral deg ree) and at other places codons
are biased to be “on the edge of this neutral set” (with low neu tral degree). It is clear that
these two cases induce different exploration densities; the prior case means low mutability
whereas the latter means high mutability. They go even furth er by giving an explanation for
these two (marginal) exploration strategies: Loci with low mutability (trivially) cause “more
resistance to the potentially destructive effect of mutatio n”, whereas loci with high mutability
might induce a “change in a neutralization epitope which has come to be recognized by the
immune system.” [ introduction, par 4 ]
Finally, several models of landscapes with tunable neutral ity have been proposed to theo-
retically investigate possible purposes of neutrality (Barnett 2000; Newman and Engelhardt
1998; Reidys and Stadler 2001).
In this paper we present a simple setup to demonstrate the dyn amics in neutral networks
in appendix B. Using Eigen’s model we show a drift towards hig h neutral degree, i.e. towards
representations of low mutability. This effect is important to understand the experiment we
present in section 5.2.
5 Paradigms of self-adaptive exploration
The goal of this section is to exemplify the principles discu ssed above by simple and trans-
parent (artificial) systems. In order to setup a running syst em we need to make some further
decisions on
(i) the problem (the space P),
(ii) the GP-map (including the choice of G),
(iii) the base density (population size, mutation rates on G, etc.),
(iv) the evaluation (implementation of E),
(v) the update rule A.
In the following Pwill simply be strings over some alphabet A; the problem is to minimize
the (Hamming) distance to a given target string. Concerning point (iv) and (v), we will use
rank-based selection, i.e. we evaluate proportionally to t he rank of each individual and update
the population by sampling this evaluation density. Point ( ii) and (iii) need more thorough
considerations:
A recursive, grammar-type GP-map. We decide to implement the GP-map as a recur-
sive mapping. More precisely, his representable as a composition of a single GP-generator5 PARADIGMS OF SELF-ADAPTIVE EXPLORATION 14
ˆh:G→G,
h=ˆh◦..◦ˆh/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
m(·) times:G→P⊂G:g∝mapsto→ˆh◦..◦ˆh/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
m(g) times(g). (13)
This inevitably requires a choice of Gsuch that P⊂G. The recursion depth mmay depend
on the point g∈G. Generically, we require that each GP-generator affects (or entangles)
only a few variables within G. The motivation is as follows: Structuredness of explorati on,
as discussed in section 3.1 and 3.2, means mutual informatio n between variables that belong
to the same phenotypic character and less mutual informatio n else. We want the generator
to represent elementary correlating effects (e.g. of intera ction), i.e. to constitute elementary
modules. For example, an elementary correlating effect is th at one character depends also on
another and a respective generator would introduce such mut ual information by mapping one
independent variable onto one which depends on other variab les. An NK-reaction network
is a basic example: the generator (the time step transformat ion) entangles Kvariables to a
new one.
Our examples will use a grammar-type recursive mapping. The space Gis organized as
G=P×/bracketleftbig
A×P/bracketrightbigr, (14)
which means that gencompasses one structure g0∈P(called axiom) and rtuples gi∈A× P
(called rules). The GP-map happlies to g∈Gby applying all rules to g0; the symbols l∈A
in each rule (actually the lhs label of a grammar rule) specif y how to apply the rule. (The GP-
generator is the single application of one rule to the axiom. ) In our examples, the recursion
depth mis always fixed (so we need no terminal symbols or other compli cated mechanisms.)
Such grammar-type encodings have been investigated in many other respects, e.g. by
Prusinkiewicz and Hanan (1989) and Prusinkiewicz and Linde nmayer (1990) discussing L-
systems as natural representation of highly regular, plant -like structures; by Kitano (1990),
Gruau (1995), Lucas (1995), and Sendhoff and Kreutz (1998) us ing grammar-encodings as
representation of neural networks. However, these approac hes are not based and motivated
on a discussion of self-adaptive exploration. Thus, althou gh in most cases the existence of
neutral sets (equivalent representations) in grammar enco dings is obvious, the importance to
introduce (neutral) variations that explore these existin g neutral sets and thereby explore dif-
ferent explorations strategies was not recognized and stre ssed. The next paragraph concerns
the introduction of such variations.
Neutral variations in grammar-type encodings. We turn to the choice of base density,
i.e. variability on G. We assume that there exist canonical mutations on P, namely flip (with
probability αper symbol), insertion, duplication and deletion (with pro bability γper string).
Since Gis composed of structures of Pthese mutations induce standard mutations on G.
However, to take all the considerations of section 4 into acc ount, we additionally introduce
neutral variations on G. These variations are supposed to allow for self-adaptabil ity as defined
above, i.e. they should allow neutral variations that vary e xploration. In our examples we5 PARADIGMS OF SELF-ADAPTIVE EXPLORATION 15
realize such variations by rule substitutions and creation s. Specifically we introduce five kinds
of variations of g∈G, which are likely to be neutral but need not always to be:
(i) Pick one rule and one structure ∈P(any rhs or the axiom) within g; then apply the
rule once to the structure.
(ii) Pick one rule and one structure; check if the rhs of the ru le is part of the structure; if
so, replace this part by applying the rule inversely.
(iii) Pick a structure and create a new rule by extracting a pa rt out of the structure and
replacing it by a symbol.
(iv) Delete a rule if it is never applied during recursion.
All of these variations will occur with probability βper rule (per structure in case (iii)).
5.1 Basic paradigm
LetPbe strings of the alphabet {0,1,x}. Consider the following two points a,b∈Gto
represent the same point 0101 in P:
a0= 01x , a 1= (x∝mapsto→01),
b0=xx , b 1= (x∝mapsto→01).
If we assume that the rhs of a1andb1have considerable mutability, the exploration kernels of
aandbare quite different: Probable (phenotypic) mutants of aare 0111 ,0100,0110, whereas
bis likely to produce mutants like 1111 ,0000,1010. The difference of these two exploration
densities is of topological nature.
In order to enable a transition between such different strate gies, the exploration of the
corresponding neutral set must be possible. In the upper exa mple it is easy to define a
neutral mutation from atob: The rule itself is to apply to the axiom. The inverse mutatio n
requires an application of the rule from right to left, i.e., see if the rhs fits somewhere and
substitute by the lhs. Our system incorporates these variat ions.
5.2 Two experiments: Variability of exploration and neutra l drift
LetPbe the strings over the alphabet {a,b,c,d,e,f,g,h}. The function fis the Hamming
distance to the fixed target string abcdeabcdeabcdeabcdeabcde , i.e. 5 times abcde . To
demonstrate a neutral drift we consider only one individual and initialize it with an axiom
equal to the target and no rule. Selection is (1+1), i.e. at ea ch time step one offspring is
produced and selected if equally good or discarded if worse. As a result of neutral variations,
the number of rules and the probability for regular mutation s in the exploration density vary
in correlation. This kind of variability of exploration is o f topological nature. The point is,
we gave an example where the topological characters of the ex ploration density vary over a
connected neutral set. See figure 3.
We enhance this example by considering a population of 100 in dividuals and non-elitist,
rank-based selection. All individuals are initialized as d escribed above. The population drifts5 PARADIGMS OF SELF-ADAPTIVE EXPLORATION 16
0 200 400 600 800 1000 1200 1400 1600 1800 2000modular exploration
rule usage
Figure 3: A single individual is tracked when drifting on a ne utral set spanned by neutral
substitutions in its grammar-encoding. Its exploration de nsity is analyzed by taking 10 000
samples at each time step. ’Modular exploration’ counts the probability for mutations that
occur equally at same positions in other blocks. These are bl ocks of 5 symbols as given by the
target string: 5×abcde . ’Rule usage’ counts how often rules are applied during recu rsion.
[Population size µ= 1; mutation probabilities α= 0.001,β= 0.1; recursion depth m= 10;
scaling of y-axes is only relative.]
towards representations (points in the neutral set of the ta rget string) with high neutrality.
This effect is explained in detail in appendix B. Here, a high n eutral degree coincides with
representations of short description length (the sum of len gths of the axiom and rhs of rules).
In order to achieve such compact representations, more rule s are extracted and included in
the representation. A visualization of the exploration den sity via mutual information maps
exhibits its clear structure that corresponds to the target string’s structure. One may interpret
that the system has “learned the problem’s structure”. See fi gure 4.5 PARADIGMS OF SELF-ADAPTIVE EXPLORATION 17
0200 400 600 8001000 1200 1400 1600 1800 200000.10.20.30.40.50.60.70.8
neutral degree
description length rule usage
5 10 15 20 25510152025
5 10 15 20 25510152025
5 10 15 20 25510152025
Figure 4: Upper plot: A population is tracked when drifting o n a neutral set. Selection is non-
elitist and rank-proportional and thus pushes the populati on towards higher neutral degree.
This is achieved by finding representations of shorter descr iption length of the modular target
string (5×abcde ). (Description length equals the sum of the lengths of axiom and all rhs).
This in turn is achieved by making use of rules. Lower plots: T he mutual information between
the 25 variables in the (phenotypic) exploration density is displayed as a matrix. The three
plots correspond to times 50, 500, and 2000. The regular, 5-m odular structure of exploration
is clearly visible. [Population size µ= 100; mutation probabilities α=β= 0.02; target: 5×
abcde ; recursion depth m= 3; scaling of y-axes is exact for neutrality, only relative for the
rest; scaling of gray-shading is only relative.]6 CONCLUSIONS 18
6 Conclusions
Major parts of this paper are concerned to develop an integra ted language for evolutionary
search based on the formalism of stochastic search and empha sizing the exploration density
and its parameterization. The benefit is a unified view on diffe rent specific approaches, their
commonness and differences. For example, at first sight it is h ard to see what a CMA evolu-
tionary strategy has in common with the codon bias in HIV sequ ences. The answer is: both
of them are concerned to model the variability of future offsp rings, the exploration density;
both of them by using a kind of genotype-phenotype mapping (a n affine transformation in the
first case). Also notions such as pleiotropy and functional p henotypic complex can properly
be defined on the basis of this language. This allows to make co ntact between biological and
computational research. The functional meaning of a genoty pe-phenotype mapping is illumi-
nated by interpreting it as a lift of an exploration density a nd topology on the search space.
We showed that a non-injective genotype-phenotype mapping can lift different exploration
strategies, different topologies to the same phenotype. Thi s is the core of how we define
self-adaptability of exploration. The definition overcome s the formal weakness of previous
definitions and is as general as the language it is based on. Th e definition opens a completely
new view on the meaning of neutrality.
In the experimental part of this paper we presented elementa ry examples of these concepts.
We illustrated the structure of exploration by a gray-shade map of the mutual information
within the exploration density, a gray-shade map of pleiotr opy. We exemplified its variability
during neutral drifts. And we demonstrated successful self -adaptability of exploration where
in the end the structure of exploration perfectly matches th e structure of the problem.
We will now discuss some further implications of the new view we have developed in this
paper:
(i) On modularity, structuredness, and evolvability. Given a system that functions
well, how should one define what a module or a functional compl ex is? One only observes that
all parts together work well as a whole. A common idea is that m odules are characterized by
high interactivity within them. By high interactivity we me an that there are high correlations
between units during the time of functioning. These are comp letely different kinds of corre-
lations than correlations between units in the evolutionar y variability. It is though possible
to draw a link: Having units that are highly interacting duri ng functioning, the fitness might
strongly depend on their teamwork. If this is the case, also t he evaluation density should
incorporate high correlations between the units (i.e. the u nits form a functional phenotypic
complex). Now, if the exploration density should approxima te the evaluation density, we also
find these correlations in the evolutionary variability.
Thus, when talking about modules, one should be aware of the i nterrelations between these
three levels of correlations: (1) during functioning, (2) i n the evaluation density, (3) in the
exploration density. Our definition of a functional phenoty pic complex refers to the 2nd level
– the evaluation density. Our hypothesis is that the advanta ge of structured systems (and6 CONCLUSIONS 19
thus the selective pressure towards structure) stems from t he 3rd level:
Systems are structured, not because this is the only possibl e way of functioning, but because
it is advantageous for variability. The advantage of struct ured variability is its capability to
explore by approximating the “problem’s structure”, the st ructure of the evaluation density.
This capability should be called evolvability .
For example, parts of a system that contribute separately to fitness should be varied and
optimized in parallel without potentially disturbing corr elations; whereas parts of a system
that only contribute to fitness when they are tuned on each oth er should be varied in corre-
lation in order to preserve this tuning.
(ii) On redundancy and neutrality Neutrality is often thought of as redundancy. From
our point of view, this is very misleading. As we pointed out i n the context of self-adaptability,
although all the genotypes in a neutral set encode the same ph enotype, they may have very
different exploration kernels. Thus, such genotypes may car ry different information. One
cannot speak of redundancy if different and relevant informa tion is encoded. If, however,
genotypes in a neutral set have identical exploration kerne ls (in the genotype space), then
they are indeed redundant. Redundancy is necessarily neutr al, but neutrality is not necessarily
redundant.
(iii) On compact representations Assume we use a Bayesian network to model the struc-
ture of exploration. Then we will explicitly encode the corr elations between all phenotypic
variables. In contrast, our second example shows how compac t representations correspond to
highly structured exploration and can be found by using recu rsive codings. The idea is that
each recursion introduces correlations in the variables. T he neutral drift towards high neutral
degree (see appendix B) induces a selective pressure toward s short representations.
(iv) On grammar-type encodings In grammar-type encodings, some single genotypic
variables (genes) might effectively represent whole groups of phenotypic variables. Thus,
when we model dependencies between variables, we can also mo del dependencies between
whole groups of phenotypic variables and not only between si ngle phenotypic variables as in
the direct modeling ansatz. This allows to introduce deep hi erarchical dependencies in the
exploration density.
Most existing approaches to grammar encoding are motivated by the fact that grammars
can represent regular structures with short description le ngth. Instead, we claim that the most
interesting point about grammars is their capability to int roduce structure in the variability,
as demonstrated in our examples. In order to explore these ca pabilities in a self-adaptive
manner, the inclusion of neutral variations in recursive or grammar-type encodings is of
crucial importance. This point seems neglected in the liter ature.
We rigorously support Kimura’s “belief that ‘neutral mutat ions’ can be the raw material
for adaptive evolution” (Kimura 1986).A EXPLORATION MODELS OF SPECIFIC APPROACHES 20
A The exploration model of different state-of-the-art evolu -
tionary algorithms
To stress the importance of the concept of exploration model ing we want to show that the
main difference between specific evolutionary algorithms is their ansatz to model exploration.
In order to do so, we embed specific algorithms in our formalis m. In particular we chose
to analyze the CMA algorithm and three recent approaches whi ch belong to the class of
“probabilistic model-building genetic algorithms” (PMBG As), see (Pelikan, Goldberg, and
Lobo 1999). All of these realize adaptive (but not self-adap tive) exploration.
Covariance Matrix Adaptation (CMA), (Hansen and Ostermaier 2000). The search
space is continuous, P=Rn. The CMA algorithm maintains as parameters qonly one
(center of mass) point p∈P, the symmetric covariance matrix C, and some adaption rate
parameters. The exploration density Mqis given by a linear transformation (via C) of a
Gaussian distribution around p. In practise, the algorithm generates λnormally distributed
mutation vectors zi∈Rn, transforms all of these vectors by multiplying the matrix C, and
adds these vectors to the center of mass pin order to generate the new λsamples. After
evaluation of the samples it is updated as follows: pis moved to the center of mass of the
selected samples and Cis adapted as
C(t+1)= (1−c)C(t)+c z⊗z . (15)
Here, cis some adaption constant and zis the average3of the selected mutations vectors.
(Hansen and Ostermaier (2000), Eq. 15, write z(z)Tinstead of z⊗z). The point is that z⊗z
is the unique symmetric matrix which maps the equally distri buted vector y= (1
n,..,1
n) toz.
Thus, the update rule for Ccorresponds to our generic approaching update whereas padopts
the new center of mass.
Dependency tree modeling, (Baluja and Davies 1997). Here, the search space is discrete ,
P=Xn. In their algorithm, the parameter qthat describes the next exploration density is
a dependency tree. Thus, the model is restricted to encode on ly pair-wise dependencies
between variables. At each time step, λsamples are generated from this exploration density;
the samples are evaluated and the best µof them are selected. A probability density A
of previously selected points is adapted by including those newly selected ones (generically
A←(1−α)A+α[Es(t)]µ). Then the dependency tree is updated by minimizing the Kull back-
Leibler divergence between AandMq. The tree’s update is an adopting since it approximates
A, whereas Aitself is updated according to an approaching update.
Factorized Distribution Algorithm (FDA), (M¨ uhlenbein, Mahnig, and Rodriguez 1999).
Again, P=Xnis discrete. The parameters qdescribe the conditional dependencies in pairs,
3More exactly an weighted average trace over time, see (Hanse n and Ostermaier 2000) Eq. 14.B ILLUSTRATING NEUTRAL DYNAMICS 21
triples, quadruples, etc. of variables. (To be exact, the al gorithm comprises also some elitists.)
The model is quite general but it relies on pre-fixed knowledg e on which pairs, triples, etc.
exactly are to be parameterized. At each time step, the depen dencies within the distribu-
tion of evaluated and selected points are calculated and ass igned to q. Therefore, this is an
adopting update.
Bayesian Optimization Algorithm (BOA), (Pelikan, Goldberg, and Cant´ u-Paz 2000).
P=Xnis discrete. Here, qis a general Bayesian dependency network that explicitly en codes
the exploration density. Thus, the model is not limited in re presenting arbitrary orders
of correlation and it is flexible in which variables are depen dent by inserting and deleting
connections in the network. After selection, the network is recalculated in order to minimize
(e.g. with a greedy algorithm) the distance (e.g. with respe ct to the Bayesian Dirichlet Metric)
between Mqand the distribution of selected. This is, except for elitis ts, also an adopting
update.
B Illustrating neutral dynamics
As an illustration of neutral dynamics we present a simple ex ample. We assume that the
search space Pis discrete and rather small, |P|=λ. Λ denotes the space of densities over P,
which actually is a simplex. Parameter q∈Qis such a density, Q= Λ, and the exploration
density Mqis a mutation τ q∈Λ of this density. This example omits sampling and thus
evaluation E: Λ→Λ directly applies to Mq=τ q. The update rule is the adopting:
q(t+1)=E τ q(t), q(t+1)
i=λ/summationdisplay
j,k=1Eijτjkq(t)
k, (16)
whereby we actually formulated Eigen’s model (see e.g. (Eig en, McCaskill, and Schuster 1989))
in our notation. Finding the eigenvectors of E τmeans finding a stationary population density.
Their eigenvalues describe their growth factor and the eige nvector with highest eigenvalue will
describe the final attractor — the quasi-species. In the pres ence of a neutral set N(here a set
of indices) we assume that only individuals on this neutral s et are evaluated positively and
without co-evolutionary (interacting) effects, i.e., Eis diagonal and
E: Λ→Λ, p i∝mapsto→/summationdisplay
jEijpj=Eiipi=/braceleftbigg0
ei(p)pii∝\e}atio\slash∈N
i∈N. (17)
We investigate two options for the evaluation factor ei(p). The first and straightforward
option is that all positions on the neutral set are evaluated equally, then
e1
i(p) =1/summationtext
j∈Npj(18)
is just the appropriate normalization factor. This option i s realized e.g. for fitness-proportion-
al evaluation (when fitness on P\Nvanishes) but also for fair ranking. For the second optionB ILLUSTRATING NEUTRAL DYNAMICS 22
neutral set N⊂P:
123456789101
2
3
4
5
6
7
8
9
10First experiment
density q∈Λ:
10−20
10−15
10−10
10−5
123456789101
2
3
4
5
6
7
8
9
10
mutated density τ q∈Λ:
10−20
10−15
10−10
10−5
123456789101
2
3
4
5
6
7
8
9
10Second experiment
density q∈Λ:
0
0.01
0.02
0.03
0.04
0.05123456789101
2
3
4
5
6
7
8
9
10
mutated density
τ q∈Λ:
0
0.01
0.02
0.03
0.04
0.05123456789101
2
3
4
5
6
7
8
9
10
Figure 5: The search space Pis represented as a 10 ×10 board. The neutral set is embedded
as depicted on the left. The exploration matrix τcorresponds to a mutation rate of 0 .1 in
each of the four directions (up,down,right,left). In the fir st experiment, when evaluation is
straightforward, e.g. fitness-proportional, it is impress ive to see how strong the attraction
towards the crossing with neutral degree 1 (with four neutra l neighbors) is. In the second
experiment, where evaluation enforces a kind of local conse rvation of population density, the
population is equally distributed on the neutral set, but ex ploration on places with high
neutral degree is proportionally higher because they have m ore neighbors from which they
“receive” offsprings.
we enforce such positions on the neutral set with low neutral degree — inverse-proportionally
to the neutral degree:
e2
i(p) =1/di/summationtext
j∈N(pj/dj), d i:=/summationdisplay
k∈Nτik. (19)
The quantity diis the probability for an offspring of individual ito be an element of N. Thus,
this option increases the evaluation of isuch that the probability to provide an offspring in
Nbecomes equal for all i∈N. This can be compared to a local conservation of population
density: Effectively, each parent in Nwill with equal probability contribute a viable offspring
to the next generation. Such a type of selection can be realiz ed by local selection mechanisms:
From each parent produce many offsprings, let only the best of these offsprings compete
with others. As a result, the quasi-species is simply consta nt on Nand vanishes elsewhere,
qi∈N= 1/|N|,qi/negationslash∈N= 0:
pi= (τ q)i=/summationdisplay
j∈Nτijqj=di
|N|, (20)
(E τ q)i∈N=1/di/summationtext
j∈N(pj/dj)pi=1
|N|=qi. (21)REFERENCES 23
The mutated density piis proportional to di(which, for individuals out of N, does not denote
the neutral degree but rather the probability for offsprings inN). Diversity is much higher
than for the first type of evaluation. See figure 5.
The first experiment is an explanation for the dynamics we obs erve in section 5.2. We
included the second experiment because it realizes what one might intuitively have expected:
on a neutral set the population is distributed equally and wi th high diversity. We showed
what kind of evaluation one has to choose to fulfill this expec tation.
The findings are conform with Nimwegen’s (1999) little examp les of random or selective
walks on a neutral set: A blind ant would try one (random) neig hboring genotype and walk
to it if it has same fitness or stay otherwise. A myopic ant woul d find all neighbors with same
fitness and walk to one (random) of those. He finds that, in temporal average , the blind ant
stays equal times at each genotype of the neutral set whereas the myopic ant stays longer at
centers of the neutral set (i.e. ∝the neutral degree). The myopic ant, since it always finds a
neutral neighbor, corresponds to our second example.
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/D9/D6/D6/CT/D2 /D8 /BT/AR/DD/D1/CT/D8/D6/CX/DC /D7/D3/CU/D8 /DB /CP/D6/CT /CX/D2/B9
/D0/D9/CS/CT /B4/CX/B5 /D8/CW/CT /CP/CQ/CX/D0/CX/D8 /DD /D8/D3 /D3/CQ/D8/CP/CX/D2 /D6/CT/D0/CX/CP/CQ/D0/CT /D7
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/D3/D6/CT/D7 /CU/D3/D6/CV/CT/D2/CT/D7 /D4/D6/D3/CQ /CT/CS /D8 /DB/CX
/CT /D3/D6 /D1/D3/D6/CT /D3/D2 /D8/CW/CT /D1/CX
/D6/D3/CP/D6/D6/CP /DD /D7/CW/D3 /DB /CP /DA /CP/D7/D8/D0/DD /CX/D2
/D6/CT/CP/D7/CT/CS
/D3/D6/D6/CT/D0/CP/B9/D8/CX/D3/D2 /B4/D8/CW/CT/D6/CT /CP/D6/CT∼700 /D7/D9
/CW /CV/CT/D2/CT/D7 /D3/D2 /D8/CW/CT
/D3/D1 /CQ/CX/D2/CT/CS /C5/D9/BD/BD/CZ /BT /CP/D2/CS /BU /D1/D3/D9/D7/CT
/CW/CX/D4/D7/CT/D6/CX/CT/D7/B5/BA /C1/D2 /D8/CW/CT /D6/CT/D1/CP/CX/D2/CX/D2/CV/B8 /DB /CT /CS/CT/D1/D3/D2/D7/D8/D6/CP/D8/CT /CW/D3 /DB /D8/CW/CX/D7 /D4/D6/D3
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/CT /CU/D3/D6 /D8/CW/CT /CX/D1/D4/D6/D3 /DA /CT/D1/CT/D2 /D8/D7/BA/CA/CP /DB /CX/D2 /D8/CT/D2/D7/CX/D8/CX/CT/D7 /CP/D2/CS /CQ/CP
/CZ/CV/D6/D3/D9/D2/CS /D7/D9/CQ/D8/D6/CP
/D8/CX/D3/D2/CC/CW/CT /D7/D8/CP/D6/D8/CX/D2/CV /D4 /D3/CX/D2 /D8 /CX/D2 /D3/D9/D6 /CP/D2/CP/D0/DD/D7/CX/D7 /D3/CU /C0/BW/C7/C6/BT/D7
/D3/D2/D7/CX/D7/D8/D7 /D3/CU /D8/CW/CT /AT/D9/D3/D6/CT/D7
/CT/D2 /D8 /CX/D2/B9/D8/CT/D2/D7/CX/D8/CX/CT/D7 /D3/CU /CP/D0/D0 /D8/CW/CT /BE/BH/B9/CQ/CP/D7/CT /D4/D6/D3/CQ /CT/D7 /D3/D2 /D8/CW/CT
/CW/CX/D4/B8 /CX/D2
/D0/D9/CS/CX/D2/CV /CQ /D3/D8/CW /D8/CW/CT /C8/C5 /CP/D2/CS /C5/C5
/CT/D0/D0/D7 /B4/BA/BV/BX/C4 /AS/D0/CT/D7/B5/BA /CC/CW/CX/D7 /CS/CP/D8/CP /CW/CP/D7 /CP/D0/D6/CT/CP/CS/DD /CV/D3/D2/CT /D8/CW/D6/D3/D9/CV/CW /D3/D2/CT /D4/D6/D3
/CT/D7/D7/CX/D2/CV /D7/D8/CT/D4 /CQ /DD/BT/AR/DD/D1/CT/D8/D6/CX/DC/B8 /D2/CP/D1/CT/D0/DD /CP/D2 /CP /DA /CT/D6/CP/CV/CT /D3/CU /D8/CW/CT /D4/CX/DC/CT/D0 /CX/D2 /D8/CT/D2/D7/CX/D8/CX/CT/D7 /B4/BF/BI /D4/CX/DC/CT/D0/D7 /D4 /CT/D6
/CT/D0/D0 /CU/D3/D6 /D8/CW/CT/C5/D9/BD/BD/CZ /D1/D3/D9/D7/CT
/CW/CX/D4/B5 /CU/D3/D6 /CT/CP
/CW /D4/D6/D3/CQ /CT
/CT/D0/D0/BA /BU/CT/CU/D3/D6/CT /D1/D3 /DA/CX/D2/CV /D3/D2 /D8/D3
/D3/D2/D7/CX/CS/CT/D6 /DB/CW/CX
/CW/D4/D6/D3/CQ /CT/D7 /CQ /CT/D0/D3/D2/CV /D8/D3 /DB/CW/CX
/CW /CV/CT/D2/CT/B8 /CX/D8 /CX/D7 /CX/D2/D7/D8/D6/D9
/D8/CX/DA /CT /D8/D3 /CX/D2/D5/D9/CX/D6/CT /CP/CQ /D3/D9/D8 /D8/CW/CT /D6/CT/D4/D6/D3 /CS/D9
/CX/CQ/CX/D0/B9/CX/D8 /DD /D3/CU /D8/CW/CT /D6/CP /DB /CS/CP/D8/CP /CX/D2 /D6/CT/D4/D0/CX
/CP/D8/CT /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/D7/BA /CC/CW/CT
/CX/CV/CP/D6 /D7/CW/CP/D4 /CT/CS
/D0/D3/D9/CS /CX/D2 /BY/CX/CV/BA /BD/B4/CP/B5/D7/CW/D3 /DB/D7 /D7/D9
/CW /CP /D8 /DD/D4/CX
/CP/D0 /CT/DC/CP/D1/D4/D0/CT/BA /C1/D2 /CP/D2 /CX/CS/CT/CP/D0 /B4/D2/D3/CX/D7/CT/D0/CT/D7/D7/B5 /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/B8 /D8/CW/CT /D7
/CP/D8/D8/CT/D6 /D4/D0/D3/D8/D3/CU /D8/CW/CT /D6/CT/D4/D0/CX
/CP/D8/CT/D7 /D7/CW/D3/D9/D0/CS /D4/D6/D3 /CS/D9
/CT /CP /D7/CX/D2/CV/D0/CT /D7/D8/D6/CP/CX/CV/CW /D8 /D0/CX/D2/CT /DB/CX/D8/CW /D9/D2/CX/D8 /D7/D0/D3/D4 /CT/B8 /D7/D3 /D8/CW/CP/D8/D8/CW/CT /CQ/D6/D3/CP/CS/CT/D2/CX/D2/CV /D3/CU /D8/CW/CT /D0/CX/D2/CT /CX/D2 /CP /D6/CT/CP/D0 /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8 /D6/CT/AT/CT
/D8/D7 /D8/CW/CT /D2/D3/CX/D7/CT/BA /C1/D2 /C0/BW/C7/C6/BT/D7/B8/D8/CW/CX/D7 /D2/D3/CX/D7/CT /CW/CP/D7 /D1 /D9/D0/D8/CX/D4/D0/CT /D7/D3/D9/D6
/CT/D7/B8 /CX/D2
/D0/D9/CS/CX/D2/CV /CX/D2 /D8/D6/CX/D2/D7/CX
/CQ/CX/D3/D0/D3/CV/CX
/CP/D0 /CP/D2/CS /D7/CP/D1/D4/D0/CT /D4/D6/D3/B9
/CT/D7/D7/CX/D2/CV /DA /CP/D6/CX/CP/CQ/CX/D0/CX/D8 /DD /B8 /CW /DD/CQ/D6/CX/CS/CX/DE/CP/D8/CX/D3/D2 /CZ/CX/D2/CT/D8/CX
/D7 /CP/D2/CS /D8/CW/CT/D6/D1/D3 /CS/DD/D2/CP/D1/CX
/D7/B8 /D2/D3/CX/D7/CT /D6/CT/D0/CP/D8/CT/CS /D8/D3/D8/CW/CT /CX/D2
/D3/D6/D4 /D3/D6/CP/D8/CX/D3/D2 /CP/D2/CS /CP/D1/D4/D0/CX/AS
/CP/D8/CX/D3/D2 /D3/CU /AT/D9/D3/D6/CT/D7
/CT/D2 /D8 /CS/DD /CT/D7/B8 /CP/D2/CS /D8/CW/CT /D1/CT/CP/D7/D9/D6/CT/D1/CT/D2 /D8 /D3/CU/D8/CW/CT /AT/D9/D3/D6/CT/D7
/CT/D2
/CT /CX/D2 /D8/CW/CT /D7
/CP/D2/D2/CX/D2/CV /D4/D6/D3
/CT/D7/D7/BA /BW/CT/D7/D4/CX/D8/CT /CP/D0/D0 /D8/CW/CT/D7/CT /D4 /D3/D8/CT/D2 /D8/CX/CP/D0 /D7/D3/D9/D6
/CT/D7/B8 /D8/CW/CT/CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/CP/D0 /D7/CX/D8/D9/CP/D8/CX/D3/D2 /CX/D7 /CT/D2
/D3/D9/D6/CP/CV/CX/D2/CV /CP/D7 /CX/D2/CS/CX
/CP/D8/CT/CS /CQ /DD /D8/CW/CT /CW/CX/CV/CW /D6/CT/D4/D6/D3 /CS/D9
/CX/CQ/CX/D0/CX/D8 /DD/CX/D2 /B4/CP/B5/BA/CC/CW/CT /CX/D2 /D8/CT/D2/D7/CX/D8 /DD /CS/CT/D4 /CT/D2/CS/CT/D2
/CT /D3/CU /D8/CW/CT /D2/D3/CX/D7/CT /CT/D2 /DA /CT/D0/D3/D4 /CT /CX/D7
/D3/D1/D1/D3/D2/D0/DD /D6/CT/CU/CT/D6/D6/CT/CS /D8/D3 /CP/D7/D8/CW/CT /D2/D3/CX/D7/CT /CU/D9/D2/D2/CT/D0 /BA /C1/D2 /BY/CX/CV/BA /BD/B4/CP/B5/B8 /D8/CW/CT /CU/D9/D2/D2/CT/D0 /CX/D7 /D3/D2/D0/DD /DA /CT/D6/DD /DB /CT/CP/CZ/D0/DD /CX/D2 /D8/CT/D2/D7/CX/D8 /DD /CS/CT/D4 /CT/D2/B9/CS/CT/D2 /D8/BA /CF /CT /D3/CQ/D7/CT/D6/DA /CT /D8/CW/CP/D8 /D8/CW/CT /D3/D2/D7/CT/D8 /D3/CU /D8/CW/CT /CX/D2 /D8/CT/D2/D7/CX/D8/CX/CT/D7 /CX/D7 /D7/CW/CX/CU/D8/CT/CS /CU/D6/D3/D1 /DE/CT/D6/D3 /D8/D3∼500 /B8/CX/D2/CS/CX
/CP/D8/CX/D2/CV /D8/CW/CP/D8
/CT/D0/D0 /CX/D2 /D8/CT/D2/D7/CX/D8/CX/CT/D7 /CW/CP /DA /CT /CP/D2 /CP/CS/CS/CX/D8/CX/DA /CT /CQ/CP
/CZ/CV/D6/D3/D9/D2/CS
/D3/D1/D4 /D3/D2/CT/D2 /D8/BA /BX/D7/D8/CX/B9/D1/CP/D8/CX/D2/CV /D8/CW/CX/D7 /CQ/CP
/CZ/CV/D6/D3/D9/D2/CS /CX/D2 /D8/CT/D2/D7/CX/D8 /DD /CX/D7 /CT/D7/D7/CT/D2 /D8/CX/CP/D0 /DB/CW/CT/D2 /D4/D6/D3
/CT/D7/D7/CX/D2/CV /D0/D3 /DB /CX/D2 /D8/CT/D2/D7/CX/D8 /DD /CS/CP/D8/CP/D4 /D3/CX/D2 /D8/D7/BA /C1/D2 /C0/BW/C7/C6/BT/D7/B8 /CX/CS/CT/D2 /D8/CX/CU/DD/CX/D2/CV /CQ/CP
/CZ/CV/D6/D3/D9/D2/CS /CX/D7 /CP /D4/D6/CX/D3/D6/CX /CP /CS/CX/AR/CT/D6/CT/D2 /D8 /D4/D6/D3/CQ/D0/CT/D1 /CU/D6/D3/D1/CX/D8/D7 /CP/D2/CP/D0/D3/CV/D9/CT /CX/D2
/BW/C6/BT /D7/D4 /D3/D8/D8/CT/CS /CP/D6/D6/CP /DD/D7/BA /CC/CW/CT/D6/CT/B8 /D3/D2/CT /D8/D6/CX/CT/D7 /D8/D3 /D1/CT/CP/D7/D9/D6/CT /D8/CW/CT /CX/D2 /D8/CT/D2/D7/CX/D8 /DD/D3/CU /D6/CT/CV/CX/D3/D2/D7 /CX/D2 /CQ /CT/D8 /DB /CT/CT/D2 /CP/CS/CY/CP
/CT/D2 /D8 /D7/D4 /D3/D8/D7 /CP/D7 /CP /D1/CT/CP/D7/D9/D6/CT /CU/D3/D6 /D8/CW/CT /D0/D3
/CP/D0 /CQ/CP
/CZ/CV/D6/D3/D9/D2/CS/BA /C1/D2
/D3/D2 /D8/D6/CP/D7/D8/B8 /D8/CW/CT /CX/D2 /D8/CT/D6/B9/CU/CT/CP/D8/D9/D6/CT /CS/CX/D7/D8/CP/D2
/CT /CX/D2 /BT/AR/DD/D1/CT/D8/D6/CX/DC /CP/D6/D6/CP /DD/D7 /CX/D7 /D8/D3 /D3 /D7/D1/CP/D0/D0 /CU/D3/D6 /CP /D7/CX/D1/B9/CX/D0/CP/D6 /D1/CT/CP/D7/D9/D6/CT/D1/CT/D2 /D8 /CP/D2/CS /D3/D2/CT /D1 /D9/D7/D8 /CT/D7/D8/CX/D1/CP/D8/CT /D8/CW/CT /CQ/CP
/CZ/CV/D6/D3/D9/D2/CS /CU/D6/D3/D1 /D8/CW/CT /D4/D6/D3/CQ /CT
/CT/D0/D0/D7/D8/CW/CT/D1/D7/CT/D0/DA /CT/D7/BA /BU/CP
/CZ/CV/D6/D3/D9/D2/CS /CX/D7 /CQ /DD /CS/CT/AS/D2/CX/D8/CX/D3/D2 /D2/D3/D2/B9/D7/D4 /CT
/CX/AS
/B8 /CP/D2/CS /D7/CW/D3/D9/D0/CS /D8/CW/CT/D6/CT/CU/D3/D6/CT /D2/D3/D8/CQ /CT /D7/CT/D2/D7/CX/D8/CX/DA /CT /D8/D3 /D8/CW/CT /D7/CX/D2/CV/D0/CT /CQ/CP/D7/CT /D7/CT/D5/D9/CT/D2
/CT /D1/D3 /CS/CX/AS
/CP/D8/CX/D3/D2 /CX/D2 /D8/CW/CT /C5/C5
/CT/D0/D0/D7/BA /BV/D3/D2/D7/CT/B9/D5/D9/CT/D2 /D8/D0/DD /B8 /DB /CT
/D3/D2/D7/CX/CS/CT/D6 /D8/CW/CT /D7/D9/CQ/D7/CT/D8 /D3/CU /D4/D6/D3/CQ /CT /D4/CP/CX/D6/D7 /DB/CW/D3/D7/CT /C8/C5 /CP/D2/CS /C5/C5 /CX/D2 /D8/CT/D2/D7/CX/D8/CX/CT/D7/CS/CX/AR/CT/D6 /CQ /DD /D0/CT/D7/D7 /D8/CW/CP/D2 /CP /CV/CX/DA /CT/D2 /D7/D1/CP/D0/D0 /D5/D9/CP/D2 /D8/CX/D8 /DD /B4/C8/C5/B9/C5/C5 /BOǫ /B5 /CP/D7 /D6/CT/D4/D6/CT/D7/CT/D2 /D8/CP/D8/CX/DA /CT /D3/CU /D8/CW/CT/CQ/CP
/CZ/CV/D6/D3/D9/D2/CS/BA /CC/CW/CT /CS/CX/D7/D8/D6/CX/CQ/D9/D8/CX/D3/D2/D7 /D3/CU /CT/CX/D8/CW/CT/D6 /D8/CW/CT /C8/C5 /D3/D6 /C5/C5
/CT/D0/D0/D7 /D3/CQ/D8/CP/CX/D2/CT/CS /CX/D2 /D8/CW/CX/D7/D1/CP/D2/D2/CT/D6 /CS/CT/D4 /CT/D2/CS /D3/D2/D0/DD /DB /CT/CP/CZ/D0/DD /D3/D2ǫ /B8 /CP/D2/CS
/CP/D2 /CQ /CT /D6/CT/CP/D7/D3/D2/CP/CQ/D0/DD /AS/D8/D8/CT/CS /D8/D3 /BZ/CP/D9/D7/D7/CX/CP/D2/D7 /CU/D6/D3/D1/D8/CW/CT/CX/D6 /D0/D3 /DB/B9/CX/D2 /D8/CT/D2/D7/CX/D8 /DD /D3/D2/D7/CT/D8 /D9/D4 /D8/D3 /D8/CW/CT/CX/D6 /D1/CP/DC/CX/D1 /D9/D1 /B4/BY/CX/CV/BA /BD/B4/CS/B5/B5/BA /CF /CT /D9/D7/CT/CSǫ= 50 /CX/D2/D9/D2/CX/D8/D7 /D3/CU /D8/CW/CT /BA/BV/BX/C4 /AS/D0/CT /CX/D2 /D8/CT/D2/D7/CX/D8/CX/CT/D7/B8 /CQ/D9/D8 /D9/D7/CX/D2/CV ǫ= 100 /D0/CT/CP/CS/D7 /D8/D3
/CW/CP/D2/CV/CT/D7 /D3/CU /D8/CW/CT /D3/D6/CS/CT/D6/D3/CU /D3/D2/D0/DD∼1% /CU/D3/D6 /D8/CW/CT /D1/CT/CP/D2 /CQ/CP
/CZ/CV/D6/D3/D9/D2/CS /angbracketleftb/angbracketright /CP/D2/CS /D7/D8/CP/D2/CS/CP/D6/CS /CS/CT/DA/CX/CP/D8/CX/D3/D2 σ /BA /BY/CX/CV/BA /BD/B4/CQ/B5/D7/CW/D3 /DB/D7 /D8/CW/CT /D6/CP /DB /CS/CP/D8/CP /CP/CU/D8/CT/D6 /CQ/CP
/CZ/CV/D6/D3/D9/D2/CS /D7/D9/CQ/D8/D6/CP
/D8/CX/D3/D2/BA /CC/CW/CT /D8 /DD/D4/CX
/CP/D0 /CQ/D6/D3/CP/CS/CT/D2/CX/D2/CV /D3/CU/D8/CW/CT /D2/D3/CX/D7/CT /CU/D9/D2/D2/CT/D0 /CP/D8 /D0/D3 /DB /CX/D2 /D8/CT/D2/D7/CX/D8/CX/CT/D7 /CT/D2/CS /D6/CT/AT/CT
/D8/D7 /D8/CW/CT /D6/CT/D7/CX/CS/D9/CP/D0 /CQ/CP
/CZ/CV/D6/D3/D9/D2/CS /B4/D8/CW/CT /CU/CP
/D8/BF/D8/CW/CP/D8σ/negationslash= 0 /B5/BA /C1/D2
/D3/D2 /D8/D6/CP/D7/D8/B8 /D8/CW/CT /BT/AR/DD/D1/CT/D8/D6/CX/DC /D4/D6/D3
/CT/CS/D9/D6/CT /CT/D7/D8/CX/D1/CP/D8/CT/D7 /angbracketleftb/angbracketright /CP/D2/CS /DA /CP/D6/CX/CP/D2
/CT/DA /CP/D6(b) /CU/D6/D3/D1 /D8/CW/CT2% /D0/D3 /DB /CT/D7/D8 /CX/D2 /D8/CT/D2/D7/CX/D8 /DD
/CT/D0/D0/D7/BA /CC/CW/CT /D1/CT/CP/D2 /CP/D2/CS /DA /CP/D6/CX/CP/D2
/CT /D3/CQ/D8/CP/CX/D2/CT/CS /D8/CW/CX/D7/DB /CP /DD /CP/D6/CT /D7/D8/D6/D3/D2/CV/D0/DD /CS/CT/D4 /CT/D2/CS/CT/D2 /D8 /D3/D2 /D8/CW/CT /CP/D6/CQ/CX/D8/D6/CP/D6/DD
/D9/D8/D3/AR /B42% /B5/BA /CC /DD/D4/CX
/CP/D0/D0/DD /B8 /DB /CT /D3/CQ/D8/CP/CX/D2 /CP
/angbracketleftb/angbracketright /D0/CP/D6/CV/CT/D6 /D8/CW/CP/D2 /D8/CW/CT /DA /CP/D0/D9/CT /D6/CT/D4 /D3/D6/D8/CT/CS /CQ /DD /D8/CW/CT /BT/AR/DD/D1/CT/D8/D6/CX/DC /D7/D3/CU/D8 /DB /CP/D6/CT (∼+15%) /B8 /D7/D3 /D8/CW/CP/D8/DB /CT /CP/D6/CT /D0/CT/CU/D8 /DB/CX/D8/CW∼86% /D3/CU /D8/CW/CT /CU/CT/CP/D8/D9/D6/CT/D7 /D0/DD/CX/D2/CV /CP/CQ /D3 /DA /CT /angbracketleftb/angbracketright+ 2σ /B8 /D6/CP/D8/CW/CT/D6 /D8/CW/CP/D8∼93% /BA/C1/D2 /CP/CS/CS/CX/D8/CX/D3/D2/B8 /D3/D9/D6 /D2/D3/CX/D7/CT /CU/D9/D2/D2/CT/D0 /CX/D7 /D7/D0/CX/CV/CW /D8/D0/DD /CQ/D6/D3/CP/CS/CT/D6/BA /C6/CT/DA /CT/D6/D8/CW/CT/D0/CT/D7/D7/B8 /D3/D9/D6 /CP/D0/CV/D3/D6/CX/D8/CW/D1 /CU/D3/D6
/D3/D1/D4 /D3/D7/CX/D8/CT /D7
/D3/D6/CT/D7 /D7/D8/CX/D0/D0 /D0/CT/CP/CS/D7 /D8/D3 /CP /D7/CX/CV/D2/CX/AS
/CP/D2 /D8 /D2/D3/CX/D7/CT /D6/CT/CS/D9
/D8/CX/D3/D2 /B4
/CU/BA /CA/CT/D7/D9/D0/D8/D7/B5/BA/BT/D7 /CP /D4/D6/CT/DA/CX/CT/DB/B8 /DB /CT /D7/CW/D3 /DB /CX/D2 /BY/CX/CV/BA /BD/B4
/B5 /DB/CW/CX
/CW /D7/D9/CQ/D7/CT/D8 /D3/CU /C8/C5 /D4/D6/D3/CQ /CT/D7 /CP/D6/CT
/D3/D2/D7/CX/CS/CT/D6/CT/CS/CQ /DD /D3/D9/D6 /CP/D0/CV/D3/D6/CX/D8/CW/D1 /DB/CW/CT/D2
/D3/D1/D4/D9/D8/CX/D2/CV /D6/CP/D8/CX/D3 /D7
/D3/D6/CT/D7 /CQ /CT/D8 /DB /CT/CT/D2 /D8/CW/CT /D8 /DB /D3 /D7/CP/D1/D4/D0/CT/D7/BA /C1/D8 /D8/D9/D6/D2/D7/D3/D9/D8
/D3/D2/D7/CX/D7/D8/CT/D2 /D8/D0/DD /D8/CW/CP/D8 /D2/D3/CX/D7/CX/CT/D6
/CT/D0/D0/D7 /CP/D6/CT /CP/D9/D8/D3/D1/CP/D8/CX
/CP/D0/D0/DD /CS/CX/D7
/CP/D6/CS/CT/CS/B8 /CW/D3 /DB /CT/DA /CT/D6/B8 /D2/D3/D8 /D3/D2/D8/CW/CT /CQ/CP/D7/CX/D7 /D3/CU /CP/D2 /CT/DA /CP/D0/D9/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /CU/D9/D2/D2/CT/D0 /D7/CW/CP/D4 /CT /B4
/CU/BA /C8/D6/D3/CQ /CT
/CT/D0/D0 /D7/CT/D0/CT
/D8/CX/D3/D2/B5/BA/BV/D3/D2/D7/D8/D6/D9
/D8/CX/D2/CV /CV/D3 /D3 /CS /CT/D7/D8/CX/D1/CP/D8/D3/D6/D7/CC/CW/CT /D4/D6/CT/DA/CX/D3/D9/D7 /CS/CX/D7
/D9/D7/D7/CX/D3/D2 /CP/CQ /D3/D9/D8 /CQ/CP
/CZ/CV/D6/D3/D9/D2/CS /CX/D2 /D8/CT/D2/D7/CX/D8 /DD /CS/CX/D7/D8/D6/CX/CQ/D9/D8/CX/D3/D2/D7 /D6/CP/CX/D7/CT/D7 /D8/CW/CT /CU/D3/D0/B9/D0/D3 /DB/CX/D2/CV /CV/CT/D2/CT/D6/CP/D0 /CX/D7/D7/D9/CT/BM /DB/CW/CP/D8 /CP/D6/CT /CV/D3 /D3 /CS /CT/D7/D8/CX/D1/CP/D8/D3/D6/D7 /CU/D3/D6 /CS/CP/D8/CP /CS/D6/CP /DB/D2 /CU/D6/D3/D1 /CP/D2 /D9/D2/CZ/D2/D3 /DB/D2/CS/CX/D7/D8/D6/CX/CQ/D9/D8/CX/D3/D2/BR /CC/CW/CT /CP/D2/D7/DB /CT/D6 /CX/D2 /DA /D3/D0/DA /CT/D7 /AS/D2/CS/CX/D2/CV /D8/CW/CT
/D3 /D3/D6/CS/CX/D2/CP/D8/CT/D7 /CX/D2 /DB/CW/CX
/CW /D8/CW/CT /CS/CX/D7/D8/D6/CX/B9/CQ/D9/D8/CX/D3/D2 /CX/D7 /D1/D3/D7/D8 /DB /CT/D0/D0 /CQ /CT/CW/CP /DA /CT/CS/BA /C1/D2 /D8/CW/CT /CQ /CT/D7/D8 /D7/CX/D8/D9/CP/D8/CX/D3/D2/B8 /CP /CS/CX/D7/D8/D6/CX/CQ/D9/D8/CX/D3/D2 /CX/D7 /D7/CW/D3/D6/D8 /D8/CP/CX/D0/CT/CS/B8/DB/CW/CX
/CW /CT/D2/D7/D9/D6/CT/D7 /D8/CW/CP/D8 /D1/D3/D1/CT/D2 /D8/D7 /CP/D6/CT /D2/D3/D8 /D3/D2/D0/DD /DB /CT/D0/D0 /CS/CT/AS/D2/CT/CS /CQ/D9/D8 /CP/D6/CT /CP/D0/D7/D3 /D6/CT/D0/CT/DA /CP/D2 /D8/D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7 /CU/D3/D6 /CP /D7/D8/CP/D8/CX/D7/D8/CX
/CP/D0 /CS/CT/D7
/D6/CX/D4/D8/CX/D3/D2 /D3/CU /D8/CW/CT /CS/CP/D8/CP /D7/CT/D8/BA /C1/D2 /CP /D7/CX/D8/D9/CP/D8/CX/D3/D2 /D3/CU /D0/D3/D2/CV/D8/CP/CX/D0/CT/CS /CS/CX/D7/D8/D6/CX/CQ/D9/D8/CX/D3/D2/D7 /B4/CT/BA/CV/BA /CS/D6/CX/DA /CT/D2 /CQ /DD /CP /D0/CP/D6/CV/CT /D2 /D9/D1 /CQ /CT/D6 /D3/CU /D3/D9/D8/D0/CX/CT/D6/D7 /CX/D2 /CP /CS/CP/D8/CP/D7/CT/D8/B5/B8/D8/CW/CT /D7/CX/D8/D9/CP/D8/CX/D3/D2 /CX/D7 /D1/D3/D6/CT
/D3/D1/D4/D0/CX
/CP/D8/CT/CS/BA /CC/CW/CT/D2/B8 /D3/D2/CT /CT/CX/D8/CW/CT/D6 /D2/CT/CT/CS/D7 /D8/D3 /CT/D7/D8/CP/CQ/D0/CX/D7/CW /CP /D1/D3 /CS/CT/D0/CS/CT/D7
/D6/CX/CQ/CX/D2/CV /CW/D3 /DB /D3/D2/CT /D7/CW/D3/D9/D0/CS /D8/D6/D9/D2
/CP/D8/CT /D8/CW/CT /CS/CP/D8/CP/D7/CT/D8 /CQ /CT/CU/D3/D6/CT
/CP/D0
/D9/D0/CP/D8/CX/D2/CV /CP /DA /CT/D6/CP/CV/CT/D7/B8 /D3/D6/DB /D3/D6/CZ /DB/CX/D8/CW /CT/D7/D8/CX/D1/CP/D8/D3/D6/D7 /DB/CW/CX
/CW /CP/D6/CT /D1/D3/D6/CT /D6/D3/CQ/D9/D7/D8 /D8/D3 /D3/D9/D8/D0/CX/CT/D6/D7/B8 /D0/CX/CZ /CT /D4 /CT/D6
/CT/D2 /D8/CX/D0/CT/D7/BA /CC /D3/CU/D3/D6/D1 /D9/D0/CP/D8/CT /D8/CW/CX/D7 /D1/D3/D6/CT /D4/D6/CT
/CX/D7/CT/D0/DD /B8 /DB /CT
/D3/D2/D7/CX/CS/CT/D6 /D8/CW/CT /CU/D3/D0/D0/D3 /DB/CX/D2/CV /D4/D6/D3/CQ/D0/CT/D1/BM /D7/D9/D4/D4 /D3/D7/CT /DB /CT/CW/CP /DA /CTn /D7/CP/D1/D4/D0/CT/D7 /CU/D6/D3/D1 /CP /D4 /D3/D7/CX/D8/CX/DA /CT /CS/CX/D7/D8/D6/CX/CQ/D9/D8/CX/D3/D2 p(x) /B8 /CP/D2/CS /D7/CP/D1/D4/D0/CT/D7 /CU/D6/D3/D1 /D8/CW/CT /D7
/CP/D0/CT/CS/CS/CX/D7/D8/D6/CX/CQ/D9/D8/CX/D3/D2 λp(x/λ) /BA /CC/CW/CT /D4/D6/D3/CQ/D0/CT/D1 /CX/D7 /D8/D3 /AS/D2/CS /D8/CW/CT /D3/D4/D8/CX/D1/CP/D0 /CT/D7/D8/CX/D1/CP/D8/D3/D6 /CU/D3/D6λ /BA /CC/CW/CT/D7/D3/D0/D9/D8/CX/D3/D2
/D0/CT/CP/D6/D0/DD /D7/CW/CP/D0/D0 /CS/CT/D4 /CT/D2/CS /D3/D2p /BA /C1/CUp /CX/D7 /CP /DB /CT/D0/D0 /CQ /CT/CW/CP /DA /CT/CS /CS/CX/D7/D8/D6/CX/CQ/D9/D8/CX/D3/D2/B8 /D8/CW/CT/D2
/angbracketleftλp(x/λ)/angbracketright//angbracketleftp(x)/angbracketright /B4/angbracketleftx/angbracketright=1
n/summationtextn
i=1xi
/CS/CT/D2/D3/D8/CT/D7 /D8/CW/CT /CP/D6/CX/D8/CW/D1/CT/D8/CX
/CP /DA /CT/D6/CP/CV/CT/B5 /CX/D7 /CP /AS/D2/CT/CT/D7/D8/CX/D1/CP/D8/D3/D6/BN /CQ/D9/D8 /CX/D8 /D1/CX/D7/CT/D6/CP/CQ/D0/DD /CU/CP/CX/D0/D7 /CX/CUp /CX/D7 /D0/D3/D2/CV /D8/CP/CX/D0/CT/CS/BA /BV/D3/D2 /DA /CT/D6/D7/CT/D0/DD /B8 /D8/CW/CT /D1/CT/CS/CX/CP/D2 /CX/D7 /CP/D7/D9/CQ /D3/D4/D8/CX/D1/CP/D0 /CT/D7/D8/CX/D1/CP/D8/D3/D6 /CX/D2 /D8/CW/CT
/CP/D7/CT /D3/CU /DB /CT/D0/D0 /CQ /CT/CW/CP /DA /CT/CS /CS/CX/D7/D8/D6/CX/CQ/D9/D8/CX/D3/D2/D7/B8 /CQ/D9/D8 /CX/D8 /CW/CP/D7 /D8/CW/CT/CP/CS/DA /CP/D2 /D8/CP/CV/CT /D3/CU /CQ /CT/CX/D2/CV /D1/D3/D6/CT /D6/D3/CQ/D9/D7/D8 /CX/D2 /D8/CW/CT /D0/D3/D2/CV /D8/CP/CX/D0/CT/CS
/CP/D7/CT/BA/BW/CP/D8/CP/D7/CT/D8/D7 /CU/D6/D3/D1 /C0/BW/C7/C6/BT/D7 /CS/D3 /CT/DC/CW/CX/CQ/CX/D8 /D7/D9
/CW /D0/D3/D2/CV /D8/CP/CX/D0/D7/B8 /CP/D7 /DB /CT /D7/CW/D3 /DB /CX/D2 /BY/CX/CV/BA /BE/BA /CC/CW/CT/CW/CX/D7/D8/D3/CV/D6/CP/D1/D7 /B4/BY/CX/CV/BA /BE/BT/B5 /D7/CW/D3 /DB /D8/CW/CTlog2
/C8/C5 /CX/D2 /D8/CT/D2/D7/CX/D8 /DD /CS/CX/D7/D8/D6/CX/CQ/D9/D8/CX/D3/D2/D7 /D3/CU /CP/D0/D0 /D4/D6/D3/CQ /CT /D7/CT/D8/D7/B8/CT/CP
/CW /C8/C5
/CT/D0/D0 /CQ /CT/CX/D2/CV /D2/D3/D6/D1/CP/D0/CX/DE/CT/CS /CQ /DD /D8/CW/CT/CX/D6 /D4/D6/D3/CQ /CT /D7/CT/D8 /D1/CT/CS/CX/CP/D2/BA /C8/D6/D3/CQ /CT /D7/CT/D8/D7 /CP/D6/CT
/D0/CP/D7/B9/D7/CX/AS/CT/CS /CX/D2 /D8/D3 /CU/D3/D9/D6 /DB/CX/D2/CS/D3 /DB/D7 /CP
/D3/D6/CS/CX/D2/CV /D8/D3 /D8/CW/CT/CX/D6 /D1/CT/CS/CX/CP/D2 /D1/CP/CV/D2/CX/D8/D9/CS/CT/BN /DB /CT /CW/CP /DA /CT /DA /CT/D6/CX/AS/CT/CS/D8/CW/CP/D8 /D6/CT/AS/D2/CT/CS /DB/CX/D2/CS/D3 /DB/D7 /CS/D3 /D2/D3/D8
/CW/CP/D2/CV/CT /D8/CW/CT /D7/CW/CP/D4 /CT /D3/CU /D8/CW/CT /CS/CX/D7/D8/D6/CX/CQ/D9/D8/CX/D3/D2/D7 /D7/CX/CV/D2/CX/AS
/CP/D2 /D8/D0/DD /BA/CC/CW/CT/D7/CT /CS/CX/D7/D8/D6/CX/CQ/D9/D8/CX/D3/D2/D7 /D7/CW/D3 /DB /D8/CW/CP/D8
/CT/D0/D0 /CX/D2 /D8/CT/D2/D7/CX/D8/CX/CT/D7 /DA /CP/D6/DD /CQ /DD /CU/CP
/D8/D3/D6/D7 /D3/CU∼25/CP/D6/D3/D9/D2/CS/D8/CW/CT/CX/D6 /D1/CT/CS/CX/CP/D2 /CX/D2 /CP/D0/D0 /D8/CW/CT /CX/D2 /D8/CT/D2/D7/CX/D8 /DD /DB/CX/D2/CS/D3 /DB/D7/B8 /CP/D2/CS /D8/CW/CP/D8 /D8/CW/CT /CS/CX/D7/D8/D6/CX/CQ/D9/D8/CX/D3/D2/D7 /CP/D6/CT /CU/CP/D6/CU/D6/D3/D1 /BZ/CP/D9/D7/D7/CX/CP/D2/BA /C6/CT/DA /CT/D6/D8/CW/CT/D0/CT/D7/D7/B8 /D0/D3/CV/B9
/D3 /D3/D6/CS/CX/D2/CP/D8/CT/D7 /D0/CT/CP/CS /D8/D3 /D6/D3/D9/CV/CW/D0/DD /D7/DD/D1/D1/CT/D8/D6/CX
/CS/CX/D7/D8/D6/CX/B9/CQ/D9/D8/CX/D3/D2/D7/B8 /CP/D8 /D0/CT/CP/D7/D8 /D9/D4 /D8/D3 /D8/CW/CT /D0/CP/D7/D8 /CX/D2 /D8/CT/D2/D7/CX/D8 /DD /DB/CX/D2/CS/D3 /DB/BA /CF /CT /D8/CW/CX/D2/CZ /D3/CU /D8/CW/CT/D7/CT /CS/CX/D7/D8/D6/CX/CQ/D9/D8/CX/D3/D2/D7/CP/D7 /D8/CW/CT /D7/D9/D1 /D3/CU /CP /DB /CT/D0/D0 /CQ /CT/CW/CP /DA /CT/CS
/D3/D1/D4 /D3/D2/CT/D2 /D8 /B4/DB/CX/D8/CW
/D3/D2 /DA /CT/D6/CV/CX/D2/CV /D1/D3/D1/CT/D2 /D8/D7 /D8/D3 /DB/CW/CX
/CW/D8/CW/CT /BV/CT/D2 /D8/D6/CP/D0 /C4/CX/D1/CX/D8 /CC/CW/CT/D3/D6/CT/D1 /CP/D4/D4/D0/CX/CT/D7/B5/B8 /D4/D0/D9/D7 /CP /D0/D3/D2/CV /D8/CP/CX/D0/CT/CS /D4/CP/D6/D8 /CS/D9/CT /D8/D3 /D3/D9/D8/D0/CX/CT/D6/D7/BA/CC/CW/CT/D7/CT /D2/CT/CT/CS /D8/D3 /CQ /CT /CX/CS/CT/D2 /D8/CX/AS/CT/CS /B4
/CU/BA /C8/D6/D3/CQ /CT
/CT/D0/D0 /D7/CT/D0/CT
/D8/CX/D3/D2/B5 /CP/D2/CS /CS/CX/D7
/CP/D6/CS/CT/CS/BA /CC/CW/CT/D2/B8 /D8/CW/CT/D1/CT/CP/D2/CX/D2/CV/CU/D9/D0 /CT/D7/D8/CX/D1/CP/D8/D3/D6/D7 /CU/D3/D6 /D8/CW/CT /D8/D6/D9/D2
/CP/D8/CT/CS /CS/CP/D8/CP /D7/CT/D8/D7
/D3/D2/D7/CX/D7/D8 /D3/CU /CP/D6/CX/D8/CW/D1/CT/D8/CX
/CP /DA /CT/D6/CP/CV/CT/D7/BG/CX/D2 /D0/D3/CV/B9
/D3 /D3/D6/CS/CX/D2/CP/D8/CT/D7 /B4/CV/CT/D3/D1/CT/D8/D6/CX
/D1/CT/CP/D2/D7/B5/BA/CC/CW/CT /D3/D6/CS/CT/D6/CT/CS /CX/D2 /D8/CT/D2/D7/CX/D8 /DD /D4/D6/D3/AS/D0/CT/D7 /D3/CU /CX/D2/CS/CX/DA/CX/CS/D9/CP/D0/B8 /D6/CP/D2/CS/D3/D1/D0/DD /D4/CX
/CZ /CT/CS /D4/D6/D3/CQ /CT /D7/CT/D8/D7 /CP/D6/CT/CP/D0/D7/D3 /D7/CW/D3 /DB/D2 /B4/BY/CX/CV/BA /BE/BU/B5/B8 /CT/CP
/CW /D3/CU /D8/CW/CT/D1 /CU/D3/D6 /CS/D9/D4/D0/CX
/CP/D8/CT /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/D7/BA /CC/CW/CT/D7/CT /CT/D1/D4/CW/CP/D7/CX/DE/CT/D8/CW/CT /D6/CT/D4/D6/D3 /CS/D9
/CX/CQ/CX/D0/CX/D8 /DD /D3/CU /D8/CW/CT /CQ/D6/D3/CP/CS /CX/D2 /D8/CT/D2/D7/CX/D8 /DD /D4/D6/D3/AS/D0/CT/D7/BA/CC/CW/CT /D1 /DD/D7/D8/CT/D6/CX/D3/D9/D7 /C5/C5
/CT/D0/D0/D7/BU/CT/CU/D3/D6/CT /CT/DC/D4/D0/CP/CX/D2/CX/D2/CV /CW/D3 /DB /D8/D3 /CS/CX/D7
/CP/D6/CS /D3/D9/D8/D0/CX/CT/D6/D7 /CP/D2/CS
/D3/D1/D4/D9/D8/CT /CP /D6/CP/D8/CX/D3 /D7
/D3/D6/CT/B8 /DB /CT /CT/DC/D4/D0/CP/CX/D2/DB/CW /DD /DB /CT /CS/D3 /D2/D3/D8 /D9/D8/CX/D0/CX/DE/CT /D8/CW/CT /C5/C5
/CT/D0/D0/D7 /CU/D3/D6 /D8/CW/CT
/CP/D0
/D9/D0/CP/D8/CX/D3/D2 /D3/CU
/D3/D1/D4 /D3/D7/CX/D8/CT /CX/D2 /D8/CT/D2/D7/CX/D8 /DD/CP/D2/CS /D6/CP/D8/CX/D3 /D7
/D3/D6/CT/D7/BA /CB/CX/D2/CV/D0/CT /D1/CX/D7/D1/CP/D8
/CW
/CT/D0/D0/D7 /D7/CT/CT/D1 /D2/D3/D8 /D8/D3 /CQ /CT
/D3/D2/D7/CX/D7/D8/CT/D2 /D8/D0/DD /CS/D3/CX/D2/CV /DB/CW/CP/D8/D8/CW/CT/DD /DB /CT/D6/CT /D3/D6/CX/CV/CX/D2/CP/D0/D0/DD /CS/CT/D7/CX/CV/D2/CT/CS /CU/D3/D6/B8 /D2/CP/D1/CT/D0/DD /D8/D3 /D7/CT/D6/DA /CT /CP/D7 /CP
/D3/D2 /D8/D6/D3/D0 /CU/D3/D6 /D2/D3/D2/B9/D7/D4 /CT
/CX/AS
/CW /DD/CQ/D6/CX/CS/CX/DE/CP/D8/CX/D3/D2/BA /C1/D2/D7/D8/CT/CP/CS/B8 /DB /CT /AS/D2/CS /D8/CW/CP/D8 /C5/C5
/CT/D0/D0/D7 /D3/CU/D8/CT/D2 /CP
/D8 /CP/D7 /CP /D4/CP/D0/CT /C8/C5/B8 /CT/D7/D7/CT/D2/B9/D8/CX/CP/D0/D0/DD /CQ/CX/D2/CS/CX/D2/CV /D8/CW/CT /D7/CP/D1/CT /D3/D0/CX/CV/D3/D2 /D9
/D0/CT/D3/D8/CX/CS/CT /CP/D7 /D8/CW/CT /C8/C5 /CS/D3/B8 /CQ/D9/D8 /D3/D2 /CP /DA /CT/D6/CP/CV/CT ∼1.8 /D8/CX/D1/CT/D7/DB /CT/CP/CZ /CT/D6 /D8/CW/CP/D2 /D8/CW/CT /C8/C5 /D4/D6/D3/CQ /CT/D7 /B4/BY/CX/CV/BA /BF/B5/BA /C6/D3/D8/CX
/CT /D8/CW/CP/D8 /B4/CP/B5 /D4/D6/CT/D7/CT/D2 /D8/D7 /D8/CW/CT /D6/CP /DB
/CT/D0/D0 /CX/D2 /D8/CT/D2/B9/D7/CX/D8/CX/CT/D7/B8 /DB/CW/CT/D6/CT/CP/D7 /B4/CQ/B5 /D6/CT/D4 /D3/D6/D8/D7 /D8/CW/CT /CS/CX/D7/D8/D6/CX/CQ/D9/D8/CX/D3/D2/D7 /D3/CU
/D3/D1/D4 /D3/D7/CX/D8/CT /CX/D2 /D8/CT/D2/D7/CX/D8/CX/CT/D7 /D3/CQ/D8/CP/CX/D2/CT/CS /CQ /DD
/D3/D2/D7/CX/CS/CT/D6/CX/D2/CV /D8/CW/CT /C8/C5 /CP/D2/CS /C5/C5 /D4/D6/D3/CQ /CT /D7/CT/D8/D7 /CP/D7 /CX/CU /D8/CW/CT/DD /DB /CT/D6/CT /D8 /DB /D3 /CS/CX/AR/CT/D6/CT/D2 /D8
/D3/D2/CS/CX/D8/CX/D3/D2/D7/CU/D3/D6 /D8/CW/CT /D7/CP/D1/CT /CV/CT/D2/CT/BA /C1/D8 /CX/D7 /D7/D3/D1/CT/DB/CW/CP/D8 /CS/CX/D7/D8/D9/D6/CQ/CX/D2/CV /D8/CW/CP/D8 /CX/D2 /D8/CW/CT /CW/CX/CV/CW /CX/D2 /D8/CT/D2/D7/CX/D8 /DD /D6/CT/CV/CX/D3/D2/B8/D8/CW/CT
/D0/D3/D9/CS /B4/CP/B5 /CT/DC/CW/CX/CQ/CX/D8/D7 /CP /DA /CP/D0/D0/CT/DD /CP/D6/D3/D9/D2/CS /D8/CW/CT /CS/CX/CP/CV/D3/D2/CP/D0/BA /CC/CW/CX/D7 /D1/CT/CP/D2/D7 /D8/CW/CP/D8 /D8/CW/CT/D6/CT /CX/D7 /CP/D7/CX/CV/D2/CX/AS
/CP/D2 /D8 /D2 /D9/D1 /CQ /CT/D6 /D3/CU /D4/D6/D3/CQ /CT/D7 /DB/CW/CT/D6/CT /D8/CW/CT /D8/CP/D6/CV/CT/D8
/CA/C6/BT/D7 /CQ/CX/D2/CS /D1/D3/D6/CT /D7/D4 /CT
/CX/AS
/CP/D0/D0/DD /D8/D3/D8/CW/CT /C5/C5/BA /CF /CT /CU/D3/D9/D2/CS /D2/D3
/D3/D2 /DA/CX/D2
/CX/D2/CV /CT/DC/D4/D0/CP/D2/CP/D8/CX/D3/D2 /CU/D3/D6 /D8/CW/CX/D7 /D1/CP/D8/D8/CT/D6/B8 /CQ/D9/D8 /CX/D7 /CX/D7 /D9/D2/D0/CX/CZ /CT/D0/DD/D8/D3 /CQ /CT /D8/CW/CT
/D3/D2/D7/CT/D5/D9/CT/D2
/CT /D3/CU /D2/D3/D2/B9/D7/D4 /CT
/CX/AS
/CW /DD/CQ/D6/CX/CS/CX/DE/CP/D8/CX/D3/D2 /B4/D2/D3/D2/B9/D7/D4 /CT
/CX/AS
/CW /DD/CQ/D6/CX/CS/CX/DE/CP/D8/CX/D3/D2/DB /D3/D9/D0/CS /D6/CT/D7/D9/D0/D8 /CX/D2 /CP /D1/CP/DC/CX/D1 /D9/D1 /D3/D2 /D8/CW/CT /CS/CX/CP/CV/D3/D2/CP/D0/B5/BA /CC/CW/CT/D6/CT/CU/D3/D6/CT/B8 /D7/D9/CQ/D8/D6/CP
/D8/CX/D2/CV /D8/CW/CT /C5/C5/CU/D6/D3/D1 /D8/CW/CT /C8/C5 /CX/D2 /D8/CT/D2/D7/CX/D8/CX/CT/D7 /CX/D7 /D0/CX/CZ /CT/D0/DD /D8/D3 /CQ /CT /D1/CX/D7/D0/CT/CP/CS/CX/D2/CV/B8 /CP/D2/CS /DB /CT /CU/D3/D9/D2/CS /CX/D8 /CU/CP /DA /D3/D6/CP/CQ/D0/CT /D8/D3/D2/D3/D8
/D3/D2/D7/CX/CS/CT/D6 /D8/CW/CT /C5/C5
/CT/D0/D0/D7 /CP/D2 /DD /CU/D9/D6/D8/CW/CT/D6/BA /BT/D7 /CP /D1/CP/D8/D8/CT/D6 /D3/CU /CU/CP
/D8/B8 /CX/D8 /CX/D7 /D2/D3/D8 /CT/D2 /D8/CX/D6/CT/D0/DD/D7/D9/D6/D4/D6/CX/D7/CX/D2/CV /D8/CW/CP/D8 /CP /D7/CX/D2/CV/D0/CT /CQ/CP/D7/CT
/CW/CP/D2/CV/CT /CS/D3 /CT/D7 /D2/D3/D8 /D4/D6/D3 /DA/CX/CS/CT /CP
/D0/CT/CP/D6
/D9/D8 /CS/CX/D7
/D6/CX/D1/CX/D2/CP/B9/D8/CX/D3/D2 /CU/D3/D6 /D2/D3/D2/B9/D7/D4 /CT
/CX/AS
/CW /DD/CQ/D6/CX/CS/CX/DE/CP/D8/CX/D3/D2 /CX/D2 /CP /D4/D6/D3
/CT/D7/D7 /CS/D3/D1/CX/D2/CP/D8/CT /CQ /DD /CZ/CX/D2/CT/D8/CX
/D7 /D6/CP/D8/CW/CT/D6 /D8/CW/CP/D8/CT/D5/D9/CX/D0/CX/CQ/D6/CX/D9/D1 /D8/CW/CT/D6/D1/D3 /CS/DD/D2/CP/D1/CX
/D7/BA/C8/D6/D3/CQ /CT
/CT/D0/D0 /D7/CT/D0/CT
/D8/CX/D3/D2 /CP/D2/CS /D6/CP/D8/CX/D3
/D3/D1/D4 /D3/D7/CX/D8/CT/D7/CF /CT /D2/D3 /DB /CS/CT/D7
/D6/CX/CQ /CT /D3/D9/D6 /CP/D0/CV/D3/D6/CX/D8/CW/D1 /CU/D3/D6 /D8/CW/CT /D7/CT/D0/CT
/D8/CX/D3/D2 /D3/CU
/CT/D0/D0/D7 /D9/D7/CT/CS /CX/D2 /D8/CW/CT
/CP/D0
/D9/D0/CP/D8/CX/D3/D2/D3/CU /D6/CP/D8/CX/D3/D7/BA /CF /CT /D3/CQ/D7/CT/D6/DA /CT/CS /D8/CW/CP/D8
/D3/D1/D4/CP/D6/CX/D2/CV /D8 /DB /D3 /CX/CS/CT/D2 /D8/CX
/CP/D0 /D4/D6/D3/CQ /CT /D7/CT/D8/D7 /CW /DD/CQ/D6/CX/CS/CX/DE/CT/CS /D8/D3/D8 /DB /D3 /CS/CX/AR/CT/D6/CT/D2 /D8 /D7/CP/D1/D4/D0/CT/D7 /D0/CT/CP/CS/D7 /D8/D3 /D7/CT/D6/CX/CT/D7 /D3/CU /D4/CP/CX/D6/DB/CX/D7/CT /C8/C5
/CT/D0/D0/D7 /D6/CP/D8/CX/D3/D7 (r1, r2, . . ., r N)/CQ /CT/CW/CP /DA/CX/D2/CV /D5/D9/CX/D8/CT /CU/CP/D6 /CU/D6/D3/D1 /CP/D2 /CX/CS/CT/CP /CW/D3/D1/D3/CV/CT/D2/CT/D3/D9/D7 /D7/CX/D8/D9/CP/D8/CX/D3/D2 /B4/CP/D0/D0ri
/CQ /CT/CX/D2/CV /D8/CW/CT /D7/CP/D1/CT/B5/BA/C1/D2/D7/D8/CT/CP/CS/B8 /D8/CW/CT /CX/D2/CS/CX/DA/CX/CS/D9/CP/D0
/CT/D0/D0 /D6/CP/D8/CX/D3/D7 /D3/CU/D8/CT/D2 /DA /CP/D6/DD /D3 /DA /CT/D6 /CP /CS/CT
/CP/CS/CT/BN /CX/D8 /CP/D0/D7/D3 /D3
/D9/D6/D7 /D8/CW/CP/D8/D7/D3/D1/CT
/CT/D0/D0/D7 /CX/D2/CS/CX
/CP/D8/CT /CP/D2 /D9/D4/B9/D6/CT/CV/D9/D0/CP/D8/CX/D3/D2 /DB/CW/CT/D6/CT/CP/D7 /D3/D8/CW/CT/D6/D7 /CX/D2/CS/CX
/CP/D8/CT /D8/CW/CT /D3/D4/D4 /D3/D7/CX/D8/CT /B4
/CU/BA/BY/CX/CV/BA /BG /B4
/B5 /CP/D2/CS /B4/CS/B5/B5/BA /C1/D2 /D8/CW/CX/D7 /D7/CX/D8/D9/CP/D8/CX/D3/D2/B8 /CP /D7/D8/D6/CP/CX/CV/CW /D8/CU/D3/D6/DB /CP/D6/CS /D0/CX/D2/CT/CP/D6 /D6/CT/CV/D6/CT/D7/D7/CX/D3/D2 /CQ /CT/D8 /DB /CT/CT/D2/C8/C5 /CX/D2 /D8/CT/D2/D7/CX/D8/CX/CT/D7 /D3/CU /D8/CW/CT /D8 /DB /D3 /D7/CP/D1/D4/D0/CT/D7 /CX/D7 /D2/D3/D8 /CP/CS/CT/D5/D9/CP/D8/CT/BA /C1/D8 /CU/D9/D6/D8/CW/CT/D6 /CW/CP/D4/D4 /CT/D2/D7 /D8/CW/CP/D8 /CW/CX/CV/CW/CX/D2 /D8/CT/D2/D7/CX/D8 /DD
/CT/D0/D0/D7 /D7/CP/D8/D9/D6/CP/D8/CT /CX/D2 /D3/D2/CT /D3/D6 /CQ /D3/D8/CW /D3/CU /D8/CW/CT /D7/CP/D1/D4/D0/CT/D7/B8 /D0/CT/CP/CS/CX/D2/CV /D8/D3 /D9/D7/CT/D0/CT/D7/D7 /B4/CT/DA /CT/D2/D1/CX/D7/D0/CT/CP/CS/CX/D2/CV/B5
/CT/D0/D0 /D6/CP/D8/CX/D3/D7/BA /CB/D9
/CW /D4/D6/D3/CQ /CT
/CT/D0/D0/D7 /CP/D6/CT /CS/CX/D7
/CP/D6/CS/CT/CS /CU/D6/D3/D1 /D3/D9/D6 /CP/D2/CP/D0/DD/D7/CX/D7/BA /CC/CW/CT/D7/CP/D8/D9/D6/CP/D8/CX/D3/D2 /D8/CW/D6/CT/D7/CW/D3/D0/CS/D7 /B4/D1/D3/D7/D8 /D0/CX/CZ /CT/D0/DD /CS/D9/CT /D8/D3 /D8/CW/CT /D4/CW/D3/D8/D3/D1 /D9/D0/D8/CX/D4/D0/CX/CT/D6/B5
/CP/D2 /CQ /CT /D6/CT/CP/CS /D3/AR /D8/CW/CT/BA/BV/BX/C4 /AS/D0/CT/D7/B8 /CQ /DD /D4/D0/D3/D8/D8/CX/D2/CV /D8/CW/CT /D1/CT/CP/D2
/CT/D0/D0 /CX/D2 /D8/CT/D2/D7/CX/D8/CX/CT/D7 /DA /CT/D6/D7/D9/D7 /D8/CW/CT /D7/D8/CP/D2/CS/CP/D6/CS /CS/CT/DA/CX/CP/D8/CX/D3/D2/D3/CU /D8/CW/CT /D4/CX/DC/CT/D0 /CX/D2 /D8/CT/D2/D7/CX/D8/CX/CT/D7/BA /C7/D9/D6 /D4/D9/D6/D4 /D3/D7/CT /CW/CT/D6/CT /CX/D7 /D2/D3/D8 /D8/D3 /CP/CS/CS/D6/CT/D7/D7 /D8/CW/CT /D5/D9/CT/D7/D8/CX/D3/D2 /D3/CU/DB/CW /DD /D7/D9
/CW /CQ/D6/D3/CP/CS /D6/CP/D8/CX/D3 /CS/CX/D7/D8/D6/CX/CQ/D9/D8/CX/D3/D2/D7 /D1/CP /DD /CP/D6/CX/D7/CT/B8 /CQ/D9/D8 /D6/CP/D8/CW/CT/D6 /CW/D3 /DB /D8/D3 /D3/D4/D8/CX/D1/CX/DE/CT /D7
/D3/D6/CT/D7/BH/CU/D3/D6 /D8/CW/CT/D1/BA /C1/D2 /CP /AS/D6/D7/D8 /D7/D8/CT/D4/B8 /DB /CT /D3/D6/CS/CT/D6 /D8/CW/CT /D7/CT/D6/CX/CT/D7 (logr1,logr2, . . . ,logrN) /BA /C6/CT/DC/D8/B8/DB /CT /CP/CX/D1 /CP/D8 /D7/D4/D0/CX/D8/D8/CX/D2/CV /D8/CW/CX/D7 /D7/CT/D8 /CX/D2 /CP/D2 /CX/D2 /D8/CT/D6/DA /CP/D0 Imed
/DB/CX/D8/CW /D3/D4/D8/CX/D1/CP/D0/D0/DD /D2/CP/D6/D6/D3 /DB /D6/CP/D2/CV/CT/B8/CP/D2/CS /CP /D7/D9/CQ/D7/CT/D8 /D3/CU /D3/D9/D8/D0/CX/CT/D6/D7 /D8/D3 /CQ /CT /D3/D1/CX/D8/D8/CT/CS /CU/D6/D3/D1 /D8/CW/CT /D6/CP/D8/CX/D3 /D7
/D3/D6/CT
/CP/D0
/D9/D0/CP/D8/CX/D3/D2/BA /CF /CT/D6/CT/D5/D9/CX/D6/CT /D8/CW/CT /D1/CT/CS/CX/CP/D2 /D8/D3 /CQ /CT /CP /D1/CT/D1 /CQ /CT/D6 /D3/CUImed
/CP/D2/CS /D3/D4/D8/CX/D1/CX/DE/CT /CU/D3/D6 /CX/D8/D7 /D0/CT/CU/D8 /CP/D2/CS /D6/CX/CV/CW /D8/CQ /D3/D9/D2/CS/CP/D6/CX/CT/D7 il
/CP/D2/CSir
/BAl= log rir−logril
/CS/CT/D2/D3/D8/CT/D7 /D8/CW/CT /D6/CP/D2/CV/CT /D3/CUImed
/BA /C1/D2 /D8/CW/CT/CP/CQ/D7/CT/D2
/CT /D3/CU /CZ/D2/D3 /DB/D0/CT/CS/CV/CT /CP/CQ /D3/D9/D8 /CW/D3 /DB /D8/CW/CT /D6/CP/D8/CX/D3/D7 /CP/D6/CT /CS/CX/D7/D8/D6/CX/CQ/D9/D8/CT/CS /DB/CX/D8/CW/CX/D2 Imed
/B8 /CT/DC
/CT/D4/D8/CU/D3/D6 /D8/CW/CT /D6/CP/D2/CV/CT /CU/D6/D3/D1 /DB/CW/CX
/CW /D8/CW/CT /D6/CP/D8/CX/D3/D7 /CP/D6/CT /CS/D6/CP /DB/D2/B8 /D8/CW/CT /D1/D3/D7/D8 /D9/D2 /CQ/CX/CP/D7/CT/CS /CP/D7/D7/D9/D1/D4/D8/CX/D3/D2/CX/D7 /D8/D3 /CP/D7/D7/D9/D1/CT /CP /D9/D2/CX/CU/D3/D6/D1 /D4/D6/D3/CQ/CP/CQ/CX/D0/CX/D8 /DD p /DB/CX/D8/CW/CX/D2 /D8/CW/CX/D7 /D6/CP/D2/CV/CT/BA /C0/CT/D2
/CT/B8 /D8/CW/CT /D4/D6/D3/CQ/CP/CQ/CX/D0/CX/D8 /DD/D3/CU /AS/D2/CS/CX/D2/CV /CP /D0/D3/CV/B9/D6/CP/D8/CX/D3 /CX/D2Imed
/CX/D7p=1
l
/CP/D2/CSp′=1
L
/CU/D3/D6 /CP/D2 /D3/D9/D8/D0/CX/CT/D6/BA /CF /CT /D8/CW/CT/D2/D6/CT/D8/CP/CX/D2 Imed
/D8/CW/CP/D8 /D1/CP/DC/CX/D1/CX/DE/CT/D7 /D8/CW/CT /D0/CX/CZ /CT/D0/CX/CW/D3 /D3 /CS /D3/CU /D8/CW/CT /CU/D9/D0/D0 /D4/D6/D3/CQ /CT /D7/CT/D8 /D6/CP/D8/CX/D3/D7 /CV/CX/DA /CT/D2 /D3/D9/D6/D1/D3 /CS/CT/D0/BA /CF /CT /D1 /D9/D7/D8 /D8/CW/CT/D6/CT/CU/D3/D6/CT /D1/CP/DC/CX/D1/CX/DE/CT L=−(N−n)logL−nlogl /B8 /DB/CW/CT/D6/CT N /CX/D7/D8/CW/CT /D8/D3/D8/CP/D0 /D2 /D9/D1 /CQ /CT/D6 /D3/CU
/CT/D0/D0/D7 /CP/D2/CSn /D8/CW/CT
/CT/D0/D0/D7 /CX/D2Imed
/BA /C1/D2 /CT/D7/D7/CT/D2
/CT/B8 /D8/CW/CX/D7 /D4/D6/D3
/CT/CS/D9/D6/CT/D4/CX
/CZ/D7 /D8/CW/CT /D3/D4/D8/CX/D1/CP/D0 /CX/D2 /D8/CT/D6/DA /CP/D0 Imed
/CP/D7 /CP /D8/D6/CP/CS/CT/D3/AR /CQ /CT/D8 /DB /CT/CT/D2 /CW/CP /DA/CX/D2/CV /D8/D3 /D3 /D1/CP/D2 /DD /D3/D9/D8/D0/CX/CT/D6/D7/B8/CP/D2/CS /D0/CT/D8/D8/CX/D2/CV /D8/CW/CT /D6/CP/D2/CV/CT /D3/CUImed
/CQ /CT
/D3/D1/CT /D8/D3 /D3 /DB/CX/CS/CT/BA /C8/D6/D3/D8/D3/D8 /DD/D4 /CT /D7/CX/D8/D9/CP/D8/CX/D3/D2/D7 /D7/CW/D3 /DB/CX/D2/CV/CW/D3 /DB /D3/D9/D6 /D1/D3 /CS/CT/D0 /D7/CT/D0/CT
/D8/D7 Imed
/CP/D6/CT /D4/D6/CT/D7/CT/D2 /D8/CT/CS /CX/D2 /BY/CX/CV/BA /BH /CU/D3/D6 /D8 /DB /D3 /CS/CX/AR/CT/D6/CT/D2 /D8
/D3/D2/CS/CX/D8/CX/D3/D2/D7/BA/BT/CU/D8/CT/D6 /CW/CP /DA/CX/D2/CV /CX/CS/CT/D2 /D8/CX/AS/CT/CS Imed
/B8 /DB /CT
/D3/D1/D4/D9/D8/CT /D7
/D3/D6/CT/D7 /CQ /DD /D8/CP/CZ/CX/D2/CV /CV/CT/D3/D1/CT/D8/D6/CX
/D1/CT/CP/D2/D7 /D3/CU
/CT/D0/D0 /D6/CP/D8/CX/D3/D7 /CP/D2/CS /CX/D2 /D8/CT/D2/D7/CX/D8/CX/CT/D7 /CU/D6/D3/D1 /C8/C5
/CT/D0/D0/D7 /CX/D2/D7/CX/CS/CT Imed
/BA/C1/D8 /CX/D7 /D2/D3 /DB /DB /D3/D6/D8/CW /DB/CW/CX/D0/CT /D0/D3 /D3/CZ/CX/D2/CV /CQ/CP
/CZ /CP/D8 /BY/CX/CV/BA /BD/B4/CQ/B5 /CP/D2/CS /B4
/B5 /D7/CW/D3 /DB/CX/D2/CV /DB/CW/CX
/CW /D4/D6/D3/CQ /CT
/CT/D0/D0/D7 /CP/D6/CT /CP
/D8/D9/CP/D0/D0/DD /D7/CT/D0/CT
/D8/CT/CS/BA /BT/D7 /CP /CU/CP
/D8/B8 /D8/CW/CT/D6/CT /CP/D6/CT /D3/D2/D0/DD /CU/CT/DB /D4/D6/D3/CQ /CT /D7/CT/D8/D7 /D8/CW/CP/D8 /CW/CP /DA /CT
/D3/D1/D1/D3/D2 /D0/D3 /DB /CX/D2 /D8/CT/D2/D7/CX/D8 /DD
/CT/D0/D0/D7/B4
/CU/BA /BY/CX/CV/BA /BE/BT/B5/BA /C1/D2/D7/D8/CT/CP/CS/B8 /D8/CW/CT /D0/D3 /DB /CX/D2 /D8/CT/D2/D7/CX/D8 /DD
/CT/D0/D0/D7 /CP/D6/CT/CS/CX/D7/D8/D6/CX/CQ/D9/D8/CT/CS /CP/D1/D3/D2/CV /D8/CW/CT /D4/D6/D3/CQ /CT /D7/CT/D8/D7/B8 /DB/CW/CX
/CW /CX/D7
/D0/CT/CP/D6/D0/DD /D6/CT/AT/CT
/D8/CT/CS /CX/D2 /BY/CX/CV/BA /BD/B4
/B5 /CQ /DD /D8/CW/CT/D0/D3 /DB /CS/CT/D2/D7/CX/D8 /DD /D3/CU /D4 /D3/CX/D2 /D8/D7 /CP/D8 /D8/CW/CT /D0/D3 /DB /CT/D2/CS/BA/CA/CT/D7/D9/D0/D8/D7/CC /D3 /CS/CT/D1/D3/D2/D7/D8/D6/CP/D8/CT /D8/CW/CT /D4 /D3/D8/CT/D2 /D8/CX/CP/D0 /D3/CU /D3/D9/D6 /D1/CT/D8/CW/D3 /CS/B8 /DB /CT /CP/D2/CP/D0/DD/DE/CT/CS /CP /D7/CT/D8 /D3/CU /C0/BW/C7/C6/BT /CW /DD/B9/CQ/D6/CX/CS/CX/DE/CP/D8/CX/D3/D2/D7 /CT/DA /CP/D0/D9/CP/D8/CX/D2/CV /D8/CW/CT /D8/D6/CP/D2/D7
/D6/CX/D4/D8/CX/D3/D2/CP/D0 /D4/D6/D3/AS/D0/CT/D7 /D3/CU /D7/CX/DC /CS/CX/AR/CT/D6/CT/D2 /D8 /D1/D3/D9/D7/CT /CQ/D6/CP/CX/D2/D6/CT/CV/CX/D3/D2/D7 /D9/D7/CX/D2/CV /D8/CW/CT /C5/D9/BD/BD/CZ /D1/D3/D9/D7/CT /BT /CP/D2/CS /BU
/CW/CX/D4 /D7/CT/D6/CX/CT/D7/BA /CC/CW/CT /CS/CX/D7/D7/CT
/D8/CX/D3/D2/D7 /CP/D2/CS /CT/D2/B9/DE/DD/D1/CP/D8/CX
/D7/D8/CT/D4/D7 /B4/D1/CP/CZ/CX/D2/CV /D8/CW/CT /D8/CP/D6/CV/CT/D8
/CA/C6/BT/B5 /DB /CT/D6/CT /D4 /CT/D6/CU/D3/D6/D1/CT/CS /CX/D2 /CS/D9/D4/D0/CX
/CP/D8/CT /CX/D2 /CP/D0/D0/CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/D7 /CP/D2/CS /D8/CW/CT /D8 /DB /D3 /D3/CQ/D8/CP/CX/D2/CT/CS /D7/CP/D1/D4/D0/CT/D7 /DB /CT/D6/CT /CW /DD/CQ/D6/CX/CS/CX/DE/CT/CS /D3/D2 /D8/D3 /D7/CT/D4/CP/D6/CP/D8/CT /CP/D6/B9/D6/CP /DD/D7/BA /BY/CX/CV/BA /BH/BT /D7/CW/D3 /DB/D7 /D8/CW/CT /D7
/CP/D8/D8/CT/D6 /D4/D0/D3/D8/D7 /D3/CU /D8/CW/CT /D6/CT/D4/D0/CX
/CP/D8/CT/D7 /CU/D6/D3/D1 /CU/D3/D9/D6 /CQ/D6/CP/CX/D2 /D6/CT/CV/CX/D3/D2/D7/B8/D8/CW/CT /BT /CP/D2/CS /BU
/CW/CX/D4/D7 /CQ /CT/CX/D2/CV /D7/D9/D4 /CT/D6/CX/D1/D4 /D3/D7/CT/CS /D3/D2 /D8/CW/CT /D7/CP/D1/CT /D4/D0/D3/D8/BA/C7/D9/D6 /D7
/D3/D6/CT/D7 /CT/DC/CW/CX/CQ/CX/D8 /CP /D1 /D9
/CW /D8/CX/CV/CW /D8/CT/D6 /D7
/CP/D8/D8/CT/D6/B8 /CT/D7/D4 /CT
/CX/CP/D0/D0/DD /CX/D2 /D8/CW/CT /D1/CX/CS /D8/D3 /D0/D3 /DB /CX/D2/B9/D8/CT/D2/D7/CX/D8 /DD /D6/CP/D2/CV/CT/BA /BY /D9/D6/D8/CW/CT/D6/B8 /DB /CT /CP/D6/CT /CP/CQ/D0/CT /D8/D3 /D6/CT/D4 /D3/D6/D8 /D7
/D3/D6/CT/D7 /CU/D3/D6 /CP/D0/D0 /D8/CW/CT /CV/CT/D2/CT/D7 /D3/D2 /D8/CW/CT/CP/D6/D6/CP /DD/D7/B8 /DB/CW/CT/D6/CT/CP/D7 /D8/CW/CT /BT/AR/DD/D1/CT/D8/D6/CX/DC /CP/D0/CV/D3/D6/CX/D8/CW/D1 /D6/CT/D4 /D3/D6/D8/D7 /D2/D3/D2/B9/D2/CT/CV/CP/D8/CX/DA /CT /DA /CP/D0/D9/CT/D7 /B4/D2/CT/CV/CP/D8/CX/DA /CT/CX/D2 /D8/CT/D2/D7/CX/D8/CX/CT/D7 /CP/D6/CT /D1/CT/CP/D2/CX/D2/CV/D0/CT/D7/D7 /CP/D2/CS /D2/D3/D8 /D4/D0/D3/D8/CP/CQ/D0/CT /D3/D2 /CP /D0/D3/CV/CP/D6/CX/D8/CW/D1/CX
/D7
/CP/D0/CT/B5 /CU/D3/D6∼70% /D3/CU/D8/CW/CT /D4/D6/D3/CQ /CT /D7/CT/D8/D7/BA /BT/D7 /D1/CT/D2 /D8/CX/D3/D2/CT/CS/B8 /D8/CW/CT /D6/CT/CP/D7/D3/D2 /DB /CT /D3/CQ/D8/CP/CX/D2 /D6/CT/D0/CP/D8/CX/DA /CT/D0/DD /CU/CT/DB /D0/D3 /DB /CX/D2 /D8/CT/D2/D7/CX/D8 /DD/CV/CT/D2/CT/D7 /B42σ /D3/CU /D6/CT/D7/CX/CS/D9/CP/D0 /CQ/CP
/CZ/CV/D6/D3/D9/D2/CS ∼100 /CX/D2 /D8/CW/CT/D7/CT /D9/D2/CX/D8/D7/B5 /CX/D7 /D8/CW/CP/D8 /D0/D3 /DB /CX/D2 /D8/CT/D2/D7/CX/D8 /DD
/CT/D0/D0/D7/D8/CT/D2/CS /D8/D3 /CQ /CT /CS/CX/D7/D8/D6/CX/CQ/D9/D8/CT/CS /CP/D1/D3/D2/CV /CS/CX/AR/CT/D6/CT/D2 /D8 /D4/D6/D3/CQ /CT /D7/CT/D8/D7 /D6/CP/D8/CW/CT/D6 /D8/CW/CP/D2 /CQ /CT/CX/D2/CV /CV/D6/D3/D9/D4 /CT/CS/BA/CC/CW/CT /CW/CX/D7/D8/D3/CV/D6/CP/D1/D7 /CX/D2 /BY/CX/CV/BA /BH/BU /D7/CW/D3 /DB /D8/CW/CT /CS/CX/D7/D8/D6/CX/CQ/D9/D8/CX/D3/D2/D7 /D3/CU /D8/CW/CTlog2
/D6/CP/D8/CX/D3/D7 /CU/D6/D3/D1 /D8/CW/CT/CU/D3/D9/D6
/D3/D1 /CQ/CX/D2/CT/CS /D6/CT/CV/CX/D3/D2/D7 /CX/D2 /CX/D2 /D8/CT/D2/D7/CX/D8 /DD /DB/CX/D2/CS/D3 /DB/D7/BA /C7/D9/D6 /CS/CX/D7/D8/D6/CX/CQ/D9/D8/CX/D3/D2/D7 /CP/D6/CT /DB /CT/D0/D0 /AS/D8/D8/CT/CS/CQ /DD /D2/CP/D6/D6/D3 /DB /BZ/CP/D9/D7/D7/CX/CP/D2/D7 /DB/CX/D8/CW /D7/D8/CP/D2/CS/CP/D6/CS /CS/CT/DA/CX/CP/D8/CX/D3/D2/D7 σ∼0.2 /CU/D3/D6 /CX/D2 /D8/CT/D2/D7/CX/D8/CX/CT/D7 >300 /BA
2σ /D8/CW/CT/D2
/D3/D6/D6/CT/D7/D4 /D3/D2/CS/D7 /D8/D3 /CP /CU/D3/D0/CS
/CW/CP/D2/CV/CT /D3/CU∼1.25 /BA /C1/D2
/D3/D2 /D8/D6/CP/D7/D8/B8 /D8/CW/CT /BT/AR/DD/D1/CT/D8/D6/CX/DC/D7
/D3/D6/CT/D7 /D0/CT/CP/CS /D8/D3 /D0/D3/D2/CV/CT/D6 /D8/CP/CX/D0/D7 /CT/D7/D4 /CT
/CX/CP/D0/D0/DD /CX/D2 /D8/CW/CT /D1/CX/CS /D8/D3 /D0/D3 /DB /CX/D2 /D8/CT/D2/D7/CX/D8 /DD /D6/CP/D2/CV/CT/BA /C6/CT/DC/D8/B8/DB /CT
/D3/D2 /D8/D6/CP/D7/D8 /CP /D6/CT/D4/D0/CX
/CP/D8/CT /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8 /BY/CX/CV/BA /BI/B4/CP/B5 /DB/CX/D8/CW /CP
/D3/D1/D4/CP/D6/CX/D7/D3/D2 /D3/CU /D8 /DB /D3 /CS/CX/AR/CT/D6/CT/D2 /D8/BI/CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/CP/D0
/D3/D2/CS/CX/D8/CX/D3/D2/D7 /BY/CX/CV/BA /BI/B4/CQ/B5/BA /BT/D7σ /CX/D7 /D2/D3/D8 /D7/D8/D6/D3/D2/CV/D0/DD /CX/D2 /D8/CT/D2/D7/CX/D8 /DD /CS/CT/D4 /CT/D2/CS/CP/D2 /D8/B8/DB /CT /CW/CP /DA /CT /D8/CT/D2 /D8/CP/D8/CX/DA /CT/D0/DD /CX/D2/CS/CX
/CP/D8/CT/CS /CX/D2 /D6/CT/CS /D8/CW/CT /CU/D3/D0/CS
/CW/CP/D2/CV/CT/D7 /D3/CU1.25 /BA /BV/D3/D2/D7/CX/D7/D8/CT/D2 /D8/D0/DD /B8 6%/D3/CU /D8/CW/CT /CV/CT/D2/CT/D7 /D0/CX/CT /CQ /CT/DD /D3/D2/CS /D8/CW/CT2σ /D0/CX/D2/CT/D7 /CX/D2 /B4/CP/B5/BA /BY /D3/D6 /D8/CW/CT
/D3/D1/D4/CP/D6/CX/D7/D3/D2 /D3/CU /D8 /DB /D3 /CS/CX/AR/CT/D6/CT/D2 /D8
/D3/D2/CS/CX/D8/CX/D3/D2 /B4/CQ/B5/B820% /D3/CU /D8/CW/CT /CV/CT/D2/CT/D7 /CP/D6/CT /CS/CX/AR/CT/D6/CT/D2 /D8/CX/CP/D0/D0/DD /CT/DC/D4/D6/CT/D7/D7/CT/CS /CQ /DD /CP /CU/CP
/D8/D3/D6 ≥1.25 /BA/CF /CT /D7/CW/D3/D9/D0/CS /CU/D9/D6/D8/CW/CT/D6 /D1/CT/D2 /D8/CX/D3/D2 /D8/CW/CP/D8 /D8/CW/CT /D0/D3
/CP/D8/CX/D3/D2 /D3/CU /D3/D9/D6 /D4 /D3/CX/D2 /D8/D7 /CX/D2 /D8/CW/CT /D7
/CP/D8/D8/CT/D6 /D4/D0/D3/D8/CX/D7 /CT/D5/D9/CX/DA /CP/D0/CT/D2 /D8 /D8/D3 /D8/CW/CT /D6/CT/D4 /D3/D6/D8/CT/CS /D6/CP/D8/CX/D3/B8 /DB/CW/CX
/CW /CX/D7 /D2/D3/D8 /D8/CW/CT
/CP/D7/CT /CU/D3/D6 /D8/CW/CT /BY /D3/D0/CS /BV/CW/CP/D2/CV/CT
/CP/D0
/D9/D0/CP/D8/CT/CS /CQ /DD /BT/AR/DD/D1/CT/D8/D6/CX/DC/BA/BY/CX/D2/CP/D0/D0/DD /B8 /DB /CT /CS/CT/D1/D3/D2/D7/D8/D6/CP/D8/CT /D8/CW/CP/D8 /D3/D9/D6 /D4/D6/D3
/CT/CS/D9/D6/CT /D0/CT/CP/CS/D7 /D8/D3 /CP /CV/D6/CT/CP/D8/D0/DD /CT/D2/CW/CP/D2
/CT/CS
/D3/D2/B9/D7/CX/D7/D8/CT/D2
/CT /CQ /CT/D8 /DB /CT/CT/D2 /D8/CW/CT /D7
/D3/D6/CT/D7 /D3/CQ/D8/CP/CX/D2/CT/CS /CU/D6/D3/D1 /D4/D6/D3/CQ /CT /D7/CT/D8/D7 /CU/D3/D6 /CX/CS/CT/D2 /D8/CX
/CP/D0 /CV/CT/D2/CT/D7/BA /CC/CW/CT/D6/CT/CP/D6/CT∼700 /CV/CT/D2/CT/D7 /D6/CT/D4/D6/CT/D7/CT/D2 /D8/CT/CS /D8 /DB/CX
/CT /D3/D6 /D1/D3/D6/CT /D3/D2 /D8/CW/CT
/D3/D1 /CQ/CX/D2/CT/CS /BT /CP/D2/CS /BU /C5/D9/BD/BD/CZ/D1/D3/D9/D7/CT
/CW/CX/D4/D7/BA /CC/CW/CT /D7/CT/D5/D9/CT/D2
/CT/D7 /CU/D3/D6 /D8 /DB /D3 /D7/D9
/CW /D7/CT/D8/D7 /D1/CP /DD /D4/D6/D3/CQ /CT /CS/CX/AR/CT/D6/CT/D2 /D8 /D0/D3
/CP/D8/CX/D3/D2/D7/D3/D2 /D8/CW/CT /D7/CP/D1/CT /CV/CT/D2/CT/B8 /D3/D6 /D3/D2/CT /D4/D6/D3/CQ /CT /D7/CT/D8 /D1/CP /DD /D6/CT/D4/D6/CT/D7/CT/D2 /D8 /CP /D7/D9/CQ/D7/CT/D5/D9/CT/D2
/CT /D3/CU /D8/CW/CT /D3/D8/CW/CT/D6/BA/C6/CT/DA /CT/D6/D8/CW/CT/D0/CT/D7/D7/B8 /D8/CW/CT/DD
/D3/D6/D6/CT/D7/D4 /D3/D2/CS /D8/D3 /D8/CW/CT /D7/CP/D1/CT /D4/CW /DD/D7/CX
/CP/D0 /CV/CT/D2/CT /CP/D2/CS /D7/CW/D3/D9/D0/CS /CX/CS/CT/CP/D0/D0/DD /D0/CT/CP/CS/D8/D3 /CX/CS/CT/D2 /D8/CX
/CP/D0 /D7
/D3/D6/CT/D7/BA /C4/CT/D8p1
/CP/D2/CSp2
/CQ /CT /D8 /DB /D3 /D7/D9
/CW /D4/D6/D3/CQ /CT /D7/CT/D8/D7 /CU/D3/D6 /CP
/D3/D1/D1/D3/D2 /CV/CT/D2/CT/B8/CP/D2/CSri
/B4i= 1,2 /B5 /D8/CW/CT /D6/CP/D8/CX/D3 /D3/CU /D8/CW/CTpi
/CX/D2 /D8/CT/D2/D7/CX/D8/CX/CT/D7 /D4/D6/D3/CQ /CT/CS /CX/D2 /D8 /DB /D3 /CS/CX/AR/CT/D6/CT/D2 /D8 /CQ/D6/CP/CX/D2/D6/CT/CV/CX/D3/D2/D7/BA /C1/D2 /BY/CX/CV/BA /BJ/B8 /DB /CT /D7/CW/D3 /DB /D8/CW/CT /CS/CX/D7/D8/D6/CX/CQ/D9/D8/CX/D3/D2/D7 /D3/CUlog2(r1
r2) /CU/D3/D6 /D8 /DB /D3
/D3/D1/D4/CP/D6/CX/D7/D3/D2/D7/BV/BD /CP/D2/CS /BV/BE /B4/BV/BE
/D3/D6/D6/CT/D7/D4 /D3/D2/CS/D7 /D8/D3 /BY/CX/CV/BA /BD/B4/CQ/B5/B5/BA /CC/CW/CT /AS/CV/D9/D6/CT /CX/D7 /D7/CT/D4/CP/D6/CP/D8/CT/CS /CX/D2 /D8/D3 /D0/CT/CU/D8/CP/D2/CS /D6/CX/CV/CW /D8 /CP
/D3/D6/CS/CX/D2/CV /D8/D3 /DB/CW/CT/D8/CW/CT/D6 /D8/CW/CT /BT/AR/DD/D1/CT/D8/D6/CX/DC /CU/D3/D0/CS
/CW/CP/D2/CV/CT /DB /CP/D7 /D6/CT/D4 /D3/D6/D8/CT/CS /DB/CX/D8/CW /CP/AG∼ /AH /CX/D2 /CP/D8 /D0/CT/CP/D7/D8 /D3/D2/CT /D3/CU /D8/CW/CT /D8 /DB /D3 /D6/CT/D4/D6/CT/D7/CT/D2 /D8/CP/D8/CX/D3/D2/D7 p1
/D3/D6p2
/B4/D8/CW/CT /AG∼ /AH /CX/D2/CS/CX
/CP/D8/CT/D7 /D8/CW/CP/D8/D8/CW/CT /CQ/CP/D7/CT/D0/CX/D2/CT /CX/D2 /D8/CT/D2/D7/CX/D8 /DD /DB /CP/D7 /DB/CX/D8/CW/CX/D2 /D8/CW/CT /D6/CT/D7/CX/CS/D9/CP/D0 /CQ/CP
/CZ/CV/D6/D3/D9/D2/CS/B8 /D7/D9/CV/CV/CT/D7/D8/CX/D2/CV /D8/CW/CP/D8 /D8/CW/CT/D6/CT/D4 /D3/D6/D8/CT/CS /DA /CP/D0/D9/CT /CX/D7 /D9/D2/D6/CT/D0/CX/CP/CQ/D0/CT/B5/BA /C1/D8 /CX/D7 /CT/DA/CX/CS/CT/D2 /D8 /D8/CW/CP/D8 /D3/D9/D6 /D6/CP/D8/CX/D3/D7 /CP/D6/CT /CU/CP/D6 /D1/D3/D6/CT
/D3/D2/D7/CX/D7/D8/CT/D2 /D8/D8/CW/CP/D2 /D8/CW/CT /BT/AR/DD/D1/CT/D8/D6/CX/DC /D7
/D3/D6/CT/D7/B8 /CT/D7/D4 /CT
/CX/CP/D0/D0/DD /CX/D2 /D8/CW/CT /D6/CX/CV/CW /D8 /D4/CP/D2/CT/D0/D7/BA /C7/D9/D6 /D7/D8/CP/D2/CS/CP/D6/CS /CS/CT/DA/CX/CP/B9/D8/CX/D3/D2/D7σ /CP/D6/CT /D7/CX/D1/CX/D0/CP/D6 /D8/CW/D6/D3/D9/CV/CW/D3/D9/D8 /CP/D0/D0 /D4/D0/D3/D8/D7/B8 /D8/CW/CTσ /D3/D2 /D8/CW/CT /D6/CX/CV/CW /D8 /D4/CP/D2/CT/D0/D7 /CQ /CT/CX/D2/CV /CQ/CP/D6/CT/D0/DD/D0/CP/D6/CV/CT/D6/BA /CC /CP/CZ/CX/D2/CV 2σ∼0.4 /CX/D1/D4/D0/CX/CT/D7 /D8/CW/CP/D895% /D3/CU /D8/CW/CT /D4/CP/CX/D6/D7 /B4r1, r2) /CS/CX/AR/CT/D6 /CX/D2 /D6/CP/D8/CX/D3/D7 /CQ /DD/CP /CU/CP
/D8/D3/D6 /D0/CT/D7/D7 /D8/CW/CP/D2∼1.3 /B8 /DB/CW/CX
/CW /CX/D7 /CP /D7/CX/CV/D2/CX/AS
/CP/D2 /D8 /D2/CP/D6/D6/D3 /DB/CX/D2/CV /CX/D2
/D3/D1/D4/CP/D6/CX/D7/D3/D2 /D8/D3 /D8/CW/CT/CS/CX/D7/D8/D6/CX/CQ/D9/D8/CX/D3/D2/D7 /D4/D6/D3 /CS/D9
/CT/CS /CQ /DD /D8/CW/CT
/D9/D6/D6/CT/D2 /D8 /BT/AR/DD/D1/CT/D8/D6/CX/DC /CP/D0/CV/D3/D6/CX/D8/CW/D1/BA/CB/D9/D1/D1/CP/D6/DD/CF /CT /D4/D6/CT/D7/CT/D2 /D8/CT/CS /CP/D2 /CX/D1/D4/D6/D3 /DA /CT/CS /CP/D4/D4/D6/D3/CP
/CW /CU/D3/D6
/D3/D1/D4/D9/D8/CX/D2/CV
/D3/D1/D4 /D3/D7/CX/D8/CT /D6/CP/D8/CX/D3 /D7
/D3/D6/CT/D7 /CU/D3/D6/CW/CX/CV/CW/B9/CS/CT/D2/D7/CX/D8 /DD /D3/D0/CX/CV/D3/D2 /D9
/D0/CT/D3/D8/CX/CS/CT /CP/D6/D6/CP /DD/D7/BA /C7/D9/D6 /D2/CT/DB /D1/CT/D8/CW/D3 /CS /CS/CX/AR/CT/D6/D7 /D7/CX/CV/D2/CX/AS
/CP/D2 /D8/D0/DD /CU/D6/D3/D1/D8/CW/CT
/D9/D6/D6/CT/D2 /D8 /BT/AR/DD/D1/CT/D8/D6/CX/DC /CP/D0/CV/D3/D6/CX/D8/CW/D1 /CX/D2 /D8/CW/CT /CU/D3/D0/D0/D3 /DB/CX/D2/CV /D1/CP/D2/D2/CT/D6/BM /B4/CX/B5 /C5/C5
/CT/D0/D0/D7 /CP/D6/CT /D2/D3/D8/CX/D2
/D0/D9/CS/CT/CS /CQ /CT
/CP/D9/D7/CT /D8/CW/CT/CX/D6 /CX/D2/CU/D3/D6/D1/CP/D8/CX/D3/D2
/D3/D2 /D8/CT/D2 /D8 /CX/D7 /D9/D2
/D0/CT/CP/D6/BN /B4/CX/CX/B5 /D6/CP/D8/CX/D3/D7 /CQ /CT/D8 /DB /CT/CT/D2 /D8 /DB /D3/CS/CX/AR/CT/D6/CT/D2 /D8 /D7/CP/D1/D4/D0/CT/D7 /CP/D6/CT /CS/CT/D6/CX/DA /CT/CS /CU/D6/D3/D1
/D3/D1/D4/CP/D6/CX/D2/CV /D8/CW/CT /C8/C5
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page 1 of 5On the beams of wavy metric tensors
A. LOINGER
Dipartimento di Fisica, Università di Milano
Via Celoria, 16 − 20133 Milano, Italy
Summary. − If we move at the same speed of a spatially limited train of an undulating metric
tensor, a famous Serini’s theorem will assure us that this train represents actually a flat
spacetime region.
PACS 04.30 − Gravitational waves and radiation: theory.
1. − The treatise of 1960 by Infeld and Plebanski contains an unsurpassed
application of the Einstein-Infeld-Hoffmann method to the problem of gravitational
radiation [1]. These authors prove that at each stage of approximation it is possible
to choose a reference system for which no emission of gravity waves does happen.
The present writer has recently published a simple, non- perturbative, general
demonstration that no motion of point masses gives origin to emission of
gravitational radiation [2].
I wish now to set forth another argument against the physical existence of
gravitational waves, which utilizes suitably a remarkable Serini’s theorem of 1919,
generalized by Einstein and Pauli in 1943, and by Lichnerowicz in 1946 [3]. A
perspicuous formulation of Serini’s result has been given independently by Fock in
his book on relativity theory [4]. In what follows I shall rest substantially upon
Fock’s ideas.
2. − For our purpose it is convenient to start from a Gaussian (“synchronous”,
according to Landau’s terminology) expression of the spacetime interval [5]:LOINGER A. , ON THE BEAMS OF WAVY METRIC TENSORS
page 2 of 5(2.1) βα
αβ− = xxtxhtcs dd),( d d222, (α, β =1,2,3) ;
let us put
(2.2) tch
∂∂
καβ
αβ=: ;
αβhand αβκare three-dimensional tensors. If jkg, (j,k =0,1,2,3), is the four-
dimensional metric tensor and jkg gdet:= , we have
(2.3) )ln( : gtctchh −
∂∂=
∂∂
=καβ αβ α
α .
If jkRis the Ricci-Einstein tensor, the field equations for a mass tensor jkTwhich
is rigorously equal to zero (with no singularities) can be written as follows:
(2.4) 04 21κκ
∂κ−≡α
c ,
0) (21
; ; 0= κ−κ≡β
αββ
βα αR ,
(2.4'') 0 ) 2 (41
21=Ρ+κκ−κκ+
∂κ∂
≡αβ βγγ
αγ
γαβαβ
αβ
t cR ,
where αβΡis the three-dimensional analogue of jkR, and we have applied tensor
analysis to the three-dimensional space whose metric tensor is αβh.
Now, Serini’s theorem concerns the interesting instances in which jkgis
time-independent and regular. Then, in lieu of (2.4), (2.4'), (2.4'') we have simply
(2.4 bis) 000≡R ,
(2.4'bis) 00≡αR ,
(2.4''bis) 0=Ρ≡αβ αβR ,LOINGER A. , ON THE BEAMS OF WAVY METRIC TENSORS
page 3 of 5A basic result of tensor analysis tells us that if αβΡis equal to zero, also the
three-dimensional curvature tensor αβγδΡ is equal to zero.
Conclusion : If the time-independent jkg, (j,k =0,1,2,3), is regular everywhere and
Minkowskian at the spatial infinity, the spatio-temporal interval sdcoincides with
Minkowski’s sd. Q.e.d.
3. − Let us now consider a rather particular, but significant, example: a finite beam
− i.e. a spatially limited train − of hypothetical gravity waves, which arrives from a
remote spatial region. Remembering that in general relativity there is no limitation
for the values of the velocities of the reference frames, we can suppose to follow
our beam in its motion by travelling at its same speed. But then the beam will be
essentially seen by us as a spatially oscillatory gravity field at rest. Accordingly, its
metric tensor jkgwill be practically time-independent − in our reference system. By
applying Serini’s theorem, we conclude that our finite undulatory train represents
actually a flat Minkowskian spacetime region, i.e. a region with four-dimensional
curvature tensor equal to zero. Our gravity radiation has revealed itself to be a
phantom, i.e. a mere coordinate undulation.
APPENDIX
On a wrong conviction
Many physicists think that without gravitational waves, one would have to
explain an instantaneous propagation of a change in the metric over the whole
universe simply obtained by changing the distribution of mass or stress in a
physical system. This conviction is erroneous. In reality, the physical non-existence
of the gravity waves is fully consistent with the fundamental principles of relativityLOINGER A. , ON THE BEAMS OF WAVY METRIC TENSORS
page 4 of 5theory: the Einstein field equations are time-symmetrical, and therefore the
discarding of time-unsymmetrical solutions is quite legitimate. Analogously,
Maxwell equations are time-symmetrical: the physical existence of electromagnetic
waves is only a theoretically valid possibility, not a theoretical necessity: the real
existence of the e.m. waves is an experimental fact. Any accelerated charge gives
out e.m. waves, whose existence can be ascertained in any reference frame.
In Einstein theory and in Maxwell theory sources and fields are inseparable
from each other, and it is impossible to distinguish cause from effect; but in
Einstein theory the physical existence of the gravity waves is not a theoretically
valid possibility.LOINGER A. , ON THE BEAMS OF WAVY METRIC TENSORS
page 5 of 5REFERENCES
[1] INFELD L. AND PLEBANSKI J., Motion and relativity (Pergamon Press, Oxford, etc.)
1960, chapter VI, in particular pp.200 and 201.
[2] LOINGER A., Nuovo Cimento , 115B (2000) 679, and
http://xxx.lanl.gov/abs/physics/0011041 (November 17th, 2000). See also my papers of
the last years on gravity waves at http://xxx.lanl.gov/abs/astro-ph/ and
http://xxx.lanl.gov/abs/gr-qc/ .
[3] See PAULI W., Teoria della Relatività (Boringhieri, Torino) 1958, sect. 62.
[4] FOCK V., The Theory of Space, Time and Gravitation , Second Revised Edition
(Pergamon Press, Oxford, etc.) 1964, sect. 56, and particularly p.209.
[5] See LANDAU L. ET LIFCHITZ E., Théorie du Champ (Éditions Mir, Moscou) 1966,
sect.99. I adopt here essentially the notations of these authors, my αβhcoincides with
their αβg. |
arXiv:physics/0102012v1 [physics.class-ph] 6 Feb 2001Ball lightning as a possible manifestation of
high-temperature superconductivity in Nature
B.L. Birbrair
Petersburg Nuclear Physics Institute
Gatchina, 188300 St. Petersburg, Russia;
birbrair@thd.pnpi.spb.ru
Abstract
In the superconducting medium the circular current support ed by its own mag-
netic field can exist giving rise to the possible underlying m echanism for the ball
lightning.
The example of such self-supporting object is provided by su perconducting circular
current around the tube of the torus as shown in Fig.1. The cur rent consists of charged
particles moving in a circle of radius r. It is worth mentioning that such a motion is two-
dimensional the necessary condition for the superconducti vity [1] thus being satisfied.
The centrifugal force is balanced by the Lorentz one so
mγv2
r=µe
cvH , γ = (1−β2)−1/2, β=v
c, (1)
where µis the magnetic permeability of the medium, mis the rest mass of the particle, e
is its charge, and γis the Lorentz factor. It must be taken into account because t he ball
lightning is luminous object the particle velocity vthus being close to the light one c(the
radiation is synchrotronic in the case under consideration ). The magnetic field within the
tube of the torus is [2]
H=2I
cR. (2)
It is practically homogeneous when r≪Rthus ensuring the circular motion of the
particles. The current strength Iis
I=Qv
2πr=ev
2πrN , Q =Ne . (3)
QandNbeing the total moving charge and the number of charged parti cles respectively.
1Putting Eqs. (2) and (3) into Eq.(1) we obtain the following c onnection between the
particle number Nand the radius of the torus
N=πmc2γ
µ e2R (4)
and the following expression for the magnetic field
H=mc2γβ
µer. (5)
The energy of the object under consideration is the sum of the magnetic field one
Em= 2π2r2Rµ
8πH2=π(γ−1)(γ+ 1)(mc2)2
4µ e2R , (6)
and the kinetic energy of the moving charges
Ek= (γ−1)mc2N=πγ(γ−1)(mc2)2
µ e2R (7)
the total energy thus being
E=Em+Ek=π(γ−1)(5γ+ 1)(mc2)2
4µ e2R . (8)
To get the expression (6) we used the relation γ2β2=γ2−1, see Eq.(1).
The Lorentz factor γcan be determined from the observed angular frequency of the
synchrotron radiation [2]
ω=2πc
λ=eH
mcγ2=βγ3c
µr=γ2(γ2−1)1/2c
µ r. (9)
In this way we get
γ2(γ2−1)1/2=2πr
λµ , (10)
where λis the wavelength of the radiation. The intensity of the radi ation is [2]
S=2ce4(γ2−1)
3(mc2)2H2=2ce2(γ2−1)2
3µ2r2. (11)
The calculations are performed assuming the charged partic les to be electrons and
putting µ= 1. The average observed diameter of the ball lightning (her eafter BL’s) is
24 cm [3, 4], but twice as large diameters are also observed ra ther often [4, 7]. For this
reason the results are obtained for both the R= 12 cm and R= 24 cm values of the
2torus radius. The tube radii are quite arbitrarily chosen as r= (1,2,3)cm. It is worth
mentioning in this connection that the sphere is not the only observed form of the BL’s:
many different forms including the torus are also observed [5 ]. The intervals of the Lorentz
factor values are determined for the visible light region ru nning from λ= 7·10−5cm (red)
toλ= 3.8·10−5cm (violet) because the observed BL colours cover all this re gion [3].
The results are shown in Table 1. Two features are important.
Table 1: Intervals for the Lorentz factors, energies, and ra diation. The left and right
bounds of each interval refer to the red and violet lights res pectively.
E ·103J
r, cm γ R= 12 cm R= 24 cm S·10−9W
144.8÷54.929÷43 58÷86 1.85÷4.18
256.4÷69.246÷69 92÷138 1.16÷2.64
364.6÷79.260÷91 120÷182 0.89÷2.01
a. The energies are rather large, being practically the same as the average value of
100 kJ for the outdoors observations of BL’s [6].
b. At the same time the intensity of the radiation is rather sm all. Both these features
are characteristic for the exploding BL’s [5].
In this way we showed that the object with the similar propert ies to those of the
exploding BL’s can exist in the superconducting medium (the superconductivity must be
high-temperature since there is no reasons to assume the tem peratures of BL’s to be low).
We do not know whether such a medium arises in the atmospheric processes leading to
the BL’s (this problem is out of the scope of the present work) , but the above results
suggest that the exploding BL’s may be a possible evidence of this phenomenon.
The author is greatly indebted to Professor O.I. Sumbaev for the permanent attention
to this work and Drs. V.L. Alexeev and A.I. Egorov for stimula ting discussions.
References
[1] L.N. Cooper, Phys.Rev. 104, 1189 (1956).
[2] B.M. Javorsky and A.A. Detlaf, ”Reference book on physics” , Nauka, M., 1968.
[3] H. Ofuruton et al., Proceedings of 5th Intern. Symp. on Ba ll Lightning (ISBL’97)
p.17, Tsugawa-Town Niigata, Japan (1997).
3[4] A. Amirov and V. Bychkov, ISBL’97, 42 (1997).
[5] A. Amirov et al., ISBL’97, 52 (1997).
[6] S.I. Stepanov, ISBL’97, 61 (1997).
[7] B.Smirnov, ISBL’97, 235 (1997).
4Figure 1: The torus of radius Rand the tube radius r. The superconducting surface
current is shown by arrows.
5 |
arXiv:physics/0102013v1 [physics.gen-ph] 6 Feb 2001
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/D3/D2/D7/CX/CS/CT/D6/CT/CS /CX/D2S /CP/D2/CSS′/BA /CB/D9
/CW /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2/D7 /CP/D6/CT /D8/CW/CT /CC/CC /CP/D2/CS /CP/D2 /CT/DC/CP/D1/D4/D0/CT /D3/CU/D8/CW/CT/D1 /CP/D6/CT/B8 /CP/D7 /CP/D0/D6/CT/CP/CS/DD /D7/CP/CX/CS/B8 /D8/CW/CT /C4 /CC /CQ /CT/D8 /DB /CT/CT/D2 /C1/BY/CA/D7/BA /CC/CW/CT /D1/CT/CP/D2/CX/D2/CV /D3/CU /D8/CW/CT /D7/CP/D1/CT /D5/D9/CP/D2 /D8/CX/D8 /DD /CA/D3/CW/D6/D0/CX
/CW /CJ/BG ℄/CT/DC/D4/D6/CT/D7/D7/CT/D7 /CX/D2 /D8/CW/CT /CU/D3/D0/D0/D3 /DB/CX/D2/CV /DB /CP /DD/BM /AH/BT /D5/D9/CP/D2 /D8/CX/D8 /DD /CX/D7 /D8/CW/CT/D6/CT/CU/D3/D6/CT /D4/CW /DD/D7/CX
/CP/D0/D0/DD /D1/CT/CP/D2/CX/D2/CV/CU/D9/D0 /B4/CX/D2 /D8/CW/CT /D7/CT/D2/D7/CT /D8/CW/CP/D8/CX/D8 /CX/D7 /D3/CU /D8/CW/CT /D7/CP/D1/CT /D2/CP/D8/D9/D6/CT /D8/D3 /CP/D0/D0 /D3/CQ/D7/CT/D6/DA /CT/D6/D7/B5 /CX/CU /CX/D8 /CW/CP/D7 /D8/CT/D2/D7/D3/D6/CX/CP/D0 /D4/D6/D3/D4 /CT/D6/D8/CX/CT/D7 /D9/D2/CS/CT/D6 /C4/D3/D6/CT/D2 /D8/DE /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/B9/D8/CX/D3/D2/D7/BA/AH /CB/CX/D1/CX/D0/CP/D6/D0/DD /BZ/CP/D1 /CQ/CP /CJ/BH ℄/B8 /DB/CW/CT/D2 /CS/CX/D7
/D9/D7/D7/CX/D2/CV /D8/CW/CT /D7/CP/D1/CT/D2/CT/D7/D7 /D3/CU /CP /D4/CW /DD/D7/CX
/CP/D0 /D5/D9/CP/D2 /D8/CX/D8 /DD /B4/CU/D3/D6 /CT/DC/CP/D1/D4/D0/CT/B8 /D8/CW/CT/D5/D9/CP/D2 /D8/CX/D8 /DDAµ(xλ, Xλ) /B8 /DB/CW/CX
/CW /CX/D7 /CP /CU/D9/D2
/D8/CX/D3/D2 /D3/CU /D8 /DB /D3 /D4 /D3/CX/D2 /D8/D7xλ
/CP/D2/CSXλ
/B5 /CU/D3/D6 /CS/CX/AR/CT/D6/CT/D2 /D8 /C1/BY/CA/D7 S /CP/D2/CSS′/B8/CS/CT
/D0/CP/D6/CT/D7/BM /AH/CC/CW/CT /D5/D9/CP/D2 /D8/CX/D8 /DDAµ(xλ, Xλ) /CU/D3/D6S /CX/D7 /D8/CW/CT /D7/CP/D1/CT /CP/D7 /D8/CW/CT /D5/D9/CP/D2 /D8/CX/D8 /DDA′
µ(x′
λ, X′
λ) /CU/D3/D6S′/DB/CW/CT/D2 /CP/D0/D0/D8/CW/CT /D4/D6/CX/D1/CT/CS /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7 /CP/D6/CT /D3/CQ/D8/CP/CX/D2/CT/CS /CU/D6/D3/D1 /D8/CW/CT
/D3/D6/D6/CT/D7/D4 /D3/D2/CS/CX/D2/CV /D9/D2/D4/D6/CX/D1/CT/CS /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7 /D8/CW/D6/D3/D9/CV/CW /C4/D3/D6/CT/D2 /D8/DE/D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2/D7 /B4/D8/CT/D2/D7/D3/D6
/CP/D0
/D9/D0/D9/D7/B5/BA/AH/C1/D8 /CW/CP/D7 /D8/D3 /CQ /CT /D2/D3/D8/CT/CS /D8/CW/CP/D8/B8 /CX/D2 /D4/D6/CX/D2
/CX/D4/D0/CT/B8 /D3/D2/CT
/CP/D2
/CW/D3 /D3/D7/CT /CP/D2 /DD /D6/CT/CU/CT/D6/CT/D2
/CT /CU/D6/CP/D1/CT /CX/D2 /DB/CW/CX
/CW /D7/D3/D1/CT /D4/CW /DD/D7/CX
/CP/D0/D5/D9/CP/D2 /D8/CX/D8 /DD /CX/D7
/D3 /DA /CP/D6/CX/CP/D2 /D8/D0/DD /CS/CT/AS/D2/CT/CS/B8 /CP/D2/CS /D8/CW/CT/D2 /D8/CW/CT /D7/CP/D1/CT /D5/D9/CP/D2 /D8/CX/D8 /DD /CX/D7
/D3/D2/D7/CX/CS/CT/D6/CT/CS /CX/D2 /CP/D0/D0 /D3/D8/CW/CT/D6 /D6/CT/CU/CT/D6/CT/D2
/CT/CU/D6/CP/D1/CT/D7/B8 /D8/D6/CP/D2/D7/CU/D3/D6/D1/CX/D2/CV /CP/D0/D0 /D4/CP/D6/D8/D7 /D3/CU /D8/CW/CP/D8 /D5/D9/CP/D2 /D8/CX/D8 /DD /CQ /DD /D8/CW/CT /CC/CC/BA /CC/CW/CT /DB/CW/D3/D0/CT /D4/CW /DD/D7/CX
/D7 /DB/CX/D0/D0 /D2/D3/D8 /CS/CT/D4 /CT/D2/CS /D3/D2/D8/CW/CT
/CW/D3/D7/CT/D2 /CU/D6/CP/D1/CT/BA /C0/D3 /DB /CT/DA /CT/D6/B8 /D8/CW/CT /D1/D3/D7/D8
/D3/D2 /DA /CT/D2/CX/CT/D2 /D8
/CW/D3/CX
/CT /CU/D3/D6 /D7/DD/D7/D8/CT/D1/D7 /DB/CX/D8/CW /D6/CT/D7/D8 /D1/CP/D7/D7 /CX/D7 /D8/CW/CT /D6/CT/D7/D8 /CU/D6/CP/D1/CT/B8/D7/CX/D2
/CT /CX/D2 /D8/CW/CP/D8 /CU/D6/CP/D1/CT /D3/D2/CT /D6/CT/D8/CP/CX/D2/D7 /D8/CW/CT /D7/CX/D1/CX/D0/CP/D6/CX/D8 /DD /DB/CX/D8/CW /D8/CW/CT /D4/D6/CT/D6/CT/D0/CP/D8/CX/DA/CX/D7/D8/CX
/D4/CW /DD/D7/CX
/D7/BA/BE/BE/BA/BD /CC /D6/D9/CT /CP/D2/CS /BT/D4/D4/CP/D6/CT/D2 /D8 /CC /D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2/D7 /D3/CU /C4/CT/D2/CV/D8/CW /B4/CE /D3/D0/D9/D1/CT/B5/CC/CW/CT /CS/CT/AS/D2/CX/D8/CX/D3/D2 /D3/CU /D0/CT/D2/CV/D8/CW /B4/DA /D3/D0/D9/D1/CT/B5 /DB/CW/CX
/CW /CX/D7 /CX/D2 /CP
/D3/D6/CS/CP/D2
/CT /DB/CX/D8/CW /CP/D0/D0 /D8/CW/CT /CP/CQ /D3 /DA /CT
/D3/D2/D7/CX/CS/CT/D6/CP/D8/CX/D3/D2 /CX/D7 /B9 /D8/CW/CT
/D3 /DA /CP/D6/CX/CP/D2 /D8 /CS/CT/AS/D2/CX/D8/CX/D3/D2 /D3/CU /D0/CT/D2/CV/D8/CW /B4/D7/CT/CT/B8 /CT/BA/CV/BA/B8 /CA/CT/CU/BA /BJ /CP/D2/CS /D6/CT/CU/CT/D6/CT/D2
/CT/D7 /D8/CW/CT/D6/CT/CX/D2/B8 /D3/D6 /CU/D3/D6 /D8/CW/CT /DA /D3/D0/D9/D1/CT /CJ/BG/B8 /BH ℄/B5/BA/CC/CW/CT /CX/D2 /DA /CP/D6/CX/CP/D2 /D8 /D7/D4/CP
/CT/D8/CX/D1/CT /D0/CT/D2/CV/D8/CW /B4/D8/CW/CT /C4/D3/D6/CT/D2 /D8/DE /D7
/CP/D0/CP/D6/B5 /CX/D7 /CU/D3/D6/D1/CT/CS /CP/D7l= (lµlµ)1/2/DB/CW/CT/D6/CT lµ
/CX/D7 /D8/CW/CT/CS/CX/D7/D8/CP/D2
/CT /BG/B9/DA /CT
/D8/D3/D6 /CQ /CT/D8 /DB /CT/CT/D2 /D8 /DB /D3 /D7/D4/CP/D8/CX/CP/D0 /D4 /D3/CX/D2 /D8/D7 /BT /CP/D2/CS /BU /D3/D2 /D8/CW/CT /B4/D1/D3 /DA/CX/D2/CV/B5 /D3/CQ /CY/CT
/D8/B8 lµ=xµ
B−xµ
A,
xµ
A,B
/CP/D6/CT /D8/CW/CT /D4 /D3/D7/CX/D8/CX/D3/D2 /BG/B9/DA /CT
/D8/D3/D6/D7 /CX/D2 /D7/D3/D1/CT /C1/BY/CAS /BA /B4/CF /CT /D9/D7/CT /BV/CP/D6/D8/CT/D7/CX/CP/D2 /D7/D4/CP
/CT
/D3 /D3/D6/CS/CX/D2/CP/D8/CT/D7 xi/CP/D2/CS /D8/CX/D1/CT
t(x0≡ct) /BA /CF/CX/D8/CW/D3/D9/D8 /D0/D3/D7/D7 /D3/CU /CV/CT/D2/CT/D6/CP/D0/CX/D8 /DD /DB /CT /DB /D3/D6/CZ /CX/D2 /CP/D2 /C1/BY/CA /DB/CX/D8/CW /D8/CW/CT /C5/CX/D2/CZ /D3 /DB/D7/CZ/CX /D1/CT/D8/D6/CX
/D8/CT/D2/D7/D3/D6
gµν
/BPdiag(−1,1,1,1) /BA/B5 /CC/CW/CT /D3/CQ/D7/CT/D6/DA /CT/D6/D7 /CX/D2 /CP/D0/D0 /D3/D8/CW/CT/D6 /C1/BY/CA/D7 /DB/CX/D0/D0 /D0/D3 /D3/CZ /CP/D8 /D8/CW/CT /D7/CP/D1/CT /CT/DA /CT/D2 /D8/D7 /CQ/D9/D8 /CP/D7/D7/D3
/CX/CP/D8/CX/D2/CV/DB/CX/D8/CW /D8/CW/CT/D1 /CS/CX/AR/CT/D6/CT/D2 /D8
/D3 /D3/D6/CS/CX/D2/CP/D8/CT/D7/BN /CX/D8 /CX/D7 /D8/CW/CT /CT/D7/D7/CT/D2
/CT /D3/CU /D8/CW/CT
/D3 /DA /CP/D6/CX/CP/D2 /D8 /CS/CT/D7
/D6/CX/D4/D8/CX/D3/D2/BA /CC/CW/CT /C4 /CC /D3/CU /D8/CW/CT/D4 /D3/D7/CX/D8/CX/D3/D2 /BG/B9/DA /CT
/D8/D3/D6/D7/B8 /D3/CU /D8/CW/CT /CS/CX/D7/D8/CP/D2
/CT /BG/B9/DA /CT
/D8/D3/D6/B8 /CP/D2/CS /D3/CU /D8/CW/CT /CX/D2 /DA /CP/D6/CX/CP/D2 /D8 /D7/D4/CP
/CT/D8/CX/D1/CT /D0/CT/D2/CV/D8/CW /CP/D6/CT /CP/D0/D0 /D8/CW/CT /CC/CC/BA/C1/D2
/D3/D2 /D8/D6/CP/D7/D8 /D8/D3 /D8/CW/CT
/D3 /DA /CP/D6/CX/CP/D2 /D8 /CS/CT/AS/D2/CX/D8/CX/D3/D2 /D3/CU /D0/CT/D2/CV/D8/CW /CP/D2/CS /D8/CW/CT /CC/CC /D3/CU /D8/CW/CT /D7/D4/CP
/CT/D8/CX/D1/CT /D8/CT/D2/D7/D3/D6/D7 /D8/CW/CT /D7/DD/D2/B9
/CW/D6 /D3/D2/D3/D9/D7 /CS/CT/AS/D2/CX/D8/CX/D3/D2 /D3/CU /D0/CT/D2/CV/D8/CW /B8 /CX/D2 /D8/D6/D3 /CS/D9
/CT/CS /CQ /DD /BX/CX/D2/D7/D8/CT/CX/D2 /CJ/BK℄/B8 /CS/CT/AS/D2/CT/D7 /D0/CT/D2/CV/D8/CW /CP/D7 /D8/CW/CT /D7/D4 /CP/D8/CX/CP/D0 /CS/CX/D7/D8/CP/D2
/CT/CQ /CT/D8/DB/CT /CT/D2 /D8/DB/D3 /D4 /D3/CX/D2/D8/D7 /D3/D2 /D8/CW/CT /B4/D1/D3/DA/CX/D2/CV/B5 /D3/CQ/CY/CT
/D8 /D1/CT /CP/D7/D9/D6 /CT /CS /CQ/DD /D7/CX/D1/D9/D0/D8/CP/D2/CT/CX/D8/DD /CX/D2 /D8/CW/CT /D6 /CT/D7/D8 /CU/D6 /CP/D1/CT /D3/CU /D8/CW/CT /D3/CQ/B9/D7/CT/D6/DA/CT/D6 /BA /C4/CT/D8l0
/CQ /CT /D8/CW/CT /D6/CT/D7/D8 /D0/CT/D2/CV/D8/CW/B8 /D3/D6 /D8/CW/CT /D8/CW/D6/CT/CT/B9/CS/CX/D1/CT/D2/D7/CX/D3/D2/CP/D0 /DA /D3/D0/D9/D1/CT dV0
/B8 /D1/CT/CP/D7/D9/D6/CT/CS /CQ /DD /D7/CX/D1 /D9/D0/D8/CP/D2/CT/CX/D8 /DD/CX/D2S0
/B8 /D8/CW/CT /D6/CT/D7/D8 /CU/D6/CP/D1/CT /D3/CU /D8/CW/CT /D3/CQ /CY/CT
/D8/B8 /CP/D8 /D7/D3/D1/CT t0=a /BA /CC/CW/CT/D2 /D8/CW/CT /D0/CT/D2/CV/D8/CW l′/B4/D8/CW/CT /BF/BW /DA /D3/D0/D9/D1/CT dV′/B5 /CS/CT/B9/D8/CT/D6/D1/CX/D2/CT/CS /D7/CX/D1 /D9/D0/D8/CP/D2/CT/D3/D9/D7/D0/DD /CX/D2 /D7/D3/D1/CT /C1/BY/CAS′/B8 /CP/D8 /D7/D3/D1/CT t′=b /B8 /CX/D7 /C4/D3/D6/CT/D2 /D8/DE
/D3/D2 /D8/D6/CP
/D8/CT/CS/B8 l′=l0(1−β2)1/2/B8/B4dV′=dV00(1−β2)1/2/B5/B8 /B4β=V/c /B8V /CX/D7 /D8/CW/CT /D6/CT/D0/CP/D8/CX/DA /CT /DA /CT/D0/D3
/CX/D8 /DD /D3/CUS0
/CP/D2/CSS′/BA /C1/D8 /CW/CP/D7 /D8/D3 /CQ /CT /CT/D1/D4/CW/CP/D7/CX/DE/CT/CS/D8/CW/CP/D8t0=a /CX/D2S0
/CP/D2/CSt′=b /CX/D2S′/CP/D6/CT /D2/D3/D8 /D6/CT/D0/CP/D8/CT/CS /CQ /DD /D8/CW/CT /C4 /CC /D3/D6 /CP/D2 /DD /D3/D8/CW/CT/D6
/D3 /D3/D6/CS/CX/D2/CP/D8/CT /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2/BA/CC/CW/CT /D8/CX/D1/CT
/D3/D1/D4 /D3/D2/CT/D2 /D8 /CX/D7 /D2/D3/D8 /D8/D6/CP/D2/D7/CU/D3/D6/D1/CT/CS /CX/D2 /D8/CW/CT /C4/D3/D6/CT/D2 /D8/DE
/D3/D2 /D8/D6/CP
/D8/CX/D3/D2/BA /CC/CW/CX/D7 /D1/CT/CP/D2/D7 /D8/CW/CP/D8 /D8/CW/CT /D3/CQ/D7/CT/D6/DA /CT/D6/D7/CX/D2S0
/CP/D2/CSS′/CS/D3 /D2/D3/D8 /D0/D3 /D3/CZ /CP/D8 /D8/CW/CT /D7/CP/D1/CT /CT/DA /CT/D2 /D8/D7/BA /C0/CT/D2
/CT /D8/CW/CT /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7 l0
/CP/D2/CSl′/D7/DD/D2
/CW/D6/D3/D2/D3/D9/D7/D0/DD /CS/CT/D8/CT/D6/B9/D1/CX/D2/CT/CS /CX/D2 /D8/CW/CT /D6/CT/D7/D8 /CU/D6/CP/D1/CT /D3/CU /D8/CW/CT /D3/CQ/D7/CT/D6/DA /CT/D6 /D6 /CT/CU/CT/D6 /D8/D3 /D8/CW/CT /D7/CP/D1/CT /D1/CT /CP/D7/D9/D6 /CT/D1/CT/D2/D8 /CX/D2S0
/CP/D2/CSS′/CP/D2/CS /D2/D3/D8 /D8/D3/D8/CW/CT /D7/CP/D1/CT /BG/BW /D8/CT/D2/D7/D3/D6 /D5/D9/CP/D2 /D8/CX/D8 /DD /BA /CC/CW/CT/D6 /CT/CU/D3/D6 /CT /D8/CW/CT /C4 /D3/D6 /CT/D2/D8/DE
/D3/D2/D8/D6 /CP
/D8/CX/D3/D2 /CP/D7 /D8/CW/CT /D8/D6 /CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2
/D3/D2/D2/CT
/D8/CX/D2/CV/D8/CW/CT/D1 /CX/D7 /B9 /D8/CW/CT /BT /CC/BA/BE/BA/BE /CE /D3/D2 /C4/CP/D9/CT/B3/D7 /D8/CW/CT/D3/D6/CT/D1/CC/CW/CT /CT/DC/CX/D7/D8/CT/D2
/CT /D3/CU /D8/CW/CT /CU/D9/D2/CS/CP/D1/CT/D2 /D8/CP/D0 /CS/CX/AR/CT/D6/CT/D2
/CT /CQ /CT/D8 /DB /CT/CT/D2 /D8/CW/CT /CC/CC /CP/D2/CS /D8/CW/CT /BT /CC /CT/D2/CP/CQ/D0/CT/D7 /D9/D7 /D8/D3 /CT/DC/CP/D1/CX/D2/CT/CP/D2 /CX/D1/D4 /D3/D6/D8/CP/D2 /D8 /D8/CW/CT/D3/D6/CT/D1 /B4/CX/D8 /CX/D7 /D7/D3/D1/CT/D8/CX/D1/CT/D7
/CP/D0/D0/CT/CS /AH/DA /D3/D2 /C4/CP/D9/CT/B3/D7 /D8/CW/CT/D3/D6/CT/D1/AH/B5/B8 /DB/CW/CX
/CW /CX/D7 /DB/CX/CS/CT/D0/DD /D9/D7/CT/CS /CX/D2/D8/CW/CT /D8/CW/CT/D3/D6/DD /D3/CU /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD /CP/D2/CS /CX/D2 /D8/CW/CT /D5/D9/CP/D2 /D8/D9/D1 /D8/CW/CT/D3/D6/DD /D3/CU /AS/CT/D0/CS/D7/BA /CC/CW/CX/D7 /D8/CW/CT/D3/D6/CT/D1 /D6/D3/D9/CV/CW/D0/DD /D7/D8/CP/D8/CT/D7/BM /CC/CW/CT/D2/CT
/CT/D7/D7/CP/D6/DD /CP/D2/CS /D7/D9Ꜷ
/CX/CT/D2 /D8
/D3/D2/CS/CX/D8/CX/D3/D2 /CU/D3/D6 /D8/CW/CT /CW /DD/D4 /CT/D6/D4/D0/CP/D2/CT /CX/D2 /D8/CT/CV/D6/CP/D0 /D3/CU /CP /D7/DD/D1/D1/CT/D8/D6/CX
/D8/CT/D2/D7/D3/D6 /D3/CU /D7/CT
/D3/D2/CS /D6/CP/D2/CZ
T /B4/integraltext
ΣTµν(x)d3σν(x) /B5 /D8/D3 /CQ /CT /CX/D2/CS/CT/D4 /CT/D2/CS/CT/D2 /D8 /D3/CU /D8/CW/CT /D3/D6/CX/CT/D2 /D8/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CP/D8 /CW /DD/D4 /CT/D6/D4/D0/CP/D2/CT /B4Σ /B5 /CX/D7 /D8/CW/CP/D8T /CQ /CT/CS/CX/DA /CT/D6/CV/CT/D2
/CT/B9/CU/D6/CT/CT/B8 ∂νTµν= 0 /B8 /B4/D0/D3
/CP/D0
/D3/D2/D7/CT/D6/DA /CP/D8/CX/D3/D2 /D0/CP /DB/B5/BN /CP/D2/CS /CX/CU /D8/CW/CP/D8 /CX/D2 /D8/CT/CV/D6/CP/D0 /CX/D7 /D3/D6/CX/CT/D2 /D8/CP/D8/CX/D3/D2/B9/CX/D2/CS/CT/D4 /CT/D2/CS/CT/D2 /D8/B8/D8/CW/CT/D2 /CX/D8
/CP/D2 /CQ /CT /DB/D6/CX/D8/D8/CT/D2 /CP/D7/integraltext
t=aTµ0(r, t)d3x /CP/D2/CS /CX/D8 /CX/D7 /BD/B5 /CX/D2/CS/CT/D4 /CT/D2/CS/CT/D2 /D8 /D3/CU /D8/CX/D1/CT/B8 /B4/integraltext
t=aTµ0(r, t)d3x=/integraltext
t=bTµ0(r, t)d3x /B5/B8 /CP/D2/CS /BE/B5 /CX/D8 /CX/D7 /CP /BG/B9/DA /CT
/D8/D3/D6/B8 /B4/D8/CW/CT /D4/D6/D3 /D3/CU /D3/CU /D8/CW/CP/D8 /D8/CW/CT/D3/D6/CT/D1
/CP/D2 /CQ /CT /CU/D3/D9/D2/CS /CX/D2/B8 /CT/BA/CV/BA/B8 /CA/CT/CU /BA/BL/B8/CB/CT
/BA /BH/BA/BK/B8 /CP/D2/CS /CA/CT/CU/BA /BD/BC/B5/BA /CF /CT /CP/D6/CT /D2/D3/D8 /CX/D2 /D8/CT/D6/CT/D7/D8/CX/D2/CV /CX/D2 /D8/CW/CT /D8/CX/D1/CT /CX/D2/CS/CT/D4 /CT/D2/CS/CT/D2
/CT /D3/CU /D8/CW/CP/D8 /CX/D2 /D8/CT/CV/D6/CP/D0 /CQ/D9/D8 /CX/D2 /CX/D8/D7/BG/B9/DA /CT
/D8/D3/D6
/CW/CP/D6/CP
/D8/CT/D6/BA /C1/D2 /D8/CW/CT /D4/D6/D3 /D3/CU /D3/CU /BE/B5 /D8/CW/CT /CX/D2 /D8/CT/CV/D6/CP/D0/D7 /CP/D6/CT /D8/CP/CZ /CT/D2 /D3 /DA /CT/D6 /D8/CW/CT /CW /DD/D4 /CT/D6/D7/D9/D6/CU/CP
/CT/D7 t=a /CX/D2S /CP/D2/CS
t′=b /CX/D2S′/BA /BT
/D3/D6/CS/CX/D2/CV /D8/D3 /D8/CW/CT /D4/D6/CT/DA/CX/D3/D9/D7 /CS/CX/D7
/D9/D7/D7/CX/D3/D2 /CP/CQ /D3/D9/D8 /D8/CW/CT /D7/CP/D1/CT/D2/CT/D7/D7 /D3/CU /CP /D4/CW /DD/D7/CX
/CP/D0 /D5/D9/CP/D2 /D8/CX/D8 /DD /CU/D3/D6/CS/CX/AR/CT/D6/CT/D2 /D8 /C1/BY/CA/D7/B8 /CP/D2/CS /D8/CW/CT /D6/CT/D0/CP/D8/CT/CS
/D3/D2/D7/CX/CS/CT/D6/CP/D8/CX/D3/D2 /CP/CQ /D3/D9/D8 /D8/CW/CT /CC/CC /CP/D2/CS /D8/CW/CT /BT /CC/B8 /DB /CT
/D3/D2
/D0/D9/CS/CT/BM /CC/CW/CT /CX/D2/D8/CT /CV/D6 /CP/D0/integraltext
t=aTµ0(r, t)d3x
/CP/D2/D2/D3/D8 /CQ /CT /CP /D8/D6/D9/CT /BG/B9/DA/CT
/D8/D3/D6 /CU/D6 /D3/D1 /D8/CW/CT /AH/CC/CC /DA/CX/CT/DB/D4 /D3/CX/D2/D8/BA/AH /C6/CP/D1/CT/D0/DD /B8 /D8/CW/CX/D7 /CX/D2/D8/CT /CV/D6 /CP/D0 /CS/D3 /CT/D7 /D2/D3/D8/D6 /CT/CU/CT/D6 /D8/D3 /D8/CW/CT /D7/CP/D1/CT /D5/D9/CP/D2/D8/CX/D8/DD
/D3/D2/D7/CX/CS/CT/D6 /CT /CS /CU/D6 /D3/D1 /CS/CX/AR/CT/D6 /CT/D2/D8 /C1/BY/CA/D7/B8 /B4/DB/CW/CX
/CW /CP/D6 /CT
/D3/D2/D2/CT
/D8/CT /CS /CQ/DD /D8/CW/CT /C4 /CC/B5/B8 /D7/CX/D2
/CT/D8/CW/CT /CW/DD/D4 /CT/D6/D4/D0/CP/D2/CT/D7 /D8/CP/CZ/CT/D2 /CP/D8t=a /CX/D2S /CP/D2/CSt′=b /CX/D2S′/CP/D6 /CT /D2/D3/D8 /D6 /CT/D0/CP/D8/CT /CS /CX/D2 /CP/D2/DD /DB/CP/DD/BA /CC/CW /D9/D7/B8 /CT/BA/CV/BA/B8 /D8/CW/CT/D4/D6/CX/D1/CT/CS /CT/D0/CT/D1/CT/D2 /D8/D7 /D3/CU /DA /D3/D0/D9/D1/CT d3x′/CX/D2 /D8/CW/CT /CX/D2 /D8/CT/CV/D6/CP/D0 /CX/D2S′/B4/integraltext
t′=bT′µ0(r′, t′)d3x′/B5 /CP/D6/CT /D2/D3/D8 /D3/CQ/D8/CP/CX/D2/CT/CS /CQ /DD /D8/CW/CT/C4 /CC /CU/D6/D3/D1S /B8 /D8/CW/CP/D2 /D8/CW/CT/DD /CP/D6/CT /D7/CX/D1/D4/D0/DD /D8/CW/CT /CT/D0/CT/D1/CT/D2 /D8/D7 /D3/CU /DA /D3/D0/D9/D1/CT /D3/CU /CP/D2 /CP/D6/CQ/CX/D8/D6 /CP/D6/DD
/CW/D3/D7/CT/D2 /CW/DD/D4 /CT/D6/D7/D9/D6/CU/CP
/CTt′=b/CX/D2S′/BA /CC/CW/CT /D8/D6 /CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2
/D3/D2/D2/CT
/D8/CX/D2/CV /D8/CW/CT /CX/D2/D8/CT /CV/D6 /CP/D0/D7 /CX/D2S /CP/D2/CSS′/CX/D7 /B9 /CP/D2 /BT /CC/B8 /D7/CX/D2
/CT /D2/D3/D8 /CP/D0 /D0 /D4 /CP/D6/D8/D7 /D3/CU/D8/CW/CP/D8
/D3/D1/D4 /D3/D9/D2/CS /D4/CW/DD/D7/CX
/CP/D0 /D5/D9/CP/D2/D8/CX/D8/DD /CP/D6 /CT /D8/D6 /CP/D2/D7/CU/D3/D6/D1/CT /CS /CQ/DD /D8/CW/CT /C4 /CC /CU/D6 /D3/D1S /D8/D3S′/BA /CC/CW/CT /D7/CX/D8/D9/CP/D8/CX/D3/D2 /DB/CX/D8/CW /D8/CW/CT/D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CP/D8 /CX/D2 /D8/CT/CV/D6/CP/D0 /CX/D7 /D5/D9/CX/D8/CT /D7/CX/D1/CX/D0/CP/D6 /D8/D3 /D8/CW/CT /CP/D0/D6/CT/CP/CS/DD /CS/CX/D7
/D9/D7/D7/CT/CS /C4/D3/D6/CT/D2 /D8/DE
/D3/D2 /D8/D6/CP
/D8/CX/D3/D2 /CP/D7 /CP/D2/BT /CC/BA /CF /CT /D7/CT/CT /D8/CW/CP/D8/B8
/D3/D2 /D8/D6/CP/D6/DD /D8/D3 /DA /D3/D2 /C4/CP/D9/CT/B3/D7 /D8/CW/CT/D3/D6/CT/D1/B8 /D8/CW/CT /DA /CP/D2/CX/D7/CW/CX/D2/CV /D3/CU /D8/CW/CT /CS/CX/DA /CT/D6/CV/CT/D2
/CT /D3/CUT /CS/D3 /CT/D7 /D2/D3/D8/CP/D7/D7/D9/D6/CT /D8/CW/CP/D8 /D8/CW/CT /D1/CT/D2 /D8/CX/D3/D2/CT/CS /CX/D2 /D8/CT/CV/D6/CP/D0 /CX/D7 /CP /D8/D6/D9/CT /BG/B9/DA /CT
/D8/D3/D6/BA /CC/CW/CT
/D3/D2/D7/CX/CS/CT/D6/CP/D8/CX/D3/D2/D7 /CP/D2/CS /D8/CW/CT /D4/D6/D3 /D3/CU/D7 /CV/CX/DA /CT/D2 /CX/D2/D8/CW/CT /D4/D6/CT/DA/CX/D3/D9/D7 /D0/CX/D8/CT/D6/CP/D8/D9/D6/CT /D2/CT/CV/D0/CT
/D8/CT/CS /D8/CW/CP/D8 /D8/CW/CT /AH/CC/CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH /CS/CT/D1/CP/D2/CS/D7 /D8/CW/CT /D7/CP/D1/CT /D4/CW /DD/D7/CX
/CP/D0 /D5/D9/CP/D2 /D8/CX/D8 /DD /D8/D3/CQ /CT
/D3/D2/D7/CX/CS/CT/D6/CT/CS /CU/D6/D3/D1 /CS/CX/AR/CT/D6/CT/D2 /D8 /C1/BY/CA/D7/B8 /CX/BA/CT/BA/B8 /D8/CW/CP/D8 /CP/D0/D0 /D4/CP/D6/D8/D7 /D3/CU /D8/CW/CP/D8 /D5/D9/CP/D2 /D8/CX/D8 /DD /CW/CP /DA /CT /D8/D3 /CQ /CT /D8/D6/CP/D2/D7/CU/D3/D6/D1/CT/CS /CQ /DD/D8/CW/CT /C4 /CC /CU/D6/D3/D1S /D8/D3S′/BA/BF/BE/BA/BF /CC/CW/CT /C8/D6/D3 /D3/CU /D8/CW/CP/D8 /D8/CW/CT /CC /D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2/D7 /D3/CUE /CP/D2/CSB /CP/D6/CT /B9 /D8/CW/CT /BT /CC/BT/D7 /CP /D2/CT/DC/D8 /CT/DC/CP/D1/D4/D0/CT /DB /CT
/D3/D2/D7/CX/CS/CT/D6 /D8/CW/CT /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2/D7 /D3/CUE /CP/D2/CSB /BA /C1/D8 /CX/D7 /CV/CT/D2/CT/D6/CP/D0/D0/DD /CQ /CT/D0/CX/CT/DA /CT/CS /D8/CW/CP/D8/D8/CW/CT
/D3 /DA /CP/D6/CX/CP/D2 /D8 /CU/D3/D6/D1 /D9/D0/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /CT/D0/CT
/D8/D6/D3 /CS/DD/D2/CP/D1/CX
/D7 /DB/CX/D8/CWFαβ/CP/D2/CS /D8/CW/CT /D9/D7/D9/CP/D0 /CU/D3/D6/D1 /D9/D0/CP/D8/CX/D3/D2 /DB/CX/D8/CWE /CP/D2/CS
B /CP/D6/CT /CT/D5/D9/CX/DA /CP/D0/CT/D2 /D8/B8 /CP/D2/CS /D8/CW/CT/D6/CT/CU/D3/D6/CT /D8/CW/CP/D8 /D8/CW/CT /D9/D7/D9/CP/D0 /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2/D7 /D3/CUE /CP/D2/CSB /CP/D6/CT /CP
/D8/D9/CP/D0/D0/DD /D8/CW/CT /CC/CC/BA/C0/D3 /DB /CT/DA /CT/D6 /DB /CT /D7/CW/D3 /DB /D8/CW/CP/D8 /D8/CW/CT/D7/CT /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2/D7 /CP/D0/D7/D3 /CQ /CT/D0/D3/D2/CV /D8/D3 /D8/CW/CT
/D0/CP/D7/D7 /D3/CU /B9 /D8/CW/CT /BT /CC/B8 /B4/D7/CT/CT /CP/D0/D7/D3/CJ/BI ℄/B5/BA /C1/D2/D8/CW/CT /D1/D3 /CS/CT/D6/D2 /CS/CT/D6/CX/DA /CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2 /D6/CT/D0/CP/D8/CX/D3/D2/D7 /CU/D3/D6E /CP/D2/CSB /B8 /B4/D7/CT/CT/B8 /CT/BA/CV/BA/B8 /CA/CT/CU/BA /BD/BD /CB/CT
/BA /BD/BD/BA/BD/BC/B8/CA/CT/CU/BA /BL /CB/CT
/BA /BF/BA/BF/BA/B5/B8 /D3/D2/CT /CX/CS/CT/D2 /D8/CX/AS/CT/D7/B8 /CX/D2 /D7/D3/D1/CT /C1/BY/CAS /B8 /D8/CW/CT
/D3/D1/D4 /D3/D2/CT/D2 /D8/D7 /D3/CU /D8/CW/CT /BF/B9/DA /CT
/D8/D3/D6/D7 Ei
/CP/D2/CSBi
/DB/CX/D8/CW/D8/CW/CT
/D3/D1/D4 /D3/D2/CT/D2 /D8/D7 /D3/CUFαβ/CP/D7Ei=F0i, /CP/D2/CSBi= (1/c)∗F0i/CX/D2 /D3/D6/CS/CT/D6 /D8/D3 /CV/CT/D8 /CX/D2 /D8/CW/CP/D8 /C1/BY/CA /D8/CW/CT /D9/D7/D9/CP/D0/C5/CP/DC/DB /CT/D0/D0 /CT/D5/D9/CP/D8/CX/D3/D2/D7/B8
∇E(r, t) = ρ(r, t)/ε0,∇ ×E(r, t) =−∂B(r, t)/∂t,
∇B(r, t) = 0 ,∇ ×B(r, t) =µ0j(r, t) +µ0ε0∂E(r, t)/∂t,/CU/D6/D3/D1 /D8/CW/CT
/D3 /DA /CP/D6/CX/CP/D2 /D8 /C5/CP/DC/DB /CT/D0/D0 /CT/D5/D9/CP/D8/CX/D3/D2/D7 /DB/CX/D8/CWFαβ/CP/D2/CS /CX/D8/D7 /CS/D9/CP/D0∗Fαβ
∂αFaβ=−jβ/ε0c, ∂ α∗Fαβ= 0/DB/CW/CT/D6/CT∗Fαβ=−(1/2)εαβγδFγδ
/CP/D2/CSεαβγδ/CX/D7 /D8/CW/CT /D8/D3/D8/CP/D0/D0/DD /D7/CZ /CT/DB/B9/D7/DD/D1/D1/CT/D8/D6/CX
/C4/CT/DA/CX/B9/BV/CX/DA/CX/D8/CP /D4/D7/CT/D9/CS/D3/D8/CT/D2/D7/D3/D6/BA/B4/C6/D3/D8/CT /D8/CW/CP/D8 /CX/D2 /D8/CW/CT
/D3 /DA /CP/D6/CX/CP/D2 /D8 /CU/D3/D6/D1 /D9/D0/CP/D8/CX/D3/D2 Fαβ/CX/D7 /D8/CW/CT /D4/D6/CX/D1/CP/D6/DD /D5/D9/CP/D2 /D8/CX/D8 /DD/BN /CX/D8 /CX/D7 /D8/CW/CT /D7/D3/D0/D9/D8/CX/D3/D2 /D3/CU /D8/CW/CT
/D3 /DA /CP/D6/CX/CP/D2 /D8 /C5/CP/DC/DB /CT/D0/D0 /CT/D5/D9/CP/D8/CX/D3/D2/D7/B8 /D3/D6 /D8/CW/CT
/D3/D6/D6/CT/D7/D4 /D3/D2/CS/CX/D2/CV /DB /CP /DA /CT /CT/D5/D9/CP/D8/CX/D3/D2 ∂σ∂σFαβ−(1/ε0c)(∂βjα−∂αjβ) =
0, /CP/D2/CS /CX/D8
/D3/D2 /DA /CT/DD/D7 /CP/D0/D0 /D8/CW/CT /CX/D2/CU/D3/D6/D1/CP/D8/CX/D3/D2 /CP/CQ /D3/D9/D8 /D8/CW/CT /CT/D0/CT
/D8/D6/D3/D1/CP/CV/D2/CT/D8/CX
/AS/CT/D0/CS/BA Fαβ/CX/D7 /CV/CT/D2/CT/D6/CP/D0/D0/DD /CV/CX/DA /CT/D2 /CP/D7
Fαβ(xµ) = (2 k/iπc)/integraldisplay/braceleftBigg/bracketleftbig
jα(x′µ)(x−x′)β−jβ(x′µ)(x−x′)α/bracketrightbig
[(x−x′)σ(x−x′)σ]2/bracerightBigg
d4x′,/DB/CW/CT/D6/CT xα, x′α/CP/D6/CT /D8/CW/CT /D4 /D3/D7/CX/D8/CX/D3/D2 /BG/B9/DA /CT
/D8/D3/D6/D7 /D3/CU /D8/CW/CT /AS/CT/D0/CS /D4 /D3/CX/D2 /D8 /CP/D2/CS /D8/CW/CT /D7/D3/D9/D6
/CT /D4 /D3/CX/D2 /D8 /D6/CT/D7/D4 /CT
/D8/CX/DA /CT/D0/DD /B8 /CP/D2/CS
k= 1/4πε0.) /BT/CU/D8/CT/D6 /D8/D6/CP/D2/D7/CU/D3/D6/D1/CX/D2/CV /CQ /DD /D8/CW/CT /C4 /CC /D8/CW/CT
/D3 /DA /CP/D6/CX/CP/D2 /D8 /C5/CP/DC/DB /CT/D0/D0 /CT/D5/D9/CP/D8/CX/D3/D2/D7 /D8/D3 /D8/CW/CTS′/CU/D6/CP/D1/CT /D3/D2/CT/AS/D2/CS/D7∂′
αF′aβ=−j′β/ε0c, ∂′
α∗F′αβ= 0. /CC/CW/CT /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2/D7 /D3/CU /CP/D0/D0 /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7 /CX/D2 /D8/CW/CT
/D3 /DA /CP/D6/CX/CP/D2 /D8/C5/CP/DC/DB /CT/D0/D0 /CT/D5/D9/CP/D8/CX/D3/D2/D7 /CP/D6/CT /B9 /D8/CW/CT /CC/CC/BA /CC/CW/CT
/D3 /DA /CP/D6/CX/CP/D2 /D8 /C5/CP/DC/DB /CT/D0/D0 /CT/D5/D9/CP/D8/CX/D3/D2/D7 /CS/D3 /D2/D3/D8
/CW/CP/D2/CV/CT /D8/CW/CT/CX/D6 /CU/D3/D6/D1 /D3/D2 /D8/CW/CT/C4 /CC /CT/D1 /CQ /D3 /CS/DD/CX/D2/CV /CX/D2 /D8/CW/CP/D8 /DB /CP /DD /D8/CW/CT /D4/D6/CX/D2
/CX/D4/D0/CT /D3/CU /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD /BA /CC/CW/CT/D2 /D3/D2/CT /CP/CV/CP/CX/D2 /CX/CS/CT/D2 /D8/CX/AS/CT/D7 /D8/CW/CT /BF/B9/DA /CT
/D8/D3/D6/D7 E′
i/CP/D2/CSB′
i
/DB/CX/D8/CW /D8/CW/CT
/D3/D1/D4 /D3/D2/CT/D2 /D8/D7 /D3/CUF′αβ/CX/D2 /D8/CW/CT /D7/CP/D1/CT /DB/CP/DD /CP/D7 /CX/D2S /B8 /CX/BA/CT/BA/B8E′
i=F′0i, /CP/D2/CSB′
i= (1/c)∗F′0i/CX/D2 /D3/D6/CS/CT/D6 /D8/D3 /D3/CQ/D8/CP/CX/D2 /D8/CW/CT /D9/D7/D9/CP/D0 /C5/CP/DC/DB /CT/D0/D0 /CT/D5/D9/CP/D8/CX/D3/D2/D7 /B4/CX/D2 /D8/CW/CT /D8/CW/D6/CT/CT/B9/DA /CT
/D8/D3/D6 /CU/D3/D6/D1/B5 /CU/D6/D3/D1 /D8/CW/CT /D8/D6/CP/D2/D7/CU/D3/D6/D1/CT/CS
/D3 /DA /CP/D6/CX/CP/D2 /D8 /C5/CP/DC/DB /CT/D0/D0 /CT/D5/D9/CP/D8/CX/D3/D2/D7/BA /CC/CW/CX/D7 /D4/D6/D3
/CT/CS/D9/D6/CT /D8/CW/CT/D2 /CV/CX/DA /CT/D7 /D8/CW/CT
/D3/D2/D2/CT
/D8/CX/D3/D2 /CQ /CT/D8 /DB /CT/CT/D2 /D8/CW/CT /BF/B9/DA /CT
/D8/D3/D6/D7 E′
i,
B′
i
/CX/D2S′/CP/D2/CSEi, Bi
/CX/D2S /CP/D7
Ei= Γ( E′
i−cεijkβjB′
k)−((Γ−1)/β2)βi(βkE′
k)
Bi= Γ( B′
i−(1/c)εijkβjE′
k)−((Γ−1)/β2)βi(βkB′
k), /B4/BD/B5/DB/CW/CT/D6/CT β=V/c /CP/D2/CSΓ = (1 −β2)−1/2. /B4/CC/CW/CT
/D3/D1/D4 /D3/D2/CT/D2 /D8/D7 /D3/CU /D8/CW/CT /BF/B9/DA /CT
/D8/D3/D6 /AS/CT/D0/CS/D7E /CP/D2/CSB /B8 /CP/D2/CS/D3/CU /D8/CW/CT /BF/B9/DA /CT/D0/D3
/CX/D8 /DDV /CP/D6/CT /DB/D6/CX/D8/D8/CT/D2 /DB/CX/D8/CW /D0/D3 /DB /CT/D6/CT/CS /B4/CV/CT/D2/CT/D6/CX
/B5 /D7/D9/CQ/D7
/D6/CX/D4/D8/D7/B8 /D7/CX/D2
/CT /D8/CW/CT/DD /CP/D6/CT /D2/D3/D8 /D8/CW/CT /D7/D4/CP/D8/CX/CP/D0
/D3/D1/D4 /D3/D2/CT/D2 /D8/D7 /D3/CU /D8/CW/CT /BG/B9/DA /CT
/D8/D3/D6/D7/BA /CC/CW/CX/D7 /D6/CT/CU/CT/D6/D7 /D8/D3 /D8/CW/CT /D8/CW/CX/D6/CS/B9/D6/CP/D2/CZ /CP/D2 /D8/CX/D7/DD/D1/D1/CT/D8/D6/CX
ε /D8/CT/D2/D7/D3/D6 /D8/D3 /D3/BA /CC/CW/CT /D7/D9/D4 /CT/D6/B9/CP/D2/CS /D7/D9/CQ/D7
/D6/CX/D4/D8/D7 /CP/D6/CT /D9/D7/CT/CS /D3/D2/D0/DD /D3/D2 /D8/CW/CT
/D3/D1/D4 /D3/D2/CT/D2 /D8/D7 /D3/CU /D8/CW/CT /BG/B9/DA /CT
/D8/D3/D6/D7 /D3/D6 /D8/CT/D2/D7/D3/D6/D7/BA/B5 /BT
/D3/D6/CS/CX/D2/CV /D8/D3 /B4/BD/B5 /D8/CW/CT
/D3/D1/D4 /D3/D2/CT/D2 /D8/D7 /D3/CU/B8 /CT/BA/CV/BA/B8 /D8/CW/CT /BF/B9/DA /CT
/D8/D3/D6E′/CX/D2S′/CP/D6/CT /CS/CT/D8/CT/D6/D1/CX/D2/CT/CS /CQ /DD /D8/CW/CT
/D3/D1/D4 /D3/D2/CT/D2 /D8/D7 /D3/CU /CQ /D3/D8/CW /BF/B9/DA /CT
/D8/D3/D6/D7 E/CP/D2/CSB /CX/D2S /BA /C7/CQ /DA/CX/D3/D9/D7/D0/DD E /CX/D2S /B8 /D1/CT /CP/D7/D9/D6 /CT /CS /CQ/DD /D8/CW/CT /D3/CQ/D7/CT/D6/DA/CT/D6/D7 /CP/D8 /D6 /CT/D7/D8 /CX/D2S /B8 /CP/D2/CSE′/CX/D2S′/B8 /D1/CT /CP/D7/D9/D6 /CT /CS /CQ/DD /D8/CW/CT/D3/CQ/D7/CT/D6/DA/CT/D6/D7 /CP/D8 /D6 /CT/D7/D8 /CX/D2S′/B8 /B4/D8/CW/CT /D7/CP/D1/CT /CW/D3/D0/CS/D7 /CU/D3/D6B /CP/D2/CSB′/B5/B8 /CS/D3 /D2/D3/D8 /D6 /CT/CU/CT/D6 /D8/D3 /D8/CW/CT /D7/CP/D1/CT /D5/D9/CP/D2/D8/CX/D8/DD
/D3/D2/D7/CX/CS/CT/D6 /CT /CS/CX/D2S /CP/D2/CSS′/B8 /CQ/D9/D8 /D8/D3 /D8/CW/CT /D7/CP/D1/CT /D1/CT /CP/D7/D9/D6 /CT/D1/CT/D2/D8 /B8 /CP/D7 /CX/D2 /D8/CW/CT /C4/D3/D6/CT/D2 /D8/DE
/D3/D2 /D8/D6/CP
/D8/CX/D3/D2/B8 /CP/D2/CS
/D3/D2/D7/CT/D5/D9/CT/D2 /D8/D0/DD /D8/CW/CT/BF/B9/DA /CT
/D8/D3/D6/D7 E /CP/D2/CSE′/B4B /CP/D2/CSB′/B5 /CP/D6/CT /D2/D3/D8
/D3/D2/D2/CT
/D8/CT/CS /CQ /DD /D8/CW/CT /CC/CC /D8/CW/CP/D2 /CQ /DD /D8/CW/CT /BT /CC/B8 /BX/D5/BA /B4/BD/B5/BA/BE/BA/BG /CC/CW/CT /BG/B9/DA /CT
/D8/D3/D6/D7 Eα/CP/D2/CSBα/CP/D2/CS /D8/CW/CT /BT /CC /D3/CUE /CP/D2/CSB/BT/D0/D8/CW/D3/D9/CV/CW Fαβ
/D3/D2 /D8/CP/CX/D2/D7 /CP/D0/D0 /D8/CW/CT /CX/D2/CU/D3/D6/D1/CP/D8/CX/D3/D2 /CP/CQ /D3/D9/D8 /D8/CW/CT /CT/D0/CT
/D8/D6/D3/D1/CP/CV/D2/CT/D8/CX
/AS/CT/D0/CS /D3/D2/CT
/CP/D2 /CX/D2 /D8/D6/D3 /CS/D9
/CT /D8/CW/CT/BG/B9/DA /CT
/D8/D3/D6/D7 Eα/CP/D2/CSBα/CX/D2/D7/D8/CT/CP/CS /D3/CU /D8/CW/CT /D9/D7/D9/CP/D0 /BF/B9/DA /CT
/D8/D3/D6/D7 E /CP/D2/CSB /BA /CC/CW/CTEα/CP/D2/CSBα/CP/D6/CT /DB /CT/D0/D0 /CS/CT/AS/D2/CT/CS/D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7 /CU/D6/D3/D1 /D8/CW/CT /AH/CC/CC /DA/CX/CT/DB/D4 /D3/CX/D2 /D8/AH /CP/D2/CS /D8/CW/CT/DD /CP/D6/CT /CS/CT/AS/D2/CT/CS /CQ /DD /D1/CT/CP/D2/D7 /D3/CUFαβ/CP/D7
Eα= (1/c)Fαβvβ,Bα= (1/c2)∗Fαβvβ. /B4/BE/B5/CC/CW/CTEα/CP/D2/CSBα/CP/D6/CT /D8/CW/CT /CT/D0/CT
/D8/D6/CX
/CP/D2/CS /D1/CP/CV/D2/CT/D8/CX
/AS/CT/D0/CS /BG/B9/DA /CT
/D8/D3/D6/D7 /D1/CT/CP/D7/D9/D6/CT/CS /CQ /DD /CP/D2 /D3/CQ/D7/CT/D6/DA /CT/D6 /D1/D3 /DA/CX/D2/CV /DB/CX/D8/CW/BG/B9/DA /CT/D0/D3
/CX/D8 /DDvα/CX/D2 /CP/D2 /CP/D6/CQ/CX/D8/D6/CP/D6/DD /D6/CT/CU/CT/D6/CT/D2
/CT /CU/D6/CP/D1/CT/BA/BG/CF /CT /D2/D3/D8/CT /D8/CW/CP/D8 /CX/D2 /D8/CW/CX/D7 /D4/CP/D4 /CT/D6 Eα/CP/D2/CSBα/CP/D6/CT /CS/CT/AS/D2/CT/CS /CX/D2 /D8/CW/CT /D7/CP/D1/CT /DB /CP /DD /CP/D7 /CX/D2 /CJ/BD/BE ℄/B8 /CX/BA/CT/BA/B8 /D8/CP/CZ/CX/D2/CV /D8/CW/CP/D8
vα/CX/D7 /D8/CW/CT /BG/B9/DA /CT/D0/D3
/CX/D8 /DD /D3/CU /CP /CU/CP/D1/CX/D0/DD /D3/CU /D3/CQ/D7/CT/D6/DA /CT/D6/D7 /DB/CW/D3 /D1/CT/CP/D7/D9/D6/CT/D7 /D8/CW/CT /AS/CT/D0/CS/D7/BA /BU/D9/D8 /CX/D8 /CX/D7
/D3/D2/D7/CX/CS/CT/D6/CT/CS /CX/D2 /CJ/BD/BE ℄ /D8/CW/CP/D8
Eα/CP/D2/CSBα/CP/D6/CT /D2/CT
/CT/D7/D7/CP/D6/DD /D3/D2/D0/DD /CU/D3/D6 /D2/D3/D2/CX/D2/CT/D6/D8/CX/CP/D0 /CP/D2/CS
/D9/D6/DA /CT/CS /D7/D4/CP
/CT/D8/CX/D1/CT/D7 /CP/D2/CS /D2/D3/D8 /CU/D3/D6 /C1/BY/CA/D7/BA /C0/D3 /DB /CT/DA /CT/D6 /D8/CW/CT/CU/D9/D2/CS/CP/D1/CT/D2 /D8/CP/D0 /D6/CT/D7/D9/D0/D8 /D8/CW/CP/D8 /D8/CW/CT /D9/D7/D9/CP/D0 /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2/D7 /D3/CUE /CP/D2/CSB /B8 /BX/D5/BA/B4/BD/B5/B8 /CQ /CT/D0/D3/D2/CV /D8/D3 /D8/CW/CT
/D0/CP/D7/D7 /D3/CU /D8/CW/CT /BT /CC/D2/CT
/CT/D7/D7/CX/D8/CP/D8/CT/D7 /D8/CW/CT /CX/D2 /D8/D6/D3 /CS/D9
/D8/CX/D3/D2 /D3/CU /D8/CW/CT /BG/B9/DA /CT
/D8/D3/D6/D7 Eα/CP/D2/CSBα/CT/DA /CT/D2 /CU/D3/D6 /C1/BY/CA/D7/BA /CC/CW/CT /CS/CT/AS/D2/CX/D8/CX/D3/D2 /B4/BE/B5
/CP/D2 /CQ /CT
/D3/D1/D4/CP/D6/CT/CS /DB/CX/D8/CW /D8/CW/CT /CS/CT/AS/D2/CX/D8/CX/D3/D2/D7 /D3/CUEα/CP/D2/CSBα/CX/D2/B8 /CT/BA/CV/BA/B8 /CA/CT/CU/BA /BD/BF/B8 /DB/CW/CT/D6/CT /D8/CW/CT
/D3 /DA /CP/D6/CX/CP/D2 /D8 /CT/D0/CT
/D8/D6/D3 /CS/DD/D2/CP/D1/CX
/D7/CX/D2 /D8/CW/CT /D1/D3 /DA/CX/D2/CV /D1/CT/CS/CX/D9/D1 /CX/D7
/D3/D2/D7/CX/CS/CT/D6/CT/CS /CP/D2/CSvα/CX/D7 /D8/CW/CT /BG/B9/DA /CT/D0/D3
/CX/D8 /DD /D3/CU /D8/CW/CT /D1/CT/CS/CX/D9/D1/B8 /D3/D6 /CX/D2 /CA/CT/CU/BA /BD/BG/B8 /DB/CW/CT/D6/CT/D8/CW/CT /D4/CW /DD/D7/CX
/CP/D0 /D1/CT/CP/D2/CX/D2/CV /D3/CUvα/CX/D7 /D9/D2/D7/D4 /CT
/CX/AS/CT/CS /B9 /CX/D8 /CX/D7 /CP/D2 /DD /D9/D2/CX/D8/CP/D6/DD /BG/B9/DA /CT
/D8/D3/D6/BA /CC/CW/CT /D6/CT/CP/D7/D3/D2 /CU/D3/D6 /D7/D9
/CW
/CW/D3/CX
/CT/D3/CUvα/CX/D2 /CJ/BD/BG ℄ /CX/D7 /D8/CW/CP/D8 /D8/CW/CT/D6/CT Eα/CP/D2/CSBα/CP/D6/CT /CX/D2 /D8/D6/D3 /CS/D9
/CT/CS /CP/D7 /D8/CW/CT /AH/CP/D9/DC/CX/D0/CX/CP/D6/DD /AS/CT/D0/CS/D7/B8/AH /DB/CW/CX/D0/CTE /CP/D2/CSB /CP/D6/CT
/D3/D2/D7/CX/CS/CT/D6/CT/CS /CP/D7 /D8/CW/CT /D4/CW /DD/D7/CX
/CP/D0 /AS/CT/D0/CS/D7/BA /C1/D2 /D3/D9/D6 /CP/D0/D8/CT/D6/D2/CP/D8/CX/DA /CT
/D3 /DA /CP/D6/CX/CP/D2 /D8 /CP/D4/D4/D6/D3/CP
/CW /D8/CW/CT /D7/CX/D8/D9/CP/D8/CX/D3/D2 /CX/D7 /CY/D9/D7/D8 /D8/CW/CT/D3/D4/D4 /D3/D7/CX/D8/CT/BN Eα/CP/D2/CSBα/CP/D6/CT /D8/CW/CT /D6/CT/CP/D0 /D4/CW /DD/D7/CX
/CP/D0 /AS/CT/D0/CS/D7/B8 /DB/CW/CX
/CW /CP/D6/CT
/D3/D6/D6/CT
/D8/D0/DD /CS/CT/AS/D2/CT/CS /CP/D2/CS /D1/CT/CP/D7/D9/D6/CT/CS /CX/D2 /BG/BW/D7/D4/CP
/CT/D8/CX/D1/CT/B8 /DB/CW/CX/D0/CT /D8/CW/CT /BF/B9/DA /CT
/D8/D3/D6/D7 E /CP/D2/CSB /CP/D6/CT /D2/D3/D8
/D3/D6/D6/CT
/D8/D0/DD /CS/CT/AS/D2/CT/CS /CX/D2 /BG/BW /D7/D4/CP
/CT/D8/CX/D1/CT /CU/D6/D3/D1 /D8/CW/CT /AH/CC/CC/DA/CX/CT/DB/D4 /D3/CX/D2 /D8/BA/AH/CC/CW/CT /CX/D2 /D8/D6/D3 /CS/D9
/D8/CX/D3/D2 /D3/CUEα/CP/D2/CSBα/CT/D2/CP/CQ/D0/CT/D7 /D9/D7 /D8/D3 /CQ /CT/D8/D8/CT/D6 /CT/DC/D4/D0/CP/CX/D2 /DB/CW /DD /D8/CW/CT /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2/D7 /D3/CUE/CP/D2/CSB /B8 /B4/BD/B5/B8 /CP/D6/CT /D8/CW/CT /BT /CC/BA /C6/CP/D1/CT/D0/DD /B8 /CX/D2 /D8/CW/CT /D1/CT/D2/D8/CX/D3/D2/CT /CS /D1/D3 /CS/CT/D6/D2 /CS/CT/D6/CX/DA/CP/D8/CX/D3/D2 /D3/CU /B4/BD/B5 /D8/DB/D3 /CS/CX/AR/CT/D6 /CT/D2/D8 /CU/CP/D1/CX/D0/CX/CT/D7/D3/CU /D3/CQ/D7/CT/D6/DA/CT/D6/D7 /DB/CW/D3 /D1/CT /CP/D7/D9/D6 /CT /D8/CW/CT /CT/D0/CT
/D8/D6/CX
/AS/CT/D0/CS /CP/D6 /CT
/D3/D2/D7/CX/CS/CT/D6 /CT /CS/B8 /D3/D2/CT /CP/D8 /D6 /CT/D7/D8 /CX/D2 /D8/CW/CT /C1/BY/CAS /B8 /CU/D3/D6 /DB/CW/CX
/CW
vα= (c,0) /B8 /CP/D2/CS /CP/D2/D3/D8/CW/CT/D6 /D3/D2/CT /CP/D8 /D6 /CT/D7/D8 /CX/D2 /D8/CW/CT /C1/BY/CA /CB/B3/B8 /CU/D3/D6 /DB/CW/CX
/CW /CP/CV/CP/CX/D2 v′α= (c,0) /B8 /CP/D2/CS /D8/CW/CT/D7/CT/D3/CQ/D7/CT/D6/DA/CT/D6/D7 /CX/D2S /CP/D2/CSS′/CP/D6 /CT /D2/D3/D8 /D6 /CT/D0/CP/D8/CT /CS /CX/D2 /CP/D2/DD /DB/CP/DD/BA /CB/D9
/CW /CP/D7/D7/D9/D1/D4/D8/CX/D3/D2/D7 /CU/D3/D6vα/CP/D2/CSv′α/D1/CT/CP/D2 /D8/CW/CP/D8 /D3/D2/CT/CS/D3 /CT/D7 /D2/D3/D8
/D3/D2/D7/CX/CS/CT/D6 /D8/CW/CT /D7/CP/D1/CT /D4/CW/DD/D7/CX
/CP/D0 /D5/D9/CP/D2/D8/CX/D8/DD /CX/D2S /CP/D2/CSS′/B8 /CQ/D9/D8 /D8/CW/CP/D8 /D8 /DB /D3 /CS/CX/AR/CT/D6/CT/D2 /D8 /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7Eα/CP/D2/CS
Ξ′α/B4/CX/D2 /DB/CW/CX
/CWvα/CX/D7 /D2/D3/D8 /D8/D6/CP/D2/D7/CU/D3/D6/D1/CT/CS/B5 /CP/D6/CT
/D3/D2/D7/CX/CS/CT/D6/CT/CS /CX/D2S /CP/D2/CSS′/B8 /D6/CT/D7/D4 /CT
/D8/CX/DA /CT/D0/DD /BA /CC/CW/CT /D5/D9/CP/D2 /D8/CX/D8 /DDΞ′α/CW/CP/D7 /D2/D3/D8/CW/CX/D2/CV /CX/D2
/D3/D1/D1/D3/D2 /DB/CX/D8/CW /D8/CW/CT /CT/D0/CT
/D8/D6/CX
/AS/CT/D0/CSEα/BA /CC/CW/CX/D7 /D1/CT/CP/D2/D7 /D8/CW/CP/D8 /D8/CW/CT /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2/D7 /B4/BD/B5 /CP/D6/CT/D8/CW/CT /BT /CC/BN /D8/CW/CT/DD /D6 /CT/CU/CT/D6 /D8/D3 /D8/CW/CT /D7/CP/D1/CT /D1/CT /CP/D7/D9/D6 /CT/D1/CT/D2/D8 /CX/D2 /D8/DB/D3 /C1/BY/CA/D7 /CP/D2/CS /D2/D3/D8 /D8/D3 /D8/CW/CT /D7/CP/D1/CT /D4/CW/DD/D7/CX
/CP/D0 /D5/D9/CP/D2/D8/CX/D8/DD /CP/D7/D6 /CT /D5/D9/CX/D6 /CT /CS /CQ/DD /D8/CW/CT /AH/CC/CC /D6 /CT/D0/CP/D8/CX/DA/CX/D8/DD/BA/AH /CC/CW/CT /CC/CC /D6/CT/CU/CT/D6/D6/CX/D2/CV /D8/D3 /D8/CW/CT /D7/CP/D1/CT /D4/CW /DD/D7/CX
/CP/D0 /D5/D9/CP/D2 /D8/CX/D8 /DD /CP/D6/CT /D8/CW/CT /C4 /CC /D3/CUEα/CP/D2/CSBα/BA/C1/D8 /CW/CP/D7 /D8/D3 /CQ /CT /D1/CT/D2 /D8/CX/D3/D2/CT/CS /CW/CT/D6/CT /D8/CW/CP/D8 /CP/D0/D8/CW/D3/D9/CV/CW /CA/D3/CW/D6/D0/CX
/CW/B8 /CJ/BG/B8 /BD℄ /CP/D2/CS /BZ/CP/D1 /CQ/CP /CJ/BH ℄
/D3/D6/D6/CT
/D8/D0/DD /CX/D2/D7/CX/D7/D8 /D3/D2
/D3 /DA /CP/D6/CX/CP/D2 /D8 /CS/CT/AS/D2/CX/D8/CX/D3/D2/D7 /D3/CU /DA /CP/D6/CX/D3/D9/D7 /D4/CW /DD/D7/CX
/CP/D0 /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7/B8 /D8/CW/CT/DD /CP/D0/D7/D3 /CS/CX/CS /D2/D3/D8 /D2/D3/D8/CX
/CT /D8/CW/CP/D8 /CT/D5/D9/CP/D8/CX/D3/D2/D7 /B4/BD/B5 /CP/D6/CT/D8/CW/CT /BT /CC /D8/CW/CP/D8 /CS/D3 /D2/D3/D8 /D6/CT/CU/CT/D6 /D8/D3 /D8/CW/CT /D7/CP/D1/CT /D4/CW /DD/D7/CX
/CP/D0 /D5/D9/CP/D2 /D8/CX/D8 /DD /CX/D2 /D8 /DB /D3 /C1/BY/CA/D7/BA/BE/BA/BH /BT/D0/D8/CT/D6/D2/CP/D8/CX/DA /CT /BV/D3 /DA /CP/D6/CX/CP/D2 /D8 /BY /D3/D6/D1 /D9/D0/CP/D8/CX/D3/D2 /DB/CX/D8/CW Eα/CP/D2/CSBα/CD/D7/CX/D2/CV Eα/CP/D2/CSBα/D3/D2/CT
/CP/D2
/D3/D2/D7/D8/D6/D9
/D8 /D8/CW/CT
/D3 /DA /CP/D6/CX/CP/D2 /D8 /CU/D3/D6/D1 /D9/D0/CP/D8/CX/D3/D2 /D3/CU /CT/D0/CT
/D8/D6/D3 /CS/DD/D2/CP/D1/CX
/D7/B8 /DB/CW/CX
/CW /CX/D7 /CT/D5/D9/CX/DA/B9/CP/D0/CT/D2 /D8 /D8/D3 /D8/CW/CT /D9/D7/D9/CP/D0
/D3 /DA /CP/D6/CX/CP/D2 /D8 /CU/D3/D6/D1 /D9/D0/CP/D8/CX/D3/D2 /DB/CX/D8/CWFαβ/BA /BY /D3/D6 /D8/CW/CP/D8 /D3/D2/CT /D2/CT/CT/CS/D7 /D8/CW/CT /CX/D2 /DA /CT/D6/D7/CT /D6/CT/D0/CP/D8/CX/D3/D2/D7 /D8/D3 /D8/CW/CT/D6/CT/D0/CP/D8/CX/D3/D2/D7 /B4/BE/B5 /CX/D2 /DB/CW/CX
/CWFαβ/DB/CX/D0/D0 /CQ /CT /CT/DC/D4/D6/CT/D7/D7/CT/CS /CQ /DD /D1/CT/CP/D2/D7 /D3/CUEα/CP/D2/CSBα/B8 /CP/D2/CSvα/B8 /B4
/D3/D1/D4/CP/D6/CT /DB/CX/D8/CW /CA/CT/CU/D7/BA/BD/BF /CP/D2/CS /BD/BG /D8/CP/CZ/CX/D2/CV /CX/D2 /D8/D3 /CP
/D3/D9/D2 /D8 /D8/CW/CT /CP/CQ /D3 /DA /CT /D1/CT/D2 /D8/CX/D3/D2/CT/CS /D6/CT/D1/CP/D6/CZ/D7 /CP/CQ /D3/D9/D8 /D8/CW/CT /D1/CT/CP/D2/CX/D2/CV /D3/CUEα, Bα/B8 /CP/D2/CS
vα/CX/D2 /D8/CW/CT/D7/CT /DB /D3/D6/CZ/D7/B5/BA /CC/CW/CT /CX/D2 /DA /CT/D6/D7/CT /D6/CT/D0/CP/D8/CX/D3/D2/D7 /CP/D6/CT
Fαβ= (1 /c)δαβ
µνvµEν+εαβµνBµvν,
∗Fαβ=δαβ
µνvµBν+ (1/c)εαβµνvµEν, /B4/BF/B5/DB/CW/CT/D6/CT δαβ
µν=δα
µδβ
ν−δα
νδβ
µ. /CC/CW/CT /BG/B9/DA /CT
/D8/D3/D6/D7 Eα/CP/D2/CSBα/D7/CP/D8/CX/D7/CU/DD /D8/CW/CT
/D3/D2/CS/CX/D8/CX/D3/D2/D7 vαEα=vβBβ= 0 /B8 /CP/D7
/CP/D2 /CQ /CT
/CW/CT
/CZ /CT/CS /CU/D6/D3/D1 /B4/BE/B5 /CP/D2/CS /B4/BF/B5/BA /CB/D9/CQ/D7/D8/CX/D8/D9/D8/CX/D2/CV /B4/BF/B5 /CX/D2 /D8/D3 /D8/CW/CT
/D3 /DA /CP/D6/CX/CP/D2 /D8 /C5/CP/DC/DB /CT/D0/D0 /CT/D5/D9/CP/D8/CX/D3/D2/D7 /DB/CX/D8/CWFαβ/DB /CT /D3/CQ/D8/CP/CX/D2 /D8/CW/CT
/D3 /DA /CP/D6/CX/CP/D2 /D8 /C5/CP/DC/DB /CT/D0/D0 /CT/D5/D9/CP/D8/CX/D3/D2/D7 /DB/CX/D8/CWEα/CP/D2/CSBα/B8
∂α(δαβ
µνvµEν) +c∂α(εαβµνBµvν) = −jβ/ε0,
∂α(δαβ
µνvµBν) + (1 /c)∂α(εαβµνvµEν) = 0 ./CC/CW/CT /D6/CT/D0/CP/D8/CX/D3/D2/D7 /B4/BF/B5 /D8/D6/CP/D2/D7/CU/D3/D6/D1 /D8/CW/CT
/D3 /DA /CP/D6/CX/CP/D2 /D8 /CU/D3/D6/D1 /D9/D0/CP/D8/CX/D3/D2 /DB/CX/D8/CWFαβ/CP/D2/CS∗Fαβ/D8/D3 /D8/CW/CT
/D3 /DA /CP/D6/CX/CP/D2 /D8 /CU/D3/D6/D1 /D9/B9/D0/CP/D8/CX/D3/D2 /DB/CX/D8/CWEα/CP/D2/CSBα/B8 /DB/CW/CX/D0/CT /D8/CW/CT /D6/CT/D0/CP/D8/CX/D3/D2/D7 /B4/BE/B5 /CS/D3 /D8/CW/CT /D6/CT/DA /CT/D6/D7/CT /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2/D7/B8 /B4/D7/CT/CT /CP/D0/D7/D3 /CJ/BI ℄/B5/BA/C1/CU /D3/D2/CT /D8/CP/CZ /CT/D7 /D8/CW/CP/D8 /CX/D2 /CP/D2 /C1/BY/CAS /D8/CW/CT /D3/CQ/D7/CT/D6/DA /CT/D6/D7 /DB/CW/D3 /D1/CT/CP/D7/D9/D6/CT Eα/CP/D2/CSBα/CP/D6/CT /CP/D8 /D6/CT/D7/D8/B8 /CX/BA/CT/BA/B8vα= (c,0) /B8/D8/CW/CT/D2E0=B0= 0 /B8 /CP/D2/CS /D3/D2/CT
/CP/D2 /CS/CT/D6/CX/DA /CT /CU/D6/D3/D1 /D8/CW/CT
/D3 /DA /CP/D6/CX/CP/D2 /D8 /C5/CP/DC/DB /CT/D0/D0 /CT/D5/D9/CP/D8/CX/D3/D2/D7 /DB/CX/D8/CWEα/CP/D2/CSBα/D8/CW/CT /C5/CP/DC/DB /CT/D0/D0 /CT/D5/D9/CP/D8/CX/D3/D2/D7 /DB/CW/CX
/CW
/D3/D2 /D8/CP/CX/D2 /D3/D2/D0/DD /D8/CW/CT /D7/D4/CP
/CT /D4/CP/D6/D8/D7 Ei/CP/D2/CSBi/D3/CUEα/CP/D2/CSBα/B8 /CT/BA/CV/BA/B8 /CU/D6/D3/D1/D8/CW/CT /AS/D6/D7/D8
/D3 /DA /CP/D6/CX/CP/D2 /D8 /C5/CP/DC/DB /CT/D0/D0 /CT/D5/D9/CP/D8/CX/D3/D2 /D3/D2/CT /CT/CP/D7/CX/D0/DD /AS/D2/CS/D7∂iEi=j0/ε0c /BA /CF /CT /D7/CT/CT /D8/CW/CP/D8 /D8/CW/CT /C5/CP/DC/DB /CT/D0/D0/CT/D5/D9/CP/D8/CX/D3/D2/D7 /D3/CQ/D8/CP/CX/D2/CT/CS /CX/D2 /D7/D9
/CW /CP /DB /CP /DD /CP/D6/CT /D3/CU /D8/CW/CT /D7/CP/D1/CT /CU/D3/D6/D1 /CP/D7 /D8/CW/CT /D9/D7/D9/CP/D0 /C5/CP/DC/DB /CT/D0/D0 /CT/D5/D9/CP/D8/CX/D3/D2/D7 /DB/CX/D8/CWE/CP/D2/CSB /BA /BY /D6/D3/D1 /D8/CW/CT /CP/CQ /D3 /DA /CT
/D3/D2/D7/CX/CS/CT/D6/CP/D8/CX/D3/D2 /D3/D2/CT
/D3/D2
/D0/D9/CS/CT/D7 /D8/CW/CP/D8 /CP/D0/D0 /D8/CW/CT /D6/CT/D7/D9/D0/D8/D7 /D3/CQ/D8/CP/CX/D2/CT/CS /CX/D2 /CP /CV/CX/DA /CT/D2 /C1/BY/CA
S /CU/D6/D3/D1 /D8/CW/CT /D9/D7/D9/CP/D0 /C5/CP/DC/DB /CT/D0/D0 /CT/D5/D9/CP/D8/CX/D3/D2/D7 /DB/CX/D8/CWE /CP/D2/CSB /D6/CT/D1/CP/CX/D2 /DA /CP/D0/CX/CS /CX/D2 /D8/CW/CT
/D3 /DA /CP/D6/CX/CP/D2 /D8 /CU/D3/D6/D1 /D9/D0/CP/D8/CX/D3/D2 /DB/CX/D8/CW/D8/CW/CT /BG/B9/DA /CT
/D8/D3/D6/D7 Eα/CP/D2/CSBα/CQ/D9/D8 /D3/D2/D0/DD /CU/D3/D6 /D8/CW/CT /D3/CQ/D7/CT/D6/DA /CT/D6/D7 /DB/CW/D3 /D1/CT/CP/D7/D9/D6/CT /D8/CW/CT /AS/CT/D0/CS/D7 Eα/CP/D2/CSBα/CP/D2/CS /CP/D6/CT/BH/CP/D8 /D6/CT/D7/D8 /CX/D2 /D8/CW/CT
/D3/D2/D7/CX/CS/CT/D6/CT/CS /C1/BY/CA/BA /CC/CW/CT/D2 /CU/D3/D6 /D7/D9
/CW /D3/CQ/D7/CT/D6/DA /CT/D6/D7 /D8/CW/CT
/D3/D1/D4 /D3/D2/CT/D2 /D8/D7 /D3/CUE /CP/D2/CSB /B8 /DB/CW/CX
/CW /CP/D6/CT /D2/D3/D8/DB /CT/D0/D0 /CS/CT/AS/D2/CT/CS /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7 /CX/D2 /D8/CW/CT /AH/CC/CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD /B8/AH
/CP/D2 /CQ /CT /D7/CX/D1/D4/D0/DD /D6/CT/D4/D0/CP
/CT/CS /CQ /DD /D8/CW/CT /D7/D4/CP
/CT
/D3/D1/D4 /D3/D2/CT/D2 /D8/D7 /D3/CU/D8/CW/CT /BG/B9/DA /CT
/D8/D3/D6/D7 Eα/CP/D2/CSBα/BA /C0/D3 /DB /CT/DA /CT/D6/B8 /CX/CU /D8/CW/CT /C4 /CC /CU/D6/D3/D1S /D8/D3 /CP/D2/D3/D8/CW/CT/D6 /C1/BY/CAS′/B8 /D1/D3 /DA/CX/D2/CV /DB/CX/D8/CWVα/D6/CT/D0/CP/D8/CX/DA /CT/D8/D3S /B8 /CX/D7 /D4 /CT/D6/CU/D3/D6/D1/CT/CS/B8 /D8/CW/CT/D2 /CX/D2S′/D3/D2/CT
/CP/D2/D2/D3/D8 /D3/CQ/D8/CP/CX/D2 /D8/CW/CT /D9/D7/D9/CP/D0 /C5/CP/DC/DB /CT/D0/D0 /CT/D5/D9/CP/D8/CX/D3/D2/D7 /DB/CX/D8/CW /D8/CW/CT /BF/B9/DA /CT
/D8/D3/D6/D7
E′/CP/D2/CSB′/B4/CS/CT/D8/CT/D6/D1/CX/D2/CT/CS /CQ /DD /D8/CW/CT /BT /CC /B4/BD/B5/B5 /CU/D6/D3/D1 /D8/CW/CT /D8/D6/CP/D2/D7/CU/D3/D6/D1/CT/CS
/D3 /DA /CP/D6/CX/CP/D2 /D8 /C5/CP/DC/DB /CT/D0/D0 /CT/D5/D9/CP/D8/CX/D3/D2/D7 /DB/CX/D8/CWE′α/CP/D2/CSB′α/BA/BF /BV/C7/C6/CE/BX/C6/CC/C1/C7/C6/BT/C4 /BW/BX/BY/C1/C6/C1/CC/C1/C7/C6/CB /C7/BY /CC/C0/BX /BX/C4/BX/BV/CC/CA /C7/B9/C5/BT /BZ/C6/BX/CC/C1/BV /C5/C7/C5/BX/C6/CC/CD/C5 /BT/C6/BW /BX/C6/BX/CA /BZ/CH/C1/D2 /D8/CW/CX/D7 /D7/CT
/D8/CX/D3/D2 /DB /CT
/D3/D2/D7/CX/CS/CT/D6 /D8/CW/CT
/D3/D2 /DA /CT/D2 /D8/CX/D3/D2/CP/D0 /D7/DD/D2
/CW/D6/D3/D2/D3/D9/D7 /CS/CT/AS/D2/CX/D8/CX/D3/D2/D7 /D3/CU /D8/CW/CT /CT/D0/CT
/D8/D6/D3/D1/CP/CV/D2/CT/D8/CX
/D1/D3/D1/CT/D2/B9/D8/D9/D1 /CP/D2/CS /CT/D2/CT/D6/CV/DD /B8 /B4/D7/CT/CT/B8 /CT/BA/CV/BA/B8 /CA/CT/CU/D7/BA /BD/B9/BF/B8 /CA/CT/CU/D7/BA /BD/BH/B8/BD/BI/B5/BA /CC/CW/CT /D8/D3/D8/CP/D0 /BG/B9/D1/D3/D1/CT/D2 /D8/D9/D1 /D3/CU /CP
/CW/CP/D6/CV/CT/CS /D7/DD/D7/D8/CT/D1
/CP/D2 /CQ /CT /DB/D6/CX/D8/D8/CT/D2 /CP/D7
Pµ
tot.= (1/c)/integraldisplay
Σ[Θµν(x) +Tµν(x)]d3σν(x), /B4/BG/B5/DB/CW/CT/D6/CT d3σµ/CP/D6/CT /D8/CW/CT
/D3/D1/D4 /D3/D2/CT/D2 /D8/D7 /D3/CU /D8/CW/CT /CX/D2/AS/D2/CX/D8/CT/D7/CX/D1/CP/D0 /DA /D3/D0/D9/D1/CT /CT/D0/CT/D1/CT/D2 /D8 /D3/CU /CP /D8/CW/D6/CT/CT/B9/CS/CX/D1/CT/D2/D7/CX/D3/D2/CP/D0 /D7/D4/CP
/CT/D0/CX/CZ /CT/CW /DD/D4 /CT/D6/D7/D9/D6/CU/CP
/CT Σ /B8Θµν/CX/D7 /CP /D7/D8/D6/CT/D7/D7/B9/CT/D2/CT/D6/CV/DD /D8/CT/D2/D7/D3/D6 /DB/CW/CX
/CW /CS/CT/D7
/D6/CX/CQ /CT/D7 /D2/D3/D2/CT/D0/CT
/D8/D6/D3/D1/CP/CV/D2/CT/D8/CX
/CU/D3/D6
/CT/D7 /CP/D2/CS /D1/CP/D8/D8/CT/D6/B4/CX/D2
/D0/D9/CS/CX/D2/CV /D8/CW/CT /C8 /D3/CX/D2
/CP/D6/GH /D7/D8/D6/CT/D7/D7/CT/D7/B8 /CJ/BD/BJ ℄ /DB/CW/CX/D0/CT Tµν/CX/D7 /D8/CW/CT /CT/D2/CT/D6/CV/DD/B9/D1/D3/D1/CT/D2 /D8/D9/D1 /CS/CT/D2/D7/CX/D8 /DD /D8/CT/D2/D7/D3/D6 /D3/CU /D8/CW/CT/CT/D0/CT
/D8/D6/D3/D1/CP/CV/D2/CT/D8/CX
/AS/CT/D0/CS/BA /CC/CW/CT /D7/D9/D1Tµν
tot.= Θµν(x) +Tµν/CX/D7 /CS/CX/DA /CT/D6/CV/CT/D2
/CT/D0/CT/D7/D7/B8 ∂νTµν
tot.= 0 /B8
/CP/D9/D7/CX/D2/CV /D8/CW/CP/D8/D8/CW/CT /CW /DD/D4 /CT/D6/D4/D0/CP/D2/CT /CX/D2 /D8/CT/CV/D6/CP/D0 /B4/BG/B5 /CX/D7 /CX/D2/CS/CT/D4 /CT/D2/CS/CT/D2 /D8 /D3/CU /D8/CW/CT /D3/D6/CX/CT/D2 /D8/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CP/D8 /CW /DD/D4 /CT/D6/D4/D0/CP/D2/CT/B8 /DB/CW/CX
/CW /CT/D2/CP/CQ/D0/CT/D7 /D3/D2/CT/D8/D3
/CW/D3 /D3/D7/CT Σ /CP/D7 /D8/CW/CT /D4/D0/CP/CX/D2 t=a /CX/D2S /CP/D2/CSt′=b /CX/D2S′/BA /BU/D3/D8/CW /D7/D8/D6/CT/D7/D7/B9/CT/D2/CT/D6/CV/DD /D8/CT/D2/D7/D3/D6/D7 /CX/D2 /B4/BG/B5/B8Θµν/CP/D2/CS
Tµν/B8 /CP/D6/CT /D8/CP/CZ /CT/D2 /D7/DD/D2
/CW/D6/D3/D2/D3/D9/D7/D0/DD /CX/D2S /CP/D2/CSS′/B8 /CX/BA/CT/BA/B8 /CP/D8 /CP /D7/CX/D2/CV/D0/CT /D8/CX/D1/CT /CX/D2 /D8/CW/CT /D3/CQ/D7/CT/D6/DA /CT/D6/B3/D7 /CU/D6/CP/D1/CT/BA /BU/D9/D8/B8 /CP/D7/D7/CW/D3 /DB/D2 /CX/D2 /D8/CW/CT /D4/D6/CT
/CT/CS/CX/D2/CV /D7/CT
/D8/CX/D3/D2/B8
/D3/D2 /D8/D6/CP/D6/DD /D8/D3 /D8/CW/CT /CP/D7/D7/CT/D6/D8/CX/D3/D2/D7 /CX/D2[2,3,15,16] /B8 /B4/DB/CW/CX
/CW /CP/D6/CT /CQ/CP/D7/CT/CS /D3/D2 /DA /D3/D2/C4/CP/D9/CT/B3/D7 /D8/CW/CT/D3/D6/CT/D1/B5/B8 /D8/CW/CT /CX/D2 /D8/CT/CV/D6/CP/D0 /D3 /DA /CT/D6 /D8/CW/CT /CW /DD/D4 /CT/D6/D4/D0/CP/D2/CT t=const. /B8 /CX/BA/CT/BA/B8Pµ
tot.
/B4/BG/B5/B8 /CX/D7 /D2/D3/D8 /CP /D8/D6/D9/CT /BG/B9/DA /CT
/D8/D3/D6/B8/D7/CX/D2
/CT /D8/CW/CT /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CP/D8 /CX/D2 /D8/CT/CV/D6/CP/D0 /CX/D7 /B9 /CP/D2 /BT /CC/BA/BF/BA/BD /CC/CW/CT /BX/D0/CT
/D8/D6/D3/D1/CP/CV/D2/CT/D8/CX
/BV/D3/D1/D4 /D3/D2/CT/D2 /D8/CD/D7/CX/D2/CV /D7/D9
/CW /D7/DD/D2
/CW/D6/D3/D2/D3/D9/D7 /CS/CT/AS/D2/CX/D8/CX/D3/D2/D7/B8 /CX/BA/CT/BA/B8 /D8/CW/CT
/CW/D3/CX
/CT t=const. /CU/D3/D6 /D8/CW/CT /CW /DD/D4 /CT/D6/D7/D9/D6/CU/CP
/CT Σ /CX/D2 /D8/CW/CT /D3/CQ/B9/D7/CT/D6/DA/CT/D6/B3/D7 /CU/D6 /CP/D1/CT /B8 /CP/D2/CS
/D3/D2/D7/CX/CS/CT/D6/CX/D2/CV /D3/D2/D0/DD /D8/CW/CT /CT/D0/CT
/D8/D6/D3/D1/CP/CV/D2/CT/D8/CX
/D3/D1/D4 /D3/D2/CT/D2 /D8 /DB/CX/D8/CWTµν/CT/DC/D4/D6/CT/D7/D7/CT/CS /CX/D2 /D8/CT/D6/D1/D7/D3/CUE /CP/D2/CSB /B8 /D3/D2/CT /AS/D2/CS/D7 /D8/CW/CT
/D3/D2 /DA /CT/D2 /D8/CX/D3/D2/CP/D0 /CU/D3/D6/D1 /CU/D3/D6Pµ/D3/CU /D8/CW/CT /AS/CT/D0/CS /CX/D2 /CP/D2 /C1/BY/CAS
Pµ
f= (1/c)/integraldisplay
t=aTµ0(r, t)d3x /B4/BH/B5/CP/D2/CS /D8/CW/CT /D8/D6/CP/CS/CX/D8/CX/D3/D2/CP/D0 /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2/D7 /CU/D3/D6 /D8/CW/CT /AS/CT/D0/CS /CT/D2/CT/D6/CV/DD /CP/D2/CS /D1/D3/D1/CT/D2 /D8/D9/D1/BM
cP0
f=Uf= (ε0/2)/integraldisplay
t=a/bracketleftbig
E2(r, t) +c2B2(r, t)/bracketrightbig
d3x=/integraldisplay
t=au(r, t)d3x,
Pf=ε0/integraldisplay
t=aE(r, t)×B(r, t)d3x=/integraldisplay
t=ag(r, t)d3x. /B4/BI/B5/B4/C6/D3/D8/CT /D8/CW/CP/D8Pµ
f
/B4/BH/B9/BI/B5 /CX/D7 /D2/D3/D8 /CP /D0/CT/CV/CX/D8/CX/D1/CP/D8/CT /BG/B9/DA /CT
/D8/D3/D6 /CP/D2/CS /D8/CW/CT /D2/D3/D8/CP/D8/CX/D3/D2 /DB/CX/D8/CW /D8/CW/CT /D7/D9/D4 /CT/D6/D7
/D6/CX/D4/D8/D7 /CX/D7 /D2/D3/D8/CP/D4/D4/D6/D3/D4/D6/CX/CP/D8/CT/B8 /CQ/D9/D8 /CW/CT/D6/CT /DB /CT /D6/CT/D8/CP/CX/D2 /D7/D9
/CW /D2/D3/D8/CP/D8/CX/D3/D2 /CS/D9/CT /D8/D3 /CW/CX/D7/D8/D3/D6/CX
/CP/D0 /D6/CT/CP/D7/D3/D2/D7/BA/B5 /BY /D3/D6 /D8/CW/CT /CS/CT/D8/CT/D6/D1/CX/D2/CP/D8/CX/D3/D2/D3/CUP′
f
/CP/D2/CSP′0
f
/CX/D2S′/D3/D2/CT /D2/CT/CT/CS/D7 /D8/D3 /D4 /CT/D6/CU/D3/D6/D1/BM /BD/B5 /D8/CW/CT /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2/D7 /D3/CU /D8/CW/CT /CX/D2 /D8/CT/CV/D6/CP/D0/D7/B8 /CX/BA/CT/BA/B8 /D3/CU /D8/CW/CT/CW /DD/D4 /CT/D6/D7/D9/D6/CU/CP
/CT/D7 t=const. /B8 /DB/CW/CX
/CW /CP/D6/CT /D8/CW/CT /BT /CC/B8 /CP/D2/CS /BE/B5 /D8/CW/CT /BT /CC /B4/BD/B5 /D3/CUE /CP/D2/CSB /BA /CC/CW/CX/D7 /D1/CT/CP/D2/D7 /D8/CW/CP/D8/D8/CW/CT
/D3/D2 /DA /CT/D2 /D8/CX/D3/D2/CP/D0 /D7/DD/D2
/CW/D6/D3/D2/D3/D9/D7 /CS/CT/AS/D2/CX/D8/CX/D3/D2/D7 /D3/CU /D8/CW/CT /CT/D0/CT
/D8/D6/D3/D1/CP/CV/D2/CT/D8/CX
/CT/D2/CT/D6/CV/DD /CP/D2/CS /D1/D3/D1/CT/D2 /D8/D9/D1 /CP/D6/CT /D2/D3/D8 /CX/D2/CP
/D3/D6/CS/CP/D2
/CT /DB/CX/D8/CW /D8/CW/CT /AH/CC/CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/BN/AH /D8/CW/CT /CT/D2/CT/D6/CV/DD/B9 /D1/D3/D1/CT/D2 /D8/D9/D1 /B4/BH/B9/BI/B5 /CP/D7 /CP /D7/DD/D2
/CW/D6/D3/D2/D3/D9/D7/D0/DD /CS/CT/D8/CT/D6/D1/CX/D2/CT/CS/D2/D3/D2/D0/D3
/CP/D0 /D4/CW /DD/D7/CX
/CP/D0 /D5/D9/CP/D2 /D8/CX/D8 /DD /CX/D2 /CP/D2 /C1/BY/CA /CW/CP/D7 /D2/D3/D8/CW/CX/D2/CV /D8/D3 /CS/D3 /DB/CX/D8/CW /D8/CW/CT
/D3/D6/D6/CT/D7/D4 /D3/D2/CS/CX/D2/CV /D5/D9/CP/D2 /D8/CX/D8 /DD /D6/CT/D0/CT/DA /CP/D2 /D8/D8/D3 /D3/CQ/D7/CT/D6/DA /CT/D6/D7 /CX/D2 /CP/D2/D3/D8/CW/CT/D6 /C1/BY/CA/BA/BF/BA/BE /CC/CW/CT /C8 /D3/CX/D2
/CP/D6/GH /CB/D8/D6/CT/D7/D7/CT/D7 /CP/D2/CS /D8/CW/CT /C6/D3/D2/CT/D0/CT
/D8/D6/D3/D1/CP/CV/D2/CT/D8/CX
/BV/D3/D1/D4 /D3/D2/CT/D2 /D8/C1/D2 /CP/CS/CS/CX/D8/CX/D3/D2/B8 /CX/D8 /CW/CP/D7 /D8/D3 /CQ /CT /D1/CT/D2 /D8/CX/D3/D2/CT/CS /D8/CW/CP/D8 /D8/CW/CT /D3/D6/CX/CV/CX/D2 /CP/D2/CS /D2/CP/D8/D9/D6/CT /D3/CU /D8/CW/CT /C8 /D3/CX/D2
/CP/D6/GH /D7/D8/D6/CT/D7/D7/CT/D7/B8 /DB/CW/CX
/CW/CP/D6/CT /CX/D2
/D0/D9/CS/CT/CS /CX/D2Θµν
/D3/D1/D4 /D3/D2/CT/D2 /D8/B8 /CP/D6/CT /D9/D2/CZ/D2/D3 /DB/D2/B8 /CP/D2/CS/B8 /CX/D2 /CU/CP
/D8/B8 /C8 /D3/CX/D2
/CP/D6/GH /D7/D8/D6/CT/D7/D7/CT/D7 /CP/D6/CT /D2/D3/D8 /D1/CT/CP/D7/D9/D6/CP/CQ/D0/CT/D4/CW /DD/D7/CX
/CP/D0 /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7/BA /CB/D9
/CW /CP /D8/CW/CT/D3/D6/DD /DB/CX/D8/CW /C8 /D3/CX/D2
/CP/D6/GH/B3/D7 /D7/D8/D6/CT/D7/D7/CT/D7 /CX/D7 /D0/CX/CZ /CT /D8/CW/CT /D8/CW/CT/D3/D6/CX/CT/D7 /DB/CW/CX
/CW /D8/D6/CX/CT/CS /CX/D2 /D8/CW/CT/BI/D7/DD/D2
/CW/D6/D3/D2/D3/D9/D7 /CU/D3/D6/D1 /D9/D0/CP/D8/CX/D3/D2 /D3/CU /D2/D3/D2/D0/D3
/CP/D0 /D4/CW /DD/D7/CX
/CP/D0 /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7 /D8/D3 /D6/CT/D7/D3/D0/DA /CT /D8/CW/CT /D4/D6/D3/CQ/D0/CT/D1 /D3/CU /CT/D5/D9/CX/D0/CX/CQ/D6/CX/D9/D1 /CX/D2/D8/CW/CT /D7/D4 /CT
/CX/CP/D0 /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD /B8 /D4/CP/D6/D8/CX
/D9/D0/CP/D6/D0/DD /D8/CW/CT /D6/CX/CV/CW /D8/B9/CP/D2/CV/D0/CT/CS /D0/CT/DA /CT/D6 /D4/D6/D3/CQ/D0/CT/D1/B8 /CQ /DD /CX/D2 /D8/D6/D3 /CS/D9
/CX/D2/CV /D8/CW/CT /AS
/D8/CX/DA /CT /CT/D2/CT/D6/CV/DD
/D9/D6/D6/CT/D2 /D8/B8 /DA /D3/D2 /C4/CP/D9/CT/B3/D7 /CT/D2/CT/D6/CV/DD
/D9/D6/D6/CT/D2 /D8 /CJ/BD/BK ℄/BN /B4/D1/CP/D2 /DD /D3/D8/CW/CT/D6/D7 /D6/CT/D4 /CT/CP/D8/CT/CS /C4/CP/D9/CT/B3/D7 /CT/DC/D4/D0/CP/D2/CP/D8/CX/D3/D2/B8 /D7/CT/CT/B8 /CT/BA/CV/BA/B8 /CA/CT/CU/BA/BD/BL/B5/BA /BU/D9/D8/B8 /CP/D7 /BT/D6/CP/D2/D3/AR [20] /D7/D8/CP/D8/CT/CS /CX/D2 /CW/CX/D7 /D7/CT/DA /CT/D6/CT
/D6/CX/D8/CX
/CX/D7/D1 /D3/CU /DA /D3/D2 /C4/CP/D9/CT/B3/D7 /CT/DC/D4/D0/CP/D2/CP/D8/CX/D3/D2/BM /AH/CC/CW/CT /CT/D2/CT/D6/CV/DD
/D9/D6/D6/CT/D2 /D8 /CX/CS/CT/CP /D3/CU /DA /D3/D2 /C4/CP/D9/CT /CW/CP/D7 /D8/D3 /CV/D3 /D8/CW/CT /DB /CP /DD /D3/CU /D4/CW/D0/D3/CV/CX/D7/D8/D3/D2/B8 /CP/D2/CS /D8/CW/CT /CT/D8/CW/CT/D6/BA /C1/D8 /CX/D7 /CX/D2 /D8/CT/D6/CT/D7/D8/CX/D2/CV /CW/D3 /DB /D1/CP/D2/CW/CP/D7 /D8/D3 /CX/D2 /DA /CT/D2 /D8 /DA /CT/D6/DD /AS/D2/CT /AT/D9/CX/CS/D7 /DB/CW/CX
/CW
/CP/D6/D6/DD /CT/D2/CT/D6/CV/DD /CQ/D9/D8 /DB/CW/CX
/CW /CP/D6/CT /D3/D8/CW/CT/D6/DB/CX/D7/CT /D9/D2/D3/CQ/D7/CT/D6/DA /CP/CQ/D0/CT/BA/AH /BX/DC/CP
/D8/D0/DD/D8/CW/CT /D7/CP/D1/CT /D7/D8/CP/D8/CT/D1/CT/D2 /D8 /CX/D7 /CP/D4/D4/D6/D3/D4/D6/CX/CP/D8/CT /CU/D3/D6 /D8/CW/CT /C8 /D3/CX/D2
/CP/D6/GH /D7/D8/D6/CT/D7/D7/CT/D7/BA/C1/D8 /CX/D7 /CX/D2 /D8/CT/D6/CT/D7/D8/CX/D2/CV /D8/D3 /CS/CX/D7
/D9/D7/D7 /CX/D2 /D1/D3/D6/CT /CS/CT/D8/CP/CX/D0 /D8/CW/CT /CT/DC/D4/D0/CP/D2/CP/D8/CX/D3/D2/D7 /CV/CX/DA /CT/D2 /CU/D3/D6 /D8/CW/CT /D2/D3/D2/CT/D0/CT
/D8/D6/D3/D1/CP/CV/D2/CT/D8/CX
/D3/D1/D4 /D3/D2/CT/D2 /D8Θµν/CX/D2 /D8/CW/CT /D8/CW/CT/D3/D6/CX/CT/D7 [2,3,15,16] /BA /BU/D3 /DD /CT/D6[2]
/D3/D2/D7/CX/CS/CT/D6/D7 /CP /D7/D4/CW/CT/D6/CX
/CP/D0 /D7/CW/CT/D0/D0 /D3/CU
/CW/CP/D6/CV/CT /B4/CP/D7/CP /D1/D3 /CS/CT/D0 /D3/CU /D8/CW/CT
/D0/CP/D7/D7/CX
/CP/D0 /CT/D0/CT
/D8/D6/D3/D2/B5 /CX/D2 /CS/CX/AR/CT/D6/CT/D2 /D8 /C1/BY/CA/D7/BA /C0/CT
/D3/D2
/D0/D9/CS/CT/D7 /D8/CW/CP/D8/B8 /CX/D2 /CP/D2 /CX/D1/D4/D6/D3/D4 /CT/D6 /C1/BY/CA/B8/D8/CW/CT/D6/CT /CX/D7 /CP /D2/CT/D8 /D8/D6/CP/D2/D7/CU/CT/D6 /D3/CU /CT/D2/CT/D6/CV/DD /CP/D2/CS /D1/D3/D1/CT/D2 /D8/D9/D1 /CU/D6/D3/D1 /D8/CW/CT /D1/CT
/CW/CP/D2/CX
/CP/D0 /D7/D8/CP/CQ/CX/D0/CX/DE/CX/D2/CV /CU/D3/D6
/CT/D7 /D8/D3 /D8/CW/CT/CT/D0/CT
/D8/D6/D3/D1/CP/CV/D2/CT/D8/CX
/AS/CT/D0/CS /CP/D7 /CP
/D3/D2/D7/CT/D5/D9/CT/D2
/CT /D3/CU /D8/CW/CT /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD /D3/CU /D7/CX/D1 /D9/D0/D8/CP/D2/CT/CX/D8 /DD /B8 /CX/BA/CT/BA/B8 /CP/D7 /CP /D6/CT/D7/D9/D0/D8 /D3/CU /D8/CW/CT/D2/D3/D2/D7/CX/D1 /D9/D0/D8/CP/D2/CT/CX/D8 /DD /D3/CU /D8/CW/CT /AH
/D0/CP/D1/D4/CX/D2/CV/AH /D3/CU /D8/CW/CT /CU/D3/D6
/CT/D7 /D3/CU
/D3/D2/D7/D8/D6/CP/CX/D2 /D8 /D3/D2 /D8/CW/CT /D1/D3 /DA/CX/D2/CV
/CW/CP/D6/CV/CT/CS /D7/CW/CT/D0/D0/BA /C7/D2 /D8/CW/CT/D3/D8/CW/CT/D6 /CW/CP/D2/CS/B8 /CP
/D3/D6/CS/CX/D2/CV /D8/D3[2] /B8 /D8/CW/CT/D6/CT /CX/D7 /D2/D3 /D7/D9
/CW /D8/D6/CP/D2/D7/CU/CT/D6 /CX/D2 /D8/CW/CT /D6/CT/D7/D8 /CU/D6/CP/D1/CT /D7/CX/D2
/CT /D8/CW/CT/D6/CT /D8/CW/CT /CU/D3/D6
/CT/D7 /D3/CU
/D3/D2/D7/D8/D6/CP/CX/D2 /D8 /DB /CT/D6/CT /CP/D4/D4/D0/CX/CT/CS /D7/CX/D1 /D9/D0/D8/CP/D2/CT/D3/D9/D7/D0/DD /BA /C0/D3 /DB /CT/DA /CT/D6/B8 /DB /CT /D2/D3/D8/CT /D8/CW/CP/D8 /D8/CW/CT /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD /D3/CU /D7/CX/D1 /D9/D0/D8/CP/D2/CT/CX/D8 /DD /CX/D7 /D2/D3/D8/CP/D2 /CX/D2 /D8/D6/CX/D2/D7/CX
/CU/CT/CP/D8/D9/D6/CT /D3/CU /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD /B8 /CQ/D9/D8 /CX/D8 /CX/D7 /CP
/D3 /D3/D6/CS/CX/D2/CP/D8/CT /CS/CT/D4 /CT/D2/CS/CT/D2 /D8 /CT/AR/CT
/D8/B8 /CX/BA/CT/BA/B8 /CX/D8 /CS/CT/D4 /CT/D2/CS/D7 /D3/D2 /D8/CW/CT/CZ/CX/D2/CS /D3/CU /D7/DD/D2
/CW/D6/D3/D2/CX/DE/CP/D8/CX/D3/D2 /D4/D6/D3
/CT/CS/D9/D6/CT /CP/CS/D3/D4/D8/CT/CS/B8 /CP/D2/CS /D8/CW /D9/D7 /D3/D2 /D8/CW/CT /CP/CS/D3/D4/D8/CT/CS
/D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2 /D4/D6/D3
/CT/CS/D9/D6/CT/D3/CU /C5/CX/D2/CZ /D3 /DB/D7/CZ/CX /D7/D4/CP
/CT/D8/CX/D1/CT/BA /CC/CW/CX/D7 /D1/CT/CP/D2/D7 /D8/CW/CP/D8 /D8/CW/CT /CP/CQ /D3 /DA /CT /D1/CT/D2 /D8/CX/D3/D2/CT/CS /D8/D6/CP/D2/D7/CU/CT/D6 /D3/CU /CT/D2/CT/D6/CV/DD /CP/D2/CS /D1/D3/D1/CT/D2 /D8/D9/D1
[2] /B8 /DB/CX/D0/D0 /CS/CT/D4 /CT/D2/CS /D3/D2 /D8/CW/CT /CP/CS/D3/D4/D8/CT/CS /D7/DD/D2
/CW/D6/D3/D2/CX/DE/CP/D8/CX/D3/D2 /D4/D6/D3
/CT/CS/D9/D6/CT /CP/D2/CS /D4/CP/D6/D8/CX
/D9/D0/CP/D6/D0/DD /CU/D3/D6 /D8/CW/CT /D7/DD/D2
/CW/D6/D3/D2/CX/DE/CP/D8/CX/D3/D2/DB/CX/D8/CW /CP/CQ/D7/D3/D0/D9/D8/CT /D7/CX/D1 /D9/D0/D8/CP/D2/CT/CX/D8 /DD /B8 /B4/D7/CT/CT/B8 /CT/BA/CV/BA/B8 /CA/CT/CU/BA /BE/BD/B5/B8 /D8/CW/CT /D2/CT/D8 /D8/D6/CP/D2/D7/CU/CT/D6 /DB /D3/D9/D0/CS /D2/CT/CT/CS /D8/D3 /CS/CX/D7/CP/D4/D4 /CT/CP/D6/BA /CC/CW /D9/D7 /D8/CW/CT
/CW/CP/D2/CV/CT/D7 /CX/D2 /CT/D2/CT/D6/CV/DD /CP/D2/CS /D1/D3/D1/CT/D2 /D8/D9/D1 /D3/CU /D8/CW/CT /D1/D3 /DA/CX/D2/CV
/CW/CP/D6/CV/CT/CS /D7/CW/CT/D0/D0 /DB/CW/CX
/CW /D3 /DB/D2 /D8/CW/CT/CX/D6 /CT/DC/CX/D7/D8/CT/D2
/CT /D8/D3 /D8/CW/CT/D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD /D3/CU /D7/CX/D1 /D9/D0/D8/CP/D2/CT/CX/D8 /DD /CP/D6/CT/B8 /CX/D2 /CU/CP
/D8/B8 /D9/D2/D4/CW /DD/D7/CX
/CP/D0/BA /CC/CW/CT/D6/CT/CU/D3/D6/CT/B8 /D8/CW/CT /CP/D7/D7/CT/D6/D8/CX/D3/D2/D7 /CV/CX/DA /CT/D2 /CX/D2[2] /CP/CQ /D3/D9/D8 /D8/CW/CT/DA /CP/D0/CX/CS/CX/D8 /DD /CP/D2/CS /D8/CW/CT /D6/CT/D0/CP/D8/CX/DA/CX/D7/D8/CX
/D3/D6/D6/CT
/D8/D2/CT/D7/D7 /D3/CU /D8/CW/CT /D8/D6/CP/CS/CX/D8/CX/D3/D2/CP/D0 /CS/CT/AS/D2/CX/D8/CX/D3/D2/D7 /B4/BH/B9/BI/B5 /CX/D2 /CT/DA /CT/D6/DD /C1/BY/CA/B8 /D8/CP/CZ /CT/D2 /CP/D8 /CP/D7/CX/D2/CV/D0/CT /D8/CX/D1/CT /CX/D2 /D8/CW/CP/D8 /CU/D6/CP/D1/CT/B8 /CP/D6/CT /D9/D2/CU/D3/D9/D2/CS/CT/CS/BA/BU/D3 /DD /CT/D6 /D9/D7/CT/CS /D8/CW/CT /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD /D3/CU /D7/CX/D1 /D9/D0/D8/CP/D2/CT/CX/D8 /DD /D8/D3 /D3/CQ/D8/CP/CX/D2 /D8/CW/CT /CP/CS/CS/CX/D8/CX/D3/D2/CP/D0
/D3/D2 /D8/D6/CX/CQ/D9/D8/CX/D3/D2/D7 /D8/D3 /D8/CW/CT /AH/D4/D9/D6/CT/CT/D0/CT
/D8/D6/D3/D1/CP/CV/D2/CT/D8/CX
/AH /CT/D2/CT/D6/CV/DD /CP/D2/CS /D1/D3/D1/CT/D2 /D8/D9/D1 /B4/BH/B9/BI/B5 /CU/D3/D6 /CP /D1/D3 /DA/CX/D2/CV
/CW/CP/D6/CV/CT/CS /D7/DD/D7/D8/CT/D1/B8 /CX/BA/CT/BA/B8 /CS/D9/D6/CX/D2/CV /CP /D8/D6/CP/D2/D7/B9/CU/D3/D6/D1/CP/D8/CX/D3/D2 /CU/D6/D3/D1 /D8/CW/CT /D6/CT/D7/D8 /CU/D6/CP/D1/CT /D8/D3 /CP/D2/D3/D8/CW/CT/D6 /C1/BY/CA/BA /C1/D2/D7/D8/CT/CP/CS /D3/CU /D8/CW/CP/D8 /D8/CW/CT /CT/D2/CT/D6/CV/DD /CX/D2
/D6/CT/D1/CT/D2 /D8 /CX/D7 /CT/DC/D4/D0/CP/CX/D2/CT/CS/CX/D2[15] /CP/D7 /CP/D6/CX/D7/CX/D2/CV /CU/D6/D3/D1 /D8/CW/CT /CU/D3/D6
/CT/D7 /D3/CU
/D3/D2/D7/D8/D6/CP/CX/D2 /D8 /CS/D9/CT /D8/D3 /DB /D3/D6/CZ /CS/D3/D2/CT /CQ /DD /D8/CW/CT/D7/CT /CU/D3/D6
/CT/D7 /CS/D9/D6/CX/D2/CV /D8/CW/CT /C4/D3/D6/CT/D2 /D8/DE
/D3/D2 /D8/D6/CP
/D8/CX/D3/D2 /D3/CU /CP /D1/D3 /DA/CX/D2/CV
/CW/CP/D6/CV/CT/CS /D7/DD/D7/D8/CT/D1/BA /CC/CW/CT /CP/CS/CS/CX/D8/CX/D3/D2/CP/D0
/D3/D2 /D8/D6/CX/CQ/D9/D8/CX/D3/D2 /D8/D3 /D8/CW/CT /CT/D0/CT
/D8/D6/D3/D1/CP/CV/D2/CT/D8/CX
/D1/D3/B9/D1/CT/D2 /D8/D9/D1 /D3/CU /CP /D1/D3 /DA/CX/D2/CV
/CW/CP/D6/CV/CT/CS /D7/DD/D7/D8/CT/D1 /CX/D7 /CP/D7/D7/D3
/CX/CP/D8/CT/CS /CX/D2[15] /DB/CX/D8/CW /D8/CW/CT /CT/D2/CT/D6/CV/DD /AT/D3 /DB /B4/CP/D7 /DA /D3/D2 /C4/CP/D9/CT/B3/D7/CT/D2/CT/D6/CV/DD
/D9/D6/D6/CT/D2 /D8/B5 /CS/D9/CT /D8/D3 /CU/D3/D6
/CT/D7 /D3/CU
/D3/D2/D7/D8/D6/CP/CX/D2 /D8/BA /BU/D9/D8/B8 /CP/D7 /D7/CW/D3 /DB/D2 /CX/D2 /CB/CT
/BA /BE/B8 /D8/CW/CT /C4/D3/D6/CT/D2 /D8/DE
/D3/D2 /D8/D6/CP
/D8/CX/D3/D2 /CX/D7 /CP/D2/BT /CC/BA /CC/CW/CX/D7 /CU/CP
/D8/B8 /D8/D3/CV/CT/D8/CW/CT/D6 /DB/CX/D8/CW /D8/CW/CT /D9/D7/CT /D3/CU /D8/CW/CT /D7/DD/D2
/CW/D6/D3/D2/D3/D9/D7 /CS/CT/AS/D2/CX/D8/CX/D3/D2/D7 /B4/BH/B9/BI/B5/B8
/CP/D9/D7/CT/D7 /D8/CW/CP/D8 /D2/CT/CX/D8/CW/CT/D6 /D8/CW/CT/D8/D6/CT/CP/D8/D1/CT/D2 /D8 /CX/D2[15] /CS/D3
/D3/D2/CU/D3/D6/D1 /DB/CX/D8/CW /D8/CW/CT /AH/CC/CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD /BA/AH/C1/D2[16] /CP /D4/CP/D6/CP/D0/D0/CT/D0 /D4/D0/CP/D8/CT/D7
/CP/D4/CP
/CX/D8/D3/D6 /CP/D2/CS /CP /D9/D2/CX/CU/D3/D6/D1/D0/DD
/CW/CP/D6/CV/CT/CS /D7/D4/CW/CT/D6/CX
/CP/D0 /D7/CW/CT/D0/D0 /DB /CT/D6/CT
/D3/D2/D7/CX/CS/CT/D6/CT/CS/BA /C1/D2/CP /D7/CX/D1/CX/D0/CP/D6 /DB /CP /DD /CP/D7 /CX/D2[2] /CP/D2/CS[15] /B8 /CX/D8 /CX/D7 /CP/D8/D8/CT/D1/D4/D8/CT/CS /CX/D2[16] /D8/D3 /D3/CQ/D8/CP/CX/D2 Tµν
tot.
/B8 /DB/CW/CX
/CW /CX/D7 /CS/CX/DA /CT/D6/CV/CT/D2
/CT/D0/CT/D7/D7/BA /BT/D7/D4 /CT
/CX/CP/D0 /CV/CP/D7/CT/D3/D9/D7 /D7/D9/CQ/D7/D8/CP/D2
/CT /B4/CP /D7/D4 /CT
/CX/CP/D0 /CZ/CX/D2/CS /D3/CU /AH/D1/D3/D0/CT
/D9/D0/CT/D7/AH /D8/CW/CP/D8 /CP/D6/CT
/D3/D2/D7/CX/CS/CT/D6/CT/CS /CP/D7 /D1/D3 /DA/CX/D2/CV /D4/CP/D6/D8/CX
/D0/CT/D7/D3/D6 /CP/D7
/D3/D2 /D8/CX/D2 /D9/D3/D9/D7 /CV/CP/D7/CT/D3/D9/D7 /D7/D9/CQ/D7/D8/CP/D2
/CT/B5 /CX/D7 /CX/D2 /D8/D6/D3 /CS/D9
/CT/CS /CX/D2 /D3/D6/CS/CT/D6 /D8/D3 /D4/D6/D3 /DA/CX/CS/CT /D8/CW/CT /CP/CS/CS/CX/D8/CX/D3/D2/CP/D0 /CT/D2/CT/D6/CV/DD /CP/D2/CS/D1/D3/D1/CT/D2 /D8/D9/D1 /D2/CT/CT/CS/CT/CS /D8/D3 /CQ/CP/D0/CP/D2
/CT /D8/CW/CT /CT/D0/CT
/D8/D6/D3/D1/CP/CV/D2/CT/D8/CX
/CT/D2/CT/D6/CV/DD /CP/D2/CS /D1/D3/D1/CT/D2 /D8/D9/D1/BA /CF /CT /D6/CT/D1/CP/D6/CZ /D8/CW/CP/D8 /D8/CW/CX/D7/CV/CP/D7/CT/D3/D9/D7 /D7/D9/CQ/D7/D8/CP/D2
/CT /CX/D7 /D8/CW/CT /D7/CP/D1/CT /CZ/CX/D2/CS /D3/CU /D7/D9/CQ/D7/D8/CP/D2
/CT /CP/D7 /D8/CW/CP/D8 /D3/D2/CT /CX/D2 /DA /D3/D0/DA /CT/CS /CX/D2 /DA /D3/D2 /C4/CP/D9/CT/B3/D7 /CT/D2/CT/D6/CV/DD
/D9/D6/D6/CT/D2 /D8/B8/CX/BA/CT/BA/B8 /CP/D7 /BT/D6/CP/D2/D3/AR [20] /D7/D8/CP/D8/CT/D7/BM /AH/BA/BA/BA/BA/DA /CT/D6/DD /AS/D2/CT /AT/D9/CX/CS/D7 /DB/CW/CX
/CW
/CP/D6/D6/DD /CT/D2/CT/D6/CV/DD /CQ/D9/D8 /DB/CW/CX
/CW /CP/D6/CT /D3/D8/CW/CT/D6/DB/CX/D7/CT /D9/D2/D3/CQ/D7/CT/D6/DA/B9/CP/CQ/D0/CT/BA/AH /CA/CT/CP/D0/D0/DD /CX/D8 /CX/D7 /CP/D7/D7/D9/D1/CT/CS /CX/D2[16] /D8/CW/CP/D8 /D8/CW/CT /D1/CT/D2 /D8/CX/D3/D2/CT/CS /AH/D1/D3/D0/CT
/D9/D0/CT/D7/AH
/CP/D6/D6/DD /D8/CW/CT /CT/D2/CT/D6/CV/DD /CP/D2/CS /D1/D3/D1/CT/D2 /D8/D9/D1/CQ/D9/D8 /D8/CW/CT/DD /CS/D3 /D2/D3/D8 /CX/D2 /D8/CT/D6/CP
/D8 /DB/CX/D8/CW /D3/D2/CT /CP/D2/D3/D8/CW/CT/D6 /CP/D2/CS/BM /AH /CS/D3 /D2/D3/D8
/CW/CP/D2/CV/CT /D8/CW/CT /CS/CX/CT/D0/CT
/D8/D6/CX
/D4/D6/D3/D4 /CT/D6/D8/CX/CT/D7 /D3/CU /D8/CW/CT/D1/CT/CS/CX/D9/D1 /CQ /CT/D8 /DB /CT/CT/D2 /D8/CW/CT /D4/D0/CP/D8/CT/D7 /DB/CW/CX
/CW /CX/D7 /D8/CW/CT /D7/CP/D1/CT /CP/D7 /D8/CW/CP/D8 /D3/CU /D8/CW/CT /DA /CP
/D9/D9/D1/BA/AH/CF /CT /D7/CT/CT /D8/CW/CP/D8 /CX/D2 /D8/CW/CT
/D3/D2 /DA /CT/D2 /D8/CX/D3/D2/CP/D0 /CS/CT/AS/D2/CX/D8/CX/D3/D2/D7 /B4/BG/B5/B8 /B4/BH/B9/BI/B5 /D8/CW/CT /D8/D6/CT/CP/D8/D1/CT/D2 /D8 /D3/CU /D8/CW/CT /CT/D0/CT
/D8/D6/D3/D1/CP/CV/D2/CT/D8/CX
/D3/D1/D4 /D3/D2/CT/D2 /D8 /CX/D2
/D0/D9/CS/CT/D7 /D8/CW/CT /BT /CC /D3/CU /D8/CW/CT /CW /DD/D4 /CT/D6/D4/D0/CP/D2/CT t=const. /CP/D2/CS /D8/CW/CT /BT /CC /D3/CUE /CP/D2/CSB /B8 /DB/CW/CX/D0/CT /D8/CW/CT/D8/D6/CT/CP/D8/D1/CT/D2 /D8 /D3/CU /D8/CW/CT /D2/D3/D2/CT/D0/CT
/D8/D6/D3/D1/CP/CV/D2/CT/D8/CX
/D3/D1/D4 /D3/D2/CT/D2 /D8 /CX/D2 /D8/D6/D3 /CS/D9
/CT/D7 /CP/D2/CS /D9/D7/CT/D7 /D9/D2/D4/CW /DD/D7/CX
/CP/D0 /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7/BA/BG /CA /C7/C0/CA/C4/C1/BV/C0/B3/CB /BW/BX/BY/C1/C6/C1/CC/C1/C7/C6/CB /C7/BY /CC/C0/BX /BX/C4/BX/BV/CC/CA /C7/C5/BT /BZ/B9/C6/BX/CC/C1/BV /C5/C7/C5/BX/C6/CC/CD/C5 /BG/B9/CE/BX/BV/CC/C7/CA/C6/CT/DC/D8/B8 /DB /CT
/D3/D2/D7/CX/CS/CT/D6 /CA/D3/CW/D6/D0/CX
/CW/B3/D7 /CS/CT/AS/D2/CX/D8/CX/D3/D2 /D3/CU /D8/CW/CT /CT/D0/CT
/D8/D6/D3/D1/CP/CV/D2/CT/D8/CX
/D1/D3/D1/CT/D2 /D8/D9/D1 /BG/B9/DA /CT
/D8/D3/D6 /CP/D2/CS /D8/CW/CT /CP/D4/D4/D6/D3/B9/D4/D6/CX/CP/D8/CT /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2/D7 /D3/CU /D8/CW/CT /CT/D0/CT
/D8/D6/D3/D1/CP/CV/D2/CT/D8/CX
/CT/D2/CT/D6/CV/DD /CP/D2/CS /D1/D3/D1/CT/D2 /D8/D9/D1/BA /C1/D2/D7/D8/CT/CP/CS /D3/CU /D8/CW/CT /D8/D6/CP/CS/CX/D8/CX/D3/D2/CP/D0/B8/D7/DD/D2
/CW/D6/D3/D2/D3/D9/D7/B8 /CS/CT/AS/D2/CX/D8/CX/D3/D2/D7 /B4/BH/B9/BI/B5 /CX/D2 /D8/CW/CT /D3/CQ/D7/CT/D6/DA /CT/D6/B3/D7 /CU/D6/CP/D1/CT /CA/D3/CW/D6/D0/CX
/CW[22,23,1] /CS/CT/AS/D2/CT/D7 /D8/CW/CT /CT/D2/CT/D6/CV/DD /CP/D2/CS/D1/D3/D1/CT/D2 /D8/D9/D1 /D3/CU /D8/CW/CT /CT/D0/CT
/D8/D6/D3/D1/CP/CV/D2/CT/D8/CX
/AS/CT/D0/CS /CX/D2 /CP /D6/CT/D0/CP/D8/CX/DA/CX/D7/D8/CX
/D3 /DA /CP/D6/CX/CP/D2 /D8 /DB /CP /DD /BA /CC/CW/CT /CV/CT/D2/CT/D6/CP/D0 /D1/CP/D2/CX/CU/CT/D7/D8/D0/DD
/D3 /DA /CP/D6/CX/CP/D2 /D8 /CS/CT/AS/D2/CX/D8/CX/D3/D2 /D3/CU /D8/CW/CT /CT/D0/CT
/D8/D6/D3/D1/CP/CV/D2/CT/D8/CX
/BG/B9/D1/D3/D1/CT/D2 /D8/D9/D1 /CX/D2 /CP/D2 /DD /C1/BY/CA /CX/D7 /CV/CX/DA /CT/D2 /CQ /DD /B4/BG/B5 /DB/CX/D8/CW/D3/D9/D8 /D8/CW/CT/BJ/D2/D3/D2/CT/D0/CT
/D8/D6/D3/D1/CP/CV/D2/CT/D8/CX
/D4/CP/D6/D8/B8 /CX/BA/CT/BA/B8 /CP/D7
Pµ
f= (1/c)/integraldisplay
ΣTµν(x)d3σν(x), /B4/BJ/B5/DB/CX/D8/CW /D8/CW/CT /D7/CP/D1/CT /D1/CT/CP/D2/CX/D2/CV /D3/CU /D7/DD/D1 /CQ /D3/D0/D7 /CP/D7 /CX/D2 /B4/BG/B5/BA /CC/CW/CT /CT/D2/CT/D6/CV/DD /D1/D3/D1/CT/D2 /D8/D9/D1 /D8/CT/D2/D7/D3/D6 Tµν/CX/D7 /CV/CX/DA /CT/D2 /CX/D2 /D8/CT/D6/D1/D7/D3/CUFµν/CP/D7
Tµν=ε0/bracketleftbig
FµαFν
α+ (1/4)gµνFαβFαβ./bracketrightbig/B4/BK/B5/CA/D3/CW/D6/D0/CX
/CW[1,22,23] /CS/CT/AS/D2/CT/CS /D8/CW/CT /CT/D0/CT
/D8/D6/D3/D1/CP/CV/D2/CT/D8/CX
/CT/D2/CT/D6/CV/DD/B9/D1/D3/D1/CT/D2 /D8/D9/D1 /CU/D3/D6 /CP /D7/DD/D7/D8/CT/D1 /CX/D2 /D9/D2/CX/CU/D3/D6/D1 /D1/D3/D8/CX/D3/D2/CP/D7 /D8/CW/CT /C4/D3/D6/CT/D2 /D8/DE /CQ /D3 /D3/D7/D8/CT/CS /D6/CT/D7/D8 /CU/D6/CP/D1/CT/B8 /CP/D2/CS /D8/CW/CT /CW /DD/D4 /CT/D6/D4/D0/CP/D2/CT /D3/CU /CX/D2 /D8/CT/CV/D6/CP/D8/CX/D3/D2 /CX/D7 /D7/D4 /CT
/CX/AS/CT/CS /D8/D3 /CQ /CT /D8/CW/CT /D4/D0/CP/D2/CT /CX/D2/DB/CW/CX
/CW /D8/CW/CT /D7/DD/D7/D8/CT/D1 /CX/D7 /CP/D8 /D6/CT/D7/D8/BA /C1/D2 /D7/D9
/CW /CP /DB /CP /DD /CX/D8 /CX/D7 /CP
/CW/CX/CT/DA /CT/CS /D8/CW/CP/D8Pµ
f
/CX/D7 /CP /BG/B9/DA /CT
/D8/D3/D6 /CT/DA /CT/D2 /D8/CW/D3/D9/CV/CW /D8/CW/CT/D6/CT /CP/D6/CT/D7/D3/D9/D6
/CT/D7 /D4/D6/CT/D7/CT/D2 /D8/BA /C1/D2 /D8/CW/CT /C1/BY/CAS(0)
/CX/D2 /DB/CW/CX
/CW /D8/CW/CT /D7/DD/D7/D8/CT/D1 /CX/D7 /CP/D8 /D6/CT/D7/D8 /CA/D3/CW/D6/D0/CX
/CW
/CW/D3 /D3/D7/CT/D7 /D8/CW/CT /CW /DD/D4 /CT/D6/D7/D9/D6/CU/CP
/CT
Σ /CX/D2 /B4/BJ/B5 /D8/D3 /CQ /CT /D8/CW/CT /D4/D0/CP/D2/CT Σ(0)=t(0)=const. /BA /CC/CW/CT /D7/CP/D1/CT Σ /CX/D7
/D3/D2/D7/CX/CS/CT/D6 /CT /CS /CU/D6 /D3/D1 /CP/D0 /D0 /D3/D8/CW/CT/D6 /C1/BY/CA/D7 /CP/D7/D6 /CT /D5/D9/CX/D6 /CT /CS /CQ/DD /D8/CW/CT
/D3/DA/CP/D6/CX/CP/D2/D8 /CP/D4/D4/D6 /D3 /CP
/CW/B8 /CX/BA/CT/BA/B8 /CX/D2 /D7/D3/D1/CT /C1/BY/CAS /D3/CQ/D8/CP/CX/D2/CT/CS /CQ /DD /D8/CW/CT /C4/D3/D6/CT/D2 /D8/DE /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2 L/CU/D6/D3/D1S(0)
/B8Σ /CX/D2 /B4/BJ/B5 /CX/D7LΣ(0)
/BAd3σν
/CX/D2 /CP/D2 /C1/BY/CAS
/CP/D2 /CQ /CT /DB/D6/CX/D8/D8/CT/D2 /CP/D7d3σν=nνd3σ /B8 /DB/CW/CT/D6/CT nν/CQ /CT/CX/D2/CV/CP /D8/CX/D1/CT/D0/CX/CZ /CT /DA /CT
/D8/D3/D6 /D2/D3/D6/D1/CP/D0 /D8/D3 /D8/CW/CT /CW /DD/D4 /CT/D6/D4/D0/CP/D2/CT Σ /B8nν= Γ(1 , β) /B8 /B4β=V/c /B8V /CX/D7 /D8/CW/CT /BF/B9/DA /CT/D0/D3
/CX/D8 /DD /D3/CU /D8/CW/CT/CU/D6/CP/D1/CT S /D6/CT/D0/CP/D8/CX/DA /CT /D8/D3S(0)
/B5/BA /C1/D2S(0)nµ
(0)= (1,0) /BAd3σ /CX/D7 /CP/D2 /CX/D2 /DA /CP/D6/CX/CP/D2 /D8 /CP/D2/CS /CX/D8 /CX/D7=dV(0)
/B8 /DB/CW/CT/D6/CT dV(0)/CX/D7 /D8/CW/CT /CX/D2/AS/D2/CX/D8/CT/D7/CX/D1/CP/D0 /CT/D0/CT/D1/CT/D2 /D8 /D3/CU /D8/CW/CT /BF/BW /D7/D4/CP/D8/CX/CP/D0 /DA /D3/D0/D9/D1/CT V(0)
/B8 /CX/BA/CT/BA/B8 /D3/CU /D8/CW/CT /BF/BW /CW /DD/D4 /CT/D6/D4/D0/CP/D2/CT t(0)=const. /B8/CX/D2 /D8/CW/CT /D6/CT/D7/D8 /CU/D6/CP/D1/CT S(0)
/D3/CU /D8/CW/CT /D7/DD/D7/D8/CT/D1/BA /CF /CT /D7/CT /CT /D8/CW/CP/D8 /CX/D2 /CA /D3/CW/D6/D0/CX
/CW/B3/D7 /CP/D4/D4/D6 /D3 /CP
/CW /D3/D2/CT /CP/D0/DB/CP/DD/D7 /CX/D2/D8/CT /CV/D6 /CP/D8/CT/D7/D3/DA/CT/D6 /D8/CW/CT /CW/DD/D4 /CT/D6/D4/D0/CP/D2/CT /DB/CW/CX
/CW /CX/D7 /D8/CW/CT /D8/D6 /CP/D2/D7/CU/D3/D6/D1/CT /CS /D8/CW/D6 /CT /CT /D7/D4 /CP
/CT /D3/CU /D8/CW/CT /D6 /CT/D7/D8 /CU/D6 /CP/D1/CT/BA /C7/CQ /DA/CX/D3/D9/D7/D0/DD /B8 /CP
/D3/D6/CS/CX/D2/CV/D8/D3 /D8/CW/CT
/D3/D2/D7/D8/D6/D9
/D8/CX/D3/D2/B8 /CX/BA/CT/BA/B8 /D7/CX/D2
/CT /CP/D0/DB /CP /DD/D7 /D8/CW/CT /D7/CP/D1/CT /D4/CW /DD/D7/CX
/CP/D0 /D5/D9/CP/D2 /D8/CX/D8 /DD /CX/D7
/D3/D2/D7/CX/CS/CT/D6/CT/CS /CU/D6/D3/D1 /CS/CX/AR/CT/D6/CT/D2 /D8 /C1/BY/CA/D7/CP/D2/CS /D3/D2/D0/DD /D8/CW/CT /CC/CC /CP/D6/CT /D9/D7/CT/CS/B8 /CX/D8 /CX/D7 /CU/D3/D9/D2/CS /D8/CW/CP/D8 /D8/CW/CT /D5/D9/CP/D2 /D8/CX/D8 /DDPµ
f
/B4/BJ/B5 /CX/D7 /CP /D0/CT/CV/CX/D8/CX/D1/CP/D8/CT /BG/B9/DA /CT
/D8/D3/D6/BA /C0/D3 /DB /CT/DA /CT/D6/B8/DB/CW/CT/D2 /CA/D3/CW/D6/D0/CX
/CW /CT/DC/D4/D0/CX
/CX/D8/D0/DD
/CP/D0
/D9/D0/CP/D8/CT/D7 /B4/BJ/B5 /CU/D3/D6 /D7/D4 /CT
/CX/AS
/D4/CW /DD/D7/CX
/CP/D0 /D7/DD/D7/D8/CT/D1 /CW/CT /CP/D0/D7/D3 /DB/D6/CX/D8/CT/D7/B8 /CP/D7 /CP/D0/D0 /D3/D8/CW/CT/D6/D7/B8 /D8/CW/CT/CT/D0/CT
/D8/D6/D3/D1/CP/CV/D2/CT/D8/CX
/CT/D2/CT/D6/CV/DD /CP/D2/CS /D1/D3/D1/CT/D2 /D8/D9/D1 /CS/CT/D2/D7/CX/D8/CX/CT/D7 /CX/D2 /D8/CT/D6/D1/D7 /D3/CU /D8/CW/CT /BF/B9/DA /CT
/D8/D3/D6/D7 E /CP/D2/CSB /CP/D2/CS /D9/D7/CT/D7 /D8/CW/CT/BT /CC /B4/BD/B5/BA/BH /BX/C4/BX/BV/CC/CA /C7/C5/BT /BZ/C6/BX/CC/C1/BV /C5/C7/C5/BX/C6/CC/CD/C5 /BG/B9/CE/BX/BV/CC/C7/CA /C1/C6/CC/BX/CA/C5/CB /C7/BYEα/BT/C6/BW Bα/C1/D2 /D3/D6/CS/CT/D6 /D8/D3 /D6/CT/D1/D3 /DA /CT /D8/CW/CT /D0/CP/D7/D8 /CT/D0/CT/D1/CT/D2 /D8 /CU/D6/D3/D1 /D8/CW/CT /D8/CW/CT/D3/D6/DD /D8/CW/CP/D8 /CX/D7 /D2/D3/D8 /CX/D2 /CP
/D3/D6/CS/CP/D2
/CT /DB/CX/D8/CW /D8/CW/CT /AH/CC/CC/D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD /B8/AH /DB /CT /DB/D6/CX/D8/CT Tµν/CP/D2/CSPµ
f
/CX/D2 /D8/CT/D6/D1/D7 /D3/CU
/D3 /DA /CP/D6/CX/CP/D2 /D8 /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7 Eα/CP/D2/CSBα/BA /CC/CW/CT/D2 /DB /CT
/D3/D1/D4/CP/D6/CT/D7/D3 /D3/CQ/D8/CP/CX/D2/CT/CS Tµν/CP/D2/CSPµ
f
/DB/CX/D8/CW /D8/D6/CP/CS/CX/D8/CX/D3/D2/CP/D0 /CS/CT/AS/D2/CX/D8/CX/D3/D2/D7 /D3/CU /D8/CW/CT /CT/D0/CT
/D8/D6/D3/D1/CP/CV/D2/CT/D8/CX
/CT/D2/CT/D6/CV/DD /CP/D2/CS /D1/D3/D1/CT/D2 /D8/D9/D1/B4/BH/B9/BI/B5 /CP/D2/CS /DB/CX/D8/CW /CA/D3/CW/D6/D0/CX
/CW/B3/D7 /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2/D7 /B4/BF/BA/BE/BF/B5 /CP/D2/CS /B4/BF/BA/BE/BG/B5 /CX/D2[23] /D3/D6 /DB/CX/D8/CW /D8/CW/CT /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2/D7 /CU/D3/D6P0
e
/CP/D2/CS
Pk
e
/D3/CQ/D8/CP/CX/D2/CT/CS /CX/D2[1] /CU/D3/D6 /D8/CW/CT
/CW/D3/CX
/CT /B4/C1 /C1/B5/BA/CD/D7/CX/D2/CV /D8/CW/CT /D6/CT/D0/CP/D8/CX/D3/D2/D7 /B4/BF/B5 /DB /CT /CT/DC/D4/D6/CT/D7/D7 Fαβ/CP/D2/CS /D8/CW /D9/D7 /CP/D0/D7/D3Tµν/B4/BJ/B5 /CX/D2 /D8/CT/D6/D1/D7 /D3/CUEα/CP/D2/CSBα/BA /CC/CW/CT/D3/CQ/D8/CP/CX/D2/CT/CS
/D3 /DA /CP/D6/CX/CP/D2 /D8 /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2 /CU/D3/D6 /D8/CW/CT /D7/DD/D1/D1/CT/D8/D6/CX
/CT/D2/CT/D6/CV/DD/B9/D1/D3/D1/CT/D2 /D8/D9/D1 /CS/CT/D2/D7/CX/D8 /DD /D8/CT/D2/D7/D3/D6 Tµν/CX/D7 /D8/CW/CT /CU/D3/D0/B9/D0/D3 /DB/CX/D2/CV
Tµν=ε0/bracketleftbig
−((gµν/2) +vµvν/c2)(EαEα+c2BαBα)
+EµEν+c2BµBν/B4/BL/B5
+(1/c)εµαβγBα(vνvγEβ−Eνvβvγ)
+(1/c)εναβγBα(vµvγEβ−Eµvβvγ)/bracketrightbig/C1/D2 /D8/D6/D3 /CS/D9
/CX/D2/CV /B4/BL/B5 /CX/D2 /D8/D3 /B4/BJ/B5 /D3/D2/CT /AS/D2/CS/D7 /D8/CW/CT /CT/DC/D4/D0/CX
/CX/D8 /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2 /CU/D3/D6 /D8/CW/CT
/D3 /DA /CP/D6/CX/CP/D2 /D8/D0/DD /CS/CT/AS/D2/CT/CS /CT/D0/CT
/D8/D6/D3/D1/CP/CV/D2/CT/D8/CX
/BG/B9/D1/D3/D1/CT/D2 /D8/D9/D1Pµ
f
/BA /BT/D0/D0 /D4/CP/D6/D8/D7 /D3/CU /D8/CW/CT /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2 /B4/BJ/B5 /DB/CX/D8/CWTµν/CU/D6/D3/D1 /B4/BL/B5 /CP/D6/CT
/D3 /DA /CP/D6/CX/CP/D2 /D8/D0/DD /CS/CT/AS/D2/CT/CS /D5/D9/CP/D2/B9/D8/CX/D8/CX/CT/D7 /DB/CW/CX
/CW /D8/D6/CP/D2/D7/CU/D3/D6/D1 /CP
/D3/D6/CS/CX/D2/CV /D8/D3 /D8/CW/CT /CC/CC/B8 /CX/BA/CT/BA/B8 /CP
/D3/D6/CS/CX/D2/CV /D8/D3 /D8/CW/CT /C4 /CC/BN /CX/D2 /CP/D2/D3/D8/CW/CT/D6 /C1/BY/CAS′/D1/D3 /DA/CX/D2/CV/DB/CX/D8/CW /D8/CW/CT /BG/B9/DA /CT/D0/D3
/CX/D8 /DDVµ/D6/CT/D0/CP/D8/CX/DA /CT /D8/D3S /D8/CW/CT /BG/B9/D1/D3/D1/CT/D2 /D8/D9/D1 /B4/BJ/B5 /DB/CX/D8/CWTµν/CS/CT/AS/D2/CT/CS /CQ /DD /B4/BL/B5 /DB/CX/D0/D0 /CW/CP /DA /CT /D8/CW/CT/D7/CP/D1/CT /CU/D3/D6/D1 /CQ/D9/D8 /DB/CX/D8/CW /D4/D6/CX/D1/CT/CS /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7 /D6/CT/D4/D0/CP
/CX/D2/CV /D8/CW/CT /D9/D2/D4/D6/CX/D1/CT/CS /D3/D2/CT/D7/BA/BH/BA/BD Pµ
f
/CX/D2S(0)
/CP/D2/CSS/C4/CT/D8 /D9/D7 /D2/D3 /DB /D9/D7/CT /D8/CW/CT /CP/CQ /D3 /DA /CT /D1/CP/D2/CX/CU/CT/D7/D8/D0/DD
/D3 /DA /CP/D6/CX/CP/D2 /D8 /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2 /CU/D3/D6Pµ
f
/DB/CX/D8/CWEα/CP/D2/CSBα/D8/D3 /CT/DC/CP/D1/CX/D2/CT /D7/D3/D1/CT/D7/D4 /CT
/CX/AS
/CP/D7/CT/D7
/D3/D2/D7/CX/CS/CT/D6/CT/CS /CX/D2[1] /CP/D2/CS[2] /BA /BY/CX/D6/D7/D8 /DB /CT /DB/D6/CX/D8/CT Tµν
(0)
/CP/D2/CSPµ
f(0)
/CX/D2S(0)
/B8 /D8/CW/CT /D6/CT/D7/D8 /CU/D6/CP/D1/CT /D3/CU /D8/CW/CT
/CW/CP/D6/CV/CT/CS /D7/D4/CW/CT/D6/CT/B8 /CX/BA/CT/BA/B8 /DB/CW/CT/D2 uα/B4/D8/CW/CT /BG/B9/DA /CT/D0/D3
/CX/D8 /DD /D3/CU /D8/CW/CT
/CW/CP/D6/CV/CT/B5 /CX/D7uα= (c,0) /B8 /DB/CW/CT/D2
/CT n(0)ν= (−1,0) /BA/BK/CC/CW/CT /D3/CQ/D7/CT/D6/DA /CT/D6/D7 /DB/CW/D3 /D1/CT/CP/D7/D9/D6/CT Eα
(0)
/CP/D2/CSBα
(0)
/CP/D6/CT /D8/CP/CZ /CT/D2 /D8/D3 /CQ /CT /CP/D8 /D6/CT/D7/D8 /CX/D2S(0)
/B8 /CP/D2/CS /D8/CW /D9/D7vα/B4/D8/CW/CT /BG/B9/DA /CT/D0/D3
/CX/D8 /DD/D3/CU /D8/CW/CT /D3/CQ/D7/CT/D6/DA /CT/D6/D7/B5 /CX/D7vα=uα= (c,0) /BA /BY /D3/D6 /D7/D9
/CW /D3/CQ/D7/CT/D6/DA /CT/D6/D7 /CX/D2S(0)
/D3/D2/CT /AS/D2/CS/D7E0
(0)= 0 /B8B0
(0)= 0 /CP/D2/CS/D3/D2/D0/DDEi
(0)/negationslash= 0 /BA /BY /D6/D3/D1
Pµ
f(0)= (1/c)/integraldisplay
Σ(0)Tµν
(0)n(0)νdV(0), /B4/BD/BC/B5/DB /CT /AS/D2/CS /D8/CW/CP/D8
P0
f(0)=−(1/c)/integraldisplay
Σ(0)T00
(0)dV(0), Pi
f(0)=−(1/c)/integraldisplay
Σ(0)Ti0
(0)dV(0)= 0, /B4/BD/BD/B5/DB/CW/CT/D6/CT /CU/D6/D3/D1 /B4/BL/B5 /D8/CW/CT
/D3/D1/D4 /D3/D2/CT/D2 /D8/D7 /D3/CUTµν
(0)
/CP/D2 /CQ /CT /CT/DC/D4/D6/CT/D7/D7/CT/CS /CX/D2 /D8/CT/D6/D1/D7 /D3/CUEα
(0)
/CP/D2/CSBα
(0)
/CP/D7T00
(0)=
−uE(0)=−(ε0/2)(Ei
(0)E(0)i) /B8Ti0
(0)=T0i
(0)= 0 /B8T11
(0)=ε0(E1
(0))2−uE(0)
/B8 /CP/D2/CS /D7/CX/D1/CX/D0/CP/D6/D0/DD /CU/D3/D6T22
(0)
/CP/D2/CS
T33
(0)
/DB/CX/D8/CW /CX/D2/CS/CT/DC/CT/D7 /BE /CP/D2/CS /BF /D6/CT/D4/D0/CP
/CX/D2/CV /D8/CW/CT /CX/D2/CS/CT/DC /BD/B8 /CP/D2/CSTij
(0)=Tji
(0)=ε0Ei
(0)Ej
(0)
/B8 /DB/CX/D8/CWi/negationslash=j /BAEi
(0)
/CP/D6/CT/D8/CW/CT /D7/D4/CP
/CT
/D3/D1/D4 /D3/D2/CT/D2 /D8/D7 /D3/CU /D8/CW/CT /BG/B9/DA /CT
/D8/D3/D6Eα
(0)
/CP/D2/CS /D8/CW/CT/DD
/D3/D6/D6/CT/D7/D4 /D3/D2/CS /D8/D3 /D8/CW/CT /D9/D7/D9/CP/D0 /BV/D3/D9/D0/D3/D1 /CQ /AS/CT/D0/CS/BA /CC/CW /D9/D7/D8/CW/CT /D7/CT/D0/CU/B9/CT/D2/CT/D6/CV/DD /CS/D9/CT /D8/D3 /D8/CW/CT /BV/D3/D9/D0/D3/D1 /CQ /AS/CT/D0/CSUf(0)
/CP/D2/CS /D8/CW/CT /D7/D4/CP
/CT /D4/CP/D6/D8Pi
f(0)
/CP/D6/CT
Uf(0)=cP0
f(0)=/integraldisplay
Σ(0)uE(0)dV(0)= (ε0/2)/integraldisplay
Σ(0)Ei
(0)E(0)idV(0),
Pi
f(0)= 0 /B4/BD/BE/B5/C1/D2 /D3/D6/CS/CT/D6 /D8/D3 /AS/D2/CS /D8/CW/CT /BG/B9/D1/D3/D1/CT/D2 /D8/D9/D1Pµ
f
/B8 /BX/D5/BA /B4/BJ/B5/B8 /CX/D2 /CP/D2/D3/D8/CW/CT/D6 /C1/BY/CAS /D1/D3 /DA/CX/D2/CV /DB/CX/D8/CW /D8/CW/CT /BG/B9/DA /CT/D0/D3
/CX/D8 /DD
Vα= (Γc,ΓV,0,0) /D6/CT/D0/CP/D8/CX/DA /CT /D8/D3S(0)
/D3/D2/CT
/CP/D2 /CT/CX/D8/CW/CT/D6 /D8/D6/CP/D2/D7/CU/D3/D6/D1 Pµ
f(0)
/CP/D7 /CP /BG/B9/DA /CT
/D8/D3/D6 /CU/D6/D3/D1S(0)
/D8/D3S /B8/D3/D6 /D8/D3 /D8/D6/CP/D2/D7/CU/D3/D6/D1 /CP/D0/D0 /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7 /D3/D2 /D8/CW/CT /D6/CX/CV/CW /D8/B9/CW/CP/D2/CS /D7/CX/CS/CT /D3/CU /B4/BD/BC/B5 /CU/D6/D3/D1S(0)
/D8/D3S /BA /CC/CW/CT /D7/CP/D1/CT /D6/CT/D7/D9/D0/D8 /CX/D7/D3/CQ/D8/CP/CX/D2/CT/CS /CP/D2/CS /CX/D8 /CX/D7
P0
f= ΓP0
f(0), P1
f=−βΓP0
f(0), P2
f=P3
f= 0. /B4/BD/BF/B5/C6/D3/D8/CT /D8/CW/CP/D8 /D8/CW/CT /D7/CP/D1/CT /CU/CP/D1/CX/D0/DD /D3/CU /D3/CQ/D7/CT/D6/DA /CT/D6/D7 /DB/CW/D3 /D1/CT/CP/D7/D9/D6/CT/D7 Eα
(0)
/CP/D2/CSBα
(0)
/CX/D2S(0)
/CX/D7
/D3/D2/D7/CX/CS/CT/D6/CT/CS /CX/D2 /CP/D0/D0/D3/D8/CW/CT/D6 /C1/BY/CA/D7/BA /CF /CT/CT /D7/CT/CT /D8/CW/CP/D8/B8 /DB/CW/CT/D2 /D6/CT/CU/CT/D6/D6/CT/CS /D8/D3 /D8/CW/CT /CX/D2 /DA /CP/D6/CX/CP/D2 /D8 /BF/BW /CX/D2 /D8/CT/CV/D6/CP/D8/CX/D3/D2 /DA /D3/D0/D9/D1/CT dV(0)
/B8 /D8/CW/CT /CT/D2/CT/D6/CV/DD/CS/CT/D2/D7/CX/D8 /DDuE
/CX/D2 /D8/CW/CT /C1/BY/CAS /D1/D3 /DA/CX/D2/CV /DB/CX/D8/CWVα= (Γc,ΓV,0,0) /D6/CT/D0/CP/D8/CX/DA /CT /D8/D3S(0)
/CX/D7ΓuE(0)
/B8
/D3/D2 /D8/D6/CP/D6/DD /D8/D3 /D8/CW/CT/D6/CT/D7/D9/D0/D8/D7 /CX/D2[1,22,23] /BA/BH/BA/BE /CC/CW/CT /B4/AH/BG/BB/BF/AH/B5 /BY /CP
/D8/D3/D6/CF /CT
/CP/D2 /D9/D7/CT /D8/CW/CT/D7/CT /D6/CT/D7/D9/D0/D8/D7 /D8/D3 /CS/CX/D7
/D9/D7/D7 /D8/CW/CT /CU/CP/D1/D3/D9/D7 /AH/BG/BB/BF/AH /CU/CP
/D8/D3/D6 /CP/D4/D4 /CT/CP/D6/CX/D2/CV /CX/D2 /D8/CW/CT /D4/D6/D3/CQ/D0/CT/D1 /D3/CU /D8/CW/CT/CT/D0/CT
/D8/D6/D3/D1/CP/CV/D2/CT/D8/CX
/D1/CP/D7/D7 /D3/CU /D8/CW/CT
/D0/CP/D7/D7/CX
/CP/D0 /CT/D0/CT
/D8/D6/D3/D2/B8 /B4/D7/CT/CT/B8 /CT/BA/CV/BA/B8 /CA/CT/CU/D7/BA /BD/B9/BF/B5/BA /C4/CT/D8 /D9/D7 /D7/D9/D4/D4 /D3/D7/CT /D8/CW/CP/D8 /CX/D2 /D8/CW/CT/D6/CT/D7/D8 /D7/DD/D7/D8/CT/D1 S(0)
/D8/CW/CT /DB/CW/D3/D0/CT /D1/CP/D7/D7 m /D3/CU /D8/CW/CT /CT/D0/CT
/D8/D6/D3/D2 /B4
/D3/D2/D7/CX/CS/CT/D6/CT/CS /CP/D7 /CP /D7/D4/CW/CT/D6/CT /D3/CU /D6/CP/CS/CX/D9/D7 R /DB/CX/D8/CW /CP/D9/D2/CX/CU/D3/D6/D1 /D7/D9/D6/CU/CP
/CT
/CW/CP/D6/CV/CT /CS/CT/D2/D7/CX/D8 /DD/B5 /CX/D7 /CS/D9/CT /D8/D3 /CT/D0/CT
/D8/D6/D3/D7/D8/CP/D8/CX
/CT/D2/CT/D6/CV/DD /D3/CU /D8/CW/CT /AS/CT/D0/CS/BA /CD/D7/CX/D2/CV /D8/CW/CT /D8/D6/CP/CS/CX/D8/CX/D3/D2/CP/D0/B8/D7/DD/D2
/CW/D6/D3/D2/D3/D9/D7/B8 /CS/CT/AS/D2/CX/D8/CX/D3/D2/D7 /B4/BH/B9/BI/B5/B8 /D3/D2/CT /AS/D2/CS/D7 /D8/CW/CP/D8 /CX/D2S(0)cP0
f(0)=mc2=Uf(0)
/B8 /CP/D2/CSPf(0)= 0 /BA /C1/D2 /CP/D2/C1/BY/CAS /CX/D2 /DB/CW/CX
/CW /D8/CW/CT /D4/CP/D6/D8/CX
/D0/CT /D1/D3 /DA /CT/D7 /DB/CX/D8/CW /D8/CW/CT /DA /CT/D0/D3
/CX/D8 /DDu /D3/D2/CT /D3/CQ/D8/CP/CX/D2/D7 /CU/D6/D3/D1 /B4/BH/B9/BI/B5P0
f= Γm(1 +u2/3) /B8/CP/D2/CSPf= (4/3)Γmu /B8 /B4/D7/CT/CT /CT/D5/CP/D8/CX/D3/D2/D7 /B4/BD/BD/B5 /CP/D2/CS /B4/BD/BE/B5 /CX/D2 /CA/D3/CW/D6/D0/CX
/CW/B3/D7
/D6/CX/D8/CX
/CX/D7/D1 [1] /D3/CU /D8/CW/CT /DB /D3/D6/CZ[2] /B5/BA /CF /CT /D7/CT/CT/D8/CW/CP/D8 /D8/CW/CT /D7/D4/D9/D6/CX/D3/D9/D7 /BG/BB/BF /CU/CP
/D8/D3/D6 /CP/D4/D4 /CT/CP/D6/D7 /CX/D2Pf
/BA /C7/CU
/D3/D9/D6/D7/CT/B8 /CP/D7 /CP/D0/D6/CT/CP/CS/DD /D7/CP/CX/CS/B8 /D8/CW/CT /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7 P0
f
/CP/D2/CSPf/CS/D3 /D2/D3/D8 /CU/D3/D6/D1 /CP /BG/B9/DA /CT
/D8/D3/D6/BA /BU/CT
/CP/D9/D7/CT /D3/CU /D8/CW/CP/D8fµ
coh
/B8 /D8/CW/CT /CU/D3/D6
/CT /CS/CT/D2/D7/CX/D8 /DD /D8/CW/CP/D8 /D4/D6/D3 /DA/CX/CS/CT/D7 /D8/CW/CT /C8 /D3/CX/D2
/CP/D6/GH /D7/D8/D6/CT/D7/D7/CT/D7/B8/CX/D7 /CX/D2 /D8/D6/D3 /CS/D9
/CT/CS /CX/D2 /D8/D3 /D8/CW/CT /D8/CW/CT/D3/D6/DD /BA /CC/CW/CT/D2 P0
coh
/CP/D2/CSPcoh
/CP/D6/CT
/CP/D0
/D9/D0/CP/D8/CT/CS /CQ /DD /D1/CT/CP/D2/D7 /D3/CUfµ
coh
/CX/D2 /D7/D9
/CW /CP /DB /CP /DD/D8/D3 /CV/CX/DA /CT /D8/CW/CP/D8 /D8/CW/CT /D7/D9/D1 /D3/CU /D8 /DB /D3 /CU/CP/D0/D7/CT /BG/B9/DA /CT
/D8/D3/D6/D7 Pµ
f
/CP/D2/CSPµ
coh
/CX/D7 /CP /CW /DD/D4 /CT/D6/D4/D0/CP/D2/CT /CX/D2 /D8/CT/CV/D6/CP/D0 /CX/D2/CS/CT/D4 /CT/D2/CS/CT/D2 /D8 /D3/CU /D8/CW/CT/D3/D6/CX/CT/D2 /D8/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CP/D8 /CW /DD/D4 /CT/D6/D4/D0/CP/D2/CT/BA /BT/D4/D4/D0/DD/CX/D2/CV /DA /D3/D2 /C4/CP/D9/CT/B3/D7 /D8/CW/CT/D3/D6/CT/D1 /CX/D8 /CX/D7
/D3/D2
/D0/D9/CS/CT/CS /CQ /DD /D8/CW/CT /D4/D6/D3/D4 /D3/D2/CT/D2 /D8/D7 /D3/CU/D8/CW/CT /D7/DD/D2
/CW/D6/D3/D2/D3/D9/D7 /CS/CT/AS/D2/CX/D8/CX/D3/D2/D7 /D8/CW/CP/D8 /D7/D9
/CW /D7/D9/D1 /CX/D7 /CP /D0/CT/CV/CX/D8/CX/D1/CP/D8/CT /BG/B9/DA /CT
/D8/D3/D6/B8 /B4/D7/CT/CT/B8 /CT/BA/CV/BA/B8 /CA/CT/CU/BA /BD /BX/D5/D7/BA /B4/BD/BG/B5 /CP/D2/CS/B4/BD/BH/B5/B5/BA /C0/D3 /DB /CT/DA /CT/D6/B8 /CS/CT/D7/D4/CX/D8/CT /D8/CW/CT /CU/CP
/D8 /D8/CW/CP/D8 /D8/CW/CT /D7/D9/D1Pµ
f+Pµ
coh
/CP/D2 /CQ /CT /DB/D6/CX/D8/D8/CT/D2 /CX/D2 /D8/CW/CT /CU/D3/D6/D1 /D3/CU /CP /BG/B9/DA /CT
/D8/D3/D6/B8/CX/BA/CT/BA/B8 /CP/D7mvµ/B8 /B4/D7/CT/CT /CT/D5/D9/CP/D8/CX/D3/D2 /B4/BD/BH/B5 /CX/D2[1] /B5/B8 /D8/CW/CX/D7 /D5/D9/CP/D2 /D8/CX/D8 /DD /CX/D7 /D2/D3/D8 /CP /D8/D6/D9/CT /BG/B9/DA /CT
/D8/D3/D6/BA /BT/D7 /DB /CT /CW/CP /DA /CT /CP/D0/D6/CT/CP/CS/DD/D7/CW/D3 /DB/D2 /D8/CW/CX/D7 /D7/D9/D1 /CS/D3 /CT/D7 /D2/D3/D8 /D6/CT/CU/CT/D6 /D8/D3 /D8/CW/CT /D7/CP/D1/CT /D5/D9/CP/D2 /D8/CX/D8 /DD
/D3/D2/D7/CX/CS/CT/D6/CT/CS /CX/D2 /CS/CX/AR/CT/D6/CT/D2 /D8 /C1/BY/CA/D7/BN /D8/CW/CT /D4/D0/CP/CX/D2/D7 t(0)=a/CX/D2S(0)
/CP/D2/CSt=b /CX/D2S /CP/D6/CT /D2/D3/D8 /D6/CT/D0/CP/D8/CT/CS /CQ /DD /D8/CW/CT /C4 /CC /D8/CW/CP/D2 /CQ /DD /D8/CW/CT /BT /CC/B8 /CP/D2/CS /D8/CW/CT /D6/CT/D0/CP/D8/CX/D3/D2/D7 /B4/BD/B5
/D3/D2/D2/CT
/D8/CX/D2/CV
E /CP/D2/CSB /CX/D2S /CP/D2/CSE(0)
/CP/D2/CSB(0)
/CX/D2S(0)
/CP/D6/CT /D2/D3/D8 /D8/CW/CT /C4 /CC /D3/CU /D8/CW/CT /D7/CP/D1/CT /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7 /CU/D6/D3/D1S(0)
/D8/D3S /B8 /D8/CW/CP/D2/D8/CW/CT/DD /CP/D6/CT /CP/D0/D7/D3 /D8/CW/CT /BT /CC/BA/C1/D2
/D3/D2 /D8/D6/CP/D7/D8 /D8/D3 /D8/CW/CT /D7/DD/D2
/CW/D6/D3/D2/D3/D9/D7 /CS/CT/AS/D2/CX/D8/CX/D3/D2/D7
/D3/D2/D7/CX/CS/CT/D6/CT/CS /CX/D2[2,3,15,16] /B8 /CA/D3/CW/D6/D0/CX
/CW/B3/D7 /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2/D7 /CU/D3/D6
P0
f
/CP/D2/CSPk
f
/CX/D2S /B8 /CS/CT/D6/CX/DA /CT/CS /CX/D2 /D8/CW/CT /BT/D4/D4 /CT/D2/CS/CX/DC /CX/D2[1] /B8 /CV/CX/DA /CT /D8/CW/CP/D8Pµ
f
/CP/D0/D3/D2/CT
/CP/D2 /CQ /CT /DB/D6/CX/D8/D8/CT/D2 /CP/D7mvµ/B8 /CP/D2/CS/BL/CP
/D3/D6/CS/CX/D2/CV/D0/DD /CX/D8 /CX/D7 /CP/D0/D7/D3
/D3/D2
/D0/D9/CS/CT/CS /CX/D2[1] /B8 /CQ /DD /D8/CW/CT /D9/D7/CT /D3/CU /DA /D3/D2 /C4/CP/D9/CT/B3/D7 /D8/CW/CT/D3/D6/CT/D1/B8 /D8/CW/CP/D8Pµ
f
/CX/D7 /CP /D8/D6/D9/CT /BG/B9/DA /CT
/D8/D3/D6/BA/C1/D2 /CA/D3/CW/D6/D0/CX
/CW/B3/D7 /CP/D4/D4/D6/D3/CP
/CW /D8/CW/CT/D6/CT /CX/D7 /D2/D3 /D7/D4/D9/D6/CX/D3/D9/D7 /BG/BB/BF /CU/CP
/D8/D3/D6 /CX/D2Pµ
f
/B8 /CP/D2/CS/B8 /CX/D2
/D3/D2 /D8/D6/CP/D7/D8 /D8/D3 /BU/D3 /DD /CT/D6/B3/D7 /CP/D4/D4/D6/D3/CP
/CW/B8/CA/D3/CW/D6/D0/CX
/CW/B3/D7Σ /CX/D2S /CX/D7
/D3/D6/D6/CT
/D8/D0/DD /CS/CT/D8/CT/D6/D1/CX/D2/CT/CS /CP/D7LΣ(0)
/BA /BU/D9/D8/B8 /CP/D7 /CP/D0/D6/CT/CP/CS/DD /D7/CP/CX/CS/B8 /D2/CT/CX/D8/CW/CT/D6 /CA/D3/CW/D6/D0/CX
/CW/B3/D7Pµ
f/CX/D7 /CP /D0/CT/CV/CX/D8/CX/D1/CP/D8/CT /BG/B9/DA /CT
/D8/D3/D6/B8 /D7/CX/D2
/CT /CW/CT /D9/D7/CT/D7 /D8/CW/CT /BT /CC /D3/CUE /CP/D2/CSB /BA /CC/CW/CT /D9/D7/CT /D3/CUE /CP/D2/CSB /CP/D2/CS /D8/CW/CT /BT /CC /B4/BD/B5/CX/D2[1] /CX/D2/D7/D8/CT/CP/CS /D3/CU /D8/CW/CT /BG/B9/DA /CT
/D8/D3/D6/D7 Eα/CP/D2/CSBα/CP/D2/CS /D8/CW/CT/CX/D6 /C4 /CC
/CP/D9/D7/CT/D7 /D8/CW/CT /CS/CX/AR/CT/D6/CT/D2
/CT /CQ /CT/D8 /DB /CT/CT/D2 /D8/CW/CT /D6/CT/D7/D9/D0/D8/D7/D3/CQ/D8/CP/CX/D2/CT/CS /CX/D2[1] /CU/D3/D6 /D8/CW/CT /CT/D2/CT/D6/CV/DD /CP/D2/CS /D1/D3/D1/CT/D2 /D8/D9/D1 /CS/CT/D2/D7/CX/D8/CX/CT/D7 /CP/D2/CS /CU/D3/D6P0
f
/CP/D2/CSPk
f
/B8 /CP/D2/CS /D8/CW/CT
/D3/D6/D6/CT/D7/D4 /D3/D2/CS/CX/D2/CV/CT/DC/D4/D6/CT/D7/D7/CX/D3/D2/D7 /D3/CQ/D8/CP/CX/D2/CT/CS /CW/CT/D6/CT/B8 /BX/D5/D7/BA /B4/BD/BE/B5 /CP/D2/CS /B4/BD/BF/B5/BA/BQ/BY /D6/D3/D1 /B4/BD/BD/B8/BD/BE/B5 /CP/D2/CS /B4/BD/BF/B5 /DB /CT /D7/CT/CT /D8/CW/CP/D8 /CX/D2 /D8/CW/CT
/D3 /DA /CP/D6/CX/CP/D2 /D8 /CP/D4/D4/D6/D3/CP
/CW /D4/D6/CT/D7/CT/D2 /D8/CT/CS /CW/CT/D6/CT /D8/CW/CT /D7/D4/D9/D6/CX/D3/D9/D7/CU/CP
/D8/D3/D6 /BG/BB/BF /CS/D3 /CT/D7 /D2/D3/D8 /CP/D4/D4 /CT/CP/D6 /CP/D2/CS /D8/CW/CP/D8 /AH/D4/D9/D6/CT /CT/D0/CT
/D8/D6/D3/D1/CP/CV/D2/CT/D8/CX
/AH Pµ
f
/CX/D7 /CP /D0/CT/CV/CX/D8/CX/D1/CP/D8/CT /BG/B9/DA /CT
/D8/D3/D6/BN /CX/D8 /D6/CT/CU/CT/D6/D7 /D8/D3/D8/CW/CT /D7/CP/D1/CT /D5/D9/CP/D2 /D8/CX/D8 /DD /CX/D2 /CP/D0/D0 /C1/BY/CA/D7 /D7/CX/D2
/CT /CP/D0/D0 /D4/CP/D6/D8/D7 /D3/CU /CX/D8 /CP/D6/CT /C4/D3/D6/CT/D2 /D8/DE /D8/D6/CP/D2/D7/CU/D3/D6/D1/CT/CS /DB/CW/CT/D2 /CV/D3/CX/D2/CV /CU/D6/D3/D1 /C1/BY/CA
S(0)
/D8/D3 /D7/D3/D1/CT /C1/BY/CAS /BA/BH/BA/BF /CA/D3/CW/D6/D0/CX
/CW/B3/D7P0
f
/CP/D2/CSPk
f
/CU/D6/D3/D1 Pµ
f
/DB/CX/D8/CW Eα/CP/D2/CSBα/CC/CW/CT /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2/D7 /CU/D3/D6Pµ
f
/B4/DB/D6/CX/D8/D8/CT/D2 /DB/CX/D8/CWEα/CP/D2/CSBα/B5
/D3/D6/D6/CT/D7/D4 /D3/D2/CS/CX/D2/CV /D8/D3 /CA/D3/CW/D6/D0/CX
/CW/B3/D7 /D6/CT/D0/CP/D8/CX/D3/D2/D7 /CU/D3/D6P0
f/CP/D2/CSPk
f
/DB/CX/D8/CWE /CP/D2/CSB[1] /B8
/CP/D2 /CQ /CT /D3/CQ/D8/CP/CX/D2/CT/CS /CX/D2 /D8/CW/CT /CU/D3/D0/D0/D3 /DB/CX/D2/CV /DB /CP /DD /BA /C4/CT/D8 /D8/CW/CT /C1/BY/CAS /CQ /CT /D8/CW/CT /CU/D6/CP/D1/CT /CX/D2/DB/CW/CX
/CW /D8/CW/CT /D4/CP/D6/D8/CX
/D0/CT /D1/D3 /DA /CT/D7 /DB/CX/D8/CW /D8/CW/CT /BG/B9/DA /CT/D0/D3
/CX/D8 /DDuα= (γuc, γuu,0,0) /B8 /CP/D2/CS /D8/CW/CT/D6/CT/CU/D3/D6/CT /D8/CW/CT /D9/D2/CX/D8 /BG/B9/DA /CT
/D8/D3/D6
nµ/CX/D7nµ= (γu, γuβu,0,0) /BA /CC/CW/CT /BG/B9/D1/D3/D1/CT/D2 /D8/D9/D1 /CX/D7 /CV/CX/DA /CT/D2 /CQ /DD /B4/BJ/B5/BA /CD/D7/CX/D2/CV /D8/CW/CT/D7/CT /D6/CT/D0/CP/D8/CX/D3/D2/D7 /DB /CT /DB/D6/CX/D8/CT /D8/CW/CT
/D3/D1/D4 /D3/D2/CT/D2 /D8/D7 /D3/CUPµ
f
/CP/D7
P0
f=−(γu/c)/integraldisplay
dV(0)/bracketleftbig
T00−βuT10/bracketrightbig
,
Pi
f=−(γu/c)/integraldisplay
dV(0)/bracketleftbig
T0i−βuT1i/bracketrightbig
. /B4/BD/BG/B5/BY /D9/D6/D8/CW/CT/D6/B8 /D0/CT/D8 /D8/CW/CT /D3/CQ/D7/CT/D6/DA /CT/D6/D7 /DB/CW/D3 /D1/CT/CP/D7/D9/D6/CT /D8/CW/CT /AS/CT/D0/CS/D7 Eα/CP/D2/CSBα/CX/D2S /CP/D6/CT /CP/D8 /D6/CT/D7/D8 /CX/D2S /B8 /CX/BA/CT/BA/B8 /D8/CW/CT/CX/D6/BG/B9/DA /CT/D0/D3
/CX/D8 /DDvα/CX/D7vα= (c,0) /BA /BY /D3/D6 /D7/D9
/CW /D3/CQ/D7/CT/D6/DA /CT/D6/D7E0=B0= 0 /BA /CC/CW/CT/D2 /CU/D6/D3/D1 /B4/BL/B5 /DB /CT /AS/D2/CS /D8/CW/CP/D8Tµν
/CP/D2/CQ /CT /DB/D6/CX/D8/D8/CT/D2 /CX/D2 /D8/CT/D6/D1/D7 /D3/CU /D8/CW/CT
/D3/D1/D4 /D3/D2/CT/D2 /D8/D7 /D3/CUEα/CP/D2/CSBα/CP/D7
T00=−uE=−(ε0/2)(EiEi+c2BiBi),
T0i=−ε0cεijkEjBk, /B4/BD/BH/B5
T11=ε0((E1)2+c2(B1)2)−uE,
T1n=ε0(E1En+c2B1Bn), n= 2,3./CF/CW/CT/D2 Tµν/CU/D6/D3/D1 /B4/BD/BH/B5 /CX/D7 /CX/D2 /D8/D6/D3 /CS/D9
/CT/CS /CX/D2 /D8/D3 /B4/BD/BG/B5 /D8/CW/CT/D2P0
f
/CP/D2/CSPi
f
/D7/CT/CT/D1 /D0/CX/CZ /CTP0
e
/CP/D2/CSPi
e
/CU/D3/D6 /D8/CW/CT
/CW/D3/CX
/CT/B4/C1 /C1/B5 /CX/D2/BA[1]. /BU/D9/D8 /D8/CW/CT /BG/B9/DA/CT
/D8/D3/D6/D7 Eα/CP/D2/CSBα/CX/D2 /B4/BD/BH/B5 /CP/D6 /CT /D1/CT /CP/D7/D9/D6 /CT /CS /CQ/DD /D8/CW/CT /D3/CQ/D7/CT/D6/DA/CT/D6/D7 /CP/D8 /D6 /CT/D7/D8 /CX/D2 /CB /B8 /CX/BA/CT/BA/B8/DB/CW/D3/D7/CT /DA /CT/D0/D3
/CX/D8 /DD /CX/D7vα= (c,0) /B8 /DB/CW/CX
/CW /D1/CT/CP/D2/D7 /D8/CW/CP/D8 /D8/CW/CT/D7/CT Eα/CP/D2/CSBα/CP/D6/CT /D2/D3/D8 /D8/CW/CT /C4 /CC /D3/CU /D8/CW/CT /D4/D6/CT/DA/CX/D3/D9/D7/D0/DD/D1/CT/D2 /D8/CX/D3/D2/CT/CS Eα
(0)
/CP/D2/CSBα
(0)
/B4/CU/D3/D6 /DB/CW/CX
/CWvα= (c,0) /CX/D2S(0)
/B5/BA /CC/CW /D9/D7 /DB /CT /AS/D2/CS /D8/CW/CP/D8/B8
/D3/D2 /D8/D6/CP/D6/DD /D8/D3 /D8/CW/CT/CS/CT/D6/CX/DA /CP/D8/CX/D3/D2 /CX/D2 /D8/CW/CT /BT/D4/D4 /CT/D2/CS/CX/DC /D3/CU[1] /B8 /BX/D5/BA /B4/BD/BG/B5 /DB/CX/D8/CWTµν/CS/CT/D8/CT/D6/D1/CX/D2/CT/CS /CQ /DD /B4/BD/BH/B5 /CX/D7 /D2/D3/D8 /D8/CW/CT /C4 /CC /D3/CUPµ
f(0)/B4/BD/BD/B8/BD/BE/B5/BA /CC/CW/CT /C4 /CC /D3/CUPµ
f(0)
/CP/D6/CT /CP
/D8/D9/CP/D0/D0/DD /CV/CX/DA /CT/D2 /CQ /DD /B4/BD/BF/B5/B8 /CP/D7 /CX/D8 /CX/D7 /D7/CW/D3 /DB/D2 /CP/CQ /D3 /DA /CT/BA /C1/CU /D3/D2/CT /D4 /CT/D6/CU/D3/D6/D1/D7 /D8/CW/CT /D7/CP/D1/CT/D4/D6/D3
/CT/CS/D9/D6/CT /CP/D7 /CX/D2 /D8/CW/CT /BT/D4/D4 /CT/D2/CS/CX/DC /D3/CU[1] /CT/DC/D4/D6/CT/D7/D7/CX/D2/CV /D8/CW/CT /BG/B9/DA /CT
/D8/D3/D6/D7 Eα/CP/D2/CSBα/CX/D2S /CQ /DD /D1/CT/CP/D2/D7 /D3/CUE′α
(0)/CP/D2/CSB′α
(0)
/B8 /D8/CW/CT /BG/B9/DA /CT
/D8/D3/D6/D7 /CX/D2 /D8/CW/CT /D6/CT/D7/D8 /CU/D6/CP/D1/CT S(0)
/D3/CU /D8/CW/CT
/CW/CP/D6/CV/CT/CS /D7/D4/CW/CT/D6/CT/B8 /DB/CW/CX
/CW /CP/D6/CT
/D3/D2/D2/CT
/D8/CT/CS /CQ /DD /D8/CW/CT/C4 /CC /DB/CX/D8/CWEα/CP/D2/CSBα/B8 /D8/CW/CT/D2 /D3/D2/CT /CS/D3 /CT/D7 /D2/D3/D8 /AS/D2/CSP0
e
/CP/D2/CSPi
e
/D3/CQ/D8/CP/CX/D2/CT/CS /CU/D3/D6 /D8/CW/CT
/CW/D3/CX
/CT /B4/C1 /C1/B5 /CX/D2/BA[1] /BA /CC/CW/CT/BG/B9/DA /CT
/D8/D3/D6/D7 E′α
(0)
/CP/D2/CSB′α
(0)
/CP/D6/CT /D2/D3/D8 /CT/D5/D9/CX/DA /CP/D0/CT/D2 /D8 /D8/D3 /D8/CW/CT /D4/D6/CT/DA/CX/D3/D9/D7/D0/DD
/D3/D2/D7/CX/CS/CT/D6/CT/CS /B4/CX/D2
/D3/D2/D2/CT
/D8/CX/D3/D2 /DB/CX/D8/CW /B4/BD/BD/B9/BD/BF/B5/B5/BG/B9/DA /CT
/D8/D3/D6/D7 Eα
(0)
/CP/D2/CSBα
(0)
/BN /D8/CW/CT /CU/D3/D6/D1/CT/D6 /D6/CT/CU/CT/D6 /D8/D3 /D8/CW/CT /CT/D0/CT
/D8/D6/CX
/CP/D2/CS /D1/CP/CV/D2/CT/D8/CX
/AS/CT/D0/CS/D7 /DB/CW/CX
/CW /CP/D6/CT /D1/CT/CP/D7/D9/D6/CT/CS /CX/D2
S /CQ /DD /D8/CW/CT /D3/CQ/D7/CT/D6/DA /CT/D6/D7 /CP/D8 /D6/CT/D7/D8 /CX/D2S /CP/D2/CS /D8/CW/CT/D2 /C4/D3/D6/CT/D2 /D8/DE /D8/D6/CP/D2/D7/CU/D3/D6/D1/CT/CS /D8/D3S(0)
/B8 /DB/CW/CX/D0/CT /D8/CW/CT /D0/CP/D8/D8/CT/D6 /D6/CT/CU/CT/D6 /D8/D3 /D8/CW/CT/CT/D0/CT
/D8/D6/CX
/CP/D2/CS /D1/CP/CV/D2/CT/D8/CX
/AS/CT/D0/CS/D7 /DB/CW/CX
/CW /CP/D6/CT /D1/CT/CP/D7/D9/D6/CT/CS /CS/CX/D6/CT
/D8/D0/DD /CX/D2S(0)
/CQ /DD /D8/CW/CT /D3/CQ/D7/CT/D6/DA /CT/D6/D7 /CP/D8 /D6/CT/D7/D8 /CX/D2S(0)
/BA/CC/CW/CT /D4/D6/CT
/CT/CS/CX/D2/CV /CS/CX/D7
/D9/D7/D7/CX/D3/D2 /D6/CT/DA /CT/CP/D0/D7 /D8/CW/CP/D8 /D8/CW/CT /CS/CX/AR/CT/D6/CT/D2
/CT /CQ /CT/D8 /DB /CT/CT/D2 /D8/CW/CT /D6/CT/D7/D9/D0/D8/D7 /CX/D2[1] /CP/D2/CS /CX/D2 /D8/CW/CX/D7 /D4/CP/D4 /CT/D6 /CX/D7/B8/CP/D7 /CP/D0/D6/CT/CP/CS/DD /D7/CP/CX/CS/B8 /CP
/D3/D2/D7/CT/D5/D9/CT/D2
/CT /D3/CU /D8/CW/CT /D9/D7/CT /D3/CU /D8/CW/CT /BF/B9/DA /CT
/D8/D3/D6/D7 E /CP/D2/CSB /CP/D2/CS /D8/CW/CT /BT /CC /B4/BD/B5 /CX/D2[1] /B8 /CP/D2/CS /D8/CW/CT/D9/D7/CT /D3/CU /D8/CW/CT /BG/B9/DA /CT
/D8/D3/D6/D7 Eα/CP/D2/CSBα/CP/D2/CS /D8/CW/CT/CX/D6 /C4 /CC /CX/D2 /D8/CW/CX/D7 /D4/CP/D4 /CT/D6/BA/BI /CB/C7/C5/BX /CA/BX/BV/BX/C6/CC /CC/CA/BX/BT /CC/C5/BX/C6/CC/CB /C7/BY /CC/C0/BX /BX/C4/BX/BV/CC/CA /C7/B9/C5/BT /BZ/C6/BX/CC/C1/BV /C5/C7/C5/BX/C6/CC/CD/C5 /BT/C6/BW /BX/C6/BX/CA /BZ/CH/C1/D2 /D8/CW/CX/D7 /D7/CT
/D8/CX/D3/D2 /DB /CT
/D3/D2/D7/CX/CS/CT/D6 /D7/D3/D1/CT /D6/CT
/CT/D2 /D8 /D8/D6/CT/CP/D8/D1/CT/D2 /D8/D7 /D3/CU /D8/CW/CT /CT/D0/CT
/D8/D6/D3/D1/CP/CV/D2/CT/D8/CX
/CT/D2/CT/D6/CV/DD /CP/D2/CS /D1/D3/D1/CT/D2 /D8/D9/D1/BA/BD/BC/BI/BA/BD /CA/D3/D1/CT/D6/B3/D7 /C9/D9/CT/D7/D8/CX/D3/D2 /CP/D2/CS /BT/D2/D7/DB /CT/D6/D7/CA/CT
/CT/D2 /D8/D0/DD /CA/D3/D1/CT/D6 [24] /D6/CT/DA/CX/DA /CT/CS /D8/CW/CT /D5/D9/CT/D7/D8/CX/D3/D2 /D3/CU /D8/CW/CT
/D3/D6/D6/CT
/D8 /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2/D7 /CU/D3/D6 /D8/CW/CT /CT/D0/CT
/D8/D6/D3/D1/CP/CV/D2/CT/D8/CX
/AS/CT/D0/CS/D1/D3/D1/CT/D2 /D8/D9/D1 /CP/D2/CS /CT/D2/CT/D6/CV/DD /CX/D2 /D8/CW/CT
/CP/D7/CT /D3/CU /AH/CQ /D3/D9/D2/CS/AH /AS/CT/D0/CS/D7/B8 /AS/CT/D0/CS/D7 /D8/CW/CP/D8 /CP/D6/CT /D8/CX/CT/CS /D8/D3 /D8/CW/CT/CX/D6 /D7/D3/D9/D6
/CT/D7/BA /C0/CT/B8 /CP/D2/CS/D1/CP/D2 /DD /D3/D8/CW/CT/D6/D7 /B4/D7/CT/CT /D6/CT/CU/CT/D6/CT/D2
/CT/D7 /CX/D2[24] /B5/B8 /D9/D7/CT/D7 /D8/CW/CT /D8/D6/CP/CS/CX/D8/CX/D3/D2/CP/D0 /D7/DD/D2
/CW/D6/D3/D2/D3/D9/D7 /CS/CT/AS/D2/CX/D8/CX/D3/D2/D7 /B4/BH/B9/BI/B5/BA /BT/D7 /DB /CT /CW/CP /DA /CT/CP/D0/D6/CT/CP/CS/DD /D7/CW/D3 /DB/D2 /D8/CW/CT /D6/CT/D0/CP/D8/CX/D3/D2/D7 /B4/BH/B9/BI/B5
/D3/D2 /D8/CP/CX/D2 /D8/CW/CT /BT /CC /D3/CU /D8/CW/CT /CW /DD/D4 /CT/D6/D4/D0/CP/D2/CT t=const. /CP/D2/CS /D8/CW/CT /BT /CC /D3/CUE/CP/D2/CSB /BA /BW/CX/AR/CT/D6/CT/D2 /D8 /CP/D2/D7/DB /CT/D6/D7 /D8/D3 /D8/CW/CX/D7 /D5/D9/CT/D7/D8/CX/D3/D2 /CW/CP /DA /CT /CQ /CT/CT/D2 /CV/CX/DA /CT/D2 /CX/D2[25] /BA /C6/CT/CX/D8/CW/CT/D6 /D8/CW/CT /CP/D2/D7/DB /CT/D6/D7[25] /D8/D3 /D8/CW/CT/D5/D9/CT/D7/D8/CX/D3/D2 /CX/D2[24] /CP/D6/CT /CX/D2 /CP
/D3/D1/D4/D0/CT/D8/CT /CP/CV/D6/CT/CT/D1/CT/D2 /D8 /DB/CX/D8/CW /D8/CW/CT /AH/CC/CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD /BA/AH /BY/CX/D6/D7/D8/B8 /D8/CW/CT/DD /CP/D0/D7/D3 /DB /D3/D6/CZ /DB/CX/D8/CW
E /CP/D2/CSB /CP/D2/CS /D8/CW/CT/CX/D6 /BT /CC /B4/BD/B5/B8 /CP/D2/CS /CQ/CP/D7/CT /D8/CW/CT/CX/D6
/D3/D2
/D0/D9/D7/CX/D3/D2/D7 /D3/D2 /DA /D3/D2 /C4/CP/D9/CT/B3/D7 /D8/CW/CT/D3/D6/CT/D1/BA/BI/BA/BE /CB
/CW /DB/CX/D2/CV/CT/D6/B3/D7 /BV/D3/D2/D7/CX/CS/CT/D6/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /AH/BG/BB/BF/AH /C8/D6/D3/CQ/D0/CT/D1/CF /CT /CW/CP /DA /CT /D8/D3 /D1/CT/D2 /D8/CX/D3/D2 /CP/D2 /CX/D2 /D8/CT/D6/CT/D7/D8/CX/D2/CV
/D3/D2/D7/CX/CS/CT/D6/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /CT/D0/CT
/D8/D6/D3/D1/CP/CV/D2/CT/D8/CX
/CT/D2/CT/D6/CV/DD /CP/D2/CS /D1/D3/D1/CT/D2 /D8/D9/D1/CP/D2/CS /D8/CW/CT /CT/D0/CT
/D8/D6/D3/D1/CP/CV/D2/CT/D8/CX
/D1/CP/D7/D7 /CV/CX/DA /CT/D2 /CX/D2[26] /CQ /DD /CB
/CW /DB/CX/D2/CV/CT/D6/BA /C0/CT /CP/D0/D7/D3 /D9/D7/CT/D7 /D8/CW/CT /D7/DD/D2
/CW/D6/D3/D2/D3/D9/D7 /CS/CT/AS/D2/CX/D8/CX/D3/D2/D3/CUPµ
f
/B4/BH/B5/BA /C1/D2 /CS/CX/AR/CT/D6/CT/D2
/CT /D8/D3 /D8/CW/CT /DB /D3/D6/CZ/D7[2,3,15,16] /CW/CT /CS/D3 /CT/D7 /D2/D3/D8 /CS/CT/CP/D0 /DB/CX/D8/CW /C8 /D3/CX/D2
/CP/D6/GH/B3/D7 /D7/D8/D6/CT/D7/D7/CT/D7/B8 /CQ/D9/D8
/CW/CP/D2/CV/CT/D7 /D8/CW/CT /CS/CT/AS/D2/CX/D8/CX/D3/D2 /D3/CU /D8/CW/CT /CT/D0/CT
/D8/D6/D3/D1/CP/CV/D2/CT/D8/CX
/CT/D2/CT/D6/CV/DD/B9/D1/D3/D1/CT/D2 /D8/D9/D1 /D8/CT/D2/D7/D3/D6 Tµν/B4/BK/B5/BA /BT/D7 /CX/D8 /CX/D7 /CP/D0/D6/CT/CP/CS/DD/D7/CP/CX/CSTµν/B4/BK/B5 /CX/D7 /D2/D3/D8 /CS/CX/DA /CT/D6/CV/CT/D2
/CT/B9/CU/D6/CT/CT /CP/D2/CS /D8/CW/CT/D6/CT/CU/D3/D6/CT /D8/CW/CT /CW /DD/D4 /CT/D6/D4/D0/CP/D2/CT /CX/D2 /D8/CT/CV/D6/CP/D0 /D3/CUTµν/B4/BJ/B5/B8 /CX/D7 /CS/CT/D4 /CT/D2/CS/CT/D2 /D8 /D3/CU/D8/CW/CT /D3/D6/CX/CT/D2 /D8/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /CW /DD/D4 /CT/D6/D4/D0/CP/D2/CT/BA /CB
/CW /DB/CX/D2/CV/CT/D6
/D3/D2/D7/D8/D6/D9/CT/D7/B8 /CQ/D9/D8 /D3/D2/D0/DD /CU/D3/D6 /CP
/D0/CP/D7/D7 /D3/CU /AS/CT/D0/CS/D7 /CP/D2/CS
/D9/D6/D6/CT/D2 /D8/D7/B8/CP/D7/D7/D3
/CX/CP/D8/CT/CS /DB/CX/D8/CW /D9/D2/CX/CU/D3/D6/D1 /D1/D3/D8/CX/D3/D2/B8 /CP /D2/CT/DB/B8
/D3/D2/D7/CT/D6/DA /CT/CS/B8 /CS/CX/DA /CT/D6/CV/CT/D2
/CT/D0/CT/D7/D7/B8 /CT/D2/CT/D6/CV/DD/B9/D1/D3/D1/CT/D2 /D8/D9/D1 /D8/CT/D2/D7/D3/D6 Tµν
Sch./D3/CU /D8/CW/CT /CT/D0/CT
/D8/D6/D3/D1/CP/CV/D2/CT/D8/CX
/AS/CT/D0/CS/BA /CB/CX/D2
/CT ∂νTµν
Sch.= 0 /CX/D8 /CX/D7 /CP
/CW/CX/CT/DA /CT/CS /CX/D2[26] /D8/CW/CP/D8 /D8/CW/CT /CW /DD/D4 /CT/D6/D4/D0/CP/D2/CT /CX/D2 /D8/CT/CV/D6/CP/D0/B4/BJ/B5/B8 /B4/DB/CX/D8/CW Tµν
Sch.
/D6/CT/D4/D0/CP
/CX/D2/CV Tµν/B4/BK/B5/B5/B8 /CX/D7 /CX/D2/CS/CT/D4 /CT/D2/CS/CT/D2 /D8 /D3/CU /D8/CW/CT /D3/D6/CX/CT/D2 /D8/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CP/D8 /CW /DD/D4 /CT/D6/D4/D0/CP/D2/CT/B8 /CP/D2/CS /D8/CW/CX/D7/CX/D7
/D3/D2/D7/CX/CS/CT/D6/CT/CS /CQ /DD /CB
/CW /DB/CX/D2/CV/CT/D6 /D8/D3 /D3 /CP/D7 /CP /D2/CT
/CT/D7/D7/CP/D6/DD /CP/D2/CS /D7/D9Ꜷ
/CX/CT/D2 /D8
/D3/D2/CS/CX/D8/CX/D3/D2 /D8/CW/CP/D8 /D8/CW/CT /CX/D2 /D8/CT/CV/D6/CP/D0 /B4/BJ/B5 /CX/D7 /CP/D8/D6/D9/CT /BG/B9/DA /CT
/D8/D3/D6/B8 /CX/BA/CT/BA/B8 /CW/CT /CP/D0/D7/D3 /CP
/CT/D4/D8/D7 /DA /D3/D2 /C4/CP/D9/CT/B3/D7 /D8/CW/CT/D3/D6/CT/D1 /CP/D7 /D8/CW/CP/D8 /CX/D8 /CX/D7 /CP
/D3/D6/D6/CT
/D8 /D3/D2/CT /CU/D6/D3/D1 /D8/CW/CT /AH/CC/CC/DA/CX/CT/DB/D4 /D3/CX/D2 /D8/BA/AH /CC/CW/CT/D2
/CW/D3 /D3/D7/CX/D2/CV /D8/CW/CP/D8Σ /CX/D2 /B4/BJ/B5 /CX/D7 /D8/CW/CT /D4/D0/CP/D2/CT t=a /CX/D2 /D7/D3/D1/CT /C1/BY/CAS /D8/CW/CT /CX/D2 /D8/CT/CV/D6/CP/D0 /B4/BJ/B5 /CQ /CT
/D3/D1/CT/D7/D8/CW/CT /D7/DD/D2
/CW/D6/D3/D2/D3/D9/D7 /CS/CT/AS/D2/CX/D8/CX/D3/D2 /D3/CUPµ
f
/B4/BH/B5/BA /BT/D0/D7/D3 /CX/D8 /CW/CP/D7 /D8/D3 /CQ /CT /D2/D3/D8/CT/CS /D8/CW/CP/D8 /D8/CW/CT /D8/CT/D2/D7/D3/D6 Tµν
Sch.
/CX/D7 /D2/D3/D8 /D9/D2/CX/D5/D9/CT/B8/CP/D2/CS /CU/D3/D6 /D8/CW/CT /D7/CP/D1/CT /AS/CT/D0/CS/B9
/D9/D6/D6/CT/D2 /D8 /CS/CX/D7/D8/D6/CX/CQ/D9/D8/CX/D3/D2 /D3/D2/CT
/CP/D2 /CW/CP /DA /CT /CS/CX/AR/CT/D6/CT/D2 /D8Tµν
Sch.
/B8 /CT/BA/CV/BA/B8 /D8/CW/CT /D8/CT/D2/D7/D3/D6 /B4/BD/B5 /CT/D5/D9/CP/D8/CX/D3/D2/B4/BG/BE/B5 /CP/D2/CS /D8/CW/CT /D8/CT/D2/D7/D3/D6 /B4/BE/B5 /CT/D5/D9/CP/D8/CX/D3/D2 /B4/BG/BG/B5 /CX/D2[26] /BA /CD/D7/CX/D2/CV /CS/CX/AR/CT/D6/CT/D2 /D8Tµν
Sch.
/B4/B4/BD/B5 /CP/D2/CS /B4/BE/B5 /CX/D2[26] /B5 /D8 /DB /D3 /CS/CX/AR/CT/D6/CT/D2 /D8/AH
/D3 /DA /CP/D6/CX/CP/D2 /D8/AH /DA /CT/D6/D7/CX/D3/D2/D7 /D3/CU /D8/CW/CT
/D3/D2
/CT/D4/D8 /D3/CU /CT/D0/CT
/D8/D6/D3/D1/CP/CV/D2/CT/D8/CX
/D1/CP/D7/D7 /DB /CT/D6/CT /D3/CQ/D8/CP/CX/D2/CT/CS /CX/D2[26] /BA /CF /CT /D2/D3/D8/CT /D8/CW/CP/D8/D8/CW/CT /D7/CP/D1/CT /D6/CT/D1/CP/D6/CZ/D7 /CP/D7 /CU/D3/D6 /D8/CW/CT /D9/D7/D9/CP/D0 /D7/DD/D2
/CW/D6/D3/D2/D3/D9/D7 /CS/CT/AS/D2/CX/D8/CX/D3/D2/D7 /CW/D3/D0/CS /CP/D0/D7/D3 /CW/CT/D6/CT/BNt(0)=const. /CX/D2S(0)
/CP/D2/CS
t=a /CX/D2S /CP/D6/CT /D2/D3/D8 /D6/CT/D0/CP/D8/CT/CS /CQ /DD /D8/CW/CT /C4 /CC /D8/CW/CP/D2 /CQ /DD /D8/CW/CT /BT /CC/BA /BY /D9/D6/D8/CW/CT/D6/B8 /D8/CW/CT /CT/D2/CT/D6/CV/DD /CP/D2/CS /D1/D3/D1/CT/D2 /D8/D9/D1 /CX/D2[26]/CP/D6/CT /D9/D0/D8/CX/D1/CP/D8/CT/D0/DD /CT/DC/D4/D6/CT/D7/D7/CT/CS /CX/D2 /D8/CT/D6/D1/D7 /D3/CUE /CP/D2/CSB /B8 /B4/D7/CT/CT /CT/D5/D9/CP/D8/CX/D3/D2/D7 /B4/BG/BF/B5/B8 /B4/BG/BH/B5 /CP/D2/CS /B4/BG/BI/B5 /CX/D2[26] /B5/BA /CC/CW /D9/D7/B8
/D3/D2 /D8/D6/CP/D6/DD /D8/D3 /D8/CW/CT /CP/D7/D7/CT/D6/D8/CX/D3/D2/D7 /CX/D2[26] /B8 /DB /CT
/D3/D2
/D0/D9/CS/CT /D8/CW/CP/D8 /CT/DA /CT/D2 /D8/CW/D3/D9/CV/CW /D8/CW/CT /CT/D2/CT/D6/CV/DD /CP/D2/CS /D1/D3/D1/CT/D2 /D8/D9/D1 /CS/CT/AS/D2/CT/CS/CQ /DD /B4/BH/B5 /B4/DB/CX/D8/CW Tµ0
Sch.
/CX/D2/D7/D8/CT/CP/CS /D3/CUTµ0/B5 /D8/D6/CP/D2/D7/CU/D3/D6/D1 /D0/CX/CZ /CT /CP /BG/B9/DA /CT
/D8/D3/D6 /B4/D7/CT/CT /CT/D5/D9/CP/D8/CX/D3/D2/D7 /B4/BI/BE/B9/BI/BF/B5 /CX/D2[26] /B5/B8 /D8/CW/CT/D7/CT/CT/D5/D9/CP/D8/CX/D3/D2/D7/B8 /DB/CW/CX
/CW /CP/D6/CT /CS/CT/D6/CX/DA /CT/CS /CU/D6/D3/D1Pµ
f
/B4/BH/B5/B8 /CS/D3 /D2/D3/D8 /CS/CT/AS/D2/CT /CP /D8/D6/D9/CT /BG/B9/DA /CT
/D8/D3/D6/B8 /CX/BA/CT/BA/B8 /D8/CW/CT/DD /CS/D3 /D2/D3/D8
/D3 /DA /CP/D6/CX/CP/D2 /D8/D0/DD/CS/CT/AS/D2/CT /D8/CW/CT /CT/D2/CT/D6/CV/DD /CP/D2/CS /D1/D3/D1/CT/D2 /D8/D9/D1 /D3/CU /D8/CW/CT /AH/CQ /D3/D9/D2/CS/AH /CT/D0/CT
/D8/D6/D3/D1/CP/CV/D2/CT/D8/CX
/AS/CT/D0/CS/BA /CC/CW/CT /BT /CC /CP/D6/CT /D9/D7/CT/CS /CX/D2 /D8/CW/CT/CS/CT/D6/CX/DA /CP/D8/CX/D3/D2 /CX/D2[26] /B8 /DB/CW/CT/D2
/CT /D3/D2/CT
/D3/D2
/D0/D9/CS/CT/D7 /D8/CW/CP/D8 /D2/CT/CX/D8/CW/CT/D6 /D8/CW/CT /CT/D5/D9/CP/D8/CX/D3/D2/D7 /B4/BI/BE/B5 /CQ/CP/D7/CT/CS /D3/D2 /D6/CT/D7/D8 /D1/CP/D7/D7m(2)/D2/D3/D6 /D8/CW/CT /D6/CT/D0/CP/D8/CX/D3/D2/D7 /B4/BI/BF/B5 /CQ/CP/D7/CT/CS /D3/D2 /D6/CT/D7/D8 /D1/CP/D7/D7 m(1)/CS/D3 /D6/CT/CU/CT/D6 /D8/D3 /D8/CW/CT /D7/CP/D1/CT /D4/CW /DD/D7/CX
/CP/D0 /D5/D9/CP/D2 /D8/CX/D8 /DD
/D3/D2/D7/CX/CS/CT/D6/CT/CS/CU/D6/D3/D1 /CS/CX/AR/CT/D6/CT/D2 /D8 /C1/BY/CA/D7/BA/BI/BA/BF /CB/D3/D1/CT /C7/D8/CW/CT/D6 /CC /D6/CT/CP/D8/D1/CT/D2 /D8/D7/CA/CT
/CT/D2 /D8/D0/DD /D8/CW/CT /CT/D0/CT
/D8/D6/D3/D1/CP/CV/D2/CT/D8/CX
/D1/CP/D7/D7 /CS/CT/D6/CX/DA /CT/CS /CU/D6/D3/D1 /D8/CW/CT /D7/CT/D0/CU/B9/CU/D3/D6
/CT /CP/D2/CS /D8/CW/CT /BG/BB/BF /CU/CP
/D8/D3/D6 /DB /CT/D6/CT /CS/CX/D7
/D9/D7/D7/CT/CS /CX/D2
[27] /BA /CC/CW/CT/D7/CT /DB /D3/D6/CZ/D7 /DB/CX/D0/D0 /D2/D3/D8 /CQ /CT
/D3/D2/D7/CX/CS/CT/D6/CT/CS /CW/CT/D6/CT/B8 /CQ/D9/D8 /DB /CT /D3/D2/D0/DD /D1/CT/D2 /D8/CX/D3/D2 /D8/CW/CP/D8/B8 /D9/D0/D8/CX/D1/CP/D8/CT/D0/DD /B8 /D8/CW/CT/DD /D9/D7/CT /D8/CW/CT/D7/DD/D2
/CW/D6/D3/D2/D3/D9/D7 /CS/CT/AS/D2/CX/D8/CX/D3/D2/D7 /CP/D2/CS /D8/CW/CT /BF/B9/DA /CT
/D8/D3/D6/D7 E /CP/D2/CSB /BA/C1/D2 /D8/CW/CT /D6/CT
/CT/D2 /D8 /DB /D3/D6/CZ[14] /D8/CW/CT
/D3 /DA /CP/D6/CX/CP/D2 /D8 /C5/CP /CY/D3/D6/CP/D2/CP /CU/D3/D6/D1 /D9/D0/CP/D8/CX/D3/D2 /D3/CU /CT/D0/CT
/D8/D6/D3 /CS/DD/D2/CP/D1/CX
/D7 /CX/D7
/D3/D2/D7/D8/D6/D9
/D8/CT/CS/BA/CC/CW/CT/D6/CT/B8 /D8/CW/CT
/D3 /DA /CP/D6/CX/CP/D2 /D8 /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2 /B4/DB/CX/D8/CW /D8/CW/CT /BG/B9/DA /CT
/D8/D3/D6/D7 Eα/CP/D2/CSBα/B5 /CU/D3/D6Tµν/CX/D7 /D3/CQ/D8/CP/CX/D2/CT/CS/B8 /CP/D2/CS /CX/D8 /CX/D7 /CT/D5/D9/CP/D0/D8/D3 /D3/D9/D6 /BX/D5/BA /B4/BL/B5/BA /BU/D9/D8/B8 /CP/D7 /DB /CT /CW/CP /DA /CT /D7/CP/CX/CS/B8 /BX/D7/D4 /D3/D7/CX/D8/D3 [14]
/D3/D2/D7/CX/CS/CT/D6/D7 /D8/CW/CT /AS/CT/D0/CS/D7 Eα/CP/D2/CSBα/CP/D7 /AH/CP/D9/DC/CX/D0/CX/CP/D6/DD/AH/AS/CT/D0/CS/D7/B8 /DB/CW/CX/D0/CT /D8/CW/CT /BF/B9/DA /CT
/D8/D3/D6 /AS/CT/D0/CS/D7E /CP/D2/CSB /CP/D6/CT
/D3/D2/D7/CX/CS/CT/D6/CT/CS /CP/D7 /D4/CW /DD/D7/CX
/CP/D0 /AS/CT/D0/CS/D7/BA /CC/CW/CT /D7/CX/D8/D9/CP/D8/CX/D3/D2 /CX/D7 /CY/D9/D7/D8 /D8/CW/CT/D3/D4/D4 /D3/D7/CX/D8/CT /CX/D2 /D3/D9/D6 /CP/D0/D8/CT/D6/D2/CP/D8/CX/DA /CT
/D3 /DA /CP/D6/CX/CP/D2 /D8 /CP/D4/D4/D6/D3/CP
/CW/BA/CC/CW/CT /D4/D6/CT
/CT/CS/CX/D2/CV /CS/CX/D7
/D9/D7/D7/CX/D3/D2 /CX/D2/CS/CX
/CP/D8/CT/D7 /D8/CW/CP/D8 /D8/CW/CT
/D3/D6/D6/CT
/D8 /CS/CT/AS/D2/CX/D8/CX/D3/D2/D7 /CU/D6/D3/D1 /D8/CW/CT /AH/CC/CC /DA/CX/CT/DB/D4 /D3/CX/D2 /D8/AH /D3/CU /D8/CW/CT/CT/D0/CT
/D8/D6/D3/D1/CP/CV/D2/CT/D8/CX
/BG/B9/D1/D3/D1/CT/D2 /D8/D9/D1 /CW/CP /DA /CT /D8/D3
/D3/D2 /D8/CP/CX/D2 /D3/D2/D0/DD
/D3 /DA /CP/D6/CX/CP/D2 /D8/D0/DD /CS/CT/AS/D2/CT/CS /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7/B8 /CX/D2
/D0/D9/CS/CX/D2/CV /D8/CW/CT/BG/B9/DA /CT
/D8/D3/D6/D7 Eα/CP/D2/CSBα/CX/D2/D7/D8/CT/CP/CS /D3/CU /D8/CW/CT /D9/D7/D9/CP/D0 /BF/B9/DA /CT
/D8/D3/D6/D7 E /CP/D2/CSB /B8 /CP/D2/CS /D8/CW/CT /CC/CC /D3/CU /BG/BW /D8/CT/D2/D7/D3/D6 /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7/CX/D2/D7/D8/CT/CP/CS /D3/CU /D8/CW/CT /BT /CC/BA /CB/D9
/CW /D6/CT/D0/CP/D8/CX/D3/D2/D7 /CP/D6/CT /D8/CW/CT /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2/D7 /B4/BJ/B9/BD/BH/B5 /CX/D2 /D8/CW/CX/D7 /D4/CP/D4 /CT/D6/BA/BD/BD/BJ /BW/C1/CB/BV/CD/CB/CB/C1/C7/C6 /BT/C6/BW /BV/C7/C6/BV/C4/CD/CB/C1/C7/C6/CB/CC/CW/CT /CU/D9/D2/CS/CP/D1/CT/D2 /D8/CP/D0 /CS/CX/AR/CT/D6/CT/D2
/CT /CQ /CT/D8 /DB /CT/CT/D2 /D8/CW/CT /CP/D4/D4/CP/D6/CT/D2 /D8 /CP/D2/CS /D8/D6/D9/CT /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2/D7 /D3/CU /D4/CW /DD/D7/CX
/CP/D0 /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7/B8/DB/CW/CX
/CW /CX/D7 /D4/D6/CT/DA/CX/D3/D9/D7/D0/DD /D1/CP/CX/D2/D0/DD /D3 /DA /CT/D6/D0/D3 /D3/CZ /CT/CS/B8 /CT/D2/CP/CQ/D0/CT/CS /D9/D7 /D8/D3 /D6/CT/DA /CT/CP/D0 /D8/CW/CP/D8 /D8/CW/CT /D9/D7/D9/CP/D0 /CU/D3/D6/D1 /D9/D0/CP/D8/CX/D3/D2 /D3/CU /CT/D0/CT
/B9/D8/D6/D3 /CS/DD/D2/CP/D1/CX
/D7 /DB/CX/D8/CW /D8/CW/CT /BF/B9/DA /CT
/D8/D3/D6/D7 E /CP/D2/CSB /CX/D7 /D2/D3/D8 /CX/D2 /CP/CV/D6/CT/CT/D1/CT/D2 /D8 /DB/CX/D8/CW /D8/CW/CT /AH/CC/CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD /BA/AH /BW/CX/AR/CT/D6/CT/D2 /D8/CS/CT/AS/D2/CX/D8/CX/D3/D2/D7 /D3/CU /D8/CW/CT /CT/D0/CT
/D8/D6/D3/D1/CP/CV/D2/CT/D8/CX
/CT/D2/CT/D6/CV/DD /CP/D2/CS /D1/D3/D1/CT/D2 /D8/D9/D1 /CP/D6/CT /D7/CW/D3 /DB/D2 /D8/D3 /CQ /CT /CX/D2 /DA /CP/D0/CX/CS /CU/D6/D3/D1 /D8/CW/CT /AH/CC/CC/DA/CX/CT/DB/D4 /D3/CX/D2 /D8/AH /D7/CX/D2
/CT /D8/CW/CT/DD
/D3/D2 /D8/CP/CX/D2 /CT/CX/D8/CW/CT/D6 /D8/CW/CT /BT /CC /D3/CU /DA /D3/D0/D9/D1/CT/B8 /D3/D6 /D8/CW/CT /BT /CC /D3/CUE /CP/D2/CSB /B8 /D3/D6 /CQ /D3/D8/CW /D3/CU /D8/CW/CT/D1/BA /CF /CT/CW/CP /DA /CT
/D3/D2/D7/D8/D6/D9
/D8/CT/CS /CP
/D3 /DA /CP/D6/CX/CP/D2 /D8 /CU/D3/D6/D1 /D9/D0/CP/D8/CX/D3/D2 /D3/CU /CT/D0/CT
/D8/D6/D3 /CS/DD/D2/CP/D1/CX
/D7 /DB/CX/D8/CW /D8/CW/CT /BG/B9/DA /CT
/D8/D3/D6/D7 Eα/CP/D2/CSBα/CT/D5/D9/CP/D0/D0/DD/CP/D7 /DA /CP/D0/CX/CS /CP/D7 /D8/CW/CT /D9/D7/D9/CP/D0
/D3 /DA /CP/D6/CX/CP/D2 /D8 /CP/D4/D4/D6/D3/CP
/CW /DB/CX/D8/CWFαβ/BA /CC/CW/CT
/D3 /DA /CP/D6/CX/CP/D2 /D8 /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2 /CU/D3/D6 /D8/CW/CT /D7/DD/D1/D1/CT/D8/D6/CX
/CT/D2/CT/D6/CV/DD/B9/D1/D3/D1/CT/D2 /D8/D9/D1 /CS/CT/D2/D7/CX/D8 /DD /D8/CT/D2/D7/D3/D6 Tµν/CX/D7 /D3/CQ/D8/CP/CX/D2/CT/CS /CQ /DD /D1/CT/CP/D2/D7 /D3/CUEα/CP/D2/CSBα/BA /CC/CW/CT /CT/D0/CT
/D8/D6/D3/D1/CP/CV/D2/CT/D8/CX
/BG/B9/D1/D3/D1/CT/D2 /D8/D9/D1Pµ
f
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/CW /DB/CX/D8/CWEα/CP/D2/CSBα/CX/D7 /D2/D3/D8 /D6/CT/D7/D8/D6/CX
/D8/CT/CS /D8/D3 /D8/CW/CT
/D0/CP/D7/D7/CX
/CP/D0 /CT/D0/CT
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/D8/D6/D3 /CS/DD/D2/CP/D1/CX
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/D8/D3/D6/D7 E /CP/D2/CSB /B8/B4/DB/CW/D3/D7/CT /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2/D7 /CP/D6/CT /D8/CW/CT /BT /CC/B5/B8
/CP/D2 /CQ /CT /D6/CT/D4/D0/CP
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/D8/D3/D6/D7 Eα/CP/D2/CSBα/B8 /B4/DB/CW/CX
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/D8/D6/D3/D1/CP/CV/D2/CT/D8/CX
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/CT/D7/CJ/BD℄ /BY/BA /CA/D3/CW/D6/D0/CX
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/CW/B8 /C6/D9/D3/DA/D3 /BV/CX/D1/CT/D2/D8/D3 /BU /BG/BH /B8 /BJ/BI /B4/BD/BL/BI/BI/B5/BA/CJ/BH℄ /BT/BA /BZ/CP/D1 /CQ/CP/B8 /BT /D1/BA /C2/BA /C8/CW/DD/D7/BA /BF/BH /B8 /BK/BF /B4/BD/BL/BI/BJ/B5/BA/CJ/BI℄ /CC/BA /C1/DA /CT/DE/CX/EI/B8 /D4/D6/CT/D4/D6/CX/D2 /D8 /CB/BV/BT/C6/B9/BL/BK/BC/BE/BC/BD/BK/BN /D4/D6/CT/D4/D6/CX/D2 /D8 /CB/BV/BT/C6/B9/BL/BK/BD/BC/BC/BJ/BK /B4/D3/D2 /D8/CW/CT /BV/BX/CA/C6 /D7/CT/D6/DA /CT/D6/B5/BA/CJ/BJ℄ /G0/BA /BZ/D6/GW/D2/B8 /BT /D1/BA /C2/BA /C8/CW/DD/D7/BA /BG/BI /B8 /BE/BG/BL /B4/BD/BL/BJ/BK/B5/BN /CC/BA /C1/DA /CT/DE/CX/EI/B8 /C8/CW/DD/D7/BA /CA /CT/DA/BA /BX /BH/BE /B8 /BH/BH/BC/BH /B4/BD/BL/BL/BH/B5/BA/CJ/BK℄ /BT/BA /BX/CX/D2/D7/D8/CT/CX/D2/B8 /BT /D2/D2/BA /C8/CW/DD/D7/CX/CZ /BD/BJ /B8 /BK/BL/BD /B4/BD/BL/BC/BH/B5/B8 /D8/D6/BA /CQ /DD /CF/BA /C8 /CT/D6/D6/CT/D8/D8 /CP/D2/CS /BZ/BA/BU/BA /C2/CT/AR/CT/D6/DD /B8 /CX/D2 /CC/CW/CT /D4/D6/CX/D2
/CX/D4/D0/CT/D3/CU /D6 /CT/D0/CP/D8/CX/DA/CX/D8/DD /B4/BW/D3 /DA /CT/D6/B8 /C6/CT/DB /CH /D3/D6/CZ/B5/BA/CJ/BL℄ /BV/BA/CF/BA /C5/CX/D7/D2/CT/D6/B8 /C3/BA/CB/BA /CC/CW/D3/D6/D2/CT /CP/D2/CS /C2/BA/BT/BA /CF/CW/CT/CT/D0/CT/D6/B8 /BZ/D6 /CP/DA/CX/D8/CP/D8/CX/D3/D2 /B8 /B4/BY /D6/CT/CT/D1/CP/D2/B8 /CB/CP/D2 /BY /D6/CP/D2
/CX/D7
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/CZ/D7/D3/D2/B8 /BV/D0/CP/D7/D7/CX
/CP/D0 /BX/D0/CT
/D8/D6 /D3 /CS/DD/D2/CP/D1/CX
/D7 /B8 /BE/D2/CS /CT/CS/D2/BA /B4/CF/CX/D0/CT/DD /B8 /C6/CT/DB /CH /D3/D6/CZ/B8 /BD/BL/BJ/BJ/B5/BA/CJ/BD/BE℄ /CA/BA /C5/BA /CF /CP/D0/CS/B8 /BZ/CT/D2/CT/D6 /CP/D0 /D6 /CT/D0/CP/D8/CX/DA/CX/D8/DD /B4/CC/CW/CT /CD/D2/CX/DA /CT/D6/D7/CX/D8 /DD /D3/CU /BV/CW/CX
/CP/CV/D3 /C8/D6/CT/D7/D7/B8 /BV/CW/CX
/CP/CV/D3/B8 /BD/BL/BK/BG/B5/BN /BW/BA/BT/BA/CC/BA/CE /CP/D2/DE/CT/D0/D0/CP/B8 /BZ/BA/BX/BA/BT/BA /C5/CP/D8/D7/CP/D7/B8 /C0/BA/CF/BA /BV/D6/CP/D8/CT/D6/B8 /BT /D1/BA /C2/BA /C8/CW/DD/D7/BA /BI/BG /B8 /BD/BC/BJ/BH /B4/BD/BL/BL/BI/B5/BA/CJ/BD/BF℄ /C2/BA/C4/BA /BT/D2/CS/CT/D6/D7/D3/D2 /CP/D2/CS /C2/BA/CF/BA /CA/DD /D3/D2/B8 /C8/CW/DD/D7/BA /CA /CT/DA/BA /BD/BK/BD /B8 /BD/BJ/BI/BH /B4/BD/BL/BI/BL/B5/BN /C8 /BA /C0/CX/D0/D0/CX/D3/D2/B8 /C8/CW/DD/D7/BA /CA /CT/DA/BA /BX /BG/BK /B8 /BF/BC/BI/BC/B4/BD/BL/BL/BF/B5/BA/CJ/BD/BG℄ /CB/BA /BX/D7/D4 /D3/D7/CX/D8/D3/B8 /BY /D3/D9/D2/CS/BA /C8/CW/DD/D7/BA /BE/BK /B8 /BE/BF/BD /B4/BD/BL/BL/BK/B5/BA/CJ/BD/BH℄ /BT/BA/C3/BA /CB/CX/D2/CV/CP/D0/B8 /C2/BA /C8/CW/DD/D7/BA /BT /BE/BH /B8 /BD/BI/BC/BH /B4/BD/BL/BL/BE/B5/BA/BD/BE/CJ/BD/BI℄ /BX/BA /BV/D3/D1/CP /DD /B8 /CI/BA /C6/CP/D8/D9/D6/CU/D3/D6/D7
/CW/BA /BG/BI/CP /B8 /BF/BJ/BJ /B4/BD/BL/BL/BD/B5/BA/CJ/BD/BJ℄ /C0/BA /C8 /D3/CX/D2
/CP/D6/GH/B8 /CA /CT/D2/CS/BA /BV/CX/D6
/BA /C5/CP/D8/BA /C8/CP/D0/CT/D6/D1/D3 /BE/BD /B8 /BD/BE/BL /B4/BD/BL/BC/BI/B5/BA/CJ/BD/BK℄ /C5/BA /DA /D3/D2 /C4/CP/D9/CT/B8 /C8/CW/DD/D7/BA /CI/CT/CX/D8/D7/BA /BD/BE /B8 /BD/BC/BC/BK /B4/BD/BL/BD/BD/B5/BA/CJ/BD/BL℄ /CF/BA /C8 /CP/D9/D0/CX/B8 /CC/CW/CT /D8/CW/CT/D3/D6/DD /D3/CU /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD /B4/C8 /CT/D6/CV/CP/D1/D3/D2 /C8/D6/CT/D7/D7/B8 /C4/D3/D2/CS/D3/D2/B8 /BD/BL/BH/BK/B5/BN /CF/BA/C3/BA/C0/BA /C8 /CP/D2/D3/CU/D7/CZ/DD /CP/D2/CS /C5/BA/C8/CW/CX/D0/D0/CX/D4/D7/B8 /BV/D0/CP/D7/D7/CX
/CP/D0 /CT/D0/CT
/D8/D6/CX
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/D3/D2/CX/B8 /C6/D9/D3/DA/D3 /BV/CX/D1/CT/D2/D8/D3 /BU /BD/BC/BG /B8 /BH/BG/BH /B4/BD/BL/BK/BL/B5/BN /BV/BA /C4/CT/D9/CQ/D2/CT/D6/B8 /C3/BA /BT/D9/AS/D2/CV/CT/D6/CP/D2/CS /C8 /BA /C3/D6/D9/D1/D1/B8 /BX/D9/D6/BA /C2/BA /C8/CW/DD/D7/BA /BD/BF /B8 /BD/BJ/BC /B4/BD/BL/BL/BE/B5/BA/CJ/BE/BE℄ /BY/BA /CA/D3/CW/D6/D0/CX
/CW/B8 /BV/D0/CP/D7/D7/CX
/CP/D0
/CW/CP/D6 /CV/CT /CS /D4 /CP/D6/D8/CX
/D0/CT/D7 /B8 /B4/BT /CS/CS/CX/D7/D3/D2/B9/CF /CT/D7/D0/CT/DD /B8 /CA/CT/CP/CS/CX/D2/CV/B8 /C5/BT/B8 /BD/BL/BI/BH/B5/BA/CJ/BE/BF℄ /BY/BA /CA/D3/CW/D6/D0/CX
/CW/B8 /BT /D1/BA /C2/BA /C8/CW/DD/D7/BA /BF/BK /B8 /BD/BF/BD/BC /B4/BD/BL/BJ/BC/B5/BA/CJ/BE/BG℄ /CA/BA/C0/BA /CA/D3/D1/CT/D6/B8 /BT /D1/BA /C2/BA /C8/CW/DD/D7/BA /BI/BF /B8 /BJ/BJ/BJ /B4/BD/BL/BL/BH/B5/BA/CJ/BE/BH℄ /C3/BA /C5
/BW/D3/D2/CP/D0/CS/B8 /BT /D1/BA /C2/BA /C8/CW/DD/D7/BA /BI/BG /B8 /BD/BH /B4/BD/BL/BL/BI/B5/BN /BY/BA /CA/D3/CW/D6/D0/CX
/CW/B8 /BT /D1/BA /C2/BA /C8/CW/DD/D7/BA /BI/BG /B8 /BD/BI /B4/BD/BL/BL/BI/B5/BN /BU/BA/CA/BA/C0/D3/D0/D7/D8/CT/CX/D2/B8 /BT /D1/BA /C2/BA /C8/CW/DD/D7/BA /BI/BG /B8 /BD/BJ /B4/BD/BL/BL/BI/B5/BA/CJ/BE/BI℄ /C2/BA /CB
/CW /DB/CX/D2/CV/CT/D6/B8 /BY /D3/D9/D2/CS/BA /C8/CW/DD/D7/BA /BD/BF /B8 /BF/BJ/BF /B4/BD/BL/BK/BF/B5/BA/CJ/BE/BJ℄ /C2/BA /BY /D6/CT/D2/CZ /CT/D0/B8 /C8/CW/DD/D7/BA /CA /CT/DA/BA /BX /BH/BG /B8 /BH/BK/BH/BL /B4/BD/BL/BL/BI/B5/BN /C0/BA /C3 /D3/D0/CQ /CT/D2/D7/D8/DA /CT/CS/D8/B8 /C8/CW/DD/D7/BA /C4 /CT/D8/D8/BA /BT /BE/BF/BG /B8 /BF/BD/BL /B4/BD/BL/BL/BJ/B5/BA/BD/BF |
arXiv:physics/0102014v1 [physics.gen-ph] 6 Feb 2001
/CA/BX/C4/BT /CC/C1/CE/BX/C4 /CH /C5/C7 /CE/C1/C6/BZ /CB/CH/CB/CC/BX/C5/CB /C1/C6 /AG/CC/CA /CD/BX /CC/CA/BT/C6/CB/BY /C7/CA/C5/BT /CC/C1/C7/C6/CB/CA/BX/C4/BT /CC/C1/CE/C1/CC/CH/AH/CC /D3/D1/CX/D7/D0/CP /DA /C1/DA /CT/DE/CX/EI/CA/D9 /CS /CT/D6 /BU/D3/EY/CZ/D3/DA/CX/EI /C1/D2/D7/D8/CX/D8/D9/D8/CT/B8 /C8/BA/C7/BA/BU/BA /BD/BK/BC/B8 /BD/BC/BC/BC/BE /CI/CP/CV/D6 /CT/CQ/B8 /BV/D6 /D3 /CP/D8/CX/CP/CX/DA /CT/DE/CX
/D6/D9/CS/CY/CT/D6/BA/CX/D6/CQ/BA/CW/D6/C1/D2 /D8/CW/CX/D7 /D4/CP/D4 /CT/D6 /D8/CW/CT /D4/CW /DD/D7/CX
/CP/D0 /D7/DD/D7/D8/CT/D1/D7
/D3/D2/D7/CX/D7/D8/CX/D2/CV /D3/CU /D6/CT/D0/CP/D8/CX/DA /CT/D0/DD /D1/D3 /DA/CX/D2/CV /D7/D9/CQ/D7/DD/D7/D8/CT/D1/D7 /CP/D6/CT
/D3/D2/D7/CX/CS/CT/D6/CT/CS/CX/D2 /D8/CW/CT /AH/D8/D6/D9/CT /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2/D7 /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH/BA /C1/D8 /CX/D7 /CU/D3/D9/D2/CS /CX/D2 /CP /D1/CP/D2/CX/CU/CT/D7/D8/D0/DD
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/D8/D6/CX
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/CW/CP/D6/CV/CT/D7 /D3/D2 /D3/D4/D4 /D3/D7/CX/D8/CT /D7/CX/CS/CT/D7 /D3/CU /CP /D7/D5/D9/CP/D6/CT /D0/D3 /D3/D4 /DB/CX/D8/CW
/D9/D6/D6/CT/D2 /D8 /CP/D2/CS /D8/CW/CT/D7/CT
/CW/CP/D6/CV/CT/D7 /CP/D6/CT /CX/D2 /DA /CP/D6/CX/CP/D2 /D8
/CW/CP/D6/CV/CT/D7/BA/C3/CT/DD /DB /D3/D6/CS/D7/BM
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/D9/D6/D6/CT/D2 /D8/B8 /CT/D0/CT
/D8/D6/CX
/AS/CT/D0/CS /CP/D2/CS
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/CW /CX/D7/CP
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/D8/D6/D3 /CS/DD/D2/CP/D1/CX
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/CT/D8/CX/D1/CT /DB/CX/D8/CW /D4/D7/CT/D9/CS/D3/B9/BX/D9
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/CT/D8/CX/D1/CT /D9/D2
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/D3/D2 /DA /CT/D2 /D8/CX/D3/D2/CP/D0 /D7/D4 /CT
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/CW/D6/D3/D2/D3/D9/D7 /CS/CT/AS/D2/CX/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D7/D4/CP/D8/CX/CP/D0 /D0/CT/D2/CV/D8/CW/B8 /CX/BA/CT/BA/B8 /D3/D2 /D8/CW/CT /BT /CC /D3/CU /D8/CW/CT/D7/D4/CP/D8/CX/CP/D0 /D0/CT/D2/CV/D8/CW /B4/D8/CW/CT /C4/D3/D6/CT/D2 /D8/DE
/D3/D2 /D8/D6/CP
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/CT /B4/D8/CW/CT
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/D8/D6/CX
/CP/D2/CS /D1/CP/CV/D2/CT/D8/CX
/D8/CW/D6/CT/CT/B9/DA /CT
/D8/D3/D6/D7 /B4/BF/B9/DA /CT
/D8/D3/D6/D7/B5 E /CP/D2/CSB /B4/D8/CW/CT
/D3/D2 /DA /CT/D2 /D8/CX/D3/D2/CP/D0 /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2/D7 /D3/CUE /CP/D2/CSB /B5/BA /CC/CW/CT /BT /CC /CP/D6 /CT/D2/D3/D8 /D8/CW/CT /D8/D6 /CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2/D7 /D3/CU /BG/BW /D7/D4 /CP
/CT/D8/CX/D1/CT /D8/CT/D2/D7/D3/D6/D7 /CP/D2/CS /D8/CW/CT/DD /CS/D3 /D2/D3/D8 /D6 /CT/CU/CT/D6 /D8/D3 /D8/CW/CT /D7/CP/D1/CT /D5/D9/CP/D2/D8/CX/D8/DD /B4/CX/D2 /BG/BW/D7/D4 /CP
/CT/D8/CX/D1/CT/B5/B8 /CQ/D9/D8/B8 /CT/BA/CV/BA/B8 /D8/CW/CT/DD /D6 /CT/CU/CT/D6 /D8/D3 /D8/CW/CT /D7/CP/D1/CT /D1/CT /CP/D7/D9/D6 /CT/D1/CT/D2/D8 /CX/D2 /CS/CX/AR/CT/D6 /CT/D2/D8 /C1/BY/CA/D7/BA/C1/D2 /D8/CW/CX/D7 /D4/CP/D4 /CT/D6 /DB /CT /CX/D2 /DA /CT/D7/D8/CX/CV/CP/D8/CT /D4/CW /DD/D7/CX
/CP/D0 /D7/DD/D7/D8/CT/D1/D7
/D3/D2/D7/CX/D7/D8/CX/D2/CV /D3/CU /D6/CT/D0/CP/D8/CX/DA /CT/D0/DD /D1/D3 /DA/CX/D2/CV /D7/D9/CQ/D7/DD/D7/D8/CT/D1/D7/B8 /CP/D7 /CX/D8/CX/D7 /CP
/D9/D6/D6/CT/D2 /D8/B9
/CP/D6/D6/DD/CX/D2/CV
/D3/D2/CS/D9
/D8/D3/D6 /B4/BV/BV/BV/B5/B8 /D9/D7/CX/D2/CV /CP
/D3 /DA /CP/D6/CX/CP/D2 /D8 /CU/D3/D6/D1 /D9/D0/CP/D8/CX/D3/D2 /D3/CU /D4/CW /DD/D7/CX
/CP/D0 /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7 /CP/D2/CS/D4/CW /DD/D7/CX
/CP/D0 /D4/CW/CT/D2/D3/D1/CT/D2/CP/B8 /CX/BA/CT/BA/B8 /D8/CW/CT /AH/CC/CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH/BA /BY/CX/D6/D7/D8 /DB /CT /CT/DC/CP/D1/CX/D2/CT /D8/CW/CT
/D3 /DA /CP/D6/CX/CP/D2 /D8 /CS/CT/AS/D2/CX/D8/CX/D3/D2 /D3/CU /D0/CT/D2/CV/D8/CW/DB/CW/CT/D2 /CS/CT/AS/D2/CT/CS /CX/D2 /CV/CT/D3/D1/CT/D8/D6/CX
/CP/D0 /D8/CT/D6/D1/D7 /CP/D2/CS /CX/D2 /CS/CX/AR/CT/D6/CT/D2 /D8
/D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/D7 /D3/CU /CP/D2 /C1/BY/CA/BA /CF /CT /CP/D0/D7/D3 /D6/CT/D4 /D3/D6/D8 /CP/D2/CT/DC/D4/D6/CT/D7/D7/CX/D3/D2 /CU/D3/D6 /D8/CW/CT /C4/D3/D6/CT/D2 /D8/DE /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2/D7/B8 /DB/CW/CX
/CW /CX/D7 /CX/D2/CS/CT/D4 /CT/D2/CS/CT/D2 /D8 /D3/CU /D8/CW/CT
/CW/D3/D7/CT/D2 /D7/DD/D2
/CW/D6/D3/D2/CX/DE/CP/D8/CX/D3/D2/B8/CX/BA/CT/BA/B8
/D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2 /D3/CU /CP/D2 /C1/BY/CA/BA /BY /D9/D6/D8/CW/CT/D6/B8 /D8/CW/CT /BT /CC /D3/CU /D8/CW/CT /D7/D4/CP/D8/CX/CP/D0 /D0/CT/D2/CV/D8/CW /B9 /D8/CW/CT /C4/D3/D6/CT/D2 /D8/DE
/D3/D2 /D8/D6/CP
/D8/CX/D3/D2 /B9 /CX/D7/CT/DC/CP/D1/CX/D2/CT/CS /CX/D2 /CS/CT/D8/CP/CX/D0/BA /CC/CW/CT/D2 /D8/CW/CT
/D3 /DA /CP/D6/CX/CP/D2 /D8 /CS/CT/AS/D2/CX/D8/CX/D3/D2 /D3/CU /D0/CT/D2/CV/D8/CW /CX/D2 /BX/CX/D2/D7/D8/CT/CX/D2/B3/D7
/D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2 /CX/D7 /CP/D4/D4/D0/CX/CT/CS/D8/D3 /D8/CW/CT
/D3/D2/D7/CX/CS/CT/D6/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /DB /CT/D0/D0/B9/CZ/D2/D3 /DB/D2 /AH/D6/CT/D0/CP/D8/CX/DA/CX/D7/D8/CX
/AH /D4/CP/D6/CP/CS/D3 /DC /AH/BV/CP/D6 /CP/D2/CS /CV/CP/D6/CP/CV/CT /D4/CP/D6/CP/CS/D3 /DC/BA/AH /C1/D8 /CX/D7 /CU/D3/D9/D2/CS/D8/CW/CP/D8 /CX/D2 /D8/CW/CT /AH/CC/CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH /CP/D2/CS/B8 /CX/CU /D3/D2/CT /DB/CP/D2/D8/D7 /D8/D3 /D6 /CT/D8/CP/CX/D2 /D8/CW/CT
/D3/D2/D2/CT
/D8/CX/D3/D2 /DB/CX/D8/CW /D8/CW/CT /D4/D6 /CT/D6 /CT/D0/CP/D8/CX/DA/CX/D7/D8/CX
/D4/CW/DD/D7/CX
/D7/CX/D2 /DB/CW/CX
/CW /D3/D2/CT /CS/CT /CP/D0/D7 /DB/CX/D8/CW /D8/CW/CT /AH/D7/D4 /CP/D8/CX/CP/D0 /D0/CT/D2/CV/D8/CW/AH /B8 /D8/CW/CT/D2 /D3/D2/D0/DD /D8/CW/CT /D6/CT/D7/D8 /D0/CT/D2/CV/D8/CW /B4/DA /D3/D0/D9/D1/CT/B5 /D3/CU /D8/CW/CT /D3/CQ /CY/CT
/D8 /CX/D7 /DB /CT/D0/D0/CS/CT/AS/D2/CT/CS /D5/D9/CP/D2 /D8/CX/D8 /DD /BA/BQ/BY /D6/D3/D1 /D8/CW/CX/D7 /D6/CT/D7/D9/D0/D8 /CP/D2/CS /D8/CW/CT
/D3 /DA /CP/D6/CX/CP/D2 /D8 /CS/CT/AS/D2/CX/D8/CX/D3/D2 /D3/CU
/CW/CP/D6/CV/CT /DB /CT /CP/D0/D7/D3 /AS/D2/CS /D8/CW/CP/D8 /CX/D2 /D8/CW/CT /AH/CC/CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH/D8/CW/CT
/CW/CP/D6/CV/CT /CS/CT/D2/D7/CX/D8 /DD /CP/D7 /D8/CW/CT /D8/CW/D6/CT/CT/B9/CS/CX/D1/CT/D2/D7/CX/D3/D2/CP/D0 /B4/BF/BW/B5 /D5/D9/CP/D2 /D8/CX/D8 /DD /CW/CP/D7 /CS/CT/AS/D2/CX/D8/CT /D4/CW /DD/D7/CX
/CP/D0 /D1/CT/CP/D2/CX/D2/CV /D3/D2/D0/DD /CU/D3/D6/BD
/CW/CP/D6/CV/CT/D7 /CP/D8 /D6/CT/D7/D8/BA /C1/D2 /D3/D6/CS/CT/D6 /D8/D3 /CP /DA /D3/CX/CS /CU/D6/D3/D1 /D8/CW/CT /CQ /CT/CV/CX/D2/D2/CX/D2/CV /D8/CW/CT /D1/CX/D7/D9/D2/CS/CT/D6/D7/D8/CP/D2/CS/CX/D2/CV /D3/CU /D8/CW/CT /AH/CC/CC /D6/CT/D0/CP/D8/CX/DA/B9/CX/D8 /DD/AH /CP/D2/CS /D3/CU /D3/D9/D6
/CW/D3/CX
/CT /D3/CU /D8/CW/CT /D6/CT/D7/D8 /CU/D6/CP/D1/CT /D3/CU /D8/CW/CT /D3/CQ /CY/CT
/D8/B8 /CP/D7 /D8/CW/CT /D7/D8/CP/D6/D8/CX/D2/CV /CU/D6/CP/D1/CT /CU/D3/D6 /D8/CW/CT /CS/CT/AS/D2/CX/D8/CX/D3/D2/D7 /D3/CU/BG/BW /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7/B8 /DB /CT /CT/D1/D4/CW/CP/D7/CX/DE/CT /D8/CW/CP/D8 /D8/CW/CT /AH/CC/CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH /CX/D7
/D3 /DA /CP/D6/CX/CP/D2 /D8 /CX/D2 /D8/CW/CT /D9/D7/D9/CP/D0 /D7/CT/D2/D7/CT/BA /C1/D2 /D8/CW/CT /AH/CC/CC/D6 /CT/D0/CP/D8/CX/DA/CX/D8/DD/AH /D3/D2/CT
/CP/D2 /CS/CT/AS/D2/CT /BG/BW /D4/CW/DD/D7/CX
/CP/D0 /D5/D9/CP/D2/D8/CX/D8/CX/CT/D7 /CP/D2/CS /CX/D2/DA/CT/D7/D8/CX/CV/CP/D8/CT /D4/CW/DD/D7/CX
/CP/D0 /D0/CP/DB/D7
/D3/D2/D2/CT
/D8/CX/D2/CV /D7/D9
/CW /BG/BW/D5/D9/CP/D2/D8/CX/D8/CX/CT/D7 /CX/D2 /CP/D2/DD /C1/BY/CA/B8 /D2/D3/D8 /D3/D2/D0/DD /CX/D2 /D8/CW/CT /D6 /CT/D7/D8 /CU/D6 /CP/D1/CT /D3/CU /D8/CW/CT /D3/CQ/CY/CT
/D8/BA /CC/CW/CT /C4 /CC /DB/CX/D0 /D0
/D3/D6/D6 /CT
/D8/D0/DD
/D3/D2/D2/CT
/D8 /D8/CW/CT/D6 /CT/D7/D9/D0/D8/D7 /D3/CU /D1/CT /CP/D7/D9/D6 /CT/D1/CT/D2/D8/D7 /D3/CU /D8/CW/CT /D7/CP/D1/CT /BG/BW /D5/D9/CP/D2/D8/CX/D8/DD /CX/D2 /D8/DB/D3/B8 /CP/D6/CQ/CX/D8/D6 /CP/D6/DD/B8 /D6 /CT/D0/CP/D8/CX/DA/CT/D0/DD /D1/D3/DA/CX/D2/CV /C1/BY/CA/D7/BA /CC/CW /D9/D7/B8/D8/CW/CT /AH/CC/CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH /CS/D3 /CT/D7 /D2/D3/D8 /D9/D7/CT /CP /D4/D6/CT/CU/CT/D6/D6/CT/CS /D6/CT/CU/CT/D6/CT/D2
/CT /CU/D6/CP/D1/CT/BA /C7/D9/D6
/CW/D3/CX
/CT /D3/CU /D8/CW/CT /D6/CT/D7/D8 /CU/D6/CP/D1/CT /D3/CU /D8/CW/CT/D3/CQ /CY/CT
/D8 /CS/D3 /CT/D7 /D2/D3/D8 /D1/CT/CP/D2 /CX/D2 /CP/D2 /DD /DB /CP /DD /D8/CW/CP/D8 /D8/CW/CX/D7 /CU/D6/CP/D1/CT /CX/D7 /CP /D4/D6/CT/CU/CT/D6/D6/CT/CS /C1/BY/CA/BA /CC/CW/CT /D6/CT/D7/D8 /CU/D6/CP/D1/CT /CX/D7/B8 /CX/D2 /CU/CP
/D8/B8/D8/CW/CT /D1/D3/D7/D8
/D3/D2 /DA /CT/D2/CX/CT/D2 /D8 /CU/D3/D6 /D8/CW/CT /D4/D9/D6/D4 /D3/D7/CT /D3/CU
/D3/D1/D4/CP/D6/CX/D7/D3/D2 /DB/CX/D8/CW /D8/CW/CT /D4/D6/CT/D6/CT/D0/CP/D8/CX/DA/CX/D7/D8/CX
/D4/CW /DD/D7/CX
/D7/B8 /CX/D2 /DB/CW/CX
/CW /D3/D2/CT/CS/D3 /CT/D7 /D2/D3/D8 /CS/CT/CP/D0 /DB/CX/D8/CW /BG/BW /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7 /CQ/D9/D8 /DB/CX/D8/CW /AH/BF/B7/BD/AH /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7 /B4/D8/CW/CT /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7 /CS/CT/AS/D2/CT/CS /CX/D2 /AH/BF/B7/BD/AH /D7/D4/CP
/CT/CP/D2/CS /D8/CX/D1/CT/B5/B8 /CP/D2/CS /DB/CX/D8/CW /D8/CW/CT /AH/BT /CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH/B8 /CX/D2 /DB/CW/CX
/CW /D3/D2/CT /DB /D3/D6/CZ/D7 /CX/D2 /BG/BW /D7/D4/CP
/CT/D8/CX/D1/CT /CQ/D9/D8 /DB/CX/D8/CW /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7/B8/CT/BA/CV/BA/B8 /D8/CW/CT /D7/D4/CP/D8/CX/CP/D0 /D0/CT/D2/CV/D8/CW/B8 /D8/CW/CT /D8/CX/D1/CT /CS/CX/D7/D8/CP/D2
/CT/B8 /D8/CW/CT /BF/B9/DA /CT
/D8/D3/D6/D7 /BX /CP/D2/CS /BU/B8 /CT/D8
/BA/B8 /D8/CW/CP/D8 /CP/D6/CT /D2/D3/D8 /BG/BW /D8/CT/D2/D7/D3/D6/D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7/BA /CC /CP/CZ/CX/D2/CV /D8/CW/CX/D7 /CX/D2 /D8/D3 /CP
/D3/D9/D2 /D8 /DB /CT /D7/CW/D3 /DB /D8/CW/CP/D8 /D8/CW/CT
/D9/D6/D6/CT/D2 /D8 /CS/CT/D2/D7/CX/D8 /DD /BG/B9/DA /CT
/D8/D3/D6 jµ/CU/D3/D6 /CP /BV/BV/BV /CX/D2/CP/D2 /CP/D6/CQ/CX/D8/D6/CP/D6/DD /C1/BY/CA /CX/D7 /CS/CT/D8/CT/D6/D1/CX/D2/CT/CS /CP/D7 /D8/CW/CT /D7/D9/D1jµ
++jµ
−
/B8 /DB/CW/CT/D6/CT /D8/CW/CT
/D9/D6/D6/CT/D2 /D8 /CS/CT/D2/D7/CX/D8 /DD /BG/B9/DA /CT
/D8/D3/D6/D7 jµ
+
/CP/D2/CS
jµ
−
/CU/D3/D6 /D4 /D3/D7/CX/D8/CX/DA /CT /CP/D2/CS /D2/CT/CV/CP/D8/CX/DA /CT
/CW/CP/D6/CV/CT/D7/B8 /D6/CT/D7/D4 /CT
/D8/CX/DA /CT/D0/DD /B8 /CW/CP /DA /CT /D8/D3 /CQ /CT /CU/D3/D9/D2/CS /CX/D2 /D8/CW/CT/CX/D6 /D3 /DB/D2 /D6/CT/D7/D8 /CU/D6/CP/D1/CT/D7/B8 /CP/D2/CS/D8/CW/CT/D2 /D8/D6/CP/D2/D7/CU/D3/D6/D1/CT/CS /CQ /DD /D8/CW/CT /C4/D3/D6/CT/D2 /D8/DE /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2 /D8/D3 /D8/CW/CT
/D3/D2/D7/CX/CS/CT/D6/CT/CS /C1/BY/CA/BA /CC/CW/CT/D2 /CX/D2 /CB/CT
/BA/BF/BA/BE /DB /CT /D5/D9/D3/D8/CT/D8/CW/CT
/D3 /DA /CP/D6/CX/CP/D2 /D8 /C5/CP/DC/DB /CT/D0/D0 /CT/D5/D9/CP/D8/CX/D3/D2/D7 /DB/CW/CT/D2 /DB/D6/CX/D8/D8/CT/D2 /CQ /DD /D8/CW/CT /CT/D0/CT
/D8/D6/D3/D1/CP/CV/D2/CT/D8/CX
/AS/CT/D0/CS /D8/CT/D2/D7/D3/D6 Fαβ/CP/D2/CS /CQ /DD /D8/CW/CT/BG/B9/DA /CT
/D8/D3/D6/D7 Eα/CP/D2/CSBα, /B4/CQ /D3/D8/CW /CU/D3/D6/D1/D7 /DB /CT/D6/CT /CP/D0/D6/CT/CP/CS/DD /CU/D3/D9/D2/CS /CX/D2 /CJ/BD℄/B5/B8 /CP/D2/CS /CP/D0/D7/D3 /DB /CT /D6/CT/D4 /D3/D6/D8 /CP /D2/CT/DB /CU/D3/D6/D1 /B9 /D8/CW/CT
/D3 /DA /CP/D6/CX/CP/D2 /D8 /C5/CP /CY/D3/D6/CP/D2/CP /CU/D3/D6/D1 /D3/CU /C5/CP/DC/DB /CT/D0/D0/B3/D7 /CT/D5/D9/CP/D8/CX/D3/D2/D7/BA /CC/CW/CT/D2 /D8/CW/CT /BG/B9/DA /CT
/D8/D3/D6/D7 Eα/CP/D2/CSBα/CP/D6/CT /CS/CT/D8/CT/D6/D1/CX/D2/CT/CS/CU/D3/D6 /CP /BV/BV/BV /B4/CX/D2/D7/D8/CT/CP/CS /D3/CU /D8/CW/CT /D9/D7/D9/CP/D0 /BF/B9/DA /CT
/D8/D3/D6/D7 E /CP/D2/CSB) /CP/D2/CS /CX/D8 /CX/D7 /D3/CQ/D8/CP/CX/D2/CT/CS /CX/D2 /D7/D9
/CW /CP
/D3 /DA /CP/D6/CX/CP/D2 /D8 /DB /CP /DD /B8/CX/BA/CT/BA/B8 /CX/D2 /D8/CW/CT /AH/CC/CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH/B8 /D8/CW/CP/D8/B8 /CU/D3/D6 /D8/CW/CT /D3/CQ/D7/CT/D6/DA /CT/D6/D7 /CP/D8 /D6/CT/D7/D8 /CX/D2 /D8/CW/CT /D6/CT/D7/D8 /CU/D6/CP/D1/CT /D3/CU /D8/CW/CP/D8 /BV/BV/BV/B8 /D8/CW/CT/D6/CT /CX/D7 /CP/D7/CT
/D3/D2/CS/B9/D3/D6/CS/CT/D6 /CT/D0/CT
/D8/D6/CX
/AS/CT/D0/CS /D3/D9/D8/D7/CX/CS/CT /D7/D8/CP/D8/CX/D3/D2/CP/D6/DD
/D3/D2/CS/D9
/D8/D3/D6 /DB/CX/D8/CW /D7/D8/CT/CP/CS/DD
/D9/D6/D6/CT/D2 /D8/BA /CB/D9
/CW /AS/CT/D0/CS/D7 /CP/D6/CT /CP/D0/D6/CT/CP/CS/DD/D8/CW/CT/D3/D6/CT/D8/CX
/CP/D0/D0/DD /D4/D6/CT/CS/CX
/D8/CT/CS /D3/D2 /CS/CX/AR/CT/D6/CT/D2 /D8 /CV/D6/D3/D9/D2/CS/D7 /CX/D2 /CJ/BH℄/B8 /D7/CT/CT /CP/D0/D7/D3 /CJ/BI℄/BA /C1/D2
/D3/D2 /D8/D6/CP/D7/D8 /D8/D3 /D4/D6/CT/DA/CX/D3/D9/D7 /DB /D3/D6/CZ/D7 /DB /CT/CP/D0/D7/D3 /AS/D2/CS /CX/D2 /D7/D9
/CW /CP
/D3 /DA /CP/D6/CX/CP/D2 /D8 /D1/CP/D2/D2/CT/D6 /D8/CW/CP/D8 /D8/CW/CT/D6/CT /CP/D6/CT /D3/D4/D4 /D3/D7/CX/D8/CT
/CW/CP/D6/CV/CT/D7 /D3/D2 /D3/D4/D4 /D3/D7/CX/D8/CT /D7/CX/CS/CT/D7 /D3/CU /CP /D7/D5/D9/CP/D6/CT/D0/D3 /D3/D4 /DB/CX/D8/CW
/D9/D6/D6/CT/D2 /D8 /CP/D2/CS /D8/CW/CT/D7/CT
/CW/CP/D6/CV/CT/D7 /CP/D6/CT /C4/D3/D6/CT/D2 /D8/DE /CX/D2 /DA /CP/D6/CX/CP/D2 /D8
/CW/CP/D6/CV/CT/D7/BA /C1/D2 /D8/CW/CT /D9/D7/D9/CP/D0 /CP/D4/D4/D6/D3/CP
/CW/B8 /CX/BA/CT/BA/B8 /CX/D2/D8/CW/CT /AH/BT /CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH/B8 /CX/D8 /CX/D7 /CU/D3/D9/D2/CS /D8/CW/CP/D8 /D8/CW/CT/D6/CT /CX/D7 /CP/D2 /CT/D0/CT
/D8/D6/CX
/D1/D3/D1/CT/D2 /D8 P /CU/D3/D6 /CP /D1/D3 /DA/CX/D2/CV /D0/D3 /D3/D4 /DB/CX/D8/CW /CP
/D9/D6/D6/CT/D2 /D8/BA/C0/D3 /DB /CT/DA /CT/D6 /DB /CT /AS/D2/CS /D8/CW/CP/D8 /D7/D9
/CW /D0/D3 /D3/D4/B8 /D6/CT/CV/CP/D6/CS/CX/D2/CV /D8/CW/CT /CT/D0/CT
/D8/D6/CX
/CT/AR/CT
/D8/D7/B8 /CP/D0/DB /CP /DD/D7 /CQ /CT/CW/CP /DA /CT/D7 /CP/D8 /D0/D3/D2/CV /CS/CX/D7/D8/CP/D2
/CT/D7 /CP/D7/CP/D2 /CT/D0/CT
/D8/D6/CX
/D5/D9/CP/CS/D6/D9/D4 /D3/D0/CT/BA/BE/BA /BV/C7 /CE /BT/CA/C1/BT/C6/CC /BT/C6/BW /CB/CH/C6/BV/C0/CA /C7/C6/C7/CD/CB /BW/BX/BY/C1/C6/C1/CC/C1/C7/C6/CB /C7/BY/C4/BX/C6/BZ/CC/C0/BT/D7 /CS/CX/D7
/D9/D7/D7/CT/CS /CX/D2 /CJ/BD℄/B8 /B4/CP/D2/CS /CJ/BF ℄/B5 /CP
/D3/D6/CS/CX/D2/CV /D8/D3 /D8/CW/CT /AH/D1/D3 /CS/CT/D6/D2/AH /D4 /D3/CX/D2 /D8 /D3/CU /DA/CX/CT/DB /D8/CW/CT /D7/D4 /CT
/CX/CP/D0 /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD /CX/D7/D8/CW/CT /D8/CW/CT/D3/D6/DD /D3/CU /BG/BW /D7/D4/CP
/CT/D8/CX/D1/CT /DB/CX/D8/CW /D4/D7/CT/D9/CS/D3/B9/BX/D9
/D0/CX/CS/CT/CP/D2 /CV/CT/D3/D1/CT/D8/D6/DD /BA /C9/D9/CP/D2 /D8/CX/D8/CX/CT/D7 /D3/CU /D4/CW /DD/D7/CX
/CP/D0 /CX/D2 /D8/CT/D6/CT/D7/D8/B8 /CQ /D3/D8/CW/D0/D3
/CP/D0 /CP/D2/CS /D2/D3/D2/D0/D3
/CP/D0/B8 /CP/D6/CT /D6/CT/D4/D6/CT/D7/CT/D2 /D8/CT/CS /CX/D2 /D8/CW/CT /D7/D4 /CT
/CX/CP/D0 /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD /CQ /DD /D7/D4/CP
/CT/D8/CX/D1/CT /D8/CT/D2/D7/D3/D6/D7/B8 /CX/BA/CT/BA/B8 /CP/D7
/D3 /DA /CP/D6/CX/CP/D2 /D8/D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7/B8 /CP/D2/CS /D8/CW/CT /D0/CP /DB/D7 /D3/CU /D4/CW /DD/D7/CX
/D7 /CP/D6/CT /DB/D6/CX/D8/D8/CT/D2 /CX/D2 /CP /D1/CP/D2/CX/CU/CT/D7/D8/D0/DD
/D3 /DA /CP/D6/CX/CP/D2 /D8 /DB /CP /DD /CP/D7 /D8/CT/D2/D7/D3/D6/CX/CP/D0 /CT/D5/D9/CP/D8/CX/D3/D2/D7/BA/CC/CW/CT /CV/CT/D3/D1/CT/D8/D6/DD /D3/CU /D8/CW/CT /D7/D4/CP
/CT/D8/CX/D1/CT /CX/D7 /CV/CT/D2/CT/D6/CP/D0/D0/DD /CS/CT/AS/D2/CT/CS /CQ /DD /D8/CW/CT /CX/D2 /DA /CP/D6/CX/CP/D2 /D8 /CX/D2/AS/D2/CX/D8/CT/D7/CX/D1/CP/D0 /D7/D4/CP
/CT/D8/CX/D1/CT /CS/CX/D7/D8/CP/D2
/CT
ds /D3/CU /D8 /DB /D3 /D2/CT/CX/CV/CW /CQ /D3/D6/CX/D2/CV /D4 /D3/CX/D2 /D8/D7/B8 ds2=dxagabdxb. /C1 /CP/CS/D3/D4/D8 /D8/CW/CT /CU/D3/D0/D0/D3 /DB/CX/D2/CV
/D3/D2 /DA /CT/D2 /D8/CX/D3/D2 /DB/CX/D8/CW /D6/CT/CV/CP/D6/CS /D8/D3/CX/D2/CS/CX
/CT/D7/BA /CA/CT/D4 /CT/CP/D8/CT/CS /CX/D2/CS/CX
/CT/D7 /CX/D1/D4/D0/DD /D7/D9/D1/D1/CP/D8/CX/D3/D2/BA /C4/CP/D8/CX/D2 /CX/D2/CS/CX
/CT/D7 a, b, c, d, ... /CP/D6/CT /D8/D3 /CQ /CT /D6/CT/CP/CS /CP
/D3/D6/CS/CX/D2/CV /D8/D3/D8/CW/CT /CP/CQ/D7/D8/D6/CP
/D8 /CX/D2/CS/CT/DC /D2/D3/D8/CP/D8/CX/D3/D2/B8 /D7/CT/CT /CJ/BJ ℄/B8 /CB/CT
/BA/BE/BA/BG/BA/BA /CC/CW/CT/DD /CS/CT/D7/CX/CV/D2/CP/D8/CT /CV/CT/D3/D1/CT/D8/D6/CX
/D3/CQ /CY/CT
/D8/D7 /CP/D2/CS /D8/CW/CT/DD /D6/D9/D2 /CU/D6/D3/D1/BC /D8/D3 /BF/BA /CC/CW /D9/D7 dxa,b/CP/D2/CSgab, /CP/D2/CS /D3/CU
/D3/D9/D6/D7/CT ds, /CP/D6/CT /CS/CT/AS/D2/CT/CS /CX/D2/CS/CT/D4 /CT/D2/CS/CT/D2 /D8/D0/DD /D3/CU /CP/D2 /DD
/D3 /D3/D6/CS/CX/D2/CP/D8/CT /D7/DD/D7/D8/CT/D1/B8/CT/BA/CV/BA/B8gab
/CX/D7 /CP /D7/CT
/D3/D2/CS/B9/D6/CP/D2/CZ
/D3 /DA /CP/D6/CX/CP/D2 /D8 /D8/CT/D2/D7/D3/D6 /B4/DB/CW/D3/D7/CT /CA/CX/CT/D1/CP/D2/D2
/D9/D6/DA /CP/D8/D9/D6/CT /D8/CT/D2/D7/D3/D6 Ra
bcd
/CX/D7 /CT/DA /CT/D6/DD/DB/CW/CT/D6/CT/DA /CP/D2/CX/D7/CW/CX/D2/CV/BN /D8/CW/CT /D7/D4/CP
/CT/D8/CX/D1/CT /D3/CU /D7/D4 /CT
/CX/CP/D0 /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD /CX/D7 /CP /AT/CP/D8 /D7/D4/CP
/CT/D8/CX/D1/CT/B8 /CP/D2/CS /D8/CW/CX/D7 /CS/CT/AS/D2/CX/D8/CX/D3/D2 /CX/D2
/D0/D9/CS/CT/D7 /D2/D3/D8/D3/D2/D0/DD /D8/CW/CT /C1/BY/CA/D7 /CQ/D9/D8 /CP/D0/D7/D3 /D8/CW/CT /CP
/CT/D0/CT/D6/CP/D8/CT/CS /CU/D6/CP/D1/CT/D7 /D3/CU /D6/CT/CU/CT/D6/CT/D2
/CT/B5/BA /BZ/D6/CT/CT/CZ /CX/D2/CS/CX
/CT/D7 /D6/D9/D2 /CU/D6/D3/D1 /BC /D8/D3 /BF/B8 /DB/CW/CX/D0/CT/D0/CP/D8/CX/D2 /CX/D2/CS/CX
/CT/D7 i, j, k, l, ... /D6/D9/D2 /CU/D6/D3/D1 /BD /D8/D3 /BF/B8 /CP/D2/CS /D8/CW/CT/DD /CQ /D3/D8/CW /CS/CT/D7/CX/CV/D2/CP/D8/CT /D8/CW/CT
/D3/D1/D4 /D3/D2/CT/D2 /D8/D7 /D3/CU /D7/D3/D1/CT /CV/CT/D3/D1/CT/D8/D6/CX
/D3/CQ /CY/CT
/D8 /CX/D2 /D7/D3/D1/CT
/D3 /D3/D6/CS/CX/D2/CP/D8/CT
/CW/CP/D6/D8/B8 /CT/BA/CV/BA/B8xµ(x0, xi) /CP/D2/CSx′µ(x′0, x′i) /CP/D6/CT /D8 /DB /D3
/D3 /D3/D6/CS/CX/D2/CP/D8/CT /D6/CT/D4/D6/CT/D7/CT/D2 /D8/CP/D8/CX/D3/D2/D7/D3/CU /D8/CW/CT /D4 /D3/D7/CX/D8/CX/D3/D2 /BG/B9/DA /CT
/D8/D3/D6 xa/CX/D2 /D8 /DB /D3 /CS/CX/AR/CT/D6/CT/D2 /D8 /CX/D2/CT/D6/D8/CX/CP/D0
/D3 /D3/D6/CS/CX/D2/CP/D8/CT /D7/DD/D7/D8/CT/D1/D7 S /CP/D2/CSS′, /CP/D2/CSgµν
/CX/D7 /D8/CW/CT
4×4 /D1/CP/D8/D6/CX/DC /D3/CU
/D3/D1/D4 /D3/D2/CT/D2 /D8/D7 /D3/CUgab
/CX/D2 /D7/D3/D1/CT
/D3 /D3/D6/CS/CX/D2/CP/D8/CT
/CW/CP/D6/D8/BA/BE/BA/BD/BA /CC/CW/CT /D7/D4/CP
/CT/D8/CX/D1/CT /D3/D6 /D8/CW/CT /CC/CC /D0/CT/D2/CV/D8/CW/C1/D2 /CV/CT/D2/CT/D6/CP/D0/B8 /CX/D2 /BG/BW /D7/D4/CP
/CT/D8/CX/D1/CT /D3/CU /D7/D4 /CT
/CX/CP/D0 /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD /CX/D8 /CX/D7 /D2/D3/D8 /D4 /D3/D7/D7/CX/CQ/D0/CT /D8/D3 /D7/CT/D4/CP/D6/CP/D8/CT /D8/CW/CT /D7/D4/CP/D8/CX/CP/D0 /CP/D2/CS/BE/D8/CT/D1/D4 /D3/D6/CP/D0 /D4/CP/D6/D8/D7 /D3/CUds /B8 /D3/D6 /CP
/D3/D6/CS/CX/D2/CV /D8/D3 /C5/CX/D2/CZ /D3 /DB/D7/CZ/CX/B3/D7 /DB /D3/D6/CS/D7/B8 /D5/D9/D3/D8/CT/CS /CW/CT/D6/CT /CP/D7 /CP /D1/D3/D8/D8/D3/B8 /D8/CW/CT /D7/D4 /CP/D8/CX/CP/D0/CP/D2/CS /D8/CT/D1/D4 /D3/D6 /CP/D0 /D4 /CP/D6/D8/D7 /D8/CP/CZ/CT/D2 /D7/CT/D4 /CP/D6 /CP/D8/CT/D0/DD /D0/D3 /D3/D7/CT /D8/CW/CT/CX/D6 /D4/CW/DD/D7/CX
/CP/D0 /D1/CT /CP/D2/CX/D2/CV /BA /CC/CW/CT/D6/CT/CU/D3/D6/CT/B8 /D3/D2/D0/DD /D8/CW/CT /CX/D2 /DA /CP/D6/CX/CP/D2 /D8/D7/D4/CP
/CT/D8/CX/D1/CT /D0/CT/D2/CV/D8/CW /B4/D8/CW/CT /C4/D3/D6/CT/D2 /D8/DE /D7
/CP/D0/CP/D6/B5 /CQ /CT/D8 /DB /CT/CT/D2 /D8 /DB /D3 /D4 /D3/CX/D2 /D8/D7 /B4/CT/DA /CT/D2 /D8/D7/B5 /CX/D2 /BG/BW /D7/D4/CP
/CT/D8/CX/D1/CT /CS/D3 /CT/D7 /CW/CP /DA /CT/CS/CT/AS/D2/CX/D8/CT /D4/CW /DD/D7/CX
/CP/D0 /D1/CT/CP/D2/CX/D2/CV /CX/D2 /D8/CW/CT /AH/CC/CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH /CP/D2/CS /CX/D8 /CX/D7 /CS/CT/AS/D2/CT/CS /CP/D7
l= (lagablb)1/2, /B4/BD/B5/DB/CW/CT/D6/CT la(lb) /CX/D7 /D8/CW/CT /CS/CX/D7/D8/CP/D2
/CT /BG/B9/DA /CT
/D8/D3/D6 /CQ /CT/D8 /DB /CT/CT/D2 /D8 /DB /D3 /CT/DA /CT/D2 /D8/D7 A /CP/D2/CSB /B8la=xa
B−xa
A
/B8xa
A,B
/CP/D6/CT /D8/CW/CT/D4 /D3/D7/CX/D8/CX/D3/D2 /BG/B9/DA /CT
/D8/D3/D6/D7/B8 /CP/D2/CSgab
/CX/D7 /D8/CW/CT /D1/CT/D8/D6/CX
/D8/CT/D2/D7/D3/D6/BA/CD/D7/CX/D2/CV /CS/CX/AR/CT/D6/CT/D2 /D8
/D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/D7 /D3/CU /CP /CV/CX/DA /CT/D2 /D6/CT/CU/CT/D6/CT/D2
/CT /CU/D6/CP/D1/CT/B8 /DB/CW/CX
/CW
/CP/D2 /CQ /CT /D6/CT/CP/D0/CX/DE/CT/CS/B8 /CU/D3/D6 /CT/DC/CP/D1/D4/D0/CT/B8/CQ /DD /D1/CT/CP/D2/D7 /D3/CU /CS/CX/AR/CT/D6/CT/D2 /D8 /D7/DD/D2
/CW/D6/D3/D2/CX/DE/CP/D8/CX/D3/D2/D7/B8 /DB /CT /AS/D2/CS /CS/CX/AR/CT/D6/CT/D2 /D8 /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2/D7/B8 /CX/BA/CT/BA/B8 /CS/CX/AR/CT/D6/CT/D2 /D8 /D6/CT/D4/D6/CT/D7/CT/D2 /D8/CP/D8/CX/D3/D2/D7/D3/CU /D8/CW/CT /D7/D4/CP
/CT/D8/CX/D1/CT /D0/CT/D2/CV/D8/CW l, /BX/D5/BA/B4/BD/B5/BA /C7/CQ /DA/CX/D3/D9/D7/D0/DD /D8/CW/CT
/D3 /D3/D6/CS/CX/D2/CP/D8/CT/D7 xµ/D3/CU /CP/D2 /CT/DA /CT/D2 /D8/B8 /DB/CW/CT/D2 /DB/D6/CX/D8/D8/CT/D2 /CX/D2/D7/D3/D1/CT
/D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2 /D3/CU /CP/D2 /C1/BY/CA/B8 /CS/D3 /D2/D3/D8 /CW/CP /DA /CT /CP/D2 /CX/D2 /D8/D6/CX/D2/D7/CX
/D1/CT/CP/D2/CX/D2/CV /CX/D2 /BG/BW /D7/D4/CP
/CT/D8/CX/D1/CT/BA /C0/D3 /DB /CT/DA /CT/D6 /D8/CW/CT/D7/D4/CP
/CT/D8/CX/D1/CT /D0/CT/D2/CV/D8/CW l /B4/BD/B5 /CS/D3 /CT/D7 /CW/CP /DA /CT /D8/CW/CT /D7/CP/D1/CT /DA /CP/D0/D9/CT /CU/D3/D6 /CP/D0/D0 /D6/CT/D0/CP/D8/CX/DA /CT/D0/DD /D1/D3 /DA/CX/D2/CV /CX/D2/CT/D6/D8/CX/CP/D0
/D3 /D3/D6/CS/CX/D2/CP/D8/CT /D7/DD/D7/D8/CT/D1/D7/CP/D2/CS /CX/D8 /D6/CT/D4/D6/CT/D7/CT/D2 /D8/D7 /CP/D2 /CX/D2 /D8/D6/CX/D2/D7/CX
/CU/CT/CP/D8/D9/D6/CT /D3/CU /D8/CW/CT /D7/D4/CP
/CT/D8/CX/D1/CT/BA/BW/CX/AR/CT/D6/CT/D2 /D8 /D7/DD/D2
/CW/D6/D3/D2/CX/DE/CP/D8/CX/D3/D2/D7 /CP/D6/CT /CS/CT/D8/CT/D6/D1/CX/D2/CT/CS /CQ /DD /D8/CW/CT /D4/CP/D6/CP/D1/CT/D8/CT/D6 ε /CX/D2 /D8/CW/CT /D6/CT/D0/CP/D8/CX/D3/D2 t2=t1+ε(t3−t1) /B8/DB/CW/CT/D6/CT t1
/CP/D2/CSt3
/CP/D6/CT /D8/CW/CT /D8/CX/D1/CT/D7 /D3/CU /CS/CT/D4/CP/D6/D8/D9/D6/CT /CP/D2/CS /CP/D6/D6/CX/DA /CP/D0/B8 /D6/CT/D7/D4 /CT
/D8/CX/DA /CT/D0/DD /B8 /D3/CU /D8/CW/CT /D0/CX/CV/CW /D8 /D7/CX/CV/D2/CP/D0/B8 /D6/CT/CP/CS /CQ /DD /D8/CW/CT
/D0/D3
/CZ /CP/D8A /B8 /CP/D2/CSt2
/CX/D7 /D8/CW/CT /D8/CX/D1/CT /D3/CU /D6/CT/AT/CT
/D8/CX/D3/D2 /CP/D8B /B8 /D6/CT/CP/CS /CQ /DD /D8/CW/CT
/D0/D3
/CZ /CP/D8B /B8 /D8/CW/CP/D8 /CW/CP/D7 /D8/D3 /CQ /CT /D7/DD/D2
/CW/D6/D3/D2/CX/DE/CT/CS/DB/CX/D8/CW /D8/CW/CT
/D0/D3
/CZ /CP/D8A /BA /CD/D7/D9/CP/D0/D0/DD /D4/CW /DD/D7/CX
/CX/D7/D8/D7 /D4/D6/CT/CU/CT/D6 /BX/CX/D2/D7/D8/CT/CX/D2/B3/D7 /D7/DD/D2
/CW/D6/D3/D2/CX/DE/CP/D8/CX/D3/D2
/D3/D2 /DA /CT/D2 /D8/CX/D3/D2 /DB/CX/D8/CWε= 1/2/CX/D2 /DB/CW/CX
/CW /D8/CW/CT /D1/CT/CP/D7/D9/D6/CT/CS
/D3 /D3/D6/CS/CX/D2/CP/D8/CT /DA /CT/D0/D3
/CX/D8 /DD /D3/CU /D0/CX/CV/CW /D8 /B4/D8/CW/CT /D3/D2/CT/B9/DB /CP /DD /D7/D4 /CT/CT/CS /D3/CU /D0/CX/CV/CW /D8/B5 /CX/D7
/D3/D2/D7/D8/CP/D2 /D8 /CP/D2/CS/CX/D7/D3/D8/D6/D3/D4/CX
/BA /BT /D2/CX
/CT /CT/DC/CP/D1/D4/D0/CT /D3/CU /CP /D2/D3/D2/B9/D7/D8/CP/D2/CS/CP/D6/CS /D7/DD/D2
/CW/D6/D3/D2/CX/DE/CP/D8/CX/D3/D2 /CX/D7 /AH/CT/DA /CT/D6/DD/CS/CP /DD/AH
/D0/D3
/CZ /D7/DD/D2
/CW/D6/D3/D2/CX/DE/CP/D8/CX/D3/D2/CJ/BK ℄ /CX/D2 /DB/CW/CX
/CW ε= 0 /CP/D2/CS /D8/CW/CT/D6/CT /CX/D7 /CP/D2 /CP/CQ/D7/D3/D0/D9/D8/CT /D7/CX/D1 /D9/D0/D8/CP/D2/CT/CX/D8 /DD/BN /D7/CT/CT /CP/D0/D7/D3 /CJ/BL ℄ /CU/D3/D6 /CP/D2 /CP/CQ/D7/D3/D0/D9/D8/CT /D7/CX/D1 /D9/D0/D8/CP/D2/CT/CX/D8 /DD/CX/D2 /D8/CW/CT /D7/D4 /CT
/CX/CP/D0 /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD /B8 /CP/D2/CS /CU/D3/D6 /D8/CW/CT /D6/CT/DA/CX/CT/DB /D3/D2 /D7/DD/D2
/CW/D6/D3/D2/CX/D7/CP/D8/CX/D3/D2 /CP/D2/CS /D8/CT/D7/D8 /D8/CW/CT/D3/D6/CX/CT/D7 /D7/CT/CT /D8/CW/CT /D6/CT
/CT/D2 /D8/CP/D6/D8/CX
/D0/CT /CJ/BD/BC ℄/BA /BT/D7 /CT/DC/D4/D0/CP/CX/D2/CT/CS /CX/D2 /CJ/BK ℄/BM /AH/BY /D3/D6 /CX/CU /DB /CT /D8/D9/D6/D2 /D3/D2 /D8/CW/CT /D6/CP/CS/CX/D3 /CP/D2/CS /D7/CT/D8 /D3/D9/D6
/D0/D3
/CZ /CQ /DD /D8/CW/CT /D7/D8/CP/D2/CS/CP/D6/CS/CP/D2/D2/D3/D9/D2
/CT/D1/CT/D2 /D8 /AH/BA/BA/BA/CP/D8 /D8/CW/CT /D7/D3/D9/D2/CS /D3/CU /D8/CW/CT /D0/CP/D7/D8 /D8/D3/D2/CT/B8 /CX/D8 /DB/CX/D0/D0 /CQ /CT /BD/BE /D3/B3
/D0/D3
/CZ/AH/B8 /D8/CW/CT/D2 /DB /CT /CW/CP /DA /CT /D7/DD/D2
/CW/D6/D3/D2/CX/DE/CT/CS/D3/D9/D6
/D0/D3
/CZ /DB/CX/D8/CW /D8/CW/CT /D7/D8/D9/CS/CX/D3
/D0/D3
/CZ /CX/D2 /CP /D1/CP/D2/D2/CT/D6 /D8/CW/CP/D8
/D3/D6/D6/CT/D7/D4 /D3/D2/CS/D7 /D8/D3 /D8/CP/CZ/CX/D2/CV ε= 0 /CX/D2t2=t1+ε(t3−t1).”/CF/CW/CT/D2 /BX/CX/D2/D7/D8/CT/CX/D2/B3/D7 /D7/DD/D2
/CW/D6/D3/D2/CX/DE/CP/D8/CX/D3/D2 /D3/CU /CS/CX/D7/D8/CP/D2 /D8
/D0/D3
/CZ/D7 /CP/D2/CS
/CP/D6/D8/CT/D7/CX/CP/D2 /D7/D4/CP
/CT
/D3 /D3/D6/CS/CX/D2/CP/D8/CT/D7 xi
e
/CP/D6/CT /D9/D7/CT/CS/CX/D2 /CP/D2 /C1/BY/CAS /B4/D8/CW/CX/D7
/D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2 /DB/CX/D0/D0 /CQ /CT /D2/CP/D1/CT/CS /BX/CX/D2/D7/D8/CT/CX/D2/B3/D7 /D3/D6 /AH/CT/AH
/D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/B5 /D8/CW/CT/D2/B8 /CT/BA/CV/BA/B8 /D8/CW/CT/CV/CT/D3/D1/CT/D8/D6/CX
/D3/CQ /CY/CT
/D8 gab
/CX/D7 /D6/CT/D4/D6/CT/D7/CT/D2 /D8/CT/CS /CQ /DD /D8/CW/CT4×4 /D1/CP/D8/D6/CX/DC /D3/CU
/D3/D1/D4 /D3/D2/CT/D2 /D8/D7 /D3/CUgab
/CX/D2 /D8/CW/CP/D8
/D3 /D3/D6/CS/CX/D2/CP/D8/CT
/CW/CP/D6/D8/B8 /CX/BA/CT/BA/B8 /CX/D8 /CX/D7 /D8/CW/CT /C5/CX/D2/CZ /D3 /DB/D7/CZ/CX /D1/CT/D8/D6/CX
/D8/CT/D2/D7/D3/D6 gµνe=diag(−1,1,1,1), /DB/CW/CT/D6/CT /AH/CT/AH /D7/D8/CP/D2/CS/D7 /CU/D3/D6 /BX/CX/D2/D7/D8/CT/CX/D2/B3/D7
/D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/BA /CF/CX/D8/CW /D7/D9
/CWgµνe
/D8/CW/CT /D7/D4/CP
/CT xi
e
/CP/D2/CS /D8/CX/D1/CTte(x0
e≡cte)
/D3/D1/D4 /D3/D2/CT/D2 /D8/D7 /D3/CUxµ
e
/CS/D3 /CW/CP /DA /CT/D8/CW/CT/CX/D6 /D9/D7/D9/CP/D0 /D1/CT/CP/D2/CX/D2/CV/BA /CC/CW/CT/D2 ds2
/CP/D2 /CQ /CT /DB/D6/CX/D8/D8/CT/D2 /DB/CX/D8/CW /D8/CW/CT /D7/CT/D4/CP/D6/CP/D8/CT/CS /D7/D4/CP/D8/CX/CP/D0 /CP/D2/CS /D8/CT/D1/D4 /D3/D6/CP/D0 /D4/CP/D6/D8/D7/B8
ds2= (dxi
edxie)−(dx0
e)2/B8 /CP/D2/CS /D8/CW/CT /D7/CP/D1/CT /CW/CP/D4/D4 /CT/D2/D7 /DB/CX/D8/CW /D8/CW/CT /D7/D4/CP
/CT/D8/CX/D1/CT /D0/CT/D2/CV/D8/CW l /B4/BD/B5, l2= (li
elie)−(l0
e)2/BA/CB/D9
/CW /D7/CT/D4/CP/D6/CP/D8/CX/D3/D2 /D6/CT/D1/CP/CX/D2/D7 /DA /CP/D0/CX/CS /CX/D2 /D3/D8/CW/CT/D6 /CX/D2/CT/D6/D8/CX/CP/D0
/D3 /D3/D6/CS/CX/D2/CP/D8/CT /D7/DD/D7/D8/CT/D1/D7 /DB/CX/D8/CW /D8/CW/CT /C5/CX/D2/CZ /D3 /DB/D7/CZ/CX /D1/CT/D8/D6/CX
/D8/CT/D2/D7/D3/D6/B8 /CP/D2/CS /CX/D2S′/D3/D2/CT /AS/D2/CS/D7l′2= (l′i
el′
ie)−(l′0
e)2, /DB/CW/CT/D6/CT l′µ
e
/CX/D2S′/CX/D7
/D3/D2/D2/CT
/D8/CT/CS /DB/CX/D8/CWlµ
e
/CX/D2S /CQ /DD /D8/CW/CT/C4 /CC/BA/C1/D2 /D8/CW/CT /D9/D7/D9/CP/D0 /CU/D3/D6/D1 /D8/CW/CT /C4 /CC
/D3/D2/D2/CT
/D8 /D8 /DB /D3
/D3 /D3/D6/CS/CX/D2/CP/D8/CT /D6/CT/D4/D6/CT/D7/CT/D2 /D8/CP/D8/CX/D3/D2/D7 /B4/CX/D2 /D8/CW/CT /AH/CT/AH
/D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/B5
xµ
e, x′µ
e
/D3/CU /CP /CV/CX/DA /CT/D2 /CT/DA /CT/D2 /D8/BA xµ
e, x′µ
e
/D6/CT/CU/CT/D6 /D8/D3 /D8 /DB /D3 /D6/CT/D0/CP/D8/CX/DA /CT/D0/DD /D1/D3 /DA/CX/D2/CV /C1/BY/CA/D7 /B4/DB/CX/D8/CW /D8/CW/CT /C5/CX/D2/CZ /D3 /DB/D7/CZ/CX /D1/CT/D8/D6/CX
/D8/CT/D2/D7/D3/D6/B5 S /CP/D2/CSS′,
x′µ
e=Lµ
ν,exν
e, L0
0,e=γe, L0
i,e=Li
0,e=−γeVi
e/c, Li
j,e=δi
j+ (γe−1)Vi
eVje/V2
e,/DB/CW/CT/D6/CT Vi
e=dxi
e/dte
/CP/D6/CT /D8/CW/CT
/D3/D1/D4 /D3/D2/CT/D2 /D8/D7 /D3/CU /D8/CW/CT /D3/D6/CS/CX/D2/CP/D6/DD /DA /CT/D0/D3
/CX/D8 /DD /BF/B9/DA /CT
/D8/D3/D6/B8 /CP/D2/CSγe≡(1−V2
e/c2)1/2/BA/BT/D7 /CT/DC/D4/D0/CP/CX/D2/CT/CS /CX/D2 /CJ/BD/BD ℄/B8 /DB/CW/CT/D2 /D7/D9
/CW /D9/D7/D9/CP/D0 /D6/CT/D4/D6/CT/D7/CT/D2 /D8/CP/D8/CX/D3/D2/D7 /D3/CU /D4/D9/D6/CT /C4/D3/D6/CT/D2 /D8/DE /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2/D7 /CP/D6/CT /CP/D4/B9/D4/D0/CX/CT/CS /D8/D3
/D3 /DA /CP/D6/CX/CP/D2 /D8 /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2/D7 /D8/CW/CT/DD /CS/CT/D7/D8/D6/D3 /DD /D8/CW/CT
/D3 /DA /CP/D6/CX/CP/D2 /D8 /CU/D3/D6/D1 /BM /AH/CQ /CT
/CP/D9/D7/CT /D8/CW/CT/DD /CT/D1/D4/D0/D3 /DD /D8/CW/D6/CT/CT/B9/DA /CT
/D8/D3/D6 /D2/D3/D8/CP/D8/CX/D3/D2/B8 /CQ /CT
/CP/D9/D7/CT /D8/CW/CT/DD /D8/D6/CT/CP/D8 /D8/CW/CT /D7/D4/CP/D8/CX/CP/D0 /CP/D2/CS /D8/CT/D1/D4 /D3/D6/CP/D0
/D3/D1/D4 /D3/D2/CT/D2 /D8/D7 /D7/CT/D4/CP/D6/CP/D8/CT/D0/DD /B8 /CP/D2/CS /CQ /CT/B9
/CP/D9/D7/CT /D8/CW/CT/DD /CP/D6/CT /D4/CP/D6/CP/D1/CT/D8/D6/CX/DE/CT/CS /CQ /DD /D8/CW/CT /D3/D6/CS/CX/D2/CP/D6/DD /DA /CT/D0/D3
/CX/D8 /DD /D8/CW/D6/CT/CT/B9/DA /CT
/D8/D3/D6 V. /AH /C1/D2 /D3/D6/CS/CT/D6 /D8/D3 /D3/CQ/D8/CP/CX/D2 /CP
/D3/B9/DA /CP/D6/CX/CP/D2 /D8 /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2 /CU/D3/D6Lµν,e
/D8/CW/CT /D3/D6/CS/CX/D2/CP/D6/DD /DA /CT/D0/D3
/CX/D8 /DD /CX/D7 /D6/CT/D4/D0/CP
/CT/CS /CX/D2 /CJ/BD/BD ℄ /CQ /DD /D8/CW/CT /D4/D6/D3/D4 /CT/D6 /DA /CT/D0/D3
/CX/D8 /DD /BG/B9/DA /CT
/D8/D3/D6
vµ
e≡dxµ
e/dτ= (γec, γevi
e), dτ≡dte/γe
/CX/D7 /D8/CW/CT /D7
/CP/D0/CP/D6 /D4/D6/D3/D4 /CT/D6/B9/D8/CX/D1/CT/B8 /D8/CW/CT /D9/D2/CX/D8 /DA /CT
/D8/D3/D6 nµ
e≡(1,0,0,0)/CP/D0/D3/D2/CV /D8/CW/CT /D8/CT/D1/D4 /D3/D6/CP/D0 /CP/DC/CX/D7 /CX/D7 /CX/D2 /D8/D6/D3 /CS/D9
/CT/CS/B8 /CP/D2/CSδi
j
/CX/D7 /D6/CT/D4/D0/CP
/CT/CS /DB/CX/D8/CW /D8/CW/CT /C5/CX/D2/CZ /D3 /DB/D7/CZ/CX /D1/CT/D8/D6/CX
/D8/CT/D2/D7/D3/D6 gµνe./CC/CW/CX/D7 /D7/CW/D3 /DB/D7 /D8/CW/CP/D8 /D8/CW/CT
/CP/D6/D8/CT/D7/CX/CP/D2 /D7/D4/CP
/CT
/D3 /D3/D6/CS/CX/D2/CP/D8/CT/D7 xi
e
/CP/D2/CS /BX/CX/D2/D7/D8/CT/CX/D2/B3/D7 /D7/DD/D2
/CW/D6/D3/D2/CX/DE/CP/D8/CX/D3/D2 /D3/CU /CS/CX/D7/D8/CP/D2 /D8
/D0/D3
/CZ/D7/CP/D6/CT /CT/DC/D4/D0/CX
/CX/D8/D0/DD
/CW/D3/D7/CT/D2 /CX/D2 /CJ/BD/BD ℄/BA /C1/D2 /D7/D9
/CW /CP /DB /CP /DD /D8/CW/CT
/D3 /DA /CP/D6/CX/CP/D2 /D8 /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2 /CU/D3/D6Lµν,e
/CX/D2 /D8/CW/CT /AH/CT/AH
/D3 /D3/D6/CS/CX/D2/CP/B9/D8/CX/DE/CP/D8/CX/D3/D2 /CX/D7 /CU/D3/D9/D2/CS /CX/D2 /CJ/BD/BD ℄/B8 /BX/D5/BA/B4/BH/B5/B8
Lµν,e≡Lµν,e(v) =gµνe−2nµ
evνe
c+(nµ
e+vµ
e/c)(nνe+vνe/c)
1−ne·ve/c./BF/CB/CX/D2
/CT /DB /CT /DB /CP/D2 /D8 /D8/D3 /D9/D7/CT /D8/CW/CT /C4 /CC /CX/D2 /CS/CX/AR/CT/D6/CT/D2 /D8
/D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/D7 /DB /CT /CV/CT/D2/CT/D6/CP/D0/CX/DE/CT /D8/CW/CT /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2 /CU/D3/D6Lµν,e/CU/D6/D3/D1 /CJ/BD/BD ℄ /CP/D2/CS /AS/D2/CS
Lab≡Lab(v) =gab−2navb
c+(na+va/c)(nb+vb/c)
1−n·v/c. /B4/BE/B5/CB/D9
/CW /CU/D3/D6/D1 /B4/BE/B5 /D3/CU /D8/CW/CT /C4 /CC
/CP/D2 /CQ /CT /CP/D4/D4/D0/CX/CT/CS /D8/D3 /CP/D2 /CP/D6/CQ/CX/D8/D6/CP/D6/DD /CX/D2/CT/D6/D8/CX/CP/D0
/D3 /D3/D6/CS/CX/D2/CP/D8/CT /D7/DD/D7/D8/CT/D1 /CX/D2 /DB/CW/CX
/CW /D8/CW/CT/D1/CT/D8/D6/CX
/D8/CT/D2/D7/D3/D6
/CP/D2 /CQ /CT /CS/CX/AR/CT/D6/CT/D2 /D8 /D8/CW/CP/D2 /D8/CW/CT /C5/CX/D2/CZ /D3 /DB/D7/CZ/CX /D1/CT/D8/D6/CX
/D8/CT/D2/D7/D3/D6/B8 /CP/D2/CS /D8/CW /D9/D7 /D8/CW/CT /CU/D3/D6/D1 /D3/CU /D8/CW/CT
/D3 /DA /CP/D6/CX/CP/D2 /D8/BG/BW /C4/D3/D6/CT/D2 /D8/DE /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2/D7 /B4/BE/B5 /CX/D7 /CX/D2/CS/CT/D4 /CT/D2/CS/CT/D2 /D8 /D3/CU /D8/CW/CT
/CW/D3/D7/CT/D2 /D7/DD/D2
/CW/D6/D3/D2/CX/DE/CP/D8/CX/D3/D2/B8 /CX/BA/CT/BA/B8
/D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/D3/CU /D6/CT/CU/CT/D6/CT/D2
/CT /CU/D6/CP/D1/CT/D7/BA /BU/D9/D8 /DB /CT /CW/CP /DA /CT /D8/D3 /D2/D3/D8/CT /D8/CW/CP/D8na/CX/D2 /B4/BE/B5 /CX/D7 /CP /D7/D4 /CT
/CX/AS
/D5/D9/CP/D2 /D8/CX/D8 /DD /BA /C6/CP/D1/CT/D0/DD /CX/D8 /CP/D0/DB /CP /DD/D7/CW/CP/D7 /D8/D3 /CQ /CT /D8/CP/CZ /CT/D2 /CP/D7 /D8/CW/CT /D9/D2/CX/D8 /DA /CT
/D8/D3/D6 /CP/D0/D3/D2/CV /D8/CW/CT /D8/CT/D1/D4 /D3/D6/CP/D0 /CP/DC/CX/D7 /CX/D2 /D8/CW/CT
/CW/D3/D7/CT/D2 /C1/BY/CA /CP/D2/CS /D8/CW/CT
/CW/D3/D7/CT/D2
/D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/BA /C6/CT/DA /CT/D6/D8/CW/CT/D0/CT/D7/D7 Lab
/D3/D6/D6/CT
/D8/D0/DD /D8/D6/CP/D2/D7/CU/D3/D6/D1/D7 /D7/D3/D1/CT /BG/BW /D8/CT/D2/D7/D3/D6 /D5/D9/CP/D2 /D8/CX/D8 /DD /CU/D6/D3/D1 /CP/D2 /C1/BY/CA /D8/D3/CP/D2/D3/D8/CW/CT/D6 /D6/CT/D0/CP/D8/CX/DA /CT/D0/DD /D1/D3 /DA/CX/D2/CV /C1/BY/CA/BA /BY /D3/D6 /CT/DC/CP/D1/D4/D0/CT/B8 /DB/CW/CT/D2 Lab
/CX/D7 /CP/D4/D4/D0/CX/CT/CS /D8/D3 /D8/CW/CT /D4 /D3/D7/CX/D8/CX/D3/D2 /BG/B9/DA /CT
/D8/D3/D6 xa/D3/D2/CT/AS/D2/CS/D7 /B4/CX/D2 /D8/CW/CT /CP/CQ/D7/D8/D6/CP
/D8 /CX/D2/CS/CT/DC /D2/D3/D8/CP/D8/CX/D3/D2/B5
x′a=xa+[n·x−(2γ+ 1)v·x/c]na+ (n·x+v·x/c)va/c
1−n·v/c. /B4/BF/B5/C4/CT/D8 /D9/D7 /CT/DC/CP/D1/CX/D2/CT /D8/CW/CT /D6/CT/D0/CP/D8/CX/D3/D2/D7 /B4/BE/B5 /CP/D2/CS /B4/BF/B5 /CX/D2 /D8 /DB /D3 /CS/CX/AR/CT/D6/CT/D2 /D8
/D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/D7/BA /BY/CX/D6/D7/D8 /CX/D2 /D8/CW/CT /AH/CT/AH
/D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/B8 /CX/D2 /DB/CW/CX
/CW /D8/CW/CT /C5/CX/D2/CZ /D3 /DB/D7/CZ/CX /D1/CT/D8/D6/CX
/D8/CT/D2/D7/D3/D6 /CX/D7 /D9/D7/CT/CS/B8 na/CQ /CT
/D3/D1/CT/D7 nµ
e≡(1,0,0,0),
vµ
e= (γec, γevi
e), /CP/D2/CSγe=−nµ
evµe/c, /CP/D7 /CX/D2 /CJ/BD/BD ℄/BA /BY /D6/D3/D1 /D8/CW/CT /CV/CT/D2/CT/D6/CP/D0 /D6/CT/D0/CP/D8/CX/D3/D2 vava=−c2/D3/D2/CT /AS/D2/CS/D7/B8/CX/D2 /D8/CW/CT /AH/CT/AH
/D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/B8 /D8/CW/CP/D8v0
e= (c2+vi
evie)1/2, /DB/CW/CX
/CW /D7/CW/D3 /DB/D7 /D8/CW/CP/D8 /D8/CW/CT /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2 /CU/D3/D6Lµν,e/CX/D7 /D4/CP/D6/CP/D1/CT/D8/D6/CX/DE/CT/CS /CT/D7/D7/CT/D2 /D8/CX/CP/D0/D0/DD /CQ /DD /D8/CW/CT /D8/CW/D6/CT/CT /D7/D4/CP/D8/CX/CP/D0
/D3/D1/D4 /D3/D2/CT/D2 /D8/D7 vi
e
/D3/CU /D8/CW/CT /D4/D6/D3/D4 /CT/D6 /DA /CT/D0/D3
/CX/D8 /DD /BG/B9/DA /CT
/D8/D3/D6 vµ
e./CC/CW/CT/D2/B8 /D9/D7/CX/D2/CV /D8/CW/CT /CP/CQ /D3 /DA /CT /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2/D7 /CU/D3/D6nµ
e, vµ
e, /CP/D2/CSγe
/D3/D2/CT /AS/D2/CS/D7 /CU/D6/D3/D1 /B4/BE/B5 /CP/D2/CS /B4/BF/B5 /D8/CW/CT /D9/D7/D9/CP/D0 /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2/D7/CU/D3/D6 /D4/D9/D6/CT /C4 /CC/B8 /CP/D7 /CX/D2 /CJ/BD/BD ℄/B8 /CX/BA/CT/BA/B8 /D8/CW/CT /CP/CQ /D3 /DA /CT /D1/CT/D2 /D8/CX/D3/D2/CT/CS Lµν,e
/CP/D2/CSx′µ
e
/B8 /CQ/D9/D8 /DB/CX/D8/CWvi
e
/D6/CT/D4/D0/CP
/CX/D2/CV /D8/CW/CT
/D3/D1/D4 /D3/D2/CT/D2 /D8/D7/D3/CU /D8/CW/CT /D3/D6/CS/CX/D2/CP/D6/DD /DA /CT/D0/D3
/CX/D8 /DD /BF/B9/DA /CT
/D8/D3/D6 V. /BT/D0/D7/D3/B8 /DB /CT /AS/D2/CS /D8/CW/CT /CP/CQ /D3 /DA /CT /D1/CT/D2 /D8/CX/D3/D2/CT/CS /D9/D7/D9/CP/D0 /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2/D7 /CX/D2 /D8/CW/CT /AH/CT/AH
/D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2 /CU/D3/D6ds2=ds2
e= (dxi
edxie)−(dx0
e)2/CP/D2/CSl2=l2
e= (li
elie)−(l0
e)2, /DB/CX/D8/CW /D8/CW/CT /D7/CT/D4/CP/D6/CP/D8/CT/CS/D7/D4/CP/D8/CX/CP/D0 /CP/D2/CS /D8/CT/D1/D4 /D3/D6/CP/D0 /D4/CP/D6/D8/D7/BA/C1/D2 /D8/CW/CT /D7/CX/D1/CX/D0/CP/D6 /DB /CP /DD /DB /CT /D9/D7/CT /D8/CW/CT /D6/CT/D0/CP/D8/CX/D3/D2/D7 /B4/BE/B5 /CP/D2/CS /B4/BF/B5 /D8/D3 /DB/D6/CX/D8/CT /D8/CW/CT
/D3/D6/D6/CT/D7/D4 /D3/D2/CS/CX/D2/CV /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2/D7 /CX/D2/CP/D2/D3/D8/CW/CT/D6
/D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/B8 /AH/D6/AH
/D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/B8 /D3/CU /CP/D2 /C1/BY/CA/B8 /DB/CW/CX
/CW /CX/D7 /CU/D3/D9/D2/CS /CX/D2 /CJ/BK ℄/B8 /DB/CW/CT/D6/CT /AH/CT/DA /CT/D6/DD/CS/CP /DD/AH/D3/D6 /AH/D6/CP/CS/CX/D3/AH /D7/DD/D2
/CW/D6/D3/D2/CX/DE/CP/D8/CX/D3/D2 /D3/CU /CS/CX/D7/D8/CP/D2 /D8
/D0/D3
/CZ/D7 /CX/D7 /D9/D7/CT/CS/BA /BY /D3/D6 /D7/CX/D1/D4/D0/CX
/CX/D8 /DD /DB /CT
/D3/D2/D7/CX/CS/CT/D6 /BE/BW /D7/D4/CP
/CT/D8/CX/D1/CT /CP/D7/CX/D2 /CJ/BK℄/BA /CC/CW/CT/D2 /D8/CW/CT /D1/CT/D8/D6/CX
/D8/CT/D2/D7/D3/D6 gab
/CQ /CT
/D3/D1/CT/D7 gµνr=/parenleftbigg
−1−1
−1 0/parenrightbigg
, /DB/CW/CT/D6/CT /AH/D6/AH /D7/D8/CP/D2/CS/D7 /CU/D3/D6 /AH/D6/CP/CS/CX/D3/AH/B4/CX/D8 /CS/CX/AR/CT/D6/D7 /CU/D6/D3/D1 /D8/CW/CP/D8 /D3/D2/CT /CX/D2 /CJ/BK℄ /D7/CX/D2
/CT /D8/CW/CT /C5/CX/D2/CZ /D3 /DB/D7/CZ/CX /D8/CT/D2/D7/D3/D6/D7 /CP/D6/CT /CS/CX/AR/CT/D6/CT/D2 /D8/B5/BA /CC/CW/CT /C4 /CCLµν,r
/CX/D2 /D8/CW/CT/AH/D6/AH
/D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2
/CP/D2 /CQ /CT /CT/CP/D7/CX/D0/DD /CU/D3/D9/D2/CS /CU/D6/D3/D1 /B4/BE/B5/B8 /CU/D6/D3/D1 /D8/CW/CT /CZ/D2/D3 /DB/D2 gµνr, /DB/CX/D8/CWnµ
r= (1,0) /CP/D2/CS
γr=−nµ
rvµr/c=γe. /CC/CW/CT/D7/CT /D6/CT/D0/CP/D8/CX/D3/D2/D7
/CP/D2 /CQ /CT /CU/D3/D9/D2/CS /CP/D7 /CX/D2 /CJ/BK ℄/B8 /D3/D6 /CQ /DD /D1/CT/CP/D2/D7 /D3/CU /D8/CW/CT /D1/CP/D8/D6/CX/DC Tµν,/DB/CW/CX
/CW /CX/D7 /CV/CX/DA /CT/D2 /CQ /CT/D0/D3 /DB/BA /CC/CW /D9/D7 /D8/CW/CT /D4/D9/D6/CT /C4/D3/D6/CT/D2 /D8/DE /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2 /D1/CP/D8/D6/CX/DC Lab
/B4/BE/B5 /CQ /CT
/D3/D1/CT/D7 /CX/D2 /D8/CW/CT /AH/D6/AH
/D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2
Lµν,r=/parenleftbiggK 0
−βr/K1/K/parenrightbigg
./BT/D0/D7/D3 /DB /CT /AS/D2/CS /D8/CW/CT /AH/D6/AH /D6/CT/D4/D6/CT/D7/CT/D2 /D8/CP/D8/CX/D3/D2 x′µ
r
/D3/CUx′a/B4/BF/B5/B8
x′0
r=Kx0
r, x′1
r= (1/K)(−βrx0
r+x1
r),/DB/CW/CT/D6/CT K= (1+2 βr)1/2/CP/D2/CSβr=dx1
r/dx0
r
/CX/D7 /D8/CW/CT /DA /CT/D0/D3
/CX/D8 /DD /D3/CU /D8/CW/CT /CU/D6/CP/D1/CT S′/CP/D7 /D1/CT/CP/D7/D9/D6/CT/CS /CQ /DD /D8/CW/CT /CU/D6/CP/D1/CT
S. /BY /D9/D6/D8/CW/CT/D6 ds2=dxagabdxb/CQ /CT
/D3/D1/CT/D7 /CX/D2 /D8/CW/CT /AH/D6/AH
/D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2 ds2=ds2
r=−/bracketleftbig
(dx0
r)2+ 2dx0
rdx1
r/bracketrightbig
./CF /CT /D7/CT/CT /D8/CW/CP/D8 /CX/D2 /D8/CW/CT /AH/D6/AH
/D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2 /D8/CW/CT /D7/D4/CP/D8/CX/CP/D0 /CP/D2/CS /D8/CT/D1/D4 /D3/D6/CP/D0 /D4/CP/D6/D8/D7 /D3/CUds2/CP/D6/CT /D2/D3/D8 /D7/CT/D4/CP/D6/CP/D8/CT/CS/B8 /D8/CW/CP/D8/CX/D7 /CS/CX/AR/CT/D6/CT/D2 /D8 /D8/CW/CP/D2 /CX/D2 /D8/CW/CT
/D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2 /DB/CX/D8/CW /D8/CW/CT /C5/CX/D2/CZ /D3 /DB/D7/CZ/CX /D1/CT/D8/D6/CX
/D8/CT/D2/D7/D3/D6/BA /CC/CW/CT /D7/CP/D1/CT /CW/D3/D0/CS/D7 /CU/D3/D6 /D8/CW/CT/D7/D4/CP
/CT/D8/CX/D1/CT /D0/CT/D2/CV/D8/CW l, /DB/CW/CX
/CW /CX/D7 /CX/D2 /D8/CW/CT /AH/D6/AH
/D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2 /CS/CT/D8/CT/D6/D1/CX/D2/CT/CS /CP/D7l2=l2
r=−/bracketleftbig
(l0
r)2+ 2l0
rl1
r/bracketrightbig
./BX/DC/D4/D6/CT/D7/D7/CX/D2/CV dxµ
r, /D3/D6lµ
r
/B8 /CX/D2 /D8/CT/D6/D1/D7 /D3/CUdxµ
e, /D3/D6lµ
e
/B4/D8/CW/CT /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2 /D1/CP/D8/D6/CX/DC /CQ /CT/D8 /DB /CT/CT/D2 /AH/D6/AH /CP/D2/CS /AH/CT/AH
/D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/D7 /CX/D7
Tµν=/parenleftbigg
1−1
0 1/parenrightbigg
,/DB/CW/CT/D2
/CT/B8 /CT/BA/CV/BA/B8x0
r=x0
e−x1
e, x1
r=x1
e, /CP/D2/CSβr=βe/(1−βe), /D7/CT/CT /CJ/BK℄/B5 /D3/D2/CT /AS/D2/CS/D7 /D8/CW/CP/D8ds2
r=ds2
e, /CP/D2/CS/CP/D0/D7/D3/B8l2
r=l2
e, /CP/D7 /CX/D8 /D1 /D9/D7/D8 /CQ /CT/BA/BG/CC/CW/CT /DB/CW/D3/D0/CT /D4/D6/CT
/CT/CS/CX/D2/CV /CS/CX/D7
/D9/D7/D7/CX/D3/D2 /CP/CQ /D3/D9/D8 /D8/CW/CT /CV/CT/D3/D1/CT/D8/D6/CX
/D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7 xa, la, ds, l, .. /CP/D2/CS /D8/CW/CT/CX/D6 /CS/CX/AR/CT/D6/CT/D2 /D8/D6/CT/D4/D6/CT/D7/CT/D2 /D8/CP/D8/CX/D3/D2/D7
/CP/D2 /CQ /CT /CX/D0/D0/D9/D7/D8/D6/CP/D8/CT/CS /CX/D2 /CP /DB /CP /DD /DB/CW/CX
/CW /CQ /CT/D8/D8/CT/D6
/D0/CP/D6/CX/AS/CT/D7 /D8/CW/CT /CS/CX/AR/CT/D6/CT/D2
/CT /CQ /CT/D8 /DB /CT/CT/D2 /D8 /DB /D3 /D7/D3/D6/D8/D7 /D3/CU/D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7/BA /BT/CV/CP/CX/D2 /DB /CT
/D3/D2/D7/CX/CS/CT/D6 /D8/CW/CT /CC/CC /D0/CT/D2/CV/D8/CW /B4/BD/B5 /B4/DB /CT /D9/D7/CT /D8/CW/CT /DB /D3/D6/CS/D7 /B9 /D8/CW/CT /CC/CC /D0/CT/D2/CV/D8/CW/B8 /D8/CW/CT /D7/D4/CP
/CT/D8/CX/D1/CT/D0/CT/D2/CV/D8/CW/B8 /CP/D2/CS /D8/CW/CT
/D3 /DA /CP/D6/CX/CP/D2 /D8/D0/DD /CS/CT/AS/D2/CT/CS /D0/CT/D2/CV/D8/CW /CP/D7 /D7/DD/D2/D3/D2 /DD/D1/D7/B5 /CX/D2 /D8 /DB /D3 /D6/CT/D0/CP/D8/CX/DA /CT/D0/DD /D1/D3 /DA/CX/D2/CV /C1/BY/CA/D7 S /CP/D2/CSS′/CP/D2/CS /CX/D2 /D8 /DB /D3
/D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/D7 /AH/CT/AH /CP/D2/CS /AH/D6/AH /CX/D2 /D8/CW/CT/D7/CT /C1/BY/CA/D7/BA /C6/D3 /DB/B8 /D0/CT/D8 /D8/CW/CT /D7/D4/CP
/CT/D8/CX/D1/CT /CQ /CT /CT/D2/CS/D3 /DB /CT/CS /DB/CX/D8/CW/CQ/CP/D7/CT /DA /CT
/D8/D3/D6/D7/B8 /D8/CW/CT /D8/CT/D1/D4 /D3/D6/CP/D0 /CP/D2/CS /D8/CW/CT /D7/D4/CP/D8/CX/CP/D0 /CQ/CP/D7/CT /DA /CT
/D8/D3/D6/D7/BA /CC/CW/CT /CQ/CP/D7/CT/D7 {eµ} /B8 /DB/CX/D8/CW /D8/CW/CT /CQ/CP/D7/CT /DA /CT
/D8/D3/D6/D7
{e0, e1} /B8 /CP/D2/CS{rµ}, /DB/CX/D8/CW /D8/CW/CT /CQ/CP/D7/CT /DA /CT
/D8/D3/D6/D7 {r0, r1}, /CP/D6/CT /CP/D7/D7/D3
/CX/CP/D8/CT/CS /DB/CX/D8/CW /AH/CT/AH /CP/D2/CS /AH/D6/AH
/D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/D7/B8/D6/CT/D7/D4 /CT
/D8/CX/DA /CT/D0/DD /B8 /D3/CU /CP /CV/CX/DA /CT/D2 /C1/BY/CA/BA /CC/CW/CT /D8/CT/D1/D4 /D3/D6/CP/D0 /CQ/CP/D7/CT /DA /CT
/D8/D3/D6 e0
/CX/D7 /D8/CW/CT /D9/D2/CX/D8 /DA /CT
/D8/D3/D6 /CS/CX/D6/CT
/D8/CT/CS /CP/D0/D3/D2/CV /D8/CW/CT /DB /D3/D6/D0/CS/D0/CX/D2/CT /D3/CU /D8/CW/CT
/D0/D3
/CZ /CP/D8 /D8/CW/CT /D3/D6/CX/CV/CX/D2/BA /CC/CW/CT /D7/D4/CP/D8/CX/CP/D0 /CQ/CP/D7/CT /DA /CT
/D8/D3/D6 /CQ /DD /CS/CT/AS/D2/CX/D8/CX/D3/D2
/D3/D2/D2/CT
/D8/D7 /D7/CX/D1/D9/D0/D8/CP/D2/CT /D3/D9/D7 /CT/DA /CT/D2 /D8/D7/B8/D8/CW/CT /CT/DA /CT/D2 /D8 /AH
/D0/D3
/CZ /CP/D8 /D6/CT/D7/D8 /CP/D8 /D8/CW/CT /D3/D6/CX/CV/CX/D2 /D6/CT/CP/CS/D7 /BC /D8/CX/D1/CT/AH /DB/CX/D8/CW /D8/CW/CT /CT/DA /CT/D2 /D8 /AH
/D0/D3
/CZ /CP/D8 /D6/CT/D7/D8 /CP/D8 /D9/D2/CX/D8 /CS/CX/D7/D8/CP/D2
/CT/CU/D6/D3/D1 /D8/CW/CT /D3/D6/CX/CV/CX/D2 /D6/CT/CP/CS/D7 /BC /D8/CX/D1/CT/AH/B8 /CP/D2/CS /D8/CW /D9/D7 /CX/D8 /CX/D7 /D7/DD/D2
/CW/D6/D3/D2/CX/DE/CP/D8/CX/D3/D2/B9/CS/CT/D4 /CT/D2/CS/CT/D2 /D8/BA /CC/CW/CT /D7/D4/CP/D8/CX/CP/D0 /CQ/CP/D7/CT /DA /CT
/D8/D3/D6
e1
/D3/D2/D2/CT
/D8/D7 /D8 /DB /D3 /CP/CQ /D3 /DA /CT /D1/CT/D2 /D8/CX/D3/D2/CT/CS /D7/CX/D1 /D9/D0/D8/CP/D2/CT/D3/D9/D7 /CT/DA /CT/D2 /D8/D7 /DB/CW/CT/D2 /BX/CX/D2/D7/D8/CT/CX/D2/B3/D7 /D7/DD/D2
/CW/D6/D3/D2/CX/DE/CP/D8/CX/D3/D2 /B4ε= 1/2 /B5/D3/CU /CS/CX/D7/D8/CP/D2 /D8
/D0/D3
/CZ/D7 /CX/D7 /D9/D7/CT/CS/BA /CC/CW/CT /D8/CT/D1/D4 /D3/D6/CP/D0 /CQ/CP/D7/CT /DA /CT
/D8/D3/D6 r0
/CX/D7 /D8/CW/CT /D7/CP/D1/CT /CP/D7e0. /CC/CW/CT /D7/D4/CP/D8/CX/CP/D0 /CQ/CP/D7/CT /DA /CT
/D8/D3/D6 r1
/D3/D2/D2/CT
/D8/D7 /D8 /DB /D3 /CP/CQ /D3 /DA /CT /D1/CT/D2 /D8/CX/D3/D2/CT/CS /D7/CX/D1 /D9/D0/D8/CP/D2/CT/D3/D9/D7 /CT/DA /CT/D2 /D8/D7 /DB/CW/CT/D2 /AH/CT/DA /CT/D6/DD/CS/CP /DD/AH
/D0/D3
/CZ /D7/DD/D2
/CW/D6/D3/D2/CX/DE/CP/D8/CX/D3/D2 /B4ε= 0 /B5/D3/CU /CS/CX/D7/D8/CP/D2 /D8
/D0/D3
/CZ/D7 /CX/D7 /D9/D7/CT/CS/BA /BT/D0/D0 /D8/CW/CT /D7/D4/CP/D8/CX/CP/D0 /CQ/CP/D7/CT /DA /CT
/D8/D3/D6/D7 r1, r′
1, .. /CP/D6/CT /D4/CP/D6/CP/D0/D0/CT/D0 /CP/D2/CS /CS/CX/D6/CT
/D8/CT/CS /CP/D0/D3/D2/CV /CP/D2/B4/D3/CQ/D7/CT/D6/DA /CT/D6/B9/CX/D2/CS/CT/D4 /CT/D2/CS/CT/D2 /D8/B5 /D0/CX/CV/CW /D8 /D0/CX/D2/CT/BA /C0/CT/D2
/CT/B8 /D8 /DB /D3 /CT/DA /CT/D2 /D8/D7 /D8/CW/CP/D8 /CP/D6/CT /CT/DA /CT/D6/DD/CS/CP /DD /B4/AH/D6/AH/B5 /D7/CX/D1 /D9/D0/D8/CP/D2/CT/D3/D9/D7 /CX/D2S/CP/D6/CT /CP/D0/D7/D3 /AH/D6/AH /D7/CX/D1 /D9/D0/D8/CP/D2/CT/D3/D9/D7 /CU/D3/D6 /CP/D0/D0 /D3/D8/CW/CT/D6 /C1/BY/CA/D7/BA /CC/CW/CT
/D3/D2/D2/CT
/D8/CX/D3/D2 /CQ /CT/D8 /DB /CT/CT/D2 /D8/CW/CT /CQ/CP/D7/CT/D7 {eµ} /CP/D2/CS{rµ}/CX/D7r0=e0, r1=e0+e1, /D7/CT/CT /CJ/BK ℄/BA /CC/CW/CT/D2 /D8/CW/CT /CV/CT/D3/D1/CT/D8/D6/CX
/CP/D0 /D5/D9/CP/D2 /D8/CX/D8 /DD /B8 /CT/BA/CV/BA/B8 /D8/CW/CT /CS/CX/D7/D8/CP/D2
/CT /BG/B9/DA /CT
/D8/D3/D6
la
AB
/CQ /CT/D8 /DB /CT/CT/D2 /D8 /DB /D3 /CT/DA /CT/D2 /D8/D7 A /CP/D2/CSB, /DB/CX/D0/D0 /CQ /CT /D6/CT/D4/D6/CT/D7/CT/D2 /D8/CT/CS /CQ /DD /D8/CW/CT /DA /CT
/D8/D3/D6 /CX/D2 /BE/BW /D7/D4/CP
/CT/D8/CX/D1/CT/B8 /DB/CW/CX
/CW /CW/CP /DA /CT/CS/CX/AR/CT/D6/CT/D2 /D8 /CS/CT
/D3/D1/D4 /D3/D7/CX/D8/CX/D3/D2/D7/B8 /D6/CT/D4/D6/CT/D7/CT/D2 /D8/CP/D8/CX/D3/D2/D7/B8 /DB/CX/D8/CW /D6/CT/D7/D4 /CT
/D8 /D8/D3{eµ},/braceleftbig
e′
µ/bracerightbig/CP/D2/CS{rµ},/braceleftbig
r′
µ/bracerightbig/CQ/CP/D7/CT/D7/BA /C6/D3/D8/CT/D8/CW/CP/D8 /CX/D2 /D8/CW/CT /AH/CC/CC /D6 /CT/D0/CP/D8/CX/DA/CX/D8/DD/AH /D8/CW/CT /D7/CP/D1/CT /CS/CX/D7/D8/CP/D2
/CT /BG/B9/DA/CT
/D8/D3/D6 la
AB
/CX/D7
/D3/D2/D7/CX/CS/CT/D6 /CT /CS /B4/D1/CT /CP/D7/D9/D6 /CT /CS/B5 /CX/D2 /CS/CX/AR/CT/D6 /CT/D2/D8/D6 /CT/D0/CP/D8/CX/DA/CT/D0/DD /D1/D3/DA/CX/D2/CV /C1/BY/CA/D7 /CP/D2/CS /CX/D2 /CS/CX/AR/CT/D6 /CT/D2/D8
/D3 /D3/D6 /CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/D7 /D3/CU /D8/CW/CT/D7/CT /C1/BY/CA/D7/BA/C1/D2 /D3/D6/CS/CT/D6 /D8/D3 /D6/CT/D8/CP/CX/D2 /D8/CW/CT
/D3/D2/D2/CT
/D8/CX/D3/D2 /DB/CX/D8/CW /D8/CW/CT /D4/D6/CT/D6/CT/D0/CP/D8/CX/DA/CX/D7/D8/CX
/D4/CW /DD/D7/CX
/D7 /CP/D2/CS /D8/D3 /CU/CP
/CX/D0/CX/D8/CP/D8/CT /D8/CW/CT
/D3/D1/D4/CP/D6/CX/D7/D3/D2/DB/CX/D8/CW /D8/CW/CT /AH/BT /CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH /DB /CT
/D3/D2/D7/CX/CS/CT/D6 /CP /D4/CP/D6/D8/CX
/D9/D0/CP/D6
/CW/D3/CX
/CT /CU/D3/D6 /D8/CW/CT /BG/B9/DA /CT
/D8/D3/D6 la
AB
/B4/CX/D2 /D8/CW/CT /D9/D7/D9/CP/D0 /AH/BF/B7/BD/AH/D4/CX
/D8/D9/D6/CT /CX/D8
/D3/D6/D6/CT/D7/D4 /D3/D2/CS/D7 /D8/D3 /CP/D2 /D3/CQ /CY/CT
/D8/B8 /CP /D6/D3 /CS/B8 /D8/CW/CP/D8 /CX/D7 /CP/D8 /D6/CT/D7/D8 /CX/D2 /CP/D2 /C1/BY/CAS /CP/D2/CS /D7/CX/D8/D9/CP/D8/CT/CS /CP/D0/D3/D2/CV /D8/CW/CT
/D3/D1/D1/D3/D2 x1
e, x′1
e− /CP/DC/CT/D7/B5/BA /C1/D2 /D8/CW/CT /AH/CT/AH
/D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2 /D8/CW/CT /D4 /D3/D7/CX/D8/CX/D3/D2 /BG/B9/DA /CT
/D8/D3/D6/D7 /D3/CU /D8/CW/CT /CT/DA /CT/D2 /D8/D7 A, xa
A, /CP/D2/CS
B, xa
B, /CX/D2S /CP/D6/CT /CS/CT
/D3/D1/D4 /D3/D7/CT/CS /DB/CX/D8/CW /D6/CT/D7/D4 /CT
/D8 /D8/D3{eµ} /CQ/CP/D7/CT /CP/D7xa
A=x0
Aee0+x1
Aee1= 0e0+ 0e1, /CP/D2/CS
xa
B=x0
Bee0+x1
Bee1= 0e0+l0e1, /CP/D2/CS /D8/CW/CT /CS/CX/D7/D8/CP/D2
/CT /BG/B9/DA /CT
/D8/D3/D6 la
AB=xa
B−xa
A
/CX/D7 /CS/CT
/D3/D1/D4 /D3/D7/CT/CS /CP/D7
la
AB=l0
ee0+l1
ee1= 0e0+l0e1./CC/CW /D9/D7 /CX/D2S /D8/CW/CT /D4 /D3/D7/CX/D8/CX/D3/D2 /BG/B9/DA /CT
/D8/D3/D6/D7 xa
A,B
/CP/D6/CT /CS/CT/D8/CT/D6/D1/CX/D2/CT/CS /D7/CX/D1 /D9/D0/D8/CP/D2/CT/D3/D9/D7/D0/DD /B8 x0
Be−x0
Ae=l0
e= 0, /CX/BA/CT/BA/B8 /D8/CW/CT/D8/CT/D1/D4 /D3/D6/CP/D0 /D4/CP/D6/D8 /D3/CUla
AB
/CX/D7 /DE/CT/D6/D3/BA /CC/CW/CT /D7/D4/CP
/CT/D8/CX/D1/CT /D0/CT/D2/CV/D8/CW l /CX/D7 /DB/D6/CX/D8/D8/CT/D2 /CX/D2 /D8/CW/CT{eµ} /CQ/CP/D7/CT /CP/D7
l=le= (lµ
elµe)1/2= (li
elie)1/2=l0,/CP/D7 /CX/D2 /D8/CW/CT /D4/D6/CT/D6/CT/D0/CP/D8/CX/DA/CX/D7/D8/CX
/D4/CW /DD/D7/CX
/D7/BN /CX/D8 /CX/D7 /CX/D2 /D8/CW/CP/D8
/CP/D7/CT /CP /D1/CT/CP/D7/D9/D6/CT /D3/CU /D8/CW/CT /D7/D4/CP/D8/CX/CP/D0 /CS/CX/D7/D8/CP/D2
/CT/B8 /CX/BA/CT/BA/B8 /D3/CU /D8/CW/CT/D6/CT/D7/D8 /D7/D4/CP/D8/CX/CP/D0 /D0/CT/D2/CV/D8/CW /D3/CU /D8/CW/CT /D6/D3 /CS/BA /CC/CW/CT /D3/CQ/D7/CT/D6/DA /CT/D6/D7 /CX/D2 /CP/D0/D0 /D3/D8/CW/CT/D6 /C1/BY/CA/D7 /DB/CX/D0/D0 /D0/D3 /D3/CZ /CP/D8 /D8/CW/CT /D7/CP/D1/CT /CT/DA /CT/D2 /D8/D7 /CQ/D9/D8/CP/D7/D7/D3
/CX/CP/D8/CX/D2/CV /DB/CX/D8/CW /D8/CW/CT/D1 /CS/CX/AR/CT/D6/CT/D2 /D8
/D3 /D3/D6/CS/CX/D2/CP/D8/CT/D7/BN /CX/D8 /CX/D7 /D8/CW/CT /CT/D7/D7/CT/D2
/CT /D3/CU /D8/CW/CT
/D3 /DA /CP/D6/CX/CP/D2 /D8 /CS/CT/D7
/D6/CX/D4/D8/CX/D3/D2/BA /CC/CW/CT/DD/CP/D0/D0 /D3/CQ/D8/CP/CX/D2 /D8/CW/CT /D7/CP/D1/CT /DA /CP/D0/D9/CT l /CU/D3/D6 /D8/CW/CT /D7/D4/CP
/CT/D8/CX/D1/CT /D0/CT/D2/CV/D8/CW/BA /C1/D8 /CW/CP/D7 /D8/D3 /CQ /CT /D4 /D3/CX/D2 /D8/CT/CS /D3/D9/D8 /D8/CW/CP/D8 /CX/D2 /D8/CW/CT /AH/CC/CC/D6 /CT/D0/CP/D8/CX/DA/CX/D8/DD/AH /CX/D8 /CX/D7 /D2/D3/D8 /D2/CT
/CT/D7/D7/CP/D6/DD /D8/D3 /D7/D8/CP/D6/D8 /CX/D2 /D8/CW/CX/D7 /CT/DC/CP/D1/D4/D0/CT /DB/CX/D8/CW /D8/CW/CT /D6 /CT/D7/D8 /CU/D6 /CP/D1/CT /D3/CU /D8/CW/CT /D3/CQ/CY/CT
/D8 /CP/D2/CS /D8/D3
/CW/D3 /D3/D7/CT/D8/CW/CT /CT/DA/CT/D2/D8/D7 A /CP/D2/CSB /D8/D3 /CQ /CT /D7/CX/D1/D9/D0/D8/CP/D2/CT /D3/D9/D7 /CX/D2 /D8/CW/CP/D8 /CU/D6 /CP/D1/CT/BA /CC/CW/CT /DB/CW/D3/D0/CT
/D3/D2/D7/CX/CS/CT/D6 /CP/D8/CX/D3/D2
/CP/D2 /CQ /CT /CS/D3/D2/CT /CX/D2 /D8/CW/CT/D7/CP/D1/CT
/D3/DA/CP/D6/CX/CP/D2/D8 /D1/CP/D2/D2/CT/D6 /CU/D3/D6 /D3/D8/CW/CT/D6
/CW/D3/CX
/CT/D7 /D3/CU /C1/BY/CA/D7 /CP/D2/CS /D3/CU /D8/CW/CT /CT/DA/CT/D2/D8/D7 A /CP/D2/CSB /CX/D2 /D8/CW/CT
/CW/D3/D7/CT/D2 /C1/BY/CA /BA /BY /D3/D6/CP/D2 /DD /D7/D8/CP/D6/D8/CX/D2/CV
/CW/D3/CX
/CT /D8/CW/CT
/D3 /DA /CP/D6/CX/CP/D2 /D8 /C4 /CC /B4/BE/B5 /DB/CX/D0/D0
/D3/D6/D6/CT
/D8/D0/DD
/D3/D2/D2/CT
/D8 /D8/CW/CT /D6/CT/D7/D9/D0/D8/D7 /D3/CU /D1/CT/CP/D7/D9/D6/CT/D1/CT/D2 /D8/D7 /D3/CU /D8/CW/CT/D7/CP/D1/CT /BG/BW /D5/D9/CP/D2/D8/CX/D8/DD /CX/D2 /D8 /DB /D3 /D6/CT/D0/CP/D8/CX/DA /CT/D0/DD /D1/D3 /DA/CX/D2/CV /C1/BY/CA/D7/BA /CC/CW/CT /D6/CT/D7/D8 /CU/D6/CP/D1/CT /D3/CU /D8/CW/CT /D3/CQ /CY/CT
/D8 /CP/D2/CS /D8/CW/CT /D7/CX/D1 /D9/D0/D8/CP/D2/CT/CX/D8 /DD/D3/CU /D8/CW/CT /CT/DA /CT/D2 /D8/D7 A /CP/D2/CSB /CX/D2 /CX/D8 /CP/D6/CT
/CW/D3/D7/CT/D2 /D3/D2/D0/DD /D8/D3 /CW/CP /DA /CT /D8/CW/CT
/D3/D2/D2/CT
/D8/CX/D3/D2 /DB/CX/D8/CW /D8/CW/CT /D4/D6/CT/D6/CT/D0/CP/D8/CX/DA/CX/D7/D8/CX
/D4/CW /DD/D7/CX
/D7/B8/DB/CW/CX
/CW /CS/CT/CP/D0/D7 /DB/CX/D8/CW /AH/BF/B7/BD/AH /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7 /CP/D2/CS /D2/D3/D8 /DB/CX/D8/CW /BG/BW /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7/BA/C4/CT/D8 /D9/D7 /D8/CW/CT/D2
/D3/D2/D7/CX/CS/CT/D6 /D8/CW/CT /D7/CP/D1/CT /BG/B9/DA /CT
/D8/D3/D6 la
AB
/CX/D2S′, /B4/DB/CW/CT/D6/CT /CX/D2 /AH/BF/B7/BD/AH /D4/CX
/D8/D9/D6/CT /D8/CW/CT /D6/D3 /CS /CX/D7 /D1/D3 /DA/CX/D2/CV/B5/BA/CC/CW/CT /D4 /D3/D7/CX/D8/CX/D3/D2 /BG/B9/DA /CT
/D8/D3/D6/D7 xa
A
/CP/D2/CSxa
B
/D3/CU /D8/CW/CT /CT/DA /CT/D2 /D8/D7 A /CP/D2/CSB /D6/CT/D7/D4 /CT
/D8/CX/DA /CT/D0/DD /CP/D6/CT /CS/CT
/D3/D1/D4 /D3/D7/CT/CS /DB/CX/D8/CW /D6/CT/D7/D4 /CT
/D8/D8/D3/braceleftbig
e′
µ/bracerightbig/CQ/CP/D7/CT /CP/D7xa
A=x′0
Aee′
0+x′1
Aee′
1= 0e′
0+ 0e′
1, /CP/D2/CSxa
B=x′0
Bee′
0+x′1
Bee′
1=−βeγel0e′
0+γel0e′
1,/CP/D2/CS /D8/CW/CT /CS/CX/D7/D8/CP/D2
/CT /BG/B9/DA /CT
/D8/D3/D6 /CX/D7 /CS/CT
/D3/D1/D4 /D3/D7/CT/CS /CP/D7
la
AB=xa
B−xa
A=l′0
ee′
0+l′1
ee′
1=−βeγel0e′
0+γel0e′
1./C6/D3/D8/CT /D8/CW/CP/D8 /CX/D2 /D8/CW/CT /AH/CT/AH
/D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/B8
/D3/D1/D1/D3/D2/D0/DD /D9/D7/CT/CS /CX/D2 /D8/CW/CT /AH/BT /CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD /B8/AH /D8/CW/CT/D6/CT /CX/D7 /CP /CS/CX/D0/CP/D8/CP/D8/CX/D3/D2 /D3/CU/D8/CW/CT /D7/D4/CP/D8/CX/CP/D0 /D4/CP/D6/D8l′1
e=γel0
/DB/CX/D8/CW /D6/CT/D7/D4 /CT
/D8 /D8/D3l1
e=l0
/CP/D2/CS /D2/D3/D8 /D8/CW/CT /C4/D3/D6/CT/D2 /D8/DE
/D3/D2 /D8/D6/CP
/D8/CX/D3/D2 /CP/D7 /D4/D6/CT/CS/CX
/D8/CT/CS /CX/D2/D8/CW/CT /AH/BT /CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD /BA/AH /C0/D3 /DA /CT/DB /CT/D6 /CX/D8 /CX/D7
/D0/CT/CP/D6 /CU/D6/D3/D1 /D8/CW/CT /CP/CQ /D3 /DA /CT /CS/CX/D7
/D9/D7/D7/CX/D3/D2 /D8/CW/CP/D8
/D3/D1/D4/CP/D6/CX/D7/D3/D2 /D3/CU /D3/D2/D0/DD /D7/D4/CP/D8/CX/CP/D0/BH/D4/CP/D6/D8/D7 /D3/CU /D8/CW/CT
/D3/D1/D4 /D3/D2/CT/D2 /D8/D7 /D3/CU /D8/CW/CT /CS/CX/D7/D8/CP/D2
/CT /BG/B9/DA /CT
/D8/D3/D6 la
AB
/CX/D2S /CP/D2/CSS′/CX/D7 /D4/CW /DD/D7/CX
/CP/D0/D0/DD /D1/CT/CP/D2/CX/D2/CV/D0/CT/D7/D7 /CX/D2 /D8/CW/CT/AH/CC/CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD /BA/AH /BT/D0/D0
/D3/D1/D4 /D3/D2/CT/D2 /D8/D7 /D3/CU /D8/CW/CT /CS/CX/D7/D8/CP/D2
/CT /BG/B9/DA /CT
/D8/D3/D6 la
AB
/CP/D6/CT /D8/D6/CP/D2/D7/CU/D3/D6/D1/CT/CS /CQ /DD /D8/CW/CT /C4 /CC /CU/D6/D3/D1S/D8/D3S′. lµ
e
/CP/D2/CSl′µ
e
/CP/D6/CT /CS/CX/AR/CT/D6/CT/D2 /D8 /D6/CT/D4/D6/CT/D7/CT/D2 /D8/CP/D8/CX/D3/D2/D7 /D3/CU /D8/CW/CT /D7/CP/D1/CT /D4/CW /DD/D7/CX
/CP/D0 /D5/D9/CP/D2 /D8/CX/D8 /DD la
AB
/D1/CT/CP/D7/D9/D6/CT/CS /CX/D2 /D8 /DB /D3/D6/CT/D0/CP/D8/CX/DA /CT/D0/DD /D1/D3 /DA/CX/D2/CV /C1/BY/CA/D7S /CP/D2/CSS′/BA /CC/CW/CT /CX/D2 /DA /CP/D6/CX/CP/D2 /D8 /D7/D4/CP
/CT/D8/CX/D1/CT /D0/CT/D2/CV/D8/CW /D3/CU /D8/CW/CP/D8 /D3/CQ /CY/CT
/D8 /CX/D2S′/CX/D7
l=l′
e= (l′µ
el′
µe)1/2=l0./C6/D3/D8/CT /D8/CW/CP/D8 /CX/CUl0
e= 0 /D8/CW/CT/D2l′µ
e
/CX/D2 /CP/D2 /DD /D3/D8/CW/CT/D6 /C1/BY/CAS′/DB/CX/D0/D0
/D3/D2 /D8/CP/CX/D2 /D8/CW/CT /D8/CX/D1/CT
/D3/D1/D4 /D3/D2/CT/D2 /D8 l′0
e∝ne}ationslash= 0. /CF /CT
/D3/D2
/D0/D9/CS/CT/CU/D6/D3/D1 /D8/CW/CT /CP/CQ /D3 /DA /CT /CS/CX/D7
/D9/D7/D7/CX/D3/D2 /D8/CW/CP/D8 /CX/CU /D3/D2/CT /DB /CP/D2 /D8/D7 /CX/D2 /D8/CW/CT /AH/CC/CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH /D8/D3
/D3/D1/D4/CP/D6/CT /CX/D2 /CP /D4/CW /DD/D7/CX
/CP/D0/D0/DD/D1/CT/CP/D2/CX/D2/CV/CU/D9/D0 /D7/CT/D2/D7/CT /D8/CW/CT /AH/D0/CT/D2/CV/D8/CW/D7/AH /D3/CU /D8 /DB /D3 /CS/CX/AR/CT/D6/CT/D2 /D8 /D3/CQ /CY/CT
/D8/D7 /D8/CW/CP/D2 /CX/D8 /CX/D7 /D4 /D3/D7/D7/CX/CQ/D0/CT /D3/D2/D0/DD /CQ /DD
/D3/D1/D4/CP/D6/CX/D2/CV /D8/CW/CT/CX/D6/CX/D2 /DA /CP/D6/CX/CP/D2 /D8 /D7/D4/CP
/CT/D8/CX/D1/CT /D0/CT/D2/CV/D8/CW/D7/BA/C1/D2 /D8/CW/CT /AH/D6/AH
/D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2 /D8/CW/CT /D4 /D3/D7/CX/D8/CX/D3/D2 /BG/B9/DA /CT
/D8/D3/D6/D7 /D3/CU /D8/CW/CT /CT/DA /CT/D2 /D8/D7 A /CP/D2/CSB, xa
A
/CP/D2/CSxa
B, /CX/D2S /CP/D6/CT/CS/CT
/D3/D1/D4 /D3/D7/CT/CS /DB/CX/D8/CW /D6/CT/D7/D4 /CT
/D8 /D8/D3{rµ} /CQ/CP/D7/CT /CP/D7xa
A=x0
Arr0+x1
Arr1= 0r0+0r1, /CP/D2/CSxa
B=x0
Brr0+x1
Brr1=
−l0r0+l0r1, /CP/D2/CS /D8/CW/CT /CS/CX/D7/D8/CP/D2
/CT /BG/B9/DA /CT
/D8/D3/D6 la
AB=xa
B−xa
A
/CX/D7 /CS/CT
/D3/D1/D4 /D3/D7/CT/CS /CP/D7
la
AB=l0
rr0+l1
rr1=−l0r0+l0r1,/CP/D2/CS /D8/CW/CT /CC/CC /D0/CT/D2/CV/D8/CW l /CX/D7
l=lr= (lµ
rlµr)1/2=le=l0/CP/D7 /CX/D8 /D1 /D9/D7/D8 /CQ /CT/BA/C1/D2S′/CP/D2/CS /CX/D2 /D8/CW/CT/braceleftbig
r′
µ/bracerightbig/CQ/CP/D7/CT /D8/CW/CT /D4 /D3/D7/CX/D8/CX/D3/D2 /BG/B9/DA /CT
/D8/D3/D6/D7 /D3/CU /D8/CW/CT /CT/DA /CT/D2 /D8/D7 A /CP/D2/CSB /CP/D6/CTxa
A= 0r′
0+ 0r′
1/CP/D2/CSxa
B=x′0
Brr′
0+x′1
Brr′
1=−Kl0r′
0+ (1 + βr)(1/K)l0r′
1
/B8 /CP/D2/CS /D8/CW/CT
/D3/D1/D4 /D3/D2/CT/D2 /D8/D7 l′µ
r
/D3/CU /D8/CW/CT /CS/CX/D7/D8/CP/D2
/CT/BG/B9/DA /CT
/D8/D3/D6 la
AB
/CP/D6/CT /CT/D5/D9/CP/D0 /D8/D3 /D8/CW/CT
/D3/D1/D4 /D3/D2/CT/D2 /D8/D7 x′µ
Br
/B8 /CX/BA/CT/BA/B8l′µ
r=x′µ
Br. /CC/CW /D9/D7 la
AB
/CX/D7 /CS/CT
/D3/D1/D4 /D3/D7/CT/CS /CP/D7
la
AB=l′0
rr′
0+l′1
rr′
1=−Kl0r′
0+ (1 + βr)(1/K)l0r′
1./C1/CU /D3/D2/D0/DD /D7/D4/CP/D8/CX/CP/D0 /D4/CP/D6/D8/D7 /D3/CUlµ
r
/CP/D2/CSl′µ
r
/CP/D6/CT
/D3/D1/D4/CP/D6/CT/CS /D8/CW/CP/D2 /D3/D2/CT /AS/D2/CS/D7 /D8/CW/CP/D8∞ ≻l′1
r≥l0
/CU/D3/D6−1/2≺βr≤0/CP/D2/CSl0≤l′1
r≺ ∞ /CU/D3/D60≤βr≺ ∞ /B8 /DB/CW/CX
/CW /D3/D2
/CT /CP/CV/CP/CX/D2 /D7/CW/D3 /DB/D7 /D8/CW/CP/D8 /D7/D9
/CW
/D3/D1/D4/CP/D6/CX/D7/D3/D2 /CX/D7 /D4/CW /DD/D7/CX
/CP/D0/D0/DD/D1/CT/CP/D2/CX/D2/CV/D0/CT/D7/D7 /CX/D2 /D8/CW/CT /AH/CC/CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD /BA/AH /C0/D3 /DA /CT/DB /CT/D6 /D8/CW/CT /CX/D2 /DA /CP/D6/CX/CP/D2 /D8 /D7/D4/CP
/CT/D8/CX/D1/CT /D0/CT/D2/CV/D8/CW /CP/D0/DB /CP /DD/D7 /D8/CP/CZ /CT/D7 /D8/CW/CT /D7/CP/D1/CT/DA /CP/D0/D9/CT
l=l′
r= (l′µ
rl′
µr)1/2=lr=l0,/CP/D2/CS /CP/D7 /CP/D0/D6/CT/CP/CS/DD /D7/CP/CX/CS/B8 /CX/D8
/CP/D2 /CQ /CT
/D3/D1/D4/CP/D6/CT/CS /CX/D2 /CP /D4/CW /DD/D7/CX
/CP/D0/D0/DD /D1/CT/CP/D2/CX/D2/CV/CU/D9/D0 /D7/CT/D2/D7/CT /CX/D2 /D8/CW/CT /AH/CC/CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD /BA/AH /C7/D2/CT
/D3/D2
/D0/D9/CS/CT/D7 /CU/D6/D3/D1 /D8/CW/CX/D7 /CS/CX/D7
/D9/D7/D7/CX/D3/D2 /D8/CW/CP/D8/B8 /CT/BA/CV/BA/B8 /D3/D9/D6 /D4/CP/D6/D8/CX
/D9/D0/CP/D6 /BG/B9/DA /CT
/D8/D3/D6 la
AB
/B4/CP /CV/CT/D3/D1/CT/D8/D6/CX
/CP/D0 /D5/D9/CP/D2 /D8/CX/D8 /DD/B5 /CX/D7/D6/CT/D4/D6/CT/D7/CT/D2 /D8/CT/CS /CX/D2 /CS/CX/AR/CT/D6/CT/D2 /D8 /CQ/CP/D7/CT/D7{eµ},/braceleftbig
e′
µ/bracerightbig
,{rµ} /CP/D2/CS/braceleftbig
r′
µ/bracerightbig/CQ /DD /CX/D8/D7
/D3 /D3/D6/CS/CX/D2/CP/D8/CT /D6/CT/D4/D6/CT/D7/CT/D2 /D8/CP/D8/CX/D3/D2/D7 lµ
e, l′µ
e, lµ
r/CP/D2/CSl′µ
r
/B8 /D6/CT/D7/D4 /CT
/D8/CX/DA /CT/D0/DD ./CF /CT /D7/CT/CT /D8/CW/CP/D8 /CX/D2 /D8/CW/CT /AH/CC/CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH /D8/CW/CT /CV/CT/D3/D1/CT/D8/D6/CX
/CP/D0 /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7/B8 /CT/BA/CV/BA/B8 /D8/CW/CT /BG/B9/DA /CT
/D8/D3/D6/D7 xa, la, /BA/BA/BA/B8 /CW/CP /DA /CT/CS/CX/AR/CT/D6/CT/D2 /D8 /D6/CT/D4/D6/CT/D7/CT/D2 /D8/CP/D8/CX/D3/D2/D7 /CS/CT/D4 /CT/D2/CS/CX/D2/CV /D3/D2 /D8/CW/CT
/CW/D3/D7/CT/D2 /C1/BY/CA /CP/D2/CS /D8/CW/CT
/CW/D3/D7/CT/D2
/D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2 /CX/D2 /D8/CW/CP/D8/C1/BY/CA/B8 /CT/BA/CV/BA/B8xµ
e,r, l′µ
e,r, .. /BA /BT/D0/D8/CW/D3/D9/CV/CW /D8/CW/CT /BX/CX/D2/D7/D8/CT/CX/D2
/D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2 /CX/D7 /D4/D6/CT/CU/CT/D6/D6/CT/CS /CQ /DD /D4/CW /DD/D7/CX
/CX/D7/D8/D7 /CS/D9/CT /D8/D3 /CX/D8/D7/D7/CX/D1/D4/D0/CX
/CX/D8 /DD /CP/D2/CS /D7/DD/D1/D1/CT/D8/D6/DD /CX/D8 /CX/D7 /D2/D3/D8/CW/CX/D2/CV /D1/D3/D6/CT /AH/D4/CW /DD/D7/CX
/CP/D0/AH /D8/CW/CP/D2 /D3/D8/CW/CT/D6/D7/B8 /CT/BA/CV/BA/B8 /D8/CW/CT /AH/D6/AH
/D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/BA/CC/CW/CT
/D3 /D3/D6/CS/CX/D2/CP/D8/CT /CS/CT/D4 /CT/D2/CS/CT/D2 /D8 /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7 /CW/CP /DA /CT /D2/D3/D8 /CP/D2 /CX/D2 /D8/D6/CX/D2/D7/CX
/D4/CW /DD/D7/CX
/CP/D0 /D1/CT/CP/D2/CX/D2/CV/BA /CC/CW/CT /D7/D4/CP
/CT/D8/CX/D1/CT /D0/CT/D2/CV/D8/CW
l /CX/D7 /CP/D2 /CT/DC/CP/D1/D4/D0/CT /D3/CU /CP /DB /CT/D0/D0 /CS/CT/AS/D2/CT/CS /D5/D9/CP/D2 /D8/CX/D8 /DD /D8/CW/CP/D8 /CX/D7 /CX/D2/CS/CT/D4 /CT/D2/CS/CT/D2 /D8 /D3/CU /D8/CW/CT
/CW/D3/D7/CT/D2 /C1/BY/CA /CP/D2/CS /CP/D0/D7/D3 /D3/CU /D8/CW/CT
/D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2 /D8/CP/CZ /CT/D2 /CX/D2 /D8/CW/CP/D8 /C1/BY/CA/BN /CX/D8 /CX/D7 /CP/D2 /CX/D2 /D8/D6/CX/D2/D7/CX
/D4/D6/D3/D4 /CT/D6/D8 /DD /D3/CU /D8/CW/CT /D7/D4/CP
/CT/D8/CX/D1/CT/BA /BY /D6/D3/D1 /D8/CW/CX/D7
/D3/D2/D7/CX/CS/CT/D6/B9/CP/D8/CX/D3/D2 /CP/D2 /CX/D1/D4 /D3/D6/D8/CP/D2 /D8
/D3/D2
/D0/D9/D7/CX/D3/D2 /CT/D1/CT/D6/CV/CT/D7/BN /D8/CW/CT /D9/D7/D9/CP/D0 /BF/BW /D0/CT/D2/CV/D8/CW /D3/CU /CP /D1/D3 /DA/CX/D2/CV /D3/CQ /CY/CT
/D8
/CP/D2/D2/D3/D8 /CQ /CT /CS/CT/AS/D2/CT/CS/CX/D2 /D8/CW/CT /BG/BW /D7/D4/CP
/CT/D8/CX/D1/CT /D3/CU /D8/CW/CT /CC/CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD /CX/D2 /CP/D2 /CP/CS/CT/D5/D9/CP/D8/CT /DB /CP /DD /B8 /D7/CX/D2
/CT /CX/D8 /CX/D7 /D3/D2/D0/DD /D8/CW/CT /D7/D4/CP/D8/CX/CP/D0 /D0/CT/D2/CV/D8/CW /CP/D2/CS/D2/D3/D8 /CP /BG/BW /D8/CT/D2/D7/D3/D6 /D5/D9/CP/D2 /D8/CX/D8 /DD /BA/BE/BA/BE/BA /CC/CW/CT /BT /CC /D3/CU /D0/CT/D2/CV/D8/CW/C1/D2
/D3/D2 /D8/D6/CP/D7/D8 /D8/D3 /D8/CW/CT
/D3 /DA /CP/D6/CX/CP/D2 /D8 /CS/CT/AS/D2/CX/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D7/D4/CP
/CT/D8/CX/D1/CT /D0/CT/D2/CV/D8/CW /CP/D2/CS /D8/CW/CT /CC/CC /D3/CU /D8/CW/CT /D7/D4/CP
/CT/D8/CX/D1/CT/D8/CT/D2/D7/D3/D6/D7
/D3/D2/D7/CX/CS/CT/D6/CT/CS /CX/D2 /D8/CW/CT /AH/CC/CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH /D8/CW/CT /D7/DD/D2
/CW/D6 /D3/D2/D3/D9/D7 /CS/CT/AS/D2/CX/D8/CX/D3/D2 /D3/CU /D7/D4 /CP/D8/CX/CP/D0 /D0/CT/D2/CV/D8/CW/B8 /CX/D2/D8/D6 /D3 /CS/D9
/CT /CS/CQ/DD /BX/CX/D2/D7/D8/CT/CX/D2 /CJ/BD/BE ℄ /CS/CT/AS/D2/CT/D7 /D0/CT/D2/CV/D8/CW /CP/D7 /D8/CW/CT /D7/D4 /CP/D8/CX/CP/D0 /CS/CX/D7/D8/CP/D2
/CT /CQ /CT/D8/DB/CT /CT/D2 /D8/DB/D3 /D7/D4 /CP/D8/CX/CP/D0 /D4 /D3/CX/D2/D8/D7 /D3/D2 /D8/CW/CT /B4/D1/D3/DA/CX/D2/CV/B5/D3/CQ/CY/CT
/D8 /D1/CT /CP/D7/D9/D6 /CT /CS /CQ/DD /D7/CX/D1/D9/D0/D8/CP/D2/CT/CX/D8/DD /CX/D2 /D8/CW/CT /D6 /CT/D7/D8 /CU/D6 /CP/D1/CT /D3/CU /D8/CW/CT /D3/CQ/D7/CT/D6/DA/CT/D6/BA /CC /D3 /D7/CT/CT /D8/CW/CT /CS/CX/AR/CT/D6/CT/D2
/CT /DB/CX/D8/CW/D6/CT/D7/D4 /CT
/D8 /D8/D3 /D8/CW/CT /AH/CC/CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH /DB /CT /CS/CT/D8/CT/D6/D1/CX/D2/CT /D8/CW/CT /D7/D4/CP/D8/CX/CP/D0 /D0/CT/D2/CV/D8/CW /D3/CU /D8/CW/CT /D6/D3 /CS
/D3/D2/D7/CX/CS/CT/D6/CT/CS /CX/D2 /D8/CW/CT /D4/D6/CT/DA/CX/D3/D9/D7/D7/CT
/D8/CX/D3/D2/BA /BT/D7 /D7/CW/D3 /DB/D2 /CP/CQ /D3 /DA /CT /CX/D2 /D8/CW/CT /AH/CC/CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD /B8/AH /CX/D2
/D3/D2 /D8/D6/CP/D7/D8 /D8/D3 /D8/CW/CT /AH/BT /CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD /B8/AH /D3/D2/CT
/CP/D2/D2/D3/D8/BI/D7/D4 /CT/CP/CZ /CP/CQ /D3/D9/D8 /D8/CW/CT /D7/D4/CP/D8/CX/CP/D0 /CS/CX/D7/D8/CP/D2
/CT/B8 /CT/BA/CV/BA/B8 /D8/CW/CT /D6/CT/D7/D8 /D7/D4/CP/D8/CX/CP/D0 /D0/CT/D2/CV/D8/CW /D3/CU /CP /D6/D3 /CS/B8 /CP/D7 /CP
/D3/D6/D6/CT
/D8/D0/DD /CS/CT/AS/D2/CT/CS/D4/CW /DD/D7/CX
/CP/D0 /D5/D9/CP/D2 /D8/CX/D8 /DD /B8/BA/CQ/D9/D8 /D3/D2/D0/DD /CP/CQ /D3/D9/D8 /BG/BW /D8/CT/D2/D7/D3/D6 /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7/B8 /D8/CW/CT /CV/CT/D3/D1/CT/D8/D6/CX
/CP/D0 /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7 /B9 /D8/CW/CT /D4 /D3/D7/CX/D8/CX/D3/D2 /BG/B9/DA /CT
/D8/D3/D6/D7 xa
A,B
/B8 /D8/CW/CT /CS/CX/D7/D8/CP/D2
/CT /BG/B9/DA /CT
/D8/D3/D6 la
AB
/B8 /D8/CW/CT /D7/D4/CP
/CT/D8/CX/D1/CT /D0/CT/D2/CV/D8/CW l /B8 /CT/D8
/BA/B8 /CP/D2/CS /D8/CW/CT/CX/D6 /BG/BW /D6/CT/D4/D6/CT/D7/CT/D2 /D8/CP/D8/CX/D3/D2/D7/B8
xµ
A,B,..e,r,.. , lµ
ABe,r,.. , le,r,.../C1/D2/D7/D8/CT/CP/CS /D3/CU /D8/D3 /DB /D3/D6/CZ /DB/CX/D8/CW /CV/CT/D3/D1/CT/D8/D6/CX
/CP/D0 /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7 xa
A,B, la
AB
/CP/D2/CSl /D3/D2/CT /CS/CT/CP/D0/D7/B8 /CX/D2 /D8/CW/CT /AH/BT /CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD /B8/AH/D3/D2/D0/DD /DB/CX/D8/CW /D8/CW/CT /D7/D4/CP/D8/CX/CP/D0/B8 /D3/D6 /D8/CT/D1/D4 /D3/D6/CP/D0/B8 /D4/CP/D6/D8/D7 /D3/CU /D8/CW/CT/CX/D6
/D3 /D3/D6/CS/CX/D2/CP/D8/CT /D6/CT/D4/D6/CT/D7/CT/D2 /D8/CP/D8/CX/D3/D2/D7 xµ
Ae,r, xµ
Be,r
/CP/D2/CSlµ
e,r./BY/CX/D6/D7/D8 /D8/CW/CT /AH/CT/AH
/D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/B8 /DB/CW/CX
/CW /CX/D7 /CP/D0/D1/D3/D7/D8 /CP/D0/DB /CP /DD/D7 /D9/D7/CT/CS /CX/D2 /D8/CW/CT /AH/BT /CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD /B8/AH /CX/D7
/D3/D2/D7/CX/CS/CT/D6/CT/CS/BA/BT
/D3/D6/CS/CX/D2/CV /D8/D3 /BX/CX/D2/D7/D8/CT/CX/D2/B3/D7 /CS/CT/AS/D2/CX/D8/CX/D3/D2 /CJ/BD/BE ℄ /D3/CU /D8/CW/CT /D7/D4/CP/D8/CX/CP/D0 /D0/CT/D2/CV/D8/CW /D8/CW/CT /D7/D4/CP/D8/CX/CP/D0 /CT/D2/CS/D7 /D3/CU /D8/CW/CT /D6/D3 /CS /D1 /D9/D7/D8 /CQ /CT/D8/CP/CZ /CT/D2 /D7/CX/D1 /D9/D0/D8/CP/D2/CT/D3/D9/D7/D0/DD /CX/D2 /D8/CW/CT
/CW/D3/D7/CT/D2
/D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/BA /C1/D2 /BG/BW /B4/CP/D8 /D9/D7 /BE/BW/B5 /D7/D4/CP
/CT/D8/CX/D1/CT /CP/D2/CS /CX/D2 /D8/CW/CT /AH/CT/AH
/D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2 /D8/CW/CT /D7/CX/D1 /D9/D0/D8/CP/D2/CT/D3/D9/D7 /CT/DA /CT/D2 /D8/D7 A /CP/D2/CSB /B4/DB/CW/D3/D7/CT /D7/D4/CP/D8/CX/CP/D0 /D4/CP/D6/D8/D7
/D3/D6/D6/CT/D7/D4 /D3/D2/CS /D8/D3 /D8/CW/CT /D7/D4/CP/D8/CX/CP/D0/CT/D2/CS/D7 /D3/CU /D8/CW/CT /D6/D3 /CS/B5 /CP/D6/CT /D8/CW/CT /CX/D2 /D8/CT/D6/D7/CT
/D8/CX/D3/D2/D7 /D3/CUx1
e
/CP/DC/CX/D7 /B4/D8/CW/CP/D8 /CX/D7 /CP/D0/D3/D2/CV /D8/CW/CT /D7/D4/CP/D8/CX/CP/D0 /CQ/CP/D7/CT /DA /CT
/D8/D3/D6 e1
/B5 /CP/D2/CS /D8/CW/CT/DB /D3/D6/D0/CS /D0/CX/D2/CT/D7 /D3/CU /D8/CW/CT /D7/D4/CP/D8/CX/CP/D0 /CT/D2/CS/D7 /D3/CU /D8/CW/CT /D6/D3 /CS /D8/CW/CP/D8 /CX/D7 /CP/D8 /D6/CT/D7/D8 /CX/D2S /CP/D2/CS /D7/CX/D8/D9/CP/D8/CT/CS /CP/D0/D3/D2/CV /D8/CW/CTx1
e
/CP/DC/CX/D7/BA/CC/CW/CT/D4 /D3/D7/CX/D8/CX/D3/D2 /BG/B9/DA /CT
/D8/D3/D6/D7 /B4/CX/D2 /D8/CW/CT /AH/CT/AH /CQ/CP/D7/CT/B5 xµ
Ae
/CP/D2/CSxµ
Be
/D3/CU /D8/CW/CT /D7/CX/D1 /D9/D0/D8/CP/D2/CT/D3/D9/D7 /B4/CP/D8te=a= 0) /CT/DA /CT/D2 /D8/D7 A/CP/D2/CSB /CX/D2S /CP/D6/CTxµ
Ae= 0e0+ 0e1, /D3/D6/B8 /CX/D2 /D7/CW/D3/D6/D8/B8 xµ
Ae= (0,0), /CP/D2/CSxµ
Be= (0, l0), /CP/D2/CS /D8/CW/CT /CS/CX/D7/D8/CP/D2
/CT/BG/B9/DA /CT
/D8/D3/D6 /B4/CX/D2 /D8/CW/CT /AH/CT/AH /CQ/CP/D7/CT/B5 lµ
ABe=xµ
Be−xµ
Ae= (0, l0). /CF /CT /CT/D1/D4/CW/CP/D7/CX/DE/CT /D8/CW/CP/D8 /CX/D8 /CX/D7 /D2/CT
/CT/D7/D7/CP/D6/DD /CX/D2 /D8/CW/CT /AH/BT /CC/D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH /D8/D3 /D8/CP/CZ /CT /D8/CW/CT /CT/D2/CS /D4 /D3/CX/D2 /D8/D7 /D3/CU /D8/CW/CT /D7/D4/CP/D8/CX/CP/D0 /D0/CT/D2/CV/D8/CW /D3/CU /D8/CW/CT /D6/D3 /CS /D8/D3 /CQ /CT /D7/CX/D1 /D9/D0/D8/CP/D2/CT/D3/D9/D7/B8 /DB/CW/CT/D6/CT/CP/D7 /CX/D2/D8/CW/CT /AH/CC/CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH /D8/CW/CT /CT/DA /CT/D2 /D8/D7 A /CP/D2/CSB
/CP/D2 /CQ /CT/B8 /CX/D2 /D4/D6/CX/D2
/CX/D4/D0/CT/B8 /D8/CP/CZ /CT/D2 /CP/D8 /CP/D6/CQ/CX/D8/D6/CP/D6/DD x0
Ae∝ne}ationslash=x0
Be. /CC/CW/CT/D2 /CX/D2
S /B8 /D8/CW/CT /D6/CT/D7/D8 /CU/D6/CP/D1/CT /D3/CU /D8/CW/CT /D3/CQ /CY/CT
/D8/B8 /D8/CW/CT /D7/D4/CP/D8/CX/CP/D0 /D4/CP/D6/D8l1
ABe=l0
/D3/CUlµ
ABe
/CX/D7
/D3/D2/D7/CX/CS/CT/D6/CT/CS /D8/D3 /CS/CT/AS/D2/CT /D8/CW/CT /D6/CT/D7/D8/D7/D4/CP/D8/CX/CP/D0 /D0/CT/D2/CV/D8/CW /B4/D8/CW/CT /D8/CT/D1/D4 /D3/D6/CP/D0 /D4/CP/D6/D8 /D3/CUlµ
ABe
/CX/D7 /D8/CP/CZ /CT/D2 /D8/D3 /CQ /CT /DE/CT/D6/D3/B5/BA /BY /D9/D6/D8/CW/CT/D6 /D3/D2/CT /D9/D7/CT/D7 /D8/CW/CT /CX/D2 /DA /CT/D6/D7/CT /C4/D3/D6/CT/D2 /D8/DE/D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2/D7 /D8/D3 /CT/DC/D4/D6/CT/D7/D7 xµ
Ae, xµ
Be, /CP/D2/CSlµ
ABe
/CX/D2S /CX/D2 /D8/CT/D6/D1/D7 /D3/CU /D8/CW/CT
/D3/D6/D6/CT/D7/D4 /D3/D2/CS/CX/D2/CV /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7 /CX/D2S′/B8/CX/D2 /DB/CW/CX
/CW /D8/CW/CT /D6/D3 /CS /CX/D7 /D1/D3 /DA/CX/D2/CV/BA /CC/CW/CX/D7 /D4/D6/D3
/CT/CS/D9/D6/CT /DD/CX/CT/D0/CS/D7 x0
A,Be=ctA,Be=γe(ct′
A,Be+βex′1
A,Be), /CP/D2/CS
x1
A,Be=γe(βect′
A,Be+x′1
A,Be), /DB/CW/CT/D2
/CT
l0
ABe=ctBe−ctAe=γe(ct′
Be−ct′
Ae) +γeβe(x′1
Be−x′1
Ae) =γel′0
ABe+γeβel′1
ABe
/B4/BG/B5/CP/D2/CS
l1
ABe=x1
Be−x1
Ae=γe(x′1
Be−x′1
Ae) +γeβe(ct′
Be−ct′
Ae) =γel′1
ABe+γeβel′0
ABe. /B4/BH/B5/C6/D3 /DB
/D3/D1/CT/D7 /D8/CW/CT /D1/CP/CX/D2 /CS/CX/AR/CT/D6/CT/D2
/CT /CQ /CT/D8 /DB /CT/CT/D2 /D8/CW/CT /D8 /DB /D3 /CU/D3/D6/D1/D7 /D3/CU /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD /BA /C1/D2/D7/D8/CT/CP/CS /D3/CU /D8/D3 /DB /D3/D6/CZ /DB/CX/D8/CW /BG/BW/D8/CT/D2/D7/D3/D6 /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7 /CP/D2/CS /D8/CW/CT/CX/D6 /C4 /CC /B4/CP/D7 /CX/D2 /D8/CW/CT /AH/CC/CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH/B5 /CX/D2 /D8/CW/CT /AH/BT /CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH /D3/D2/CT /CU/D3/D6/CV/CT/D8/D7 /CP/CQ /D3/D9/D8/D8/CW/CT /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D8/CT/D1/D4 /D3/D6/CP/D0 /D4/CP/D6/D8l0
ABe, /BX/D5/BA/B4/BG/B5/B8 /CP/D2/CS
/D3/D2/D7/CX/CS/CT/D6/D7 /D3/D2/D0/DD /D8/CW/CT /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT/D7/D4/CP/D8/CX/CP/D0 /D4/CP/D6/D8l1
ABe, /BX/D5/BA/B4/BH/B5 . /BY /D9/D6/D8/CW/CT/D6/B8 /CX/D2 /D8/CW/CP/D8 /D6/CT/D0/CP/D8/CX/D3/D2 /CU/D3/D6l1
ABe
/D3/D2/CT /CP/D7/D7/D9/D1/CT/D7 /D8/CW/CP/D8t′
Be=t′
Ae=t′
e=b,/CX/BA/CT/BA/B8 /D8/CW/CP/D8x′1
Be
/CP/D2/CSx′1
Ae
/CP/D6/CT /D7/CX/D1 /D9/D0/D8/CP/D2/CT/D3/D9/D7/D0/DD /CS/CT/D8/CT/D6/D1/CX/D2/CT/CS /CP/D8 /D7/D3/D1/CT /CP/D6/CQ/CX/D8/D6/CP/D6/DD t′
e=b /CX/D2S′. /C0/D3 /DB /CT/DA /CT/D6/B8 /CX/D2 /BG/BW/B4/CP/D8 /D9/D7 /BE/BW/B5 /D7/D4/CP
/CT/D8/CX/D1/CT /D7/D9
/CW /CP/D2 /CP/D7/D7/D9/D1/D4/D8/CX/D3/D2 /D1/CT/CP/D2/D7 /D8/CW/CP/D8 /CX/D2S′/D3/D2/CT /CP
/D8/D9/CP/D0/D0/DD /CS/D3 /CT/D7 /D2/D3/D8
/D3/D2/D7/CX/CS/CT/D6 /D8/CW/CT /D7/CP/D1/CT/CT/DA /CT/D2 /D8/D7 A /CP/D2/CSB /CP/D7 /CX/D2S /CQ/D9/D8 /D7/D3/D1/CT /D3/D8/CW/CT/D6 /D8 /DB /D3 /CT/DA /CT/D2 /D8/D7 C /CP/D2/CSD, /DB/CW/CT/D2
/CT t′
Be=t′
Ae
/CW/CP/D7 /D8/D3 /CQ /CT /D6/CT/D4/D0/CP
/CT/CS/DB/CX/D8/CWt′
De=t′
Ce=b. /CC/CW/CT /CT/DA /CT/D2 /D8/D7 C /CP/D2/CSD /CP/D6/CT /D8/CW/CT /CX/D2 /D8/CT/D6/D7/CT
/D8/CX/D3/D2/D7 /D3/CU /D8/CW/CT /D0/CX/D2/CT /B4/D8/CW/CT /CW /DD/D4 /CT/D6/D7/D9/D6/CU/CP
/CT t′
e=b/DB/CX/D8/CW /CP/D6/CQ/CX/D8/D6/CP/D6/DD b /B5 /D4/CP/D6/CP/D0/D0/CT/D0 /D8/D3 /D8/CW/CT /D7/D4/CP/D8/CX/CP/D0 /CP/DC/CX/D7x′1
e
/B4/DB/CW/CX
/CW /CX/D7 /CP/D0/D3/D2/CV /D8/CW/CT /D7/D4/CP/D8/CX/CP/D0 /CQ/CP/D7/CT /DA /CT
/D8/D3/D6 e′
1
/B5 /CP/D2/CS /D3/CU /D8/CW/CT/CP/CQ /D3 /DA /CT /D1/CT/D2 /D8/CX/D3/D2/CT/CS /DB /D3/D6/D0/CS /D0/CX/D2/CT/D7 /D3/CU /D8/CW/CT /D7/D4/CP/D8/CX/CP/D0 /CT/D2/CS /D4 /D3/CX/D2 /D8/D7 /D3/CU /D8/CW/CT /D6/D3 /CS/BA /CC/CW/CT/D2 /CX/D2 /D8/CW/CT /CP/CQ /D3 /DA /CT /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2/CU/D3/D6l1
ABe
/B4/BH/B5 /D3/D2/CT /CW/CP/D7 /D8/D3 /DB/D6/CX/D8/CT x′1
De−x′1
Ce=l′1
CDe
/CX/D2/D7/D8/CT/CP/CS /D3/CUx′1
Be−x′1
Ae=l′1
ABe. /CC/CW/CT /D7/D4/CP/D8/CX/CP/D0 /D4/CP/D6/D8/D7 l1
ABe/CP/D2/CSl′1
CDe
/CP/D6/CT /D8/CW/CT /D7/D4 /CP/D8/CX/CP/D0 /CS/CX/D7/D8/CP/D2
/CT/D7 /CQ /CT/D8 /DB /CT/CT/D2 /D8/CW/CT /CT/DA /CT/D2 /D8/D7 A, B /CP/D2/CSC /B8D, /D6/CT/D7/D4 /CT
/D8/CX/DA /CT/D0/DD /BA /CC/CW/CT /D7/D4 /CP/D8/CX/CP/D0/CS/CX/D7/D8/CP/D2
/CT l1
ABe=x1
Be−x1
Ae
/CS/CT/AS/D2/CT/D7 /CX/D2 /D8/CW/CT /AH/BT /CC /D6 /CT/D0/CP/D8/CX/DA/CX/D8/DD/B8/AH /CP/D2/CS /CX/D2 /D8/CW/CT /AH/CT/AH /CQ /CP/D7/CT /B8 /D8/CW/CT /D7/D4 /CP/D8/CX/CP/D0 /D0/CT/D2/CV/D8/CW /D3/CU/D8/CW/CT /D6 /D3 /CS /CP/D8 /D6 /CT/D7/D8 /CX/D2S, /DB/CW/CX/D0/CT l′1
CDe=x′1
De−x′1
Ce
/CX/D7
/D3/D2/D7/CX/CS/CT/D6 /CT /CS /CX/D2 /D8/CW/CT /AH/BT /CC /D6 /CT/D0/CP/D8/CX/DA/CX/D8/DD/B8/AH /CP/D2/CS /CX/D2 /D8/CW/CT /AH/CT/AH/CQ /CP/D7/CT/B8 /D8/D3 /CS/CT/AS/D2/CT /D8/CW/CT /D7/D4 /CP/D8/CX/CP/D0 /D0/CT/D2/CV/D8/CW /D3/CU /D8/CW/CT /D1/D3/DA/CX/D2/CV /D6 /D3 /CS /CX/D2S′. /CF/CX/D8/CW /D8/CW/CT/D7/CT /CS/CT/AS/D2/CX/D8/CX/D3/D2/D7 /DB /CT /AS/D2/CS /CU/D6/D3/D1 /D8/CW/CT/CT/D5/D9/CP/D8/CX/D3/D2 /CU/D3/D6l1
ABe
/B4/BH/B5 /D8/CW/CT /D6/CT/D0/CP/D8/CX/D3/D2 /CQ /CT/D8 /DB /CT/CT/D2 l′1
e=l′1
CDe
/CP/D2/CSl1
e=l1
ABe=l0
/CP/D7 /D8/CW/CT /CU/CP/D1/D3/D9/D7 /CU/D3/D6/D1 /D9/D0/CP/CT/CU/D3/D6 /D8/CW/CT /C4/D3/D6/CT/D2 /D8/DE
/D3/D2 /D8/D6/CP
/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D1/D3 /DA/CX/D2/CV /D6/D3 /CS
l′1
e=x′1
De−x′1
Ce=l0/γe= (x1
Be−x1
Ae)(1−β2
e)1/2, /B4/BI/B5/DB/CX/D8/CWt′
Ce=t′
De, /CP/D2/CStBe=tAe, /DB/CW/CT/D6/CT βe=Ve/c /B8Ve
/CX/D7 /D8/CW/CT /D6/CT/D0/CP/D8/CX/DA /CT /DA /CT/D0/D3
/CX/D8 /DD /D3/CUS /CP/D2/CSS′. /C6/D3/D8/CT /D8/CW/CP/D8/D8/CW/CT /D7/D4/CP/D8/CX/CP/D0 /D0/CT/D2/CV/D8/CW/D7 l0
/CP/D2/CSl′1
e
/D6/CT/CU/CT/D6 /D2/D3/D8 /D8/D3 /D8/CW/CT /D7/CP/D1/CT /BG/BW /D8/CT/D2/D7/D3/D6 /D5/D9/CP/D2 /D8/CX/D8 /DD /B8 /CP/D7 /CX/D2 /D8/CW/CT /AH/CC/CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD /B8/AH /CQ/D9/D8/D8/D3 /D8 /DB /D3 /CS/CX/AR/CT/D6/CT/D2 /D8 /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7 /CX/D2 /BG/BW /D7/D4/CP
/CT/D8/CX/D1/CT/BA /CC/CW/CT/D7/CT /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7 /CP/D6/CT /D3/CQ/D8/CP/CX/D2/CT/CS /CQ /DD /D8/CW/CT /D7/CP/D1/CT /D1/CT/CP/D7/D9/D6/CT/B9/D1/CT/D2 /D8/D7 /CX/D2S /CP/D2/CSS′; /D8/CW/CT /D7/D4/CP/D8/CX/CP/D0 /CT/D2/CS/D7 /D3/CU /D8/CW/CT /D6/D3 /CS /CP/D6/CT /D1/CT/CP/D7/D9/D6/CT/CS /D7/CX/D1 /D9/D0/D8/CP/D2/CT/D3/D9/D7/D0/DD /CP/D8 /D7/D3/D1/CTte=a /CX/D2S /CP/D2/CS/CP/D0/D7/D3 /CP/D8 /D7/D3/D1/CTt′
e=b /CX/D2S′, /CP/D2/CSa /CX/D2S /CP/D2/CSb /CX/D2S′/CP/D6/CT /D2/D3/D8 /D6/CT/D0/CP/D8/CT/CS /CQ /DD /D8/CW/CT /C4 /CC /D3/D6 /CP/D2 /DD /D3/D8/CW/CT/D6
/D3 /D3/D6/CS/CX/D2/CP/D8/CT/D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2/BA /CF/CW/CX/D0/CT /CX/D2 /D8/CW/CT /AH/CC/CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH /D3/D2/CT /CS/CT/CP/D0/D7 /DB/CX/D8/CW /CT/DA /CT/D2 /D8/D7 /CP/D7
/D3/D6/D6/CT
/D8/D0/DD /CS/CT/AS/D2/CT/CS /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7/CX/D2 /BG/BW /D7/D4/CP
/CT/D8/CX/D1/CT /CX/D2 /BX/CX/D2/D7/D8/CT/CX/D2/B3/D7 /CP/D4/D4/D6/D3/CP
/CW /CJ/BD/BE ℄ /D8/CW/CT /D7/D4/CP/D8/CX/CP/D0 /CP/D2/CS /D8/CT/D1/D4 /D3/D6/CP/D0 /D4/CP/D6/D8/D7 /D3/CU /CT/DA /CT/D2 /D8/D7 /CP/D6/CT /D8/D6/CT/CP/D8/CT/CS/D7/CT/D4/CP/D6/CP/D8/CT/D0/DD /B8 /CP/D2/CS /D1/D3/D6/CT/D3 /DA /CT/D6 /D8/CW/CT /D8/CX/D1/CT
/D3/D1/D4 /D3/D2/CT/D2 /D8 /CX/D7 /D2/D3/D8 /D8/D6/CP/D2/D7/CU/D3/D6/D1/CT/CS /CX/D2 /D8/CW/CT /C4/D3/D6/CT/D2 /D8/DE
/D3/D2 /D8/D6/CP
/D8/CX/D3/D2/BA/CC/CW/CT /C4 /CC /B4/BE/B5 /CX/D7 /D8/CW/CT /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2 /CX/D2 /BG/BW /D7/D4/CP
/CT/D8/CX/D1/CT /CP/D2/CS /CX/D8 /D8/D6/CP/D2/D7/CU/D3/D6/D1/D7 /D7/D3/D1/CT /BG/BW /D8/CT/D2/D7/D3/D6 /D5/D9/CP/D2 /D8/CX/D8 /DD
Qa..
b..(xc, xd, ..) /CU/D6/D3/D1S /D8/D3Q′a..
b..(x′c, x′d, ..) /CX/D2S′, /B4/CP/D0/D0 /D4/CP/D6/D8/D7 /D3/CU /D8/CW/CT /D5/D9/CP/D2 /D8/CX/D8 /DD /CP/D6/CT /D8/D6/CP/D2/D7/CU/D3/D6/D1/CT/CS/B5/B8 /DB/CW/CX
/CW/BJ/D1/CT/CP/D2/D7 /D8/CW/CP/D8 /CX/D2 /BG/BW /D7/D4/CP
/CT/D8/CX/D1/CT /CX/D7 /D2/D3/D8 /D4 /D3/D7/D7/CX/CQ/D0/CT /D8/D3 /D2/CT/CV/D0/CT
/D8 /D8/CW/CT /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2 /D3/CUl0/CP/D7 /CP /D4/CP/D6/D8 /D3/CUlµ,/CP/D7 /CS/D3/D2/CT /CX/D2 /D8/CW/CT /CS/CT/D6/CX/DA /CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /C4/D3/D6/CT/D2 /D8/DE
/D3/D2 /D8/D6/CP
/D8/CX/D3/D2 /B4/BI/B5/BA /C0/D3 /DB /CT/DA /CT/D6/B8 /CX/CU /D3/D2/CT /CS/D3 /CT/D7 /D2/D3/D8 /CU/D3/D6/CV/CT/D8 /D8/CW/CT/D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D8/CT/D1/D4 /D3/D6/CP/D0 /D4/CP/D6/D8l0
ABe, /BX/D5/BA/B4/BG/B5/B8 /CP/D2/CS /D8/CP/CZ /CT/D7 /CX/D2 /CX/D8 /D8/CW/CP/D8t′
Be=t′
Ae, tBe=tAe
/B4/CP/D7 /CX/D2/D8/CW/CT /CS/CT/D6/CX/DA /CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /C4/D3/D6/CT/D2 /D8/DE
/D3/D2 /D8/D6/CP
/D8/CX/D3/D2/B5/B8 /D8/CW/CT/D2 /D3/D2/CT /AS/D2/CS/D7 /CU/D6/D3/D1 /B4/BG/B5 /D8/CW/CP/D8x′1
Be=x′1
Ae, /DB/CW/CX
/CW /CX/D7 /CX/D2 /D8/CW/CT/D3/CQ /DA/CX/D3/D9/D7
/D3/D2 /D8/D6/CP/D7/D8 /DB/CX/D8/CW /D8/CW/CT /CU/D3/D6/D1 /D9/D0/CP/CT /CU/D3/D6 /D8/CW/CT /C4/D3/D6/CT/D2 /D8/DE
/D3/D2 /D8/D6/CP
/D8/CX/D3/D2/BA/C4/CT/D8 /D9/D7 /CP/D0/D7/D3 /D7/CT/CT /CS/D3 /CT/D7 /D8/CW/CT /C4/D3/D6/CT/D2 /D8/DE
/D3/D2 /D8/D6/CP
/D8/CX/D3/D2/B8 /CP/D7 /D8/CW/CT
/D3 /D3/D6/CS/CX/D2/CP/D8/CT /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2/B8
/CW/CP/D2/CV/CT /D8/CW/CT/CX/D2 /D8/CT/D6/DA /CP/D0 ds, /DB/CW/CX
/CW /CS/CT/AS/D2/CT/D7 /D8/CW/CT /CV/CT/D3/D1/CT/D8/D6/DD /D3/CU /D8/CW/CT /D7/D4/CP
/CT/D8/CX/D1/CT/BA /C1/D2S /CP/D2/CS /CX/D2 /D8/CW/CT /AH/CT/AH /CQ/CP/D7/CT /D8/CW/CT /CX/D2 /D8/CT/D6/DA /CP/D0 ds/CX/D7ds2=ds2
e= (dx1
e)2−(c2dte)2, /CP/D2/CS /DB/CX/D8/CWdte= 0, /CP/D7 /CP/D7/D7/D9/D1/CT/CS /CX/D2 /D8/CW/CT /CS/CT/D6/CX/DA /CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /C4/D3/D6/CT/D2 /D8/DE
/D3/D2 /D8/D6/CP
/D8/CX/D3/D2/B8 /CX/D8 /CQ /CT
/D3/D1/CT/D7/B8 /CX/D2S ds2= (dx1
e)2. /C1/D2S′, /DB/CW/CT/D6/CT /CX/D8 /CX/D7 /CP/D7/D7/D9/D1/CT/CS /D8/CW/CP/D8dt′
e= 0, /CP/D2/CS /DB/CX/D8/CW /D8/CW/CT/D6/CT/D0/CP/D8/CX/D3/D2 /CU/D3/D6 /D8/CW/CT /C4/D3/D6/CT/D2 /D8/DE
/D3/D2 /D8/D6/CP
/D8/CX/D3/D2 /B4/BI/B5/B8dx′1
e=dx1
e/γe, /D8/CW/CT /CX/D2/AS/D2/CX/D8/CT/D7/CX/D1/CP/D0 /D7/D4/CP
/CT/D8/CX/D1/CT /CS/CX/D7/D8/CP/D2
/CT ds′/CQ /CT
/D3/D1/CT/D7/B8 /CX/D2S′ds′2= (dx1
e)2/γ2
e, /CP/D2/CS /D8/CW /D9/D7ds′∝ne}ationslash=ds./C4/CT/D8 /D9/D7 /D2/D3 /DB
/D3/D2/D7/CX/CS/CT/D6 /D8/CW/CT /C4/D3/D6/CT/D2 /D8/DE /AH
/D3/D2 /D8/D6/CP
/D8/CX/D3/D2/AH /CX/D2 /D8/CW/CT /AH/D6/AH
/D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/BA /BT
/D3/D6/CS/CX/D2/CV /D8/D3 /BX/CX/D2/B9/D7/D8/CT/CX/D2/B3/D7 /CS/CT/AS/D2/CX/D8/CX/D3/D2 /CJ/BD/BE ℄ /D3/CU /D8/CW/CT /D7/D4/CP/D8/CX/CP/D0 /D0/CT/D2/CV/D8/CW /D8/CW/CT /D7/D4/CP/D8/CX/CP/D0 /CT/D2/CS/D7 /D3/CU /D8/CW/CT /D6/D3 /CS /D1 /D9/D7/D8 /CQ /CT /D8/CP/CZ /CT/D2 /D7/CX/D1 /D9/D0/D8/CP/D2/CT/D3/D9/D7/D0/DD/CX/D2 /D8/CW/CT
/CW/D3/D7/CT/D2
/D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/BA /C1/D2 /BG/BW /B4/CP/D8 /D9/D7 /BE/BW/B5 /D7/D4/CP
/CT/D8/CX/D1/CT /CP/D2/CS /CX/D2 /D8/CW/CT /AH/D6/AH /CQ/CP/D7/CT /D8/CW/CT /D7/D4/CP/D8/CX/CP/D0 /CT/D2/CS/D7 /D3/CU/D8/CW/CT
/D3/D2/D7/CX/CS/CT/D6/CT/CS /D6/D3 /CS/B8 /D8/CW/CP/D8 /CX/D7 /CP/D8 /D6/CT/D7/D8 /CX/D2S, /D1 /D9/D7/D8 /D0/CX/CT /D3/D2 /D8/CW/CT /D0/CX/CV/CW /D8 /D0/CX/D2/CT/B8 /CX/BA/CT/BA/B8 /D3/D2 /D8/CW/CTx1
r
/CP/DC/CX/D7 /B4/D8/CW/CP/D8 /CX/D7 /CP/D0/D3/D2/CV/D8/CW/CT /D7/D4/CP/D8/CX/CP/D0 /CQ/CP/D7/CT /DA /CT
/D8/D3/D6 r1
/B5/BA /C0/CT/D2
/CT /D8/CW/CT /D7/CX/D1 /D9/D0/D8/CP/D2/CT/D3/D9/D7 /CT/DA /CT/D2 /D8/D7 E /CP/D2/CSF /B4/DB/CW/D3/D7/CT /D7/D4/CP/D8/CX/CP/D0 /D4/CP/D6/D8/D7
/D3/D6/D6/CT/D7/D4 /D3/D2/CS/D8/D3 /D8/CW/CT /D7/D4/CP/D8/CX/CP/D0 /CT/D2/CS/D7 /D3/CU /D8/CW/CT /D6/D3 /CS/B5 /CP/D6/CT /D8/CW/CT /CX/D2 /D8/CT/D6/D7/CT
/D8/CX/D3/D2/D7 /D3/CUx1
r
/CP/DC/CX/D7 /CP/D2/CS /D8/CW/CT /DB /D3/D6/D0/CS /D0/CX/D2/CT/D7 /D3/CU /D8/CW/CT /D7/D4/CP/D8/CX/CP/D0/CT/D2/CS/D7 /D3/CU /D8/CW/CT /D6/D3 /CS/BA /C6/D3/D8/CT /D8/CW/CP/D8 /CX/D2 /D3/D9/D6 /BE/BW /D7/D4/CP
/CT/D8/CX/D1/CT /D8/CW/CT /CT/DA /CT/D2 /D8/D7 E /CP/D2/CSF /CP/D6/CT /D2/D3/D8 /D8/CW/CT /D7/CP/D1/CT /CT/DA /CT/D2 /D8/D7 /CP/D7 /D8/CW/CT/CT/DA /CT/D2 /D8/D7 A /CP/D2/CSB,
/D3/D2/D7/CX/CS/CT/D6/CT/CS /CX/D2 /D8/CW/CT /AH/CT/AH /CQ/CP/D7/CT /CU/D3/D6 /D8/CW/CT /D7/CP/D1/CT /D6/D3 /CS /CP/D8 /D6/CT/D7/D8 /CX/D2S, /D7/CX/D2
/CT /D8/CW/CT /D7/CX/D1 /D9/D0/D8/CP/D2/CT/CX/D8 /DD /D3/CU/D8/CW/CT /CT/DA /CT/D2 /D8/D7 /CX/D7 /CS/CT/AS/D2/CT/CS /CX/D2 /CS/CX/AR/CT/D6/CT/D2 /D8 /DB /CP /DD/D7/BA /CC/CW/CT{rµ} /D6/CT/D4/D6/CT/D7/CT/D2 /D8/CP/D8/CX/D3/D2/D7 /D3/CU /D8/CW/CT /D4 /D3/D7/CX/D8/CX/D3/D2 /BG/B9/DA /CT
/D8/D3/D6/D7 xa
E
/CP/D2/CS
xa
F
/D3/CU /D8/CW/CT /CT/DA /CT/D2 /D8/D7 E /CP/D2/CSF /CX/D2S /CP/D6/CTxµ
Er= (0,0) /CP/D2/CSxµ
Fr= (0, l0), /CP/D2/CS /D3/CU /D8/CW/CT /CS/CX/D7/D8/CP/D2
/CT /BG/B9/DA /CT
/D8/D3/D6
la
EF
/CX/D7lµ
EFr=xµ
Fr−xµ
Er= (0, l1
r) = (0 , l0). /C0/D3 /DB /CT/DA /CT/D6/B8 /CP/D7 /D2/D3/D8/CX
/CT/CS /CP/CQ /D3 /DA /CT/B8 /CX/D2 /BG/BW /D7/D4/CP
/CT/D8/CX/D1/CT /D8/CW/CT /D7/D4/CP/D8/CX/CP/D0/D0/CT/D2/CV/D8/CW /CX/D2 /D8/CW/CT /AH/D6/AH /CQ/CP/D7/CTl1
r=l0
/B4/DB/CX/D8/CW x0
Fr=x0
Er
/B5 /CX/D7 /D2/D3/D8 /D8/CW/CT /D7/CP/D1/CT /BG/BW /D5/D9/CP/D2 /D8/CX/D8 /DD /CP/D7 /D8/CW/CT /D7/D4/CP/D8/CX/CP/D0 /D0/CT/D2/CV/D8/CW /CX/D2/D8/CW/CT /AH/CT/AH /CQ/CP/D7/CTl1
e=l0
/B4/DB/CX/D8/CW x0
Be=x0
Ae), /D7/CX/D2
/CT /D8/CW/CT /D7/CX/D1 /D9/D0/D8/CP/D2/CT/CX/D8 /DD /CX/D7 /CS/CT/AS/D2/CT/CS /CX/D2 /CP /CS/CX/AR/CT/D6/CT/D2 /D8 /DB /CP /DD /BA /BT/D4/D4/D0/DD/CX/D2/CV/D8/CW/CT /D7/CP/D1/CT /D4/D6/D3
/CT/CS/D9/D6/CT /CP/D7 /CX/D2 /D8/CW/CT
/CP/D7/CT /D3/CU /D8/CW/CT /CS/CT/D6/CX/DA /CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /C4/D3/D6/CT/D2 /D8/DE
/D3/D2 /D8/D6/CP
/D8/CX/D3/D2 /CX/D2 /D8/CW/CT /AH/CT/AH /CQ/CP/D7/CT /DB /CT/AS/D2/CS /D8/CW/CT /D6/CT/D0/CP/D8/CX/D3/D2/D7 /CU/D3/D6l0
r
/CP/D2/CSl1
r
/D3/D6/D6/CT/D7/D4 /D3/D2/CS/CX/D2/CV /D8/D3 /B4/BG/B5 /CP/D2/CS /B4/BH/B5/B8 /D6/CT/D7/D4 /CT
/D8/CX/DA /CT/D0/DD /B8
l0
EFr=x0
Fr−x0
Er= (1/K)(x′0
Fr−x′0
Er) = (1 /K)l′0
EFr, /B4/BJ/B5
l1
EFr=x1
Fr−x1
Er= (βr/K)(x′0
Fr−x′0
Er) +K(x′1
Fr−x′1
Er) = (βr/K)l′0
EFr+Kl′1
EFr, /B4/BK/B5
K= (1 + 2 βr)1/2. /BY /D9/D6/D8/CW/CT/D6/B8 /CX/D2 /D8/CW/CT /AH/D6/AH /CQ/CP/D7/CT/B8 /D3/D2/CT /CP/CV/CP/CX/D2 /CU/D3/D6/CV/CT/D8/D7 /D8/CW/CT /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D8/CT/D1/D4 /D3/D6/CP/D0/D4/CP/D6/D8l0
EFr
/B4/BJ/B5 /D3/CUlµ
EFr
/CP/D2/CS /CP/D7/D7/D9/D1/CT/D7 /D8/CW/CP/D8 /CX/D2 /D8/CW/CT /D6/CT/D0/CP/D8/CX/D3/D2 /CU/D3/D6l1
EFr
/B4/BK/B5x′1
Fr
/CP/D2/CSx′1
Er
/CP/D6/CT /D7/CX/D1 /D9/D0/D8/CP/D2/CT/D3/D9/D7/D0/DD/CS/CT/D8/CT/D6/D1/CX/D2/CT/CS /CP/D8 /D7/D3/D1/CT x′0
Fr=x′0
Er=b /CX/D2S′. /C0/D3 /DB /CT/DA /CT/D6/B8 /CX/D2 /D8/CW/CT /D7/CP/D1/CT /DB /CP /DD /CP/D7 /CX/D2 /D8/CW/CT /AH/CT/AH /CQ/CP/D7/CT/B8 /CX/D2 /BG/BW /B4/CP/D8/D9/D7 /BE/BW/B5 /D7/D4/CP
/CT/D8/CX/D1/CT /D7/D9
/CW /CP/D2 /CP/D7/D7/D9/D1/D4/D8/CX/D3/D2 /D1/CT/CP/D2/D7 /D8/CW/CP/D8 /CX/D2S′/D3/D2/CT /CP
/D8/D9/CP/D0/D0/DD /CS/D3 /CT/D7 /D2/D3/D8
/D3/D2/D7/CX/CS/CT/D6 /D8/CW/CT /D7/CP/D1/CT/CT/DA /CT/D2 /D8/D7 E /CP/D2/CSF /CP/D7 /CX/D2S /CQ/D9/D8 /D7/D3/D1/CT /D3/D8/CW/CT/D6 /D8 /DB /D3 /CT/DA /CT/D2 /D8/D7 G /CP/D2/CSH, /CP/D2/CS /D8/CW/CP/D8 /D8/CW/CT /CT/D5/D9/CP/D0/CX/D8 /DD x′0
Fr=x′0
Er=b/CW/CP/D7 /D8/D3 /CQ /CT /D6/CT/D4/D0/CP
/CT/CS /CQ /DDx′0
Hr=x′0
Gr=b. /CC/CW/CT /CT/DA /CT/D2 /D8/D7 G /CP/D2/CSH /CP/D6/CT /D8/CW/CT /CX/D2 /D8/CT/D6/D7/CT
/D8/CX/D3/D2/D7 /D3/CU /D8/CW/CT /D0/CX/D2/CT /B4/D8/CW/CT/CW /DD/D4 /CT/D6/D7/D9/D6/CU/CP
/CT x′0
Hr=x′0
Gr=b /DB/CX/D8/CW /CP/D6/CQ/CX/D8/D6/CP/D6/DD b /B5 /D4/CP/D6/CP/D0/D0/CT/D0 /D8/D3 /D8/CW/CT /D7/D4/CP/D8/CX/CP/D0 /CP/DC/CX/D7x′1
r
/B4/DB/CW/CX
/CW /CX/D7 /CP/D0/D3/D2/CV /D8/CW/CT/D7/D4/CP/D8/CX/CP/D0 /CQ/CP/D7/CT /DA /CT
/D8/D3/D6 r′
1
/B5 /CP/D2/CS /D3/CU /D8/CW/CT /CP/CQ /D3 /DA /CT /D1/CT/D2 /D8/CX/D3/D2/CT/CS /DB /D3/D6/D0/CS /D0/CX/D2/CT/D7 /D3/CU /D8/CW/CT /D7/D4/CP/D8/CX/CP/D0 /CT/D2/CS /D4 /D3/CX/D2 /D8/D7 /D3/CU /D8/CW/CT /D6/D3 /CS/BA/CC/CW/CT/D2 /CX/D2 /D8/CW/CT /CP/CQ /D3 /DA /CT /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2 /CU/D3/D6l1
EFr
/B4/BK/B5l′0
GHr= 0 /B8 /CP/D2/CSl′1
EFr=x′1
Fr−x′1
Er
/CX/D7 /D6/CT/D4/D0/CP
/CT/CS /CQ /DD
l′1
GHr=x′1
Hr−x′1
Gr. /C6/D3 /DB/B8 /CX/D2 /D8/CW/CT /AH/D6/AH /CQ /CP/D7/CT /B8 /D8/CW/CT /D7/D4 /CP/D8/CX/CP/D0 /CS/CX/D7/D8/CP/D2
/CT l1
EFr=x1
Fr−x1
Er
/CS/CT/AS/D2/CT/D7 /CX/D2 /D8/CW/CT /AH/BT /CC/D6 /CT/D0/CP/D8/CX/DA/CX/D8/DD/AH /D8/CW/CT /D7/D4 /CP/D8/CX/CP/D0 /D0/CT/D2/CV/D8/CW /D3/CU /D8/CW/CT /D6 /D3 /CS /CP/D8 /D6 /CT/D7/D8 /CX/D2S, /DB/CW/CX/D0/CT l′1
GHr=x′1
Hr−x′1
Gr
/CS/CT/AS/D2/CT/D7 /D8/CW/CT /D7/D4 /CP/D8/CX/CP/D0/D0/CT/D2/CV/D8/CW /D3/CU /D8/CW/CT /D1/D3/DA/CX/D2/CV /D6 /D3 /CS /CX/D2S′. /CC/CW/CT/D2/B8 /CU/D6/D3/D1 /D8/CW/CT /CT/D5/D9/CP/D8/CX/D3/D2 /CU/D3/D6l1
EFr
/B4/BK/B5/B8 /CP/D2/CS /DB/CX/D8/CW /D8/CW/CT/D7/CT /CS/CT/AS/D2/CX/D8/CX/D3/D2/D7/B8/DB /CT /AS/D2/CS /D8/CW/CT /D6/CT/D0/CP/D8/CX/D3/D2 /CQ /CT/D8 /DB /CT/CT/D2 l′1
r=l′1
GHr
/CP/D2/CSl1
r=l1
EFr=l0
/CP/D7 /D8/CW/CT /C4/D3/D6/CT/D2 /D8/DE /AH
/D3/D2 /D8/D6/CP
/D8/CX/D3/D2/AH /D3/CU /D8/CW/CT/D1/D3 /DA/CX/D2/CV /D6/D3 /CS /CX/D2 /D8/CW/CT /AH/D6/AH /CQ/CP/D7/CT/B8
l′1
r=x′1
Hr−x′1
Gr=l0/K= (1/K)(x1
Fr−x1
Er), /B4/BL/B5/DB/CX/D8/CWx′0
Hr=x′0
Gr
/CP/D2/CSx0
Fr=x0
Er. /C1/D2
/D3/D2 /D8/D6/CP/D7/D8 /D8/D3 /D8/CW/CT /AH/CT/AH
/D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2 /DB /CT /AS/D2/CS /D8/CW/CP/D8 /CX/D2 /D8/CW/CT /AH/D6/AH /CQ/CP/D7/CT/D8/CW/CT/D6/CT /CX/D7 /CP /D0/CT/D2/CV/D8/CW /CS/CX/D0/CP/D8/CP/D8/CX/D3/D2 ∞ ≻l′1
r≻l0
/CU/D3/D6−1/2≺βr≺0 /CP/D2/CS /D8/CW/CT /D7/D8/CP/D2/CS/CP/D6/CS /AH/D0/CT/D2/CV/D8/CW
/D3/D2 /D8/D6/CP
/D8/CX/D3/D2/AH
l0≻l′1
r≻0 /CU/D3/D6 /D4 /D3/D7/CX/D8/CX/DA /CT βr, /DB/CW/CX
/CW
/D0/CT/CP/D6/D0/DD /D7/CW/D3 /DB/D7 /D8/CW/CP/D8 /D8/CW/CT /AH/C4/D3/D6/CT/D2 /D8/DE
/D3/D2 /D8/D6/CP
/D8/CX/D3/D2/AH /CX/D7 /D2/D3/D8 /D4/CW /DD/D7/CX
/CP/D0/D0/DD
/D3/D6/D6/CT
/D8/D0/DD /CS/CT/AS/D2/CT/CS /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2/BA/CF /CT /D7/CT/CT /CU/D6/D3/D1 /D8/CW/CT /D4/D6/CT
/CT/CS/CX/D2/CV /CS/CX/D7
/D9/D7/D7/CX/D3/D2 /D8/CW/CP/D8 /B9 /D8/CW/CT /C4 /D3/D6 /CT/D2/D8/DE
/D3/D2/D8/D6 /CP
/D8/CX/D3/D2 /CX/D7 /D8/CW/CT /D8/D6 /CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2
/D3/D2/B9/D2/CT
/D8/CX/D2/CV /CS/CX/AR/CT/D6 /CT/D2/D8 /D5/D9/CP/D2/D8/CX/D8/CX/CT/D7 /B4/CX/D2 /BG/BW /D7/D4 /CP
/CT/D8/CX/D1/CT/B5 /CX/D2 /CS/CX/AR/CT/D6 /CT/D2/D8 /C1/BY/CA/D7 /CP/D2/CS /CS/CX/AR/CT/D6 /CT/D2/D8
/D3 /D3/D6 /CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/D7/B8/CP/D2/CS /CP/D0/D7/D3 /CX/D8
/CW/CP/D2/CV/CT/D7 /D8/CW/CT /CX/D2/AS/D2/CX/D8/CT/D7/CX/D1/CP/D0 /D7/D4 /CP
/CT/D8/CX/D1/CT /CS/CX/D7/D8/CP/D2
/CT ds /CP/D2/CS
/D3/D2/D7/CT /D5/D9/CT/D2/D8/D0/DD /D8/CW/CT /D4/D7/CT/D9/CS/D3/B9/BX/D9
/D0/CX/CS/CT /CP/D2/CV/CT /D3/D1/CT/D8/D6/DD /D3/CU /D8/CW/CT /BG/BW /D7/D4 /CP
/CT/D8/CX/D1/CT/BA /CB/D9
/CW
/CW/CP/D6 /CP
/D8/CT/D6/CX/D7/D8/CX
/D7 /D3/CU /D8/CW/CT /C4 /D3/D6 /CT/D2/D8/DE
/D3/D2/D8/D6 /CP
/D8/CX/D3/D2 /CP/D7 /D8/CW/CT
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/D3/D9/D2/D8 /D8/CW/CT/D2 /D8/CW/CT/D6 /CT /CX/D7 /D2/D3 /D4/D0/CP
/CT /CU/D3/D6 /D8/CW/CT /C4 /D3/D6 /CT/D2/D8/DE
/D3/D2/D8/D6 /CP
/D8/CX/D3/D2 /CU/D3/D6/D1/D9/D0/CP/CT/B8 /CP/D2/CS /D3/D2/D0/DD/D8/CW/CT /D7/D4 /CP
/CT/D8/CX/D1/CT /D0/CT/D2/CV/D8/CW /CP/D2/CS /D8/CW/CT /D7/D4 /CP
/CT/D8/CX/D1/CT /D5/D9/CP/D2/D8/CX/D8/CX/CT/D7 /CP/D6 /CT /DB/CT/D0 /D0 /CS/CT/AS/D2/CT /CS /D5/D9/CP/D2/D8/CX/D8/CX/CT/D7/BA/BE/BA/BF/BA /AH/BV/CP/D6 /CP/D2/CS /CV/CP/D6/CP/CV/CT /D4/CP/D6/CP/CS/D3 /DC/AH/C1/D2 /D8/CW/CT /D4/D6/CT/DA/CX/D3/D9/D7 /D7/CT
/D8/CX/D3/D2/D7 /DB /CT /CW/CP /DA /CT /CT/DC/CP/D1/CX/D2/CT/CS /D8/CW/CT /D1/CP/CX/D2
/CW/CP/D6/CP
/D8/CT/D6/CX/D7/D8/CX
/D7 /D3/CU /CQ /D3/D8/CW /CU/D3/D6/D1/D7 /D3/CU /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD /BA/C6/D3 /DB /DB /CT /DB /CP/D2 /D8 /D8/D3 /D7/CW/D3 /DB /D8/CW/CT /CS/CX/AR/CT/D6/CT/D2
/CT /CQ /CT/D8 /DB /CT/CT/D2 /D8/CW/CT /D8/D6/CT/CP/D8/D1/CT/D2 /D8/D7 /D3/CU /D6/CT/D0/CP/D8/CX/DA /CT/D0/DD /D1/D3 /DA/CX/D2/CV /D7/DD/D7/D8/CT/D1/D7 /CX/D2 /D8/CW/CT/AH/BT /CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH /CP/D2/CS /D8/CW/CT /AH/CC/CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD /BA/AH /CD/D7/D9/CP/D0/D0/DD /D7/D9
/CW /D7/DD/D7/D8/CT/D1/D7 /CP/D6/CT /D8/D6/CT/CP/D8/CT/CS /CX/D2 /CP /D2/D3/D2
/D3 /DA /CP/D6/CX/CP/D2 /D8 /D1/CP/D2/D2/CT/D6/B8/CX/BA/CT/BA/B8 /CX/D2 /D8/CW/CT /AH/BT /CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD /B8/AH /CQ/D9/D8 /CW/CT/D6/CT /DB /CT /D7/CW/CP/D0/D0 /D4/D6/CT/D7/CT/D2 /D8 /D8/CW/CT /D8/D6/CT/CP/D8/D1/CT/D2 /D8 /D3/CU /D7/D9
/CW /D7/DD/D7/D8/CT/D1/D7 /CX/D2 /CP /D1/CP/D2/CX/CU/CT/D7/D8/D0/DD
/D3 /DA /CP/D6/CX/CP/D2 /D8 /D1/CP/D2/D2/CT/D6/B8 /CX/BA/CT/BA/B8 /CX/D2 /D8/CW/CT /AH/CC/CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD /BA/AH /C1/D2 /D3/D6/CS/CT/D6 /D8/D3 /D7/CT/CT /D8/CW/CT /CS/CX/AR/CT/D6/CT/D2
/CT/D7 /CQ /CT/D8 /DB /CT/CT/D2 /CQ /D3/D8/CW /D8/D6/CT/CP/D8/B9/D1/CT/D2 /D8/D7 /DB /CT /CS/D3 /D2/D3/D8 /D2/CT/CT/CS /D8/D3 /DB /D3/D6/CZ
/D3/D1/D4/D0/CT/D8/CT/D0/DD /CX/D2 /CV/CT/D3/D1/CT/D8/D6/CX
/CP/D0 /D8/CT/D6/D1/D7/B8 /CQ/D9/D8 /DB /CT
/CP/D2
/CW/D3 /D3/D7/CT /D7/D3/D1/CT /D7/D4 /CT
/CX/AS
/D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/B8 /CT/BA/CV/BA/B8 /D8/CW/CT /D7/CX/D1/D4/D0/CT/D7/D8 /D3/D2/CT/B8 /D8/CW/CT /AH/CT/AH
/D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/BA /CC/CW/CT/D6/CT/CU/D3/D6/CT/B8 /CX/D2 /D8/CW/CT /CU/D3/D0/D0/D3 /DB/CX/D2/CV/B8 /DB /CT/D6/CT/D7/D8/D6/CX
/D8 /D3/D9/D6/D7/CT/D0/DA /CT/D7 /D8/D3 /D8/CW/CT /AH/CT/AH
/D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2 /CP/D2/CS/B8 /CU/D3/D6 /D7/CX/D1/D4/D0/CX
/CX/D8 /DD /CX/D2 /D2/D3/D8/CP/D8/CX/D3/D2/B8 /DB /CT /D3/D1/CX/D8 /D8/CW/CT /D7/D9/CQ/D7
/D6/CX/D4/D8/B9/CT/B9 /CX/D2 /CP/D0/D0 /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7/BA /C0/D3 /DB /CT/DA /CT/D6/B8 /DB /CT /D7/CW/CP/D0/D0 /D3/CU/D8/CT/D2 /DB/D6/CX/D8/CT /D8/CW/CT /CX/D1/D4 /D3/D6/D8/CP/D2 /D8 /D6/CT/D0/CP/D8/CX/D3/D2/D7 /CX/D2 /CV/CT/D3/D1/CT/D8/D6/CX
/CP/D0 /D8/CT/D6/D1/D7/B8 /CP/D2/CS/CP/D0/D7/D3 /DB /CT /D7/CW/CP/D0/D0 /CT/DC/D4/D0/CP/CX/D2 /DB/CW/CX
/CW /D6/CT/D7/D9/D0/D8/D7 /CP/D2/CS
/D3/D2
/D0/D9/D7/CX/D3/D2/D7 /CP/D6/CT /CX/D2/CS/CT/D4 /CT/D2/CS/CT/D2 /D8 /D3/CU /D8/CW/CT
/CW/D3/D7/CT/D2
/D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/BA/BL/BT/D7 /CP/D0/D6/CT/CP/CS/DD /CS/CX/D7
/D9/D7/D7/CT/CS/B8 /CX/CU /CX/D2 /CP/D2 /C1/BY/CAS′/CX/D2 /DB/CW/CX
/CW /D8/CW/CT /D8/CX/D1/CT
/D3/D1/D4 /D3/D2/CT/D2 /D8 /D3/CU /D8/CW/CT /CS/CX/D7/D8/CP/D2
/CT /BG/B9/DA /CT
/D8/D3/D6
l′µ
AB
/B8 /CX/BA/CT/BA/B8l′0
AB, /CX/D7 /DE/CT/D6/D3 /B4/D7/CX/D1 /D9/D0/D8/CP/D2/CT/D3/D9/D7/D0/DD /CS/CT/D8/CT/D6/D1/CX/D2/CT/CS /CT/DA /CT/D2 /D8/D7 A /CP/D2/CSB /B5/B8 /D8/CW/CT/D2l′µ
AB
/D3/D1/D4/D6/CX/D7/CT/D7 /D3/D2/D0/DD /D7/D4/CP/D8/CX/CP/D0
/D3/D1/D4 /D3/D2/CT/D2 /D8/D7/BA /C0/CT/D2
/CT/B8 /CX/D2S′/D8/CW/CT /CX/D2 /DA /CP/D6/CX/CP/D2 /D8 /D7/D4/CP
/CT/D8/CX/D1/CT /D0/CT/D2/CV/D8/CW l′/CX/D7 /CV/CX/DA /CT/D2 /CP/D7 /D8/CW/CT /D9/D7/D9/CP/D0 /BF/BW /CS/CX/D7/D8/CP/D2
/CT/CQ /CT/D8 /DB /CT/CT/D2 A /CP/D2/CSB /BA /BU/D9/D8/B8 /CX/D2 /D7/D9
/CW /CP
/CP/D7/CT/B8lµ
AB
/CX/D2 /D8/CW/CT /D6/CT/D7/D8 /CU/D6/CP/D1/CT S /D3/CU /D8/CW/CT /D3/CQ /CY/CT
/D8 /CS/D3 /CT/D7 /CW/CP /DA /CTl0
AB∝ne}ationslash= 0 /B8/CP/D2/CS /D8/CW/CT /D7/D4/CP
/CT/D8/CX/D1/CT /D0/CT/D2/CV/D8/CW l /CX/D2S /B4/CX/D8 /CX/D7=l′/B5
/D3/D9/D0/CS /D8/CP/CZ /CT /CS/CX/AR/CT/D6/CT/D2 /D8 /DA /CP/D0/D9/CT/D7 /CS/CT/D4 /CT/D2/CS/CX/D2/CV /D3/D2 /D8/CW/CT
/CW/D3/D7/CT/D2 /C1/BY/CA
S′/BA /CB/D9
/CW /CP/D2 /CP/D6/CQ/CX/D8/D6/CP/D6/CX/D2/CT/D7/D7 /CX/D2l /B8 /CP/D0/D8/CW/D3/D9/CV/CW /D2/D3/D8 /CU/D3/D6/CQ/CX/CS/CS/CT/D2 /CQ /DD /CP/D2 /DD /D4/CW /DD/D7/CX
/CP/D0 /D0/CP /DB/B8 /DB /D3/D9/D0/CS
/D3/D1/D4/D0/CX
/CP/D8/CT /CQ /D3/D8/CW/D8/CW/CT/D3/D6/DD /CP/D2/CS /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/BA /BY /D9/D6/CW/CT/D6/D1/D3/D6/CT/B8 /D3/D2/CT /DB /CP/D2 /D8/D7 /D8/D3 /D6/CT/D8/CP/CX/D2 /D8/CW/CT
/D3/D2/D2/CT
/D8/CX/D3/D2 /DB/CX/D8/CW /D8/CW/CT /D4/D6/CT/D6/CT/D0/CP/D8/CX/DA/CX/D7/D8/CX
/D3/D2
/CT/D4/D8 /D3/CU /D8/CW/CT /D7/D4/CP/D8/CX/CP/D0 /D0/CT/D2/CV/D8/CW/BA /CC/CW/CT/D6/CT/CU/D3/D6/CT/B8 /D8/CW/CT /D1/D3/D7/D8
/D3/D2 /DA /CT/D2/CX/CT/D2 /D8
/CW/D3/CX
/CT /CU/D3/D6 /D8/CW/CT /CU/D6/CP/D1/CT /CX/D2 /DB/CW/CX
/CW /D8/CW/CT /D8/CX/D1/CT
/D3/D1/D4 /D3/D2/CT/D2 /D8 l′0
AB
/D3/CUl′µ
AB
/CX/D7 /DE/CT/D6/D3 /CX/D7 /D8/CW/CTS /CU/D6/CP/D1/CT/BA /CC/CW /D9/D7 /CX/D2S /D8/CW/CT /D4 /D3/D7/CX/D8/CX/D3/D2 /BG/B9/DA /CT
/D8/D3/D6/D7 xµ
A,B
/CP/D6/CT /CS/CT/D8/CT/D6/D1/CX/D2/CT/CS/D7/CX/D1 /D9/D0/D8/CP/D2/CT/D3/D9/D7/D0/DD /B8 x0
B−x0
A=l0
AB= 0, /CP/D2/CS /D8/CW/CT /D7/D4/CP
/CT/D8/CX/D1/CT /D0/CT/D2/CV/D8/CW l /CQ /CT
/D3/D1/CT/D7 /CX/D2S, /D8/CW/CT /D6/CT/D7/D8 /CU/D6/CP/D1/CT /D3/CU/D8/CW/CT /D3/CQ /CY/CT
/D8/B8 /D8/CW/CT /D6/CT/D7/D8 /D7/D4/CP/D8/CX/CP/D0 /D0/CT/D2/CV/D8/CW l0, /CX/BA/CT/BA/B8l= (li
ABlABi)1/2=l0
/B8 /CP/D7 /CX/D2 /D8/CW/CT /D4/D6/CT/D6/CT/D0/CP/D8/CX/DA/CX/D7/D8/CX
/D4/CW /DD/D7/CX
/D7/BA/CC/CW/CT /D3/CQ/D7/CT/D6/DA /CT/D6/D7 /CX/D2 /CP/D0/D0 /D3/D8/CW/CT/D6 /C1/BY/CA/D7 /DB/CX/D0/D0 /D0/D3 /D3/CZ /CP/D8 /D8/CW/CT /D7/CP/D1/CT /CT/DA /CT/D2 /D8/D7 /CQ/D9/D8 /CP/D7/D7/D3
/CX/CP/D8/CX/D2/CV /DB/CX/D8/CW /D8/CW/CT/D1 /CS/CX/AR/CT/D6/CT/D2 /D8
/D3 /D3/D6/CS/CX/D2/CP/D8/CT/D7/BN /D8/CW/CT/DD /CP/D0/D0 /AS/D2/CS /B4/D1/CT/CP/D7/D9/D6/CT/B5 /D8/CW/CT /D7/CP/D1/CT /DA /CP/D0/D9/CTl=l0
/CU/D3/D6 /D8/CW/CT /D7/D4/CP
/CT/D8/CX/D1/CT /D0/CT/D2/CV/D8/CW/BA /C6/D3/D8/CT/B8 /CP/D7 /DB /CT /CW/CP /DA /CT/D1/CT/D2 /D8/CX/D3/D2/CT/CS/B8 /D8/CW/CP/D8 /CX/CUl0
AB= 0 /D8/CW/CT/D2l′µ
AB
/CX/D2 /CP/D2 /DD /D3/D8/CW/CT/D6 /C1/BY/CAS′/DB/CX/D0/D0
/D3/D2 /D8/CP/CX/D2 /D8/CW/CT /D8/CX/D1/CT
/D3/D1/D4 /D3/D2/CT/D2 /D8 l′0
AB∝ne}ationslash= 0 /BA/CF /CT /D3/D2
/CT /CP/CV/CP/CX/D2 /CT/D1/D4/CW/CP/D7/CX/DE/CT /D8/CW/CP/D8 /D8/CW/CT
/CW/D3/CX
/CT /D3/CU /D8/CW/CT /D6/CT/D7/D8 /CU/D6/CP/D1/CT /D3/CU /D8/CW/CT /D3/CQ /CY/CT
/D8 /CP/D7 /D8/CW/CT /D7/D8/CP/D6/D8/CX/D2/CV /CU/D6/CP/D1/CT /CU/D3/D6/D8/CW/CT
/D3/D2/D7/CX/CS/CT/D6/CP/D8/CX/D3/D2 /CX/D7 /D2/D3/D8 /CS/CX
/D8/CP/D8/CT/CS /CQ /DD /D4/CW /DD/D7/CX
/CP/D0 /D6/CT/D5/D9/CX/D6/CT/D1/CT/D2 /D8/D7/BA /CC/CW/CX/D7
/CW/D3/CX
/CT /CX/D7/B8 /CX/D2 /CU/CP
/D8/B8 /CS/CT/D8/CT/D6/D1/CX/D2/CT/CS /D3/D2/D0/DD/CQ /DD /D3/D9/D6 /CS/CT/D7/CX/D6/CT /D8/D3 /CW/CP /DA /CT /CP /D5/D9/CP/D2 /D8/CX/D8 /DD /D8/CW/CP/D8
/D3/D6/D6/CT/D7/D4 /D3/D2/CS/D7 /CX/D2 /D8/CW/CT /BG/BW /D7/D4/CP
/CT/D8/CX/D1/CT /D8/D3 /D8/CW/CT /D4/D6/CT/D6/CT/D0/CP/D8/CX/DA/CX/D7/D8/CX
/D7/D4/CP/D8/CX/CP/D0/D0/CT/D2/CV/D8/CW/BA /BT/D7 /CP/D0/D6/CT/CP/CS/DD /D7/CP/CX/CS /CP/D8 /D8/CW/CT /CT/D2/CS /D3/CU /CB/CT
/BA/BE/BA/BD /D8/CW/CT /D9/D7/D9/CP/D0 /BF/BW /D0/CT/D2/CV/D8/CW /D3/CU /CP /D1/D3 /DA/CX/D2/CV /D3/CQ /CY/CT
/D8
/CP/D2/D2/D3/D8 /CQ /CT/CS/CT/AS/D2/CT/CS /CX/D2 /D8/CW/CT /BG/BW /D7/D4/CP
/CT/D8/CX/D1/CT/B8 /CX/BA/CT/BA/B8 /CX/D2 /D8/CW/CT /AH/CC/CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH/B8 /CX/D2 /CP/D2 /CP/CS/CT/D5/D9/CP/D8/CT /DB /CP /DD /BA /C7/D2/D0/DD /D8/CW/CT /D7/D4/CP
/CT/D8/CX/D1/CT/D0/CT/D2/CV/D8/CW l /CS/D3 /CT/D7 /CW/CP /DA /CT /CP /CS/CT/AS/D2/CX/D8/CT /D8/CW/CT/D3/D6/CT/D8/CX
/CP/D0 /CP/D2/CS /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/CP/D0 /D1/CT/CP/D2/CX/D2/CV /CP/D2/CS /CX/D8 /CX/D7 /CP/D2 /CX/D2 /DA /CP/D6/CX/CP/D2 /D8 /D5/D9/CP/D2 /D8/CX/D8 /DD /BA/CC/CW/CX/D7 /CW/D3/D0/CS/D7 /CU/D3/D6 /CP/D0/D0 /D4 /D3/D7/D7/CX/CQ/D0/CT /D7/DD/D2
/CW/D6/D3/D2/CX/DE/CP/D8/CX/D3/D2/D7/BA /CF/CX/D8/CW /D3/D9/D6
/CW/D3/CX
/CT /D3/CU /D8/CW/CT /D6/CT/D7/D8 /CU/D6/CP/D1/CT /D3/CU /D8/CW/CT /D3/CQ /CY/CT
/D8 /CP/D7 /D8/CW/CT/D7/D8/CP/D6/D8/CX/D2/CV /CU/D6/CP/D1/CT/B8 /CX/BA/CT/BA/B8 /DB/CX/D8/CWl=l0, /D8/CW/CT /D7/D4/CP/D8/CX/CP/D0 /D6/CT/D7/D8 /D0/CT/D2/CV/D8/CW /CS/CT/D8/CT/D6/D1/CX/D2/CT/CS /D7/CX/D1 /D9/D0/D8/CP/D2/CT/D3/D9/D7/D0/DD /CX/D2S /D3/CQ/D8/CP/CX/D2/D7 /D8/CW/CT/D4/D6/D3/D4 /CT/D6/D8/CX/CT/D7 /D3/CU /D8/CW/CT /D7/D4/CP
/CT/D8/CX/D1/CT /D0/CT/D2/CV/D8/CW l. /CC/CW/CT/D2/B8 /D8/CW/CT
/D3 /D3/D6/CS/CX/D2/CP/D8/CT /D1/CT/CP/D7/D9/D6/CT/D1/CT/D2 /D8/D7 /D3/CUx′µ
A,B
/CX/D2 /CP/D2 /C1/BY/CAS′/CX/D2/DB/CW/CX
/CW /CP/D2 /D3/CQ /CY/CT
/D8 /CX/D7 /D1/D3 /DA/CX/D2/CV /CP/D6/CT /D2/D3/D8 /D3/CU /CX/D2 /D8/CT/D6/CT/D7/D8 /CX/D2 /D8/CW/CT/CX/D6 /D3 /DB/D2 /D6/CX/CV/CW /D8 /CQ/D9/D8 /D8/CW/CT/DD /CW/CP /DA /CT /D8/D3 /CT/D2/CP/CQ/D0/CT /D3/D2/CT /D8/D3/AS/D2/CS /D8/CW/CT /D6/CT/D7/D8 /D7/D4/CP/D8/CX/CP/D0 /D0/CT/D2/CV/D8/CW l0
/BA /C1/D2 /D8/CW/CT /D4/D6/CT/D6/CT/D0/CP/D8/CX/DA/CX/D7/D8/CX
/AH/BF/B7/BD/AH /D4/CX
/D8/D9/D6/CT/B8 /CP/D2/CS /CX/D2 /D8/CW/CT /AH/BT /CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD /B8/AH /D3/D2/CT
/CP/D2
/D3/D1/D4/CP/D6/CT /D8/CW/CT /D7/D4 /CP/D8/CX/CP/D0 /D0/CT/D2/CV/D8/CW/D7 /D3/CU /D8 /DB /D3 /D6/CT/D0/CP/D8/CX/DA /CT/D0/DD /D1/D3 /DA/CX/D2/CV /D3/CQ /CY/CT
/D8/D7/BA /BU/D9/D8 /CX/D2 /D8/CW/CT /AH/CC/CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH /D8/CW/CT/D7/D4 /CP
/CT/D8/CX/D1/CT /D0/CT/D2/CV/D8/CW/D7 /B4/D8/CW/CP/D8
/D3/D2 /D8/CP/CX/D2 /CQ /D3/D8/CW /D7/D4/CP/D8/CX/CP/D0 /CP/D2/CS /D8/CT/D1/D4 /D3/D6/CP/D0 /D4/CP/D6/D8/D7/B5 /CP/D6/CT /DB /CT/D0/D0 /CS/CT/AS/D2/CT/CS /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7 /CX/D2 /BG/BW/D7/D4/CP
/CT/D8/CX/D1/CT /CP/D2/CS /D8/CW/CT/DD /B8 /D3/D6 /D8/CW/CT /D6/CT/D7/D8 /D7/D4/CP/D8/CX/CP/D0 /D0/CT/D2/CV/D8/CW/D7/B8
/CP/D2 /CQ /CT
/D3/D1/D4/CP/D6/CT/CS /CX/D2 /CP /D4/CW /DD/D7/CX
/CP/D0/D0/DD /D1/CT/CP/D2/CX/D2/CV/CU/D9/D0 /DB /CP /DD /BA/C4/CT/D8 /D9/D7 /CX/D0/D0/D9/D7/D8/D6/CP/D8/CT /D8/CW/CT /D4/D6/CT
/CT/CS/CX/D2/CV /CS/CX/D7
/D9/D7/D7/CX/D3/D2
/D3/D2/D7/CX/CS/CT/D6/CX/D2/CV /D8/CW/CT /DB /CT/D0/D0/B9/CZ/D2/D3 /DB/D2 /AH/BV/CP/D6 /CP/D2/CS /CV/CP/D6/CP/CV/CT /D4/CP/D6/CP/CS/D3 /DC/AH/B4/D7/CT/CT/B8 /CT/BA/CV/BA/B8 /CJ/BJ ℄/B8 /D4/BA/BL/B5/BA /CC/CW/CT
/D3/D1/D1/D3/D2 /CP/D7/D7/CT/D6/D8/CX/D3/D2 /CP/CQ /D3/D9/D8 /D8/CW/CX/D7 /AH/D4/CP/D6/CP/CS/D3 /DC/AH /CX/D7 /D8/CW/CP/D8 /CX/D8
/D3/D1/CT/D7 /D3/D9/D8 /CS/D9/CT /D8/D3/B8 /CJ/BJ ℄/BM/AH/CC/CW/CT /D0/CP
/CZ /D3/CU /CP /D2/D3/D8/CX/D3/D2 /D3/CU /CP/CQ/D7/D3/D0/D9/D8/CT /D7/CX/D1 /D9/D0/D8/CP/D2/CT/CX/D8 /DD /CX/D2 /D7/D4 /CT
/CX/CP/D0 /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD /BA/BA/BA/BA/B8/AH /CP/D2/CS
/D3/D2/D7/CT/D5/D9/CT/D2 /D8/D0/DD /CS/D9/CT /D8/D3/D8/CW/CT /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD /D3/CU /D8/CW/CT /C4/D3/D6/CT/D2 /D8/DE
/D3/D2 /D8/D6/CP
/D8/CX/D3/D2/BA /C0/D3 /DB /CT/DA /CT/D6/B8 /CP/D7 /CS/CX/D7
/D9/D7/D7/CT/CS /CP/CQ /D3 /DA /CT/B8 /D8/CW/CT /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD /D3/CU /D7/CX/D1 /D9/D0/D8/CP/D2/CT/B9/CX/D8 /DD /CX/D7 /CP
/D3 /D3/D6/CS/CX/D2/CP/D8/CT /CS/CT/D4 /CT/D2/CS/CT/D2 /D8 /CT/AR/CT
/D8 /CP/D2/CS/B8 /CU/D3/D6 /CT/DC/CP/D1/D4/D0/CT/B8 /CU/D3/D6ε= 0 /B8 /CJ/BK ℄/B8 /D8/CW/CT /CP/CQ/D7/D3/D0/D9/D8/CT /D7/CX/D1 /D9/D0/D8/CP/D2/CT/CX/D8 /DD /CX/D7/D4/D6/CT/D7/CT/D6/DA /CT/CS/BA /BT/D0/D7/D3/B8 /D8/CW/CT /C4/D3/D6/CT/D2 /D8/DE
/D3/D2 /D8/D6/CP
/D8/CX/D3/D2 /CX/D7 /CP/D2 /BT /CC /CP/D2/CS /CX/D8 /CW/CP/D7 /D2/D3/D8/CW/CX/D2/CV /D8/D3 /CS/D3 /DB/CX/D8/CW /D8/CW/CT /BG/BW /D4/D7/CT/D9/CS/D3/B9/BX/D9
/D0/CX/CS/CT/CP/D2 /CV/CT/D3/D1/CT/D8/D6/DD /D3/CU /D8/CW/CT /D7/D4 /CT
/CX/CP/D0 /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD /BA /CC/CW/CT/D6/CT/CU/D3/D6/CT/B8 /DB /CT /CS/CX/D7
/D9/D7/D7 /D8/CW/CX/D7 /AH/D4/CP/D6/CP/CS/D3 /DC/AH /D9/D7/CX/D2/CV
/D3 /DA /CP/D6/CX/CP/D2 /D8/BG/BW /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7/B8 /CX/BA/CT/BA/B8 /CX/D2 /D8/CW/CT /AH/CC/CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH/BA /BU/D9/D8 /CU/D3/D6 /D3/D9/D6 /D4/D9/D6/D4 /D3/D7/CT/D7/B8 /CP/D7 /CX/D8 /CX/D7 /CP/D0/D6/CT/CP/CS/DD /CT/DC/D4/D0/CP/CX/D2/CT/CS/B8 /D8/CW/CT/D6/CT/CX/D7 /D2/D3 /D2/CT/CT/CS /D8/D3 /CS/CX/D7
/D9/D7/D7 /D8/CW/CT /AH/D4/CP/D6/CP/CS/D3 /DC/AH /CX/D2 /CV/CT/D3/D1/CT/D8/D6/CX
/CP/D0 /D8/CT/D6/D1/D7/B8 /D8/CW/CP/D2 /CX/D8
/CP/D2 /CQ /CT
/D3/D2/D7/CX/CS/CT/D6/CT/CS /CX/D2 /D8/CW/CT /CX/D2/CT/D6/D8/CX/CP/D0
/D3 /D3/D6/CS/CX/D2/CP/D8/CT /D7/DD/D7/D8/CT/D1/D7 /DB/CX/D8/CW /D8/CW/CT /C5/CX/D2/CZ /D3 /DB/D7/CZ/CX /D1/CT/D8/D6/CX
/D8/CT/D2/D7/D3/D6/D7/B8 /D8/CW/CP/D8 /CX/D7 /CX/D2 /D8/CW/CT /AH/CT/AH
/D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/BA /CC/CW/CT/CU/D6/CP/D1/CT /CX/D2 /DB/CW/CX
/CW /CP /CV/CP/D6/CP/CV/CT /CX/D7 /CP/D8 /D6/CT/D7/D8 /CX/D7 /CS/CT/D2/D3/D8/CT/CS /CQ /DDS /DB/CW/CX/D0/CT /D8/CW/CP/D8 /D3/D2/CT /CX/D2 /DB/CW/CX
/CW /CP
/CP/D6 /CX/D7 /CP/D8 /D6/CT/D7/D8 /CQ /DDS′/BA/CC/CW/CT /D9/D2/D4/D6/CX/D1/CT/CS /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7 /CP/D6/CT /CX/D2S /CP/D2/CS /D8/CW/CT /D4/D6/CX/D1/CT/CS /D3/D2/CT/D7 /CX/D2S′/BA /C1/D2/D7/D8/CT/CP/CS /D3/CU /BG/BW /D7/D4/CP
/CT/D8/CX/D1/CT /DB /CT /DB /D3/D6/CZ/CW/CT/D6/CT /DB/CX/D8/CW /BE/BW /D7/D4/CP
/CT/D8/CX/D1/CT/BA /CC/CW/CT /CV/CP/D6/CP/CV/CT /CP/D2/CS /D8/CW/CT
/CP/D6 /CP/D6/CT /CP/D0/D3/D2/CV /D8/CW/CT
/D3/D1/D1/D3/D2 x1, x′1− /CP/DC/CT/D7/B8 /CP/D2/CS /D8/CW/CT/DD/CP/D6/CT /D3/CU /CT/D5/D9/CP/D0 /D4/D6/D3/D4 /CT/D6 /D0/CT/D2/CV/D8/CW/D7 lg=l′
c=l /BA /BY/CX/D6/D7/D8 /DB /CT
/D3/D2/D7/CX/CS/CT/D6 /D8/CW/CT /AH/D4/CP/D6/CP/CS/D3 /DC/AH /CX/D2 /D8/CW/CTS /CU/D6/CP/D1/CT/BA /CC/CW/CT/CS/CX/D7/D8/CP/D2
/CT /BG/B9/DA /CT
/D8/D3/D6 lµ
g
/CU/D3/D6 /D8/CW/CT /CV/CP/D6/CP/CV/CT /CX/D7 /CS/CT/D8/CT/D6/D1/CX/D2/CT/CS /CS/CX/D6/CT
/D8/D0/DD /CX/D2S /BA /C0/D3 /DB /CT/DA /CT/D6 /D8/CW/CT /CS/CX/D7/D8/CP/D2
/CT /BG/B9/DA /CT
/D8/D3/D6 lµ
c/CU/D3/D6 /D8/CW/CT /D1/D3 /DA/CX/D2/CV
/CP/D6 /CX/D2S /CW/CP/D7 /D8/D3 /CQ /CT /CS/CT/D8/CT/D6/D1/CX/D2/CT/CS /CX/D2 /D7/D9
/CW /CP /DB /CP /DD /D8/CW/CP/D8 /D3/D2/CT /AS/D6/D7/D8 /AS/D2/CS/D7 /CX/D8 /CX/D2 /D8/CW/CT
/CP/D6 /D3 /DB/D2/D6/CT/D7/D8 /CU/D6/CP/D1/CT S′/B8 /CP/D2/CS /D8/CW/CT/D2 /D8/D6/CP/D2/D7/CU/D3/D6/D1/D7 /CX/D8 /CQ /DD /D8/CW/CT /C4 /CC /D8/D3 /D8/CW/CT /D6/CT/D7/D8 /CU/D6/CP/D1/CT /D3/CU /D8/CW/CT /CV/CP/D6/CP/CV/CT S /BA /CC/CW/CX/D7 /D4/D6/D3
/CT/CS/D9/D6/CT/CU/D3/D0/D0/D3 /DB/D7 /CU/D6/D3/D1 /D8/CW/CT /D4/D6/CT
/CT/CS/CX/D2/CV /CV/CT/D2/CT/D6/CP/D0 /CS/CX/D7
/D9/D7/D7/CX/D3/D2 /DB/CW/CT/D6/CT /CX/D8 /CX/D7 /D7/CW/D3 /DB/D2 /D8/CW/CP/D8 /D3/D2/D0/DD /D8/CW/CT /D6/CT/D7/D8 /D0/CT/D2/CV/D8/CW/B8 /CX/BA/CT/BA/B8 /D8/CW/CT/CX/D2 /DA /CP/D6/CX/CP/D2 /D8 /D7/D4/CP
/CT/D8/CX/D1/CT /D0/CT/D2/CV/D8/CW/B8 /D3/CU /CP /D1/D3 /DA/CX/D2/CV /D3/CQ /CY/CT
/D8 /CX/D7 /DB /CT/D0/D0 /CS/CT/AS/D2/CT/CS /D5/D9/CP/D2 /D8/CX/D8 /DD /CX/D2 /D8/CW/CT /AH/CC/CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH/BA /CC/CW/CT/D4 /D3/D7/CX/D8/CX/D3/D2 /BG/B9/DA /CT
/D8/D3/D6/D7 /D3/CU /D8/CW/CT /D7/D4/CP/D8/CX/CP/D0 /CT/D2/CS /D4 /D3/CX/D2 /D8/D7 A /CP/D2/CSB /D3/CU /D8/CW/CT /CV/CP/D6/CP/CV/CT /CX/D2S /CP/D6/CT /D8/CP/CZ /CT/D2 /D8/D3 /CQ /CTxµ
Ag= (0,0)/CP/D2/CSxµ
Bg= (0, l), /CP/D2/CS /D8/CW/CT /CS/CX/D7/D8/CP/D2
/CT /BG/B9/DA /CT
/D8/D3/D6 /CX/D7lµ
g=xµ
Bg−xµ
Ag= (0, l). /B4/C1/D2 /D8/CW/CT /D6/CT/D1/CP/CX/D2/CX/D2/CV /D4/CP/D6/D8 /D3/CU/D8/CW/CX/D7 /D4/CP/D4 /CT/D6 /DB /CT /D7/CW/CP/D0/D0 /D9/D7/CTA, B, C, D, ... /CP/D2/CSA′, B′, ... /D8/D3 /CS/CT/D2/D3/D8/CT /D8/CW/CT /D7/D4/CP/D8/CX/CP/D0 /D4 /D3/CX/D2 /D8/D7 /D3/D2 /D8/CW/CT /B4/D1/D3 /DA/CX/D2/CV/B5/D3/CQ /CY/CT
/D8/BA/B5 /CC/CW/CT /CX/D2 /DA /CP/D6/CX/CP/D2 /D8 /D7/D4/CP
/CT/D8/CX/D1/CT /D0/CT/D2/CV/D8/CW /D3/CU /D8/CW/CT /CV/CP/D6/CP/CV/CT /CX/D7lg= (lµ
glµg)1/2=l. /CC/CW/CT /CU/D6/CP/D1/CT S′/B8 /D8/D3/CV/CT/D8/CW/CT/D6/DB/CX/D8/CW /D8/CW/CT
/CP/D6/B8 /D1/D3 /DA /CT/D7 /D6/CT/D0/CP/D8/CX/DA /CT /D8/D3S /DB/CX/D8/CW /BG/B9/DA /CT/D0/D3
/CX/D8 /DD vµ= (γc, γV ), γ= (1−β2)1/2, β=V/c. /C4/CT/D8 /D8/CW/CT/D3/D6/CX/CV/CX/D2 /D3/CUS′/B4/DB/CX/D8/CW /D8/CW/CT /D0/CT/CU/D8 /CT/D2/CS /D3/CU /D8/CW/CT
/CP/D6A′/CP/D8/D8/CP
/CW/CT/CS /D8/D3 /CX/D8/B5
/D3/CX/D2
/CX/CS/CT/D7 /DB/CX/D8/CW /D8/CW/CT /D3/D6/CX/CV/CX/D2 /D3/CUS /B4/DB/CX/D8/CW /D8/CW/CT/D0/CT/CU/D8 /CT/D2/CS /D3/CU /D8/CW/CT /CV/CP/D6/CP/CV/CT A /CP/D8/D8/CP
/CW/CT/CS /D8/D3 /CX/D8/B5 /CP/D8 /D8/CW/CT /D1/D3/D1/CT/D2 /D8 t=t′= 0 /BA /CC/CW/CT /D4 /D3/D7/CX/D8/CX/D3/D2 /BG/B9/DA /CT
/D8/D3/D6/D7 /D3/CU /D8/CW/CT/CT/D2/CS /D4 /D3/CX/D2 /D8/D7 A′/CP/D2/CSB′/D3/CU /D8/CW/CT
/CP/D6 /CX/D2S′/CP/D6/CTx′µ
A′c= (0,0) /CP/D2/CSx′µ
B′c= (0, l) /B8 /CP/D2/CS /D8/CW/CT /CS/CX/D7/D8/CP/D2
/CT /BG/B9/DA /CT
/D8/D3/D6/BD/BC/CX/D7l′µ
c=x′µ
B′c−x′µ
A′c= (0, l) /BA /CC/CW/CT /CX/D2 /DA /CP/D6/CX/CP/D2 /D8 /D7/D4/CP
/CT/D8/CX/D1/CT /D0/CT/D2/CV/D8/CW /D3/CU /D8/CW/CT
/CP/D6 /CX/D7l′
c= (l′µ
cl′
µc)1/2=l /BA /CC /D3/AS/D2/CS /D8/CW/CT /D4 /D3/D7/CX/D8/CX/D3/D2 /BG/B9/DA /CT
/D8/D3/D6 /D3/CU /D8/CW/CT /D6/CX/CV/CW /D8 /CT/D2/CSB′/D3/CU /D8/CW/CT
/CP/D6 /CP/D2/CS /D8/CW/CT /CS/CX/D7/D8/CP/D2
/CT /BG/B9/DA /CT
/D8/D3/D6 /D3/CU /D8/CW/CT
/CP/D6 /CX/D2S/D3/D2/CT /CP/D4/D4/D0/CX/CT/D7 /D8/CW/CT /C4 /CC /D8/D3x′µ
B′c
/CP/D2/CSl′µ
c
/BA /CC/CW/CT/D2 /D3/D2/CT /AS/D2/CS/D7xµ
B′c=lµ
c= (γβl, γl ) /BA /CC/CW/CT /D7/D4/CP
/CT/D8/CX/D1/CT /D0/CT/D2/CV/D8/CW/D3/CU /D8/CW/CT
/CP/D6 /CX/D7 /D2/D3 /DB /CS/CT/D8/CT/D6/D1/CX/D2/CT/CS /CX/D2S /B8 /CP/D2/CS /CX/D7 /CP/D7 /CQ /CT/CU/D3/D6/CT lc=l /B8 /CT/D5/D9/CP/D0 /D8/D3 /CX/D8/D7 /D6/CT/D7/D8 /D0/CT/D2/CV/D8/CW/BA /C7/CQ /DA/CX/D3/D9/D7/D0/DD /B8/CX/D2 /D8/CW/CT /AH/CC/CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH /D2/D3/D8/CW/CX/D2/CV
/D3/D9/D0/CS /CQ /CT /D7/CP/CX/CS /DB/CW/CT/D8/CW/CT/D6 /D8/CW/CT /D1/D3/DA/CX/D2/CV
/CP/D6 /AS/D8 /CX/D2 /D8/D3 /D8/CW/CT /CV/CP/D6/CP/CV/CT /D3/D6 /D2/D3/D8/B8/CP/D7 /D8/CW/CT
/D3/D1/D4/CP/D6/CX/D7/D3/D2 /D3/CU /D8/CW/CT /D7/D4/CP/D8/CX/CP/D0 /D4/CP/D6/D8/D7 /D3/CU /D8/CW/CT /CS/CX/D7/D8/CP/D2
/CT /BG/B9/DA /CT
/D8/D3/D6/D7 /CU/D3/D6 /D8/CW/CT /CV/CP/D6/CP/CV/CT /CP/D2/CS /D8/CW/CT
/CP/D6 /CW/CP/D7 /D2/D3/D4/CW /DD/D7/CX
/CP/D0 /D1/CT/CP/D2/CX/D2/CV/BA /B4/C6/D3/D8/CT /D8/CW/CP/D8 /D8/CW/CT/D6/CT /CX/D7 /CP /CS/CX/D0/CP/D8/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D7/D4/CP/D8/CX/CP/D0 /D4/CP/D6/D8 /D3/CU /D8/CW/CT /D1/D3 /DA/CX/D2/CV
/CP/D6/B8 /CP/D2/CS /D2/D3/D8/D8/CW/CT /C4/D3/D6/CT/D2 /D8/DE
/D3/D2 /D8/D6/CP
/D8/CX/D3/D2 /CP/D7 /CX/D2 /D8/CW/CT /AH/BT /CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH/BA/B5 /C7/D2/D0/DD /CX/CU /CQ /D3/D8/CW /D3/CQ /CY/CT
/D8/D7 /CP/D6/CT /D6/CT/D0/CP/D8/CX/DA /CT/D0/DD /CP/D8 /D6/CT/D7/D8 /D8/CW/CT/CX/D6/D7/D4/CP/D8/CX/CP/D0 /D4/CP/D6/D8/D7
/CP/D2 /CQ /CT
/D3/D1/D4/CP/D6/CT/CS /CX/D2 /D4/CW /DD/D7/CX
/CP/D0/D0/DD /D1/CT/CP/D2/CX/D2/CV/CU/D9/D0 /D7/CT/D2/D7/CT/BA /CC/CW/CT /D7/CP/D1/CT
/D3/D2
/D0/D9/D7/CX/D3/D2/D7 /CW/D3/D0/CS /CU/D6/D3/D1 /D8/CW/CT/D4 /D3/CX/D2 /D8 /D3/CU /DA/CX/CT/DB /D3/CU /D8/CW/CT /D3/CQ/D7/CT/D6/DA /CT/D6 /CX/D2S′/B8 /D3/D6 /CP/D2 /DD /D3/D8/CW/CT/D6 /C1/BY/CA/BA /CC/CW/CT/D7/CT
/D3/D2
/D0/D9/D7/CX/D3/D2/D7 /CW/D3/D0/CS /D2/D3/D8 /D3/D2/D0/DD /CU/D3/D6 /BX/CX/D2/D7/D8/CT/CX/D2/B3/D7
/D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2 /B4/DB/CW/CX
/CW /CX/D7 /D9/D7/CT/CS /CW/CT/D6/CT/B5 /CQ/D9/D8 /CU/D3/D6 /CP/D2 /DD /D3/D8/CW/CT/D6 /D4 /D3/D7/D7/CX/CQ/D0/CT
/D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2 /D3/CU /C1/BY/CA/D7/BA/BF/BA /BV/CD/CA/CA/BX/C6/CC/B9/BV/BT/CA/CA /CH/C1/C6/BZ /BV/C7/C6/BW/CD/BV/CC/C7/CA /BT/C6/BW /BX/CG/CC/BX/CA/C6/BT/C4 /BX/C4/BX/BV/CC/CA/C1/BV /BY/C1/BX/C4/BW/CB/C1/C6 /CC/C0/BX /AH/CC/CC /CA/BX/C4/BT /CC/C1/CE/C1/CC/CH/AH/C4/CT/D8 /D9/D7 /D2/D3 /DB /CP/D4/D4/D0/DD /D8/CW/CT/D7/CT /CX/CS/CT/CP/D7 /D8/D3 /D8/CW/CT
/D3/D2/D7/CX/CS/CT/D6/CP/D8/CX/D3/D2 /D3/CU /CP /BV/BV/BV /CX/D2 /D8/CW/CT /AH/CC/CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH/BA /BT/D2 /CX/D2/AS/D2/CX/D8/CT/D7/D8/D6/CP/CX/CV/CW /D8 /DB/CX/D6/CT /DB/CX/D8/CW /CP /D7/D8/CT/CP/CS/DD
/D9/D6/D6/CT/D2 /D8 /CX/D7 /D7/CX/D8/D9/CP/D8/CT/CS /CP/D0/D3/D2/CV /D8/CW/CTx1/CP/DC/CX/D7/BA /BT
/D9/D6/D6/CT/D2 /D8 /CX/D7 /AT/D3 /DB/CX/D2/CV /CX/D2−x1/CS/CX/D6/CT
/D8/CX/D3/D2 /CP/D2/CS /CP
/D3/D6/CS/CX/D2/CV/D0/DD /CT/D0/CT
/D8/D6/D3/D2/D7 /D1/D3 /DA /CT /CX/D2+x1/CS/CX/D6/CT
/D8/CX/D3/D2/BA /CF /CT /D7/D9/D4/D4 /D3/D7/CT /D8/CW/CP/D8 /D4 /D3/D7/CX/D8/CX/DA /CT /CP/D2/CS /D2/CT/CV/CP/D8/CX/DA /CT
/CW/CP/D6/CV/CT /CS/CT/D2/D7/CX/D8/CX/CT/D7 /CP/D6/CT /D3/CU /CT/D5/D9/CP/D0 /D1/CP/CV/D2/CX/D8/D9/CS/CT /DB/CW/CT/D2 /CQ /D3/D8/CW /D7/D9/CQ/D7/DD/D7/D8/CT/D1/D7 /CP/D6/CT /D6/CT/D0/CP/D8/CX/DA /CT/D0/DD /CP/D8 /D6/CT/D7/D8/B8 /CX/BA/CT/BA/B8 /CQ /CT/CU/D3/D6/CT /CP
/D9/D6/D6/CT/D2 /D8 /CX/D7 /CT/D7/D8/CP/CQ/D0/CX/D7/CW/CT/CS /CX/D2 /D8/CW/CT /DB/CX/D6/CT/BA /C1/D2 /CP /BV/BV/BV /D8/CW/CT /DB/CX/D6/CT /B4/CX/BA/CT/BA/B8 /D8/CW/CT /CX/D3/D2/D7/B5 /CX/D7 /D7/D9/D4/D4 /D3/D7/CT/CS /D8/D3 /CQ /CT /CP/D8 /D6/CT/D7/D8 /CX/D2S /B8/DB/CW/CX/D0/CT /D8/CW/CT /CT/D0/CT
/D8/D6/D3/D2/D7 /CP/D6/CT /CP/D8 /D6/CT/D7/D8 /CX/D2S′/BA/BU/CT/CU/D3/D6/CT /CS/CT/D8/CT/D6/D1/CX/D2/CX/D2/CV /D8/CW/CT
/D9/D6/D6/CT/D2 /D8 /CS/CT/D2/D7/CX/D8 /DD /BG/B9/DA /CT
/D8/D3/D6/D7 jµ/CX/D2S /CP/D2/CSS′/DB /CT /CV/CX/DA /CT /D8/CW/CT /D1/CP/D2/CX/CU/CT/D7/D8/D0/DD
/D3 /DA /CP/D6/CX/CP/D2 /D8/CS/CT/AS/D2/CX/D8/CX/D3/D2 /D3/CU /CP
/CW/CP/D6/CV/CT /DB/CX/D8/CW/CX/D2 /CP /CQ /D3/D9/D2/CS/CP/D6/DD δH /D3/CU /CP/D2 /CP/D6/CQ/CX/D8/D6/CP/D6/DD /CW /DD/D4 /CT/D6/D7/D9/D6/CU/CP
/CT H /B4/D7/CT/CT/B8 /CT/BA/CV/BA/B8 /CJ/BD/BI ℄/B8 /CJ/BD/BJ ℄/B5/B8
QδH=/integraldisplay
Hjµdσµ,
/B4/BD/BC/B5/DB/CW/CT/D6/CT dσµ/CX/D7 /D8/CW/CT /BG/B9/DA /CT
/D8/D3/D6 /D3/CU /CP/D2 /CT/D0/CT/D1/CT/D2 /D8 /D3/CU /D8/CW/CT /CW /DD/D4 /CT/D6/D7/D9/D6/CU/CP
/CT H /BA /B4/CC/CW/CX/D7 /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2
/CP/D2 /CQ /CT /DB/D6/CX/D8/D8/CT/D2/CX/D2 /CP /D1/D3/D6/CT /CV/CT/D2/CT/D6/CP/D0 /CU/D3/D6/D1/B8 /CX/BA/CT/BA/B8 /CX/D2 /CV/CT/D3/D1/CT/D8/D6/CX
/CP/D0 /D8/CT/D6/D1/D7/B8 /D6/CT/D4/D0/CP
/CX/D2/CV /BZ/D6/CT/CT/CZ /CX/D2/CS/CT/DC µ /CQ /DD /D8/CW/CT /CP/CQ/D7/D8/D6/CP
/D8 /CX/D2/CS/CT/DC
a. /B5 /CC/CW/CT /CX/D2 /DA /CP/D6/CX/CP/D2
/CT /D3/CU
/CW/CP/D6/CV/CT /CS/CT/AS/D2/CT/CS /CQ /DD /B4/BD/BC/B5 /CX/D7 /D4/D6/D3 /DA /CT/CS /CX/D2 /CJ/BD/BI ℄ /CU/D3/D6 /CP /D0/CX/D2/CT/CP/D6 /BV/BV/BV/B8 /CP/D2/CS /CP/D0/D7/D3 /CU/D3/D6 /D8/CW/CT/CV/CT/D2/CT/D6/CP/D0
/CP/D7/CT /D3/CU /BG/BW /D7/D4/CP
/CT/D8/CX/D1/CT /CX/D2 /CJ/BD/BJ ℄/BA /C1/CU /D8/CW/CT /CW /DD/D4 /CT/D6/D7/D9/D6/CU/CP
/CT H /CX/D7
/CW/D3/D7/CT/D2 /CX/D2 /D8/CW/CT /D6/CT/D7/D8 /CU/D6/CP/D1/CT /D3/CU
/CW/CP/D6/CV/CT/D7/CX/D2 /D7/D9
/CW /CP /DB /CP /DD /D8/CW/CP/D8 /CX/D8 /CX/D7 /D8/CW/CT /D7/D4/CP
/CT/B9/D0/CX/CZ /CT /D4/D0/CP/D2/CT t=const., /D8/CW/CT/D2 /D8/CW/CT
/CW/CP/D6/CV/CT dQ /CX/D7 /CV/CX/DA /CT/D2 /CP/D7 /CX/D2 /D8/CW/CT/D4/D6/CT/D6/CT/D0/CP/D8/CX/DA/CX/D7/D8/CX
/D4/CW /DD/D7/CX
/D7 dQ=ρdV /BN /D8/CW/CT /BG/B9/DA /CT
/D8/D3/D6 jµ/CW/CP/D7 /D3/D2/D0/DD /D8/CW/CT /D8/CX/D1/CT
/D3/D1/D4 /D3/D2/CT/D2 /D8 j0=cρ /B8 /D7/CX/D2
/CT /DB /CT/CP/D6/CT /CX/D2 /D8/CW/CT /D6/CT/D7/D8 /CU/D6/CP/D1/CT /D3/CU
/CW/CP/D6/CV/CT/D7/B8 /CP/D2/CSdσµ
/CP/D0/D7/D3 /CW/CP/D7 /D3/D2/D0/DD /D8/CX/D1/CT
/D3/D1/D4 /D3/D2/CT/D2 /D8 /DB/CW/CX
/CW /CX/D7 /D8/CW/CT /D7/DD/D2
/CW/D6/D3/D2/D3/D9/D7/D0/DD/CS/CT/AS/D2/CT/CS /D6/CT/D7/D8 /DA /D3/D0/D9/D1/CT dV /BA /BV/D3/D2/D7/CT/D5/D9/CT/D2 /D8/D0/DD /B8 /D8/CW/CT
/CW/CP/D6/CV/CT /CS/CT/D2/D7/CX/D8 /DD ρ /CX/D7 /CS/CT/AS/D2/CT/CS /CP/D7 /D8/CW/CT /D6/CP/D8/CX/D3 /D3/CUdQ /CV/CX/DA /CT/D2/CQ /DD /B4/BD/BC/B5/B8 /CQ/D9/D8 /D8/CP/CZ /CT/D2 /D7/CX/D1 /D9/D0/D8/CP/D2/CT/D3/D9/D7/D0/DD /CX/D2 /D8/CW/CT /D6/CT/D7/D8 /CU/D6/CP/D1/CT /D3/CU /D8/CW/CT
/CW/CP/D6/CV/CT/D7/B8 /CP/D2/CS /D8/CW/CT /D7/DD/D2
/CW/D6/D3/D2/D3/D9/D7/D0/DD /CS/CT/AS/D2/CT/CS/D6/CT/D7/D8 /DA /D3/D0/D9/D1/CT dV, ρ =dQ/dV, /CP/D2/CS /CX/D8 /CX/D7 /DB /CT/D0/D0 /CS/CT/AS/D2/CT/CS /D5/D9/CP/D2 /D8/CX/D8 /DD /CU/D6/D3/D1 /D8/CW/CT /AH/CC/CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH /DA/CX/CT/DB/D4 /D3/CX/D2 /D8/BA/C7/CQ /DA/CX/D3/D9/D7/D0/DD /D8/CW/CT
/CW/CP/D6/CV/CT /CS/CT/D2/D7/CX/D8 /DD /CW/CP/D7 /D8/CW/CT
/D3/D1/D1/D3/D2 /D4/D6/CT/D6/CT/D0/CP/D8/CX/DA/CX/D7/D8/CX
/D1/CT/CP/D2/CX/D2/CV /D3/D2/D0/DD /CX/D2 /D8/CW/CT /D6/CT/D7/D8 /CU/D6/CP/D1/CT /D3/CU /D8/CW/CT
/CW/CP/D6/CV/CT/D7/BA /CC/CW/CT
/CW/CP/D6/CV/CT /CS/CT/D2/D7/CX/D8 /DD /D3/CU /D1/D3 /DA/CX/D2/CV
/CW/CP/D6/CV/CT/D7 /CX/D7 /D2/D3/D8 /CP /DB /CT/D0/D0 /CS/CT/AS/D2/CT/CS /D5/D9/CP/D2 /D8/CX/D8 /DD /CU/D6/D3/D1 /D8/CW/CT /AH/CC/CC /D6/CT/D0/CP/D8/CX/DA/B9/CX/D8 /DD/AH /D4 /D3/CX/D2 /D8 /D3/CU /DA/CX/CT/DB /CX/D2 /D8/CW/CT /D7/CP/D1/CT /DB /CP /DD /CP/D7 /D8/CW/CT /D7/D4/CP/D8/CX/CP/D0 /D0/CT/D2/CV/D8/CW /D3/D6 /D8/CW/CT /DA /D3/D0/D9/D1/CT /D3/CU /CP /D1/D3 /DA/CX/D2/CV /D3/CQ /CY/CT
/D8 /CP/D6/CT /D2/D3/D8
/D3/D6/D6/CT
/D8/D0/DD /CS/CT/AS/D2/CT/CS /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7 /CX/D2 /D8/CW/CT /AH/CC/CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH/BA /CC/CW/CX/D7 /CX/D7 /CX/D2
/D3/D2 /D8/D6/CP/D7/D8 /DB/CX/D8/CW /D8/CW/CT /AH/BT /CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH /CP/D2/CS/D8/CW/CT /C4/D3/D6/CT/D2 /D8/DE
/D3/D2 /D8/D6/CP
/D8/CX/D3/D2/B8 /DB/CW/CT/D6/CT /D8/CW/CT
/CW/CP/D6/CV/CT /CS/CT/D2/D7/CX/D8 /DD /D3/CU /D8/CW/CT /D1/D3 /DA/CX/D2/CV
/CW/CP/D6/CV/CT/D7 /CX/D7 /CS/CT/AS/D2/CT/CS/BN /CX/D8 /CX/D7 /CT/D2/CW/CP/D2
/CT/CS/CQ /DDγ= (1−β2)1/2/D6/CT/D0/CP/D8/CX/DA /CT /D8/D3 /D8/CW/CT /D4/D6/D3/D4 /CT/D6
/CW/CP/D6/CV/CT /CS/CT/D2/D7/CX/D8 /DD /CS/D9/CT /D8/D3 /D8/CW/CT /C4/D3/D6/CT/D2 /D8/DE
/D3/D2 /D8/D6/CP
/D8/CX/D3/D2 /D3/CU /D8/CW/CT/D1/D3 /DA/CX/D2/CV /DA /D3/D0/D9/D1/CT/BA /CC/CW/CT/D6/CT/CQ /DD /B8 /DB/CW/CT/D2 /CS/CT/D8/CT/D6/D1/CX/D2/CX/D2/CV /D8/CW/CT
/D9/D6/D6/CT/D2 /D8 /CS/CT/D2/D7/CX/D8 /DD /BG/B9/DA /CT
/D8/D3/D6 jµ/CX/D2 /D7/D3/D1/CT /C1/BY/CA /CX/D2 /DB/CW/CX
/CW/D8/CW/CT
/CW/CP/D6/CV/CT/D7 /CP/D6/CT /D1/D3 /DA/CX/D2/CV /D3/D2/CT /AS/D6/D7/D8 /CW/CP/D7 /D8/D3 /AS/D2/CS /D8/CW/CP/D8 /DA /CT
/D8/D3/D6 /CX/D2 /D8/CW/CT /D6/CT/D7/D8 /CU/D6/CP/D1/CT /D3/CU /D8/CW/CT
/CW/CP/D6/CV/CT/D7/B8 /DB/CW/CT/D6/CT/D8/CW/CT /D7/D4/CP
/CT
/D3/D1/D4 /D3/D2/CT/D2 /D8 j= 0 /CP/D2/CSγ= 1, /CP/D2/CS /D8/CW/CT/D2 /D8/D3 /D8/D6/CP/D2/D7/CU/D3/D6/D1 /CQ /DD /D8/CW/CT /C4 /CC /D7/D3 /CS/CT/D8/CT/D6/D1/CX/D2/CT/CS jµ/D8/D3 /D8/CW/CT
/D3/D2/D7/CX/CS/CT/D6/CT/CS /C1/BY/CA/BA /BT
/D3/D6/CS/CX/D2/CV /D8/D3 /D8/CW/CX/D7
/D3/D2/D7/CX/CS/CT/D6/CP/D8/CX/D3/D2 /D8/CW/CT /D7/CX/D1/D4/D0/CT/D7/D8 /CP/D2/CS /D8/CW/CT
/D3/D6/D6/CT
/D8 /DB /CP /DD /B8 /CU/D6/D3/D1 /D8/CW/CT /AH/CC/CC/D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH /DA/CX/CT/DB/D4 /D3/CX/D2 /D8/B8 /D8/D3 /CS/CT/D8/CT/D6/D1/CX/D2/CT /D8/CW/CT
/D9/D6/D6/CT/D2 /D8 /CS/CT/D2/D7/CX/D8 /DD /BG/B9/DA /CT
/D8/D3/D6 jµ/CX/D2 /D7/D3/D1/CT /C1/BY/CA /CU/D3/D6 /CP /BV/BV/BV /CX/D7 /D8/CW/CT/CU/D3/D0/D0/D3 /DB/CX/D2/CV/BM /CC/CW/CT
/D9/D6/D6/CT/D2 /D8 /CS/CT/D2/D7/CX/D8 /DD /BG/B9/DA /CT
/D8/D3/D6/D7 jµ
+
/CP/D2/CSjµ
−
/CU/D3/D6 /D4 /D3/D7/CX/D8/CX/DA /CT /CP/D2/CS /D2/CT/CV/CP/D8/CX/DA /CT
/CW/CP/D6/CV/CT/D7/B8 /D6/CT/D7/D4 /CT
/D8/CX/DA /CT/D0/DD /B8/CW/CP /DA /CT /D8/D3 /CQ /CT /CS/CT/D8/CT/D6/D1/CX/D2/CT/CS /CX/D2 /D8/CW/CT/CX/D6 /D6/CT/D7/D8 /CU/D6/CP/D1/CT/D7 /CP/D2/CS /D8/CW/CT/D2 /D8/D6/CP/D2/D7/CU/D3/D6/D1/CT/CS /CQ /DD /D8/CW/CT /C4 /CC /D8/D3 /D8/CW/CT /CV/CX/DA /CT/D2 /C1/BY/CA/BA/C1/D8 /CW/CP/D7 /D8/D3 /CQ /CT /D2/D3/D8/CT/CS /D8/CW/CP/D8 /CX/D2 /D8/CW/CT /AH/CC/CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH /CX/D8 /CX/D7 /D2/D3/D8 /D2/CT
/CT/D7/D7/CP/D6/DD /D8/D3 /CS/CT/D8/CT/D6/D1/CX/D2/CT jµ/CU/D3/D6 /CP /BV/BV/BV/B8 /CX/D2/CP/D2 /CP/D6/CQ/CX/D8/D6/CP/D6/DD /C1/BY/CA/B8 /CX/D2 /D8/CW/CT /D1/CT/D2 /D8/CX/D3/D2/CT/CS /DB /CP /DD /BA /C1/D2 /D8/CW/CP/D8 /CU/D6/CP/D1/CT /DB /CT
/D3/D9/D0/CS /D7/D8/CP/D6/D8 /CX/D2 /B4/BD/BC/B5 /DB/CX/D8/CW /CP/D2 /CP/D6/CQ/CX/D8/D6/CP/D6/DD/D7/D4/CP
/CT/B9/D0/CX/CZ /CT /CW /DD/D4 /CT/D6/D7/D9/D6/CU/CP
/CT H /CP/D2/CS /CS/CT/D8/CT/D6/D1/CX/D2/CT ja/CP/D2/CSdσa/CX/D2 /D7/D3/D1/CT
/D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2 /D8/CW/CP/D8 /CX/D7 /CS/CX/AR/CT/D6/CT/D2 /D8 /D8/CW/CP/D2/D8/CW/CT /AH/CT/AH
/D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/BA /BU/D9/D8 /D8/CW/CT/D2 /DB /CT /D0/D3 /D3/D7/CT /D8/CW/CT
/D3/D2/D2/CT
/D8/CX/D3/D2 /DB/CX/D8/CW /D8/CW/CT /D4/D6/CT/D6/CT/D0/CP/D8/CX/DA/CX/D7/D8/CX
/D2/D3/D8/CX/D3/D2/D7/B8 /D8/CW/CT
/CW/CP/D6/CV/CT /CS/CT/D2/D7/CX/D8 /DD ρ, /D8/CW/CT
/D9/D6/D6/CT/D2 /D8 /CS/CT/D2/D7/CX/D8 /DD j /B4/BF/B9/DA /CT
/D8/D3/D6/B5/B8 /D8/CW/CT /D7/D4/CP/D8/CX/CP/D0 /D0/CT/D2/CV/D8/CW /CP/D2/CS /DA /D3/D0/D9/D1/CT/B8 /CT/D8
/BA/B8 /CP/D2/CS /DB/CX/D8/CW /D8/CW/CT/BD/BD/D4/D6/CT/D6/CT/D0/CP/D8/CX/DA/CX/D7/D8/CX
/D6/CT/D0/CP/D8/CX/D3/D2 dQ=ρdV./BF/BA/BD/BA /CC/CW/CT
/D9/D6/D6/CT/D2 /D8 /CS/CT/D2/D7/CX/D8 /DD jµ/CX/D2 /D8/CW/CT /CX/D3/D2/D7/B3 /D6/CT/D7/D8 /CU/D6/CP/D1/CT /CB/C0/CT/D2
/CT/B8 /D8/CW/CT
/D9/D6/D6/CT/D2 /D8 /CS/CT/D2/D7/CX/D8 /DD /BG/B9/DA /CT
/D8/D3/D6 /CX/D2S /B8 /CU/D3/D6 /D8/CW/CT
/D3/D2/D7/CX/CS/CT/D6/CT/CS /DB/CX/D6/CT /DB/CX/D8/CW
/D9/D6/D6/CT/D2 /D8/B8 /CX/D7jµ=jµ
++jµ
−,/DB/CW/CT/D6/CT jµ
+= (cρ0,0). /CC/CW/CT /D4 /D3/D7/CX/D8/CX/DA /CT
/CW/CP/D6/CV/CT /CS/CT/D2/D7/CX/D8 /DD ρ+
/CX/D7=ρ0, /DB/CW/CT/D6/CT ρ0
/CX/D7 /D8/CW/CT /D4 /D3/D7/CX/D8/CX/DA /CT
/CW/CP/D6/CV/CT /CS/CT/D2/D7/CX/D8 /DD/CU/D3/D6 /D8/CW/CT /DB/CX/D6/CT /CP/D8 /D6/CT/D7/D8 /CQ/D9/D8 /DB/CX/D8/CW/D3/D9/D8 /CP
/D9/D6/D6/CT/D2 /D8/BA /CC /D3 /AS/D2/CSjµ
−
/CX/D2S /D3/D2/CT /CW/CP/D7/B8 /CP/D7 /CP/D0/D6/CT/CP/CS/DD /D7/CP/CX/CS/B8 /D8/D3 /AS/D2/CS /D8/CW/CT/CT/D0/CT
/D8/D6/D3/D2/D7/B3
/CW/CP/D6/CV/CT /CS/CT/D2/D7/CX/D8 /DD ρ′
−
/B8 /CP/D2/CS /D8/CW/CT
/D9/D6/D6/CT/D2 /D8 /CS/CT/D2/D7/CX/D8 /DD /BG/B9/DA /CT
/D8/D3/D6 /D3/CU /D8/CW/CT /CT/D0/CT
/D8/D6/D3/D2/D7 j′µ
−
/CX/D2 /D8/CW/CT/CX/D6 /D6/CT/D7/D8 /CU/D6/CP/D1/CT
S′/B8 /DB/CW/CT/D6/CT ρ′
−
/CX/D7 /DB /CT/D0/D0 /CS/CT/AS/D2/CT/CS /D5/D9/CP/D2 /D8/CX/D8 /DD /B8 /CP/D2/CS /D8/CW/CT/D2 /D8/D3 /D8/D6/CP/D2/D7/CU/D3/D6/D1 /D8/CW/CT/D1 /D8/D3 /D8/CW/CT /CX/D3/D2/D7/B3 /D6/CT/D7/D8 /CU/D6/CP/D1/CT S /BA /C1/D2/D8/CW/CT /D6/CT/D7/D8 /DB/CX/D6/CT/B8 /CQ/D9/D8 /DB/CX/D8/CW/D3/D9/D8 /CP
/D9/D6/D6/CT/D2 /D8/B8 /D8/CW/CT
/CW/CP/D6/CV/CT /CS/CT/D2/D7/CX/D8 /DD /D3/CU /D8/CW/CT /CT/D0/CT
/D8/D6/D3/D2/D7/B8 /DB/CW/CX
/CW /CP/D6/CT /CP/D8 /D6/CT/D7/D8 /D8/CW/CT/D6/CT/B8 /CX/D7
−ρ0
/BA /CC/CW/CT/D2/B8 /CX/D8 /CU/D3/D0/D0/D3 /DB/D7 /CU/D6/D3/D1 /D8/CW/CT /D4/D6/CT/DA/CX/D3/D9/D7
/D3/D2/D7/CX/CS/CT/D6/CP/D8/CX/D3/D2 /D8/CW/CP/D8 /CX/D2S′/B8 /DB/CW/CT/D6/CT /D8/CW/CT /CT/D0/CT
/D8/D6/D3/D2/D7 /CX/D2 /D8/CW/CP/D8 /DB/CX/D6/CT/B8/CQ/D9/D8 /DB/CX/D8/CW /CP
/D9/D6/D6/CT/D2 /D8/B8 /CP/D6/CT /CP/D8 /D6/CT/D7/D8/B8 /D8/CW/CT /D4/D6/D3/D4 /CT/D6
/CW/CP/D6/CV/CT /CS/CT/D2/D7/CX/D8 /DD ρ′
−
/D3/CU /D8/CW/CT /CT/D0/CT
/D8/D6/D3/D2/D7 /D1 /D9/D7/D8 /CP/CV/CP/CX/D2 /CQ /CT /CT/D5/D9/CP/D0/D8/D3−ρ0, /CX/BA/CT/BA/B8
ρ′
−=−ρ0, j′µ
−= (−cρ0,0). /B4/BD/BD/B5/BU/DD /D1/CT/CP/D2/D7 /D3/CU /B4/BD/BD/B5 /CP/D2/CS /D8/CW/CT /C4 /CC /DB /CT /AS/D2/CS /D8/CW/CT
/D9/D6/D6/CT/D2 /D8 /CS/CT/D2/D7/CX/D8/CX/CT/D7 /CX/D2S /CP/D7
jµ
−= (−cγρ0,−cγβρ 0), jµ= (c(1−γ)ρ0,−cγβρ 0). /B4/BD/BE/B5/BX/D5/D7/BA /B4/BD/BD/B5 /CP/D2/CS /B4/BD/BE/B5 /CP/D6/CT /CX/D2
/D3/D2 /D8/D6/CP/D7/D8 /D8/D3 /CP/D0/D0 /D4/D6/CT/DA/CX/D3/D9/D7 /DB /D3/D6/CZ/D7 /CU/D6/D3/D1 /D8/CW/CT /D8/CX/D1/CT /D3/CU /BV/D0/CP/D9/D7/CX/D9/D7/B8 /BV/D0/CP/D9/D7/CX/D9/D7 /CW /DD/B9/D4 /D3/D8/CW/CT/D7/CX/D7/B8 /D7/CT/CT /CJ/BD/BK ℄/B8 /D9/D2 /D8/CX/D0 /D8/D3 /CS/CP /DD /BA /C1/D2 /D8/CW/CT /BV/D0/CP/D9/D7/CX/D9/D7 /CW /DD/D4 /D3/D8/CW/CT/D7/CX/D7 /CX/D8 /CX/D7 /D7/CX/D1/D4/D0/DD /D7/D9/D4/D4 /D3/D7/CT /CS /D8/CW/CP/D8 /CX/D2 /D8/CW/CT /CX/D3/D2/D7/B3 /D6/CT/D7/D8/CU/D6/CP/D1/CT S /D8/CW/CT
/CW/CP/D6/CV/CT /CS/CT/D2/D7/CX/D8 /DD /D3/CU /D8/CW/CT /D1/D3 /DA/CX/D2/CV /CT/D0/CT
/D8/D6/D3/D2/D7 ρ−=−ρ0. /C0/D3 /DB /CT/DA /CT/D6 /D8/CW/CT /D7/CP/D1/CT /CT/D5/D9/CP/D8/CX/D3/D2/D7 /DB /CT/D6/CT/CP/D0/D6/CT/CP/CS/DD /D3/CQ/D8/CP/CX/D2/CT/CS /CX/D2 /CJ/BH℄/B8 /DB/CW/CT/D6/CT /D8/CW/CT /AH/BT /CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH /DB/CX/D8/CW /D8/CW/CT /C4/D3/D6/CT/D2 /D8/DE
/D3/D2 /D8/D6/CP
/D8/CX/D3/D2 /CX/D7 /D9/D7/CT/CS/BA /CC/CW/CX/D7 /D1/CP /DD/D7/CT/CT/D1 /D7/D9/D6/D4/D6/CX/D7/CX/D2/CV /D8/CW/CP/D8 /D8/CW/CT /D7/CP/D1/CT /CT/D5/D9/CP/D8/CX/D3/D2/D7 /CT/DC/CX/D7/D8 /CX/D2 /CJ/BH ℄ /B4/DB/CX/D8/CW /D8/CW/CT /AH/BT /CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH/B5 /CP/D2/CS /CW/CT/D6/CT/B8 /DB/CW/CT/D6/CT /D8/CW/CT/AH/CC/CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH /CX/D7
/D3/D2/D7/CX/CS/CT/D6/CT/CS /CP/D2/CS /D8/CW /D9/D7 /D3/D2/D0/DD /D8/CW/CT
/D3 /DA /CP/D6/CX/CP/D2 /D8 /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7 /CP/D6/CT /D9/D7/CT/CS/BA /BU/D9/D8/B8 /DB /CT /D1 /D9/D7/D8 /D2/D3/D8/CT/D8/CW/CP/D8 /D8/CW/CT /D6/CT/D7/D9/D0/D8/D7 /D3/CQ/D8/CP/CX/D2/CT/CS /CX/D2 /CJ/BH℄ /CP/D6/CT /D2/D3/D8 /CP
/D8/D9/CP/D0/D0/DD /CQ/CP/D7/CT/CS /D3/D2 /D8/CW/CT /BT /CC/B8 /CX/BA/CT/BA/B8 /D3/D2 /D8/CW/CT /C4/D3/D6/CT/D2 /D8/DE
/D3/D2 /D8/D6/CP
/D8/CX/D3/D2/B8/D8/CW/CP/D2 /D3/D2 /D8/CW/CT /CP/D7/D7/D9/D1/D4/D8/CX/D3/D2 /D8/CW/CP/D8 /CX/D2 /D8/CW/CT /CT/D0/CT
/D8/D6/D3/D2/D7/B3 /D6/CT/D7/D8 /CU/D6/CP/D1/CT S′/D8/CW/CT /CT/D0/CT
/D8/D6/D3/D2/D7/B3
/CW/CP/D6/CV/CT /CS/CT/D2/D7/CX/D8 /DD ρ′
−
/CX/D7
=−ρ0. /C1/D2 /CP
/D3 /DA /CP/D6/CX/CP/D2 /D8 /CP/D4/D4/D6/D3/CP
/CW/B8 /CX/BA/CT/BA/B8 /CX/D2 /D8/CW/CT /AH/CC/CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD /B8/AH /BX/D5/BA /B4/BD/BD/B5 /CX/D7 /D2/CT/CX/D8/CW/CT/D6 /CW/DD/D4 /D3/D8/CW/CT/D7/CX/D7 /B4/CP/D7 /CX/D2/D8/CW/CT /D8/D6/CP/CS/CX/D8/CX/D3/D2/CP/D0 /CP/D4/D4/D6/D3/CP
/CW/B5 /D2/D3/D6 /D8/CW/CT /CP/D7/D7/D9/D1/D4/D8/CX/D3/D2 /B4/CP/D7 /CX/D2 /CJ/BH ℄/B5/B8 /CQ/D9/D8 /CX/D8 /CX/D7 /CP
/D3/D2/D7/CT/D5/D9/CT/D2
/CT /D3/CU /D8/CW/CT
/D3 /DA /CP/D6/CX/CP/D2 /D8/CS/CT/AS/D2/CX/D8/CX/D3/D2 /D3/CU /CP/D2 /CX/D2 /DA /CP/D6/CX/CP/D2 /D8
/CW/CP/D6/CV/CT /B4/BD/BC/B5 /CP/D2/CS /D3/CU /D8/CW/CT /CX/D2 /DA /CP/D6/CX/CP/D2
/CT /D3/CU /D8/CW/CT /D6/CT/D7/D8 /D0/CT/D2/CV/D8/CW/B8 /CX/BA/CT/BA/B8 /CX/D8 /D6/CT/D7/D9/D0/D8/CT/CS /CU/D6/D3/D1/D8/CW/CT /D9/D7/CT /D3/CU
/D3/D6/D6/CT
/D8/D0/DD /CS/CT/AS/D2/CT/CS
/D3 /DA /CP/D6/CX/CP/D2 /D8 /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7/BA/BF/BA/BE/BA /CC/CW/CTFαβ/CP/D2/CS /D8/CW/CTEα, Bα/CU/D3/D6/D1 /D9/D0/CP/D8/CX/D3/D2/D7 /D3/CU /CT/D0/CT
/D8/D6/D3 /CS/DD/D2/CP/D1/CX
/D7/C0/CP /DA/CX/D2/CV /CS/CT/D8/CT/D6/D1/CX/D2/CT/CS /D8/CW/CT /D7/D3/D9/D6
/CT/D7 jµ/DB /CT /AS/D2/CS /D8/CW/CT /CT/D0/CT
/D8/D6/CX
/CP/D2/CS /D1/CP/CV/D2/CT/D8/CX
/AS/CT/D0/CS/D7 /CU/D3/D6 /D8/CW/CP/D8 /CX/D2/AS/D2/CX/D8/CT /DB/CX/D6/CT/DB/CX/D8/CW
/D9/D6/D6/CT/D2 /D8/BA /C7/D2/CT /DB /CP /DD /CX/D7 /D8/D3 /D7/D8/CP/D6/D8 /DB/CX/D8/CW /D8/CW/CT
/D3 /DA /CP/D6/CX/CP/D2 /D8 /C5/CP/DC/DB /CT/D0/D0 /CT/D5/D9/CP/D8/CX/D3/D2/D7 /DB/CX/D8/CWFαβ/CP/D2/CS /CX/D8/D7 /CS/D9/CP/D0
∗Fαβ
∂αFaβ=−jβ/ε0c, ∂ α∗Fαβ= 0 /B4/BD/BF/B5/DB/CW/CT/D6/CT∗Fαβ=−(1/2)εαβγδFγδ
/CP/D2/CSεαβγδ/CX/D7 /D8/CW/CT /D8/D3/D8/CP/D0/D0/DD /D7/CZ /CT/DB/B9/D7/DD/D1/D1/CT/D8/D6/CX
/C4/CT/DA/CX/B9/BV/CX/DA/CX/D8/CP /D4/D7/CT/D9/CS/D3/D8/CT/D2/D7/D3/D6/BA/C1/D2 /D7/D9
/CW /CP
/D3 /DA /CP/D6/CX/CP/D2 /D8 /CU/D3/D6/D1 /D9/D0/CP/D8/CX/D3/D2 /BYαβ/CX/D7 /D8/CW/CT /D4/D6/CX/D1/CP/D6/DD /D5/D9/CP/D2 /D8/CX/D8 /DD/BN /CX/D8 /CX/D7 /D8/CW/CT /D7/D3/D0/D9/D8/CX/D3/D2 /D3/CU /B4/BD/BF/B5/B8 /D3/D6 /D8/CW/CT
/D3/D6/D6/CT/D7/D4 /D3/D2/CS/CX/D2/CV /DB /CP /DA /CT /CT/D5/D9/CP/D8/CX/D3/D2
∂σ∂σFαβ−(1/ε0c)(∂βjα−∂αjβ) = 0, /B4/BD/BG/B5/CP/D2/CS /CX/D8
/D3/D2 /DA /CT/DD/D7 /CP/D0/D0 /D8/CW/CT /CX/D2/CU/D3/D6/D1/CP/D8/CX/D3/D2 /CP/CQ /D3/D9/D8 /D8/CW/CT /CT/D0/CT
/D8/D6/D3/D1/CP/CV/D2/CT/D8/CX
/AS/CT/D0/CS/BA /CC/CW/CT/D6/CT /CX/D7 /D2/D3 /D2/CT/CT/CS /D8/D3 /CX/D2 /D8/D6/D3 /CS/D9
/CT/CT/CX/D8/CW/CT/D6 /D8/CW/CT /CX/D2 /D8/CT/D6/D1/CT/CS/CX/CP/D8/CT /CT/D0/CT
/D8/D6/D3/D1/CP/CV/D2/CT/D8/CX
/BG/B9/D4 /D3/D8/CT/D2 /D8/CX/CP/D0 Aµ/D3/D6 /D8/CW/CT
/D3/D2/D2/CT
/D8/CX/D3/D2 /D3/CU /D8/CW/CT
/D3/D1/D4 /D3/D2/CT/D2 /D8/D7 /D3/CUFαβ/DB/CX/D8/CW /D8/CW/CT /D9/D7/D9/CP/D0 /BF/B9/DA /CT
/D8/D3/D6/D7 E /CP/D2/CSB. /CC/CW/CT /CV/CT/D2/CT/D6/CP/D0 /D7/D3/D0/D9/D8/CX/D3/D2 /CX/D2 /D8/CW/CT /D6/CT/D8/CP/D6/CS/CT/CS /D6/CT/D4/D6/CT/D7/CT/D2 /D8/CP/D8/CX/D3/D2 /D3/CU /B4/BD/BG/B5 /D3/D6/B4/BD/BF/B5 /CX/D7
Fαβ(xµ) = (2 k/iπc)/integraldisplay/braceleftBigg/bracketleftbig
jα(x′µ)(x−x′)β−jβ(x′µ)(x−x′)α/bracketrightbig
[(x−x′)σ(x−x′)σ]2/bracerightBigg
d4x′, /B4/BD/BH/B5/DB/CW/CT/D6/CT xα, x′α/CP/D6/CT /D8/CW/CT /D4 /D3/D7/CX/D8/CX/D3/D2 /BG/B9/DA /CT
/D8/D3/D6/D7 /D3/CU /D8/CW/CT /AS/CT/D0/CS /D4 /D3/CX/D2 /D8 /CP/D2/CS /D8/CW/CT /D7/D3/D9/D6
/CT /D4 /D3/CX/D2 /D8 /D6/CT/D7/D4 /CT
/D8/CX/DA /CT/D0/DD /B8 k=
1/4πε0. /BT/CU/D8/CT/D6 /D8/D6/CP/D2/D7/CU/D3/D6/D1/CX/D2/CV /CQ /DD /D8/CW/CT /C4 /CC /B4/BD/BF/B5 /D8/D3 /D8/CW/CTS′/CU/D6/CP/D1/CT /D3/D2/CT /AS/D2/CS/D7 /D8/CW/CT /D7/CP/D1/CT /CT/D5/D9/CP/D8/CX/D3/D2/D7 /DB/CX/D8/CW/BD/BE/D4/D6/CX/D1/CT/CS /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7 /D6/CT/D4/D0/CP
/CX/D2/CV /D8/CW/CT /D9/D2/D4/D6/CX/D1/CT/CS /D3/D2/CT/D7/B8 /D7/CX/D2
/CT /D8/CW/CT /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2/D7 /D3/CU /CP/D0/D0 /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7 /CX/D2 /B4/BD/BF/B5/CP/D6/CT /AL /D8/CW/CT /CC/CC/BA/C1/D2/D7/D8/CT/CP/CS /D3/CU /D8/D3 /DB /D3/D6/CZ /DB/CX/D8/CWFαβ/B9 /CU/D3/D6/D1 /D9/D0/CP/D8/CX/D3/D2 /D3/D2/CT
/CP/D2 /CT/D5/D9/CX/DA /CP/D0/CT/D2 /D8/D0/DD /D9/D7/CT /D8/CW/CTEα, Bα/B9 /CU/D3/D6/D1 /D9/D0/CP/D8/CX/D3/D2/B8/DB/CW/CX
/CW /CX/D7 /D4/D6/CT/D7/CT/D2 /D8/CT/CS /CX/D2 /CJ/BD℄ /CP/D2/CS /CJ/BF℄/BA /C1/D8 /CX/D7 /D7/CW/D3 /DB/D2 /D8/CW/CT/D6/CT /D8/CW/CP/D8 /CX/D2 /D8/CW/CT /AH/CC/CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH /D3/D2/CT /CW/CP/D7 /D8/D3 /D9/D7/CT/D8/CW/CT /BG/B9/DA /CT
/D8/D3/D6/D7 Eα/CP/D2/CSBα/CX/D2/D7/D8/CT/CP/CS /D3/CU /D8/CW/CT /D9/D7/D9/CP/D0 /BF/B9/DA /CT
/D8/D3/D6/D7 E /CP/D2/CSB. /CC/CW/CT /D9/D7/D9/CP/D0 /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2/D7 /D3/CUE/CP/D2/CSB, /D3/CQ/D8/CP/CX/D2/CT/CS /CQ /DD /D8/CW/CT /CX/CS/CT/D2 /D8/CX/AS
/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT
/D3/D1/D4 /D3/D2/CT/D2 /D8/D7 /D3/CUE /CP/D2/CSB /DB/CX/D8/CW /D8/CW/CT
/D3/D1/D4 /D3/D2/CT/D2 /D8/D7 /D3/CUFαβ/B4Ei=F0i/CP/D2/CSBi=∗F0i/B5/B8 /CP/D6/CT /D7/CW/D3 /DB/D2 /D8/D3 /CQ /CT /D8/CW/CT /BT /CC /D6/CT/CU/CT/D6/D6/CX/D2/CV /D8/D3 /D8/CW/CT /D7/CP/D1/CT /D1/CT/CP/D7/D9/D6/CT/D1/CT/D2 /D8/D7 /CX/D2 /CS/CX/AR/CT/D6/CT/D2 /D8/C1/BY/CA/D7 /CP/D2/CS /D2/D3/D8 /D8/D3 /D8/CW/CT /D7/CP/D1/CT /D5/D9/CP/D2 /D8/CX/D8 /DD /B8 /D7/CT/CT /CJ/BD℄ /CP/D2/CS /CJ/BF ℄/BA /C1/D2 /D8/CW/CP/D8 /DB /CP /DD /CX/D8 /CX/D7 /CU/D3/D9/D2/CS /CX/D2 /CJ/BD℄ /B4/CP/D2/CS /CJ/BF ℄/B5 /D8/CW/CP/D8
/D3/D2 /D8/D6/CP/D6/DD /D8/D3 /D8/CW/CT
/D3/D1/D1/D3/D2 /CQ /CT/D0/CX/CT/CU /D8/CW/CT /D9/D7/D9/CP/D0 /D2/D3/D2
/D3 /DA /CP/D6/CX/CP/D2 /D8 /CU/D3/D6/D1 /D9/D0/CP/D8/CX/D3/D2 /DB/CX/D8/CW /D8/CW/CT /BF/B9/DA /CT
/D8/D3/D6/D7 E /CP/D2/CSB /CX/D7/D2/D3/D8 /CT/D5/D9/CX/DA /CP/D0/CT/D2 /D8 /D8/D3 /D8/CW/CT
/D3 /DA /CP/D6/CX/CP/D2 /D8 /CU/D3/D6/D1 /D9/D0/CP/D8/CX/D3/D2/D7/BA Eα/CP/D2/CSBα/CP/D6/CT /CS/CT/D8/CT/D6/D1/CX/D2/CT/CS /CQ /DD /D8/CW/CT
/D3 /DA /CP/D6/CX/CP/D2 /D8 /C5/CP/DC/DB /CT/D0/D0/CT/D5/D9/CP/D8/CX/D3/D2/D7 /CS/CT/D6/CX/DA /CT/CS /CX/D2 /CJ/BD℄ /B4/CP/D2/CS /CJ/BF℄/B5/B8
∂α(δαβ
µνvµEν) +c∂α(εαβµνBµvν) = −jβ/ε0,
∂α(δαβ
µνvµBν) + (1 /c)∂α(εαβµνvµEν) = 0 , /B4/BD/BI/B5/DB/CW/CT/D6/CT Eα/CP/D2/CSBα/CP/D6/CT /D8/CW/CT /CT/D0/CT
/D8/D6/CX
/CP/D2/CS /D1/CP/CV/D2/CT/D8/CX
/AS/CT/D0/CS /BG/B9/DA /CT
/D8/D3/D6/D7 /D1/CT/CP/D7/D9/D6/CT/CS /CQ /DD /CP /CU/CP/D1/CX/D0/DD /D3/CU /D3/CQ/D7/CT/D6/DA /CT/D6/D7/D1/D3 /DA/CX/D2/CV /DB/CX/D8/CW /BG/B9/DA /CT/D0/D3
/CX/D8 /DD vµ/B8 /CP/D2/CSδαβ
µν=δα
µδβ
ν−δα
νδβ
µ. /BY /D3/D6 /D8/CW/CT /CV/CX/DA /CT/D2 /D7/D3/D9/D6
/CT/D7 jµ/D3/D2/CT
/D3/D9/D0/CS /D7/D3/D0/DA /CT /D8/CW/CT/D7/CT/CT/D5/D9/CP/D8/CX/D3/D2/D7 /CP/D2/CS /AS/D2/CS /D8/CW/CT /CV/CT/D2/CT/D6/CP/D0 /D7/D3/D0/D9/D8/CX/D3/D2/D7 /CU/D3/D6Eα/CP/D2/CSBα./CF /CT /D2/D3/D8/CT /D8/CW/CP/D8 /CX/D8 /CX/D7 /D4 /D3/D7/D7/CX/CQ/D0/CT /D8/D3 /DB/D6/CX/D8/CT /D8/CW/CT /CT/D5/D9/CP/D8/CX/D3/D2/D7 /B4/BD/BI/B5 /CX/D2 /CP /D7/D3/D1/CT/DB/CW/CP/D8 /D7/CX/D1/D4/D0/CT/D6 /CU/D3/D6/D1/B8 /D8/CW/CT
/D3 /DA /CP/D6/CX/CP/D2 /D8/C5/CP /CY/D3/D6/CP/D2/CP /CU/D3/D6/D1/B8 /CX/D2 /D8/D6/D3 /CS/D9
/CX/D2/CV Ψα=Eα−icBα. /CC/CW/CT/D2 /D8/CW/CT
/D3 /DA /CP/D6/CX/CP/D2 /D8 /C5/CP /CY/D3/D6/CP/D2/CP /CU/D3/D6/D1 /D3/CU /C5/CP/DC/DB /CT/D0/D0/B3/D7/CT/D5/D9/CP/D8/CX/D3/D2/D7 /CQ /CT
/D3/D1/CT/D7
(γµ)βα∂µΨα=−jβ/ε0, /B4/BD/BJ/B5/DB/CW/CT/D6/CT /D8/CW/CTγ /B9/D1/CP/D8/D6/CX
/CT/D7 /CP/D6/CT
(γµ)βα=δµβ
ργvρgγ
α+iεµβαγvγ. /B4/BD/BK/B5/C1/D2 /D8/CW/CT
/CP/D7/CT /D8/CW/CP/D8jµ= 0 /BX/D5/BA /B4/BD/BJ/B5 /CQ /CT
/D3/D1/CT/D7 /BW/CX/D6/CP
/B9/D0/CX/CZ /CT /D6/CT/D0/CP/D8/CX/DA/CX/D7/D8/CX
/DB /CP /DA /CT /CT/D5/D9/CP/D8/CX/D3/D2 /CU/D3/D6 /CU/D6/CT/CT /D4/CW/D3/D8/D3/D2/D7
(γµ)βα∂µΨα= 0. /B4/BD/BL/B5/CF /CT /D7/CW/CP/D0/D0 /D2/D3/D8 /CU/D9/D6/D8/CW/CT/D6 /CS/CX/D7
/D9/D7/D7 /D8/CW/CT
/D3 /DA /CP/D6/CX/CP/D2 /D8 /C5/CP /CY/D3/D6/CP/D2/CP /CU/D3/D6/D1 /D9/D0/CP/D8/CX/D3/D2 /D7/CX/D2
/CT /CX/D8 /DB/CX/D0/D0 /CQ /CT /D6/CT/D4 /D3/D6/D8/CT/CS /CT/D0/D7/CT/DB/CW/CT/D6/CT/BA/BF/BA/BF/BAEα/CU/D3/D6 /CP /BV/BV/BV /CX/D2 /D8/CW/CT /AH/CC/CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH/C1/D2/D7/D8/CT/CP/CS /D3/CU /D8/D3 /D7/D3/D0/DA /CT /B4/BD/BI/B5 /D3/D6 /B4/BD/BJ/B5 /D8/D3 /AS/D2/CSEα/CP/D2/CSBα/CU/D3/D6 /CP /BV/BV/BV /DB /CT /AS/D6/D7/D8 /AS/D2/CSFαβ/CU/D6/D3/D1 /B4/BD/BH/B5/CX/D2/D7/CT/D6/D8/CX/D2/CV /CX/D2 /D8/D3 /CX/D8jµ/CU/D6/D3/D1 /B4/BD/BE/B5 /CP/D2/CS /D4 /CT/D6/CU/D3/D6/D1/CX/D2/CV /D8/CW/CT /CX/D2 /D8/CT/CV/D6/CP/D8/CX/D3/D2/BA /CC/CW/CT/D2 /DB /CT /D9/D7/CT /D8/CW/CT /D6/CT/D0/CP/D8/CX/D3/D2/D7 /CV/CX/DA /CT/D2 /CX/D2/CJ/BD ℄ /B4/CP/D2/CS /CJ/BF ℄/B5/B8 /DB/CW/CX
/CW
/D3/D2/D2/CT
/D8 Fαβ/CP/D2/CSEα, Bα/B9
/D3 /DA /CP/D6/CX/CP/D2 /D8 /CU/D3/D6/D1 /D9/D0/CP/D8/CX/D3/D2/D7/B8
Eα= (1/c)Fαβvβ,Bα= (1/c2)∗Fαβvβ. /B4/BE/BC/B5/CC/CW/CT /CX/D2 /DA /CT/D6/D7/CT /D6/CT/D0/CP/D8/CX/D3/D2/D7
/D3/D2/D2/CT
/D8/CX/D2/CV /D8/CW/CTEα, Bα/CP/D2/CSFαβ/B9
/D3 /DA /CP/D6/CX/CP/D2 /D8 /CU/D3/D6/D1 /D9/D0/CP/D8/CX/D3/D2/D7 /CP/D6/CT /CP/D0/D7/D3 /CV/CX/DA /CT/D2 /CX/D2 /CJ/BD ℄/B4/CP/D2/CS /CJ/BF℄/B5 /CP/D2/CS /D8/CW/CT/DD /CP/D6/CT
Fαβ= (1/c)δαβ
µνvµEν+εαβµνBµvν,∗Fαβ=δαβ
µνvµBν+ (1/c)εαβµνvµEν. /B4/BE/BD/B5/CC /CP/CZ/CX/D2/CV /D8/CW/CP/D8 /D8/CW/CT /CU/CP/D1/CX/D0/DD /D3/CU /D3/CQ/D7/CT/D6/DA /CT/D6/D7 /DB/CW/D3 /D1/CT/CP/D7/D9/D6/CT/D7 Eα/CX/D7 /CP/D8 /D6/CT/D7/D8 /CX/D2 /D8/CW/CT /CB /CU/D6/CP/D1/CT/B8 /CX/BA/CT/BA/B8 /D8/CW/CP/D8vµ= (−c,0)/D3/D2/CT /AS/D2/CS/D7 /CU/D6/D3/D1 /B4/BE/BC/B5 /D8/CW/CP/D8E0= 0, Ei=F0i/DB/CW/CT/D2
/CT
E1= 0, E2= 2k(1−γ)ρ0y(y2+z2)−1, E3= 2k(1−γ)ρ0z(y2+z2)−1. /B4/BE/BE/B5/CC/CW/CT /CT /D5/D9/CP/D8/CX/D3/D2 /B4/BE/BE/B5 /D7/CW/D3/DB/D7 /D8/CW/CP/D8 /D8/CW/CT /D3/CQ/D7/CT/D6/DA/CT/D6 /DB/CW/D3 /CX/D7 /CP/D8 /D6 /CT/D7/D8 /D6 /CT/D0/CP/D8/CX/DA/CT /D8/D3 /CP /DB/CX/D6 /CT /DB/CX/D8/CW /D7/D8/CT /CP/CS/DD
/D9/D6/D6 /CT/D2/D8 /DB/CX/D0 /D0/D7/CT /CT/B8 /CX/BA/CT/BA/B8 /D1/CT /CP/D7/D9/D6 /CT/B8 /D8/CW/CT /D7/CT
/D3/D2/CS /D3/D6 /CS/CT/D6 /CT/D0/CT
/D8/D6/CX
/AS/CT/D0/CS /D3/D9/D8/D7/CX/CS/CT /D7/D9
/CW /CP /BV/BV/BV /BA /CC/CW /D9/D7 /D8/CW/CT /D6/CT/D7/D9/D0/D8 /DB/CW/CX
/CW /CX/D7/CU/D3/D6 /D7/D9
/CW /AS/CT/D0/CS/D7 /D4/D6/CT/CS/CX
/D8/CT/CS /D3/D2 /CS/CX/AR/CT/D6/CT/D2 /D8 /CV/D6/D3/D9/D2/CS/D7 /CX/D2 /CJ/BH℄ /CX/D7 /D4/D6/D3 /DA /CT/CS /D8/D3 /CQ /CT
/D3/D6/D6/CT
/D8 /CX/D2 /D8/CW/CT /AH/CC/CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH/D8/D3 /D3/B8 /CX/BA/CT/BA/B8 /DB/CW/CT/D2 /CP/D0/D0 /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7 /CP/D6/CT /D8/D6/CT/CP/D8/CT/CS /CX/D2 /CP
/D3 /DA /CP/D6/CX/CP/D2 /D8 /D1/CP/D2/D2/CT/D6/B8 /D7/CT/CT /D8/CW/CT /CS/CX/D7
/D9/D7/D7/CX/D3/D2 /CP/D8 /D8/CW/CT /CT/D2/CS /D3/CU/CB/CT
/BA/BF/BA/BD/BA /CF /CT /D8/CW /D9/D7 /AS/D2/CS /D8/CW/CP/D8 /D8/CW/CT/D7/CT /CT/D0/CT
/D8/D6/CX
/AS/CT/D0/CS/D7 /D2/CP/D8/D9/D6/CP/D0/D0/DD
/D3/D1/CT /D3/D9/D8 /CX/D2 /D8/CW/CT /AH/CC/CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH /D8/D6/CT/CP/D8/D1/CT/D2 /D8/D3/CU /D4/CW /DD/D7/CX
/CP/D0 /D7/DD/D7/D8/CT/D1/D7
/D3/D2/D7/CX/D7/D8/CX/D2/CV /D3/CU /D6/CT/D0/CP/D8/CX/DA /CT/D0/DD /D1/D3 /DA/CX/D2/CV /D7/D9/CQ/D7/DD/D7/D8/CT/D1/D7/BA /CB/D9
/CW /AS/CT/D0/CS/D7 /DB /CT/D6/CT /D2/D3/D8 /D7/CT/CP/D6
/CW/CT/CS /CU/D3/D6/CP/D2/CS/B8 /CX/D8 /D7/CT/CT/D1/D7/B8 /DB /CT/D6/CT /D2/D3/D8 /D3/CQ/D7/CT/D6/DA /CT/CS /CT/CP/D6/D0/CX/CT/D6 /CS/D9/CT /D8/D3 /D8/CW/CT/CX/D6 /CT/DC/D8/D6/CT/D1/CT /D7/D1/CP/D0/D0/D2/CT/D7/D7/BA /B4/CC /D3 /CQ /CT /D1/D3/D6/CT /D4/D6/CT
/CX/D7/CT/B8 /D8/CW/CT/BD/BF/D7/CX/D1/CX/D0/CP/D6 /D7/CT
/D3/D2/CS/B9/D3/D6/CS/CT/D6 /CT/D0/CT
/D8/D6/CX
/AS/CT/D0/CS/D7 /B4∝v2/c2/B5 /CW/CP /DA /CT /CQ /CT/CT/D2 /CS/CT/D8/CT
/D8/CT/CS /CX/D2 /CJ/BD/BK ℄ /CP/D2/CS /CJ/BD/BL ℄/B8 /CQ/D9/D8 /CX/D8 /CX/D7 /D2/D3/D8 /D7/D9/D6/CT/D8/CW/CP/D8 /D8/CW/CT/DD /CP/D6/CT
/CP/D9/D7/CT/CS /CQ /DD /D8/CW/CT /CT/AR/CT
/D8 /D4/D6/CT/CS/CX
/D8/CT/CS /CX/D2 /CJ/BH℄ /CP/D2/CS /CW/CT/D6/CT/BA/B5 /C0/D3 /DB /CT/DA /CT/D6/B8 /C1 /D7/D9/D4/D4 /D3/D7/CT /D8/CW/CP/D8 /D7/D9
/CW /AS/CT/D0/CS/D7/D1 /D9/D7/D8 /D4/D0/CP /DD /CP/D2 /CX/D1/D4 /D3/D6/D8/CP/D2 /D8 /D6/D3/D0/CT /CX/D2 /D1/CP/D2 /DD /D4/CW /DD/D7/CX
/CP/D0 /D4/CW/CT/D2/D3/D1/CT/D2/CP /DB/CX/D8/CW /D7/D8/CT/CP/CS/DD
/D9/D6/D6/CT/D2 /D8/D7/B8 /D4/CP/D6/D8/CX
/D9/D0/CP/D6/D0/DD /CX/D2/D8/D3/CZ /CP/D1/CP/CZ/D7 /CP/D2/CS /CP/D7/D8/D6/D3/D4/CW /DD/D7/CX
/D7/B8 /DB/CW/CT/D6/CT /CW/CX/CV/CW
/D9/D6/D6/CT/D2 /D8/D7 /CT/DC/CX/D7/D8/B8 /CP/D2/CS /CX/D2 /D7/D9/D4 /CT/D6
/D3/D2/CS/D9
/D8/D3/D6/D7/B8 /DB/CW/CT/D6/CT /D8/CW/CT /CT/D0/CT
/D8/D6/CX
/AS/CT/D0/CS/D7 /D3/CU /DE/CT/D6/D3/D8/CW /D3/D6/CS/CT/D6 /D3/D9/D8/D7/CX/CS/CT /BV/BV/BV/D7 /CP/D6/CT /CP/CQ/D7/CT/D2 /D8/B8 /B4/D7/CT/CT /CJ/BE/BC ℄/B5/BA/CB/CX/D1/CX/D0/CP/D6/D0/DD /B8 /D8/CW/CT /D1/CP/CV/D2/CT/D8/CX
/AS/CT/D0/CS /BG/B9/DA /CT
/D8/D3/D6 Bµ
/CP/D2 /CQ /CT /CP/D0/D7/D3 /D3/CQ/D8/CP/CX/D2/CT/CS /CU/D6/D3/D1 /B4/BE/BC/B5 /CP/D2/CS /D8/CW/CT /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2 /CU/D3/D6
Fαβ/B4/BD/BH/B5/BA /BY /D3/D6 /D8/CW/CT /D3/CQ/D7/CT/D6/DA /CT/D6/D7 /DB/CX/D8/CWvµ= (−c,0) /D3/D2/CT /AS/D2/CS/D7B0= 0 /CP/D2/CSBi= (−1/c)∗Fi0. /C1/D2 /D8/CT/D6/D1/D7/D3/CU /D8/CW/CT /CZ/D2/D3 /DB/D2∗Fi0/DB /CT /AS/D2/CS /CU/D3/D6Bi/D8/CW/CT /D9/D7/D9/CP/D0 /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2 /CU/D3/D6 /D8/CW/CT /D1/CP/CV/D2/CT/D8/CX
/AS/CT/D0/CS /D3/CU /CP/D2 /CX/D2/AS/D2/CX/D8/CT /D7/D8/D6/CP/CX/CV/CW /D8/DB/CX/D6/CT /DB/CX/D8/CW
/D9/D6/D6/CT/D2 /D8/B8 /B4/D3/D2/D0/DD /D8/CW/CT
/D9/D6/D6/CT/D2 /D8 /CS/CT/D2/D7/CX/D8 /DD /CX/D7γ /D8/CX/D1/CT/D7 /CQ/CX/CV/CV/CT/D6/B5/BA/BG/BA /BV/C0/BT/CA /BZ/BX/CB /C7/C6 /BT /BV/CD/CA/CA/BX/C6/CC /C4/C7/C7/C8 /C1/C6 /CC/C0/BX /AH/CC/CC /CA/BX/C4/BT /CC/C1/CE/C1/CC/CH/AH/C1/D2 /D8/CW/CX/D7 /D7/CT
/D8/CX/D3/D2 /DB /CT /CS/CX/D7
/D9/D7/D7 /D8/CW/CT /D1/CP
/D6/D3/D7
/D3/D4/CX
/CW/CP/D6/CV/CT /D3/CU /CP /D7/D5/D9/CP/D6/CT /D0/D3 /D3/D4 /DB/CX/D8/CW /CP /D7/D8/CT/CP/CS/DD
/D9/D6/D6/CT/D2 /D8 /CX/D2 /D8/CW/CT/AH/CC/CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH/BA /CC/CW/CX/D7 /CX/D7 /CP/D0/D6/CT/CP/CS/DD /CS/CX/D7
/D9/D7/D7/CT/CS /CX/D2 /D2 /D9/D1/CT/D6/D3/D9/D7 /D4/D6/CT/DA/CX/D3/D9/D7 /DB /D3/D6/CZ/D7 /CQ/D9/D8 /CU/D6/D3/D1 /D8/CW/CT /AH/BT /CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH/DA/CX/CT/DB/D4 /D3/CX/D2 /D8 /CQ /DD /D9/D7/CX/D2/CV /D8/CW/CT /D7/DD/D2
/CW/D6/D3/D2/D3/D9/D7 /CS/CT/AS/D2/CX/D8/CX/D3/D2 /D3/CU /D0/CT/D2/CV/D8/CW /CP/D2/CS /D8/CW/CT /C4/D3/D6/CT/D2 /D8/DE
/D3/D2 /D8/D6/CP
/D8/CX/D3/D2/BA /C4/CT/D8 /CP /D7/D5/D9/CP/D6/CT/D0/D3 /D3/D4 /CX/D7 /CP/D8 /D6/CT/D7/D8 /CX/D2 /D8/CW/CTx, y− /D4/D0/CP/D2/CT /CX/D2 /CP/D2 /C1/BY/CAS /BA /CC/CW/CT /D7/D4/CP/D8/CX/CP/D0 x, y−
/D3 /D3/D6/CS/CX/D2/CP/D8/CT/D7 /D3/CU /D8/CW/CT
/D3/D6/D2/CT/D6/D7 /CP/D6/CT/BM
A(0,0), B(1,0), E(1,1), F(0,1). /CC/CW/CT /CT/D0/CT
/D8/D6/D3/D2/D7 /D1/D3 /DA /CT /CU/D6/D3/D1A /D8/D3B /CX/D2 /D8/CW/CTAB /D7/CX/CS/CT/BA /CF /CT /AS/D6/D7/D8
/D3/D2/D7/CX/CS/CT/D6/D8/CW/CT
/CW/CP/D6/CV/CT /CX/D2 /D8/CW/CTAB /D7/CX/CS/CT /CX/D2 /D8 /DB /D3 /CU/D6/CP/D1/CT/D7/BN /CX/D2S /B8 /D8/CW/CT /CX/D3/D2/D7/B3 /D6/CT/D7/D8 /CU/D6/CP/D1/CT/B8 /CP/D2/CSS′
AB
/B8 /D8/CW/CT /CT/D0/CT
/D8/D6/D3/D2/D7/B3 /D6/CT/D7/D8/CU/D6/CP/D1/CT/BA /CC/CW/CT /CT/DA /CP/D0/D9/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT
/CW/CP/D6/CV/CT /CX/D2 /D8/CW/CTAB /D7/CX/CS/CT /CX/D2S /CP/D2/CSS′
AB
/D9/D7/CX/D2/CV /B4/BD/BC/B5 /CX/D7 /CP/D0/D6/CT/CP/CS/DD
/D3/D6/D6/CT
/D8/D0/DD/D4 /CT/D6/CU/D3/D6/D1/CT/CS /CX/D2 /CJ/BD/BI ℄/BA /C0/D3 /DB /CT/DA /CT/D6 /D8/CW/CT/D6/CT /CX/D7 /CP/D2 /CX/D1/D4 /D3/D6/D8/CP/D2 /D8 /CS/CX/AR/CT/D6/CT/D2
/CT /CQ /CT/D8 /DB /CT/CT/D2 /D8/CW/CT
/CP/D0
/D9/D0/CP/D8/CX/D3/D2 /D3/CUQAB
/CX/D2 /CJ/BD/BI ℄/CP/D2/CS /CX/D2 /D8/CW/CX/D7 /D4/CP/D4 /CT/D6/BA /C1/D2 /CJ/BD/BI ℄ /D8/CW/CT
/CW/CP/D6/CV/CT /CS/CT/D2/D7/CX/D8 /DD /D3/CU /D8/CW/CT /CT/D0/CT
/D8/D6/D3/D2/D7 /CX/D2S′
AB
/CX/D7 /D2/D3/D8 /D7/D4 /CT
/CX/AS/CT/CS /CQ/D9/D8 /D7/CX/D1/D4/D0/DD/D8/CP/CZ /CT/D2 /D8/D3 /CW/CP /DA /CT /D7/D3/D1/CT /D9/D2/CS/CT/D8/CT/D6/D1/CX/D2/CT/CS /DA /CP/D0/D9/CT−λ /CP/D2/CS /CX/D8/D7 /DA /CP/D0/D9/CT /CX/D2S /CX/D7 /CU/D3/D9/D2/CS /CQ /DD /D1/CT/CP/D2/D7 /D3/CU /D8/CW/CT /C4 /CC/BA /CC/CW/CT
/D3 /DA /CP/D6/CX/CP/D2
/CT /D3/CU /D8/CW/CT /CS/CT/AS/D2/CX/D8/CX/D3/D2 /B4/BD/BC/B5 /B4/D8/CW/CT /C4/D3/D6/CT/D2 /D8/DE /D7
/CP/D0/CP/D6/B5 /DB/CX/D0/D0 /DD/CX/CT/D0/CS /D8/CW/CP/D8QAB
/CX/D2S /CX/D7 /CT/D5/D9/CP/D0 /D8/D3Q′
AB
/CX/D2
S′
AB
/CU/D3/D6 /CP/D2 /DD
/CW/D3/CX
/CT /D3/CU−λ /BA /C7/D9/D6 /CS/CX/D7
/D9/D7/D7/CX/D3/D2 /D3/CU /CP/D2 /CX/D2/AS/D2/CX/D8/CT /DB/CX/D6/CT /DB/CX/D8/CW /CP
/D9/D6/D6/CT/D2 /D8 /D6/CT/DA /CT/CP/D0/D7 /D8/CW/CP/D8ρ′
−
/CP/D2/CS
j′µ
−
/CX/D2S′
AB
/CP/D6/CT /CS/CT/D8/CT/D6/D1/CX/D2/CT/CS /CQ /DD /B4/BD/BD/B5/BA /CC/CW/CT/D2 jµ
−
/CX/D2S /CX/D7 /D3/CQ/D8/CP/CX/D2/CT/CS /CQ /DD /D8/CW/CT /C4 /CC/B8 /CP/D2/CS /CX/D8 /CX/D7 /CV/CX/DA /CT/D2 /CQ /DD /B4/BD/BE/B5/BA/C0/CT/D2
/CT/B8 /DB /CT /AS/D2/CS /CU/D6/D3/D1 /B4/BD/BC/B5 /D8/CW/CP/D8
QAB= (1/c)/integraldisplayl
0(j0
++j0
−)dx= (1−γ)ρ0l /B4/BE/BF/B5/CC/CW/CX/D7 /D6/CT/D7/D9/D0/D8 /CX/D7 /CP/D0/D6/CT/CP/CS/DD /CU/D3/D9/D2/CS /CX/D2 /CJ/BH℄/BA /CB/CX/D1/CX/D0/CP/D6/D0/DD /B8 /D8/D3 /AS/D2/CSQ′
AB
/CX/D2S′
AB
/D3/D2/CT /AS/D6/D7/D8 /CS/CT/D8/CT/D6/D1/CX/D2/CT/D7 jµ
+
/CX/D2S,
jµ
+= (cρ0,0), /CP/D2/CS /D8/CW/CT/D2 /CQ /DD /D8/CW/CT /C4 /CC /D3/D2/CT /D3/CQ/D8/CP/CX/D2/D7 j′µ
+= (cγρ0,−cγβρ 0) /CX/D2S′
AB
/BAQ′
AB
/B8 /CU/D3/D6 /D8/CW/CT /D1/D3 /DA/CX/D2/CV/D0/D3 /D3/D4 /CX/D2S′
AB
/B8 /CX/D7 /CU/D3/D9/D2/CS /CU/D6/D3/D1 /B4/BD/BC/B5 /CX/D2 /D8/CW/CT /D7/CP/D1/CT /DB /CP /DD /CP/D7 /CX/D2 /CJ/BD/BI ℄ /CP/D2/CS /CX/D8 /CX/D7/B8 /D3/CU
/D3/D9/D6/D7/CT/B8 /CT/D5/D9/CP/D0 /D8/D3QAB
/BA /CC/CW/CT/CT/DA /CP/D0/D9/CP/D8/CX/D3/D2 /D3/CUQEF
/CX/D2 /D8/CW/CTEF /D7/CX/CS/CT/B8 /DB/CW/CX
/CW /CX/D7 /D4/CP/D6/CP/D0/D0/CT/D0 /D8/D3 /D8/CW/CTAB /D7/CX/CS/CT/B8 /D4/D6/D3
/CT/CT/CS/D7 /CX/D2 /D8/CW/CT /D7/CP/D1/CT /DB /CP /DD /CP/D7/CU/D3/D6QAB
/BA /C0/D3 /DB /CT/DA /CT/D6 /D8/CW/CT /D6/CT/D0/CP/D8/CX/DA /CT /DA /CT/D0/D3
/CX/D8 /DD /D3/CU /D8/CW/CTS′
EF
/CU/D6/CP/D1/CT/B8 /D8/CW/CT /D6/CT/D7/D8 /CU/D6/CP/D1/CT /CU/D3/D6 /D8/CW/CT /CT/D0/CT
/D8/D6/D3/D2/D7 /CX/D2 /D8/CW/CT
EF /D7/CX/CS/CT/B8 /CP/D2/CS /D8/CW/CTS /CU/D6/CP/D1/CT /CX/D7 /D2/D3 /DBvµ= (γc,−γv) /BA /C0/CT/D2
/CT/B8 jµ
−
/CX/D7 /D2/D3 /DBjµ
−= (−cγρ0, cγβρ 0) /CP/D2/CS
QEF= (1/c)/integraldisplay0
l(j0
++j0
−)dx=−QAB. /B4/BE/BG/B5/CC/CW /D9/D7 /DB /CT /AS/D2/CS /D8/CW/CP/D8 /D8/CW/CT/D6/CT /CP/D6/CT
/CW/CP/D6/CV/CT/D7 QAB
/CP/D2/CS−QAB
/D3/D2 /D8/CW/CT /D7/CX/CS/CT/D7AB /CP/D2/CSEF /B8 /D6/CT/D7/D4 /CT
/D8/CX/DA /CT/D0/DD /B8 /CX/D2/D8/CW/CTS /CU/D6/CP/D1/CT /CX/D2 /DB/CW/CX
/CW /D8/CW/CT /D0/D3 /D3/D4 /CX/D7 /CP/D8 /D6/CT/D7/D8/BA /BT
/D3/D6/CS/CX/D2/CV /D8/D3 /B4/BD/BC/B5QEF
/CX/D7 /CP/D2 /CX/D2 /DA /CP/D6/CX/CP/D2 /D8
/CW/CP/D6/CV/CT/B8 /CW/CT/D2
/CT
Q′
EF
/CU/D3/D6 /D8/CW/CT /D1/D3 /DA/CX/D2/CV /D0/D3 /D3/D4 /CX/D7=QEF=−QAB
/BA /C5/D3/D6/CT/D3 /DA /CT/D6/B8 /CX/D8
/CP/D2 /CQ /CT /CX/D1/D1/CT/CS/CX/CP/D8/CT/D0/DD
/D3/D2
/D0/D9/CS/CT/CS /D8/CW/CP/D8/D8/CW/CT
/CW/CP/D6/CV/CT QBE
/B8 /CX/D2S /B8 /CX/D2 /D8/CW/CT /DA /CT/D6/D8/CX
/CP/D0 /D7/CX/CS/CTBE /D1 /D9/D7/D8 /CQ /CT /D8/CW/CT /D7/CP/D1/CT /CP/D7QAB
/BN /CX/D8 /CX/D7 /D8/CW/CT /D7/CX/D1/D4/D0/CT
/CW/CP/D2/CV/CT/D3/CU /D8/CW/CT /D7/D4/CP/D8/CX/CP/D0 /CP/DC/CT/D7 /DC /CP/D2/CS /DD /BA /CC/CW /D9/D7 QBE=QAB
/BA /CB/CX/D1/CX/D0/CP/D6/D0/DD /D8/CW/CT
/CW/CP/D6/CV/CT QFA
/B8 /CX/D2S /B8 /CX/D2 /D8/CW/CT /DA /CT/D6/D8/CX
/CP/D0/D7/CX/CS/CTFA /CX/D7QFA=−QAB
/BA /CC/CW/CT /D8/D3/D8/CP/D0
/CW/CP/D6/CV/CT /CX/D2S /B8 /D8/CW/CT /D6/CT/D7/D8 /CU/D6/CP/D1/CT /D3/CU /D8/CW/CT /D0/D3 /D3/D4 /DB/CX/D8/CW
/D9/D6/D6/CT/D2 /D8/B8 /CX/D7 /DE/CT/D6/D3
Q=QAB+QBE+QEF+QFA= 0 /B8 /CP/D7 /CX/D8 /D1 /D9/D7/D8 /CQ /CT/BA /C1/D8 /D6/CT/D1/CP/CX/D2/D7 /DE/CT/D6/D3 /CX/D2 /CT/DA /CT/D6/DD /C1/BY/CA /D7/CX/D2
/CT /D8/CW/CT
/CW/CP/D6/CV/CT/D7/CX/D2 /CP/D0/D0 /D7/CX/CS/CT/D7 /CP/D6/CT /CX/D2 /DA /CP/D6/CX/CP/D2 /D8
/CW/CP/D6/CV/CT/D7 /CP
/D3/D6/CS/CX/D2/CV /D8/D3 /B4/BD/BC/B5/BA /CC/CW /D9/D7/B8 /DB /CT /AS/D2/CS /D8/CW/CT /D7/CP/D1/CT /CQ /CT/CW/CP /DA/CX/D3/D9/D6 /CU/D3/D6 /CP /D1/D3 /DA/CX/D2/CV/D0/D3 /D3/D4 /DB/CX/D8/CW
/D9/D6/D6/CT/D2 /D8 /CP/D2/CS /CU/D3/D6 /D8/CW/CT /D7/CP/D1/CT /D0/D3 /D3/D4 /CQ/D9/D8 /CP/D8 /D6/CT/D7/D8 /CX/D2 /CP /CV/CX/DA /CT/D2 /C1/BY/CA/BA/C1/D2 /D8/CW/CT /D8/D6/CP/CS/CX/D8/CX/D3/D2/CP/D0 /AH/BT /CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH /CP/D4/D4/D6/D3/CP
/CW /DB/CX/D8/CW /D8/CW/CT /D7/DD/D2
/CW/D6/D3/D2/D3/D9/D7 /CS/CT/AS/D2/CX/D8/CX/D3/D2 /D3/CU /D0/CT/D2/CV/D8/CW /CP/D2/CS /D8/CW/CT/C4/D3/D6/CT/D2 /D8/DE
/D3/D2 /D8/D6/CP
/D8/CX/D3/D2 /B4/D7/CT/CT/B8 /CT/BA/CV/BA/B8 /CJ/BE/BD ℄/B5 /D8/CW/CT
/CW/CP/D6/CV/CT/D7 /D3/D2 /CP/D0/D0 /D7/CX/CS/CT/D7 /CP/D6/CT /D7/D9/D4/D4 /D3/D7/CT/CS /D8/D3 /CQ /CT /DE/CT/D6/D3 /CX/D2 /D8/CW/CT /D6/CT/D7/D8/CU/D6/CP/D1/CT /D3/CU /CP /D0/D3 /D3/D4 /DB/CX/D8/CW
/D9/D6/D6/CT/D2 /D8/BA /C1/D2 /D8/CW/CT /C1/BY/CA /CX/D2 /DB/CW/CX
/CW /D8/CW/CT /D0/D3 /D3/D4 /DB/CX/D8/CW
/D9/D6/D6/CT/D2 /D8 /CX/D7 /D1/D3 /DA/CX/D2/CV /CX/D8 /CX/D7 /CU/D3/D9/D2/CS/D8/CW/CP/D8Q′
AB∝ne}ationslash=QAB, /CS/D9/CT /D8/D3 /D8/CW/CT /C4/D3/D6/CT/D2 /D8/DE
/D3/D2 /D8/D6/CP
/D8/CX/D3/D2/B8 /CP/D2/CS /D8/CW/CP/D8 /D8/CW/CT
/CW/CP/D6/CV/CT /D3/D2 /D8/CW/CTEF /D7/CX/CS/CT /CX/D7 /AH/B9/AH /D3/CU/D8/CW/CP/D8 /D3/D2/CT /D3/D2 /D8/CW/CTAB /D7/CX/CS/CT/BA /BY /D9/D6/D8/CW/CT/D6/D1/D3/D6/CT/B8 /CX/D8 /CX/D7 /D3/CQ/D8/CP/CX/D2/CT/CS /D8/CW/CP/D8 /D8/CW/CT
/CW/CP/D6/CV/CT/D7 /D3/D2 /D8/CW/CT /DA /CT/D6/D8/CX
/CP/D0 /D7/CX/CS/CT/D7 /D3/CU/CP /D0/D3 /D3/D4 /DB/CX/D8/CW
/D9/D6/D6/CT/D2 /D8 /CP/D6/CT /DE/CT/D6/D3 /CU/D3/D6 /CQ /D3/D8/CW /D8/CW/CT /D0/D3 /D3/D4 /CP/D8 /D6/CT/D7/D8 /CP/D2/CS /D1/D3 /DA/CX/D2/CV /D0/D3 /D3/D4/B8 /D7/CX/D2
/CT /CU/D3/D6 /DA /CT/D6/D8/CX
/CP/D0 /D7/CX/CS/CT/D7/BD/BG/D8/CW/CT/D6/CT /CX/D7 /D2/D3 /C4/D3/D6/CT/D2 /D8/DE
/D3/D2 /D8/D6/CP
/D8/CX/D3/D2/BA /CB/D9
/CW /D6/CT/D7/D9/D0/D8/D7 /D3/CQ/D8/CP/CX/D2/CT/CS /CX/D2 /D8/CW/CT
/D3/D1/D1/D3/D2 /CP/D4/D4/D6/D3/CP
/CW /D0/CT/CS /D8/CW/CT /D4/CW /DD/D7/CX
/D7
/D3/D1/D1 /D9/D2/CX/D8 /DD /B4/C1 /CP/D1 /D2/D3/D8 /CP /DB /CP/D6/CT /D3/CU /CP/D2 /DD /CT/DC
/CT/D4/D8/CX/D3/D2/B5 /D8/D3
/D3/D2
/D0/D9/CS/CT /D8/CW/CP/D8 /D8/CW/CT/D6/CT /CX/D7 /CP/D2 /CT/D0/CT
/D8/D6/CX
/D1/D3/D1/CT/D2 /D8 P /CU/D3/D6/CP /D1/D3 /DA/CX/D2/CV /D0/D3 /D3/D4 /DB/CX/D8/CW
/D9/D6/D6/CT/D2 /D8/B8 /B4/D7/CT/CT /CJ/BE/BD℄/B8 /BX/D5/BA/B4/BD/BK/B9/BH/BK/B5/B5/BA /CC/CW/CT /CP/D4/D4 /CT/CP/D6/CP/D2
/CT /D3/CU /D8/CW/CX/D7 /AH/D6/CT/D0/CP/D8/CX/DA/CX/D7/D8/CX
/AH /CT/AR/CT
/D8 /CP/D2/CS/CX/D8/D7
/D3/D2/D7/CT/D5/D9/CT/D2
/CT/D7 /CP/D6/CT /CS/CX/D7
/D9/D7/D7/CT/CS /CX/D2 /D2 /D9/D1/CT/D6/D3/D9/D7 /D4/CP/D4 /CT/D6/D7 /CP/D2/CS /CQ /D3 /D3/CZ/D7/BA /BV/D3/D2 /D8/D6/CP/D6/DD /D8/D3 /CP/D0/D0 /D8/CW/CT/D7/CT /DB /D3/D6/CZ/D7 /DB /CT /AS/D2/CS/CX/D2 /D8/CW/CT /AH/CC/CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH /D8/CW/CP/D8 /CP/D8 /D0/D3/D2/CV /CS/CX/D7/D8/CP/D2
/CT/D7 /CP /D0/D3 /D3/D4 /DB/CX/D8/CW
/D9/D6/D6/CT/D2 /D8 /CP/D0/DB /CP /DD/D7 /B4/CX/D2 /CP/D2 /DD /C1/BY/CA/B5 /CQ /CT/CW/CP /DA /CT/D7 /CP/D7 /CP/D5/D9/CP/CS/D6/D9/D4 /D3/D0/CT/BA /CC/CW/CT /CT/DC/CX/D7/D8/CT/D2
/CT /D3/CU /CP /D5/D9/CP/CS/D6/D9/D4 /D3/D0/CT /CT/D0/CT
/D8/D6/CX
/D1/D3/D1/CT/D2 /D8 /CX/D2/D7/D8/CT/CP/CS /D3/CU /CS/CX/D4 /D3/D0/CT /CT/D0/CT
/D8/D6/CX
/D1/D3/D1/CT/D2 /D8 /DB/CX/D0/D0/CX/D2/AT/D9/CT/D2
/CT /D1/CP/D2 /DD /D4/D6/CT/DA/CX/D3/D9/D7 /D6/CT/D7/D9/D0/D8/D7 /CU/D6/D3/D1 /D8/CW/CT
/D0/CP/D7/D7/CX
/CP/D0 /CT/D0/CT
/D8/D6/D3 /CS/DD/D2/CP/D1/CX
/D7 /D9/D4 /D8/D3 /D8/CW/CT /CX/D2 /D8/CT/D4/D6/CT/D8/CP/D8/CX/D3/D2 /D3/CU /CP/D2/CT/D0/CT
/D8/D6/CX
/D1/D3/D1/CT/D2 /D8 /CU/D3/D6 /CP /D1/D3 /DA/CX/D2/CV /D2/CT/D9/D8/D6/D3/D2/BA/BH/BA /BV/C7/C6/BV/C4/CD/CB/C1/C7/C6/CB/C1/D2 /D8/CW/CX/D7 /D4/CP/D4 /CT/D6 /D8/CW/CT /AH/CC/CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH /CX/D7
/D3/D2/D7/CX/D7/D8/CT/D2 /D8/D0/DD /CP/D4/D4/D0/CX/CT/CS /D8/D3 /D8/CW/CT /CX/D2 /DA /CT/D7/D8/CX/CV/CP/D8/CX/D3/D2 /D3/CU /D6/CT/D0/CP/D8/CX/DA /CT/D0/DD /D1/D3 /DA/CX/D2/CV/D7/DD/D7/D8/CT/D1/D7/BA /CF /CT /D4/D6/CT/D7/CT/D2 /D8/CT/CS /D8/CW/CT /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2/D7 /CU/D3/D6 /D8/CW/CT
/D3 /DA /CP/D6/CX/CP/D2 /D8/D0/DD /CS/CT/AS/D2/CT/CS /D7/D4/CP
/CT/D8/CX/D1/CT /D0/CT/D2/CV/D8/CW /B4/BD/B5 /CP/D2/CS /CU/D3/D6
/D3/B9/DA /CP/D6/CX/CP/D2 /D8 /BG/BW /C4/D3/D6/CT/D2 /D8/DE /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2/D7 /B4/BE/B5 /DB/CW/CT/D2 /CQ /D3/D8/CW /CP/D6/CT /DB/D6/CX/D8/D8/CT/D2 /CX/D2 /CV/CT/D3/D1/CT/D8/D6/CX
/CP/D0 /D8/CT/D6/D1/D7/B8 /CP/D2/CS /CX/D2 /D8/CW/CT /AH/CT/AH/CP/D2/CS /AH/D6/AH
/D3 /D3/D6/CS/CX/D2/CP/D8/CX/DE/CP/D8/CX/D3/D2/D7/BA /C1/D8 /CX/D7 /CP/D0/D7/D3 /D7/CW/D3 /DB/D2 /D8/CW/CP/D8 /CX/D8 /CX/D7 /D2/D3/D8 /D8/CW/CT /D7/D4/CP/D8/CX/CP/D0 /D0/CT/D2/CV/D8/CW /CQ/D9/D8 /D8/CW/CT /D7/D4/CP
/CT/D8/CX/D1/CT /D0/CT/D2/CV/D8/CW/B4/BD/B5 /DB/CW/CX
/CW /CX/D7 /DB /CT/D0/D0 /CS/CT/AS/D2/CT/CS /D5/D9/CP/D2 /D8/CX/D8 /DD /CU/D6/D3/D1 /D8/CW/CT /AH/CC/CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH /DA/CX/CT/DB/D4 /D3/CX/D2 /D8/BN /CX/D2 /D8/CW/CT /D6/CT/D7/D8 /CU/D6/CP/D1/CT /D3/CU /D8/CW/CT /D3/CQ /CY/CT
/D8/CP/D2/CS /DB/CW/CT/D2 /D8/CW/CT /D8/CT/D1/D4 /D3/D6/CP/D0 /D4/CP/D6/D8 /D3/CUla
AB
/CX/D7 /DE/CT/D6/D3 /D8/CW/CT /D7/D4/CP
/CT/D8/CX/D1/CT /D0/CT/D2/CV/D8/CW l /CQ /CT
/D3/D1/CT/D7 /D8/CW/CT /D6/CT/D7/D8 /D7/D4/CP/D8/CX/CP/D0 /D0/CT/D2/CV/D8/CW /D3/CU/D8/CW/CT /D3/CQ /CY/CT
/D8/BA /CD/D7/CX/D2/CV /D8/CW/CX/D7 /D6/CT/D7/D9/D0/D8 /CP/D2/CS /D8/CW/CT
/D3 /DA /CP/D6/CX/CP/D2 /D8 /CS/CT/AS/D2/CX/D8/CX/D3/D2 /D3/CU
/CW/CP/D6/CV/CT /B4/BD/BC/B5 /D8/CW/CT /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2/D7 /B4/BD/BE/B5 /CU/D3/D6 /D8/CW/CT
/D9/D6/D6/CT/D2 /D8 /CS/CT/D2/D7/CX/D8 /DD /BG/B9/DA /CT
/D8/D3/D6/D7 /D3/CU /CP /BV/BV/BV /DB /CT/D6/CT /CU/D3/D9/D2/CS /CX/D2 /D8/CW/CT /CX/D3/D2/D7/B3 /D6/CT/D7/D8 /CU/D6/CP/D1/CT/BA /CC/CW/CT/D2 /D8/CW/CT /BG/B9/DA /CT
/D8/D3/D6/D7 Eα/CP/D2/CS
Bα/CU/D3/D6 /CP /BV/BV/BV /CP/D6/CT /CS/CT/D8/CT/D6/D1/CX/D2/CT/CS /CQ /DD /D1/CT/CP/D2/D7 /D3/CU /D8/CW/CT /CZ/D2/D3 /DB/D2 Fαβ/B4/BD/BH/B5 /CP/D2/CS /D8/CW/CT /D6/CT/D0/CP/D8/CX/D3/D2/D7 /B4/BE/BC/B5 /CP/D2/CS /B4/BE/BD/B5/B8/DB/CW/CX
/CW
/D3/D2/D2/CT
/D8 /D8/CW/CTFαb/CP/D2/CSEα, Bα/CU/D3/D6/D1 /D9/D0/CP/D8/CX/D3/D2/D7 /D3/CU /CT/D0/CT
/D8/D6/D3 /CS/DD/D2/CP/D1/CX
/D7/BA /CC/CW/CX/D7 /DD/CX/CT/D0/CS/D7 /BX/D5/BA /B4/BE/BE/B5/B8 /DB/CW/CX
/CW/CX/D7 /D3/D2/CT /D3/CU /D8/CW/CT /D1/CP/CX/D2 /D6/CT/D7/D9/D0/D8/D7 /CU/D3/D9/D2/CS /CX/D2 /D8/CW/CX/D7 /D4/CP/D4 /CT/D6/BA /C1/D8 /D7/CW/D3 /DB/D7 /D8/CW/CP/D8 /CU/D3/D6 /D8/CW/CT /D3/CQ/D7/CT/DA /CT/D6/D7 /CP/D8 /D6/CT/D7/D8 /CX/D2 /D8/CW/CT /CX/D3/D2/D7/B3/D6/CT/D7/D8 /CU/D6/CP/D1/CT /D8/CW/CT /D7/D4/CP/D8/CX/CP/D0
/D3/D1/D4 /D3/D2/CT/D2 /D8/D7 Ei/D3/CUEα/CP/D6/CT /CS/CX/AR/CT/D6/CT/D2 /D8 /CU/D6/D3/D1 /DE/CT/D6/D3 /D3/D9/D8/D7/CX/CS/CT /CP /BV/BV/BV/BA /CC/CW/CT /D7/CT
/D3/D2/CS/CX/D1/D4 /D3/D6/D8/CP/D2 /D8 /D6/CT/D7/D9/D0/D8/B8 /DB/CW/CX
/CW /CX/D7 /CP/D0/D7/D3 /CU/D3/D9/D2/CS /CX/D2 /CP
/D3/D1/D4/D0/CT/D8/CT/D0/DD
/D3 /DA /CP/D6/CX/CP/D2 /D8 /D1/CP/D2/D2/CT/D6/B8 /CX/BA/CT/BA/B8 /CX/D2 /D8/CW/CT /AH/CC/CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH/D8/D6/CT/CP/D8/D1/CT/D2 /D8/B8 /CX/D7 /D8/CW/CT /CT/DC/CX/D7/D8/CT/D2
/CT /D3/CU /CX/D2 /DA /CP/D6/CX/CP/D2 /D8
/CW/CP/D6/CV/CT/D7 /D3/D2 /CP /D0/D3 /D3/D4 /DB/CX/D8/CW
/D9/D6/D6/CT/D2 /D8/BA /CC/CW/CT/D6/CT /CP/D6/CT /D3/D4/D4 /D3/D7/CX/D8/CT
/CW/CP/D6/CV/CT/D7/D3/D2 /D3/D4/D4 /D3/D7/CX/D8/CT /D7/CX/CS/CT/D7 /D3/CU /CP /D7/D5/D9/CP/D6/CT /D0/D3 /D3/D4 /DB/CX/D8/CW
/D9/D6/D6/CT/D2 /D8/B8 /CQ/D9/D8 /D8/CW/CT /D8/D3/D8/CP/D0
/CW/CP/D6/CV/CT /D3/CU /D8/CW/CP/D8 /D0/D3 /D3/D4 /CX/D7 /DE/CT/D6/D3/BA /CC/CW/CT/D7/CT/D6/CT/D7/D9/D0/D8/D7 /CP/D6/CT /CS/CX/AR/CT/D6/CT/D2 /D8 /CU/D6/D3/D1 /D8/CW/D3/D7/CT /CU/D3/D9/D2/CS /CX/D2 /CP/D0/D0 /D4/D6/CT/DA/CX/D3/D9/D7 /DB /D3/D6/CZ/D7 /CX/D2 /DB/CW/CX
/CW /D1/CP/CX/D2/D0/DD /D8/CW/CT /AH/BT /CC /D6/CT/D0/CP/D8/CX/DA/CX/D8 /DD/AH /CX/D7/D9/D7/CT/CS/BA/BT /BV/C3/C6/C7 /CF/C4/BX/BW/BZ/BX/C5/BX/C6/CC/CB/C1 /CP/D1 /CX/D2/CS/CT/CQ/D8/CT/CS /D8/D3 /C8/D6/D3/CU/BA /BY/BA /CA/D3/CW/D6/D0/CX
/CW /CU/D3/D6 /D6/CT/CP/CS/CX/D2/CV /D8/CW/CT /AS/D6/D7/D8 /DA /CT/D6/D7/CX/D3/D2 /D3/CU /D8/CW/CX/D7 /D4/CP/D4 /CT/D6 /CP/D2/CS /CU/D3/D6 /D9/D7/CT/CU/D9/D0/D7/D9/CV/CV/CT/D7/D8/CX/D3/D2/D7 /CP/D2/CS /CT/D2
/D3/D9/D6/CP/CV/CT/D1/CT/D2 /D8/B8 /CP/D2/CS /CP/D0/D7/D3 /D8/D3 /CP/D2 /CP/D2/D3/D2 /DD/D1/D3/D9/D7 /D6/CT/CU/CT/D6/CT/CT /CU/D3/D6 /D9/D7/CT/CU/D9/D0
/D3/D1/D1/CT/D2 /D8/D7/BA/CA/CT/CU/CT/D6/CT/D2
/CT/D7/CJ/BD℄ /CC/BA/C1/DA /CT/DE/CX/EI/B8 /BY /D3/D9/D2/CS/BA /C8/CW/DD/D7/BA /C4 /CT/D8/D8 /BA /BD/BE /B8 /BD/BC/BH /B4/BD/BL/BL/BL/B5/BA/CJ/BE℄ /BY/BA /CA/D3/CW/D6/D0/CX
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arXiv:physics/0102015v1 [physics.comp-ph] 6 Feb 2001Quantum Monte Carlo Method for Attractive Coulomb Potentia ls
J.S. Kole and H. De Raedt
Institute for Theoretical Physics and Materials Science Ce ntre,
University of Groningen, Nijenborgh 4, NL-9747 AG Groninge n, The Netherlands
(DRAFT: February 2, 2008)
Starting from an exact lower bound on the imaginary-time pro pagator, we present a Path-
Integral Quantum Monte Carlo method that can handle singula r attractive potentials. We illustrate
the basic ideas of this Quantum Monte Carlo algorithm by simu lating the ground state of hydrogen
and helium.
PACS numbers: 05.10.-a, 05.30-d, 05.10.Ln
I. INTRODUCTION
Quantum Monte Carlo (QMC) simulation is a powerful method fo r computing the ground state and non-zero
temperature properties of quantum many-body systems [1,2] . There are two fundamental problems that limit the
application of these methods. The first and most important is the minus-sign problem on which we have nothing to
say in this paper, see however [3,4]. The second problem aris es if one would like to simulate systems with attractive
singular potentials, the Coulomb interaction being the pri me example. The purpose of this paper is to present an
approach that solves the latter problem, in a form that fits ra ther naturally in the standard Path Integral QMC
(PIQMC) approach and leaves a lot of room for further systema tic improvements.
Let us first recapitulate the basic steps of the procedure to s et up a PIQMC simulation. Writing KandVfor the
kinetic and potential energy respectively the first step is t o approximate the imaginary-time propagator by a product
of short-time imaginary-time propagators. The standard ap proach is to invoke the Trotter-Suzuki formula [5,6]
e−β(K+V)= lim
m→∞/parenleftBig
e−βK/me−βV/m )/parenrightBigm
, (1)
to construct a sequence of systematic approximations Zmto the partition function Z[6,7]:
Z=Trexp(−βH) = lim
m→∞Zm (2)
Zm=/integraldisplay
dr1· · ·drmm/productdisplay
n=1/an}b∇acketle{trn|e−βK/m|rn+1/an}b∇acket∇i}hte−βV(rn+1)/m, (3)
where rm+1=r1and use has been made of the fact that the potential energy is d iagonal in the coordinate represen-
tation. Taking the limit m→ ∞, (3) yields the Feynman path integral [9] for a system with Ha miltonian H=K+V.
Expression (3) is the starting point for PIQMC simulation.
In the case the attractive Coulomb interaction, it is easy to see why the standard PIQMC approach fails. Let us
take the hydrogen atom as an example. The Hamiltonian reads
H=−¯h2
2M∇2−q2
r, (4)
where qdenotes the charge of the electron and M=me/(1 +me/mp),me(mp) being the mass of the electron
(proton). Replacing the imaginary-time free-particle pro pagator in (3) by its explicit, exact expression
/an}b∇acketle{tr|e−βK/m|r′/an}b∇acket∇i}ht=/parenleftbiggmM
2πβ¯h2/parenrightbigg3/2
exp/parenleftbigg
−mM|x−x′|2
2β¯h2/parenrightbigg
, (5)
we obtain
Zm=/parenleftbiggmM
2πβ¯h2/parenrightbigg3m/2/integraldisplay
dr1· · ·drmexp/bracketleftBigg
−mM
2β¯h2m/summationdisplay
n=1(rn−rn+1)2/bracketrightBigg
exp/bracketleftBigg
+βq2
mm/summationdisplay
n=11
rn/bracketrightBigg
. (6)
PIQMC calculates ratio of integrals such as (6) by using a Mon te Carlo procedure to generate the coordinates
{r1, . . . , r m}. The integrand in (6) serves as the weight for the importance sampling process. As the latter tends to
1maximize the integrand, it is clear that because of the facto rs exp/parenleftbig
+βq2m−1r−1
n/parenrightbig
, the points {r1, . . . , r m}will, after
a few steps, end up very close to the origin. In the case of a sin gular, attractive potential importance sampling based
on (6) fails. Using instead of the simplest Trotter-Suzuki f ormula (1) a more sophisticated one [8] only makes things
worse because these hybrid product formulae contain deriva tives of the potential with respect to the coordinates.
The problem encountered in setting up a PIQMC scheme for mode ls with a singular, attractive potential is just a
signature of the fundamental difficulties that arise when one tries to define the Feynman path integral for the hydrogen
atom [10]. The formal solution to this problem is known [10,1 1]. It is rather complicated and not easy to incorporate
in a PIQMC simulation.
In spirit the method proposed in this paper is similar to the o ne used to solve the hydrogen path integral: Use
the quantum fluctuations to smear out the singularity of the p otential. Mathematically we implement this idea by
applying Jensen’s inequality to the propagator [12]. Appli cations of the Feynman path-integral formalism are often
based on a combination of Jensen’s inequality and a variatio nal approach [9,10] so it is not a surprise that similar
tricks may work for PIQMC as well.
The paper is organized as follows. In Section 2 we give a simpl e derivation of an exact lower bound on the
imaginary-time propagator. This inequality naturally defi nes a sequence of systematic approximations ˆZmto the
partition function. Although each ˆZmlooks very similar to Zm, the former can be used for PIQMC with attractive,
singular potentials. For pedagogical reasons, in Section 3 we illustrate the approach by presenting an analytical
treatment of the harmonic oscillator. In Section 4 we give th e explicit form of the approximate propagator for the
attractive Coulomb potential and present PIQMC results for the ground state of the hydrogen and helium atom.
II. LOWER BOUND ON THE PROPAGATOR
Consider a system with Hamiltonian H=K+Vand a complete set of states {|x/an}b∇acket∇i}ht}that diagonalizes the hermitian
operator V. In the case that Vcontains a singular attractive part we replace V= lim ǫ→0Vǫby a regular Vǫ(x)>−∞
and take the limit ǫ→0 at the end of the calculation. Using the Trotter-Suzuki for mula we can write
/an}b∇acketle{tx|e−τ(K+Vǫ)|x′/an}b∇acket∇i}ht= lim
m→∞/an}b∇acketle{tx|/parenleftBig
e−τK/me−τVǫ/m/parenrightBigm
|x′/an}b∇acket∇i}ht, (7)
= lim
m→∞/integraldisplay
dx1· · ·dxnm/productdisplay
i=1/an}b∇acketle{txi|e−τK/m|xi+1/an}b∇acket∇i}hte−τVǫ(xi)/m, (8)
= lim
m→∞/integraltextdx1· · ·dxn/producttextm
i=1/an}b∇acketle{txi|e−τK/m|xi+1/an}b∇acket∇i}hte−τVǫ(xi)/m
/integraltext
dx1· · ·dxn/producttextm
i=1/an}b∇acketle{txi|e−τK/m|xi+1/an}b∇acket∇i}ht/integraldisplay
dx1· · ·dxnm/productdisplay
i=1/an}b∇acketle{txi|e−τK/m|xi+1/an}b∇acket∇i}ht.(9)
If/an}b∇acketle{tx|e−τK|x′/an}b∇acket∇i}ht ≥0 for all τ,xandx′, the function
ρ({xi}) =m/productdisplay
i=1/an}b∇acketle{txi|e−τK/m|xi+1/an}b∇acket∇i}ht/slashBigg/integraldisplay
dx1· · ·dxnm/productdisplay
i=1/an}b∇acketle{txi|e−τK/m|xi+1/an}b∇acket∇i}ht, (10)
is a proper probability density. Clearly (10) is of the form/integraltextdx1· · ·dxnρ({xi})f({xi}) so that we can apply Jensen’s
inequality
/integraldisplay
dx1· · ·dxnρ({xi})eg({xi})≥exp/parenleftbigg/integraldisplay
dx1· · ·dxnρ({xi})g({xi})/parenrightbigg
, (11)
and obtain
/an}b∇acketle{tx|e−τ(K+Vǫ)|x′/an}b∇acket∇i}ht ≥ /an}b∇acketle{t x|e−τK|x′/an}b∇acket∇i}htlim
m→∞exp/parenleftBigg
−τ
mm/summationdisplay
i=1/integraldisplay
dx1· · ·dxmVǫ(xi)/producttextm
n=1/an}b∇acketle{txn|eτK/m|xn+1/an}b∇acket∇i}ht
/an}b∇acketle{tx|e−τK|x′/an}b∇acket∇i}ht/parenrightBigg
, (12)
≥ /an}b∇acketle{tx|e−τK|x′/an}b∇acket∇i}htlim
m→∞exp/parenleftBigg
−τ
mm/summationdisplay
i=1/integraldisplay
dxi/an}b∇acketle{tx|eτK/m|xi/an}b∇acket∇i}htVǫ(xi)/an}b∇acketle{txi|eτK/m|x′/an}b∇acket∇i}ht
/an}b∇acketle{tx|e−τK|x′/an}b∇acket∇i}ht/parenrightBigg
. (13)
Form→ ∞, the sum over ncan be replaced by an integral over imaginary time. Finally w e letǫ→0 and obtain [12]
2/an}b∇acketle{tx|e−τ(K+V)|x′/an}b∇acket∇i}ht ≥ /an}b∇acketle{t x|e−τK|x′/an}b∇acket∇i}htexp/braceleftbigg
−/integraldisplayτ
0du/an}b∇acketle{tx|e−uKV e−(τ−u)K|x′/an}b∇acket∇i}ht
/an}b∇acketle{tx|e−τK|x′/an}b∇acket∇i}ht/bracerightbigg
. (14)
Note that l.h.s of (14) reduces to the standard, symmetrized Trotter-Suzuki formula approximation [13,14] if we
replace the integral over uby a two-point trapezium-rule approximation. This replace ment also changes the direction
of inequality as can been seen directly from the upperbound [ 12]
/an}b∇acketle{tx|e−τ(K+V)|x′/an}b∇acket∇i}ht ≤ /an}b∇acketle{t x|e−τK|x′/an}b∇acket∇i}htexp/braceleftbigg
−/integraldisplayτ
0duln/parenleftbigg/an}b∇acketle{tx|e−uKe−τVe−(τ−u)K|x′/an}b∇acket∇i}ht
/an}b∇acketle{tx|e−τK|x′/an}b∇acket∇i}ht/parenrightbigg/bracerightbigg
≤ /an}b∇acketle{tx|e−τK|x′/an}b∇acket∇i}hte−τV(x).(15)
Expression (14) can be used to define a new type of approximant to the partition function namely
ˆZm=/parenleftbiggM
2πτ¯h2/parenrightbigg3m/2/integraldisplay
dr1. . . dr mm/productdisplay
n=1exp/bracketleftbigg
−M
2τ¯h2(rn−rn+1)2−/integraldisplayτ
0du/an}b∇acketle{trn|e−uKV e−(τ−u)K|rn+1/an}b∇acket∇i}ht
/an}b∇acketle{trn|e−τK|rn+1/an}b∇acket∇i}ht/bracketrightbigg
. (16)
where τ=β/m. The simplest approximant ˆZ1corresponds to the Feynman’s variational approximation to the full
Feynman path integral [9,10]. The main difference between (3 ) and (16) is that the bare potential e−τV(x)is replaced
by an effective potential that is obtained by convoluting the bare potential and free-particle propagators e−uKand
e−(τ−u)K. Convolution smears out singularities. As we show below, in the case of the attractive Coulomb interaction
expression (14) is finite, for any choice of xandx′. For the approximants ˆZmto be useful in PIQMC, it is necessary
that the integral over ucan be done efficiently. In the next two sections we show how thi s can be done.
III. ILLUSTRATIVE EXAMPLE
It is instructive to have at least one example for which the de tails can be worked out analytically, without actually
using PIQMC. Not surprisingly this program can be carried ou t for the harmonic oscillator. For notational convenience
we will consider the one-dimensional model Hamiltonian H=K+V, with K=−(¯h2/2M)d2/dx2andV=Mω2x2.
Calculating the matrix element /an}b∇acketle{tx|e−uKV e−(τ−u)K|x′/an}b∇acket∇i}htin (16) is a straightforward excercise in perfoming Gaussia n
integrals [15]. We obtain
ˆZm=/parenleftbiggmM
2πβ¯h2/parenrightbiggm/2/integraldisplay
dx1. . . dx mm/productdisplay
n=1exp/bracketleftbigg
−mM
2β¯h2(xn−xn+1)2−βMω2
6m(x2
n+x2
n+1+xnxn+1+β¯h2
2mM)/bracketrightbigg
.(17)
The integrand in (17) is a quadratic form and can be diagonali zed by a Fourier transformation with respect to the
index n. Evaluation of the resulting Gaussian integrals yields
ˆZm= 2−m/2exp/parenleftbigg
−β2¯h2ω2
12m/parenrightbiggm−1/productdisplay
n=0/bracketleftbigg
1 +β2¯h2ω2
3m−/parenleftbigg
1−β2¯h2ω2
6m/parenrightbigg
cos/parenleftbigg2πn
m/parenrightbigg/bracketrightbigg−1/2
. (18)
Taking the partial derivative of −lnˆZmwith respect to βgives the corresponding approximation to the energy:
ˆEm=β¯h2ω2
6m/bracketleftBigg
1 +m−1/summationdisplay
n=02 + cos(2 πn/m )
1−cos(2πn/m ) +β2¯h2ω2(2 + cos(2 πn/m ))/6m/bracketrightBigg
. (19)
For comparison, if we use of the standard Trotter-Suzuki for mula we obtain [7]
Em=β¯h2ω2
2m2m−1/summationdisplay
n=01
1−cos(2πn/m ) +β2¯h2ω2/2m2(20)
In Table 1 we present numerical results obtained from (19) an d (20) and compare with the exact value of the energy
E= (¯hω/2)coth( β¯hω/2)). Note that the average of the two approximations, i.e. ( ˆEm+Em)/2, is remarkably close
to the exact value E, an observation for which we have no mathematical justificat ion at this time.
3TABLE I. Numerical results for the exact energy of the harmon ic oscillator ( E), and approximations based on (19) ( ˆEm)
and (20) ( Em). We use units such that ¯ hω= 1 and βis dimensionless.
β m Em E ˆEm
1 1 1.00000 1.08198 1.16668
10 1.08101 1.08198 1.08292
50 1.08194 1.08198 1.08202
100 1.08197 1.08198 1.08199
500 1.08198 1.08198 1.08198
5 1 0.20000 0.50678 1.03333
10 0.49199 0.50678 0.51938
50 0.50617 0.50678 0.50694
100 0.50678 0.50678 0.50679
500 0.50678 0.50678 0.50679
10 1 0.10000 0.50005 1.76667
10 0.44273 0.50005 0.54316
50 0.49757 0.50005 0.50234
100 0.49942 0.50005 0.50064
500 0.50002 0.50005 0.50007
4IV. ATTRACTIVE COULOMB POTENTIAL
As a second example we will consider a neutral system consist ing of two electrons with opposite spin and a nucleus.
The Hamiltonian reads [16,17]
H=−¯h2
2M1∇2
1−¯h2
2M2∇2
2−q2
|r1|−q2
|r2|+2q2
|r1−r2|, (21)
where the vectors r1andr2describe the position of the two electrons, with the nucleus placed in the origin. It is
convenient to introduce the notation Ki=−Di∇2
i,Di= ¯h2/2Mi,Vi=V(ri),V12=V(r1−r2), and V(r) =q2/|r|,
fori= 1,2. Application of inequality (14) requires the evaluation o f
I(r1, r2, r′
1, r′
2) =−/integraltextτ
0du/an}b∇acketle{tr1r2|e−u(K1+K2)(V1+V2−2V12)e−(τ−u)(K1+K2)|r′
1r′
2/an}b∇acket∇i}ht
/an}b∇acketle{tr1r2|e−β(K1+K2)|r′
1r′
2/an}b∇acket∇i}ht(22)
=−/integraltextτ
0du/an}b∇acketle{tr1|e−uK1V1e−(τ−u)K1|r′
1/an}b∇acket∇i}ht
/an}b∇acketle{tr1|e−τK1)|r′
1/an}b∇acket∇i}ht−/integraltextτ
0du/an}b∇acketle{tr2|e−uK2V2e−(τ−u)K2|r′
2/an}b∇acket∇i}ht
/an}b∇acketle{tr2|e−τK2|r′
2/an}b∇acket∇i}ht
+2/integraltextτ
0du/an}b∇acketle{tr1r2|e−u(K1+K2)V12e−(τ−u)(K1+K2)|r′
1r′
2/an}b∇acket∇i}ht
/an}b∇acketle{tr1r2|e−τ(K1+K2)|r′
1r′
2/an}b∇acket∇i}ht,
where we made use of the fact that [ K1, V2] = [K2, V1] = 0. It is sufficient to consider the last term of (22). Inserti ng
a complete set of states for both particles we obtain
I12(r1, r2, r′
1, r′
2) =/integraltextτ
0du/integraltext
dr′′
1/integraltext
dr′′
2/an}b∇acketle{tr1r2|e−u(K1+K2)|r′′
1r′′
2/an}b∇acket∇i}htV(r′′
1−r′′
2)/an}b∇acketle{tr′′
1r′′
2|e−(τ−u)(K1+K2)|r′
1r′
2/an}b∇acket∇i}ht
/an}b∇acketle{tr1r2|e−τ(K1+K2)|r′
1r′
2/an}b∇acket∇i}ht. (23)
Inserting the explicit expression for the free-particle pr opagator (5), a straightforward manipulation of the Gaussi an
integrals in (23) gives
I12(r1, r2, r′
1, r′
2, D) =/integraldisplayτ
0du/integraldisplay
dr/parenleftbiggτ
4πu(τ−u)D/parenrightbigg3/2
V(r)exp/braceleftbigg
−[τr−(τ−u)(r1−r2)−u(r′
1−r′
2)]2
4uτ(τ−u)D/bracerightbigg
,(24)
where D=D1+D2
In the case of the Coulomb potential, the integral over rcan be evaluated analytically by changing to spherical
coordinates. The remaining integral over uis calculated numerically. In practice, it is expedient to r eplace the
integration over uby an integration over an angle. An expression that is adequa te for numerical purposes is
I12(r1, r2, r′
1, r′
2, D) = 2τq2/integraldisplayπ/2
0dφerf/bracketleftbig
(4τD)−1/2|(r1−r2)tanφ+ (r′
1−r′
2)cotφ|/bracketrightbig
|(r1−r2)tanφ+ (r′
1−r′
2)cotφ|. (25)
It is easy to check that I12(r1, r2, r′
1, r′
2, D) is finite. The expressions for the first and second contribut ions in (22) can
be obtained from (25) by putting ( D2,r2,r′
2) and ( D1,r1,r′
1) equal to zero, i.e. I1(r1, r′
1, D1) =I12(r1,0, r′
1,0, D1)
andI2(r2, r′
2, D2) =I12(0, r2,0, r′
2, D2).
For the helium atom M=M1=M2, and the m-th approximant to the partition function reads
ˆZHe
m=/parenleftbiggM
2πτ¯h2/parenrightbigg3m/integraldisplay
dr1. . . dr mdr′
1. . .dr′
mexp/braceleftBigg
−M
2τ¯h2m/summationdisplay
n=1/bracketleftbig
(rn−rn+1)2+ (r′
n−r′
n+1)2/bracketrightbig/bracerightBigg
(26)
×exp/braceleftBigg
τm/summationdisplay
n=1/bracketleftBig
I1(rn, rn+1, D1) +I2(r′
n, r′
n+1, D1)−2I12(rn, rn+1, r′
n, r′
n+1,2D1)/bracketrightBig/bracerightBigg
,
whereas in the case of the hydrogen atom we have
5ˆZH
m=/parenleftbiggM
2πτ¯h2/parenrightbigg3m/2/integraldisplay
dr1. . . dr mexp/braceleftBigg
−M
2τ¯h2m/summationdisplay
n=1(rn−rn+1)2+τm/summationdisplay
n=1I1(rn, rn+1, D1)/bracerightBigg
, (27)
withτ=β/m. As the integrands in (26) and (27) are always finite, express ions (26) and (27) can be used perform
PIQMC simulations.
In the path integral formalism the ground state energy is obt ained by letting β→ ∞ andβ/m →0, i.e. E=
limβ→∞limβ/m→0ˆEm. Of course, in numerical work, taking one or both these limit s is impossible. In Tables 2
and 3 we present numerical results of PIQMC estimates of the g round state energy Eof the hydrogen and helium
atom. These results have been obtained from five statistical ly independent simulations of 100000 Monte Carlo steps
per degree of freedom each. The systematic errors due to the d iscretization of the path integral are hidden in the
statistical noise. The PIQMC procedure we have used is stand ard [1,7] except for a trick we have used to improve
the efficiency of sampling the paths, details of which are give n in the appendix. Although a ground state calculation
pushes the PIQMC method to point of becoming rather inefficien t, the numerical results are in satisfactory agreement
with the known values.
V. DISCUSSION
We have show that is possible to perform PIQMC simulations fo r quantum systems with attractive Coulomb
potentials. Instead of the conventional Trotter-Suzuki fo rmula approach one can use (16) to construct a path integral
that is free of singularities. In practice, a numerical calc ulation of the latter requires only minor modifications of a
standard PIQMC code.
The efficiency of the PIQMC method describe above can be improv ed with relatively modest efforts. Instead of using
the free-particle propagator K, we are free to pick any other model Hamiltonian H0for which the matrix elements of
e−τH0are positive and integrals involving these matrix elements are known analytically. An obvious choice would be
to take for H0a set of harmonic oscillators. The matrix elements of e−τH0are Gaussians and hence the conditions
used to derive (14) are satisfied. If necessary the approxima ntˆZmcan be improved further by optimization of the
parameters of the oscillators. For m= 1 this approach is identical to the variational method prop osed by Feynman
and Kleinert [18–21] and independently by Giachetti and Tog netti [22,23]. Extending the PIQMC method in this
direction is left for future research.
TABLE II. Path-integral Quantum Monte Carlo results for the ground state energy of the hydrogen Hamiltonian, in units
ofq2/a0(a0= ¯h2/Mq2). The exact value is E=−0.5.
β m ˆEH
m
20 400 -0.496 ( ±0.004)
20 800 -0.503 ( ±0.005)
40 800 -0.498 ( ±0.006)
TABLE III. Path-integral Quantum Monte Carlo results for th e ground state energy of the helium Hamiltonian, in units of
q2/a0. The experimental value is E=−2.904.
β m ˆEHe
m
10 400 -2.84 (±0.02)
10 800 -2.88 (±0.02)
10 1200 -2.92 (±0.03)
6APPENDIX
In PIQMC the simplest mehod for sampling paths is to change on e degree of freedom at each Monte Carlo step.
Usually this is rather inefficient and one adds Monte Carlo mov es that make global changes of the path, e.g. moves that
resembles the classical motion. In this appendix we present a more sophisticated scheme which we found performed
very well at very low temperature. The basic idea is to change variables such that the kinetic energy term in the path
integral becomes a diagonal quadratic form, i.e.
m/summationdisplay
k=1(xk−xk+1)2=m/summationdisplay
k=2y2
k, (28)
where xm+1=x1. After some straightforward algebra one finds that the trans formation from the {xi}to the {yi}is
given by
y2
k=m−k+ 2
m−k+ 1/parenleftbigg
xk−(m−k+ 1)xk−1+xm+1
m−k+ 2/parenrightbigg2
. (29)
The expression for xkin terms of the {ui}reads
xk=y1+k/summationdisplay
j=2m−k+ 1
m−j+ 1/parenleftbiggm−j+ 1
m−j+ 2/parenrightbigg1/2
yj,1< k≤m, (30)
withx1=y1. From (30) we conclude that the computational work for makin g a global change of the path (i.e.
simultaneously changing all yi) is linear in m, hence optimal. It is also clear that the variable y1plays the role of the
“classical” position. The variables y2, . . . , y mdescribe the quantum fluctuations.
[1] K.E. Schmidt and D.M. Ceperley, in: The Monte Carlo Method in Condensed Matter Physics , ed. K. Binder, Springer,
Berlin, 203 (1992).
[2] H. De Raedt and W. von der Linden, in: The Monte Carlo Method in Condensed Matter Physics , ed. K. Binder, Springer,
Berlin, 249 (1992).
[3] H. De Raedt and M. Frick, Phys. Rep. 231, 107 (1993).
[4] H. De Raedt, W. Fettes, and K. Michielsen, in: “Quantum Monte Carlo Methods in Physics and Chemistry” , eds. M.P.
Nightingale and C.J. Umrigar, NATO-ASI Series, 37 (Kluwer, The Netherlands 1999).
[5] S. Lie and F. Engel, Theorie der Transformationgruppen , Teubner, Leipzig, 1888.
[6] M. Suzuki, S. Miyashita, and A. Kuroda, Prog. Theor. Phys .58, 1377 (1977).
[7] H. De Raedt, and A. Lagendijk, Phys. Rep. 127, 233 7 (1985).
[8] M. Suzuki, Phys. Lett. A 201, 425 (1995).
[9] R.P. Feynman and A.R. Hibbs, Quantum Mechanics and Path Integrals , McGraw-Hill, New York, 1965.
[10] H. Kleinert, Path integrals in Quantum Mechanics, Statistics and Polyme r Physics , World Scientific, London, 1990.
[11] I.H. Duru and H. Kleinert, Phys. Lett. B 84, 30 (1979)
[12] K. Symanzik, J. Math. Phys. 6, 1155 (1965).
[13] H. De Raedt and B. De Raedt, Phys. Rev. A 28, 3575 (1983).
[14] M. Suzuki, J. Math. Phys. 26, 601 (1985).
[15] This is the case for all V(x) that are polynomial in x.
[16] L.I. Schiff, Quantum Mechanics McGraw-Hill, New York, (1968).
[17] G. Baym, Lectures on Quantum Mechanics , W.A. Benjamin, Reading MA, (1969).
[18] R.P. Feynman and H. Kleinert, Phys. Rev. A 34, 5080 (1986).
[19] H. Kleinert, Phys. Lett. B 181, 324 (1986).
[20] H. Kleinert, Phys. Lett. A 118, 195 (1986).
[21] W. Janke and B.K. Chang, Phys. Lett. B 129, 140 (1988).
[22] R. Giachetti and V. Tognetti, Phys. Rev. Lett. 55, 912 (1985).
[23] R. Giachetti, V. Tognetti, and R. Vaia, Physica Scripta 40, 451 (1989).
7 |
EXACT ENUMERATION OF TWO -DIMENSIONAL CLOSED RANDOM
PATHS USING A DSP PROCESSOR
B. Afsari†, N. Sadeghi -Meybodi†, S. Rouhani†‡
†School of intelligent Systems, Institute for studies in Theoretical Physics and Mathematics (IPM), Tehran/Iran
‡ Department of Ph ysics, Sharif University of Technology, Tehran/ Iran
ABSTRACT
The aim of this paper is to show that Digital
Signal Processors (DSPs) can be used to
efficiently implement complex algorithms. As an
example we have chosen the problem of
enumerating closed two-dimensional random
paths. An Evaluation Module Board (EVM) for
TMS320C6201 fixed -point processor is used. The
algorithm is implemented in hand -written
parallel assembly language. Some techniques are
used to fit the algorithm to the parallel structure
of the processor and also to avoid Branch and
Condition -Checking tasks. Common optimization
methods are also employed to improve the
execution speed of the code. These methods are
shown to yield a good efficiency in using the
maximum computation power of th e processor.
We use these results to obtain the area
distribution of the paths.
1.INTRODUCTION
Conformations of polymers and proteins are an
interesting and rapidly evolving subject of
research. [1] Some of the question posed such
as designer drugs and polymers are
consequently of great commercial significance.
More fundamental questions such as whether
the native shapes of proteins are accidental, in
the sense that they are results of an evolving
system or that any other shape would have
worked just as well, are hotly debated. [2]
The geometrical attributes of a certain protein
shape may make mechanisms of folding
operate better in the sense that they are better
candidates for the ground state of the complex
inter monomer potential.
A vital tool in analy zing the geometry of the
problem of protein folding is the study of
random walks. When stripped of all its
complex interactions a polymer may be thought
of as a random walk in three dimensions. To
study this problem, analytical tools can be used
to a certa in context, but finally numerical
methods are the only tools viable.
Some efforts on the derivation of distribution
for closed self -avoiding walks have been
successful. Bellisard has shown that the
distribution of areas for closed self -avoiding
random walk s in two dimensions is related to
non-commutative geometry. [3] Using this
connection, he has obtained a distribution that can be iteratively expanded in power of N1,
where N is the length of the walk.
Total enumeration of random walks , or the
distribution of walks with given geometric
properties are among question which are
interesting to the researchers of this field but
with high complexity. The performance of such
tasks takes very long time on the average PC.
Use of Digital Signal P rocessors may help the
situation. The very high rates of process will
lessen the execution time. We have chosen a
specific example in order to test the efficiency
of TMS320C6201 EVM for performing such
calculations. Namely the problem of finding the
distri bution of areas for a closed random path
of fixed length. The advantage being that
• Partial analytic results exist against
which numerical calculation may be
tested.
• The problem is notoriously hard to
perform numerically, due to its very
high complexity.
• Many applications exist, as well as
applications in the polymer
conformations: the distribution of
closed random paths for example is
used in calculation of magneto
resistance and other properties of a
two-dimensional electron system. [4]
DSP processors are similar to general -purpose
microprocessors except that they are more
optimized to perform multiplication and
addition operations. In the core of our
algorithm also there exists a sum of products
for computing the area of paths. In addition the
'C6x’s para llel structure seems to be promising
to handle such high complexity task.
TMS320C6201 is the first device in ‘C6x family of
DSP processors that uses Texas Instruments’ (TI)
Very Long Instruction Word (VLIW) architecture,
called VelociTI [5]. 'C6201 is bas ed on the fixed -
point 32 -bit TMS320C62xx ('C62x) CPU [6]. The
device contains a CPU, a 2K -32bit Internal Program
Memory and Internal Data Memory and other
controlling and interfacing structures. The CPU is
connected to the Internal Program Memory via an 8 -
32bit bus . The CPU contains 8 Functional Units
(ALUs) and 32 -32bit general purpose registers
evenly shared between two Data Paths; Data Path 1
contains the A register file and units indexed by
1(.D1, .L1, .M1, .S1) and Data Path 2 contains the B
register file and units indexed by 2(.D2, .L2, .M2,
.S2). This architecture enables the CPU to fetch 8 instructions per cycle and attain a possible peak
MIPS power of 8 × Clock frequencies.
The instruction set of the C6x is a RISC -like one [7].
Most of the instructi ons are executed in single cycle
and use registers as operands, so programs are
designed on a Store/Load basis. This means that data
should be first transferred to the CPU registers, be
manipulated there by functional units and the result
should be, again, stored in the registers.
2.ALGORITHM
The task then is to enumerate all closed random
walks of length N on a two dimensional square
lattice. Evidently, the result will aid the calculation
of entities such as the area of closed random paths.
The apparent solution is first generating a closed
path, and then calculates its area; repeat this until all
closed paths are exhausted. Developing a suitable
path generation algorithm seems to be difficult. This
problem is a serial one in its nature, and will not be
compatible with parallel architecture of 'C6201
EVM.
Therefore it was decided to generate all random
walks of a given length, followed by checking for
closure. This works because a good proportion of
paths are closed.
An algorithm was devised that concurre ntly
calculates the area partially, and checks each path for
closure (a blind search). This method can effectively
make use of the parallel structure of the processor.
Therefore, the computation of area does not impose
an overload on the checking algorithm and reduces
the memory space needed for keeping necessary data
for calculation of the area upon validation of each
closed path. In other words, by the time a closed path
is recognized and established, the area of that closed
path is ready to be logged.
A random path is a random sequence of four
possible steps( →,←,↑,↓). Each of which can be
represented by two bits. Using this representation we
can store a full path in a register or concatenation of
registers (e.g. A15 or A15 -A14). This register
representati on can also be used easily to generate a
new path: by incrementing by one of the current
value a new path is obtained. For each path all steps
(2 bits per step) are extracted in turn.
Each step is used to calculate (update) 3 parameters:
x(n), y(n), S(n), where (x(n),y(n)) is the coordinate
of the nth node of the path and S(n) is the partial
area:
The algorithm is:
do
{ S(0)=x(0)=y(0)=0
for(n=1;n ≤N;n=n+1)
{
if( Step n == ↑ or ↓ ),
{
x(n) = x(n - 1),
y(n) = y(n - 1) ± 1, ( + for ↑ and for ↓ )
S(n) = S(n – 1) ± x(n)
}/ end of if
if (Step n == → or ←) {
y(n) = y(n - 1),
S(n) = S(n - 1),
x(n)=x(n - 1) ± 1 (+ for → and – for ←).
}/ end of if
} / end of for
if (x(N)==0 & y(N)==0)
{the path is closed and S(N) is its area}
} while (all paths are checked)
An instruction named EXT (Extraction) can extract a
specified bit field in register and stores the sign
extended v ersion of the extracted field in a register.
This instruction is very useful for efficient
implementation. Using this instruction and a suitable
bit allocation for steps we achieve an efficient way
for implementing the updating task. If we chose 01
for ↑ and 11 for ↓ then EXT (extraction) of these
steps results in +1 and -1, respectively. Thus this
eliminates the need to discriminate between adding 1
and subtracting 1 in updating formulas when the step
is in the vertical direction. In other words we can
avoid Branch or another checking (conditional)
instruction. An almost similar way can be used for
the horizontal steps: we chose 00 for → and 10 for
←, but we must logically OR the extracted value
with 1 to yield +1 and –1 respectively.
In order to speed up the calculation a recursive
method can be employed. Each path of length N is
divided into two sections of length M and P. We
shall then calculate x(M), y(M) and S(M) for section
M. We call these the end data of section M. These
values can then be given as input to the above
routine for a path of length P. In this manner
enumeration of a path of length (M + P) is achieved.
At first it appears that this approach does not reduce
the complexity of PM+4 , but there exists many
redundant paths. Furthermore this method allows
one to use the already calculated data for the shorter
paths, to be used in the calculation of longer paths!
In this way computation is highly accelerated.
It must be noted that this can be done due to the
memoryless nature of the computation.
4.WHY WRITING IN PARALLEL
ASSEMBLY?
A program for C6x can be written in C, Linear
Assembly or Parallel Assembly. As far as using
parallel structure of the processor is concerned, the
most efficient method is Parallel Assembly.
Develo ping a large program in Parallel and even
Linear Assembly is a time -consuming task. TI has
introduced an optimizing C compiler [7], which
performs some optimization methods to increase the
efficiency of the output Parallel Assembly code. In
many cases the compiler cannot produce a code as
efficient as hand written Parallel Assembly, so it
may be needed to write some time -critical parts of a
program in Parallel Assembly [8].
In the case of our program two main reasons for
writing in Parallel Assembly are: that first the
algorithm is a Register -level algorithm and uses
hardware directly; second, the program is very short
and it is relatively easy to impose software ∑=−+ =n
iiy iyix nS
1))( )1()(( )(pipelining, Loop Unrolling and completely filling
the delay slots.
5.CONSIDERATIONS FOR
PROGR AM DEVELOPMENT
There are some strategies, that due to parallel
structure of the ‘C6201, make it possible to produce
highly efficient parallel assembly code. Among
them, we have utilized software pipelining, loop
unrolling and filling delay slots [9]. The first two
aim at increasing the efficiency of loops and the
latter maximizes parallelism.
5.1. Software pipelining
Software pipelining is a technique used to schedule
instructions from a loop so that multiple iterations
execute in parallel. The main po int here is that the
parallel resources on the 'C6x make it possible to
initiate loop iteration before previous iteration finish.
The goal of the software pipelining is to start new
loop iterations as soon as possible. We calculated the
average number of i nstructions executed per cycle
and after normalizing it the maximum unit usage per
cycle was found. (Table 1)
5.2. Loop Unrolling
Loop Unrolling is a technique to reduce the
overhead of Branch instruction in a loop. This is
accomplished by replacing all or some iterations of a
loop with individual copies of the loop itself. As an
example, to insert three individual copies of the loop
to replace a loop of 6 iterations in a program to act as
a loop with the iteration of 2. For loops of low
iteration number and of short body, this can result in
great reduction of Branch overhead. Loop Unrolling
increases the code size and reduces the flexibility of
the code.
Due to the capacity of program memory on the
processor we have been able to completely unroll the
loops and therefore avoiding all the Branch
instructions that were necessary for looping.
5.3. Filling Delay Slots
The ultimate measure of efficiency in using the
processor is the number of delay slots, which are
filled with instructions and how much the
instructions are executed in parallel. In serial
algorithms, it is possible to move some instructions
and steps further up in the program primarily to fill
the delay slots, and secondly to induce a parallel
appearance to the program.
We have done this by wr iting parallel hand -written
assembly, but it is also possible to use the assembly
optimizer, which requires the freedom of the
optimizer to assign different units to different tasks.
6.IMPLEMENTATION
There are two loops to be implemented in this
algori thm. First, the inner loop that checks a specified path and calculates its area (this loop is
iterated N times where N is the random walk’s
length.). Second, the outer loop which sweeps all the
possible paths and is iterated N4 times. L oop
Unrolling technique is imposed on the inner loop. As
mentioned the Loop Unrolling technique reduces
software flexibility, accordingly for each path length
N a new program should be developed. Thus for
specified N, Loop Unrolling, Software Pipelining
and Filling Delay Slots are applied.
For avoiding cross -path in using processor’s two
data paths and register files, each data path (path
1&2) works separately on half of the possible
random walk paths. In other words processor’s two
data paths execute the s ame algorithm in parallel, but
on different walk paths.
In parallel assembly implementation we tried to
avoid Branch instructions as much as possible. Only
one Branch instructions is used, i.e. for the outer
loop. Other condition checking are realized by
Conditional Instructions. This increases speed at the
expense of code size.
The Load/Store instructions also should be avoided
as much as possible. For each random walk path one
Store and one Load is used to store the final result of
each random walk path .( If the checked path is
closed a counter in the Data Memory corresponding
to the path’s area is read to a register, its value is
incremented and the new value is stored in the Data
Memory again).That is, no intermediate variable is
load/stored from/in th e data memory.
To use the processor’s complete power, the whole
program and data should be stored on the processors
internal data and program memory.
7.CONCLUSION
We developed programs in 'C6201 hand -written
Parallel Assembly for closure checking and
calculating the area of closed paths in two -
dimensional random walk (on square lattice). Due to
the exponential complexity of the algorithm larger
values of N are not feasible to be handled by EVM,
unless using a recursive method to use calculated
data for the shorter paths, to find the areas for the
longer paths.
Sample of the areas obtained, are logged in an area
histogram for 20,18,16=N (Table 2). We
succeeded in completing the calculation up to N=28.
The results are too lengthy to present he re and are
available on request.
The performance of the developed programs can be
evaluated based on different factors. It took 9 hours
to calculate the areas for N = 28 using EVM board
(where 'C6201 had the clock frequency of
160MHz[11]), where as the sa me calculation on a
Pentium ?? with 233 clock frequency, would have
taken more than 28 years to implement.
The other possibility is to see how efficient is a
program as far as using parallel structure of the
processor is concerned. We can calculate the
average number of instructions executed per cycle,
especially in loops. The closer this number is to
eight; the more efficient is the loop. We can
normalize this figure to the maximum unit usage per
cycle (8 units per cycle). We call the latter in percent as Unit Usage Factor (UUF). UUF is computed for
the outer loop (Table1).
8. REFERENCES
[1].C.Chothia, Nature 357, 543(1992); C.A Orengo,
D.T.Jones , J.M.Thornton, Nature 372, 631(1994)
[2].V.Shahrezaei, N. Hamedani and M.R.
Ejtehadi,Protein Ground State Candidates in a
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[4]G.M Minkov, A.V. Germanenku, S.A. Negashev
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[5]. Nat Sehan, High VelociTI Processing, IEEE
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[6].TMS320C6201/C6701 Peripherals Reference
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[7].TMS320C62X/C67X CPU and Instruction Set,
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Texas Instruments, March 1998.
N
Number of
Clock Cycles
for the outer
loop
Number of
Instructions
in the outer
loop
UUF for
the outer
loop
16 36 232 80.55%
18 39 263 84.3%
20 43 287 83.4%
Table 1. UUF for the outer loop for different N
AREA N= 20 N= 18 N= 16
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
20
120
440
1360
3740
10680
26920
66960
155560
359760
805280
1749200
3699300
7803420
15887160
32101920
63687440
124781340
239017700
452521560
827935484
1469700900
2445203460
3718483560
4920045800
5486681368
0
0
0
0
0
36
144
540
1584
4788
13392
34560
84312
208728
478008
1085724
2398752
5208372
10920456
22761228
45306288
88145244
159762240
263462220
369612648
424925880
0
0
0
0
0
0
0
0
0
16
96
352
1088
3760
10336
28064
73056
184104
435040
1036368
2289760
5015108
10127744
18569808
28133728
33820044
Table 2. Histogram of the number of closed paths with
the corresponding areas, for different v alues of N. Due
to symmetry the histogram is not shown for negative
values of area.
|
arXiv:physics/0102017v1 [physics.atom-ph] 7 Feb 2001Relativistically extended Blanchard recurrence relation
for hydrogenic matrix elements
R. P. Mart´ ınez-y-Romero *
Facultad de Ciencias, Universidad Nacional Aut´ onoma de M´ exico,
Apartado Postal 50-542, M´ exico City, Distrito Federal C. P . 04510.
H. N. N´ u˜ nez-Y´ epez †
Departamento de F´ ısica, Universidad Aut´ onoma Metropoli tana-Iztapalapa
Apartado Postal 55–534, Iztapalapa, Distrito Federal C. P. 09340 M´ exico.
A. L. Salas-Brito ‡
Laboratorio de Sistemas Din´ amicos, Departamento de Cienc ias B´ asicas,
Universidad Aut´ onoma Metropolitana-Azcapotzalco.
Apartado Postal 21–726, Coyoac´ an, Distrito Federal C. P. 0 4000 M´ exico.
Abstract.
General recurrence relations for arbitrary non-diagonal, radial hydrogenic ma-
trix elements are derived in Dirac relativistic quantum mec hanics. Our approach
is based on a generalization of the second hypervirial metho d previously employed
in the non-relativistic Schr¨ odinger case. A relativistic version of the Pasternack-
Sternheimer relation is thence obtained in the diagonal ( i.e.total angular momentum
and parity the same) case, from such relation an expression f or the relativistic virial
theorem is deduced. To contribute to the utility of the relat ions, explicit expressions
for the radial matrix elements of functions of the form rλandβrλ—whereβis a
Dirac matrix— are presented.
Keywords: Relativistic hydrogen atom, recurrence relatio ns, non-diagonal radial ma-
trix elements, relativistic Pasternack-Sternheimer rela tion.
PACS: 3.65.Ca
* E-mail: rodolfo@dirac.fciencias.unam.mx
†E-mail: nyhn@xanum.uam.mx
‡E-mail: asb@correo.azc.uam.mx
1I. Introduction
Recurrence relations for matrix elements are very useful to ols in quantum calcu-
lations [1–5] since the direct computation of such elements is generally very cumber-
some. An interesting example is the Blanchard’s relation wh ich is a useful recurrence
formula for arbitrary ( i.e.not necessarily diagonal) non-relativistic matrix elemen ts
of the form /angbracketleftn1l1|rλ|n2l2/angbracketright, where the |nl/angbracketrightstand for non-relativistic hydrogenic radial
energy eigenstates; according to this relation, once any th ree successive matrix ele-
ments of powers of the radial coordinate, r, are known, any other can be deduced in
terms of the previously known ones. The Blanchard recurrenc e relation was derived
more than twenty five years ago using a calculation-intensiv e method [6], which is
not surprising since in general recurrence relations are ra ther difficult to obtain.
Trying to overcome such difficulties, different approaches ha ve been proposed for
obtaining general recurrence relations; some of them are ba sed on algebraic methods,
others use sum rules and hypervirial theorems [4,7–10]. In p articular, a hypervirial
result has been employed to obtain the Blanchard relation in a more compact way
than in the original deduction [11]. In relativistic quantu m mechanics, on the other
hand, and despite its physical and its possible chemical int erest [7,8,12,13], excepting
for the results reported in [14], there are not as yet general recurrence relations which
could be used for calculating matrix elements of powers of th e radial coordinate in
terms of previously known ones. We should mention though tha t there are previous
efforts in such direction, in which closed forms for certain r elativistic matrix elements
have been evaluated [16,17] and certain, mostly diagonal, r ecurrence relations for
relativistic and quasirelativistic states have been calcu lated [3,9,12,17] .
In this paper we employ a relativistic calculation inspired on the hypervirial
method [11] to deduce a recurrence relation for the, in gener al, non-diagonal radial
matrix elements of succesive powers of rλand ofβrλ—whereβis a 4×4 Dirac matrix
[18–20]— for relativistic hydrogenic states in the energy b asis. The assumptions we
use are that the nucleus is point-like and fixed in space, and t hat a description
using the Dirac equation is valid. We first study the recurren ce relations in the
general case, in which the matrix elements are taken between states with different
principal quantum numbers n1/negationslash=n2,, different total angular momentum quantum
numbersj1/negationslash=j2,mj1/negationslash=mj2, and, as we use the quantum number ǫ≡(−1)j+l−1/2
instead of parity for labelling the hydrogenic eigenstates , whereǫ1/negationslash=ǫ2. We find
that in general the recurrence relations depend on matrix el ements of both powers
ofrand ofβr. In practical terms this means that we need two recurrence re lations
as the relativistic version of the single-equation Blancha rd relation [Eqs. (6) and (7)
of section II]. Given its special interest, we study in parti cular the case where the
total angular momentum and parity become equal, j1=j2andǫ1=ǫ2, in the two
states—not mattering the relative values of the principal q uantum number n. We
2also address the completely diagonal case where n1=n2,j1=j2, andǫ1=ǫ2.
Both of the particular cases mentioned above require specia l treatment for avoiding
possible divisions by zero in the general expressions; such results are immediately
used to obtain a relativistic version of the Pasternack-Ste rnheimer rule [21] and to
obtain an expression for the relativistic virial theorem [9 ,14,22].
This paper is organized as follows. In section II we review th e second hypervirial
scheme used for deriving the non-relativistic Blanchard re lation. In section III, after
obtaining the radial Hamiltonian useful for implementing t he hypervirial result in
relativistic quantum mechanics, we proceed to use it to dedu ce by a long but direct
calculation the relativistic recurrence formulae. In sect ion IV we study in particular
the diagonal case ( j2=j1,ǫ2=ǫ1) to derive the relativistic Pasternak-Sternheimer
rule and use it (when n1=n2) to obtain a version of the relativistic virial theorem.
In the Appendix, we obtain explicit expressions for diagona l and non-diagonal matrix
elements for any power of rand ofβrbetween radial relativistic hydrogenic states.
As it becomes evident, such results are rather cumbersome fo r relatively large values
of the power; for small values, on the other hand, they are bet ter regarded as starting
values for the recurrence relations derived in section III o f this article. Furthermore,
these results can be of utility not only for relativistic ato mic and molecular studies
but also for evaluating matrix elements of interactions des igned to test Lorentz and
CPT invariance in hydrogen [14,29].
II. The non-relativistic recurrence relation
Both the Blanchard relation and its predecesor the Kramers s election rule, were
originally obtained employing directly the Schr¨ odinger e quation together with ap-
propriate boundary conditions and, at least in the former ca se, a great deal of com-
putations [1,6]. A much simpler approach based on a generali zed hypervirial result
and certain Hamiltonian identities has been developed to si mplify the computations
leading to the Blanchard relation [11]. This technique seem ed to us an appropriate
starting point for deriving relativistic recurrence formu lae. It is with such relativis-
tic extension in mind that we review in this section the hyper virial method as it
is applied in non-relativistic quantum mechanics. In this s ection, as in most of the
paper, we employ atomic units ¯ h=m=e= 1.
The idea is to start with the radial Schr¨ odinger equation fo r a central potential
V(r) written in the form
Hk|nklk/angbracketright=Enklk|nklk/angbracketright, (1)
where |nklk/angbracketright=ψnklk(r) andEnklkare an energy eigenfunction and its corresponding
energy eigenvalue with principal and angular momentum quan tum numbers, nkand
lk, respectively; kis just a label, and Hk, the non-relativistic radial Hamiltonian, is
given by
3Hk=−1
2d2
dr2−1
rd
dr+lk(lk+ 1)
2r2+V(r). (2)
Although we want to calculate the radial matrix elements of t erms of the form
rλ, it is best for our purposes to consider first matrix elements of an arbitrary radial
functionf(r). With such choice we can readily show [11] that
(Ei−Ek)/angbracketleftnili|f(r)|nklk/angbracketright=/angbracketleftnili|/parenleftig
−1
2f′′−f′d
dr−1
rf′+∆−
ik
2f
r2/parenrightig
|nklk/angbracketright, (3)
where we use ∆−
ik≡li(li+ 1)−lk(lk+ 1),Ek≡Enklk, and the primes stand
for radial derivatives. Please recall that the matrix eleme nt of an arbitrary radial
functionf(r) is
/angbracketleftnili|f(r)|nklk/angbracketright=/integraldisplay∞
0r2ψ∗
nili(r)f(r)ψnklk(r)dr. (4)
To establish the result we are after, we apply the previous re sult (3) to the radial
functionξ(r)≡Hif(r)−f(r)Hk, to find
2(Ei−Ek)2/angbracketleftnili|f(r)|nklk/angbracketright=
/angbracketleftnili|/parenleftbig
Hi(Hif(r)−f(r)Hk)−(Hif(r)−f(r)Hk)Hk+
Hi(Hif(r)−f(r)Hk)−(Hif(r)−f(r)Hk)Hk/parenrightbig
|nklk/angbracketright.(5)
This is the generalized second hypervirial valid for arbitr ary radial potential energy
functions,V(r), introduced in Eq. (8) of Ref. 11.
The second hypervirial takes a particularly simple form whe nf(r) is a power
of the position, let us say f(r) =rλ+2; using this expression for f(r) and restricting
ourselves to the Coulomb potential, V(r) =−Z/r, we obtain [11], after a long —but
much shorter than in [6]— direct calculation, the Blanchard relation
λ(Ei−Ek)2/angbracketleftnili|rλ+2|nklk/angbracketright=c0/angbracketleftnili|rλ|nklk/angbracketright+c1/angbracketleftnili|rλ−1|nklk/angbracketright
+c2/angbracketleftnili|rλ−2|nklk/angbracketright;(6)
where the hydrogenic energy eigenvalues are Ea=−Z2/2n2
a, independent of l, and
c0=Z2(λ+ 1)/bracketleftbigg
(li−lk)(li+lk+ 1)/parenleftbigg1
n2
i−1
n2
k/parenrightbigg
+λ(λ+ 2)/parenleftbigg1
n2
k+1
n2
i/parenrightbigg/bracketrightbigg
c1=−2Zλ(λ+ 2)(2λ+ 1)
c2=1
2(λ+ 2)/bracketleftbig
λ2−(lk−li)2/bracketrightbig /bracketleftbig
(lk+li+ 1)2−λ2/bracketrightbig
.(7)
4From this result we can also obtain, as special cases of the Bl anchard recurrence
relation (6), first the Pasternack-Sternheimer selection r ule [21]:
/angbracketleftnili|Z
r2|nklk/angbracketright= 0, (8)
saying that the matrix element of the potential 1 /r2vanishes between radial states of
central potentials when their angular momenta coincide and when the corresponding
energy eigenvalues depend on the principal quantum number o nly. Second, in the
completely diagonal case ( i.e.ni=nk,li=lk), we can further obtain the non-
relativistic quantum virial theorem [9]
/angbracketleftV/angbracketright=−Z/angbracketleft1
r/angbracketright= 2/angbracketleftE/angbracketright. (9)
As we exhibit in section IV, we can obtain analogous results u sing our recurrence
relations in relativistic quantum mechanics.
III. The relativistic recurrence relations.
In this section we apply the method sketched in section II to t he relativistic
Dirac case. We clearly need to start with a radial Dirac Hamil tonian analogous to
(2). To obtain such Hamiltonian we start with the Dirac Hamil tonianHDand the
corresponding time-independent Dirac equation for a centr al potential
HD=cα αα·p+βc2+V(r), H DΨ(r) =EΨ(r); (10)
where we are using atomic units, α ααandβare the 4 ×4 Dirac matrices [18–20], which
in the Dirac representation are given by
α αα=/parenleftbigg
0σ σσ
σ σσ0/parenrightbigg
, β =/parenleftbigg
1 0
0−1/parenrightbigg
, (11)
where the 1’s and 0’s stand respectively, for 2 ×2 unit and zero matrices and the
σ σσis the vector composed by the three 2 ×2 Pauli matrices σ σσ= (σx,σy,σz). Please
notice that, despite the selection of natural units we shall , where it aids interpreta-
tion, reinsert the appropriate dimensional factors in cert ain equations. The energy
eigenvalues are given explicitly in Eq. (63) of section V. Th e Hamiltonian HDis
rotationally invariant, hence the solutions of the Dirac eq uation (10) can be written
in the alternative but entirely equivalent forms [19,23]
Ψ(r,θ,φ) =1
r
Fnjǫ(r)Yjmz(θ,φ)
iGnjǫ(r)Y′
jmz(θ,φ)
=1
r
Fnκ(r)χκmz(θ,φ)
iGnκ(r)χ−κmz(θ,φ)
, (12)
whereχκmzandχ−κmz, orYjmandY′
jm, are spinor spherical harmonics of opposite
parity, and κ=−ǫ(j+1/2) is the eigenvalue of the operator Λ ≡β(1 +Σ ΣΣ·L) which
5commutes with HD(where Σ Σ Σ≡σ σσ⊗I= diag(σ σσ,σ σσ)). The second form in (12) is the
preferred in Ref. 18. Parity is a good quantum number in the pr oblem because central
potentials are invariant under reflections; parity varies a s (−1)land, according to
the triangle’s rule of addition of momenta, the orbital angu lar momentum is given
byl=j±1/2. But, instead of working directly with parity or with κ, we prefer the
quantum numbers jandǫ, introduced above, which can be shown also to be
ǫ=
1 Ifl=j+1
2,
−1 Ifl=j−1
2,(13)
thusl=j+ǫ/2 in all cases. We also define l′=j−ǫ/2; accordingly, the spherical
spinor Yjmdepends on lwhereas the spherical spinor Y′
jm, which has the opposite
parity, depends on l′. Writing the solutions in the form (12) completely solves th e
angular part of the problem.
To construct the radial Hamiltonian, we use the relation
(α αα·r)(α αα·p) = (Σ ΣΣ·r)(Σ ΣΣ·p) =r·p+iΣ ΣΣ·L; (14)
we then use J2= [L+ (1/2)Σ ΣΣ]2=L2+ Σ ΣΣ·L+ 3/4 but for expressing the term
L·Σ ΣΣ, we also need an expression for L2acting on the eigenfunctions (12). Directly
from this equation we see that when L2is applied to any central potential state, the
big component of the state function behaves with the orbital quantum number l=
j+ǫ/2, whereas the small one does so with the orbital quantum numb erl′=j−ǫ/2;
we have then,
l(l+ 1) =j(j+ 1) +ǫ(j+1
2) +1
4, (15)
for the big component, and
l′(l′+ 1) =j(j+ 1)−ǫ(j+1
2) +1
4, (16)
for the small one. The action of L2upon a solution of the form (12) is therefore
always of the form
L2=j(j+ 1) +βǫ(j+1
2) +1
4, (17)
whereβis the Dirac matrix (11). From this result we obtain the term L·Σ ΣΣand,
substituting it into ( α αα·p), we finally obtain
(α αα·p) =αr[pr−iβǫ
r(j+1
2)], (18)
where
6αr≡1
rα αα·r, p r=−i
r/parenleftbigg
1 +rd
dr/parenrightbigg
. (19)
We are now ready to write the relativistic radial Hamiltonia n, and the corre-
sponding radial Dirac equation, as
Hk=cαr/bracketleftbigg
pr−iβǫk
r/parenleftbigg
jk+1
2/parenrightbigg/bracketrightbigg
+βc2+V(r),
Hkψk(r) =Ekψk(r),(20)
where we introduced the purely radial eigenfunctions
ψk(r)≡1
r/parenleftbigg
Fnkjkǫk(r)
iGnkjkǫk(r)/parenrightbigg
(21)
in a 2 ×2 representation where, β= diag(+1,−1),αr=/parenleftbigg
0−1
−1 0/parenrightbigg
, and the radial
Dirac equation becomes then [14,19]
c2+ (Vk(r)−Ek) −c(−ǫk(jk+ 1/2)/r+d/dr)
c(ǫk(jk+ 1/2)/r+d/dr) −c2+ (Vk(r)−Ek)
Fnkjkǫk(r)
Gnkjkǫk(r)
= 0.(22)
Though this explicit representation can be used for our prob lem [24,25], it is not
really necessary since all our results are representation i ndependent.
The relativistic recurrence relation we are after, can be de duced using a similar
reasoning as the used in section II for the non-relativistic case. Let us first calculate
the non-diagonal matrix element of an arbitrary radial func tionf(r)
(E2−E1)/angbracketleftn2j2ǫ2|f(r)|n1j1ǫ1/angbracketright=
/angbracketleftn2j2ǫ2|H2f(r)−f(r)H1|n1j1ǫ1/angbracketright
=−ic/angbracketleftn2j2ǫ2|αr/parenleftbigg
f′(r) +∆−
21
2rβf(r)/parenrightbigg
|n1j1ǫ1/angbracketright,(23)
where from now on the labelling in the kets stand for the three quantum numbers
nk,jk, andǫk, we have defined ∆−
21≡ǫ2(2j2+ 1)−ǫ1(2j1+ 1), and the matrix
elements of radial functions are calculated as
/angbracketleftn2j2ǫ2|f(r)|n1j1ǫ1/angbracketright=/integraldisplay
f(r)(F∗
2(r)F1(r) +G∗
2(r)G1(r))dr,
/angbracketleftn2j2ǫ2|βf(r)|n1j1ǫ1/angbracketright=/integraldisplay
f(r)(F∗
2(r)F1(r)−G∗
2(r)G1(r))dr.(24)
7where the subscripts stand for the 3 quantum numbers specify ing the state.
We next proceed to calculate a “second order iteration” by su bstitutingf(r)→
ξ(r) =H2f(r)−f(r)H1in the last expression. Let us calculate first H2ξandξH1,
H2ξ=−c2/parenleftbiggf′(r)
r+f′′(r) +f′(r)d
dr/parenrightbigg
−c2∆−
21
2rβ/parenleftbigg
f′(r) +f(r)d
dr/parenrightbigg
+
c2ǫ2(2j2+ 1)
2rβ/parenleftbigg
f′(r) +∆−
21
2rβf(r)/parenrightbigg
−icαr/parenleftbigg
f′(r) +∆−
21
2rβf(r)/parenrightbigg/parenleftbig
V(r)−βc2/parenrightbig
,(25)
and
ξH1=−c2
r/parenleftbigg
f′(r)−∆−
21
2rβf(r)/parenrightbigg
−c2/parenleftbigg
f′(r)−∆−
21
2rβf(r)/parenrightbiggd
dr+
−c2ǫ1(2j1+ 1)
2rβ/parenleftbigg
f′(r)−∆−
21
2rβf(r)/parenrightbigg
−icαr/parenleftbigg
f′(r) +∆−
21
2rβf(r)/parenrightbigg/parenleftbig
V(r) +βc2/parenrightbig
.
(26)
Then, we write down the difference of the matrix elements asso ciated with Eqs. (25)
and (26)
(E2−E1)2/angbracketleftn2j2ǫ2|f(r)|n1j1ǫ1/angbracketright=
/angbracketleftn2j2ǫ2| −c2∆−
21
2r2βf(r)−c2f′′(r)−c2∆−
21
2rβf′(r)−c2∆−
21
rβf(r)d
dr+
c2∆+
21
2rβf′(r) +c2/parenleftbigg∆−
21
2r/parenrightbigg2
f(r) + 2ic3αrβ/parenleftbigg
f′(r) +∆−
21
2rβf(r)/parenrightbigg
|n1j1ǫ1/angbracketright.(27)
where we have defined ∆+
21≡ǫ2(2j2+ 1) +ǫ1(2j1+ 1). Please notice that here and
in what follows we are always assuming ∆−
21/negationslash= 0.
This last expression (27) is the direct relativistic equiva lent of the generalized
second hypervirial [Cf. Eq. (5) above]. The expression invo lves the operator d/dr,
but here, due to the presence of Dirac matrices in the result, we cannot use the trick
employed in the non relativistic case where we took advantag e of the Hamiltonian
to simplify the calculation [11]. Instead, let us calculate the following second order
iteration for non-diagonal matrix elements
/angbracketleftn2j2ǫ2|H2ξ+ξH1|n1j1ǫ1/angbracketright= (E2
2−E2
1)/angbracketleftn2j2ǫ2|f(r)|n1j1ǫ1/angbracketright=
/angbracketleftn2j2ǫ2|/parenleftbigg
−2c2f′(r)
r+c2∆−
21
2r2βf(r)−c2f′′(r)−2c2f′(r)d
dr+
c2∆+
21∆−
21
4r2f(r)−2icαr/bracketleftig
f′(r) +∆−
21
2rβf(r)/bracketrightig
V(r)/parenrightbigg
|n1j1ǫ1/angbracketright;(28)
8due to the presence of Dirac matrices in our results, we also r equire to calculate
non-diagonal matrix elements for expressions involving αrf(r) andβf(r), namely
H2(−iαrf(r)) =
c/bracketleftbigg
−f(r)
r−f′(r)−f(r)d
dr+ǫ2
2r(2j2+ 1)βf(r)/bracketrightbigg
+ic2αrβf(r)−iαrV(r)f(r),(29)
and
(−iαrf(r))H1=
−cf(r)/bracketleftbigg1
r/parenleftbigg
1 +rd
dr/parenrightbigg
+ǫ1
2r(2j1+ 1)β/bracketrightbigg
−ic2αrβf(r)−iαrV(r)f(r);
(30)
adding up these two last expressions, we get
(E2+E1)/angbracketleftn2j2ǫ2| −iαrf(r)|n1j1ǫ1/angbracketright=
/angbracketleftn2j2ǫ2| −2cf(r)
r−cf′(r)−2cf(r)d
dr+c∆−
21
2rβf(r)−2iαrV(r)f(r)|n1j1ǫ1/angbracketright.(31)
From the matrix element of H2(−iαrcf(r))−(−iαrcf(r))H1, we can obtain
(E2−E1)/angbracketleftn2j2ǫ2| −iαrf(r)|n1j1ǫ1/angbracketright=
/angbracketleftn2j2ǫ2| −cf′(r) +c∆+
21
2rβf(r) + 2c2iαrβf(r)|n1j1ǫ1/angbracketright;(32)
proceeding in a similar way for H2(βf(r)) + (βf(r))H1, we get
(E2+E1)/angbracketleftn2j2ǫ2|βf(r)|n1j1ǫ1/angbracketright=/angbracketleftn2j2ǫ2|icβα rf′(r)−icαr∆−
21
2rf(r)
+ 2/bracketleftbig
c2+βV(r)/bracketrightbig
f(r)|n1j1ǫ1/angbracketright.(33)
Equations (23–33) are the basic equations of our problem. To proceed, we
consider, as in the non-relativistic case, radial function s of the form f(r) =rλand
insert the explicit expression for the Coulomb potential: V(r) =−Z/r. Let us
mention though that our results can be generalized to other p ower of potentials,
such as the Lennard-Jones potentials [26].
Substituting f(r) =rλin (28), it follows
/parenleftbig
E2
2−E2
1/parenrightbig
/angbracketleftn2j2ǫ2|rλ|n1j1ǫ1/angbracketright=
/angbracketleftn2j2ǫ2|c2/bracketleftbigg∆+
21∆−
21
4−λ(λ+ 1)/bracketrightbigg
rλ−2+c2∆−
21
2βrλ−2−2c2λrλ−1d
dr+
−2icαr/parenleftbigg
λ+∆−
21
2β/parenrightbigg
rλ−1V(r)|n1j1ǫ1/angbracketright;(34)
9hence, we can eliminate the term containing the derivative o perator in this last
equation, using f(r) =rλ−1in Eq. (31), to get the result
/parenleftbig
E2
2−E2
1/parenrightbig
/angbracketleftn2j2ǫ2|rλ|n1j1ǫ1/angbracketright=
/angbracketleftn2j2ǫ2|c2∆+
21∆−
21
4rλ−2+c2∆−
21
2β(1−λ)rλ−2−icαrβ∆−
21rλ−1V(r)+
+ (E2+E1)λ(−icαr)rλ−1|n1j1ǫ1/angbracketright;(35)
in this last equation, we use Eq. (23) to eliminate the term wi th−icαr∆−
21βrλ−1, to
get
(E2
2−E2
1)/angbracketleftn2j2ǫ2|rλ|n1j1ǫ1/angbracketright=/angbracketleftn2j2ǫ2|c2/bracketleftbigg∆−
21∆+
21
4+∆−
21
2(1−λ)β/bracketrightbigg
rλ−2+
2Z/bracketleftbig
icαrrλ−2(1−λ)−(E2−E1)rλ−1/bracketrightbig
−(E2+E1)λicα rrλ−1|n1j1ǫ1/angbracketright.(36)
Now, from Eq. (32) with f(r) =rλ−1we get
(E2−E1)/angbracketleftn2j2ǫ2| −iαrrλ−1|n1j1ǫ1/angbracketright=
/angbracketleftn2j2ǫ2| −c(λ−1)rλ−2+c∆+
21
2βrλ−2+ 2ic2αrβmrλ−1|n1j1ǫ1/angbracketright(37)
and, usingf(r) =rλin Eq. (33) to eliminate the term 2 icαrβmrλ−1from the above
equation, we obtain
(E2−E1)/angbracketleftn2j2ǫ2| −iαrrλ−1|n1j1ǫ1/angbracketright=
/angbracketleftn2j2ǫ2| −c(λ−1)rλ−2+c∆+
21
2βrλ−2−2c
λ(E2+E1)βrλ+c2
λ(−iαr)∆−
21rλ−1+
+4c3
λrλ−4cZ
λβrλ−1|n1j1ǫ1/angbracketright;
(38)
which can be written as
/bracketleftbigg
(E2−E1)−∆−
21c2
λ/bracketrightbigg
/angbracketleftn2j2ǫ2|(−iαrrλ−1)|n1j1ǫ1/angbracketright=/angbracketleftn2j2ǫ2| −c(λ−1)rλ−2+
4c3
λrλ+c∆+
21
2βrλ−2−4Zc
λβrλ−1−2c
λ(E2+E1)βrλ|n1j1ǫ1/angbracketright.(39)
We can also obtain a new relationship for the matrix elements of−iαrrλ−1,
using Eq. (23) with f(r) =rλ, and substitute the result in Eq. (37) to eliminate the
term 2iαrβmrλ−1
10(E2−E1)/angbracketleftn2j2ǫ2| −iαrrλ−1|n1j1ǫ1/angbracketright=
/angbracketleftn2j2ǫ2| −c(λ−1)rλ−2+c∆+
21
2βrλ−2+4c2λ
∆−
21(−iαr)rλ−1
−4c
∆−
21(E2−E1)rλ|n1j1ǫ1/angbracketright.(40)
Rearranging terms, we obtain
/bracketleftbigg
(E2−E1)−4c2λ
∆−
21/bracketrightbigg
/angbracketleftn2j2ǫ2|(−iαrrλ−1)|n1j1ǫ1/angbracketright=
/angbracketleftn2j2ǫ2| −c(λ−1)rλ−2−4c
∆−
21(E2−E1)rλ+c∆+
21
2βrλ−2|n1j1ǫ1/angbracketright.(41)
The relation we are looking for follows from this last result and Eq. (36). We use
succesively rλ−1andrλ−2from Eq. (41) to eliminate the terms 2( E2+E1)λicα rrλ−1
and 2icαrrλ−2(1−λ) that appear in Eq. (36) to finally get [14]
c0/angbracketleftn2j2ǫ2|rλ|n1j1ǫ1/angbracketright=3/summationdisplay
i=1ci/angbracketleftn2j2ǫ2|rλ−i|n1j1ǫ1/angbracketright+3/summationdisplay
i=2di/angbracketleftn2j2ǫ2|βrλ−i|n1j1ǫ1/angbracketright,
(42)
where the numbers ci,i= 0,...3 are given by
c0=(E2
2−E2
1)(E2−E1)∆−
21
(E2−E1)∆−
21−4c2λ,
c1=−2Z(E2−E1)2∆−
21
(E2−E1)∆−
21−4c2(λ−1),
c2=c2∆−
21∆+
21
4−c2λ(λ−1)(E1+E2)∆−
21
(E2−E1)∆−
21−4c2λ,
c3=−2Zc2(λ−1)(λ−2)∆−
21
(E2−E1)∆−
21−4c2(λ−1),(43)
and the numbers di,i= 2 and 3, by
d2=c2∆−
21
2/bracketleftbigg
(1−λ) +λ(E2+E1)∆+
21
(E2−E1)∆−
21−4c2λ/bracketrightbigg
,
d3=Zc2(λ−1)∆−
21∆+
21
(E2−E1)∆−
21−4c2(λ−1).(44)
As we may have expected, we need to know six matrix coefficients instead of
only three as in the non-relativistic case. This is a consequ ence of the fact that, in
11the Dirac case, we have to deal with the big and the small compo nents in the state
function, doubling in this sense the “degrees of freedom” of the system.
It does not seems to be possible to avoid the β-dependency in Eq. (44), and thus,
taken on its own, Eq. (41) does not allow the computation of /angbracketleftn2j2ǫ2|rλ|n1j1ǫ1/angbracketrightin
terms of the /angbracketleftn2j2ǫ2|rλ−a|n1j1ǫ1/angbracketright,a= 1,2,3. The situation is not hopeless though
because it is still possible to obtain another recurrence re lation for non-diagonal
matrix elements of βrλsimply by eliminating the term −iαrrλ−1between Eqs. (39)
and (41). In such a way we get
e0/angbracketleftn2j2ǫ2|βrλ|n1j1ǫ1/angbracketright=b0/angbracketleftn2j2ǫ2|rλ|n1j1ǫ1/angbracketright+b2/angbracketleftn2j2ǫ2|rλ−2|n1j1ǫ1/angbracketright
+e1/angbracketleftn2j2ǫ2|βrλ−1|n1j1ǫ1/angbracketright+e2/angbracketleftn2j2ǫ2|βrλ−2|n1j1ǫ1/angbracketright,(45)
where the numbers biandeii= 1,2,3 are given by
b0=4λ/bracketleftbig
(E2−E1)2−4c4/bracketrightbig
,
b2=c2(1−λ)/bracketleftbig
(∆−
21)2−4λ2/bracketrightbig
,
e0=2(E2+E1)[(E2−E1)∆−
21−4c2λ],
e1=4Z[4c2λ−(E2−E1)∆−
21],
e2=c2∆+
21
2[(∆−
21)2−4λ2].(46)
Equations (42) and (45) together are the useful recurrence r elations in the relativistic
Dirac case.
IV. The diagonal case ∆−
21= 0(j2=j1,ǫ2=ǫ1).
In the results of the last section we always assume ∆−
21/negationslash= 0, but in order to study
the diagonal case we must have ǫ1=ǫ2andj1=j2; this in turn imply ∆−
21= 0 and
(as always!) ∆+
21≡∆+/negationslash= 0. To deal with the diagonal case we start all over again.
The equation set for this case is particularly simple, first f rom Eq. (23) we have
(E2−E1)/angbracketleftn2jǫ|f(r)|n1jǫ/angbracketright=/angbracketleftn2jǫ|(−icαrf′(r)|n1jǫ/angbracketright, (47)
then we can procced to calculate the second order iteration b y substituting, as in
the previous section, f(r)→ξ−=H2f(r)−f(r)H1in (47) to obtain
(E2−E1)2/angbracketleftn2jǫ|f(r)|n1jǫ/angbracketright=/angbracketleftn2jǫ| −c2f′′(r) +c2∆+
2rf′(r)β+
2ic3αrβf′(r)|n1jǫ/angbracketright;(48)
and then substitute f(r)→ξ+=H2f(r) +f(r)H1again in (47) to get instead
12(E2
2−E2
1)/angbracketleftn2jǫ|f(r)|n1jǫ/angbracketright=/angbracketleftn2jǫ| −2c2f′(r)
r−c2f′′(r)−2c2f′(r)d
dr
−2icαrf′(r)V(r)|n1jǫ/angbracketright.(49)
The equations equivalent of Eqs. (31–33) are in this case
(E2+E1)/angbracketleftn2jǫ|(−iαrf(r))|n1jǫ/angbracketright=/angbracketleftn2jǫ| −2cf(r)
r−cf′(r)−2cf(r)d
dr
−2iαrV(r)f(r)|n1jǫ/angbracketright,(50)
and
(E2−E1)/angbracketleftn2jǫ|(−iαrf(r))|n1jǫ/angbracketright=/angbracketleftn2jǫ| −cf′(r) +c∆+
2rβf(r)+
2iαrβc2f(r)|n1jǫ/angbracketright.(51)
We also have, for the matrix elements of βf(r),
(E2+E1)/angbracketleftn2jǫ|βf(r)|n1jǫ/angbracketright=/angbracketleftn2jǫ| −icαrβf′(r)+
2/bracketleftbig
c2+βV(r)/bracketrightbig
f(r)|n1jǫ/angbracketright.(52)
These expressions are the basic equations for the case ∆−
21= 0.
We can now obtain a recurrence relation valid in the diagonal case. First, let
us usef(r) =rλin Eq. (48) to get
(E2−E1)2/angbracketleftn2jǫ|rλ|n1jǫ/angbracketright=λ/angbracketleftn2jǫ| −c2(λ−1)rλ−2+c2∆+
2βrλ−2+
2ic3αrβrλ−1|n1jǫ/angbracketright.(53)
Evaluating now equation (52) with f(r) =rλ, we obtain
(E2+E1)/angbracketleftn2jǫ|βrλ|n1jǫ/angbracketright=/angbracketleftn2jǫ| −icαrβλrλ−1
+ 2/parenleftbigg
c2−Zβ
r/parenrightbigg
rλ|n1jǫ/angbracketright,(54)
and eliminating the icαrβλrλ−1between Eqs. (53) and (54), we finally get
/bracketleftbig
(E2−E1)2−4c4/bracketrightbig
/angbracketleftn2jǫ|rλ|n1jǫ/angbracketright=λc2∆+
21
2/angbracketleftn2jǫ|βrλ−2|n1jǫ/angbracketright
−4Zc2/angbracketleftn2jǫ|βrλ−1|n1jǫ/angbracketright −2c2(E2+E1)/angbracketleftn2jǫ|βrλ|n1jǫ/angbracketright
−c2λ(λ−1)/angbracketleftn2jǫ|rλ−2|n1jǫ/angbracketright.(55)
13This is the only recurrence relation we get in the diagonal ca se. To “close” the
relation we can use the diagonal recurrence relations given in [9].
The special case when λ= 0 is of particular interest
/bracketleftbig
(E2−E1)2−4c4/bracketrightbig
δn1n2=−4Zc2/angbracketleftn2jǫ|β
r|n1jǫ/angbracketright
−2c2(E2+E1)/angbracketleftn2jǫ|β|n1jǫ/angbracketright.(56)
This expression could be considered as a relativistic gener alization of the Pasternak-
Sternheimer rule of non relativistic quantum mechanics (Eq uation (8) of section II)
[21], which says that the expectation value between hydroge nic states of the 1 /r2
potential, vanishes when the orbital angular momenta of the states 1 and 2 coincide,
i.e.whenl1=l2. In the relativistic case the expectation value of the β/rpotential
(which could be regarded as the square root of 1 /r2including bothsigns), does not
necessarily vanish even when the total angular momenta of th e two states coincide:
i.e.it does not vanish when j1=j2. Again, this agrees with the fact that the non-
relativistic Pasternack-Sternheimer rule is applicable t o eigenfunctions of potentials
whose energy eigenvalues depend only on the principal quant um number—which is
not the case for the hydrogen atom in Dirac relativistic quan tum mechanics [14].
Moreover, two special cases are immediately deduced from th is last expression
(56):
1) The first case, when n1/negationslash=n2, is
/angbracketleftn2jǫ|Zβ
r|n1jǫ/angbracketright=−1
2(E2+E1)/angbracketleftn2jǫ|β|n1jǫ/angbracketright. (57)
2) The other case follows when n1=n2
c2=− /angbracketleftβV(r)/angbracketright+E/angbracketleftβ/angbracketright=Z/angbracketleftbiggβ
r/angbracketrightbigg
+E/angbracketleftβ/angbracketright, (58)
which is the relativistic virial theorem [22]; from the rela tionc2< β > =E[9], we
can also put it in the alternative form
E2=c2/angbracketleftβV(r)/angbracketright+c4=−c2Z/angbracketleftbiggβ
r/angbracketrightbigg
+c4. (59)
V. The values of <rλ>and<βrλ>.
The recurrence relations found above, involve in principle simple expressions
(since they involve only matrix elements of Dirac hydrogeni c states) that can be
burdensome to handle. Given such situation, we have also cal culated explicit formu-
las that are needed to evaluate the diagonal and the non-diag onal matrix elements
of interest. The expressions are related to the hypergeomet ric function and can be
deduced from the two differential equations that follow from the Hamiltonian (20),
14as it is shown in the Appendix. In particular, from Eq. (A.15) we calculate <rλ>
and<βrλ>. We quote the results here and refer to the Appendix for the de tails.
<rλ>=mc2|C|2
(2k)λ+12s−1/bracketleftbig
I2s
nn(λ)u2+I2s
n−1n−1(λ)v2+EuvI2s
nn−1(λ)/bracketrightbig
, (60a)
and
<βrλ>=E|C|2
(2k)λ+12s−1/bracketleftbig
I2s
nn(λ)u2+I2s
n−1n−1(λ)v2+mc2uvI2s
nn−1(λ)/bracketrightbig
; (60b)
in these expressions n= 0,1,2,· · ·, and [23,27]
k≡1
¯hc/radicalbig
m2c4−E2, ζ≡Ze2
¯hc=ZαF, τ j≡ǫ(j+1
2),
ν≡/radicalbigg
mc2−E
mc2+E, s≡/radicalig
τ2
j−ζ2,(61)
whereαF≃1/137 is the fine structure constant and the I2s
nm(λ) symbols are defined
in equation (A.15) of the Appendix. The numbers uandvareconstants such that
u= (τj+s+n−ζν−1)1/2, v= (n+ 2s)(τj+s+n−ζν−1)−1/2; (62)
in the Appendix we give a simple proof of this result. Notice t hat in this section
we have explicitly written ¯ h,e, andcin our results. Finally, to obtain C, we use
relations (61) to get ( τj+s+n−ζν−1)−1= (n+s−τj−ζν−1)/n(n+ 2s); we
need also (n+s) =ζE/√
m2c4−E2, which is obtained from the expression for the
energy eigenvalues of the Dirac hydrogen atom:
E=mc2
1 +Z2α2
F/parenleftig
n−j−1/2 +/radicalbig
(j+ 1/2)2−Z2α2
F/parenrightig2
−1/2
; (63)
elementary algebra gives then the result
|C|=¯h2s−1
Zαc2/radicalbigg
n!k
2m3[Γ(n+ 2s+ 1)]−1/2. (64)
where we have written explicitly the dimensional factors.
15Acknowledgements.
This work has been partially supported by CONACyT. It is a ple asure to thank C.
Cisneros for all the collaboration, and V. M. Shabaev for mak ing us aware of Ref.
15. ALSB and HNNY acknowledge the help of F. C. Minina, B. Caro , M. X’Sac,
M. Osita, Ch. Dochi, F. C. Bonito, G. Abdul, C. Sabi, C. F. Quim o, S. Mahui, R
Sammi, M. Mati, U. Becu, Q. Chiornaya, E. Hera and M. Sieriy. L ast but not least,
this paper is dedicated to the memory of our beloved friends Q . Motita, B. Kuro,
M. Mina, Ch Cori, C. Ch. Ujaya, Ch. Mec, F. Cucho, R. Micifuz an d U. Kim.
Appendix. Explicit expressions for relativistic matrix elements ofrλand
βrλ
It is possible to obtain explicit expressions for the diagon al and non diagonal
matrix elements in the case V(r) =−Z/r.The purpose of this appendix is to give the
basic relation that is needed for such evaluation. As we heav ily draw from results
previously obtained, in this section we use the notation of R ef. 23, in particular
¯h=c=e= 1, though we sometimes write all the dimensional constants .
We are interested in the bound states of the problem, so the qu antityk≡√
m2−E2is positive. We can write the differential equations for the r adial part of
any central problem in terms of the dimensionless variable ρ≡kr[23,25] and the
symbols defined in (60)
/parenleftbigg
−d
dρ+τj
ρ/parenrightbigg
G(ρ) =/parenleftbigg
−ν+ζ
ρ/parenrightbigg
F(ρ),
/parenleftbigg
+d
dρ+τj
ρ/parenrightbigg
F(ρ) =/parenleftbigg
ν−1+ζ
ρ/parenrightbigg
G(ρ);(A1)
where we look for solutions of the form
F(ρ) =√
m+E[ψ−(ρ) +ψ+(ρ)], (A2)
G(ρ) =√
m−E[ψ−(ρ)−ψ+(ρ)]. (A3)
The solution to these coupled differential equations can be w ritten in terms of
the Laguerre polynomials of non-integer index [25,27,28]
ψ+(ρ) =aρsexp(−ρ)L2s
n−1(2ρ),
ψ−(ρ) =bρsexp(−ρ)L2s
n(2ρ),(A4)
where the Laguerre polynomials Lα
n(ρ) are related to both the hypergeometric fun-
ction, 1F1(−n,α+1;ρ), and the Sonine polynomials, T(n)
α(ρ) [28], through the rela-
tion
16Lα
n(ρ) =Γ(α+n+ 1)
n!Γ(α+ 1)1F1(−n;α+ 1;ρ) = (−1)nΓ(α+n+ 1)T(n)
α(ρ), (A5)
andaandbare constants. Substitution of these results in Eq. (A1) giv es the condi-
tion
a(τj+s−ζν−1+n) +b(n+ 2s) = 0,
b(τj−s+ζν−1−n)−an= 0.(A6)
Solving these last two equations give us a relationship betw eennandν. From Eq.
(45) we see that we can solve for the energy Eand obtain the relativistic energy
spectrum (63), provided we first introduce the principal qua ntum number N≡
j+ 1/2 +n. To proceed further, we take
b=−a(τj+s+n−ζν−1)/(n+ 2s), (A7)
and write the result in a symmetrized form:
F(ρ) =/radicalbig
mc2+ECρse−ρ/bracketleftbig
uL2s
n(2ρ) +vL2s
n−1(2ρ)/bracketrightbig
,
G(ρ) =−/radicalbig
mc2−ECρse−ρ/bracketleftbig
uL2s
n(2ρ)−vL2s
n−1(2ρ)/bracketrightbig
,(A8)
where
u= (τj+s+n−ζν−1)1/2, v= (n+ 2s)(τj+s+n−ζν−1)−1/2, (A9)
Cis a normalization constant that can be obtained from
/integraldisplay∞
0e−xxαLα
n(x)Lα
m(x) =δmnΓ(n+α+ 1)
n!; ( A10)
after some work we obtain
|C|=¯h2s−1
Zαc2/radicalbigg
n!k
2m3[Γ(n+ 2s+ 1)]−1/2. (A11)
We can also calculate the expectation values for diagonal an d non diagonal
matrix elements. For diagonal, arbitrary power matrix elem ents of the form <rλ>
and<βrλ>, we need to calculate the expression
Iα
nm(λ) =/integraldisplay∞
0e−xxα+λLα
n(x)Lα
m(x)dx. (A12)
This expression converges for Re(α+λ+ 1)>0, and is zero if λis an integer such
thatm−n>λ ≥0,where without loss of generality, we assume that m>n . From
17Rodrigues formula and ( dm/dxm)xk+λ= (−1)m[−k−λ]mxk+λ−m, where [n],nan
integer, is a Pochhammer symbol [28], we find, after a m-times partial integration,
Iα
nm(λ) =1
m!n/summationdisplay
k=0(−1)kΓ(n+α+ 1)Γ(α+k+λ+ 1)[−k−λ]m
k! (n−k)! Γ(α+k+ 1). (A13)
We use now the identity [ −k−λ]m= [−k−λ]k[−λ]m−k,change the order of sum-
mationk→n−kand use the identities
[−λ]m−n+k= [−λ]m−n[−λ+m−n]k,
Γ(n+α+ 1) = ( −1)kΓ(α+n−k+ 1)[−α−n]k,
Γ(α+λ+ +n+ 1) = ( −1)kΓ(α+λ+n−k+ 1)[−α−λ−n]k,
[k−n−λ]n−k= (−1)nΓ(λ+n+ 1)
Γ(λ+ 1)1
[−λ−n]k,(A14)
to obtain that
Iα
nm(λ) =
[−λ]m−nΓ(α+λ+n+ 1)Γ(λ+n+ 1)
m!n! Γ(λ+ 1)
3F2(−α−n,−λ+m−n,−n;−λ−n,−α−λ−n; 1).(A15)
We consider two cases for the general matrix elements /angbracketleftn2j2ǫ2|rλ|n1j1ǫ1/angbracketrightand
/angbracketleftn2j2ǫ2|βrλ|n1j1ǫ1/angbracketright; the first one, when k1=k2, where we need to evaluate
Ks1s2nn(λ) =/integraldisplay∞
0xs1+s2+λe−xL(2s1)
n(x)L(2s2)
m(x)dr, (A16)
and the second one, when k1/negationslash=k2, where we need
Ks1s2nm(λ) =/integraldisplay∞
0rs1+s2+λe−(k1+k2)rL(2s1)
n(2k1r)L(2s2)
m(2k2r)dr. (A17)
In the first case, we see that integral (A16) is convergent if Re(s1+s2+λ+ 1)>0,
and vanishes when s1−s2+λis an integer such that m−n>s 1−s2+λ≥0. Using
a similar reasoning as in the diagonal case, we get
Ks1s2
nm(λ) =
[−λ+s2−s1]m−nΓ(s1+s2+ +λ+n+ 1)Γ(λ+s1−s2+n+ 1)
m!n! Γ(λ+s1−s2+ 1)
3F2(−2s1−n,−λ+s2−s1+m−n,−n;−λ+s2−s1−n,−λ−s1−s2−n; 1).
(A18)
18In the second case, the integral converges for Re(s1+s2+λ+ 1)>0,and is
not zero provided k1/negationslash=k2.A straightforward calculation by parts shows that
Ks1s2nm(λ) =n/summationdisplay
j=0m/summationdisplay
i=0(−1)j(k2−k1)m−i(k1+k2)(i−m−s1−s2−λ−1)
i!j! (m−i)! (n−j)!
Γ(n+ 2s1+ 1)[s2−s1−λ−j]i
Γ(2s1+j+ 1)Γ(m+s2+s1+λ−i+ 1),(A19)
wherek1/negationslash=k2.Although less practical than the other expressions found he re, we
still can rewrite Eq. (A19) in a different form using the well k nown identities [28]
[p]m−i= (−1)m−i Γ(−p+ 1)
[−p−m+ 1] iΓ(−p−m+ 1),
Γ(−p−m+i+ 1) = [ −p−m+ 1] iΓ(−p−m+ 1),
(m−i)! =m! (−1)i
[−m]i,
Γ(p+ 1) = [p−m+ 1] mΓ(p−m+ 1).(A20)
for any number pandmandiintegers. After some algebra, we finally get
Ks1s2nm(λ)
=(−1)mΓ(n+ 2s1+ 1)Γ(s1+s2+λ+ 1)
m! (k2+k1)s1+s2+λ+1n/summationdisplay
j=0(−1)j[s1−s2+λ+j−m+ 1] m
j! (n−j)! Γ(2s1+j+ 1)
×2F1(−m,s1+s2+λ+ 1;s1−s2+λ+j−m+ 1;k2−k1
k2+k1).
(A21)
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New York, 1964).
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21 |
arXiv:physics/0102018v1 [physics.atom-ph] 8 Feb 2001HEP/123-qed
Problems connected with electrons trajectory in separated atom
Evgueni V. Kovarski
e-mail: ekovars@netscape.net
(November 02, 2000)
Abstract
The time-dependent electromagnetic field can results both p air waves and
pair particles. It can be for mathematical relations betwee n two functions
with identical argument and difference of phases equal to π. Two examples
both the opportunity of phase synchronism for different freq uencies of at
atom -field interaction with known abnormal dispersion of re fraction and the
possible trajectories for pair particles are considered.
0155+b
Typeset using REVT EX
1INTRODUCTION
At atom-field interaction frequency-time splitting of a spe ctral line of spontaneous radi-
ation can occurs due to given the dressed atom theory at high i ntensity of external field or
for explained splitting each of two energy atomics levels to some sublevels at low intensity of
external field as a result of quantum interference effect, as i t was supposed and demonstrated
with help of 3D pictures titled as Probability-Time-Freque ncy (PTF) manifolds [1]. On the
other hand for such observable characteristics as the spont aneous radiation is possible to use
the transformation from trigonometrical functions of prob ability of a finding a particle at a
concrete energy level inside atom to the exponential charac teristics in space outside of atom.
The rate of emission for the quantum transition depend on the derivative of probability P(τ)
R=∞/integraldisplay
0γ·[exp(−γt)]·/parenleftBig˙P/parenrightBig
·dt (1)
The well known formula for the transition probability P(τ) given in single frequency exci-
tation of the two level atom is:
P1=/bracketleftBigg4Ω2
4Ω2+ ∆ω2/bracketrightBigg
sin2/parenleftbiggτ
2√
4Ω2+ ∆ω2/parenrightbigg
(2)
where Ω is the Rabi frequency and ∆ ω=ω0−ω1,are the laser frequency detuning values.
Then the procedure of finding the rate of the power absorption leads to the correspondent
profile:
R1=2Ω2γ
∆ω2+ 4Ω2+γ2(3)
At a denominator of the formula there are three members, one o f which can be incorpo-
rated with other of two stayed members. In this case this form ula can be compared with a
standard structure of Lorentz profile, at which denominator always there are only two mem-
bers. Therefore communication of two spaces both the atomic and the external are carried
out except the amplitude in unique parameter known as a full w idth on half of maximum
2(FWHM) of spectral line measured in a scale of frequencies or time. This FWHM for a
Lorentz profile is designated as (2 γ). The connection between this size of FWHM and a
damping constant of oscillator same designated, but descri bing losses of energy of a particle
at interaction with other particles in environment is espec ially brightly shown for wide of
FWHM spectral lines due to collisions with other atoms. It is much more difficult to ex-
plain such connection between FWHM and ( 2γ) for isolated oscillator or for radiation from
stopped atom by laser cooling. Despite of it isolated oscill ator has spontaneous radiation
and therefore is characterized by so-called natural width o f a spectral line connected with
the same ( 2γ) as the damping constant from classical equation of oscilla tor. It is possible to
tell that it is one of difficult examples when the classical per formances need to be combined
with the quantum, if to mean that the energy level can be split as in the dressed theory. If
such splitting is not connected to a magnetic field, therefor e does not exist rules of selection
for optical transitions and then there is a question about th at occurs with a particle on line
interacted with EM field if it is with identical probability a t different energy statuses of sub
levels. The time of spontaneous emission tSfrom upper energy level is well known:
tS=3π¯hǫ0c3
0
ω3
0d2
21(4)
The probability that an electron with life, having unknown t ime (t), will leave the upper
energy level, and its spontaneous radiation will fade with k nown constant ( γ−1
S) is defined
by function
G1=γS·exp(−γS·t) (5)
The probability that the upper level will become empty with a known damping constant ( γ)
during the unknown time of spontaneous radiation ( tS) is:
G2=γ·exp(−γ·tS) (6)
At definition of complete probability of time of life it is nec essary to take into account both
probabilities [1]
3G=G1·G2=1
t·tS·exp/bracketleftBigg
−t2+t2
S
t·tS/bracketrightBigg
(7)
The lifetime tgoing from function G2is oversized the lifetime tL=2tsfrom the first
distribution G1due to the contribution of no radiative decay that is importa nt for a separated
atom.
Other classical example is under consideration of the phase and group speed of the waves
interaction with a separated atom, because for only one wave of an external EM field, as
against a single pulse mode or mode of several EM waves, the gr oup speed of the single
wave exist only as a mathematics value and therefore does not influence on phase speed, if
of course is to consider reception of a single external wave f rom single external oscillator.
It is well known that the dispersion of an index of refraction n(ω) named as the abnormal
refraction is in a vicinity of resonant frequency ( ω0) at interaction of an electromagnetic wave
(EM) with two levels atoms E=E0·cos(ω·t) , where the wave number is k=ω·n/c0, (E0)
is the amplitude of an external EM wave, ( c0) is the speed of the light in vacuum. Phase
(vf) and group ( vg) speeds of a wave involved for an explanation of this effect ar e defined
by the known relations due to the wave number can be spread out in a number on degrees
of frequency is in a vicinity of resonant frequency ( ω0):
vf=c0
n(ω)=ω
k(ω)(8)
vg=1
dk/dω=c0
n+ (ω−ω0)·(dn/dω)(9)
On the other hand such single wave can be submitted from exter nal space as a set waves
and then the physical sense for application of group speed is kept for single wave. However
it is possible to assume and return situation, when single ex ternal wave cooperates with set
of particles having for example a parallel trajectories.
4I. WHETHER THERE IS A PROBLEM OF AN ELECTRONS TRAJECTORY IN
ATOM?
Last example can be served by application of mutual orientat ion of a trajectory of a
particle and field. Because the phase speed ( vf) depends on ( ω−ω0), therefore, in particular,
for two separated EM waves with frequencies detuned symmetr ical respect to the central
frequency of quantum transition in atom, named as sidebands modes of EM field, as known,
that is possible to supply different phase speeds of EM waves i n same environment. Is
possible spatially to divide these waves and to direct them t owards each other. So, the atom
placed in such field, will test action of different forces f1,2=dp/dt, where p=m·dx/dt,
where ( m) is the mass of the particle,( x) is the displacement of the atom along the EM field.
When an atom is under perturbation of two symmetrical and coh erent laser frequencies [1],
therefore the frequency detuning conditions for such bichr omate laser waves are:
E(τ) =E0[cos(ω1τ) + cos(ω2τ)] (10)
∆ω2=ω2−ω0 (11)
∆ω1=ω0−ω1 (12)
For the special perturbation of the upper energy level there are two symmetrical frequencies
and we can use the relation:
∆ω2= ∆ω1= ∆ω (13)
The transition probability is:
P2=/bracketleftbigg
sin/parenleftbigg
Ω·sin ∆ωτ
∆ω/parenrightbigg/bracketrightbigg2
(14)
With Bessel functions the probability P2can be written:
P2= [2J1(ρ) sin ∆ωτ+ 2J3(ρ) sin 3∆ωτ+ 2J5(ρ) sin 5∆ωτ+..]2(15)
5ρ=Ω
∆ω(16)
The rate of radiation R2can be obtained by the same way as above for a single frequency
excitation. After some algebra with Bessel functions and by use the known relation
sin[sinΘ] =2·∞/summationdisplay
k=0J2k+1(z)·sin[(2k+1)·Θ] (17)
there are two presentations of the rate R2:
R2=γ
2(∆ω)2/summationdisplay/bracketleftBigg
2J2k+2(2ρ)(2n+ 2)2
(2n+ 2)2∆ω2+γ2/bracketrightBigg
(18)
R2=γ
2(∆ω)2/summationdisplay
2J2k+2(2ρ)1
∆ω2+/parenleftBig
γ
2n+2/parenrightBig2
(19)
Here numbers ( k/negationslash=0) worth at Bessel functions ( J2k) coincide with numbers ( 2n+2=2k),
where ( n=0,1,2,..) There are series with Lorentzian profiles. If the condition for detuning
is: ∆ω≫γ, then the rate of the power absorption R2is simply :
R2=Γ
2[1−J0(2ρ)], (20)
Thus the trajectory of an atom plays the important role, beca use the atom can be stopped
by sidebands method that results in known now results on cool ing atoms. Contrary an atom
can be accelerated.
Let us note, that is interesting to consider the problem of th e movement of an electron
within atom with the trajectory along and against a directio n of a wave vector of the EM field,
because with the rotation inside atom the electron can addit ional braking or acceleration.
So both the trajectory of electrons within atom especially i n a case of quantum transition
and spatial orientation of a field inside atom are important t oo. In this work it is offered to
discuss only some of these problems due to this question is co nnected with the important
presence of a damping coefficient ( γ) in equation of the oscillator and because such modes
of movement should be shown in radiation of a driven charge.
6The quantum mechanics does not recognize exact definition of coordinates or trajectory of
movement. If the force in classical physics is defined as f=dp/dt, in the quantum mechanics
it is (F=−∂U/∂t), where ( U) represents known potential. At the same time potential
depends on coordinate ( x), which by virtue of known uncertainty relation in the quant um
mechanics cannot be defined. However, a wave function ( ψ) in the quantum mechanics
describes wave with not observable characteristics as the ” wave frequency” ( H/¯h) and is
presented through the operators of coordinate ( X) and moment (Λ), because the hamiltonian
2H=X2+Λ2:
ψ(x,t) =/bracketleftbigg
exp/parenleftbigg
−ıH
¯ht/parenrightbigg/bracketrightbigg
·ψ(x,0) (21)
Application of the operators is connected with two linear sp aces, one of which is the space
of laboratory where classical physics is work and coordinat e is measured as x:
Xψ=xψ (22)
Λψ=−ı·¯h∂ψ
∂x(23)
The information about space of an atom can be received only in space of classical physics
named in this work as the space of laboratory where for the kno wn classical model of the
oscillator the exam of the differential equation gives infini te on size value of displacement
(x) of the oscillator for both conditions the absence of the dam ping constant ( γ) and the
exact resonance condition ( ω=ω0):
¨x+2γ˙ x+ω2
0x=e
2m·E0·[exp(ıωt) +exp(−ıωt)] (24)
x=/bracketleftBigge
2m·E2
0
ω2
0−ω2+2ıγω/bracketrightBigg
+c.c. (25)
Therefore it is obvious, that it is simultaneously impossib le to simulate both conditions
(ω=ω0) and (γ= 0). Despite of it, it is possible to explain increase of disp lacement up to
infinite size by display of the appeared acceleration. Reall y, the well known equation of a
7clasical oscillator can be considered as a threshold condit ion at balance of several acceleration
mechanisms for one connected electron. When the frequency o f a field comes nearer to
resonant frequency, then in this vicinity of frequencies or in the appropriate vicinity of time
in system necessarily there should be an the damping, differe ntly irrespective of energy of
an electromagnetic field oscillations will simply be stoppe d when the electron becomes free,
because increase of displacement up to infinite size.
Therefore there is a vicinity of time, when there are process es with several types of
the damping mechanisms braking the electron. From known obs ervable processes of these
mechanisms, connected to presence, first of all, it is necess ary to allocate the absorption
and both spontaneous and compelled radiation. The importan t problem for one atom is the
understanding of a nature of the spontaneous radiation damp ing constant. This problem
concerns to a question on so-called optical friction and is g eneral for the classical and quan-
tum approach, as is connected that the damping is present at p rocesses, which are not for
separately taken atom, because there is no interaction with environment.
In the quantum mechanics the damping ( γ) is entered in the equations provided that
the electron already is at the exited level, and the field at th is moment has no energy or is
switched simply off like a case for a short pulse of a EM field. Th e presence of the damping
is entered in the assumption, that probability of a finding at om in the exited status is . In
case of unlimited quantity of oscillators, the system of the equations results for complete
probability of spontaneous transition from the exited stat us [2]. Thus the damping has
the imaginary part and gives the amendment on displacement o f own frequency, neglecting
with which, however, it is possible to receive, that is define d by summation on all radiated
quantum.
The important note is about the time as the complex number, wh ich has not only size
opposite to the frequency of an external field, but also size, opposite to the complex damping
coefficient.
The connection between a spectral contour for probability a nd its FWHM is determined
as well as by Lorentz function with inknown value of the ampli tude:
8L=const·γ
∆ω2+γ2/4(26)
Let’s note, that the size (∆ ω=|ω−ω0|) is variable and depends on adjustment of frequency
of an external field ( ω) in relation to resonant frequency of quantum transition ( ω0) in atom
between two energy levels. Let’s consider the norm one of the two typical functions, which
make refraction index:
kA=const·b
a2+b2(27)
kB=const·a
a2+b2(28)
It is possible to fix in them one of two variable ( a) or (b). The norm can be used in three
cases
const·/integraldisplayda
a2+b2=const·/bracketleftbigg1
barctana
b+C/bracketrightbigg
(29)
A/integraldisplaydy
y2+1=A·π=1 (30)
A·π=/integraldisplaysin2(A·y)
y2·dy (31)
Anyone here is taken variable ( y) and constants ( A) and a free constant ( C) . It is clear, that
it is possible to solve this system of the equations and to rec eive a number of the decisions,
instead of one known ( y).
Considering only one analytical function of complex variab le (z) on a complex plane one
can observe the power numbers ( q) [3]:
f(z) =1
z2+1(32)
In a circle ( z<1), except for points ( z/negationslash=ı), this function is an indefinitely decreasing
geometrical progression with the radius of convergence ( r=1):
f(z) =∞/summationdisplay
q=0(−1)q·z2q(33)
9In the other circle |z−1|<√
2, the radius of convergence ( r=√
2):
f(z) =∞/summationdisplay
0(−1)q·sin(q+1)·π
4
2q+1
2·(z−1)q(34)
Therefore any quantum process observable in space of labora tory and described by the
formula passes through a number ( q/negationslash=n0), where ( n0) is the main quantum number of
discrete energy levels. Consequently the picture of energy levels distribution in the space
of laboratory should correspond to this picture. It is possi ble to try to show importance
of distinction of performances about the main quantum numbe r in different spaces on the
following example which is connected with classical parame ter of refraction.
The direct transition from consideration of one atom to cons ideration of environment
consisting many ( N) atoms and back requires a physical explanation of applicab ility for
separated atom ( N=1) not only concept of the damping constant, but also such conc epts
as a parameter of refraction ( n) or permeability ( ǫ). As is known, the polarization of
environment ( P) is defined by the dipole moment ( d=e·x). On the other hand, the
polarization is defined by a susceptibility ( α) and electrical component of an electromagnetic
field (E):
P=N·α·E=N·e·x=N·d (35)
ǫ=1+4π·α (36)
n=√ǫ (37)
However permeability ( ǫ) and parameter of refraction ( n) can not be simply transferred from
space of environment into the space of separate atom. In this work is possible to assume,
that there are features of structure of atom, which are shown that the connected electron
in atom and the free electron in space of laboratory are the sa me particle which are taking
place in different spaces. Therefore they can have difference s in such characteristics, as a
charge ( e) and the mass ( m), because they have differences on dynamics of the movement
10driven by EM field. The space of separate atom can have especia l structure, in which the
mass and the charges of the connected electron interacted wi th a EM field differ among
themselves. Thus connected electrons with a negative charg e in different atoms are con-
sidered taking place in different spaces, and free electrons with a negative charge leaving
of energy limits of different atoms in space of laboratory are considered identical (standard
electron). The change of particles mass ( m) during movement should result in the certain
conformity between complete energy of a particle and energy of the appropriate quantum
transition. It is natural to assume, that the electromagnet ic wave crosses set of spaces with
different phase speed in each of them. Let’s name as resonant s paces such spaces, which are
connected among themselves by any interaction to occurrenc e of a particle or information
electromagnetic field. As an information electromagnetic fi eld we shall name such electro-
magnetic field, which satisfies with the Maxwell equations an d bears in the characteristics
the information on investigated space. Let’s assume, that t he resonant spaces having, at
least, linear communication between one vector in one space with other vectors in the other
space cooperate only. Let’s assume, that the birth of an elec tromagnetic field occurs on bor-
der of resonant spaces. The communication of two linear spac es of laboratory and atom is
carried out by conformity between dependence of amplitude o f a field on time and frequency
in one space and in the another space with dependence of proba bility of finding electron at a
energy level from time and frequency. Factor of proportiona lity defines a time scale in each
space, i.e. the frequency of a EM field ( ω) and Rabi frequency (Ω):
Ω =d12·E
¯h(38)
where the dipole matrix element is ( d12=e·x12). It is possible to assume, that the vectors
(/vectord) and (/vectorx) characterize the space not as linearly dependent pair vect ors, and as linear -
independent pair vectors, therefore is possible to write do wn:
/vectore=e1·/vectord+e2·/vectorx (39)
Thus, the charge ( e) of the connected electron is a vector ( /vectore). This vector has projections to
the allocated directions and, hence, brings in charging sym metry to space of atom. Therefore,
11the application of the classical physical characteristics such as ( n), (ǫ) for separate atom is
possible.
Assuming, that trajectory of an electron with speed ( V) is the circle, in which centre is
nucleus, is necessary to return to a hypothesis of Bohr and De Brogle. On one orbit with
length ( 2πR), where radius ( R) is defined by:
R=n0·¯h
m·V, (40)
is possible to lay an integer of lengths of waves ( λ·n0), where (λ) is characterizes a wave
nature of a material particle and ( n0) is the main quantum number:
λ
n0=h
m·V(41)
On the other hand the same radius ( R) of an orbit can be determined from known relation:
1
8πǫ0·e2
R=m·V2
2(42)
One can to observe that the charge has the same vector nature t hat the velocity. Let’s copy
the formula in view of a known ratio ( λ·ν=c0), where the frequency is ( 2πν=ω), the
speed of light ( c0) is connected to phase speed of electrons wave through ( c0=vf·n), where
(n) - is a parameter of refraction.
n=¯hω
m·1
vf·V·n0 (43)
Thus, the parameter of refraction ( n) of an atom should be discrete size due to the main
quantum number.
Let’s return to the decision of the classical equation of osc illator. It is possible to notice
what exactly from here follows, that the permeability ( ǫ) is complex number. For this reason
the parameter of refraction ( n=√ǫ) is defined by a root of the second degree from complex
number. However parameter of refraction defines a phase spee d of an electrons wave. It
is easy to show, that this phase speed is split on two waves dis tinguished on a phase on
number (π). This implies, that the main quantum number ( n0) also can result in processes
with the same phase displacement.
12In physics some cases are known, when the difference of phases is equal to ( π). First it
is shown at reflection of a EM wave. Secondly it is possible to s earch for such difference in
phases at display of linear polarization of a wave. In third, such difference exists between
positive and negative parts of periodic sine wave function.
II. POSSIBLE PAIR PROCESSES FOR WAVES AND PARTICLES
Let’s consider as an example probable reaction of an atom spa ce to action of an external
field. For this purpose we shall lead analogy between space of an optical crystal and the two
levels atom. Let external field has given, for example, ordin ary linear polarization named
as a (o) -wave, which in turn, as is known, can be submitted consisti ng from both the right
and the left rotating vectors of polarization. Such two vect ors, opposite on rotation can
be considered, as two waves having the identical module and a rguments distinguished on
numberπ. On an entrance of a crystal such external ( o)-field is usually represented as two
identical ( o)-waves having identical frequency ωand two identical amplitudes. Actually
by such consideration the one wave goes about two waves havin g the identical module and
different arguments. As a result of interaction, a unusual ( e)-wave with the polarization
revolved on corner of 90 degrees is left the crystal. As is kno wn, there is such spatial
direction inside a crystal, in which the so-called ( oo−e) interaction results in identical
factors of a parameter of refraction n0(ω) =ne(2ω) on the frequency ωfor (o)- wave and
on the frequency 2 ωfor (e) -wave for known procedure of the second harmonic generatio n
(SHG). Both fields are external in relation to space of a cryst al, as their characteristics are
measured in space of classical physics named as a space of lab oratory. Thus, the phase
synchronism of speeds vf=c0/nis a reaction of interaction of two spaces.
In gas environment of atoms the reaction of interaction both spaces the laboratory space
and the space of atoms shown as similar SHG of 2 ωshould not essentially differ from reaction
by the space of a crystal. Near to a line of absorption always t here is an opportunity of
a choice of two frequencies with an identical parameter of re fraction due to the known
13abnormal dependence of the refraction parameter. In the spe cific case frequencies can be
multiple, as SHG, but the square-law dependence for low inte nsity external EM field of SHG
intensity completely not necessarily should be observed. T hus it is necessary to distinguish,
that at such low intencities the SHG in atomic environment di ffers from known process of the
two photon absorption within an atom with the subsequent rad iation. Generally is possible
to think that at high intencities the multi photons generati on of radiation differs from multi
photons of absorption with the subsequent radiation, becau se can occur at conditions when
the determining factor is the dependence n(ω) near to a line of resonant absorption.
External in relation to atom and periodically time-depende nt electromagnetic (EM) field
E=Eocosω·tin space of laboratory is usually represented as analytical continuation of
function valid variable ( ω·t) in complex area. In this work is paid attention to some featu res
of such transition in a complex plane and back from the point o f view of interpretation of
physics of model connected to generation of an electromagne tic field.
When on some given piece [ a,b], where a=ω1t1andb=ω2t2of the valid real axis
chosen along any direction of the space, that for length [ a,b], included in complex area,
there is a unique function of the complex variable z=a+ı·b. Accepting the same value
that function real variable the electromagnetic field can be submitted on this piece by a
converging sedate number in a known kind
/summationdisplay
cn[ωt−ω0t0]n(44)
From the mathematical point of view on definition of complex n umbers follows, that one
complex number can characterizes one physical value of pair real numbers with the estab-
lished order of following of one real number behind another, where thus the only imaginary
number z=ı·bis equivalent to a=0. In particular, the EM field in space of laboratory
is submitted as
E0
2[exp(ı·ωt) +exp(−ı·ωt)] (45)
Also this EM field incorporates the sum of two functions compl ex variable
14ζ1,2=exp(z1,2) =exp(a±ı·b) =exp(±ı·ωt) (46)
It means, that two variable z1,2are correspond to a separately taken wave where the module
of each is identical, but the arguments differ is familiar
E0|ζ|=E0exp(a) =E0 (47)
argζ1,2=±b=±ωt (48)
It is understandable, that interpretation of different mark s of arguments of a field can serve
performance about two vectors directed along the allocated direction of a numerical axis in
opposite directions and distinguished on a corner equal π.
It is known, that the loss of energy of an external field commun icates, for example,
with such effects, as effect of saturation in two-level atom ( N1=N2) or with other known
nonlinear effects occurring to a plenty connected electrons in different atoms at a large
intensity of the external field. At the same time, in separate ly taken atom connected electron
increases the energy only for the account (¯ hω). Therefore according to the quantum theory
for excite separately taken oscillator it is enough to have v aguely weak on intensity external
EM field.
Actually a threshold condition should exist as a ratio betwe en losses of the electron on
radiationγRand losses of a field on absorption γA. At such understanding of processes
the connected electron always radiates, including at the mo ment of transition to the exited
energy level. For example, considering one-dimensional cl assical oscillator at the included
external field, according to the theory of Einstein, it is nec essary to exam processes of
absorption, and also processes of the compelled and spontan eous radiation. These processes
are characterized in known coefficients A21,B21andB12. In this connection, it is possible
to write down the balance equation for various damping const ants of classical oscillator,
because the damping constant of spontaneous radiation γSis equal to A21:
A21+E2
x
8πB21=N1
N2E2
x
8πB12 (49)
15γS+γB=γA (50)
The process of absorption is a resulting process and consequ ently the return size of γA
corresponds to time of life γA=t−1
L. Therefore the parameter γ=γAshould be present at
the equation of the oscillator. Therefore becomes obvious, that the time of life should not
be equaled of time spontaneous radiation γ−1
S.
If there is an absorption of energy of EM field in space of atom, this implies classical
performance that the external field should cross a surface of atom under the laws by and
large connected with the known laws of Fresnel for environme nts. Visible light absorbed by
the majority of known atoms, does not test of diffraction on su ch small on the sizes object
as atom. However, for shorter lengths of waves or for atoms wi th the large sizes occurring
processes of reflection and (or) the absorption would result in occurrence of a diffraction
shadow from atom. As the sizes of atom are small in comparison with length of a EM wave,
such sizes allow to make transition to a limiting case of disa ppearance of objects in classical
diffraction tasks, when the external field can be submitted on a complex plane as a spiral,
including in a limiting case with a resulting vector along an imaginary axis. Agreeing with
application of a known principle of Optics it is possible to a ssume, that the separate atom
is a source of secondary waves. Apparently, the secondary wa ves in interpretation of this
principle simply should mean the fact of an output from atom o f pair waves with a difference
of phases distinguished on number π.
The EM wave can be presented as a continuous analytical funct ionf(z) of complex
variable z=x+ı·ywith˙f/negationslash= 0, through partial derivative of functions ( u) and ( v):
f(z) =u(x,y) +ı·v(x,y) (51)
The continuity of function is carried out during each period , when there are limiting values
lim[f(z)] = [f(z0)] in a vicinity z→z0, when the inverse function is determined as
z=φ|f(z)| (52)
˙f=1
˙φ|f(z)|(53)
16/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleuxuy
vxvy/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/negationslash= 0 (54)
Therefore at return transition to the valid plane the functi on EM field as the function of
complex variable can be presented as two functions valid var iable ( u) and ( v).
For presentation it is possible to consider possible interp retation of occurrence of a usual
periodic wave with constant amplitude E=Eo·sin(ωt). Such recording for both electrical
components of a field means that at increase of size of product of frequency at time there
is an increase of a corner between these components of a field. Here analogy to periodic
movement on a circle for a particle having an electrical char ge first of all is pertinent. In
this case it is possible to speak that between the center of a c ircle where there is a positively
charged nucleus and the particle with a negative charge exis ts electromagnetic interaction
resulting to such movement. For a wave with a wave vector k=ω/c, where cis a speed of
light, such periodic decision corresponds to the wave equat ion
∂2E
∂t2+c2·k·E=0 (55)
If the particle is at fixed energy level, its trajectory can be connected to a trajectory of other
pair particles taking place on other energies levels of the s ame atom. There is an example
for a particle fixed on a circle with a radius ( a) and driven together with a circle along the
allocated direction ( x) in space of an atom in a plane ( x,y) with a parameter ( T) of the
corner of turn acts:
x=a·[T−sin(T)] (56)
y=a·[1−cos(T)] (57)
x=Arccosa−y
a·/radicalBig
2ay−y2 (58)
z=x+ı·y=a·[T+ı·exp(−ı·T)] (59)
17It is known, that such trajectory is a projection of movement on a spiral. Radius of curvature
monotonously changes in conformity with a sine function:
ρ=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/bracketleftBig
1 +/parenleftBigdy
dx/parenrightBig/bracketrightBig3
2
d
dx/parenleftBig
dy
dx/parenrightBig/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle=4a·sinT
2(60)
It is easy to notice connection between interpretation of a t rajectory for the simple and
double frequency of a wave. It is possible, that Radius of cur vatureρof this trajectory for
pair fields is variable size and can corresponds to amplitude (E) of EM wave, the radius
of the circle ( a) can corresponds to ( E0), the corner Tcan corresponds to ωt. For pair
particles it is possible to connect these characteristics w ith the Rabi frequency.
In each point such curve radius of curvature is perpendicula r trajectories and all together
such radiuses form other curve, of what it is possible to be co nvinced if to replace T=τ−π.
This will be the same trajectory, but with other coordinates :
X+a·π=a·[τ−sin(τ)] (61)
Y+2a=a·[1−cos(τ)] (62)
Therefore it is possible to assume, that there is one more fiel d or particle carrying out the
same movement with the same frequency and time, but taking pl ace on other circle with
same radius. The trajectories of these pair processes will b e moved with a phase difference
on number ( π) and together they will correspond to one function with vari able on a mark by
amplitude. Such movement for a particle can be simulated as m ovement at a fixed energy
level with friction ( γ), which is present at the equations of classical and quantum oscillators.
Continuing this logic it is possible to enter other parallel trajectories for charges of particles
and it is possible to investigate new EM processes.
In connection with the proposed model is possible to approve , that if the order of following
functions in investigated physical process is established to within a constant difference of
the phase, thus is established possibility to observe a pair physical processes for waves and
particles not only at space of the atom or the space of laborat ory, but for different spaces
18where the flat part of space in general probably may be divided on inverse sub spaces with
in parallel current direct and return processes.
Probably here there should be balanced restrictions on imag ination, because it is possible
too to assume, that the physical laws containing mathematic al functions from even degrees
of the arguments should correspond to pair processes and par ticles ”koquark”.
19REFERENCES
[1] please see Evgueni V. Kovarski Pra Aw6329, September 08, 1998, aps1999jun22-001 and
aps1999feb10-003
[2] B.I.Stepanov, V.P.Gribkovski, Introduction in the the ory of radiation, Minsk, 1963, p.
191.
[3] A.G.Sveshnikov, A.N.Tihonov, Theory of functions comp lex variable,Science, 1999, p.
73
20 |
arXiv:physics/0102019v1 [physics.ed-ph] 8 Feb 2001Bose-Einstein condensates, optical black holes and Fermat s
principle
M. Marklund, D. Anderson, F. Cattani, M. Lisak and L. Lundgre n
Department of Electromagnetics, Chalmers University of Te chnology, SE–41296 Gteborg, Sweden
((December 7, 2012))
Fermat’s principle and variational analysis is used to anal yze the trajec-
tories of light propagating in a radially inhomogeneous med ium with a sin-
gularity in the center. It is found that the trajectories are similar to those
around a black hole, including the feature of an unstable pho ton orbit, from
which light spirals into the optical black hole.
I. INTRODUCTION
There has recently been much interest focused on the extra-o rdinary properties associated
with Bose-Einstein condensates - clouds of atoms cooled dow n to nano-Kelvin temperatures
where all atoms are in the same quantum state and macroscopic quantum conditions prevail,
[1,2].
One of the most dramatic experiments which has been performe d in Bose-Einstein conden-
sates is the demonstration of optical light pulses travelin g at extremely small group velocities
e.g. velocities as small as 17 m/s have been reported, [2].
In a physically suggestive application of the optical prope rties of Bose-Einstein condensates,
it was recently suggested that they could be used to create, i n the laboratory, dielectric
analogues of relativistic astrophysical phenomena like th ose associated with a black hole,
[3].
The astrophysical concept of a black hole is one of the most fa scinating and intriguing phe-
nomena related to the interaction between light and matter w ithin the framework of general
relativity. Black holes are believed to form when compact st ars undergo complete gravita-
1tional collapse due to, e.g., accretion of matter from its su rroundings (in general relativity,
the pressure contributes to the gravitational mass of a fluid , and an increased pressure will,
after a certain point, therefore only help to accelerate the collapse phase). The picture of
a black hole swallowing light and matter without conveying a ny direct information of its
existence is an awesome physical illustration of Shakespea r’s famous metaphor of Death –
“The undiscovered country from which no traveler returns.”
In ordinary dielectric media, unrealistic physical condit ions would be required in order
to demonstrate any of the spectacular effects of general rela tivity. However, in dielectric
media characterized by very small light velocities, new pos sibilities appear. In particular,
it was recently suggested that by creating a vortex structur e in such a dielectric medium
it is possible to mimic the properties of an optical black hol e [3]. What this study in
fact showed was that one could construct, by using the aforem entioned vortex, an unstable
photon orbit, much like the orbit at a radius r= 3Min the Schwarschild geometry [4]. In
order to construct an event horizon, one would need to supple ment the vortex flow by a
radial motion of the fluid [5,6]. In this paper, though, we wil l on the grounds of simplicity
only analyze the fluid with vortex motion. One could generali ze this study by including an
angular dependence in the refractive index of the dielectri c medium (see Sec.II).
The vortex, which involves a rotating cylindrical velocity field, tends to attract light
propagating perpendicular to its axis of rotation and to mak e it deviate from its straight
path. In the model considered in Refs. [1,3], the increasing , in fact diverging, rotational
velocity field of the vortex core attracts light by the optica l Aharanov–Bohm effect and causes
a bending of the light ray similar to that of the gravitationa l field in general relativity. It is
also shown that there exists a critical radius, rcrit, from the vortex core with the properties
that, if light rays come closer to the core than rcrit,the light will fall inescapably towards
the singularity of the vortex core. In this sense, the critic al radius, rcrit, plays the role
of an optical unstable photon orbit, analogous to the unstab le orbit in the Schwarzschild
geometry.
The basic physical effect involved in the analysis of the ray p ropagation in Refs. [1,3]
2is the fact that the refractive index, n of a medium changes wh en the medium is moving.
In Refs. [1,3], the medium is assumed to have a cylindrical vo rtex velocity field V=V(r)
given by
V(r) =W
r/hatwideϕ (1)
where ris the radius from the vortex core centre, /hatwideϕis the azimuthal unit vector and 2 πW
is the vorticity.
Advanced physical concepts like Bose-Einstein condensate s, vortex velocity fields and
black holes are fascinating physical effects, which however may be difficult to present in a
simplified manner for undergraduate students. Nevertheles s, it is important to try to convey
inspiration for and arouse curiosity in such phenomena. An i nteresting example of such an
effort was recently made in Ref. [7], where the basic physical mechanism, in the form of
a classical two level system, was used to demonstrate the pos sibility of the very low light
velocities observed in Bose- Einstein condensates. The pur pose of the present work is similar
to that of Ref. [7]. We will try to give a simple classical exam ple of a medium where the
refractive index has a divergence at r= 0 and which exhibits some of the dramatic properties
of the light behaviour around a black hole as discussed in Ref s. [1,3,5,6]. In addition we
want to illustrate the power and beauty of the classical prin ciple of Fermat and of variational
methods in connection with these new concepts.
II. FERMAT’S PRINCIPLE
It is well known that light tends to be deflected towards regio ns with higher refractive
index. Consider light propagating in a cylindrically symme tric medium where the refractive
index increases towards the centre. In such a situation we ex pect the light path to look
qualitatively as in Fig. 1. The actual light path is determin ed by Fermat’s principle, i.e.,
δ/integraldisplay
n(r)ds= 0 (2)
where n(r) is the refractive index and dsis an infinitesimal element along the light ray.
3We will consider light propagation in a medium where the refr active index is of the
qualitative form, cf Eq. (1)
n(r)≃
1as r≫r0
r0/r as r ≪r0(3)
One possible realization of n(r) with the desired properties is given by
n2(r) = 1 +/parenleftbiggr0
r/parenrightbigg2
(4)
Using Fermat’s principle and expressing the light path as th e relation θ=θ(r) where θis
the polar angle, Eq. (2) implies
δ/integraldisplay
n(r)/radicaltp/radicalvertex/radicalvertex/radicalbt1 +r2/parenleftBiggdθ
dr/parenrightBigg2
dr= 0. (5)
The Euler-Lagrange variational equation corresponding to Eq. (5) reads
d
dr
n(r)r2dθ/dr/radicalBig
1 +r2(dθ/dr )2
= 0 (6)
which determines the trajectory of the light.
III. SOLUTION OF THE LIGHT TRAJECTORY
Equation (6) directly implies that
r2n(r)dθ/dr/radicalBig
1 +r2(dθ/dr )2=b (7)
where the constant bis determined by initial conditions, i.e., the properties o f the incident
light. Equation (7) is easily inverted to read
dθ
dr=±b/radicalBig
r4n2(r)−b2r2(8)
For a light ray incident as shown in Fig. 1, we clearly have dθ/dr < 0 (at least up to some
minimum radius r=rmin). It is illustrative to first consider the trivial case of a ho mogeneous
medium with n(r)≡1. In this case Eq. (8) becomes
4dθ
dr=−b√
r4−b2r2(9)
which can easily be solved to yield the light path in the form
θ= arccot/radicalBigg
r2
b2−1 (10)
or, simpler,
r=b
sinθ(11)
Clearly this is the straight line solution
y=rsinθ (12)
where the parameter bplays the role of “impact parameter” or minimum distance fro m the
centre. In order to emphasize this property we redefine ri≡b. Let us now consider the
model variation for n(r) as given by Eq. (4). Within this model Eq. (8) becomes
dθ
dr=−b/radicalBig
r4(1 +r2
0/r2)−b2r2=−ri/radicalBig
r4−(r2
i−r2
0)r2(13)
Clearly the solution of Eq. (13) will depend crucially on the relative magnitude of riandr0
i.e. the impact parameter relative to the characteristic ra dial extension of the inhomogeneity.
Consider first the case when r0< ri. Equation (13) can then be rewritten as
b1
ridθ
dr=−b1/radicalBig
r4−b2
1r2, (14)
b1≡/radicalBig
r2
i−r2
0. (15)
Equation (14) is of the same form as Eq. (9) and we directly inf er the following solution
r=/radicalBig
r2
i−r2
0
sin/bracketleftBig
θ/radicalBig
1−r2
0/r2
i/bracketrightBig (16)
Asθ→0, we still have asymptotically
r≃ri
sinθ(17)
5However, the trajectory is now bending towards the origin an d the minimum distance occurs
at the polar angle θ=θmgiven by
θm=π
21/radicalBig
1−r2
0/r2
i(18)
The corresponding minimum distance, rmis
rm≡r(θm) =/radicalBig
r2
i−r2
0 (19)
We also note that the trajectory is symmetrical around the an gleθmand that the asymptotic
angle of the outgoing light ray is
θ∞≡limr→∞θ(r) =π/radicalBig
1−r2
0/r2
i= 2θm (20)
The solution given by Eq. (16) describes a trajectory which i s bent towards the centre of
attraction at r= 0. Depending on the ratio r0/ri, the trajectory is either more or less bent
or may even perform a number of spirals towards the centre bef ore again turning outwards
and escaping, cf Fig. 2. The number of turns, N,which the light ray does around the origin
before escaping is simply
N=/floorleftBigg2θm
2π/floorrightBigg
=1
2/radicalBig
1−r2
0/r2
i (21)
where ⌊x⌋denotes the largest integer less than x. Let us now consider the special case
when the impact parameter equals the characteristic width o f the refractive index core, i.e.,
ri=r0. The equation for the trajectory now simplifies to
dθ
dr=−ri
r2, (22)
with the simple solution
θ=ri
r; (23)
therefore, the trajectory describes a path in the form of Ark imede’s spiral as the light falls
into oblivion in the core centre, cf Fig. 3. The ultimate fate of the light in the situation
6when ri< r0is now obvious, it is doomed to fall into the black hole more or less directly
depending on the ratio r0/ri>1. The actual trajectory in this case is given by a slight
generalization of Eq. (16) viz
r=/radicalBig
r2
0−r2
i
sinh[θ/radicalBig
1−r2
i/r2
0](24)
This solution does indeed convey the expected behaviour, th e trajectory spirals
monotonously into the optical singularity.
IV. FINAL COMMENTS
The present analysis is inspired by recent discoveries and d iscussions about light propaga-
tion in Bose-Einstein condensates where extremely low ligh t velocities can be obtained. This
has triggered speculations about possible laboratory demo nstrations of effects, which nor-
mally are associated with general relativistic conditions . In particular, it has been suggested
that it should be possible to create the analogue of a black ho le using a divergent inspiral of
a Bose-Einstein condensate. In the present work we have anal yzed a simple classical exam-
ple of light propagation as determined by Fermat’s principl e in a medium characterized by
a radially symmetric refractive index. In analogy with the v ariation of the vortex velocity
field, the refractive index is assumed to diverge towards the centre. This classical example
exhibit some of the characteristic properties of light prop agating around a black hole where
the gravitational attraction deflects the light and where, u nder certain conditions, the light
may be “swallowed” by the black hole. In the example analyzed here, the unstable photon
orbit of the Schwarzschild black hole corresponds to the cha racteristic radius, r0, of the re-
fractive index variation, which together with the impact pa rameter, ri, of the incident light
completely determines the light trajectory. If r0< ri, the light is more or less deflected, but
ultimately escapes. However, if r0≥ri, the light spirals into the singularity. One should
also point out that this result is dependent on the singulari ty in the refractive index (4). If
this is removed, the unstable photon orbit also disappears. As noted earlier, to solve this
7problem, a radial flow in the condesate has to be introduced. T he investigation is based
on Fermat’s principle and variational analysis, in this way illustrating the use of classical
methods in connection with very new and fascinating concept s at the front line of modern
research.
[1] U. Leonhardt and P. Piwnicki, “Optics of nonuniformly mo ving media,” Phys. Rev. A 60,
4301–4312 (1999).
[2] L. V. Hau, S. E. Harris, Z. Dutton and C. H. Behroozi, “Ligh t speed reduction to 17 metres
per second in an ultracold atomic gas,” Nature (London) 397, 594–598 (1999).
[3] U. Leonhardt and P. Piwnicki, “Relativistic effects of li ght in moving media with extremely low
group velocity,” Phys. Rev. Lett. 84, 822–825 (2000).
[4] C. W. Misner, K. S. Thorne and J. A. Wheeler, Gravitation ( W H Freeman & Co, 1973).
[5] M. Visser, “Comment on ‘Relativistic effects of light in m oving media with extremely low group
velocity’,” Phys. Rev. Lett. 85, 5252–5252 (2000).
[6] U. Leonhardt and P. Piwnicki, “Comment on ‘Relativistic effects of light in moving media with
extremely low group velocity’ — U. Leonhardt and P. Piwnicki reply,” Phys. Rev. Lett. 85,
5253–5253 (2000).
[7] K. T. Mc Donald, “Slow light,” Am. J. Phys. 68, 293–294 (2000).
8/Minus4/Minus2024
x/Minus0.500.51yr
Θ
FIG. 1. Qualitative plot of a light ray trajectory in a cylind rically symmetric medium with a
refractive index, which increases towards the centre.
/Minus2/Minus1012
x/Minus0.500.51y
FIG. 2. Light trajectories for the refractive index model of Eq. (4) for different impact radii
ri> r0(r0/ri= 0.4,0.9, and 0 .98).
/Minus0.3/Minus0.2/Minus0.100.10.2
x/Minus0.200.20.4yr0/Slash1ri/EΘual1
FIG. 3. Light trajectory in the case of ri=r0, the spiral of Arkimedes.
9 |
arXiv:physics/0102020v1 [physics.optics] 8 Feb 2001DSF−6/2001
SISSA−12/2001/EP
physics/0102020
On a universal photonic tunnelling time
Salvatore Esposito∗
Dipartimento di Scienze Fisiche, Universit` a di Napoli “Fe derico II”
and
Istituto Nazionale di Fisica Nucleare, Sezione di Napoli
Complesso Universitario di Monte S. Angelo, Via Cinthia, I- 80126 Napoli, Italy
∗Visiting scientist at SISSA–ISAS, Via Beirut 4, I-34013 Tri este, Italy
E-mail: Salvatore.Esposito@na.infn.it
We consider photonic tunnelling through evanescent region s and obtain general analytic expressions
for the transit (phase) time τ(in the opaque barrier limit) in order to study the recently p roposed
“universality” property according to which τis given by the reciprocal of the photon frequency. We
consider different physical phenomena (corresponding to pe rformed experiments) and show that such
a property is only an approximation. In particular we find tha t the “correction” factor is a constant
term for total internal reflection and photonic bandgap, whi le it is frequency-dependent in the case
of undersized waveguide. The comparison of our predictions with the experimental results shows
quite a good agreement with observations and reveals the ran ge of applicability of the approximated
“universality” property.
I. INTRODUCTION
In recent times, some photonic experiments [1]- [4] deal-
ing with evanescent mode propagation have drawn some
attention because of their intriguing results. All such ex-
periments have measured the time required for the light
to travel through a region in which only evanescent prop-
agation occurs, according to classical Maxwell electrody-
namic. If certain conditions are fulfilled (i.e. in the limit
of opaque barriers), the obtained transit times are usually
shorter than the corresponding ones for real (not evanes-
cent) propagation through the same region. Due to the
experimental setups, this has been correctly interpreted
in terms of group velocities [5] greater than cinside the
considered region. Although there has been some confu-
sion in the scientific community, leading also to several
different definitions of the transit time [6], these results
are not at odds with Einstein causality since, according
to Sommerfeld and Brillouin [7], the front velocity rather
than the group velocity is relevant for this. Waves which
are solutions of the Maxwell equations always travel in
vacuum with a front velocity equal to cwhile, in certain
conditions, their phase and group velocities can be differ-
ent fromc[8]. It is worthwhile to observe that the quoted
experiments are carried out studying different phenom-
ena (undersized waveguide, photonic bandgap, total in-
ternal reflection) and exploring different frequency ranges
(from optical to microwave region).
The interest in such experiments is driven by the fact that
evanescent mode propagation through a given region can
be viewed as a photonic tunnelling effect through a “po-
tential” barrier in that region. This has been shown, for
example, in Ref. [9] using the formal analogy between the
(classical) Helmholtz wave equation and the (quantummechanical) Schr¨ odinger equation (see also Ref. [10]). In
this respect, the photonic experiments are very useful
to study the question of tunnelling times, since experi-
ments involving charged particle (e.g. electrons) are not
yet sensible enough to measure transit times due to some
technical difficulties [11].
From an experimental point of view, the transit time τ
for a wave-packet propagating through a given region is
measured as the interval between the arrival times of the
signal envelope at the two ends of that region whose dis-
tance isD. In general, if the wave-packet has a group
velocityvg, this means that τ=D/v g. Sincevg=dω/dk
(kwave-vector, ωangular frequency), then we can write
[12]:
τ=dφ
dω, (1)
wheredφ=Ddkis the phase difference acquired by the
packet in the considered region. The above argument
works as well for matter particles in quantum mechanics,
changing the role of angular frequency and wave-vector
into the corresponding ones of energy and momentum
through the Planck - de Broglie relations.
However, difficulties arise when we deal with tunnelling
times, since inside a barrier region the wave-vector (or
the momentum) is imaginary, and hence no group veloc-
ity can be defined. As a matter of fact, different defini-
tions of tunnelling time exist. While we refer the read to
the quoted literature [6], here we use the simple defini-
tion of phase time which coincides with Eq. (1). In fact,
althoughvgseems meaningless in this case, nevertheless
Eq. (1) is meaningful also for evanescent propagation.
The adopted point of view takes advantage of the fact
that experimental results [1]- [4] seem to confirm the def-
inition of phase time for the tunnelling transit time.
1a 0III
zV(z) n(z)
I II
FIG. 1. A barrier potential V(z) for a particle or a barrier
refractive index n(z) for an electromagnetic wave.
Recently, Haibel and Nimtz [4] have noted that, regard-
less of the different phenomena studied, all experiments
have measured photonic tunnelling times which are ap-
proximately equal to the reciprocal of the frequency of
the radiation used in the given experiment. Such a “uni-
versal” behaviour is quite remarkable in view of the fact
that, although photonic barrier traversal takes place in
all the quoted experiments, nevertheless the boundary
conditions are peculiar of each experiment.
In the present paper we carefully study the proposed uni-
versality starting from a common feature of tunnelling
phenomena and, in the following section, derive a gen-
eral expression for the transit (phase) time. Different ex-
periments manifest themselves into different dispersion
relations for the barrier region. We then analyze each
peculiar experiment in Sects. III,IV,V and compare the-
oretical predictions with experimental observations. Fi-
nally, in Sect. VI, we discuss our results and give conclu-
sions.
Note that, differently from other possible analysis (see,
for example, the comparison with a photonic bandgap
experiment in [13]), we deal with only tunnelling times,
which have been directly observed, and not with veloci-
ties which, in the present case, are derived from transit
times.
II. PHASE TIME AND DISPERSION RELATION
In this paper we study one-dimensional problems or,
more in general, phenomena in which evanescent prop-
agation takes place along one direction, say z. Let us
then consider a particle or a wave-packet moving along
thez-axis entering in a region [0 ,a] with a potential bar-
rierV(z) or a refractive index n(z), as depicted in Figure
1. The energy/frequency of the incident particle/wave is
below the maximum of the potential or cutoff frequency.
For all experiments we’ll consider, the barrier can be
modelled as a square one, in which V(z) orn(z) is con-
stant in regions I,II,III but different from one region to
another. We also assume that V(z) orn(z) is equal in I
and II and take this value as the reference one.
The propagation of the particle/wave through the bar-
rier is described a by a scalar field ψrepresenting theSchr¨ odinger wave function in the particle case or some
scalar component of the electric or magnetic field in the
wave case. (The precise meaning of ψin the case of wave
propagation depends on the particular phenomenon we
consider. However, the aim of this paper is to show that a
common background for all tunnelling phenomena exist).
Given the formal analogy between the Schr¨ odinger equa-
tion and the Helmholtz equation [9], [10], this function
takes the following values in regions I,II,III, respective ly:
ψI=eikz+Re−ikz(2)
ψII=Ae−χz+Beχz(3)
ψIII=Teik(z−a), (4)
wherekandk2=iχare the wave-vectors ( p= ¯hkis
the momentum) in regions I (or III) and II, respectively.
Note that we have suppressed the time dependent factor
eiωt. Obviously, the physical field is represented by a
wave-packet with a given spectrum in ω:
ψ(z,t) =/integraldisplay
dωη(ω)ei(kz−ωt). (5)
whereη(ω) is the envelope function. Keeping this in
mind we use, however, for the sake of simplicity, the sim-
ple expressions in Eqs. (2), (3), (4). Furthermore, for the
moment, we disregard the explicit expression fro kand
χin terms of the angular frequency ω(or the relation
betweenpandE= ¯hω). As well known, the coefficients
R,T,A,B can be calculated from the matching condi-
tions at interfaces:
ψI(0) =ψII(0), ψ II(a) =ψIII(a) (6)
ψ′
I(0) =ψ′
II(0), ψ′
II(a) =ψ′
III(a), (7)
where the prime denotes differentiation with respect to
z. Substituting Eqs. (2), (3), (4) into (6), (7) we are then
able to find R,T,A,B and thus the explicit expression for
the function ψ. Here we focus only on the transmission
coefficientT; its expression is as follows:
T=/bracketleftbig
1−r2e−2χa/bracketrightbig−1/parenleftbig
1−r2/parenrightbig
e−χa(8)
with:
r=χ+ik
χ−ik. (9)
The interesting limit is that of opaque barriers, in which
χa≫1. All photonic tunnelling experiments have
mainly dealt with this case, in which “superluminal”
propagation is predicted [14]. Taking this limit into Eq.
(8) we have:
T≃2/bracketleftbigg
1−ik2−χ2
2kχ/bracketrightbigg−1
e−χa. (10)
The quantity φin Eq. (1), relevant for the tunnelling
time, is just the phase of T:
2φ≃arctank2−χ2
2kχ. (11)
The explicit evaluation of τin Eq. (1) depends, clearly,
from the dispersion relations k=k(ω) andχ=χ(ω).
However, by substituting Eq. (11) into (1) we are able
to write:
τ= 2/bracketleftBigg
1 +/parenleftbiggk
χ/parenrightbigg2/bracketrightBigg−1
d
dωk
χ, (12)
showing that τdepends only on the ratio k/χ. We can
also obtain a particularly expressive relation by introduc -
ing the quantities:
k1
v1=kdk
dω,k2
v2=−χdχ
dω. (13)
In fact, in this case we get:
τ=2
χk/bracketleftbiggχ2
k2+χ2k1
v1+k2
k2+χ2k2
v2/bracketrightbigg
. (14)
Note that while k1andk2are the real or imaginary wave-
vectors in regions I (or III) and II, v1andv2represent
the “real” or “imaginary” group velocities in the same
regions. Obviously, an imaginary group velocity (which
is the case for v2) has no physical meaning, but we stress
that in the physical expression for the time τin (14) only
the ratiok2/v2enters, which is a well-defined real quan-
tity.
Equations (12) and (14) are very general ones (holding in
the limit of opaque barriers): they apply to alltunnelling
phenomena. It is nevertheless clear that peculiarities of a
given experiment enter in τonly through the dispersion
relationsk=k(ω) andχ=χ(ω) or, better, k(ω)/χ(ω).
As an example of application of the obtained general
formula, we here consider the case of tunnelling of non
relativistic electrons with mass mthrough a potential
square barrier of height V0. (In the next sections we then
study in detail the three types of experiment already per-
formed). The electron energy is E= ¯hω(withE < V 0)
while the momenta involved in the problem are p= ¯hk
andiq= ¯hk2=i¯hχ. In this case, the dispersion relations
read as follows:
k=/radicalbigg
2mω
¯h(15)
χ=/radicalBigg
2m(V0−¯hω)
¯h2(16)
and thus:
k
χ=/radicalbigg
¯hω
V0−¯hω. (17)
By substituting into Eq. (12) we immediately find:
τ=¯h/radicalbig
E(V0−E)=1
¯h2m
χk. (18)2
n1n1n az
x
II
IIII
<n1
θ
FIG. 2. Frustrated total internal reflection in a double
prism.
III. TOTAL INTERNAL REFLECTION
The first photonic tunnelling phenomenon we consider is
that of frustrated total internal reflection [15]. This is
a two-dimensional process, but tunnelling proceeds only
in one direction. With reference to Figure 2, a light
beam impinges from a dielectric medium (typically a
prism) with index n1onto a slab with index n2< n1.
If the incident angle is greater than the critical value
θc= arcsinn2/n1, most of the beam is reflected while
part of it tunnels through the slab and emerges in the
second dielectric medium with index n1. Note that wave-
packets propagate along the xdirection, while tunnelling
occurs in the zdirection.
The wave-vectors k1,k2in regions I (or III) and II satisfy:
k2
1=k2
x+k2(19)
k2
2=k2
x−χ2, (20)
wherekxis thexcomponent of k1ork2andk,χare as
defined in the previous section. The dispersion relations
in regions I (or III) and II are, respectively:
k1=ω
cn1 (21)
k2=ω
cn2. (22)
These equations also define the introduced quantities:
v1=c
n1(23)
v2=c
n2. (24)
It is now very simple to obtain the tunnelling time in the
opaque barrier limit for this process; in fact, by substi-
tuting Eqs. (21)-(24) into Eq. (14) we find:
τ=1
ω2k2
x
χk. (25)
Furthermore, using the obvious relations:
3kx=k1sinθ=ω
cn1sinθ (26)
k=k1cosθ=ω
cn1cosθ (27)
χ=/radicalBig
k2
1sin2θ−k2
2=ω
c/radicalBig
n2
1sin2θ−n2
2,(28)
we finally get:
τ=1
νn1sin2θ
πcosθ/radicalBig
n2
1sin2θ−n2
2. (29)
This formula can be directly checked with experiments.
However, we firstly observe the interesting feature of this
expression which does satisfy the property pointed out
by Haibel and Nimtz [4]. In fact, the time τin Eq. (29)
is just given, apart from a numerical factor depending on
the geometry and construction of the considered experi-
ment, by the reciprocal of the frequency of the radiation
used. In a certain sense, the numerical factor can be
regarded as a “correction” factor to the “universality”
property of Haibel and Nimtz.
Several experiments measuring the tunnelling time in the
considered process have been performed [3].
In the experiment carried out by Balcon and Dutriaux [3],
two fused silica prisms with n1= 1.403 and an air gap
(n2= 1) are used. They employed a gaussian laser beam
of wave-length 3 .39µmwith an incident angle θ= 45.5o.
Using these values into Eq. (29) we predict a tunnelling
time of 36.8fs, to be compared with the experimental re-
sult of about 40 fs. As we can see, the agreement is good
and the “correction” factor in (29) is quite important for
this to occur (compare with the Haibel and Nimtz pre-
diction of 11 .3fs).
In the measurements by Mugnai, Ranfagni and Ronchi
[3], the microwave region is explored, with a signal whose
frequency is in the range 9 ÷10GHz. They used two
paraffin prisms ( n1= 1.49) with an air gap ( n2= 1),
while the incidence angle is about 60o. For this exper-
iment we predict a tunnelling time of 87 .2ps, while the
experimental result is 87 ±7ps∗.
Finally, we consider the recent experiment performed
by Haibel and Nimtz [4] with a microwave radiation at
ν= 8.45GHz and two perspex prisms ( n1= 1.605) sep-
arated by an air gap ( n2= 1). For an incident angle of
45o, from (29) we predict τ= 80.8ps. The observed ex-
perimental result is, instead, 117 ±10ps. In this case, the
agreement is not very good (while, dropping the “correc-
tion” factor, Haibel and Nimtz find a better agreement);
probably this is due to the fact that the condition of
opaque barrier is not completely fulfilled.
∗Note that the value of 134 psused by Haibel and Nimtz
refers to the gap filled with paraffin. In this case no tunnellin g
effect is present. We observe that also for this experiment
the “correction” factor in (29) plays a crucial role for the
tunnelling times0zaI
IIIII
FIG. 3. A waveguide with an undersized region.
IV. UNDERSIZED WAVEGUIDE
Let us now consider propagation through undersized rect-
angular waveguides as observed in [1]. Also in this
case, evanescent propagation proceeds along one direc-
tion (sayz) and the results obtained in Sect. II may
apply. With reference to Figure 3, a signal propagating
inside a “large” waveguide at a certain point undergoes
through a “smaller” waveguide for a given distance a. As
well known [16], the signal propagation inside a waveg-
uide is allowed only for frequencies higher than a typical
value (cutoff frequency) depending on the geometry of
the waveguide. In the considered setup, the two differ-
ently sized waveguides I (or III) and II have, then, differ-
ent cutoff frequencies (the first one, ω1, is smaller than
the second one, ω2), and we consider the propagation of a
signal whose frequency (or range of frequencies) is larger
thanω1but smaller than ω2:ω1< ω < ω 2. In such a
case, in the region 0 <z <a only evanescent propagation
is allowed and, thus, the undersized waveguide acts as a
barrier for the photonic signal. With the same notation
of Sect. II, the dispersion relations in the large and small
waveguide are, respectively:
ck=/radicalBig
ω2−ω2
1 (30)
cχ=/radicalBig
ω2
2−ω2, (31)
so that:
k
χ=/radicalBigg
ω2−ω2
1
ω2
2−ω2, (32)
By substituting this expression into Eq. (12), we imme-
diately find the tunnelling time in the regime of opaque
barrier (χa≫1):
τ=1
ν·1
π/radicalBigg
ν4
(ν2−ν2
1)(ν2
2−ν2). (33)
On the contrary to what happens for tunnelling in to-
tal internal reflection setups, the coefficient of the term
1/νisn’t constant but depends itself on frequency. Thus,
in the case of undersized waveguides, the assumed “uni-
versality” property of Haibel and Nimtz cannot apply in
general; depending on the cutoff frequencies, it is only
a partial approximate property for frequencies far way
from the cutoff values (i.e. when the term in the square
root does not strongly depend on ν).
Let us now compare the prediction (33) with the ex-
perimental results obtained in [1]. In the performed
4n1n2n1n2zn1n2 n0
0 d 2d Ndn0d d1 2
FIG. 4. An ideal photonic bandgap device.
experiment we have microwave radiation along waveg-
uides whose cutoff frequencies are ν1= 6.56GHz and
ν2= 9.49GHz, respectively. The radiation frequencies
are around ν= 8.7GHz, so that tunnelling phenom-
ena occur in the undersized waveguide. By substituting
these values into Eq. (33), we predict a tunnelling time
of 128ps, confronting the observed time of about 130 ps.
As it is evident, also for an undersized waveguide setup
the theory matches quite well with experiments. Note
that, despite of the rich frequency dependence in Eq.
(33), the Haibel and Nimtz property also works quite
well (although some correction needs), since the central
frequency value of the radiation used in the experiment
is far enough from the cutoff values.
V. PHOTONIC BANDGAP
The last phenomenon we consider is that of light prop-
agation through photonic bandgap materials. The ideal
setup is depicted in Figure 4. Light impinges on a succes-
sion of thin plane-parallel films composed of Ntwo-layer
unit cells of thicknesses d1,d2and constant, real refrac-
tive indices n1,n2, embedded into a medium of index n0.
It is known [17] that such a multilayer dielectric mirror
possesses a (one-dimensional) “photonic bandgap”, that
is a range of frequencies corresponding to pure imaginary
values of the wave-vector. In practice, it is the optical
analog of crystalline solids possessing bandgaps. Increas -
ing the number of periods will result in an exponential
increase of the reflectivity, and thus the opaque barrier
condition can be fulfilled. In general, the study of elec-
tromagnetic properties of such materials is very compli-
cated, and the dispersion relation we need to evaluate the
phase time in the proposed formalism is quite involved
for physical situations. This study was performed ana-
lytically in [13] where the dispersion relation (and other
useful quantities) was derived starting from the complex
transmission coefficient of the considered barrier. It is,
then, quite a meaningless issue to get the tunnelling time
from the dispersion relation obtained from the transmis-
sion coefficient, while it is easier to have directly the
phase time τfrom Eq. (1), where φis the phase of the
complex transmission coefficient. We consider only the
relevant case in which each layer is designed so that the
optical path is exactly 1 /4 of some reference wave-lengthλ0:n1d1=n2d2=λ0/4. In such a case, λ0corresponds
to the midgap frequency ω0(λ0= 2πc/ω 0). This condi-
tion is fulfilled in the considered experiments [2]. Finally ,
we further assume normal incidence of the light on the
photonic bandgap material.
From [13] we then obtain the following expression for the
transmission coefficient:
T= [(AC−B) +iAD]−1, (34)
whereA,B,C,D are real quantities given by:
A=sinNβ
sinβ(35)
B=sin(N−1)β
sinβ(36)
C=acosπω
ω0+b (37)
D=csinπω
ω0(38)
a=1−r2
02
t02t21t12(39)
b=r2
12(r2
02−1)
t02t21t12(40)
c=2r02r12−r2
02−1
t02t21t12(41)
rij=ni−nj
ni+nj(42)
tij=2nj
ni+nj(43)
sinβ=1
t12t21/radicalBigg
r2
12/parenleftbigg
cosπω
ω0−1/parenrightbigg
+ sin2πω
ω0(44)
(i,j= 1,2).The phase φof the transmission coefficient
thus satisfies:
tanφ=AD
B−AC. (45)
By substituting into Eq. (1), we finally get an analytic
expression for the tunnelling tome of light with frequency
νclose to the midgap one ν0forNlayers:
τ=1
ν0·1
2csinhNθ
sinh(N−1)θ+ (b−a) sinhNθ,(46)
whereθis simply obtained from:
sinhθ=1
2/parenleftbiggn2
n1−n1
n2/parenrightbigg
. (47)
For future reference, we also report the appropriate
formula for N=k+ (1/2) (integer k) multilayer di-
electric mirrors. In practice, this models the case
5of a stratified medium whose structure has the form
n1n2n1n2...n1n2n1(note, however, this is an approx-
imation since, in general, d/2 is not equal to a). In such
a case, Eq. (46) is just replaced by:
τ=1
ν0·1
2ccoshNθ
cosh(N−1)θ+ (b−a) coshNθ,(48)
Let us observe that, similarly to total internal reflection,
at midgap the time τin Eq. (46) or (48) is again given
by the reciprocal of the frequency times a “correction”
constant factor.
We now analyze experimental results [2] in the light of
our theoretical speculations.
In the experiment performed by Steinberg, Kwiat and
Chiao, the authors used a quarter-wave multilayer di-
electric mirror with a ( HL)5Hstructure with a total
thickness of d= 1.1µmattached on one side of a sub-
strate and immersed in air ( n0= 1). Here, Hrepre-
sents a titanium oxide film with n1= 2.22, whileLis a
fused silica layer with n2= 1.41. Thus, we have approxi-
matelyN= 5+(1/2). As incident light, they employed a
wave-packet centred at a wave-length λ0= 702nm, corre-
sponding to the midgap frequency ν0of about 427 GHz.
By substituting these numbers in our formula (48) we
predict a tunnelling time τ= 2.51fs, corresponding to
a delay time ∆ t, with respect to non tunnelling pho-
tons propagating at the speed of light the distance d, of
−1.09fs. This has to be compared with the experimental
result of ∆ t=−(1.47±0.2)fs. However, we point out
that our analytical prediction is affected by two major
approximations. The first one is, as already remarked,
that the experimental sample is not really a 5 + (1 /2)
periodic structure. Since, in the considered case, the re-
maining “half-period” is made of high refractive index,
corresponding to a quarter-wave thickness d1=λ0n1/4
smaller than d2=λ0n2/4, a better approximation is
achieved by using Eq. (46) with N= 5. In this case
we haveτ= 2.42fsor a delay time ∆ t=−1.18fs,
which is in better agreement with the experimental re-
sult. Furthermore, in our analysis (leading to Eq, (46) or
(48)) there is no room for considering the substrate on a
side of which the photonic bandgap material is attached.
Such an asymmetric structure cannot be taken into ac-
count in an analytic framework, but has to be studied us-
ing numerical matrix transfer method which would give
quite a good agreement with observations [13]. However,
this is beyond the aim of this paper, and we judge our
2σprediction quite satisfactory.
Finally, we consider the experiment carried out by Spiel-
mann et al. [2] on alternated quarter-wave layers of fused
silicaLand titanium dioxide Hhaving the structure
of (substrate )(HL)n(air) withN= 3,5,7,9,11. They
used optical pulses of frequency 375 THz corresponding
to the midgap frequency of their photonic bandgap ma-
terial. Obviously, increasing Nwe have a better re-
alization of opaque barrier condition. From Eq. (46)
withN= 11 (note, however, that for N≥5 the factorsinh(N−1)θ/sinhNθis almost constant) we have a tun-
nelling time of 2 .81fsto be compared with the observed
value of 2.71fs. We address the fact that, apart the pres-
ence of the substrate which introduces some approxima-
tion as discussed above, in the considered experiment the
incidence of the light on the sample is not normal, being
≈20othe angle between the axis of the sample and the
beam propagation direction. In this case, the described
computations are only approximated ones and, again, the
exact result can be obtained only through numerical im-
plementation. Nevertheless, also within the limits of our
calculations, the agreement between theory and experi-
ment is quite good.
A final comment regards the predictions of the “univer-
sality” property proposed by Haibel and Nimtz. Neglect-
ing the “correction” factor in Eq. (46) would yield the
values of ∆ t=−1.26fsandτ= 2.67fsfor the de-
lay time in the Steinberg, Kwiat and Chiao experiment
and the transit time for the Spielmann et al. experi-
ment, respectively. In both cases, the agreement with the
observed values is better than our approximated predic-
tions, showing that the presence of the substrate (and the
non normal incidence in the second experiment) pushes
up the “correction” factor in Eq. (46).
VI. CONCLUSIONS
In this paper we have scrutinized the recently proposed
[4] “universality” property of the photonic tunnelling
time, according to which the barrier traversal time for
photons propagating through an evanescent region is ap-
proximately given by the reciprocal of the photonic fre-
quency, irrespective of the particular setup employed. To
this end, the transit time in the relevant region, defined
here as in Eq. (1), needs to be computed for the different
explored phenomena, and in Sect. II we have given gen-
eral expressions for this time in the opaque barrier limit.
The peculiarities of a given photonic setup enter in these
expression only through the dispersion relation relating
the wave-vector and the frequency. More in detail, we
have shown how the knowledge of the ratio between the
wave-vectors in the barrier region and outside it, as a
function of the photon frequency, is sufficient to evaluate
the transit time τin Eq. (12).
Several specific cases, corresponding to the different
classes of experimentally investigated phenomena, have
then been considered. In particular, in Sect. III we have
studied light propagation in a setup in which the evanes-
cent region is provided by total internal reflection, while
in Sect. IV the propagation through undersized waveg-
uides has been considered and, finally, in Sect. V the case
of a photonic bandgap has been analyzed. The relevant
results for the three mentioned phenomena are given in
Eqs. (29), (33) and (46), respectively. As can be easily
seen from these expressions, the frequency dependence
of the tunnelling time for the cases of total internal re-
6TABLE I. Comparison between predicted and observed
tunnelling times for several experiment (FTIR, UWG and
PBG stands for frustrated total internal reflection, unders ized
waveguide and photonic bandgap, respectively). τexpis the
experimental result while τthis our prediction as from Eqs.
(29), (33) and (46).For reference to the Haibel and Nimtz
property, we also report the value 1 /ν.
Phenomenon Experiment 1/ν τ th τexp
FTIR Balcon et al. [3] 11 .3fs36.8fs∼40fs
FTIR Mugnai et al [3] 100 ps87.2ps87±7ps
FTIR Haibel et al. [4] 120 ps81ps117±10ps
UWG Enders et al. [1] 115 ps128ps∼130ps
PBG Steinberg et al. [2] 2 .34fs2.42fs2.13±0.2fs
PBG Spielmann et al. [2] 2 .67fs2.81fs∼2.71fs
flection and photonic bandgap is just as predicted by the
property outlined by Haibel and Nimtz [4], although we
have derived a “correction” factor depending on the ge-
ometry and on the intrinsic properties of the sample (this
factor is not far from unity). On the contrary, such a
factor is frequency dependent for undersized waveguides,
revealing a more rich dependence of τonνthan the sim-
ple 1/νone (see Eq. (33)). We can then conclude that
the “universality” property of Haibel and Nimtz is only
an approximation, but it gives the right order of magni-
tude (and, in some case, even more) for the tunnelling
time. This conclusion holds also for undersized waveg-
uide propagation, provided that the photon frequency is
far enough from the cutoff frequencies.
We have then calculated the tunnelling times for the dif-
ferent existing experiments and compared the theoretical
values with the observed ones. Results are summarized in
Table 1, where we also report the Haibel and Nimtz pre-
diction 1/ν. From these we can see that, in general, the
agreement of our prediction with the experimental values
is satisfactory. As pointed out in the previous section, the
calculations performed here for photonic bandgap mate-
rials assume some approximations in treating the com-
plex sample, which are nevertheless required to obtain
analytical expressions. Our prediction then suffer of this,
and the simple 1 /νrule fits better with experiments. In
all other cases, the “correction” factor introduced in this
paper is quite relevant for the agreement with observa-
tions to be good.
ACKNOWLEDGMENTS
The author is indebted with Prof. E. Recami for many
fruitful discussions and useful informations about the
subject of this paper. He also gratefully acknowledges
Prof. A. Della Selva for discussions.[1] A. Enders and G. Nimtz, J. Phys. I(France) 2(1992)
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F. Kraus, Phys. Rev. Lett. 73(1994) 2308; G. Nimtz, A.
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851; D. Mugnai, A. Ranfagni and L. Ronchi, Phys. Lett.
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ski and K. Wynne, Phys. Rev. Lett. 84(2000) 1431.
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[5] See the discussion in W. Heitmann and G. Nimtz, Phys.
Lett.A 196 (1994) 154.
[6] V.S. Olkhovsky and E. Recami, Phys. Report 214(1992)
339 and Refs. therein.
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7 |
arXiv:physics/0102021v1 [physics.bio-ph] 9 Feb 2001Codon Distributions in DNA
A.Som1,a, S.Chattopadhyay1,b, J.Chakrabarti1,cand D.Bandyopadhyay2,d
1Department of Theoretical Physics
Indian Association for the Cultivation of Science
Calcutta 700 032, INDIA
2Linguistics Research Unit
Indian Statistical Institute
Calcutta 700 035, INDIA
1Abstract
The codons, sixtyfour in number, are distributed over the co ding parts of DNA sequences.
The distribution function is the plot of frequency-versus- rank of the codons. These distribu-
tions are characterised by parameters that are almost unive rsal, i.e., gene independent. There
is but a small part that depends on the gene. We present the the ory to calculate the universal
(gene-independent) part. The part that is gene-specific, ho wever, has undetermined overlaps
and fluctuations.
PACS number(s): 87.10.+e, 85.15.-v, 05.40.+j, 72.70+m
E-mails:
atpas@mahendra.iacs.res.in
btpsc@mahendra.iacs.res.in
ctpjc@mahendra.iacs.res.in
dbandyo@isical.ac.in
21 Introduction
The methods of statistical linguistics are used in recent ye ars to study DNA sequences[1]. The genome
projects generate large volumes of data on DNA. Fast and reli able computational tools to analyse
this huge data of billlions of bases are required. The idea is to identify features in the sequences and
to correlate them with known biological functions. The meth ods of statistical linguistics[2] could
provide reliable computational algorithms. This is what we investigate here.
The sequences are made of the nucleotide bases A, C, G and T. Th e arrangement of the bases over
the linear chain determines all the information there is in D NA. The regions that code for proteins,
the coding regions (or the exons), have bases working in grou ps of three to make proteins. These
triplets are called codons. The biologically meaningful wo rds are these codons. The noncoding parts
consist of the introns and the flanks. These are presumed impo rtant in regulatory and promotional
activities. The biologically meaningful word structures i n these regions are not known. A gene
generally comprises of a number of exon regions separated by introns. Since the biological functions
thus far are associated with the triplet codons, we concern o urselves only with these triplet words, the
codons. Therefore, in our analysis, instead of an entire gen e, we consider the coding DNA sequence
(CDS) region of the gene, where the exon segments are put toge ther, splicing the introns out.
Natural languages are characterised by structures determi ned by rules of grammar. The words
put together with these rules carry sense. The rules give coh erence and meaning to long texts. The
3languages have this long-range order. The frequency spectr a show the presence of the long periods.
These are identified by the1
fβtype behaviour in the low frequency region[3]. Words placed at random
will have quite different frequency spectrum with no long-ra nge behavior. The early work on natural
languages dealing with the statistical distributions of wo rds, done by Zipf [4], assigned ranks to the
words. The word most frequent has rank=1; the next most has ra nk=2 and so on. Zipf showed that
for natural languages the plot of frequency, fn, versus rank, n, is of the power-law form:
fn=f1
nα(1)
where f1is the frequency of rank 1. In the Zipf’s original analysis th e power-index αwas assumed
to be one. Subsequent studies have allowed for deviations fr om one.
The DNA sequence of the letters A, C, G and T does have1
fβfrequency spectrum[5]. It is possible,
therefore, that the sequences have long-range order and und erlying grammer rules. The opinion on
this issue remains divided[6]. Some have taken the view that DNA is language-like[7]. In the coding
regions the long periods have lower incidence than in the non -coding parts. The Zipf-type fits in
DNA regions (with overlapping n-tuples) have shown that the index αis higher in the non-coding
segments over the coding ones. The averaged αover several overlapping n-tuples is nearer to the
value for natural languages for non-coding segments than th e coding ones[1,7].
The body of evidence presented in support of the language-li ke features of DNA has remained
ambiguous[8]. For one it is not known how the power-law Zipf- behaviour of natural languages is
4connected to the long-range correlations[9]. It is known, f or instance, that pseudorandom sequences
satisfy Zipf- behaviour. Further, it is known that the frequ encies of A, C, G and T vary somewhat
more for the introns and the flanks over the exons[10]. The “lo ng-range” order that is observed for
these noncoding regions may be an outcome of the frequency di fferences. The higher value of the
Zipf index for the noncoding segments may again be ascribed t o these differences in the frequencies
of the bases.
The importance of statistical linguistics as a computation al tool remains insufficiently explored for
DNA sequences. While the Zipf law is probably not connected t o the deeper features of languages
such as the universal grammar, the coherence and the long per iods, it could still be useful. For
instance, the index αof languages could be (and is) used in computer algorithms to identify authors.
The texts generated by authors vary slightly in their Zipf in dex. The index, therefore, identifies the
author. Could one use similar algorithms to identify region s from the genome segments and relate
them to their biological functions?
As precision and reliability are important we have weighed t he merits of power-law fits over
exponential fits. Since we are solely concerned with non-ove rlapping 3-tuples (i.e. the codons), we
find the exponential fits have consistently lower χ2. [Chi-square ( χ2) is the sum of the ratio of the
squared difference between observed value at the ithpoint (o i) and the expected value at the ithpoint
(ei) to the expected value at the ithpoint (e i), i.e., χ2=/summationtext
i(oi−ei)2
ei; where the sum i runs over the
number of points of the fit. The value of χ2depends on the total number of points to be fit minus one,
5sometimes called the degree of freedom, df.] The exponentia ls, therefore, provide better fits. That
the power-law fits for DNA sequences are worse than the expone ntials have also been observed by
others[11]. The power law of Zipf is characterised by two par ameters, the index αand the frequency
of rank one, i.e. f1. The number of parameters for the exponential fit is of intere st to us. The Zipf’s
law is used to find the relationship connecting vocabulary to the text-length. Such connection does
exist for the exponential fit as well.
The parameters of the exponential rank-frequency relation depend crucially on the text-length.
Once this parameter is known, the approximate length of the s egment gets known as well. Indeed,
the exponential fits are largely determined by two quantitie s, the frequency of rank 1, i.e., f1and the
text-length of the sequence. There is however a small part th at is characteristic of the gene. This
signature of the gene is potentially useful in generating al gorithms to identify the gene and relate to
the biological functions.
2 The Approach
Out of the four bases A, C, G & T we have 4 ×4×4 = 64 possible triplets. Three combinations,
namely, TAA, TAG & TGA are the stop condons. Thus 64 - 3=61 is th e meaningful vocabulary.
The codon most frequent has rank n=1, the next most has n=2 and so on. We define frequency,
f, of a particular codon as the number of times it appears in the sequence. [Note this definition is
6different from some of the references where fn=Number of words of rank n
Total number of words. The frequency of rank n is fn.
Here both frequency( f) and rank(n) are dimensionless.
Observations on the CDS reveal that many codons may have the s ame frequency. Note that the
CDS we are dealing with are relatively short sequences of sev eral hundred to several thousand bases.
This problem of multiple codons having the same frequency is called frequency degeneracy.
First, as we consider only codons, 61 in number, the problem o f saturation of vocabulary for
large text-length is clear. However, for most genes we obser ve that the actual usage of codons is
smaller than 61. The codon usage is sometimes referred to as t he vocabulary, i.e. the total number
of different codons, used in the CDS.
From the Zipf’s law [equation(1)] with α=1 we have
ln(fn) =ln(f1)−ln(n)
If we plot ln( fn) vs ln(n) we have a straight line with slope -1 and intercept o n the y-axis at ln( f1).
Clearly, the maximum rank is just equal to f1. When αdeviates from 1, f1and the maximum rank
are connected to each other through α. The maximum rank (i.e. the vocabulary) along with f1(or
α) determine the text-length l, i.e., the total number of trip lets, as follows :
l=f1+f2+f1+f3+....+fn
=f1(1 +1
2α+1
3α+....+1
nα)
7Thus, αmay be thought of as a function of f1and the text-length l. We want to arrive at the
corresponding relation for our exponential fits.
3 The Exponential Fit
All the degenerate frequencies are assigned different rank n umber. Thus if CCG and CAG have the
same frequency of occurrence they belong to two different ran ks (one following the other) in our
work. Therefore, here too, the codon usage, maximum rank and vocabulary are synonymous. The
exponential function that connects frequency to rank is
fn=f1exp{−β(n−1)} (2)
where β, a dimensionless constant for a particular gene, is to be det ermined from the fit.
We have tried this fit function on over 300 CDS. The CDS are sour ced from the EMBL[12] and
the GenBank[13] data bases. Table 1 gives the values of βfor some of the sequences under study.
The plots showing the fit is figure(1).
The index βin the exponential of equation(2) takes different values for the genes. It turns out,
however, that βis not completely a free parameter. Indeed, from Table 1, we n otice that CDS that
have text-lengths and also f1that are close have similar, though not identical, βvalues. Notice, for
instance, the β-globin CDS from the chicken and the clawed frog have the same l and f1, 147 and
9 respectively; whereas the lysozyme CDS from the fish, Cyprinus carpio has 146 as l and 9 as f1.
8Theβvalues for the β-globin CDS of the chicken and the frog are 0.05773 and 0.0577 2; while the
lysozyme CDS, though functionally quite unrelated to the β-globin, has the βvalue of 0.06056. So
the value of βis determined to a considerable extent by f1and the text-length of the sequence, l.
There is but a part in βthat is characteristic of the gene.
4 Plot of βvs.f1
Figure(2) gives plots of βvsf1for four complete CDS coding for α-globin, β-globin, phosphoglycerate
kinase and globulin proteins. The χ2values indicate that the relationship between βandf1is linear
to a good approximation. The plot for each CDS involves data o n the gene from different species.
These are sourced from GenBank. Each of the linear plots are s pecific to the gene. The evolution
of the genes, as we move higher in the evolutionary hierarchy , does not significantly alter the overall
text-length of the CDS regions.
The slope of the globin CDS, the αand the β, are nearly equal. As we show in the subsequent
pages the value of βis considerably determined by f1and l. There is but a small part that is unique
to the gene. For the case of the αand the βglobins notice that the text-lengths of these CDS vary
in a small range between 143 and 147. Table 1 shows that any two quite unrelated CDS can have β
values that are close provided their text-lengths and the f1are nearly equal.
The plots in figure(3) of βvsf1keep the text-length l fixed at 140 for the same four genes.
9Though the closeness in the values of the slope indeed show th e influence of l on the βvalue, the
small differences indicate the presence of the l-independen t part in the βvalue.
That the βvalues are not completely determined by f1and l, but do have a component, albeit
small, coming from the genes is illustrated in our next plot, figure(4). A number of different CDS,
each from a different organism, were chosen and cut at three di fferent text-lengths 30, 140 and 300,
i.e., we considered only the first 30, 140 and 300 triplets res pectively out of the whole CDS. The
plot of βvsf1for these three different text-lengths indicates that when t he text-length is held fixed,
but the genes are varied, the exponential gives a better fit ov er the linear. It is noteworthy that
even though the genes are unrelated in as far as their biologi cal functions are concerned, the codon
distributions, described by the experimental fit of figure(4 ), are not completely unrelated.
Taken together, the two plots, figure(3) and figure(4), tell u s:
(i) When the text-length, l, is held fixed, and the genes are no t varied, the plot of βvsf1is linear
and
(ii) When the text-length, l, is held fixed, and the genes are v aried, the plot of βvsf1is exponential.
Thus, we conclude that the value of βdoes have a part that is gene specific.
105 Plot of βvs l
β, as we have observed from Table 1, depends on f1and l. Beyond that there is the part that is gene
specific. In other words the parameters of the functional fit d o depend, in a small way, on the gene.
This dependence we discuss later. Here, in this section, we c oncern ourselves with the dependence
ofβon the text-length of the CDS.
We plot βvs l keeping f1fixed. The plots in figure(5) show the dependence for four diffe rent
values of f1, namely f1=7,f1=9,f1=20, and f1=38.
In plotting figure(5) we considered the f1values of the natural CDS. We had the option to cut
the CDS into fragments to suit our value of f1. This procedure turned out to be arbitrary as the
f1value may remain fixed over some hundred bases. Cutting into f ragments is nonunique. It was,
therefore, difficult to restrict our study of βvs l for a particular gene. For a specific CDS (from
different species) the text-length does not vary significant ly in most cases. Therefore for a fixed
value of f1the CDS were searched over different genes. Thus f1is held fixed, but genes vary.
Though more data for each gene could have improved the result , nevertheless the relationship
between βand l for fixed f1has a linear trend. As the text-length increases βdecreases. However,
the plots for different values of f1are not parallel. They depend on f1. The slope reaches a maximum
at around f1= 10 and tend to decrease as we go away from f1=10 on either side. For large values
off1, the slopes tend to become parallel.
116 Theory of β
We have seen βdepends on the text-length, l, and the frequency of rank 1, f1.
(1) When the text-length l is held fixed, genes not varied, βdepends linearly on f1. The plot of βvs
f1shows that∆β
∆f1is positive.
(2) When the text-length is kept fixed, but the genes are varie d, the plot of βvsf1show deviations
from linearity. An exponential fit appears more appropriate .
(3) When f1is held fixed (genes are varied as well) the plot of βvs l shows an approximate linear
behaviour.∆β
∆lis negative. Note that, because of the points mentioned earl ier, the variations in l (in
figure 5) are over a rather small range. As a result the full l-d ependence is not clear from figure(5).
In this section we investigate βtheoretically. Let us denote the maximum rank by nmax. Since
the frequency of nmaxis almost always one, we get
1 =f1exp{ −β(nmax−1)} (3)
Or,
nmax=lnf1
β+ 1 (4)
The text-length l is just the sum over all the frequencies. Th us,
l=nmax/summationdisplay
n=1f1.e−β(n−1)(5)
=f1(1−e−β(nmax−1))
1−e−β(6)
12Substituting for nmaxfrom equation(4), we get
l=f1−1
1−e−β(7)
Thus,
β=−ln[1−1
l(f1−1)] (8)
Since, the quantityf1
lis small compared to one, we get, to the first approximation
β=f1−1
l+higher orders (9)
Equation(9) tells us
(i)βvsf1, when l is kept fixed, is linear; the slope is positive.
(ii)βvs l, with f1fixed, is hyperbolic. If the text-length variation is small w e expect an approximate
linear relation with negative slope (as observed in figure(5 )). How good the relation(9) is checked in
Table 1.
While the relation(9) tells us that βis entirely determined by the ratio of f1-1 to l, figure(3)
tells us that this quantity does have a characteristic depen dence on the gene family. We conclude,
therefore, that the relation(9) does not determine βentirely. There is a part that is gene specific.
The theoretical values of β, equation(9), is reasonably close to the values obtained fr om the CDS.
The dependence of βonf1and l of equation(9) is gene-independent. It is the universa l part of β.
The deviation from this universal part, even though small, i s established in figure(3) and figure(4).
13We define the quantity β′that gives a measure of this deviation through the relation:
β= [f1−1
l+1
2(f1−1)2
l2]β′
=βTh. β′(10)
where βTh= [f1−1
l+1
2(f1−1)2
l2].
We have retained the first two orders inf1
l[ of equation(8)]. This is to make sure the higher-orders
inf1
ldo not account for the deviations. The values of β′appear in the last columm of Table 1.
7β,β′and Evolution
We get back to Table 1 for the CDS of α-globin, β-globin, insulin and globulin. We notice the
value of f1increases as we walk up along the ladder of evolution. The inc rease in f1increases β
while the text-length of the CDS does not change significantl y in evolution. The results for insulin
and the globulin CDS [Table 1] carry at least one exception. I nterestingly, for both these CDS, the
exceptional species is the same, the rabbit. The rabbit has f1andβvalues greater than the human
for these two CDS. The number of exceptions increase for the t wo globins. Some fishes show greater
f1(and hence β) values than the amphibian species, the African clawed frog . If we average βfor the
mammals we find it always exceeds the other groups.
On the other hand, if we compare the β′values for each of these four CDS, α-globin and globulin
do not show any clear pattern. In insulin, the β′values increase as we move from fish to mammals
14through amphibia. But the syrian hamster CDS is found to have lower β′than the clawed frog CDS.
Besides the rat has greater β′compared to the human. In β-globin, the Atlantic salmon fish stands
as an exception. Otherwise, the β′value increases from amphibia, bird to mammals. But here the
representatives of amphibia and bird have the same value, an d the lemur exceeds the value of human.
We conclude that the value of β′, though independent of l and f 1, is less species specific; whereas the
value of βdoes have evolutionary content.
8 Gene-Specific Signatures
In figure(2) we showed that βvsf1is a straight line when the genes are not varied. When the gene s
are varied, but the text-length is held constant, the relati onship of βtof1is no longer linear. The
exponential fit is appropriate for this case. This led us to co nclude that there is a part to βthat is
gene-specific.
In figure(3) we plotted βvsf1keeping the genes fixed for different organisms. The slope∆β
∆f1is
a characteristic of the gene. There is a variation in the slop e as we go from one gene to another.
The regular, namely exponential form, obtained in figure(4) in the plot of βvsf1, l being kept
constant, tells us that the variations of β, as we go from one gene to another, is orderly.
βhas a part that is gene independent. We isolate this universa l component of βtheoretically.
This part comes out to be a function of the text-length of the s equence and the frequency of rank 1,
15i.e.f1. The quantity β′, defined in equation(10), measures the deviation of the actu alβfrom this
universal, gene-independent, contribution given in equat ion(10). If the gene specific features are not
dominant, β′should be close to one. Table 1 gives us the values of β′. Clearly, the gene specific
components in βcould be as high as 40% (as in insulin). We are led to conclude t hat the methods of
statistical linguistics, of the Zipf variety, has the poten tial in algorithms to identify genes from the
databases.
The quantity β′that isolates the gene-specific components of βis however not unique to genes.
Observations on β′(Table 1) show that the range of variations in β′do overlap for different genes.
There continues to be undetermined fluctuations in the value s ofβ′. Work is currently in progress
to isolate the unique gene-identifying signatures in the Zi pf-approach.
16References
[1] R. N. Mantegna, S. V. Buldyrev, A. L. Goldberger, S. Havli n, C.-K. Peng, M. Simons, and H.
E. Stanley, Phys. Rev. Lett. 73, 3169 (1994).
[2] D. Welsh, Codes and Cryptography (Oxford University Press, Oxford 1988).
[3] E. W. Montroll and M. F. Shlesinger, in Nonequilibrium Phenomena II From Stochastics to
Hydrodynamics , edited by J. L. Lebowitz and E. W. Montroll (North Holland, A msterdam,
1984).
[4] G. K. Zipf, Human behavior and the Principle of Least Effort (Addison-Wesely Press, Cambridge
MA, 1949).
[5] R. F. Voss, Phys. Rev. Lett. 68, 3805 (1992); W. Li and K. Kaneko, Europhys. Lett. 17, 655
(1992); C.-K. Peng, S. V. Buldyrev, S. Havlin, M. Simons, H. E . Stanley, and A. L. Goldberger,
Nature 365, 168 (1992).
[6] S. Karlin and V. Brendel, Science 259, 677 (1993); D. Larhammar and C. A. Chatzidimitriou-
Dreissman, Nucl. Acid Res. 21, 5167 (1993); S. Nee, Nature 357, 450 (1992).
[7] A. Czirok, R. N. Mantegna, S. Havlin, and H. E. Stanley, Ph ys. Rev. E 52, 446 (1995).
17[8] B. Mandelbrot, in Structures of Language , edited by R. Jacobson (AMS, New York 1961); N.
E. Israeloff, M. Kagalenko, and K. Chan, Phys. Rev. Lett. 76, 1976 (1996); R. F. Voss, Phys.
Rev. lett. 76, 1978 (1996); S. Bonhoeffer, A. V. M. Herz, Maarten C. Boerlij st, S. Nee, M. A.
Nowak, and R. M. May Phys. Rev. Lett. 76, 1977 (1996).
[9] I. Kanter and D. A. Kessler, Phys. Rev. Lett. 74, 4559 (1995).
[10] See S. Bonhoeffer et al in ref. (8).
[11] A. A. Tsonis, J. B. Elsner, and P. A. Tsonis, J. Theor. Bio l.76, 25 (1997).
[12] URL for EMBL database: http://www.embl-heidelberg.d e/
[13] URL for GenBank database: http://www.ncbi.nlm.nih.g ov/Entrez/
18Figure Legends
Figure 1. The plots of frequency (f) vs. rank (n) are the expon ential functions (equation 2). Here
different codons with the same frequency of occurrence are gi ven consecutive ranks. The data cor-
responds to the α-globin CDS from Duck (Acc. No. J00923). The βvalue comes out to be 0.06801.
The text-length, l of the CDS is 143; f 1is 10.
Figure 2. βis plotted as a function of f 1for the natural CDS of 4 different proteins from vari-
ous species. The relationship turns out to be linear.
symbol CDS range of l m c sd
⋆ α -globin : 142-151 0.0083 -0.0136 0.0029
◦ β-globin : 146-149 0.0092 -0.0258 0.0014
△ phosphoglycerate kinase : 417-418 0.0031 -0.0169 0.0008
▽ Globulin : 399-413 0.0036 -0.0277 0.0022
[Keys: m →slope; c →constant; sd →standard deviation]
19Figure 3. The text-length (l) is kept fixed at 140 to plot βas a function of f 1for the CDS of the
same 4 proteins as in figure 2. The best fit here is a linear one.
symbol CDS m c sd
⋆ α -globin : 0.0080 -0.0093 0.0015
◦ β-globin : 0.0095 -0.0239 0.0013
△ phosphoglycerate kinase : 0.0094 -0.0167 0.0029
▽ Globulin : 0.0097 -0.0250 0.0007
[Keys: m →slope; c →constant; sd →standard deviation]
20Figure 4. βis plotted as a function of f 1at 3 different values of l. Here a number of different CDS
from various species are chosen and cut at 3 text-lengths 30, 140 and 300. For text-lengths 30 and
140, 15 CDS were chosen (GenBank accession numbers are AF007 570, L37416, M16024, AF053332,
AF001310, M15387, V00410, M15052, L47295, X07083, M59772, J05118, AF056080, AF170848 and
M64656), while for text-length 300, 13 CDS were chosen (GenB ank accession numbers are U02504,
AF000953, M73993, AF054895, AF076528, AF053332, M15052, U 65090, Z54364, U53218, AB013732,
M15668 and U69698). Unlike figure 2 and figure 3, the exponenti al gives the better fit over the linear.
The fit function: Y=Y0 + A.e(X/t).
symbol l Y0 A t
⋆ 30 0.0236 0.0357 2.7704
◦ 140 0.0324 0.0481 12.8086
△ 300 0.0018 0.0133 12.4689
21Figure 5. βis plotted as a function of l for 4 different values of f 1. For each f 1, natural CDS
of that particular f 1, are considered. The relationship between βand l for fixed f 1comes out to be
linear.
symbol f 1 m c sd
⋆ 7 -4.84 ×10−40.1154 6.89 ×10−4
◦ 9 -8.54 ×10−40.1841 0.0021
△ 20 -1.63 ×10−40.1133 7.14 ×10−4
▽ 38 -1.33 ×10−40.1458 8.85 ×10−4
[Keys: m →slope; c →constant; sd →standard deviation]
22Table 1: Theβvalues for some CDS from different organisms. The l and f1stand for
the total number of the triplet codons and the frequency of th e most frequent codon
respectively. The χ2value signifies how good the fit is and the degrees of freedom,
denoted by df, is simply one less than the total number of rank s. The βThandβ′are
explained in equation (10).
Protein Organism Accession no. lf1 β χ2df βTh β′
α-globin Ark Clam X71386 151 70.04221 0.137 520.0405 1.0415
Rainbow Trout D88114 144 90.05893 0.202 430.0571 1.0321
Cyprinus carpio AB004739 144 100.06890 0.450 450.0645 1.0691
Black Rockcod AF049916 144 110.07649 0.594 410.0719 1.0646
Duck J00923 143 100.06801 0.105 400.0645 1.0551
Pigeon X56349 143 100.06872 0.155 400.0649 1.0584
23Protein Organism Accession no. lf1 β χ2df βTh β′
α-globin Chicken V00410 142 100.07251 0.893 460.0654 1.1089
House Mouse V00714 142 90.06037 0.192 450.0579 1.0421
Rhesus Monkey J004495 143 100.06568 0.353 370.0649 1.0117
Rabbit M11113 143 100.06661 0.188 380.0649 1.0260
Norway Rat U62315 143 100.06897 0.386 430.0649 1.0624
Otolemur M29648 143 130.09286 0.727 380.0874 1.0620
Grevy’s Zebra U70191 143 130.09678 0.272 400.0874 1.1068
Human V00488 143 140.10045 0.007 350.0950 1.0569
Orangutan M12157 143 150.11022 0.487 370.1027 1.0732
Horse M17902 143 150.11385 0.399 400.1027 1.1086
24Protein Organism Accession no. lf1 β χ2df βTh β′
α-globin Sheep X70215 143 170.13269 1.153 380.1182 1.1231
Goat J00043 143 170.13675 1.432 410.1182 1.1574
Salamander M13365 144 90.06240 0.489 510.0571 1.0928
Clawed Frog X14260 142 100.07394 0.411 480.0654 1.1308
β-globin Atlantic Salmon X69958 149 110.07382 0.543 430.0694 1.0643
Clawed Frog Y00501 147 90.05772 0.196 450.0559 1.0326
Chicken V00409 147 90.05773 0.324 460.0559 1.0327
House Mouse V00722 147 80.05075 0.099 460.0488 1.0410
Rabbit V00882 146 90.06091 0.133 460.0563 1.0817
Rat X06701 147 100.06849 0.545 430.0631 1.0856
25Protein Organism Accession no. lf1 β χ2df βTh β′
β-globin Oppossum J03643 148 120.08164 2.183 450.0771 1.0592
Sheep X14727 146 120.08413 0.351 390.0782 1.0761
Goat M15387 146 130.09558 0.406 420.0856 1.1170
Lemur M15734 148 140.10743 1.375 420.0917 1.1715
Human AF007546 148 150.11245 1.530 390.0991 1.1349
Insulin Salmon J00936 106 70.06425 0.490 450.0582 1.1040
Clawed Frog M24443 107 80.07922 0.841 460.0676 1.1726
Syrian Hamster M26328 111 90.08656 0.703 420.0747 1.1592
Guinea Pig K02233 111 90.09220 0.815 450.0747 1.2348
Owl Monkey J02989 109 130.14189 1.667 390.1162 1.2216
Octodon degus M57671 110 120.14122 1.322 440.1050 1.345
26Protein Organism Accession no. lf1 β χ2df βTh β′
Insulin Rat J00747 111 120.14785 2.192 440.1040 1.4216
Human J00265 111 130.17379 2.795 420.1240 1.4012
Rabbit U03610 111 180.21253 2.940 320.1648 1.2890
Globulin Pig AF204929 413 180.03901 0.860 580.0420 0.9286
Bovine AF204928 412 190.04173 1.227 570.0446 0.9348
Djungarian Hamster U16673 400 250.06195 5.871 590.0618 1.0024
Norway Rat NM012650 404 260.06505 7.256 590.0638 1.0196
House Mouse NM011367 404 280.07215 9.484 580.0691 1.0447
Human NM001040 403 330.09463 18.202 600.1112 0.8511
Rabbit AF144711 399 390.12568 19.189 600.0998 1.2596
27Protein Organism Accession no. lf1 β χ2df βTh β′
Heat shock Babesia microti U53448 646 350.05127 0.867 550.0540 0.9491
protein 70 Pacific Oyster AF144646 660 360.05235 1.576 580.0544 0.9616
Human U56725 640 400.06454 3.140 590.0628 1.0277
Mouse L27086 642 380.06131 2.627 600.0593 1.0341
Chinook Salmon U35064 645 420.06640 1.533 600.06559 1.0124
Rat L16764 642 480.07369 6.523 400.0759 0.9710
Phospho- Human X80497 1236 510.03709 10.391 610.0413 0.8990
rylase Rabbit X60421 1236 580.04458 7.694 610.0472 0.9449
kinase Mouse X74616 1242 470.03244 8.927 610.0377 0.8598
Glycogen Human J04501 738 440.05968 6.984 600.0599 0.9952
synthase Mouse U53218 739 370.04718 7.113 600.0499 0.9455
28Protein Organism Accession no. lf1 β χ2df βTh β′
Glycogen Rabbit AF017114 736 490.06603 3.001 590.0674 0.9804
synthase Rat J05446 704 280.03483 1.945 600.0391 0.8910
Troponin C Chicken M16024 162 170.12374 1.577 450.1037 1.1938
Human M22307 161 230.19581 3.333 400.1460 1.3413
Mouse M57590 161 210.17806 4.565 420.1319 1.3496
Rabbit J03462 161 240.19294 3.964 360.1531 1.2606
Clawed Frog AB003080 162 160.12250 1.370 470.0969 1.2645
Albumin Bovine M73993 608 380.06437 9.754 590.0627 1.0265
Human NM001133 600 340.05643 9.235 580.0565 0.9986
Clawed Frog M18350 607 410.06845 15.699 560.0681 1.0056
29Protein Organism Accession no. lf1 β χ2df βTh β′
Lysozyme Anopheles gambiae U28809 141 110.08073 0.561 450.0734 1.0993
Bovine M95099 148 70.04359 0.094 510.0414 1.0539
Cyprinus carpio AB027305 146 90.06056 0.390 470.0563 1.0757
Human M19045 149 70.04341 0.122 520.0411 1.0567
Pig U44435 149 80.04946 0.503 510.0481 1.0287
Lactate Alligator L79952 334 160.05460 0.441 580.0459 1.1890
dehydro- Cyprinus carpio AF076528 334 230.0708 2.166 530.0680 1.0401
genase Human U13680 333 200.05961 3.075 570.0587 1.0157
Pig U95378 333 190.05461 2.347 570.0555 0.9838
Pigeon L79957 334 190.05536 2.110 560.0553 1.0003
Clawed Frog AF070953 333 200.05831 2.010 530.0586 0.9935
30Protein Organism Accession no. lf1 β χ2df βTh β′
Phospho- Candida albicans U25180 418 340.08126 2.388 380.0821 0.9901
glycerate Leishmania major L25120 418 340.08677 1.132 560.0821 1.0573
kinase Mouse M15668 418 230.05298 1.155 580.0540 0.9807
Rat M31788 418 230.05374 1.825 600.0540 0.9948
Schistosoma mansoni L36833 417 290.07284 5.498 600.0694 1.0494
Carboxy- Aedes aegypti AF165923 428 200.04373 1.785 610.0454 0.9636
peptidase Bovine M61851 420 220.05170 0.417 590.0512 1.0088
A Human M27717 418 200.04477 1.128 590.0465 0.9630
Mouse J05118 418 230.05124 6.547 580.0540 0.9485
310 10 20 30 400246810
Figure 1f
n68101214161820222426283032340.030.040.050.060.070.080.090.100.110.12
Figure 2β
F16 8 10 12 140.040.050.060.070.080.090.100.110.12
Figure 3β
F10 510152025300.050.100.150.200.25
Figure 4β
F11002003004005006007000.0400.0450.0500.0550.0600.0650.0700.0750.0800.0850.0900.095
Figure 5β
l |
arXiv:physics/0102022v1 [physics.atom-ph] 9 Feb 2001Three-potential formalism for the three-body scattering p roblem with attractive
Coulomb interactions
Z. Papp1,2, C-.Y. Hu1, Z. T. Hlousek1, B. K´ onya2and S. L. Yakovlev3
1Department of Physics and Astronomy, California State Univ ersity, Long Beach, CA 90840, USA
2Institute of Nuclear Research of the Hungarian Academy of Sc iences, Debrecen, Hungary
3Department of Mathematical and Computational Physics, St. Petersburg State University, St. Petersburg, Russia
(February 2, 2008)
A three-body scattering process in the presence of Coulomb i nteraction can be decomposed formally
into a two-body single channel, a two-body multichannel and a genuine three-body scattering. The
corresponding integral equations are coupled Lippmann-Sc hwinger and Faddeev-Merkuriev integral
equations. We solve them by applying the Coulomb-Sturmian s eparable expansion method. We
present elastic scattering and reaction cross sections of t hee++Hsystem both below and above
theH(n= 2) threshold. We found excellent agreements with previous calculations in most cases.
PACS number(s): 31.15.-p, 34.10.+x, 34.85.+x, 21.45.+v, 0 3.65.Nk, 02.30.Rz, 02.60.Nm
The three-body Coulomb scattering problem is one
of the most challenging long-standing problems of non-
relativistic quantum mechanics. The source of the dif-
ficulties is related to the long-range character of the
Coulomb potential. In the standard scattering theory
it is supposed that the particles move freely asymptoti-
cally. That is not the case if Coulombic interactions are
involved. As a result the fundamental equations of the
three-body problems, the Faddeev-equations, become ill-
behaved if they are applied for Coulomb potentials in a
straightforward manner.
The first, and formally exact, approach was proposed
by Noble [1]. His formulation was designed for solv-
ing the nuclear three-body Coulomb problem, where all
Coulomb interactions are repulsive. The interactions
were split into short-range and long-range Coulomb-like
parts and the long-range parts were formally included in
the ”free” Green’s operator. Therefore the corresponding
Faddeev-Noble equations become mathematically well-
behaved and in the absence of Coulomb interaction they
fall back to the standard equations. However, the asso-
ciated Green’s operator is not known. This formalism,
as presented at that time, was not suitable for practical
calculations.
In Noble’s approach the separation of the Coulomb-like
potential into short-range and long-range parts were car-
ried out in the two-body configuration space. Merkuriev
extended the idea of Noble by performing the splitting
in the three-body configuration space. This was a cru-
cial development since it made possible to treat attrac-
tive Coulomb interactions on an equal footing as repul-
sive ones. This theory has been developed using in-
tegral equations with connected (compact) kernels and
transformed into configuration-space differential equa-
tions with asymptotic boundary conditions [2]. In prac-
tical calculations, so far only the latter version of the
theory has been considered. The primary reason is that
the more complicated structure of the Green’s operators
in the kernels of the Faddeev-Merkuriev integral equa-tions has not yet allowed any direct solution. However,
use of integral equations is a very appealing approach
since no boundary conditions are required.
Recently, one of us has developed a novel method for
treating the three-body problem with repulsive Coulomb
interactions in three-potential picture [3]. In this ap-
proach a three-body Coulomb scattering process can be
decomposed formally into a two-body single channel, a
two-body multichannel and a genuine three-body scat-
tering. The corresponding integral equations are coupled
Lippmann-Schwinger and Faddeev-Noble integral equa-
tions, which were solved by using the Coulomb-Sturmian
separable expansion method. The approach was tested
first for bound-state problems [4] with repulsive Coulomb
plus nuclear potential. Then it was extended to calculate
p−dscattering at energies below the breakup threshold
[3] and more recently we have used the method to cal-
culate resonances of three- αsystems [5]. Also atomic
bound-state problems with attractive Coulomb interac-
tions have been considered [6]. These calculations showed
an excellent agreement with the results of other well es-
tablished methods. The efficiency and the accuracy of
the method was demonstrated.
The aim of this paper is to generalize this method
for solving the three-body Coulomb problem with re-
pulsive and attractive Coulomb interactions. We com-
bine the concept of three-potential formalism with
the Merkuriev’s splitting of the interactions and solve
the resulting set of Lippmann-Schwinger and Faddeev-
Merkuriev integral equations by applying the Coulomb-
Sturmian separable expansion method. In this paper
we restrict ourselves to energies below the three-body
breakup threshold.
1I. INTEGRAL EQUATIONS OF THE
THREE-POTENTIAL PICTURE
We consider a three-body system with Hamiltonian
H=H0+vC
α+vC
β+vC
γ, (1)
whereH0is the three-body kinetic energy operator and
vC
αdenotes the Coulomb-like interaction in subsystem
α. The potential vC
αmay have repulsive or attractive
Coulomb tail and any short-range component. We use
the usual configuration-space Jacobi coordinates xαand
yα;xαis the coordinate between the pair ( β,γ) andyα
is the coordinate between the particle αand the center
of mass of the pair ( β,γ). Thus the potential vC
α, the
interaction of the pair ( β,γ), appears as vC
α(xα). We
also use the notation X={xα,yα} ∈R6.
A. Merkuriev’s cut of the Coulomb potential
The Hamiltonian (1) is defined in the three-body
Hilbert space. The two-body potential operators are for-
mally embedded in the three-body Hilbert space
vC=vC(x)1y. (2)
Merkuriev introduced a separation of the three-body con-
figuration space into different asymptotic regions. The
two-body asymptotic region Ω αis defined as a part of
the three-body configuration space where the conditions
|xα|<x0
α(1 +|yα|/y0
α)1/ν, (3)
withx0
α,y0
α>0 andν >2, are satisfied. He proposed to
split the Coulomb interaction in the three-body configu-
ration space into short-range and long-range terms
vC
α=v(s)
α+v(l)
α, (4)
where the superscripts sandlindicate the short- and
long-range attributes, respectively. The splitting is car -
ried out with the help of a splitting function ζ,
v(s)(x,y) =vC(x)ζ(x,y), (5)
v(l)(x,y) =vC(x)[1−ζ(x,y)]. (6)
The function ζis defined such that
ζ(x,y)X→∞− − − − →/braceleftbigg
1, X∈Ωα
0 otherwise.(7)
In practice, in the configuration-space differential equa-
tion approaches, usually the functional form
ζ(x,y) = 2//braceleftbig
1 +/bracketleftbig
(x/x0)ν/(1 +y/y0)/bracketrightbig/bracerightbig
,(8)
was used.The long-range Hamiltonian is defined as
H(l)=H0+v(l)
α+v(l)
β+v(l)
γ, (9)
and its resolvent operator is
G(l)(z) = (z−H(l))−1. (10)
Then, the three-body Hamiltonian takes the form
H=H(l)+v(s)
α+v(s)
β+v(s)
γ. (11)
In the conventional Faddeev theory the wave function
components are defined by
|ψα/an}bracketri}ht= (z−H0)−1vα|Ψ/an}bracketri}ht, (12)
wherevαis a short-range potential and |ψα/an}bracketri}htis the Fad-
deev component of the total wave function |Ψ/an}bracketri}ht. While
the total wave function |Ψ/an}bracketri}ht, in general, has three dif-
ferent kind of two-body asymptotic channels, |ψα/an}bracketri}htpos-
sesses only α-type two-body asymptotic channel. The
other channels are suppressed by the short-range poten-
tialvα. This procedure is called asymptotic filtering and
it guarantees the asymptotic orthogonality of the Fad-
deev components [7].
The aim of the Merkuriev procedure was to formally
obtain a three-body Hamiltonian with short-range poten-
tialsv(s)and long-range Hamiltonian H(l)in order that
we can repeat the procedure of the conventional Faddeev
theory. The total wave function |Ψ/an}bracketri}htis split into three
components,
|Ψ/an}bracketri}ht=|ψα/an}bracketri}ht+|ψβ/an}bracketri}ht+|ψγ/an}bracketri}ht, (13)
with components defined by
|ψα/an}bracketri}ht=G(l)v(s)
α|Ψ/an}bracketri}ht. (14)
This procedure is an example of asymptotic filtering. The
short-range potential v(s)
αacting on |Ψ/an}bracketri}htsuppresses the
possibleβandγasymptotic two-body channels, pro-
videdG(l)itself does not introduce any new two-body
asymptotic channels. With the Merkuriev splitting this
is avoided because H(l)does not have two-body asymp-
totic channels even if some of the long-range potentials
have attractive Coulomb tail. In the attractive case v(l)
appears as a valley along the y=xνparabola-like curve
with Coulomb-like asymptotic behavior in xat any finite
y. (See Figs. 1 and 2 for the short- and long-range parts,
respectively). However, as y→ ∞ the depth of the valley
goes to zero, consequently the two-body bound states are
pushed up, and finally the system does not have any two-
body asymptotic channels. We note that the Merkuriev
formalism contains the Noble’s in the limit y0→ ∞.
B. The three-potential picture
In Ref. [3] the three body scattering problem with
repulsive Coulomb interactions were considered in the
2three-potential picture. In this picture the scattering
process can be decomposed formally into three consec-
utive scattering processes: a two-body single channel, a
two-body multichannel and a genuine three-body scatter-
ing. This formalism also provides the integral equations
and the method of constructing the S-matrix. Below we
adapt this formalism to attractive Coulomb interactions
along the Merkuriev approach.
The asymptotic Hamiltonian is defined as
Hα=H0+vC
α, (15)
and the asymptotic states are the eigenstates of Hα
Hα|Φα/an}bracketri}ht=E|Φα/an}bracketri}ht, (16)
where /an}bracketle{txαyα|Φα/an}bracketri}ht=/an}bracketle{tyα|χα/an}bracketri}ht/an}bracketle{txα|φα/an}bracketri}htis a product of a
scattering state in coordinate yαand a bound state in
the two-body subsystem xα.
We define the two asymptotic long-range Hamiltonians
as
H(l)
α=H0+vC
α+v(l)
β+v(l)
γ (17)
and
/tildewideHα=H0+vC
α+u(l)
α, (18)
whereu(l)
αis an auxiliary potential in coordinate yα, and
it is required to have the asymptotic form
u(l)
α∼Zα(Zβ+Zγ)/yα (19)
asyα→ ∞. In fact, u(l)
αis an effective Coulomb-like
interaction between the center of mass of the subsystem
α(with charge Zβ+Zγ) and the third particle (with
chargeZα). We introduced this potential in order that
we compensate the long range Coulomb tail of v(l)
β+v(l)
γ
in Ω α.
Let us introduce the resolvent operators:
G(z) = (z−H)−1, (20)
G(l)
α(z) = (z−H(l)
α)−1, (21)
/tildewideGα(z) = (z−/tildewideHα)−1. (22)
The operator G(l)
αis the long-range channel Green’s oper-
ator and/tildewideGαis the channel distorted long-range Green’s
operator. These operators are connected via the follow-
ing resolvent relations:
G(z) =G(l)
α(z) +G(l)
α(z)VαG(z), (23)
G(l)
α(z) =/tildewideGα(z) +/tildewideGα(z)UαG(l)
α(z), (24)whereVα=v(s)
β+v(s)
γandUα=v(l)
β+v(l)
γ−u(l)
α.
The scattering state, which evolves from the asymp-
totic state |Φα/an}bracketri}htunder the influence of H, is given as
|Ψ(±)
α/an}bracketri}ht= lim
ε→0iεG(Eα±iε)|Φα/an}bracketri}ht. (25)
Similarly, we can define the following auxiliary scattering
states
|Φ(l)(±)
α/an}bracketri}ht= lim
ε→0iεG(l)
α(E±iε)|Φα/an}bracketri}ht (26)
and
|/tildewideΦ(±)
α/an}bracketri}ht= lim
ε→0iε/tildewideGα(E±iε)|Φα/an}bracketri}ht, (27)
which describe scattering processes due to Hamiltonians
H(l)
αand/tildewideHα, respectively.
The S-matrix elements of scattering processes are ob-
tained from the resolvent of the total Hamiltonian by the
reduction technique [8]
Sβj,αi= lim
t→∞lim
ε→0iεei(Eβj−Eαi)t/an}bracketle{tΦβj|G(Eαi+ iε)|Φαi/an}bracketri}ht.
(28)
The subscript iandjdenotes the i-th andj-th eigen-
states of the corresponding subsystems, respectively. If
we substitute (23) into (28) we get the following two
terms:
S(1,2)
βj,αi= lim
t→∞lim
ε→0iεei(Eβj−Eαi)t/an}bracketle{tΦβj|G(l)
α(Eαi+ iε)|Φαi/an}bracketri}ht
(29)
S(3)
βj,αi= lim
t→∞lim
ε→0iεei(Eβj−Eαi)t/an}bracketle{tΦβj|G(l)
α(Eαi+ iε)
VαG(Eαi+ iε)|Φαi/an}bracketri}ht. (30)
Substituting Eq. (24) into (29), the first term yields two
more terms
S(1)
βj,αi= lim
t→∞lim
ε→0iεei(Eβj−Eαi)t/an}bracketle{tΦβj|/tildewideGα(Eαi+ iε)|Φαi/an}bracketri}ht
(31)
S(2)
βj,αi= lim
t→∞lim
ε→0iεei(Eβj−Eαi)t/an}bracketle{tΦβj|/tildewideGα(Eαi+ iε)
UαG(l)
α(Eαi+ iε)|Φαi/an}bracketri}ht. (32)
Using of the properties of the resolvent operators the
limits can be performed and we arrive at the following,
physically plausible, result. The first term, S(1)
βj,αi, is the
S-matrix of a two-body single channel scattering on the
potentialu(l)
α
S(1)
βj,αi=δβαδjiS(u(l)
α). (33)
Ifu(l)
αis a pure Coulomb interaction S(u(l)
α) falls back
to the S-matrix of the Rutherford scattering, if u(l)
αis
identically zero S(1)
βj,αiequals to unity. The second term,
3S(2)
βj,αi, describes a two-body multichannel scattering on
the potential Uα
S(2)
βj,αi=−2πiδβαδ(Eβj−Eαi)/an}bracketle{t/tildewideΦ(−)
βj|Uα|Φ(l)(+)
αi/an}bracketri}ht.(34)
The third term gives account of the complete three-body
dynamics
S(3)
βj,αi=−2πiδ(Eβj−Eαi)/an}bracketle{tΦ(l)(−)
βj|Vα|Ψ(+)
αi/an}bracketri}ht.(35)
C. Lippmann-Schwinger integral equation for |Φ(l)
α/angbracketright
Starting from the definition of |Φ(l)
α/an}bracketri}ht, Eq. (26), by uti-
lizing the resolvent relation (24) and the definition (27),
we easily derive a Lippmann-Schwinger equation
|Φ(l)(±)
α/an}bracketri}ht=|/tildewideΦ(±)
α/an}bracketri}ht+/tildewideGα(E±iǫ)Uα|Φ(l)(±)
α/an}bracketri}ht,(36)
where |/tildewideΦ(±)
α/an}bracketri}htare given by
|/tildewideΦ(±)
α/an}bracketri}ht=|/tildewideχ(±)
α/an}bracketri}ht|φα/an}bracketri}ht. (37)
The state |/tildewideχ(±)
α/an}bracketri}htis a scattering state in the Coulomb-like
potentialu(l)
α(yα).
D. Faddeev-Merkuriev integral equations for the
wave function components
The integral equations for the wave function |Ψ(±)
α/an}bracketri}htare
arrived at by combining the resolvent relation (23) and
Eq. (25). In this case however we have three resolvent
relations and therefore we obtain a triad of Lippmann-
Schwinger equations
|Ψ(±)
α/an}bracketri}ht=|Φ(l)(±)
α/an}bracketri}ht+G(l)
α(E±i0)Vα|Ψ(±)
α/an}bracketri}ht (38)
|Ψ(±)
α/an}bracketri}ht= G(l)
β(E±i0)Vβ|Ψ(±)
α/an}bracketri}ht (39)
|Ψ(±)
α/an}bracketri}ht= G(l)
γ(E±i0)Vγ|Ψ(±)
α/an}bracketri}ht. (40)
Although these three equations together provide unique
solutions [9], their kernels are not connected therefore
they cannot be solved by iterations. The way out of
the problem is to use the Faddeev decomposition which
leads to equations with connected kernels, thus they are
effectively Fredholm-type integral equations.
Multiplying each elements of the triad from left by
G(l)v(s)
αand utilizing (14) we get the set of Faddeev-
Merkuriev integral equations for the components
|ψ(±)
α/an}bracketri}ht=|Φ(l)(±)
α/an}bracketri}ht+G(l)
α(E±i0)v(s)
α[|ψ(±)
β/an}bracketri}ht+|ψ(±)
γ/an}bracketri}ht] (41)
|ψ(±)
β/an}bracketri}ht= G(l)
β(E±i0)v(s)
β[|ψ(±)
γ/an}bracketri}ht+|ψ(±)
α/an}bracketri}ht] (42)
|ψ(±)
γ/an}bracketri}ht= G(l)
γ(E±i0)v(s)
γ[|ψ(±)
α/an}bracketri}ht+|ψ(±)
β/an}bracketri}ht].(43)Merkuriev showed that after a certain number of iter-
ations these equations were reduced to Fredholm inte-
gral equations of the second kind with compact kernels
for all energies, including energies below ( E < 0) and
above (E > 0) the three-body breakup threshold [2].
Thus all the nice properties of the original Faddeev equa-
tions established for short-range interactions remain val id
also for the case of Coulomb-like potentials. We note
that the triad of Lippmann-Schwinger equations and the
set of Faddeev equations describe the same physics, the
equations have identical spectra and in fact, the Faddeev
equations are the adjoint representations of the triad of
Lippmann-Schwinger equations [10].
Utilizing the properties of the Faddeev components the
matrix elements in (35) can be rewritten in a form better
suited for numerical calculations
/an}bracketle{tΦ(l)(−)
βj|Vα|Ψ(+)
αi/an}bracketri}ht=/summationdisplay
γ/negationslash=β/an}bracketle{tΦ(l)(−)
βj|v(s)
β|ψ(+)
γi/an}bracketri}ht.(44)
Summarizing, in the three-potential formalism, start-
ing from |/tildewideΦ(±)
α/an}bracketri}ht, by solving a Lippmann-Schwinger equa-
tion, we determine |Φ(l)(±)
α/an}bracketri}ht. Then from |Φ(l)(+)
α/an}bracketri}ht, by
solving the set of Faddeev-Merkuriev equations, we de-
termine the components |ψ(+)
α/an}bracketri}ht. Finally using Eqs. (34)
and (44) we construct the S-matrix.
II. COULOMB-STURMIAN SEPARABLE
EXPANSION APPROACH TO THE
THREE-BODY INTEGRAL EQUATIONS
In order to solve operator equations in quantum me-
chanics one needs a suitable representation for the oper-
ators. For solving integral equations it is especially ad-
vantageous if one uses a representation where the Green’s
operator is simple. For the two-body Coulomb Green’s
operator there exists a Hilbert-space basis in which its
representation is very simple. This is the Coulomb-
Sturmian (CS) basis. In this representation-space the
Coulomb Green’s operator can be given by simple and
well-computable analytic functions [11]. This basis forms
a countable set. If we represent the interaction term on
a finite subset of the basis it looks like a kind of sep-
arable expansion of the potential, and so the integral
equation becomes a set of algebraic equations which can
then be solved without any further approximation. The
completeness of the basis ensures the convergence of the
method.
This approximation scheme has been thoroughly tested
in two-body calculations. Bound- and resonant-state cal-
culations were presented first [11]. Then the method
was extended to scattering states [12]. Since only the
asymptotically irrelevant short-range interaction is ap-
proximated, the correct Coulomb asymptotic is guaran-
teed [13]. A recent account of this method is presented
in Ref. [14]. The method also proved to be very efficient
in solving three-body Faddeev-Noble integral equations
4for bound- [4] and scattering-state [3] problems with re-
pulsive Coulomb interactions.
In subsection A we define the basis states in two- and
three-particle Hilbert space. In subsection B we review
some of the most important formulae of the two-body
problem. In subsections C and D we describe the calcu-
lation of the S-matrix and the solution of the Faddeev-
Merkuriev integral equations. We follow the line pre-
sented in Ref. [3].
A. Basis states
The Coulomb-Sturmian functions [15] in some angular
momentum state lare defined as
/an}bracketle{tr|nl/an}bracketri}ht=/bracketleftbiggn!
(n+ 2l+ 1)!/bracketrightbigg1/2
(2br)l+1exp(−br)L2l+1
n(2br),
(45)
n= 0,1,2,.... Here,Lrepresents the Laguerre polyno-
mials andbis a fixed parameter. In an angular momen-
tum subspace they form a complete set
1= lim
N→∞N/summationdisplay
n=0|/tildewidenl/an}bracketri}ht/an}bracketle{tnl|= lim
N→∞1N, (46)
where |/tildewidenl/an}bracketri}htin configuration-space representation reads
/an}bracketle{tr|/tildewidenl/an}bracketri}ht=/an}bracketle{tr|nl/an}bracketri}ht/r.
The three-body Hilbert space is a direct sum of two-
body Hilbert spaces. Thus, the appropriate basis in an-
gular momentum representation should be defined as a
direct product
|nνlλ/an}bracketri}htα=|nl/an}bracketri}htα⊗ |νλ/an}bracketri}htα,(n,ν= 0,1,2,...),(47)
with the CS states of Eq. (45). Here landλdenote the
angular momenta associated with Jacobi coordinates x
andy, respectively. In our three-body Hilbert space basis
we take bipolar harmonics in the angular variables and
CS functions in the radial coordinates. The complete-
ness relation takes the form (with angular momentum
summation implicitly included)
1= lim
N→∞N/summationdisplay
n,ν=0|/tildewidestnνlλ/an}bracketri}htα α/an}bracketle{tnνlλ|= lim
N→∞1α
N,(48)
where /an}bracketle{txαyα|/tildewidestnνlλ/an}bracketri}htα=/an}bracketle{txαyα|nνlλ/an}bracketri}htα/(xαyα). It should
be noted that in the three-particle Hilbert space we can
introduce three equivalent basis sets which belong to frag-
mentationα,βandγ.
B. Coulomb-Sturmian separable expansion in
two-body scattering problems
Let us study a two-body case of short-range plus
Coulomb-like interactions
vl=v(s)
l+vC(49)and consider the inhomogeneous Lippmann-Schwinger
equation for the scattering state |ψl/an}bracketri}htin some partial wave
l
|ψl/an}bracketri}ht=|φC
l/an}bracketri}ht+gC
l(E)v(s)
l|ψl/an}bracketri}ht. (50)
Here|φC
l/an}bracketri}htis the regular Coulomb function, gC
l(E) is the
two-body Coulomb Green’s operator
gC
l(E) = (E−h0
l−vC)−1(51)
with the free Hamiltonian h0
l. We make the following
approximation on Eq. (50)
|ψl/an}bracketri}ht=|ϕC
l/an}bracketri}ht+gC
l(E)1Nv(s)
l1N|ψl/an}bracketri}ht, (52)
i.e. we approximate the short-range potential v(s)
lby a
separable form
v(s)
l= lim
N→∞1Nv(s)
l1N≈1Nv(s)
l1N=N/summationdisplay
n,n′=0|/tildewidenl/an}bracketri}htv(s)
l/an}bracketle{t/tildewidern′l|
(53)
where the matrix
v(s)
lnn′=/an}bracketle{tnl|v(s)
l|n′l/an}bracketri}ht. (54)
These matrix elements can always be calculated (numer-
ically) for any reasonable short-range potential. In prac-
tice we use Gauss-Laguerre quadrature, which is well-
suited to the CS basis.
Multiplied with the CS states /an}bracketle{t/tildewidenl|from the left,
Eq. (52) turns into a linear system of equations for the
wave-function coefficients ψln=/an}bracketle{t/tildewidenl|ψl/an}bracketri}ht
[(gC
l(E))−1−v(s)
l]ψl= (gC
l(E))−1ϕC
l, (55)
where the underlined quantities are matrices with the
following elements
ϕC
ln=/an}bracketle{t/tildewidenl|ϕC
l/an}bracketri}ht (56)
and
gC
lnn′(E) =/an}bracketle{t/tildewidenl|gC
l(E)|/tildewidern′l/an}bracketri}ht. (57)
1. The matrix elements /angbracketleft/tildewidenl|gC
l(z)|/tildewidern′l/angbracketright
The key point in the whole procedure is the exact
and analytic calculation of the CS matrix elements of
the Coulomb Green’s operator and of the overlap of the
Coulomb and CS functions. For the Green’s matrix we
have developed two independent, analytic approaches.
Both are based on the observation that the Coulomb
Hamiltonian possesses an infinite symmetric tridiagonal
(Jacobi) matrix structure on CS basis.
5Let us consider the radial Coulomb Hamiltonian
hC
l=−¯h2
2m/parenleftbiggd2
dr2−l(l+ 1)
r2/parenrightbigg
+Z
r, (58)
wherem,landZstands for the mass, angular mo-
mentum and charge, respectively. The matrix JC
nn′=
/an}bracketle{tn|(z−hC
l)|n′/an}bracketri}htpossesses a Jacobi structure,
JC
nn= 2(n+l+ 1)(k2−b2)¯h2
4mb−Z (59)
and
JC
nn−1=−[n(n+ 2l+ 1)]1/2(k2+b2)¯h2
4mb,(60)
wherek= (2mz/¯h2)1/2is the wave number. The main
result of Ref. [16] is that for Jacobi matrix systems the
N’th leading submatrix gC(N)
nn′of the infinite Green’s ma-
trix can be determined by the elements of the Jacobi
matrix
gC(N)
nn′= [JC
nn′+δnNδn′NJC
NN+1C]−1, (61)
whereCis a continued fraction
C=−uN
dN+uN+1
dN+1+uN+2
dN+2+· · ·, (62)
with coefficients
un=−JC
n,n−1/JC
n,n+1, d n=−JC
n,n/JC
n,n+1.(63)
In Ref. [16] it was shown that although the continued
fractionCis convergent only on the upper-half kplane it
can be continued analytically to the whole kplane. This
is because the unanddncoefficients satisfy the limit
properties
u≡lim
n→∞un=−1 (64)
d≡lim
n→∞dn= 2(k2−b2)/(k2+b2). (65)
Then the continued fraction appears as
C=−uN
dN+uN+1
dN+1+· · ·+u
d+u
d+· · ·. (66)
Therefore the tail wofCsatisfies the implicit relation
w=u
d+w, (67)
which is solved by
w±= (b±ik)2/(b2+k2). (68)Replacing the tail of the continued fraction by its explicit
analytical form w±, we can speed up the convergence
and, more importantly turn a non-convergent continued
fraction into a convergent one [17]. Analytic continuation
is achieved by using w±instead of the non-converging
tail. In Ref. [16] it was shown that w+provides an ana-
lytic continuation of the Green’s matrix to the physical,
whilew−to the unphysical Riemann-sheet. This way
Eq. (62) together with (61) provides the CS basis repre-
sentation of the Coulomb Green’s operator on the whole
complexkplane. We note here that with the choice of
Z= 0 the Coulomb Hamiltonian (58) reduces to the ki-
netic energy operator and our formulas provide the CS
basis representation of the Green’s operator of the free
particle as well. We emphasize that this procedure does
not truncate the Coulomb Hamiltonian, because all the
higherJnn′matrix elements are implicitly contained in
the continued fraction.
We note that gChas already been calculated before
[11]. From the J-matrix structure a three-term recursion
relation follows for the matrix elements gC
nn′. This recur-
sion relation is solvable if the first element gC
00is known.
It is given in a closed analytic form
gC
00=4mb
¯h21
(b−ik)21
l+ iη+ 1
×2F1/parenleftBigg
−l+ iη,1;l+ iη+ 2,/parenleftbiggb+ ik
b−ik/parenrightbigg2/parenrightBigg
,(69)
whereη=Zm/(¯h2k) is the Coulomb parameter and 2F1
is the hypergeometric function. For those cases where the
first or the second index of 2F1is equal to unity, there
exists a continued fraction representation, which is very
efficient in practical calculations. It was shown that the
two methods lead to numerically identical results for all
energies and our numerical continued fraction represen-
tation possesses all the analytic properties of gC. The
exact analytic knowledge of gCallows us to calculate the
matrix elements of the full Green’s operator in the whole
complex plane
gl(z) = ((gC
l(z))−1−v(s)
l)−1. (70)
The overlap vector of CS and the Coulomb functions
/an}bracketle{t/tildewidenl|ϕC
l/an}bracketri}htis known analytically [12]. It can be calculated
by a three-term recursion, derived from the J-matrix, us-
ing the starting value
/an}bracketle{t/tildewide0l|ϕC
l/an}bracketri}ht= exp(2ηarctan(k/b))/radicalBigg
2πη
exp(2πη)−1
×/parenleftbigg2k/b
1 +k2/b2/parenrightbiggl+1l/productdisplay
i=1/parenleftbiggη2+i2
i(i+ 1/2)/parenrightbigg1/2
.(71)
6C. Calculation of the three-body S-matrix
The aim of any scattering calculation is to determine
the S-matrix elements. In our case we need to calcu-
late the terms (33), (34) and (44) of the three-potential
picture.
The termS(1)
βj,αiis trivial because it is just the two-
body S-matrix of the Coulomb-like potential u(l)
α.
To calculate the second term, S(2)
βj,αiof Eq. (34), the
matrix elements /an}bracketle{t/tildewideΦ(−)
αj|Uα|Φ(l)(+)
αi/an}bracketri}htare needed. Since
/an}bracketle{t/tildewideΦ(−)
αj|contains a two-body bound-state wave function
in coordinate xαthis matrix element is confined to Ω α,
whereUαis of short-range type. Therefore a separable
approximation is justified
/an}bracketle{t/tildewideΦ(−)
αj|Uα|Φ(l)(+)
αi/an}bracketri}ht ≈ /an}bracketle{t/tildewideΦ(−)
αj|1α
NUα1α
N|Φ(l)(+)
αi/an}bracketri}ht,(72)
i.e, in this matrix element, we can approximate Uαby a
separable form
Uα= lim
N→∞1α
NUα1α
N≈1α
NUα1α
N
≈N/summationdisplay
n,ν,n′,ν′=0|/tildewidestnνlλ/an}bracketri}htαUα
α/an}bracketle{t/tildewidestn′ν′l′λ′| (73)
where
Uα
nνlλ,n′ν′l′λ′=α/an}bracketle{tnνlλ|Uα|n′ν′l′λ′/an}bracketri}htα. (74)
The matrix element appears as
/an}bracketle{t/tildewideΦ(−)
αj|Uα|Φ(l)(+)
αi/an}bracketri}ht ≈N/summationdisplay
/an}bracketle{t/tildewideΦ(−)
αj|/tildewidestnνlλ/an}bracketri}htαUα
α/an}bracketle{t/tildewidestn′ν′l′λ′|Φ(l)(+)
αi/an}bracketri}ht.
(75)
In calculating the third term, S(3)
βj,αiof (44), we have
matrix elements of the type /an}bracketle{t/tildewideΦl(−)
αj|v(s)
α|ψ(+)
βi/an}bracketri}ht. Here we
can again approximate the short-range potential v(s)
αin
the three-body Hilbert space by a separable form
v(s)
α= lim
N→∞1α
Nv(s)
α1β
N≈1α
Nv(s)
α1β
N
≈N/summationdisplay
n,ν,n′,ν′=0|/tildewidestnνlλ/an}bracketri}htαv(s)
αββ/an}bracketle{t/tildewidestn′ν′l′λ′| (76)
where
v(s)
αβnνlλ,n ′ν′l′λ′=α/an}bracketle{tnνlλ|v(s)
α|n′ν′l′λ′/an}bracketri}htβ.(77)
In (76) the ket and bra states belong to different frag-
mentations depending on the neighbors of the potential
operators in the matrix elements. Finally, the matrix
elements take the form
/an}bracketle{tΦl(−)
αj|v(s)
α|ψ(+)
βi/an}bracketri}ht ≈N/summationdisplay
/an}bracketle{tΦl(−)
αj|/tildewidestnνlλ/an}bracketri}htαv(s)
αββ/an}bracketle{t/tildewidestn′ν′l′λ′|ψ(+)
βi/an}bracketri}ht.
(78)We conclude that to calculate the S-matrix of the
three-potential formulae we need the CS matrix elements
(74) and (77), which can always be evaluated numeri-
cally by using the transformation of Jacobi coordinates
[18]. In addition we need the CS wave function com-
ponents α/an}bracketle{t/tildewidestnνlλ|/tildewideΦ(±)
αi/an}bracketri}ht,α/an}bracketle{t/tildewidestnνlλ|Φl(±)
αi/an}bracketri}htand α/an}bracketle{t/tildewidestnνlλ|ψ(+)
α/an}bracketri}ht.
We determine them in the following section by solv-
ing Lippmann-Schwinger and Faddeev-Merkuriev inte-
gral equations.
It should be noted that the approximations (73) and
(76) used in calculating the matrix elements (75) and
(78) become equalities as Ngoes to infinity. In practical
calculations we increase Nuntil we observe numerical
convergence in scattering observables.
D. Solution of the three-body integral equations
In the set of Faddeev-Merkuriev equations (41-43) we
make the approximation of (76)
|ψα/an}bracketri}ht=|Φ(l)
αi/an}bracketri}ht+G(l)
α[1α
Nv(s)
α1β
N|ψβ/an}bracketri}ht+1α
Nv(s)
α1γ
N|ψγ/an}bracketri}ht] (79)
|ψβ/an}bracketri}ht=G(l)
β[1β
Nv(s)
β1γ
N|ψγ/an}bracketri}ht+1β
Nv(s)
β1α
N|ψα/an}bracketri}ht] (80)
|ψγ/an}bracketri}ht=G(l)
γ[1γ
Nv(s)
γ1α
N|ψα/an}bracketri}ht+1γ
Nv(s)
γ1β
N|ψβ/an}bracketri}ht].(81)
Multiplied by the CS states α/an}bracketle{t/tildewidestnνlλ|,β/an}bracketle{t/tildewidestnνlλ|and γ/an}bracketle{t/tildewidestnνlλ|,
respectively, from the left the set of integral equations
turn into a linear system of algebraic equations for
the coefficients of the Faddeev components ψαnνlλ=
α/an}bracketle{t/tildewidestnνlλ|ψα/an}bracketri}ht:
[(G(l))−1−v(s)]ψ= (G(l))−1Φ(l), (82)
with
G(l)
αnνlλ,n ′ν′l′λ′=α/an}bracketle{t/tildewidestnνlλ|G(l)
α|/tildewidestn′ν′l′λ′/an}bracketri}htα,(83)
and
Φ(l)
αnνlλ=α/an}bracketle{t/tildewidestnνlλ|Φ(l)
α/an}bracketri}ht. (84)
Notice that the matrix elements of the Green’s operator
are needed only between the same partition αwhereas
the matrix elements of the potentials occur only between
different partitions αandβ.
1. The matrix elements α/angbracketleft/tildewidestnνlλ |G(l)
α|/tildewidestn′ν′l′λ′/angbracketrightαand
α/angbracketleft/tildewidestnνlλ |Φ(l)
α/angbracketright
Unfortunately neither the matrix elements (83) nor the
overlaps (84) are known. The appropriate Lippmann-
Schwinger equation for G(l)
αwas proposed by Merkuriev
[2]
G(l)
α(z) =Gas
α(z) +Gas
α(z)Vas
αG(l)
α(z), (85)
7whereGas
αandVas
αare the asymptotic channel Green’s
operator and potential, respectively. A similar equation
is valid for |Φ(l)
α/an}bracketri}ht
|Φ(l)
α/an}bracketri}ht=|Φas
α/an}bracketri}ht+Gas
α(z)Vas
α|Φ(l)
α/an}bracketri}ht. (86)
BothG(l)
αand|Φ(l)
α/an}bracketri}htare genuine three-body quantities.
One may wonder why a single Lippmann-Schwinger equa-
tion suffices. The Hamiltonian H(l)
αhas a peculiar prop-
erty - it has only α-type two-body asymptotic channels.
For such systems a single Lippmann-Schwinger equation
provides a unique solution [19].
The objects Gas
α,Vas
αand Φas
αare very complicated.
Their leading order terms were constructed in configura-
tions space in the different asymptotic regions. The po-
tentialVas, as|X| → ∞ , decays faster than the Coulomb
potential in all directions of the three-body configuration
space:Vas∼ O(|X|−1−ǫ), ǫ > 0 [2]. Therefore we may
express the solutions of Eqs. (85) and (86) formally as
(G(l)
α)−1= (Gas
α)−1−Vas
α (87)
and
[(Gas
α)−1−Vas
α]Φ(l)
α= (Gas
α)−1Φas
α, (88)
respectively, where
Gas
αnνlλ,n ′ν′l′λ′=α/an}bracketle{tnνlλ|Gas
α|n′ν′l′λ′/an}bracketri}htα, (89)
Vas
αnνlλ,n ′ν′l′λ′=α/an}bracketle{tnνlλ|Vas
α|n′ν′l′λ′/an}bracketri}htα (90)
and
Φas
αnνlλ=α/an}bracketle{t/tildewidestnνlλ|Φas
α/an}bracketri}ht. (91)
Here,Gas
α,Vas
αand Φas
αappear between finite number of
of square-integrable CS states, which confine the domain
of integration to Ω α. In this region, however, Gas
αcoin-
cides with/tildewideGα,Vas
αwithUαand Φas
αwith/tildewideΦα[2]. Finally
we have
(G(l)
α)−1= (/tildewideGα)−1−Uα, (92)
where
/tildewideGαnνlλ,n ′ν′l′λ′=α/an}bracketle{tnνlλ|/tildewideGα|n′ν′l′λ′/an}bracketri}htα (93)
and
Uα
nνlλ,n′ν′l′λ′=α/an}bracketle{tnνlλ|Uα|n′ν′l′λ′/an}bracketri}htα. (94)
And in a similar way
[(/tildewideGα)−1−Uα]Φ(l)
α= (/tildewideGα)−1/tildewideΦα, (95)
where
/tildewideΦαnνlλ=α/an}bracketle{t/tildewidestnνlλ|/tildewideΦα/an}bracketri}ht. (96)
We note that from Eq. (92) follows that the left side
of Eq. (95) is just the inhomogeneous term of Eq. (82).
Both Eqs. (95) and (82) are solved with the same inho-
mogeneous term.2. The matrix elements α/angbracketleftnνlλ |/tildewideGα|n′ν′l′λ′/angbracketrightαand
α/angbracketleft/tildewidestnνlλ |/tildewideΦα/angbracketright
The three-particle free Hamiltonian can be written as
a sum of two-particle free Hamiltonians
H0=h0
xα+h0
yα. (97)
Then the Hamiltonian /tildewideHαof Eq. (18) appears as a sum
of two Hamiltonians acting on different coordinates
/tildewideHα=hxα+hyα, (98)
withhxα=h0
xα+vC
α(xα) andhyα=h0
yα+u(l)
α(yα),
which, of course, commute. The state |/tildewideΦα/an}bracketri}ht, which is
an eigenstate of /tildewideHα, is a product of a two-body bound-
state wave function in coordinate xαand a two-body
scattering-state wave function in coordinate yα. Their
CS representations are known from the two-particle case
described before.
The matrix elements of /tildewideGαcan be determined by mak-
ing use of the convolution theorem
/tildewideGα(z) = (z−hxα−hyα)−1
=1
2πi/contintegraldisplay
Cdz′(z−z′−hxα)−1(z′−hyα)−1.(99)
The contour Cshould encircle, in positive direction, the
spectrum of hyαwithout penetrating into the spectrum
ofhxα.
The convolution theorem follows from a more general
formula. A function of a self adjoint operator his defined
as
f(h) =1
2πi/contintegraldisplay
Cdzf(z)(z−h)−1, (100)
whereCis a contour around the spectrum of handf
should be analytic on the region encircled by C.
In the following we suppose that u(l)either vanishes
or is a repulsive Coulomb-like potential. This assump-
tion is not necessary but it greatly simplifies the analysis
below. Numerical examples show that there are a great
many physical three-body systems where this condition
is satisfied. This condition ensures that hydoes not have
bound states.
To examine the analytic structure of the integrand (99)
let us shift the spectrum of gxαby takingz=E+ iε
with positive ε. In doing so, the two spectra become
well separated and the spectrum of gyαcan be encircled.
The contour Cis deformed analytically in such a way
that the upper part descends to the unphysical Riemann
sheet ofgyα, while the lower part of Ccan be detoured
away from the cut [see Fig. 3]. The contour still encircles
the branch cut singularity of gyα, but in the ε→0 limit
avoids the singularities of gxα. Thus, the mathemati-
cal conditions for the contour integral representation of
8/tildewideGα(z) in Eq. (99) is met. The matrix elements /tildewideGαcan
be cast in the form
/tildewideGα(z) =1
2πi/contintegraldisplay
Cdz′gxα(z−z′)gyα(z′), (101)
where the corresponding CS matrix elements of the two-
body Green’s operators in the integrand are known ana-
lytically for all complex energies.
III. TEST OF THE METHOD
We demonstrate the power of this new method by cal-
culating elastic phase shifts of e++Hscattering below
thePs(n= 1) threshold and cross sections of the e++H
elastic scattering as well as p++Psreaction channels up
to thePs(n= 2) threshold. In all examples we have to-
tal angular momentum L= 0 and we have taken angular
momentum channels up to l= 10. We use atomic units.
Let us numerate the particles e+,pande−, with
massesme±= 1meandmp= 1836.1527me, by 1, 2 and
3, respectively. In the channel 3 there are no two-body
asymptotic channels since the particles e+andpdo not
form bound states. Therefore, we can take v(s)
3≡0 and
include the total vC
3in the long range Hamiltonian
H=H(l)+v(s)
1+v(s)
2, (102)
H(l)=H0+v(l)
1+v(l)
2+vC
3. (103)
In this case |ψ3/an}bracketri}ht ≡ 0 and we have the set of two-
component Faddeev-Merkuriev equations
|ψ1/an}bracketri}ht=|φ(l)
1/an}bracketri}ht+G(l)
1v(s)
1|ψ1/an}bracketri}ht (104)
|ψ2/an}bracketri}ht=G(l)
2v(s)
2|ψ2/an}bracketri}ht. (105)
The parameters of the splitting function ζof Eq. (8) are
rather arbitrary. The final converged results should be
insensitive to their values; our numerical experiences con -
firm this expectation. For the parameters of ζwe have
takenν= 2.1,x0= 3 andy0= 10, whereas for the
parameters of CS functions we have taken b= 0.9. We
have experienced that the rate of convergence is rather
insensitive on the choice of bover a broad interval.
First we examine the convergence of the results for
cross sections at incident wave numbers k1= 0.71,
k1= 0.75 andk1= 0.8, which correspond to scatter-
ing states in the Ore gap. Table I shows the conver-
gence ofe++H−>e++Helastic scattering ( σ11) and
e++H−> p++Pspositronium formation ( σ12) cross
sections (in πa2
0) with respect to N, the number of CS
functions in the expansion, and with respect to increas-
ing the angular momentum channels in the bipolar ex-
pansion. For comparison we provide the results of Ref.
[20]. We can see that very good accuracy is achieved even
with relatively low Nin the expansion.
In Table II we compare our converged results for phase
shifts (in radians) below the Ps(n= 1) threshold to thatof other methods. Ref. [21] is the best variational calcu-
lation. In Ref. [22] the Schr¨ odinger equation was solved
by means of finite-element method. In Refs. [23] and [20]
the configurations space Faddeev-Merkuriev differential
equations were solved using the bipolar harmonic expan-
sion method and in total angular momentum represen-
tation, respectively. We can report perfect agreements
with previous calculations.
In Table III we present partial cross sections in
theH(n= 2) −Ps(n= 2) gap (threshold energies
0.7496-0.8745 Ry). In Ref. [24] the configurations space
Faddeev-Merkuriev differential equations were solved us-
ing the bipolar harmonic expansion in the angular vari-
ables an quintic spline expansion in the radial coordi-
nates. We can report fairly good agreements.
IV. CONCLUSION
We have extended the three-potential formalism for
treating the three-body scattering problem with all kinds
of Coulomb interactions including attractive ones. We
adopted Merkuriev’s approach and split the Coulomb po-
tentials in the three-body configuration space into short-
range and long-range terms. In this picture the three-
body Coulomb scattering process can be decomposed into
a single channel Coulomb scattering, a two-body multi-
channel scattering on the intermediate-range polarizatio n
potential and a genuinely three-body scattering due to
the short-range potentials. The formalism provides us a
set of Lippmann–Schwinger and Faddeev-Merkuriev in-
tegral equations.
These integral equations are certainly too complicated
for the most of the numerical methods available in the
literature. The Coulomb-Sturmian separable expansion
method can be successfully applied. It solves the three-
body integral equations by expanding only the short-
range terms in a separable form on Coulomb-Sturmian
basis while treating the long-range terms in an exact
manner via a proper integral representation of the three-
body channel distorted Coulomb Green’s operator. The
use of the Coulomb-Sturmian basis is essential as it allows
an exact analytic representation of the two-body Green’s
operator, and thus the contour integral for the channel
distorted Coulomb Green’s operator can be calculated.
The method provides solutions which are asymptotically
correct, at least in Ω α, which is sufficient if the scattering
process starts from a two-body asymptotic state. Since
the two-body Coulomb Green’s operator is exactly calcu-
lated all thresholds are automatically in the right locatio n
irrespective of the rank of the separable approximation.
The method possesses good convergence properties and
in practice it can be made arbitrarily accurate by em-
ploying an increasing number of terms in the expansion.
Certainly, there is plenty of room for improvement but
we are convinced that this method can be a very power-
ful tool for studying three-body systems with Coulomb
9interactions.
ACKNOWLEDGMENTS
This work has been supported by the NSF Grant
No.Phy-0088936 and by the OTKA Grant No. T026233.
We also acknowledge the generous allocation of computer
time at the NPACI, formerly San Diego Supercomputing
Center, by the National Resource Allocation Commit-
tee and at the Department of Aerospace Engineering of
CSULB.TABLE I. Convergence of e++H−> e++Helastic scat-
tering ( σ11) and e++H−> p+Pspositronium formation
(σ12) cross sections (in πa2
0) with respect to N, the number of
CS functions in the expansion, and with respect to increasin g
the angular momentum channels ( lmax) in the bipolar basis.
lmax= 6 lmax= 8 lmax= 10
N σ11 σ12 σ11 σ12 σ11 σ12
k1= 0.71, Ref. [20]: σ11= 0.025,σ12= 0.0038
120.02662 0.00423 0.02664 0.00397 0.02665 0.00393
130.02608 0.00424 0.02609 0.00398 0.02610 0.00394
140.02581 0.00423 0.02582 0.00398 0.02583 0.00394
150.02562 0.00424 0.02561 0.00398 0.02562 0.00395
160.02548 0.00425 0.02546 0.00400 0.02547 0.00396
170.02541 0.00426 0.02539 0.00401 0.02539 0.00397
180.02532 0.00427 0.02529 0.00401 0.02530 0.00398
190.02528 0.00427 0.02524 0.00402 0.02525 0.00398
200.02522 0.00428 0.02517 0.00403 0.02518 0.00399
k1= 0.75, Ref. [20]: σ11= 0.044,σ12= 0.0043
120.04412 0.00441 0.04412 0.00424 0.04413 0.00422
130.04345 0.00440 0.04344 0.00422 0.04345 0.00421
140.04318 0.00440 0.04317 0.00423 0.04318 0.00421
150.04280 0.00440 0.04278 0.00423 0.04279 0.00421
160.04269 0.00440 0.04265 0.00423 0.04266 0.00422
170.04252 0.00441 0.04248 0.00424 0.04249 0.00423
180.04246 0.00442 0.04240 0.00425 0.04241 0.00423
190.04238 0.00442 0.04232 0.00426 0.04232 0.00424
200.04232 0.00442 0.04225 0.00426 0.04226 0.00424
k1= 0.80, Ref. [20]: σ11= 0.063,σ12= 0.0047
120.06572 0.00475 0.06571 0.00467 0.06572 0.00467
130.06573 0.00481 0.06571 0.00473 0.06572 0.00473
140.06518 0.00483 0.06515 0.00475 0.06517 0.00475
150.06488 0.00485 0.06484 0.00477 0.06486 0.00477
160.06457 0.00486 0.06452 0.00478 0.06453 0.00478
170.06440 0.00487 0.06433 0.00479 0.06435 0.00479
180.06427 0.00487 0.06420 0.00479 0.06422 0.00480
190.06418 0.00487 0.06409 0.00480 0.06411 0.00480
200.06412 0.00488 0.06402 0.00480 0.06404 0.00480
TABLE II. Phase shifts (in radians) of e++H−> e++H
elastic scattering below the positronium formation thresh old.
k Ref. [21] Ref. [22] Ref. [23] Ref. [20] This work
0.1 0.1483 0.152 0.149 0.149 0.1480
0.2 0.1877 0.188 0.188 0.189 0.1876
0.3 0.1677 0.166 0.166 0.169 0.1673
0.4 0.1201 0.118 0.120 0.121 0.1199
0.5 0.0624 0.061 0.060 0.062 0.0625
0.6 0.0039 0.003 0.003 0.0038
0.7 -0.0512 -0.053 -0.050 -0.0513
10TABLE III. Partial cross sections (in πa2
0) in the
H(n= 2) −Ps(n= 2) gap (threshold energies 0.7496-8745
Ry). Numbers 1,2,3 and 4 denote the channels e++H(1s),
e++H(2s),e++H(2p) and p++Ps(1s), respectively.
E1(Ry) σ11 σ12 σ13 σ14
0.77 Ref. [24] 0.090 0.000702 0.000454 0.00572
0.77 This work 0.0951 0.000673 0.000331 0.00558
0.80 Ref. [24] 0.096 0.00115 0.000364 0.00585
0.80 This work 0.1010 0.00127 0.000371 0.00563
0.83 Ref. [24] 0.0993 0.00170 0.000885 0.00581
0.83 This work 0.1063 0.00163 0.000813 0.00566
0.84 Ref. [24] 0.101 0.00190 0.00113 0.00580
0.84 This work 0.1080 0.00173 0.00105 0.00566
0204060
x 050100
y-0.4-0.20
FIG. 1. The short-range part v(s)of the −1/xattractive
Coulomb potential.
0 10 20 30 40 50 60
x 020406080100
y-0.1-0.08-0.06-0.04-0.02
FIG. 2. The long range part v(l)of the −1/xattractive
Coulomb potential.
Cz'g_x(E+iε−z')
g_y(z')
FIG. 3. Analytic structure of gxα(z−z′)gyα(z′) as a func-
tion of z′withz=E+ iε,E <0,ε >0. The contour C
encircles the continuous spectrum of hyα. A part of it, which
goes on the unphysical Riemann-sheet of gyα, is drawn by
broken line.
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12 |
arXiv:physics/0102023v1 [physics.atom-ph] 9 Feb 2001Harmonic generation in ring-shaped molecules
F. Ceccherini and D. Bauer
Theoretical Quantum Electronics (TQE), Darmstadt Univers ity of Technology,
Hochschulstr. 4A, D-64289 Darmstadt, Germany
(December 20, 2012)
ABSTRACT
We study numerically the interaction between an intense cir cularly polarized laser field and an
electron moving in a potential which has a discrete cylindri cal symmetry with respect to the laser
pulse propagation direction. This setup serves as a simple m odel, e.g., for benzene and other
aromatic compounds. From general symmetry considerations , within a Floquet approach, selection
rules for the harmonic generation [O.Alon et al. Phys. Rev. Lett. 803743 (1998)] have been
derived recently. Instead, the results we present in this pa per have been obtained solving the time-
dependent Schr¨ odinger equation ab initio for realistic pulse shapes. We find a rich structure which
is not always dominated by the laser harmonics.
PACS numbers: 31.15.Ar, 33.80.Wz, 42.65.Ky
I. INTRODUCTION
The generation of harmonics (HG) through the interaction of atoms with intense laser fields is a topic that has been
broadly studied from both a theoretical and an experimental point of view. The big interest in the HG is due to
the possible use as a source of short-wavelength radiation. In fact, through the harmonic emission it is possible to
generate coherent XUV radiation using table-top lasers. Re cently, harmonics of wavelength as short as 67 ˚A have
been reported [1]. The harmonics spectra obtained from a sin gle atom in a monochromatic laser field present some
common and well known features: (i) only linearly polarized odd harmonics are generated [2], (ii) the spectrum has
a plateau structure , (iii) the plateau is extended up to a cut -off that is located around Ip+ 3.17Up, where Ipis the
ionization energy of the atom and Upis the ponderomotive energy [3]. The presence of only odd har monics is due
to symmetry reasons (a more detailed argument will be discus sed in the next section) and the location of the cut-off
can be explained, at least qualitatively, with the so-calle d “simple man’s theory” [4]. The interaction of a single atom
with two circularly polarized lasers of frequencies ωand 2 ωhas been investigated recently [5]; it has been found that
while the harmonic orders 3 n−1 and 3 n+ 1 are allowed, the harmonic order 3 nis forbidden by the selection rules
of the dipole emission, where nis any positive integer. These results agreed with a previou s experiment [6].
More recently, the generation of harmonics in more complex s ystems than the single atom has become a strongly
addressed topic. A model for harmonic emission from atomic c lusters has been proposed [7]. The harmonics generated
by electrons moving in the periodic potential of a crystal ha ve been investigated also [8,9].
In this work we want to study the generation of harmonics in ri ng-shaped molecules, like benzene and other aromatic
compounds. This kind of molecules exhibits an invariance un der a rotation of a certain angle around the axis that is
orthogonal to the molecule plane and goes through its center . In this case the potential is periodic in the azimuthal
direction.
The HG from ring-like molecules interacting with a circular ly polarized field presents many different features with
respect to the single-atom case in a linearly polarized field : (i) within the same harmonic range fewer lines are emitted
and the higher is the number of atoms in the molecule, the lowe r is the number of emitted lines, (ii) odd and even
harmonics are equally possible, (iii) the harmonics are alt ernately left or right circularly polarized. In our opinion ,
all these peculiar properties make this topic challenging a nd worth to be studied in detail.
The paper is organized as following: in Section II we summari ze the derivation of the selection rules for the ring-shaped
molecules obtained by Alon et al. [11]. In Section III the numerical model used in our simulati ons is presented and
discussed. In Section IV we describe the interaction betwee n the ring molecule and the laser field. In Section V we
show the results obtained for different intensities and freq uencies together with a broad discussion. Finally, in Secti on
VI we give a summary and an outlook. Atomic units (a.u.) are us ed throughout the paper.
1II. SELECTION RULES
In the case of an atom or a molecule which are shone by a laser fie ld of frequency ω, the Hamiltonian is periodic in
time with a period τ= 2π/ω:H(t+τ) =H(t). The time-dependent Schr¨ odinger equation (TDSE) for suc h a system
can be written as
/bracketleftbigg
H(t)−i∂
∂t/bracketrightbigg
ΨW(/vector r, t) = 0. (1)
where the operator between the square brackets is called Flo quet operator. The solutions are of the form:
ΨW(/vector r, t) = Φ( /vector r, t)e−iWtwith Φ( /vector r, t+τ) = Φ( /vector r, t), (2)
where Wis the quasi-energy and Φ( /vector r, t) is a square integrable function.
Because the set of all the functions that are square-integra ble in a certain interval and have a finite norm over a cycle,
forms a composite Hilbert space, we can apply the extended Hi lbert space formalism. The probability to get the nth
harmonic from a system in a state Ψ W(/vector r, t) is [10]
σ(n)
W∝n4/vextendsingle/vextendsingle/vextendsingle/vextendsingle∝angb∇acketleft∝angb∇acketleftΦ(/vector r, t)|ˆµe−inωt|Φ(/vector r, t)∝angb∇acket∇ight∝angb∇acket∇ight/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
, (3)
where ˆ µis the dipole operator and the double bracket stands for the i ntegration over space and time.
In the case of an atom (in the dipole approximation) the Floqu et operator is invariant under the so-called second
order dynamical symmetry operator (DSO) [11],
P2= (/vector r→ −/vector r, t→t+π/ω). (4)
Therefore, the states |Φ∝angb∇acket∇ight∝angb∇acket∇ightare simultaneous eigenfunctions of the Floquet operator an d the second order DSO with
eigenvalues ±1. The nth harmonic is therefore emitted only if
∝angb∇acketleft∝angb∇acketleftΦ|ˆµe−inωt|Φ∝angb∇acket∇ight∝angb∇acket∇ight=∝angb∇acketleft∝angb∇acketleftP2Φ|P2ˆµe−inωtP−1
2|P2Φ∝angb∇acket∇ight∝angb∇acket∇ight ∝negationslash= 0, (5)
leading to
ˆµ(/vector r)e−inωt= ˆµ(−/vector r)e−inω(t+π/ω)(6)
that is fulfilled only with oddn’s. Instead, where the Hamiltonian is invariant under a rota tion around an N-fold
symmetry axis P2can be replaced by [11]
PN=/parenleftbigg
ϕ→ϕ+2π
N, t→t+2π
Nω/parenrightbigg
, (7)
where ϕis the angular coordinate around the symmetry axis. With an a lgebra similar to the single atom case we
derive
e±i(ϕ+2π
N)e−inω(t+2π
Nω)=e±iϕe−inωt=⇒e−i2π(n±1)
N= 1, (8)
from which n=kN±1, k∈ Nfollows. That means that the higher is the symmetry order Nthe less are the generated
harmonics within a fixed frequency interval. In the limit of a continuous symmetry C∞a circularly polarized laser does
not generate any harmonics. The two harmonics of each couple have opposite polarization, clockwise and anticlockwise
[12].
III. NUMERICAL MODEL
In order to keep the numerical effort manageable we restrict o urselves to a two-dimensional (2D) model where the
molecule plane and the rotating electric field are properly r epresented. We study a single active electron in a ring-
shaped potential of Nions. Different kinds of “smoothed” potentials can be used fo r this purpose [13]. The potential
used in our simulations reads
2V(ρ, ϕ) =−A/radicalbig
(ρ−r0)2+β(αcos(N ϕ) + 2−α) (9)
where r0is the radius of the molecule, and ρandϕare the polar coordinates. βis the parameter which gives the
“degree of smoothness” of the potential and determines the w idth of the ground state along the ring. αmoves the
local maxima of the potential keeping the minima constant (t he parameter αis introduced in order to avoid the
presence of points where V= 0 for finite ρ, because that could generate non-physical ionization). Fi nally, Ais the
“strength” of the potential. For our simulations we chose α= 1.075 and β= 0.3. Once the values of αandβ, which
lead to reasonable model properties, have been found and fixe d, we varied Afor choosing the ionization energy of the
molecule. The potential has Noscillations in the azimuthal direction, each minimum repr esents the location of one
of the Nions. The potential goes to zero for ρ→ ∞.
It has been tested that the generation of harmonics is very we akly dependent on the fine details of the atomic potential.
Instead, it is strongly dependent on its geometry and symmet ries. It is therefore worth to look for a model potential
that, keeping the proper symmetry, can be quite easily numer ically optimized, i.e., requiring as few as possible grid
points. In order to achieve this minimization of the number o f grid points and to not break any physical symmetry
we used a polar grid. Moreover, we paid attention to use alway s a number of points in the azimuthal direction that
is an integer multiple of N. We will take N= 6, like benzene, and therefore the potential will exhibit a C6symmetry
around the orthogonal axis.
For a good understanding of the harmonic spectra it is essent ial to study the level scheme in detail. Therefore, in order
to characterize our model we have calculated the energy of th e first six states for different potential strengths Ain
the interval between 0.2 and 1.6. In this interval the energy of the ground state of the molecule decreases from −0.13
down to −2.66. The first, the fourth and the fifth states are non degenerat e, the others have a double degeneracy. In
Fig. 1 the energetic behavior of those states versus Ais shown. Contour plots of the six states for an intermediate
value of A(A= 0.80) are shown in Fig.2. The pattern shape and the symmetries d o not change with A, but the
“average radius”, i.e., the spatial extension, does. Clear ly, for low values of Athe probability density is more loosely
bound than for high A. In particular this is true for the upper states. For the sake of easy reference later on, we name
the six states as 0,1a,1b,2a,2b,3,4,5aand5b. The subscripts are used just to distinguish among the degen erate
states. The non-degenerate states 0,4and5have the full C6symmetry.
As one can infer from Fig.1 the six states can be divided in two branches: the first one, containing the first four states,
0−3, decreases very fast as Aincreases, and a second one, containing the two upper states ,4−5, that decreases
much more slowly and lies even in the continuum for A <0.4. As a result, for an increasing Aan increasing gap
between the two branches appears. To obtain the ionization p otential of real benzene ( −0.34 a.u.) we have to choose
A= 0.37. Surprisingly, at that position the level scheme of our si mple model resembles the molecular orbital (MO)
scheme of the real benzene very well [14]. In particular, the states 4and5are still in the continuum so that only
the four states 0−3, possessing the same degeneracies as the MO, are bound. A mag nification of the region around
A= 0.37 is also shown in Fig. 1. Another parameter that will play a r ole in the HG is the level spacing Ω between
the ground state and the first excited state. For an increasin gA, i.e., a decreasing ground state energy, Ω decreases.
IV. MOLECULE-FIELD INTERACTION
In dipole approximation the time-dependent Schr¨ odinger e quation for a single electron in a laser field /vectorE(t) and under
the influence of an effective ionic potential V(/vector r) is given in length gauge by
i∂
∂tΨ(/vector r, t) =/parenleftbigg
−1
2/vector∇2+V(/vector r) +/vectorE(t)·/vector r/parenrightbigg
Ψ(/vector r, t). (10)
In our case the dipole approximation is excellent because th e molecule has a size much smaller than the wavelength
of the laser field. We used a circularly polarized laser field t hat in cartesian coordinates is described by
/vectorE(t) =E(t)√
2/parenleftBig
cos(ωt)/vector ex+ sin( ωt)/vector ey/parenrightBig
, (11)
where E(t) is a slowly varying envelope and ωis the laser frequency. In polar coordinates we obtain the TD SE
i∂
∂tΨ(ρ, ϕ, t) =/parenleftbigg
−1
2ρ∂
∂ρ/parenleftbigg
ρ∂Ψ
∂ρ/parenrightbigg
−1
2ρ2∂2
∂ϕ2+V(ρ, ϕ) +E(t)ρcos(ϕ−ωt)/parenrightbigg
Ψ(ρ, ϕ, t). (12)
3This TDSE can be solved ab initio on a PC. We did this by propagating the wavefunction in time wi th a Crank-
Nicholson approximant to the propagator U(t, t+ ∆t) = exp[ −i∆tH(t+ ∆t/2)] where H(t) is the explicitly time-
dependent Hamiltonian corresponding to the TDSE (12). Our a lgorithm is fourth order in the grid spacing ∆ ρ, ∆ϕ
and second order in the time step ∆ t. The boundary condition is Ψ(0 , t) = Ψ(2 π, t) for all ρandt. Probability density
which approaches the grid boundary is removed by an imaginar y potential.
V. RESULTS
Here we discuss the results obtained from our 2D simulations and we compare them with previous results from
a one dimensional model (1D) presented elsewhere [15]. Our s tudies were mainly focused on the structure of the
harmonic spectrum for different values of the ionization ene rgy. In general, our findings show that, together with
the harmonics we expected from the selection rules other lin es are present and their location can be, in most of the
cases, explained with the help of the level scheme. In partic ular, we observed that for higher Athe gap between the
two branches of Fig. 1 plays an important role. For each Avarious simulations with pulses of the same frequency
and length but different intensities were performed. We used sine-square pulses of 30 cycles duration and a frequency
ω= 0.0942, unless noted otherwise. In order to better understand the additional lines which appear besides the
expected harmonics it is useful to study the low intensity re gime first.
A. Low Fields
In the case of a single atom, when the intensity of the field is n ot high enough for generating harmonics efficiently, the
Fourier transform of the dipole shows only the fundamental. Clearly, the threshold of the field strength to observe any
harmonics depends on the ionization potential. In Fig. 3 two spectra, for different A, obtained from the interaction
of the ring molecule with a low field pulse are shown. For the di pole emission spectrum of Fig.3a an electric field
amplitude E= 0.02 a.u. was used and A= 1.6. In this case apart from the fundamental two additional lin es are
present although there are not harmonics. The positions of t he two lines are at 16 .3ωand 18 .5ω, respectively. We
refer to them as Λ aand Λ bhere after. These two lines correspond to two resonances bet ween the states 0→5and
3→4and therefore they move towards the red if Adecreases. This is confirmed in Fig.3b, where A= 1.4 and
E= 0.01. In Fig. 3b also the 5th harmonic is present, this is due to t he fact that when the ionization potential is
lower the generation of harmonics requires weaker fields. A s ubfundamental line at ω−Ω, corresponding to a virtual
transition from ωto the first excited state 1, is also present in both cases. We name this line Υ. Looking cl oser,
one observes that, with respect to the positions one would ex pect from the unperturbed level scheme, Λ aand Λ bare
blue-shifted, whereas Υ is slightly red-shifted. This oppo site shift can be explained by the dynamical Stark effect.
The ground state remains almost uneffected and the lower-lyi ng states move slightly. The higher-lying states instead,
experience a relatively strong shift. Therefore, with incr easing laser intensity the gap between the two branches in
Fig. 1 increases, leading to a blue shift of Λ aand Λ b. The level spacing ω−Ω decreases, giving rise to a red-shifted
Υ. The three lines Λ a, Λband Υ are the first lines to appear in the low field regime. The su bfundamental line can be
considered as characteristic for the low field regime. Indee d, with increasing field strength, it moves towards the red
and finally becomes very difficult to be resolved.
B. High Fields
With increasing laser intensity the actual harmonic spectr um develops. In Fig. 4 we show HG spectra for four different
electric field amplitudes and A= 1.6. In Fig. 4a E= 0.03 a.u. and the situation is, at first sight, quite similar to
that one of Fig. 3a: Λ aand Λ bare present, but also the first allowed harmonic is there, i.e . the 5th. Increasing the
field to E= 0.15 a.u., Fig. 4b, also the 7th, the 11th and the 13th appear, an d the two lines Λ aand Λ bcannot be
distinguished anymore but they merge into a single structur e not resolved in our plot. This last phenomenon can be
explained taking into account at least three different effect s: (i) in general, the width of Λ aand Λ bincreases for higher
fields, (ii) other channels, i.e. other resonance lines betw een the two branches, can be opened if the field becomes
intense, and (iii) through a removal of the degeneracy of the states by the electric field more possible resonance lines
are obtained. These factors generate a kind of broad “hill” i n the harmonic spectrum, the position of which is function
a ofA. Moreover, in Fig.4b other satellite lines around the expec ted harmonics are present. Those lines are decays
from virtual states to real states, like the subfundamental Υ. In Fig.4c, E= 0.24 a.u., more couples of harmonics
are present and the hill is reduced to a background modulatio n of the main HG spectrum. This becomes even clearer
4in Fig.4d where harmonics up to the 47th are observed and the 1 7th and 19th are located just on the hill. We have
also studied in more detail how the strength of the harmonics increases (or decreases) in function of the electric field.
The results for harmonics up to the 31st are shown in Fig. 5. Fo r each couple of harmonics there is a minimum field
threshold, below which the harmonic lines cannot be picked o ut from the background. All the harmonic strengths
of Fig. 5 are normalized to the fundamental. What is worth to s tress is that for the 5th harmonic we can have an
efficiency up to 14%.
Repeating the sequence of simulations shown in Fig.4 for a lo werAgives results that are quite similar to those shown
in Fig. 4 as long as the gap between the two branches is quite la rge. When the gap becomes of the order of about
ten photons the two lines Λ aand Λ bplay a role that is less important. This is due to the fact that , as the gap is
smaller, the two lines are expected to be located in a low freq uency region and therefore they are easily hidden by the
background of the main HG spectrum that in the low frequency r egion is higher. Furthermore, as the intensity of the
harmonic lines is strongly enhanced with increasing field, t he strength of Λ aand Λ bis not. Also the extension of the
harmonic spectrum is dependent on A, for lower Aless harmonics can be generated (for a fixed frequency).
As already mentioned, Ω is the distance between 0and1. For high A(or high ω) we have ω >Ω but for decreasing
A(or decreasing ω), Ω approaches the laser frequency and overtakes it. The rat io ofωand Ω strongly affects the
harmonic emission by the ring molecule. In particular, we ob served that when ω≈Ω a very complex spectrum is
generated and together with the expected harmonics many oth er lines of similar intensities are present. This effect
can be seen in Fig. 6. A very similar behavior was also observe d in the 1D model [15]. In particular, the shape of the
additional satellite structures around each allowed harmo nic are in the two cases alike. It seems that in this resonant
case the system is not in a single non-degenerate Floquet sta te as assumed for the proof of the selection rules [11].
In that derivation the pulse was assumed as infinite. Therefo re, the particular behavior of the dipole emission could
be also due to a pulse shape effect. However, pulse shape effect s should be not dependent on the frequency, and in
fact, we have additional lines for all the frequencies, but t hose lines play always a minor role respect to the expected
harmonics. Instead, when the laser frequency becomes nearl y resonant many new strong lines appear.
Keeping the same pulse parameters and decreasing the parame terAthe ionization increases. When we want to study
a model that is closer to the real benzene, we have to take A= 0.37. With this condition the physical scheme is very
different from those of the cases previously discussed; the i onization energy is reduced from 2 .68 to 0 .34 and, as we
already mentioned, only four states are present and the “gap ” does not exist at all. Under these conditions a pulse
with the frequency ω= 0.0942, which we used so far, leads to emission spectra without any harmonic structure. This is
mainly for the reason that the frequency is relatively high w ith respect to the ionization potential and therefore just a
very few photons are sufficient for reaching the continuum. Th erefore, unless the field is very low the ionization would
prevail at soon. Also making a comparison with the rule for th e cut-off position used in the atomic case ( Ip+3.17Up)
we should not expect any harmonics due to the low value of Up. Therefore, we made a series of simulations with the
benzene model but using a lower frequency, ω= 0.0314. With this frequency eleven photons are required for re aching
the continuum, i.e., the molecule can be ionized only with a h igh-order multiphoton process. A spectrum obtained
with this low frequency and E= 0.035 is shown in Fig. 7. Like in the highly-bound high frequenc y case the emission
spectrum exhibits the harmonics allowed by the selection ru les. The efficiency of the harmonics in Fig. 7 is not as high
as that one of the harmonics of Fig. 4 but we belief that this is mainly just a problem of optimization. Another line,
located around 3 ωis also present in the spectrum of Fig.7. This line is a resona nce between the states 0and1(for
A= 0.37, Ω = 0 .0914≈3ω). It is interesting to note that in this case of a weakly bound electron the results from
the 2D simulations are different with respect to those from th e 1D simulations [15]. In the latter case no harmonic
structure was observed for ω <Ω. This, in our opinion, could be due to the reason that in the 1 D model the level
scheme is qualitatively different. In particular, there is n o continuum in the 1D case.
VI. CONCLUSIONS
In this work we have studied the harmonic emission in a ring mo lecule. We have shown that when a ring molecule
interacts with a laser pulse, together with the series of har monics predicted by the selection rules, other lines are
present. Under certain conditions the strength of these lin es can be comparable with that one of the harmonics. Our
HG spectra present a structure and a complexity that is absen t in the numerical results shown in [11]. This is due to
the reason that while there a 1D Floquet simulation was perfo rmed in our studies a realistic pulse (i.e., a finite pulse
with a certain envelope) and a 2D model with ionization inclu ded were used, and the TDSE was solved ab initio .
What is worth to note is the scaling of the TDSE (12) with respe ct to the size of the molecule. If one scales the
molecule radius like ρ′=αρ, the TDSE (12) remains invariant if t′=α2t,V′=V/α2,E′=E/α3,ω′=ω/α2are
chosen. Therefore our results for high Acan reproduce the results that would have been obtained for a bigger molecule
with a lower ionization potential interacting with a field of lower frequency. Moreover, because of the generality of
5the hypotheses taken into account, one can think about the di fferent cases of Fig. 1 as the level scheme of positively
charged molecules as well.
To our knowledge, so far harmonics in ring-shaped molecules have been investigated experimentally only in gaseous
samples [16]. Studying a gaseous sample is very different wit h respect to what we did in our simulations. Because
the molecules in a gas have a random orientation it is not poss ible to apply the symmetry properties discussed. If
the propagation direction of the circularly polarized field is not orthogonal to the plane of the molecule, the molecule
“sees” a field that is elliptically polarized. This breaks th e discrete rotational symmetry and other harmonics become
possible. Nonetheless, even if it is not possible to make a di rect comparison it is useful to note that the results
presented in [16] confirm that ring molecules, like benzene a nd cyclohexane, can tolerate short pulses of high intensity
and phenomena like fragmentation and Coulomb explosion do n ot play a big role. In order to reproduce in a real
experiment the results we presented, it is fundamental to pr epare a sample where most of the molecules lie in the
same plane. This could be done with some orientation techniq ues or, considering the particular shape of the organic
molecules, preparing a very thin layer.
We described and discussed results for a molecules with N= 6, but cases with higher Nare as well possible. Moreover,
the higher is Nthe higher is the frequency of the first generated harmonic; i n the limit of very high N, harmonics of
very short wavelength could be generated.
In this work we a took into account a single active electron, b ut also including correlation through both, a full
description [11] or an appropriate treatment like time-dep endent density functional theory, would not not change the
symmetry properties of the Hamiltonian describing the syst em. Therefore, the selection rules would apply as well.
More complex molecules which produces similar selection ru les [17] are the nanotubes [19]. They can be very long
in the longitudinal direction and exhibit a discretized cyl indrical symmetry. A semiclassical approach to harmonic
generation from nanotubes was also investigated [18]. Unfo rtunately, because of their size, the dipole approximation
would not be accurate enough. Therefore, an ab initio numerical simulation in 3D would be at the limit, or probably
beyond, the calculation capabilities of even the fastest co mputers now available.
ACKNOWLEDGEMENTS
This work was supported in part by the Deutsche Forschungsge meinschaft within the SPP “Wechselwirkung intensiver
Laserfelder mit Materie”.
6FIGURES
FIG. 1. Energetic behavior of the first six states versus A. For lower Asome of the states belong to the continuum;
the magnification shows the region around A= 0.37 where four states are bound. For higher Athe set of states is
split in two branches and a gap between those branches appear s.
Fig. 2. Contour plots of the first six states. For each double d egenerate state, two linearly independent state are
shown. The non degenerate states present a fully C6symmetry.
Fig. 3. Emission spectrum in the low field regime. In (a) A= 1.6 and E= 0.02, in (b) A= 1.4 and E= 0.01. The
lines Λ a, Λband Υ are present in both pictures. When Adecreases the two lines Λ aand Λ bare red-shifted. This can
be observed comparing (b) with (a).
Fig. 4. Evolution of the harmonic spectrum with increasing fi eld for A= 1.6. In (a) E= 0.03, in (b) E= 0.15, in (c)
E= 0.24 and in (d) E= 0.30. When the field increases additional satellite lines appe ar with the low order harmonics.
The highest allowed resolved harmonic is the 47th.
Fig. 5. Strength of each harmonic line versus the electric fie ldE. It is worth to note that the first allowed harmonic,
i.e., the 5th, can reach an efficiency up to 14%. In the low field r egion the strength of the the 17th and 19th harmonic
is particularly high, this is due to the presence of the “hill ”. The value of Ais constant, A= 1.6.
Fig. 6. Emission spectrum for A= 1.00 and ω= 0.0942. The value of Ω approaches the laser frequency ωand the
spectrum exhibits a complex structure with many additional strong lines.
Fig. 7. Emission spectrum for A= 0.37 and ω= 0.0314. Also in this case, where the ionization potential corr esponds
to that one of the benzene molecule, the spectrum exhibits th e same structure as in the highly-bound high frequency
case. The line around 3 ωis given by a decay from the state 1to the state 0.
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8-3-2.5-2-1.5-1-0.50
0.20.40.60.811.21.41.6-0.3-0.2-0.10
0.3 0.325 0.35 0.375 0.4
/BU/CT/D2/DE/CT/D2/CT/BT /CJ/CP/BA/D9/BA℄/BX/D2/CT/D6/CV/DD /CJ/CP/BA/D9/BA℄
Fig. 1: F.Ceccherini and D.Bauer, “Harmonic generation in . ..”/BC /BD/CP
/BD/CQ/BE/CP
/BE/CQ
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/BH/CQ/DC /CJ/CP/BA/D9/BA℄/DD /CJ/CP/BA/D9/BA℄
Fig. 2: F.Ceccherini and D.Bauer, “Harmonic generation in . ..”/B4/CP/B5/B4/CQ/B5
/CA
/CA
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/D3/D6/CS/CT/D6/CB/D8/D6/CT/D2/CV/D8/CW /CJ/CP/D6/CQ/BA /D9/BA℄
Fig. 3: F.Ceccherini and D.Bauer, “Harmonic generation in . ..”/B4/CP/B5 /B4/CQ/B5/B4
/B5 /B4/CS/B5/C0/CP/D6/D1/D3/D2/CX
/D3/D6/CS/CT/D6/CB/D8/D6/CT/D2/CV/D8/CW /CJ/CP/D6/CQ/BA /D9/BA℄
Fig. 4: F.Ceccherini and D.Bauer, “Harmonic generation in . ..”1e-201e-181e-161e-141e-121e-101e-081e-060.00010.011
0.03 0.075 0.12 0.165 0.21 0.255 0.3/BX/D0/CT
/D8/D6/CX
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/BE/BL/BF/BD/C6/D3/D6/D1/CP/D0/CX/DE/CT/CS /D7/D8/D6/CT/D2/CV/D8/CW
Fig. 5: F.Ceccherini and D.Bauer, “Harmonic generation in . ..”/C0/CP/D6/D1/D3/D2/CX
/D3/D6/CS/CT/D6/CB/D8/D6/CT/D2/CV/D8/CW /CJ/CP/D6/CQ/BA /D9/BA℄
Fig. 6: F.Ceccherini and D.Bauer, “Harmonic generation in . ..”/C0/CP/D6/D1/D3/D2/CX
/D3/D6/CS/CT/D6/CB/D8/D6/CT/D2/CV/D8/CW /CJ/CP/D6/CQ/BA /D9/BA℄
Fig. 7: F.Ceccherini and D.Bauer, “Harmonic generation in . ..” |
arXiv:physics/0102024v1 [physics.bio-ph] 9 Feb 2001DESY–01–013
physics/0102024
On the Compatibility Between Physics and
Intelligent Organisms
John C. Collins,∗†
DESY, Notkestraße 85, D-22603 Hamburg, Germany,
and
II Institut f¨ ur Theoretische Physik, Universit¨ at Hambur g,
Luruper Chaussee 149, D-22761 Hamburg, Germany
9 February 2001
Abstract
It has been commonly argued, on the basis of G¨ odel’s theorem
and related mathematical results, that true artificial inte lligence can-
not exist. Penrose has further deduced from the existence of human
intelligence that fundamental changes in physical theorie s are needed.
I provide an elementary demonstration that these deduction s are mis-
taken.
Is real artificial intelligence possible? Are present-day t heories of physics
sufficient for a reductionist explanation of consciousness? Among the long
history of discussions of these questions [1–4], the eloque nt writings of Pen-
rose [2] stand out for strong mathematical arguments that gi ve negative an-
swers to both questions.
For a physicist, Penrose’s result is quite striking. He clai ms that under-
standing the human brain entails big changes in current micr oscopic theories
∗E-mail: collins@phys.psu.edu
†On leave from: Physics Department, Penn State University, 1 04 Davey Laboratory,
University Park PA 16802, U.S.A.
1of physics (e.g., quantum mechanics in general and quantum g ravity in par-
ticular). This is contrary to our normal scientific experien ce. The enormous
progress in both elementary particle physics and in molecul ar biology during
the last 30 years has had no direct mutual influence on the two fi elds, except
on our students’ career choices. Now we have an eminent theor etical physicist
telling elementary particle physicists to look to neurons r ather than multi-
billion-dollar accelerators for progress in physics. Shou ld we believe him?
In this paper, I provide an elementary argument that this cha in of rea-
soning fails.
1 Penrose’s argument
Penrose observes that current microscopic theories of phys ics are compu-
tational, and that they appear to underlie all chemical and b iological phe-
nomena. It follows that it is possible to simulate all proper ties of biological
organisms by a computer program. Of course, the computer pro gram will
be impractically big to implement, but this fact does not affe ct Penrose’s
mathematics. Among the biological organisms are mathemati cians, so the
computer program provides artificial mathematicians that a re completely
equivalent to human mathematicians. This runs afoul of Turi ng’s halting
theorem, which, taken at face value, implies that artificial mathematicians
are always less powerful than human mathematicians.
From this contradiction, Penrose deduces that better theor ies of physics
are needed and that the new theories must be non-computation al, unlike
current theories, such as the “Standard Model”, which are al l particular
quantum mechanical theories. On the way he also demolishes a ll hope for
true artificial intelligence.
Of course, this argument attracted much comment [3, 4]. The c ritics
observe that real computer programs appear to have a much ric her range
of behavior than the kind of computation used in Turing’s the orem. This
theorem applies to strictly non-interactive computer prog rams (technically
known as Turing machines), whereas real intelligent entiti es are obviously
much more like interactive computer programs. But Penrose a lways appears
to have a comeback. For example, if an intelligent computer n eeds to be
trained by an environment, then he tells us to simulate the en vironment by a
computer, just as one might hook up an aircraft control compu ter to a flight
simulator instead of a real crashable jumbo jet.
2Thus Penrose counters the criticism by observing that from a n interactive
program one can construct a non-interactive program, i.e., one that does not
depend on repeated interaction with other beings or with an e nvironment. I
will show that this construction fails. The construction of a non-interactive
program satisfying a particular precise specification of th e kind needed in
the Turing theorem inevitably loses access to the full power s of a putative
intelligent interactive program.
2 Turing’s halting theorem
The technical results use the concept of a “Turing machine”. Now a Turing
machine is simply any device for performing a defined computa tion. Turing’s
achievement was to characterize this mathematically, i.e. , to define in general
what a computer program is. Hence, instead of Turing machine s, we can
equally well discuss realistic computers and actual progra mming languages.
The halting theorem — see Penrose’s excellent treatment [2] — concerns
a subroutine Tk(n) which takes one argument and which obeys the following
specification:
Tk(n) halts if and only if it has constructed a correct proof that
the one-argument subroutine defined by ndoes not halt when
presented with data n.
Here, the subscript in Tkrepresents the source code for the subroutine. The
argument nis the source code for another subroutine, and Tkconcerns itself
with proving properties of this second subroutine.
Turing’s halting theorem is obtained when one sets n=k, i.e., when
Tkis asked to prove a theorem about itself. There is a contradic tion unless
Tk(k) does not halt; this result is exactly the halting theorem. F rom the
subroutine’s specification, we see that the subroutine is un able to prove this
same theorem, the one stated in the previous sentence.
But humans can prove the theorem. From this follow the conclu sions
about the impossibility of artificial intelligence, etc.
3 Non-Turing computations
What is the relation between a dry abstract theorem-proving subroutine and
a computer program simulating biological organisms? The be havior of the
3organisms (even the mathematicians) is clearly a lot richer and more varied
than that of the theorem prover. Basically the answer is in th e common
assertion that all computers and computer programs are exam ples of Turing
machines; that is, they can each be viewed as some subroutine Tk. To obtain
the Turing machine used in the halting theorem, one simply ha s to ask the
simulated mathematician to prove an appropriate theorem.
However, the common assertion, of the equivalence between T uring ma-
chines and computer programs, is not exactly correct. The id ea of a Turing
machine is that it is given some definite input, it runs, and th en it returns
the results of the computation. This is appropriate for the c alculation of a
trigonometric function, for example. But a real computer pr ogram may be
interactive; it may repeatedly send output to the outside wo rld and receive
input in response, word processors and aircraft control pro grams being ob-
vious examples. Such computer programs are not Turing machi nes, strictly
speaking. As Geroch and Hartle [5] have observed, a Turing ma chine is
equivalent to a particular kind of computer program, a progr am whose input
is all performed in a single statement that is executed once.
The legalistic distinction between Turing and non-Turing c omputations
matters critically for Penrose’s results. Software that at tempts to mimic
real intelligence must surely be in the more extended class o f interactive
programs. Moreover, if one is to avoid programming all the de tails of its be-
havior, the program must learn appropriate responses in int eraction with an
environment. The prototypical case is unsupervised learni ng by an artificial
neural network, an idea with obvious and explicit inspirati on from biological
systems.
To be able to using the halting theorem, one must demonstrate that, given
some software that genuinely reproduces human behavior, on e can construct
from it a subroutine of the Turing type suitable for use in the theorem.
4 Interactive programs
Penrose [2] gives a number of examples, that appear to show th at it is easy
to construct the requisite non-interactive subroutine usi ng the interactive
program as a component.
However, there is a big problem in figuring out how to present t he input
to the program, to tell it what theorem is to be proved. Now the program,
which we can call an artificial mathematician, is in the posit ion of a research
4scientist whose employer specifies a problem to be worked on. To be effective,
such a researcher must be able to question the employer’s ord ers at any point
in the project. The researcher’s questions will depend on th e details of the
progress of the research. (“What you suggested didn’t quite work out. Did
you intend me to look at the properties of XXYZ rather than XYZ ?”) As
every scientist knows, if the researcher does not have the fr eedom to ask
unanticipated questions, the whole research program may fa il to achieve its
goals.
Therefore to construct the non-interactive program needed by Penrose
one must discover the questions the artificial mathematicia n will ask and
attach a device to present the answers in sequence.1The combination of the
original computer and the answering machine is the entity to which Turing’s
halting theorem is to be applied.
How does one discover ahead of time the questions that will be asked?
(Remember that the program is sufficiently complex that one do es not de-
sign it by planning ahead of time the exact sequence of instru ctions to be
executed.) One obvious possibility is simply to run the prog ram interactively
to discover the questions. Then one programs the answering m achine with
the correct answers and reruns the program.
This is exactly what a software manufacturer might do to prov ide a
demonstration of a graphical design program. Both the graph ical design
program and the answering machine are interactive programs ; but the com-
bination receives no input from the outside world and is ther efore an actual
Turing machine.
Here comes a difficulty that as far as I can see is unsolvable. Th e first
input to the program was a request to prove a particular theor em about a
particular computing system. This computing system happen ed to be the
program itself, together with all its ancillary equipment . When one reruns
the program after recording the answers to its questions, th e theorem under
consideration has changed. The theorem is now about the orig inal computing
system including the answering machine, and, most importan tly, the answers
recorded on it.
The answers were recorded when the program was asked to prove a theo-
1In the case of a complete microscopic simulation of the real w orld, one must also
figure out how to present mathematics research problems to th e beings that are created
by the simulation. This is quite non-trivial given that the a ctual programming concerned
itself exclusively with the interactions of quarks, gluons and electrons. Nevertheless let us
assume that this problem has been solved.
5rem about the computing system with no answers on the answeri ng machine.
Why should the questions remain the same when the theorem has changed?
If they don’t, then the recorded answers can easily be wildly inappropriate.
Of course, the theorem has not changed very much. However, in a com-
plex computing system the output often depends sensitively on the details
of the input. Indeed, a system that is intended to be intellig ent and creative
should show just such unpredictable behavior.
No matter how one goes about it, to discover the exact questio ns that the
artificial mathematician will ask requires us to know the ans wers. But one
doesn’t know which answers are needed until one knows the que stions. And
one must know the exact questions and answers, for otherwise one cannot
set up the subroutine used in Turing’s halting theorem. The s ubroutine is
asked to prove a theorem Kabout a certain subroutine. The proof of Turing’s
halting theorem is inapplicable if even one bit of machine co de differs between
the subroutine that attempts to prove the theorem Kand the subroutine that
is the subject of the theorem.
Once one realizes that the exact information on the question s and answers
cannot be found, the applicability of the halting theorem to a simulation of
biological organisms fails, and with it Penrose’s chain of a rgument.
5 Conclusion
Intelligent software must behave much more like a human than the kinds of
software that are encompassed by the strict definition of a Tu ring machine.
Penrose’s conclusion requires taking absolutely literall y the idea that every
computation can be reduced to some Turing machine, so that he can use
Turing’s halting theorem. The proof of the theorem requires perfect equality
between a certain subroutine that proves theorems and the su broutine that
is the subject of a theorem to be proved. But the practicaliti es of convert-
ing intelligent software to the non-interactive software u sed in the halting
theorem preclude one from achieving this exact equality.
We see here an example of a common phenomenon in science: any s tate-
ment we make about the real world is at least slightly inaccur ate. When
we employ logical and mathematical reasoning to make predic tions, the rea-
soning is only applicable if it is robust against likely devi ations between the
mathematics and the real world. This does not seem to be the ca se for
Penrose’s reasoning.
6Acknowledgments
I would like to thank A. Ashtekar, S. Finn, J. Hartle, D. Jacqu ette, R.
Penrose and L. Smolin for useful discussions, and I am partic ularly grateful
to J. Banavar for a careful critique of this paper. I would als o like to thank
the U.S. Department of Energy for financial support, and the A lexander von
Humboldt foundation for an award.
References
[1] For example, J.R. Lucas, Philosophy 36, 120 (1961).
[2] R. Penrose, Shadows of the Mind (Oxford, 1994).
[3] Discussion about R. Penrose’s The Emperor’s New Mind , inBehavioural
and Brain Sciences 13(4) 655 (1990).
[4] Symposium on Roger Penrose’s Shadows of the Mind inPsyche 2(1995),
athttp://psyche.cs.monash.edu.au/psyche-index-v2.html .
[5] R. Geroch, J.B. Hartle, Found. Phys. 16, 533 (1986).
7 |
arXiv:physics/0102026v1 [physics.flu-dyn] 10 Feb 2001On the thermocapillary motion of deformable droplets
V.Berejnov
Department of Chemical Engineering, Technion, Haifa 32000 , Israel
February 2, 2008
Abstract
In studies on Marangoni type motion of particles the surface tension is often
approximated as a linear function of temperature. For defor mable particles in a
linear external temperature gradient far from the referenc e point this approximation
yields a negative surface tension which is physically unrea listic. It is shown that
H. Zhou and R. H. Davis ( J. Colloid Interface Sci. ,181, 60, (1996)) presented
calculation where the leading deformable drop moved into a r egion of negative
surface tension. With respect numerical studies the restri ction of the migration of
two deformable drops is given in terms of the drift time.
The bulk fluid motion induced by an interface has been studied for over a century.
One of the most interesting phenomena is the capillary motio n of particles through a
viscous fluid. Young, Goldstein and Block (1) and later Bratu khin (2) performed the
first systematic study of the migration of bubbles and drople ts. As noted in review (3)
the capillary motion arises due to gradient of the surface te nsion γat the interface as a
result of a non–uniform temperature or surfactant distribu tion in the surrounding media.
The surface tension gradient results in a tangential stress on the interface which causes
1the motion of the surrounding liquid by viscous traction. Th en, the droplet or bubble
will move in the direction of decreasing interfacial tensio n. It is necessary to note that
the normal component of the capillary forces arising during the motion may deform the
shape of a particle (2). Young, Goldstein and Block (1) and ot hers have shown that in
the limit of high surface tension (undeformed spherical par ticle) its motion is controlled
by surface tension gradients only. Note that the motion of a d eformable particle also
depends on the surface tension itself.
If the particle moves with constant velocity the transforma tion of a laboratory co-
ordinate system to a coordinate system moving with the parti cle frame will essentially
simplify the solution (2), (3), (5). Let us denote the partic le coordinate system moving
with the droplet velocity UbyO′and the laboratory coordinate system by Orespec-
tively. We consider the coordinate transform from OtoO′in the case of a drop moving
in the uniform external temperature gradient Aex(2), see Fig.1. For an arbitrary point
Fwe obtain,
R=R′+Ut,Vi(R, t) =V′
i(R′) +U, T i(R, t) =T′
i(R′) +A U t, [1]
where i= 1,2 correspond to the inner and outer liquid phase, respective ly,Vis the fluid
velocity, T′denotes the difference between the temperature TinOand a undisturbed
temperature AUtat the center of O′,Ris a radius vector which points from OtoFand
tis the time.
In the limit of an infinitely large surface tension the normal stress boundary condition
is not modified under the above transformation [8]. However, in the case of finite surface
tension, this boundary condition requires special attenti on. Usually, γis assumed to be
linearly dependent on temperature or on concentration is li nearized (6),
γ(R, T) =γ0(T0) +∂γ
∂T/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
T=T0(T(R)−T0), [2]
where ∂γ/∂T is a constant and T0andγ0correspond to the reference values of temper-
ature and surface tension, respectively. Note that for many cases ∂γ/∂T < 0. Due to
2the transformation of Tthe surface tension γ(R, T) is also transformed in the moving
coordinate system,
γ′(R′, T′) =γ0(T0)−∂γ
∂TA U t +∂γ
∂T/parenleftBig
T′(R′)−T0/parenrightBig
. [3]
The surface tension γ′is time dependent now. Recall that the surface tension must b e
positive (6),
γ≥0, γ′≥0. [4]
From [3] and [4] follows an upper bound of the drift distance Utin system Oor an upper
bound of the time of particle migration in the moving system O′.
Ignoring the above restrictions results in the appearance o f a negative surface tension
in the course of the particle migration in finite time and thus may lead to a physically
unrealistic behavior of the particle. This restriction is r elaxed in the case of the un-
deformed drop (1), (4) and (3) where the normal stress bounda ry condition is always
satisfied. However, this is not true when the surface tension has a finite value. We noted
that in the literature on thermocapillary migration of drop s and bubbles no attention
was paid to this point. For example, Zhou and Davis (7) first co nsidered the problem of
axisymmetric thermocapillary migration of two deformable viscous drops . The authors
assumed a linear dependence of surface tension on temperatu re. In terms of (7) we have
γ(xs) =γ0+∂γ
∂T(T(xs)−T0(xr)), [5]
where T(xs) is the temperature at a point xson the interface and T0(xr) is a reference
temperature. In an attempt to obtain a solution which is inde pendent of the choice of
xr, Zhou and Davis fix xrto be the intersection point of the axis of symmetry with the
surface of the leading drop, see Fig.1 and their Fig.1 in (7). It is important to note
that this choice of xrmeans a coordinate transform from the laboratory frame to the
coordinate system moving with the leading droplet. Hence, t he normal stress balance is
modified. The other boundary conditions and the governing eq uations remain the same
due to the linearity of Stokes and Laplace equations (7). For more details see (2) and (4).
3Zhou and Davis (7) give for the dimensionless surface tensio n in the moving coordinate
system
¯γ(xs) = 1−q¯T(xs), [6]
where ¯ γ=γ/γ0is the dimensionless surface tension, q=aA(−∂γ/∂T ) is the di-
mensionless rate of change of the interfacial tension due to temperature variation,
¯T(xs) = ( T(xs)−T0(xr))/(aA) is the dimensionless temperature difference and ais
the radius of the first drop. It can readily be seen that Eq.[6] defines surface tension
which is positive for any time or migration distances. As we s howed before, the correct
transformation of the linear approximation [5] leads to a ne gative surface tension in finite
time. The previous conclusion that physically acceptable s olutions must be restricted by
migration time contradicts Eq. [6].
Let us derive the correct form of the transformed surface ten sion in terms of (7). The
problem of the migration of two droplets is evolutionary and it must be accomplished
by a kinematic condition applied on the droplets’ surfaces. The transformation from the
laboratory coordinate system to the particle coordinate sy stem are given by (5):
R=R′+/integraldisplayt2
t1U(t)dt,Vi(R, t) =V′
i(R′) +U(t), [7]
Ti(R, t) = T′
i(R′) +A/integraldisplayt2
t1U(t)dt. [8]
The migration velocity of the droplet now depends on time and therefore the migration
distance Uton the right hand side of [3] is given as an integral term,
γ′(R′, T′) =γ0(T0)−∂γ
∂TA/integraldisplayt2
t1U(t)dt+∂γ
∂T/parenleftBig
T′(R′)−T0/parenrightBig
. [9]
In terms of (7) we have for the dimensionless surface tension
¯γ(xs) = 1 +q
a/integraldisplay
U(t)dt−q¯T(xs). [10]
The integral term in Eq. [10] changes the scenario of a numeri cal calculation. The surface
tension changes with time and it is necessary to keep ¯ γpositive.
4We shall now proceed to estimate the time when the surface ten sion of some point
xron the leading drop will not satisfy [4]. For simplicity let u s stay in the laboratory
coordinate system, for ¯ γ= 0 we obtain the relation1
q¯T(xs) = 1, [11]
where ¯T(xs) = (T(xs)−T0(x0))/(aA) andx0is a reference point in system O. It is
readily seen that the dimensionless length of a spatial fram e is given by X=1
q. Then
the maximum transformation distance of the leading drop is t he difference between X
and the initial position xr. For the case of equal material parameters considered by (7)
we have a= 1 and the surface separation distance on the axes is ∼1. Hence the length
of the drops’ drift is also ∼1. Let us assume that the lower bound of the velocities for
moving deformable drops is the velocity of non–deformable d rops. For slightly unequal
drops and a large separation distance between their centers the velocities are nearly the
same and equal to the Young–Bratukhin value of 0.133.. Follo wing Eq. [12] in (7) we
normalize this value with 2 /15 because for inner and outer liquids the viscosity and the
thermal diffusivity are equal. From this normalization proc edure we obtain that the
migration velocities are ∼1. As a result the critical value of the migration time is ∼1.
We developed a numerical code for solving the problem of the m otion of two de-
formable viscous drops in an external temperature gradient (9). Restrictions [4] were
considered in the laboratory coordinate system O. In Fig.2 we plotted the evolution of
the minimum separation distance dbetween the droplets’ surfaces in time. We chose
a= 1,α= 0.5,q= 0.2 in terms of (7), where αis the droplets radii ratio. The dotted
curve confines the physical region where [4] is satisfied. The curves 1 −10 correspond to
different initial separations. Curve 2 is in agreement with t he results given by Zhou and
Davis (see Fig.4 in (7)) and with the asymptotics for the non– deformable drops (8).
Our computations show that the patterns of drops deformatio ns are similar to those
described by (7) but correspond to smaller separation dista ncesd. Note that our analysis
1On physical grounds this limit corresponds to phase transit ion.
5is restricted by [4] while the results of (7) lie in the physic ally unrealistic region. For
initially spherical drops and an initial separation distan ced= 0.01 Fig.3 depicts the
series of drops’ profiles corresponding to the points a,bandcin Fig.2.
The author wish to thank A. M. Leshansky and T. Loimer for help ful discussions.
References
1. Young, N.O., Golsdtein, J.S., and Block, M.J., J. Fluid Mech. ,6, 350, (1959).
2. Bratukhin, Yu.K., Fluid Dyn. ,10, 833, (1975)
3. Subramanian, R.S., in “Transport Processes in Bubbles, Drops and Particles” ,
(P.Chhabra, D.DeKee), 1, Hemisphere, New York, 1992
4. Subramanian, R.S., AIChe J. ,27, 646, (1981)
5. Antanovskii, L.K., and Kopbosinov, B.K., J.Prikl.Meh. and Teh.Fis ,2, 59, (1986)
6. Adamson A.W., “Physical Chemistry of Surfaces”, 3-Ed., W iley-Interscience Pub-
lication, New York, (1976)
7. Zhou, H., and Davis, R.H, J. Colloid Interface Sci. ,181, 60, (1996)
8. Keh, H.J., and Chen, S.H., Int. J. Multiphase flow ,16, 515, (1990).
9. Berejnov, V., Lavrenteva, O.M, and Nir, A., J. Colloid Interface Sci. , (submitted),
(2001)
6Figures
xr''γ
T
FO'
'O Ut
RRX
'γ
Figure 1: Geometric sketch of a drop immersed in an external t emperature gradient ∇T
parallel to its axis of symmetry.
700.5 11.5 22.5 3
t0.00010.0010.010.11d1
3
4
5
6782
9
10a
b
c
Figure 2: Evolution of the separation distances for two defo rmable viscous drops under
a linear external temperature gradient as a function of the i nitial separation (9). The
values of the parameters are the same as in (7): a= 1,α= 0.5,q= 0.2.
8a
b
c
Figure 3: Deformation patterns for the initial separation 0 .01 and a= 1,α= 0.5,
q= 0.2. Figures a, b, c correspond to the respective points on curve 10 of the Fig.2.
9 |
arXiv:physics/0102027v1 [physics.gen-ph] 12 Feb 2001A Reconciliation of
Collision Theory and Transition State Theory
Yong-Gwan Yi∗
February 20, 2014
Abstract
A statistical-mechanical treatment of collision leads to a formal connec-
tion with transition-state theory. This paper suggests tha t collision theory
and transition-state theory might ultimately be joined as a collision induced
transition state theory.
Collision theory and transition-state theory are alternat ive approaches to chem-
ical reaction rates [1]. There have been many important exte nsions of the kinetic
theory of collision and modifications of the transition-sta te theory. In this paper, I
should like to point out that above all collision theory and t ransition-state theory
could have been joined at their early stages as a collision in duced transition state
theory. I shall sketch a statistical-mechanical treatment of collision and its formal
connection with transition-state theory.
Consider a collision process between two molecules of AandB. We can discuss
in the coordinate system of the center of mass the collision t hat occurs between
the two molecules. All of the energy which goes into exciting the activated complex
must come from the energy of relative motion of the reactants . Energy in the center
of mass motion cannot contribute. According to the kinetic t heory of collision, the
rate constant has to be weighted by the Maxwell-Boltzmann di stribution function
f(u) of relative speed u, with integration over speeds from zero to infinity, to give
the overall average rate constant:
kC=/integraldisplay∞
0σuf(u)du, (1)
where σis the collision cross section. The rate constant in this exp ression is given
by M. Trautz in 1916 and by W. C. M. Lewis in 1918. It is convenie nt to integrate
over the translational energy instead of the speed u.
It is instructive to evaluate the rate constant in terms of en ergy states instead
of direct integration. We now consider the basic method of st atistical mechanics of
evaluating partition function [2]. Statistical mechanics states:
∗Geo-Sung furume Apt. 101-401, Gebong-1-dong Guro-ku, Seou l, 152-091 Korea
1The partition function is a sum over all states Ω, very many of which
have the same energy. One can perform the sum by first summing o ver
all the Ω( E) states in the energy range between EandE+δE, and then
summing over all such possible energy ranges. Thus
Q=/summationdisplay
ne−En/kBT=/summationdisplay
EΩ(E)e−E/k BT. (2)
The summand here is just proportional to the probability tha t the sys-
tem has an energy between EandE+δE. Since Ω( E) increases very
rapidly while exp( −E/k BT) decreases very rapidly with increasing E,
the summand Ω( E)exp(−E/k BT) exhibits a very sharp maximum at
some value E∗of the energy. The mean value of the energy must then
be equal to E∗, and the summand is only appreciable in some narrow
range ∆ E∗surrounding E∗. The partition function must be equal to
the value Ω( E∗)exp(−E∗/kBT) of the summand at its maximum mul-
tiplied by a number of the order of ∆ E∗/δE, this being the number of
energy intervals δEcontained in the range ∆ E∗. Thus
Q= Ω(E∗)e−E∗/kBT/parenleftbigg∆E∗
δE/parenrightbigg
,so lnQ= ln Ω( E∗)−E∗
kBT+ln/parenleftbigg∆E∗
δE/parenrightbigg
.
(3)
But, if the system has fdegrees of freedom, the last term on the right is
at most of the order of ln fand is thus utterly negligible compared to the
other terms which are of the order of f. Hence, the result agrees with
the general definition S=kBln Ω(E∗) for the entropy of a macroscopic
system of mean energy E∗.
We have seen the basic method of statistical mechanics of eva luating the parti-
tion function. If we apply this to the integration of Eq. (1), we expect an expression
for the rate constant to be roughly
kC=σu∗/parenleftbigg∆E∗
δE/parenrightbigg
Ω(E∗)e−E∗/kBT, (4)
where u∗is a relative velocity for reaching the activated state. Thi s summation
indicates that the integration over the translational ener gy has a very sharp maxi-
mum at the activation energy E∗. The width ∆ E∗of the maximum, given by the
square root of the dispersion, is very small relative to E∗for a macroscopic system.
The Maxwell-Boltzmann distribution function we have used i s the one normalized
to unity on integration over all states. For the results of mo re realistic calculation
the normalization should be expressed in its explicit form. The following expression
is then obtained:
kC=σu∗/parenleftbigg∆E∗
δE/parenrightbigg/parenleftbiggΩ(E∗)
QAQB/parenrightbigg
e−E∗/kBT. (5)
This equation may also be written in terms of an entropy chang e in reaching
the activated state. As (∆ E∗/δE)Ω(E∗) represents a number of energy states in
the activated state, the expression for the rate constant in terms of entropy change
will be
kC=σu∗e∆S∗/kBe−E∗/kBT. (6)
2The basic method of statistical mechanics shows how an entro py term can be intro-
duced in the kinetic theory expression. It becomes evident t hat the kinetic theory of
collision does not lack the entropy term that should appear i n the expression for the
equilibrium constant. In a system of chemical reaction the e ntropy of the system is
a function of energy E, volume V, and the number of molecules N:S=S(E, V, N ).
Here ∆ S∗represents the change in entropy due to the change in energy i n reaching
the activated state. Hence, we can replace ∆ S∗in Eq. (6), using the thermodynamic
relations, by its generalization
∆S∗−→∆S∗−P∆V∗
T+µ∆N∗
T, (7)
for a system of chemical reaction. The rate constant can then be written in a general
form
kC=σu∗e−∆G∗/kBT, (8)
where ∆ G∗is the Gibbs energy change in going from the initial to the act ivated
state. The evaluation of collision in terms of energy states disproves that collision
theory of reaction rates is not consistent with the fact that at equilibrium the
ratio of rates in the forward and reverse directions is the eq uilibrium constant:
The kinetic theory expression provides us with a kinetic the oretical derivation of
the expression for the equilibrium constant, relating the m acroscopic equilibrium
constant to quantities that describe the situation on the mo lecular scale.
The transition-state theory was published almost simultan eously by H. Eyring
and by M. G. Evans and M. Polanyi in 1935. The rate equation for a bimolecular
reaction derived by this theory of reaction is
kTS=Q‡
QAQBe−E∗/kBT=kBT
h/parenleftbiggQ‡
QAQB/parenrightbigg
e−E∗/kBT. (9)
The partition functions QAandQBrelate to the two reactants, and Q‡is a special
type of partition function for the activated complex. It is j ust like a partition
function for a normal molecule, except that one of its vibrat ional degrees of freedom
is in the act of passing over to the translation along the reac tion coordinate.
Equation (5) is very suggestive in relating collision theor y to transition-state the-
ory. The kinetic theory expression leads us to an idea of conn ecting with transition-
state theory formula. Identifying Q‡with (∆ E∗/δE)Ω(E∗), we can put both theo-
ries into some perspective. From their formal expressions t he reaction can be viewed
as a succession of two steps −collision and transition state. The overall rate is then
given by the sum of two average lifetimes: rate = ( k−1
C+k−1
TS)−1. The rate reads
explicitly
rate =/bracketleftbigg/braceleftbigg
σu∗/parenleftbiggQ‡
QAQB/parenrightbigg/bracerightbigg−1
+/braceleftbiggkBT
h/parenleftbiggQ‡
QAQB/parenrightbigg/bracerightbigg−1/bracketrightbigg−1
e−E∗/kBT. (10)
The kinetic theory counts every sufficiently energetic colli sion as an effective one.
Equation (10) suggests, however, correcting the collision frequency by involving the
translation along the reaction coordinate in the evaluatio n of the partition function
over the translational energy states. The essential featur e of the argument is that
transition state is brought about by energetic collisions a nd that the rate of a
3reaction is determined by the frequency of these collisions and by the resulting
translations along the reaction coordinate.
The dependence of the overall reaction rate on the relative r ates of collision
and transition state reflects the most important aspects of u nimolecular reactions
[3]. Equation (10) is in exact agreement in form with the rate equation given by
Rice-Ramsperger-Kassel-Marcus (RRKM) theory of unimolec ular reactions. The
distribution function that has been used in RRKM theory is eq ual in expression
to that given by the basic method of statistical mechanics of evaluating partition
function. But the present discussion has shown the rate equa tion in a general for-
mulation of bimolecular reactions, and thus has given it a mu ch wider applicability.
The formalism provides a framework in terms of which molecul ar reactions can be
understood in a qualitative way. Usually, the kinetic theor y values are too high
for all except atom −atom reactions. Hence, the transition-state theory values can
be regarded as exerting important control over the rates of m olecular reactions. It
might be due to this high-pressure limit that has led kTSto much closer agreement
with experiment.
References
[1] K. J. Laidler, Chemical Kinetics (Harper & Row, 1987), 3rd ed.
[2] F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill, 1965);
R. H. Fowler, Statistical Mechanics (Cambridge, 1936), 2nd ed.
[3] K. A. Holbrook, Chem. Soc. Rev. 12, 163 (1983).
4 |
arXiv:physics/0102029v1 [physics.med-ph] 11 Feb 2001Extraction of137Cs by alcohol-water solvents
from plants containing cardiac glycosides
February 2, 2008
S.N. Dzyubak, Yu.I. Gubin
State Scientific Centre of Medicines,
Astronomicheskaya st., 61085 Kharkov, Ukraine
O.P. Dzyubak1, P.V. Sorokin, V.F. Popov
Institute of High Energy Physics and Nuclear Physics,
Akademicheskaya st., 61108 Kharkov, Ukraine
A.A. Orlov, V.P. Krasnov
Ukrainian Research Institute of Forestry and Forest Melior ation,
Poushkinskaya st, 61024 Kharkov, Ukraine
Abstract
As a result of nuclear power plant accidents, large areas rec eive ra-
dioactive inputs of137Cs. This cesium accumulates in herbs growing in
such territories. The problem is whether the herbs contamin ated by radio-
cesium may be used as a raw material for medicine. The answer d epends
on the amount of137Cs transfered from the contaminated raw material
to the medicine. We have presented new results of the transfe r of137Cs
from contaminated Digitalis grandiflora Mill. andConvallaria majalis L.
to medicine. We found that the extraction of137Cs depends strongly on
the hydrophilicity of the solvent. For example 96.5%( vol.) ethyl alcohol
extracts less137Cs (11.6 %) than 40%( vol.) ethyl alcohol or pure water
(66.2 %). The solubility of the cardiac glycosides is invers e to the solu-
bility of cesium, which may be of use in the technological pro cesses for
manufacturing ecologically pure herbal medicine.
Key words: Herbal raw material; Herbal medicine;137Cs contamina-
tion;137Cs transfer.
PACS : 87.52.-g, 87.53.Dq, 87.90.+y
1E-mail address: dzyubak@fnal.gov
11 Introduction
At the present time, herbal medicines containing cardiac gl ycosides are widely
used in the medical treatment of heart disease as the equival ent synthetic ana-
logues are not available yet. The Convallaria andDigitalis species are used as
raw material for the herbal cardiac medicine.
As a result of nuclear power plant accidents, large areas rec eive radioactive in-
puts of137Cs. This cesium accumulates in herbs growing in such territo ries.
The problem is whether the herbs contaminated by radiocesiu m may be used
as a raw material for medicine. The answer depends on the amou nt of137Cs
transfered from the contaminated raw material to the medici ne.
After the Chernobyl catastrophe a large area, growing many s pecies of medicinal
plants (such as Convallaria majalis L., Rhamnus catartica L., Acorus calam us
L., Vaccinium vitisidea L., Vaccinium myrtillus L. etc. ) was contaminated with
a high concentration of radionuclides. The main contaminan t (more then 90%
of the overall radioactivity) of the Ukrainian Polessie is137Cs [1]. The majority
(85 to 97% ) of137Cs is located in the soil layer at a depth of 0 to 10 cm, where
the roots of the medicinal herbs take up the137Cs [2].
Various solvents are used in the pharmaceutical industry to prepare the herbal
medicinal products. The poperties of these solvents define a quantitative and
qualitative structure of substances that can be extracted. The literature gives
some often contradictory information on the transfer of137Cs from medicinal
plant raw material to water and alcohol. Several studies [3, 4, 5, 6, 7] have
shown that the amount of radionuclide transferred from the s oil to the raw ma-
terial varied in wide range (from 10 to 250 fold). This differe nce was mainly
determined by specific features of the plants and depended st rongly on the type
of soil and climatic conditions which occurred during the ve getative period. The
transfer of137Cs from the plant raw material is 24-75% for the aqueous medic -
inal products and is 20-30% for the alcoholic ones [8, 9, 10, 1 1]. Grodzinskii
et al. [12] found that the specific activity of137Cs in water extracts was three
orders of magnitude less than in the initial plant raw materi al.
One can conclude, from the above publications that experime nts have been
mainly devoted to studying the transfer of137Cs from plant raw material to
water and some galenical preparations. Systematic studies of the dependency
of radionuclide extraction efficiency on various types of sol vents have not been
done yet. Therefore the study of radionuclide transfer from soil to herbs and
from herbs to the herbal medicinal products is very importan t.
In this paper we present our experimental results on the effec t of solvent
type (pure water, 40 %(vol.), 70%(vol.)and 96.5 %(vol.)aqueous ethyl alcohol )
on137Cs extraction efficiency.
2 Methods
22.1 Raw material
We studied the medicinal plant species (the herbs Digitalis grandiflora Mill ,
flowers and leaves of Convallaria majalis L. ) containing cardiac glycosides. The
raw material was taken from an experimental plot of the Povch ansk forest area,
Luginy district, Zhitomir region where the soil contaminat ion by137Cs ranged
from 296 to 925kBq
m2and the radioactivity of the tested plant raw material was
1.17 – 50.83kBq
kg(see Table 1). All raw plant material was rolled three times i n
preparation for the extraction process.
2.2 Tinctures and aqueous extracts
We used the following solvents: pure water, 40 %(vol.), 70%(vol.)and 96.5 %(vol.)
aqueous ethyl alcohol without heating and vaporization of solvent. Samples (10.0
galiquots) of raw material were put in glass extractors, fille d up with the ap-
propriate concentration of solvent and let stand for extrac tion. After 48 hours,
the extracts were discharged to a flask and the extractors wer e filled up with a
new portion of the solvent for further extraction. The opera tion was repeated
after 24 hours and 48 hours to obtain the second and third extr acts. Then the
solvent residuals were removed using vacuum and added to the extracts. All
extracts were combined to obtain the primary tinctures of 10 0mlvolume.
2.3 Decoctions
The solvent that we used was hot water. The 10.0 gsamples of raw material were
put in glass vessels, filled up with pure water and boiled for 1 5 minutes. After
that these vessels were cooled and kept at room temperature f or 10 minutes.
The filtered extracts and the solvent residuals were then com bined to give the
decoctions of 100 mlvolume.
2.4 Measurement of radioactivity
Theγactivity of both the initial raw material and the extracts we re measured
during the experiment. The standards of the Health Ministry of Ukraine state
that the specific activity of137Cs in medicinal plants must be less than 600Bq
kg
[13]. We needed to measure a small activity and commercial de vices (like the
LP – 4900B ) have insufficient sensitivity, therefore we used the γdetector from
the Institute for Single Crystals (Kharkov, Ukraine). The s cintillator ( BGO)
was a cylinder with d=40 mmand h=40 mm. The detector was calibrated with
60Co(Eγ=1.173 MeV,Eγ=1.332 MeV) and137Cs (Eγ=0.662 MeV) sources
from a standard calibration set before measurements. For th eEγ= 0.662 MeV
photon the energy resolution of the detector was determined to be 13.4%. The
photopeak efficiency of photon registration for the point sou rce was ε=0.42. The
detector and samples were placed inside a 5 cmthick lead shield to decrease
the background. The background at the 0.662 MeV peak region was about 0.2
[count
sec]. The samples were measured directly inside Marinelli’s ve ssel (100 ml
3volume). The detector efficiency was 31.5 [Bq×sec
count]. The activity of137Cs in the
samples was determined by formula A [Bq
kg] = 31 .5×Nph.p., where Nph.p.[1
sec×kg]
is the number of counts in the photopeak of the measured subst ance.
The data acquisition system was assembled in CAMAC standard . The γspec-
trum parameters were calculated with the MINUIT program fro m the ROOT
package [14].
All samples were measured over a wide range of photon energy ( 0.1 – 2.0 MeV).
Only one peak at 0.662 MeV was detected which meant that137Cs was present
and therefore we only studied the transfer of137Cs. The statistical error of the
measurements did not exceed 5%. The absolute error was 15% an d depended
on the standard calibrated source of137Cs.
3 Results and discussion
The results of the determination of137Cs transfer from the herbs to alcohol-
water and pure water extracts are presented in Table 1. From t he table one can
see that for the plants investigated, the137Cs transfer does not depend on the
type of raw material (within the error limits). The reason fo r certain scattering
of the data for137Cs transfer for 96.5%( vol.) alcohol can be due to saturation
of solvent by cesium.
Studying the dependence of the137Cs transfer on the solvent hydrophilicity
shows that for 96.5%( vol.) alcohol the transfer of137Cs to the extract is minimal,
ranging from 4.6 to 17.6%. For 70%( vol.) alcohol the transfer reaches 45.3 to
66.5%. In our experiments the maximal amount of137Cs was extracted by
40%(vol.) alcohol (62.8 to 83.2%) and pure water at room temperature ( 63.0 to
73.0%). Decoctions extracted radiocesium similar to 70%( vol.) alcohol (49.3 to
60.8%) but less than 40%( vol.) alcohol or pure water at room temperature. The
reason is that in the process of heating some of the cesium che mically interacts
with the raw material and can not be extracted.
As follows from our results the 96.5%( vol.) alcohol extracts less (about 6
times)137Cs than 40%( vol.). From Table 2 one can see that solubility of the
cardiac glycosides strongly depends on solvent type and the solubility tendency
of the cardiac glycosides is inverse to cesium [15]. Methano l extracts the car-
diac glycosides much more (a factor of 570 for Digitoxin and a factor of 18 for
Convallatoxin ) as compared to water. Thus the ratio of137Cs to the cardiac
glycosides extracted strongly depends on solvent type.
4 Conclusion
We have presented new results on the transfer of137Cs from raw material to
medicine. We have found that the extraction of137Cs from Digitalis grandi-
flora Mill. andConvallaria majalis L. containing cardiac glycosides strongly
depends on solvent hydrophilicity and where 96.5%( vol.) alcohol extracts less
137Cs (about 6 times) than 40%( vol.) orpure water . The solubility tendency of
4the cardiac glycosides is inverse to that of cesium and this f act can be of use in
the technological processes for manufacturing ecological ly pure herbal medicine.
Acknowledgements
The authors are grateful to S.F. Burachos for use of the BGO scintillator.
References
[1] Buzun V.O., Vozniuk V.M., Davydov I.M. at al., 1999. The b asis of forest
radioecology, 1999. Kiev (in Ukrainian).
[2] Pushkarev, A.V., Primachenko, V.M., Yu.Ia., Sushik et a l. 1997. Integral
characteristic of storage distribution of technogenic rad iocesium in soil level
(Ukrainian Polessie), ISSN 1025-6415. Reports of National Academy of Sci-
ences of Ukraine 6, 187-192 (in Russian).
[3] Orlov, A.A., Krasnov, V.P., Irklienko, S.P. et al., 1996 . Study of radioactive
contamination of herbs from the forests of Ukrainian Poless ie. Problemy
ecologii lesiv i lisokorystuvannia na Polissi Ukrainy. Nau k. praci Polis’koi
ALNDS, Zhitomir 3, 55-64 (in Ukrainian).
[4] Krasnov V.P., Orlov A.A., Irklienko S.P. et al., 1996.137Cs contamination
of herbs of Ukrainian Polessie. Rast. resursy 3, 36-43 (in Ru ssian).
[5] Orlov, A.A., Krasnov, V.P., Shelest, Z.M., Kurbet, T.V. , 1997.137Cs accu-
mulation by herbs in various forest’s cenosis of Ukrainian P olessie. In: 3th
Congress on radiation studies 2. Puschino, 364-365 (in Russ ian).
[6] Sanarov, E.M., Balandovich, B.A., Kuz’min, E.V. et al., 1998. Ecological
eva-luation of radionuclide contamination of medicinal ra w material at Altai
region and problem of a regulation. Journal Chemistry of pla nt material
2(1), Altai State University, 19- 24 (in Russian).
[7] Grischenko, E.N., Grodzinskii, D.M., Moskalenko, V.N. et al., 1990. Ra-
dionuclide contamination of herbal raw material in various areas of Ukraine
after failure on ChAES. In: Ekologicheskie aspekty v farmat zii, Moskva,
p.56 (in Russian).
[8] Antonova, V.A., Seditzkaia, Z.L., 1989. Influence of pre paration technology
of medicinal products on137Cs transition to the liquid medicinal products.
Gigiena i sanitaria 7, 87-88 (in Russian).
[9] Prokofiev, O.N., Antonova, V.A., Seditzkaia, Z.L. 1992. Assessment of allow-
able levels of the total specific activity of a mixture of radi onuclides in liquid
medicaments and medicinal raw material. Gigiena i sanitari a 5-6, 31-34 (in
Russian).
5[10] Prokofiev, O.N., Antonova, V.A., Seditzkaia, Z.L., 199 3. The approach to
determination of test objective levels of activity of a mixt ure of radionuclides
in medicinal raw material and in liquid medicaments. In: Rad iacionnye as-
pekty Chernobyl’scoi avarii. Gidrometizdat, Obninsk 2 (in Russian).
[11] S.V. Dmitriev, A.A. Fetisov, V.A. Percev et al. 1991. Ab out contamination
of wild medicinal plants by137Cs. Gigiena i sanitaria 12, 51-53 (in Russian).
[12] Grodzinskii D.M., Kolomietz K.D., Kutlahmedov Yu.A. e t al., 1991. Antro-
pogeneous radionuclide anomaly and plants. Lybid’, Kiev, p .160 (in Rus-
sian).
[13] Allowable levels of contents of137Cs and90Sr radionuclides in feed products
and potable water. (DR-97), 1997. Kiev (in Ukrainian).
[14] Brun R., Rademakers F., 1997. ROOT - An Object Oriented D ate Analysis
Framework. Nucl.Inst.Meth. in Phys. Res. A389, 81-86.
[15] Baumgarten G. In Buch: Herz- und Kreislaufwirksame Pha rmaka Halle-
Wittenberg, 1969, S.331.
6Table 1
137Cs transfer from medicinal plant raw material to alcohol and water extracts
Specific Specific137Cs transfer137Cs transfer
137Cs137Cs to tinctures, to water,
activity activity % %
Herbs in soil, in raw extracts at
material, 96 .50700400decoctions room
kBq×m−2kBq×kg−1temperature
Herb of Digitalis No1 925 2.66 11.6 46.6 66.2 56.4 66.2
Herb of Digitalis No2 814 1.48 17.6 45.3 62.8 49.3 68.9
Average 14.6 46.0 64.5 52.9 67.6
Leaves of Convallaria No1 296 1.49 10.7 63.1 67.1 60,0 74.0
Leaves of Convallaria No2 777 50.83 6.3 59.7 63.9 59.1 73.0
Average 8.5 61.4 65.5 59.6 73.5
Flowers of Convallaria No1 740 11.73 4.6 58.9 66.0 60.8 69.1
Flowers of Convallaria No2 296 2.81 10.0 66.5 69.4 67.3 71.5
Flowers of Convallaria No3 407 3.64 14.0 52.7 83.2 59.6 63.0
Average 9.5 59.4 72.9 62.6 67.8
7Table 2
Solubility of some cardiac glycosides
Part of solvent needed to dissolve
Glycoside Herb one part of glycoside
water methanol
Digitoxin Digitalis 40 000 70
Convallatoxin Convallaria 1000 56
8 |
arXiv:physics/0102030v1 [physics.atom-ph] 12 Feb 2001High-precision calculations of van der Waals coefficients fo r
heteronuclear alkali-metal dimers
A. Derevianko∗, J. F. Babb, and A. Dalgarno
Institute for Theoretical Atomic and Molecular Physics
Harvard-Smithsonian Center for Astrophysics, Cambridge, Massachusetts 02138
(February 2, 2008)
Abstract
Van der Waals coefficients for the heteronuclear alkali-meta l dimers of Li,
Na, K, Rb, Cs, and Fr are calculated using relativistic ab initio methods aug-
mented by high-precision experimental data. We argue that t he uncertainties
in the coefficients are unlikely to exceed about 1%.
PACS: 34.20.Cf, 32.10.Dk, 31.15.Ar
Typeset using REVT EX
∗Permanent Address: Department of Physics, University of Ne vada, Reno, Nevada 89557
1Considerable attention has been given to the determination of the coefficients of the
leading term of the van der Waals attractions of two alkali me tal atoms because of their
importance in the simulation, prediction, and interpretat ion of experiments on cold atom
collisions, photoassociation and fluorescence spectrosco py [1–6]. There is strong interest in
heteronuclear molecules formed by pairs of different alkali metal atoms. Experiments have
been carried out on trap loss in mixtures of Na with K [7,8], Rb [9,10], and Cs [11] and on
optical collisions [12] in a Na-Cs mixture and on molecular f ormation [13]. The mixtures
of magnetically-trapped alkali metal atoms Na-Cs and Na-K h ave been proposed [14] as a
means to search for evidence of an electric dipole moment to t est for violation of parity
and time reversal symmetry. We extend here previous studies [15] of the van der Waals
coefficient between pairs of identical ground state alkali me tal atoms to unlike ground state
atoms.
The leading team of the van der Waals interaction is given at a n atom separation R
by [16,17],
VAB(R) =−CAB
6
R6, (1)
where CAB
6is the van der Waals coefficient. We use atomic units throughou t.
The van der Waals coefficient may be expressed as
CAB
6=2
3/summationdisplay
st|/angbracketleftvA|DA|sA/angbracketright|2|/angbracketleftvB|DB|tB/angbracketright|2
(EA
s−EA
v) + (EB
t−EB
v), (2)
where |vA/angbracketrightis the ground state atomic wave function of atom A with energy EA
v, and similarly
for atom B, and |sA/angbracketrightand|tB/angbracketrightrepresent complete sets of intermediate atomic states with ,
respectively, energies EA
sandEB
t. The electric dipole operators are DA=/summationtextNA
i=1rA
i, where
rA
iis the position vector of electron imeasured from nucleus A, NAis the total number of
atomic electrons for atom A, and similarly for atom B.
At this point the two-center molecular-structure problem i s reduced to the determination
ofatomic matrix elements and energies. The dependence on one-center atomic properties
becomes explicit when Eq. (2) is cast into the Casimir-Polde r form
CAB
6=3
π/integraldisplay∞
0αA(iω)αB(iω)dω , (3)
where αA(iω) is the dynamic polarizability of imaginary argument for at om A given by
αA(iω) =2
3/summationdisplay
s/parenleftBig
EA
s−EA
v/parenrightBig
|/angbracketleftvA|DA|sA/angbracketright|2
(EAs−EAv)2+ω2, (4)
andα(ω= 0) is the ground-state static dipole polarizability. In th e limit of infinite frequency
the function αA(iω) satisfies
αA(iω)→NA
ω2, (5)
as a consequence of the nonrelativistic Thomas-Reiche-Kuh n sum rule.
2Modern all-order many-body methods are capable of predicti ng electric-dipole matrix
elements for principal transitions and energies in alkali- metals to within errors approaching
0.1% [18]. Many-body methods augmented by high-precision e xperimental data for principal
transitions, similar to those employed in PNC calculations [19], have led to a high-precision
evaluation of dynamic dipole polarizabilities for alkali- metal atoms [15]. The values of C6
previously calculated for homonuclear dimers [15] are in excellent agreement with analyses
of cold-atom scattering of Na [20], Rb [2], and Cs [6,21]. Her e we employ the same methods
to compute the van der Waals coefficients for heteronuclear al kali-metal dimers.
Precise nonrelativistic variational calculations of C6for Li 2have been carried out [22].
They provide a critical test of our procedures. We separate t he dynamic polarizability
into valence and core contributions, which correspond resp ectively to valence-electron and
core-electron excited intermediate states in the sum, Eq. ( 4). In our calculations for Li we
employ high-precision experimental values for the princip al transition 2 s−2pJ, all-order
many-body data and experimental energies for 3 pJand 4pJintermediate states, and Dirac-
Hartree-Fock values for higher valence-electron excitati ons. The high-precision all-order
calculations were performed using the relativistic linear ized coupled-cluster method trun-
cated at single and double excitations from a reference dete rminant [18,23]. Contributions
of valence-excited states above 4 pJwere obtained by a direct summation over a relativistic
B-spline basis set [24] obtained in the “frozen-core” ( VN−1) Dirac-Hartree-Fock potential.
Core excitations were treated with a highly-accurate relat ivistic configuration-interaction
method applied to the two-electron Li+ion. For the heavier alkali-metals [15] the random-
phase approximation [25] was used to calculate this contrib ution.
The principal transition 2 s−2pJaccounts for 99% of the static polarizability and 96%
of the Li 2dispersion coefficient. In accurate experiments McAlexande ret al. [26] reported
a lifetime of the 2 pstate of 27.102(9) ns (an accuracy of 0.03%) and Martin et al. [27]
reported 27.13(2) ns. In our calculations we employ the more precise value from Ref. [26]; in
the subsequent error analysis we arbitrarily assigned an er ror bar of twice the quoted value
of Ref. [26], so that the two experiments are consistent.
The dynamic core polarizability of Li was obtained in the fra mework of the relativistic
configuration-interaction (CI) method for helium-like sys tems. This CI setup is described
by Johnson and Cheng [28], who used it to calculate precise re lativistic static dipole po-
larizabilities. We extended their method to calculate the dynamic polarizability α(iω) for
two-electron systems. The numerical accuracy was monitore d by comparison with results of
Ref. [28] for the static polarizability of Li+and with the sum rule, Eq. (5), in the limit of
large frequencies. Core-excited states contribute only 0. 5% to C6and 0.1% to α(0) for Li.
Their contribution becomes much larger for heavier alkali m etals.
We calculated static and dynamic polarizabilities and used quadrature, Eq. (3), to obtain
the dispersion coefficient. The results are C6= 1390 and α(0) = 164 .0. There are two major
sources of uncertainties in the final value of C6— experimental error in the dipole matrix
elements of the principal transition, and theoretical erro r related to higher valence-electron
excitations. The former results in a uncertainty of 0.12%, a nd the latter much less. The
result C6= 1390(2) is in good agreement with the nonrelativistic variational result of Yan
et al. [22],C6= 1393 .39. The slight discrepancy between the two values may arise b ecause
in our formulation, the correlations of core-excited state s with the valence electron were
disregarded as were intermediate states containing simult aneous excitation of the valence
3electron with one or both core electrons. On the other hand, R ef. [22] did not account for
relativistic corrections. Relativistic contractions lea d to a smaller value of C6and to better
agreement between the present result and that of Ref. [22]. S imilar error analysis for the
static polarizability of Li leads to α(0) = 164 .0(1), which agrees with the numerically precise
nonrelativistic result of 164.111 [22]. An extensive compa rison with other published data for
the values of α(0) and C6for lithium is given in Ref. [22]. For the heavier alkali meta l atoms
we followed the procedures of Ref. [15] to calculate α(iω). The results for Cs are illustrated
in Fig. 1. They indicate that while most of the contribution t oC6comes from the resonant
transition at ω∼0.05 a.u. the core excitations are significant.
Results and Conclusions —We evaluated the dispersion coefficients for various hetero nu-
clear alkali-metal dimers with the quadrature Eq. (3). The c alculated values are presented in
Table I. Most of the contributions to CAB
6come from the principal transitions of each atom.
An analysis of the dispersion coefficient of unlike atoms yiel ds the approximate formula
CAB
6≈1
2/radicalBig
CAA
6CBB
6∆EA+ ∆EB√∆EA∆EB, (6)
where the energy separations of the principal transitions a re designated as ∆ EAand ∆ EB.
Eq. (6) combined with the high-accuracy values of C6for homonuclear dimers [15] gives
accurate approximations to our results based on Eq. (3). For example, Eq. (6) overestimates
our accurate value from Table I for Li-Na by 0.4% and for Cs-Li by 2%. We may use Eq. (6)
to estimate the uncertainties δCAB
6in the heteronuclear cases from the uncertainties δCAA
6
andδCBB
6in the homonuclear dispersion coefficients,
δCAB
6
CAB
6≈1
2
/parenleftBiggδCAA
6
CAA
6/parenrightBigg2
+/parenleftBiggδCBB
6
CBB
6/parenrightBigg2
1/2
.
The accuracy of C6for homonuclear dimers was assessed in Ref. [15] and a detail ed discus-
sion for the Rb dimer is given in Ref. [29]. Analyzing the erro r in this manner using the
quoted coefficients and their uncertainties from Ref. [15] we find that most of the disper-
sion coefficients reported here have an estimated uncertaint y below 1%. The corresponding
values are given in parentheses in Table I.
In Fig. 2 we present for the dispersion coefficients of the dime rs involving Cs a com-
parison between our calculated values and the most recent de terminations [6,30]. We give
the percentage deviation from our calculations. It is appar ent that the other calculations
that employed one-electron model potentials and according ly omitted contributions from
core-excited states yield values systematically smaller t han ours.
The discrepancies are most significant for Cs 2where the number of electrons is greatest.
Fig. 2 also compares the values for the Cs 2dimer with values deduced from ultracold-collision
data [6,30]. The agreement of our prediction 6851(74) [15] w ith their values for C6in Cs 2is
close. Core-excited states contribute 15% [31,15] to the va lue of the C6coefficient for the Cs
dimer and are needed to fulfill the oscillator strength sum ru le, Eq. (5). In the present ap-
proach the contributions of core-excited states to dynamic polarizabilities are obtained using
the random-phase approximation, which nonrelativistical ly satisfies the oscillator strength
sum rule exactly [25]. In the inset of Fig. 1, it is illustrate d that our calculated α(iω)
approaches N/ω2asωbecomes asymptotically large, where N= 55 for Cs. While the
4deviation between the present calculations and the model po tential calculations are smaller
for dimers involving lighter atoms, an accurate accounting of core-excited states is essential
to achieve high accuracy in dispersion coefficient calculati ons for heavy atoms [31–33].
Few experimental data are available for comparison in the he teronuclear case, except for
NaK. The results from investigations of NaK molecular poten tials based on spectral analy-
sis [34] are compared to our value in Table II. Our value is sma ller than the experimental
values. Earlier theoretical calculations of dispersion co efficients for NaK have been tabu-
lated and evaluated by Marinescu and Sadeghpour [35] and by Z emke and Stwalley [36].
Those values are generally lower than our value of 2447(6) ex cept for that of Maeder and
Kultzelnigg [32] who give 2443.
The present study extends the application of modern relativ istic atomic structure meth-
ods to calculations of ground state van der Waals coefficients of Li 2and of the heteronuclear
alkali-metal atoms. We argue that the uncertainty of the coe fficients is unlikely to exceed
1%. Additional experimental data from future cold-collisi on experiments or spectroscopy
would provide further tests of the present calculations.
This work was supported by the Chemical Sciences, Geoscienc es and Biosciences Division
of the Office of Basic Energy Sciences, Office of Science, U.S. De partment of Energy and by
the National Science Foundation under grant PHY97-24713. T he Institute for Theoretical
Atomic and Molecular Physics is supported by a grant from the NSF to Harvard University
and the Smithsonian Institution.
5FIGURES
10
0
10
1
10
2
10
3
10
4
ω
0
10
20
30
40
50
60
ω
2
α
(i
ω
)
0
0.5
1
ω
0
50
100
150
200
250
300
350
400
450
α
(i
ω
)
FIG. 1. The dependence of the dynamic dipole polarizability α(iω) with frequency ωfor Cs.
The inset illustrates the behavior of the quantity ω2α(iω) at asymptotically large ω, where the
dashed line represents the contribution of the core-excite d states to the total ω2α(iω) (solid line)
and the arrow marks the non-relativistic limit N= 55 following from the sum rule, Eq. (5). All
quantities are in atomic units.
6-8
-6
-4
-2
0
2
Deviation, %
LiCs
NaCs
KCs
RbCs
Cs
2
FIG. 2. Percentage deviation of results of recent calculati ons [35,37] from our values for van
der Waals coefficients C6for Cs-Li, Cs-Na, Cs-K,Cs-Rb, and Cs-Cs. The values with err or bars
placed along the horizontal line at 0 correspond to our resul ts with the estimated uncertainties.
Circles represent the results of Ref. [35] and triangles the results of Ref. [37]. For Cs-Cs, to the
right of the vertical dotted line, we show the difference betw een our present prediction, our earlier
prediction [15] and the values deduced from cold-collision data in Ref. [6] (square) and Ref. [30]
(diamond).
7TABLES
TABLE I. Dispersion coefficients C6and their estimated uncertainties (parentheses) for al-
kali-metal atom pairs in atomic units. Coefficients for Na 2, K2, Rb 2, Cs2, and Fr 2are from
Ref. [15].
Li Na K Rb Cs Fr
Li 1389(2) 1467(2) 2322(5) 2545(7) 3065(16) 2682(23)
Na 1556(4) 2447(6) 2683(7) 3227(18) 2842(24)
K 3897(15) 4274(13) 5159(30) 4500(39)
Rb 4691(23) 5663(34) 4946(44)
Cs 6851(74) 5968(60)
Fr 5256(89)
8TABLE II. Comparision of present theoretical and experimen tal values for the dispersion
coefficient for NaK.
Reference C6
This work 2447(6)
Russier-Antoine et al., [34] 2519(10)a
Ishikawa et al., [38] 2646(31)a
Rosset al., [39] 2669.4(20)a
aExperiment.
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arXiv:physics/0102031v1 [physics.ed-ph] 12 Feb 2001Two particle states, lepton mixing and oscillations
M. Kachelrieß1, E. Resconi2and S. Sch¨ onert3,4
1TH Division, CERN, CH–1211 Geneva 23
2Dipartimento di Fisica, Universit´ a di Genova, Via Dodecan eso, 33, I–16146 Genova
3∗Max-Planck-Institut f¨ ur Kernphysik, Saupfercheckweg 1, D–69117 Heidelberg
4Research Center for Cosmic Neutrinos, Institute for Cosmic Ray Research, University of Tokyo
Discussions of lepton mixing and oscillations consider gen erally only flavor oscillations of neu-
trinos and neglect the accompanying charged leptons. In cas es of experimental interest like pion or
nuclear beta decay an oscillation pattern is expected indee d only for neutrinos if only one of the
two produced particles is observed. We argue that flavor osci llations of neutrinos without detecting
the accompanying lepton is a peculiarity of the two-particl e states |lν/angbracketrightproduced in pion or nuclear
beta decay. Generally, an oscillation pattern is only found if both particles are detected. We discuss
in a pedagogical way how this distinction of the neutrinos ar ises, although on the level of the La-
grangian lepton mixing does not single them out against char ged leptons. As examples, we discuss
the difference between the state |lν/angbracketrightproduced by the decay of real Wboson and a Woriginating
from pion decay.
14.60Pq
I. INTRODUCTION
The Standard Model (SM) of particle physics has
proven to be a firm basis on which all our knowledge of
this field rests since its construction 30 years ago [1]. Pre-
cision tests performed in the last decade demonstrated
in particular that it is also correct at the quantum level.
Novel phenomena such as neutrino masses or supersym-
metric particles, which cannot be accommodated within
the SM, should not be thought to contradict it, but rather
to guide us as to new physics beyond it. Nowadays, this
hunt for new phenomena is the main topic in particle
physics. In contrast to the search for supersymmetry, for
which there is no positive signal so far1, there is mounting
experimental evidence for neutrino oscillations. On the
one hand, there are five solar neutrino experiments using
different techniques that see a deficit in the solar neu-
trino flux [4]. Although this deficit could have its origin
in principle also in non-standard solar- or nuclear physics ,
it can be shown that these explanations are experimen-
tally excluded [5]. On the other hand, the case for an os-
cillation solution to the atmospheric neutrino deficit has
become recently even stronger in the general perception.
This is largely because both the zenith-angle distribu-
tion and the dependence of the ratio νe/νµas function of
the ratio (oscillation length)/(neutrino energy) found bythe Superkamiokande collaboration support the neutrino
oscillation hypothesis [6].
Most aspects of neutrino oscillations have been dis-
cussed extensively in the literature. The usual deriva-
tion, presented e.g. in Ref. [7], of the probability Pthat
a relativistic neutrino with momentum pand flavor αhas
the flavor βafter the time t,
Pνα→νβ=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay
lU(ν)
αlU(ν)∗
βlexp (−iElt)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
, (1)
uses only basic facts of quantum mechanics. Here, El=/radicalbig
m2
l+p2is the energy of the neutrino mass eigenstate
land the matrix Uis commonly chosen to represent the
unitary transformation matrix between weak and mass
eigenstates of the neutrino. In spite (or, perhaps, be-
cause) of the simplicity of its derivation, Eq. (1) raises
several conceptual questions. The most prominent ones
are if for the neutrino mass eigenstates a definite energy
or momentum should be used, under which conditions
the use of wave packets with smeared energy and/or mo-
mentum is necessary, the problem of coherence, and the
connection between the quantum mechanical treatment
and quantum field theory [8].
In this article we want to discuss a more basic ques-
tion, namely why mixing in the lepton sector reveals it-
self experimentally in flavor oscillations of neutrinos but
∗Permanent address
1We should however mention that the annual modulation
signal seen by the DAMA experiment can be consistently
explained as the scattering of supersymmetric dark matter
particles on their target nucleons [2]. Also the experimen-
tal evidence for a light Higgs found at LEPII points towards
low-energy supersymmetry [3].
1not of charged leptons. This question is motivated by
the following simple fact: At the level of the Lagrangian
describing the charged-current interactions of leptons,
LCC=−g√
2/summationdisplay
i,j¯lL,iγµVijνL,jW−
µ+ h.c. , (2)
the mixing V=U(ch)†U(ν)between charged leptons and
neutrinos has two different sources: It could be ascribed
either completely to mixing in the neutrino ( V=U(ν))
or in the charged lepton sector ( V=U(ch)†), or most
probably to some superposition of both. Physical results
do not depend on the particular decomposition of Vin
the Standard Model with lepton mixing2. Thus, knowing
only the charged-current Lagrangian, one would not ex-
pect any fundamental difference between neutrinos and
charged leptons in oscillation experiments. Main pur-
pose of this article is to clarify the exact reason for this
difference.
Another formulation of this question is to ask for which
conditions it is allowed to neglect the charged lepton pro-
duced together with the neutrino in a two-particle state,
e.g., in the decay of a pion or a real Wboson. We will
show that there is a crucial difference between these two
cases. While in the first one the charged lepton plays
only a “spectator rˆ ole” and can be neglected to a good
approximation, in the second case the two-particle state
has to be considered.
Furthermore, we shall address the question whether
charged leptons can show an oscillation pattern. Usu-
ally it is argued that their oscillation frequencies are too
high to be observable and moreover, that the coherence of
wave packets corresponding to different mass eigenstates
is lost under experimental conditions.
In general, an oscillation pattern is observed if 1) the
distance between the source and the detector is smaller
than the coherence length lcohand 2) the size of the
source and of the detector are smaller than the oscil-
lation length losc. In analogy to neutrino oscillations [9],
the coherence length for charged leptons can be written
as
lcoh=σ2E2
∆m2, (3)
with the width of the wave packet σ, the energy Eand
mass difference ∆ m2=m2
i−m2
jof the charged lep-
tons. As an example we estimate lcohfor pion decay in
flight with an energy of Eπ≈40 GeV. Using σ≈γτ
and the pion half life τ= 2.6×10−8s one obtainslcoh=O(108m). The oscillation length losc= 4πE/∆m2
would amount to O(10−11m). Consequently, the coher-
ence condition could be met experimentally while the os-
cillation pattern would be smeared out. If this were the
complete argument, one still could hope to derive infor-
mation about Vmeasuring only charged leptons. We will
show however, that it is necessary to measure the accom-
panying neutrino simultaneously in order to obtain any
information on V.
II. MIXING IN THE LEPTON SECTOR
Let us first recall how fermion mass matrices are di-
agonalized. Generally, the mass terms in the Lagrangian
are given by
Lmass=−/summationdisplay
α,β¯νL,αM(ν)
αβνR,β−/summationdisplay
α,β¯lL,αM(ch)
αβlR,β+ h.c.,
(4)
and the mass matrices Mαβare not hermitian or even di-
agonal in the basis of the weak eigenstates. (We denote
the weak eigenstates να={νe, νµ, ντ}andlα={e, µ, τ}
by greek indices α, β . . . , and the mass eigenstates by latin
indices i, j, . . . Furthermore, we have assumed that the
neutrinos have only Dirac mass terms to simplify the for-
mulas). Since the mass matrices are not hermitian, they
cannot be diagonalized by a simple unitary transforma-
tion. However, arbitrary mass matrices can be diagonal-
ized by a biunitary transformation [10],
M(ν)
diag=U(ν)†M(ν)T(ν), (5)
where U(ν)U(ν)†=T(ν)T(ν)†=1. Then, the connection
between weak and mass eigenstates is given by
νR,α=/summationdisplay
iT(ν)
αiνR,i, ν L,α=/summationdisplay
iU(ν)
αiνL,i,(6)
and similar equations hold for the charged leptons.
Inserting the transformations of Eq. (6) into the
charged current Lagrangian of the SM,
LCC=−g√
2/summationdisplay
α¯lL,αγµνL,αW−
µ+ h.c., (7)
results in
LCC=−g√
2/summationdisplay
i,j¯lL,iγµVijνL,jW−
µ+ h.c., (8)
2We call SM with lepton mixing any model that allows non-
zero neutrino masses but reproduces otherwise in the low-
energy limit the SM. This means in particular that we do not
consider sterile neutrinos.
2where we introduced the analogue of the CKM matrix in
the lepton sector [11], V=U(ch)†U(ν). Since the charged
current interaction involves only left-chiral fields of bot h
charged leptons and neutrinos, the product of the two
mixing matrices of the right-handed leptons, T(ch)†T(ν),
is unobservable.
In the case of massless neutrinos we can choose the
neutrino mass eigenstates arbitrarily. In particular, we
can set U(ν)=U(ch)for any given U(ch), hereby rotat-
ing away the mixing. This shows that neutrino masses
are a necessary condition for non-trivial consequences of
mixing in the lepton sector.
Finally, we recall that there are no flavor-changing neu-
tral currents within the standard model: The neutral cur-
rent Lagrangian is diagonal both in the weak eigenbasis,
LNC=−g
2 cosθW/summationdisplay
α[¯νL,αγµνL,α+¯lα(gV+gAγ5)γµlα]Zµ
(9)
and in the mass basis
LNC=−g
2 cosθW/summationdisplay
i[¯νL,iγµνL,i+¯li(gV+gAγ5)γµli]Zµ
(10)
due to the unitarity conditions, U(ν)†U(ν)=1and
U(ch)†U(ch)=1.
III. PION DECAY AND LEPTON MIXING
Many neutrino oscillations experiments use as source
for the initial lepton-neutrino state charged pions. In the
SM without lepton mixing, a tree-level calculation gives
for the ratio Rofπ→eνeandπ→µνµdecay rates
R=Γ(π→eνe)
Γ(π→µνµ)=m2
e
m2µ(m2
π−m2
e)2
(m2π−m2µ)2≈1.28×10−4.
(11)
Since angular momentum conservation in the pion rest
frame requires a helicity flip of the lepton, the S-matrix
elements of these decays are proportional to the lepton
masses mαand, therefore, the branching into electrons is
suppressed. Hence, the two-particle state |l+ν/an}bracketri}htπcreated
by a decaying positively charged pion is given by
|l+ν/an}bracketri}htπ=1√
N/summationdisplay
α=e,µmα(1−m2
α/m2
π)|l+
ανα/an}bracketri}ht,(12)
where Nis a normalization constant. We have included
the phase space factor 1 −m2
α/m2
πinto Eq. (12) because
we assume that the state |l+ν/an}bracketri}htπlives for a macroscopic
time between its creation and detection. Therefore, boththe lepton and the neutrino are approximately on their
mass-shell.
Let us now examine what are the necessary changes if
we want to account for lepton mixing. Inserting Eq. (6)
into Eq. (12) we obtain
|l+ν(t)/an}bracketri}htπ=1√
N2/summationdisplay
i=13/summationdisplay
j=1aiVij|l+
iνj/an}bracketri}hte−i(Ei+Ej)t,(13)
where ai=mi(1−m2
i/m2
π).
We are ordering the mass eigenstates according to the
value of mi, i.e.ml1< m l2< m l3. Thus the state l3can-
not be populated in pion decay and, therefore, is omitted
in the summation. Furthermore, we have assumed that
all three neutrinos masses are extremely small compared
to the electron mass as it is suggested by the currently
favored interpretation of neutrino oscillation experimen ts
and cosmology. Therefore, we could omit safely new
terms in the S-matrix element proportional to mνi, that
in principle change the branching ratio Eq. (11) from its
SM value [12]. Note that the relative phases Vijof the
different components |l+
iνj/an}bracketri}htare fixed by the Lagrangian,
while the aiare real numbers. The time evolution of
|l+ν(t)/an}bracketri}htπis trivial, because we have expressed |l+ν(t)/an}bracketri}htπ
as a sum over mass eigenstates.
In Eq. (13), we have not displayed explicitly the finite
lifetimes τ= 1/Γ of the states l2andν2,3, because this
point is not essential for our discussion. However, the
finite lifetimes can be restored treating the energy as a
complex number, E= (m2+p2)−iΓ/2 and noting that
the decay products of |l+ν/an}bracketri}htπdo not interfere with it.
Apart from the large difference between the lifetime
ofl2and of ν2,3, there is another, more important, dis-
tinction between neutrinos and charged leptons. If one
decomposes |l+ν/an}bracketri}htπexplicitly into its basis states, then
|l+ν/an}bracketri}htπ=1√
N3/summationdisplay
α=1/parenleftBigg2/summationdisplay
i=1aiU(ch)∗
αi|l+
i/an}bracketri}ht/parenrightBigg
⊗
3/summationdisplay
j=1U(ν)
αj|νj/an}bracketri}ht
.
(14)
Defining a new basis appropriate for |l+ν/an}bracketri}htπby
|l+ν/an}bracketri}htπ=3/summationdisplay
α=1|l+
α/an}bracketri}htπ⊗ |να/an}bracketri}htπ (15)
and comparing with Eq. (14), it follows that the neu-
trino state |να/an}bracketri}htπproduced in pion decay is just a
usual weak eigenstate, |να/an}bracketri}htπ=/summationtext3
j=1U(ν)
αj|νj/an}bracketri}ht=|να/an}bracketri}ht.
By contrast, the charged lepton state is |l+
α/an}bracketri}htπ=/summationtext2
i=1ai/√
N U(ch)∗
αi|l+
i/an}bracketri}ht /ne}ationslash=|l+
α/an}bracketri}ht. We will see below that it
is the presence of the prefactors a1/ne}ationslash=a2which allows the
observation of neutrino oscillations in pion decay with-
out detecting the charged lepton. Note however also that
|l+
α/an}bracketri}htπ/ne}ationslash=|l+
α/an}bracketri}hteven for a1=a2, because the component l3
is missing.
3Using a2≫a1, we can approximate |l+ν(t)/an}bracketri}htπas
|l+ν(t)/an}bracketri}htπ≈3/summationdisplay
j=1V2j|l+
2νj/an}bracketri}hte−i(E2+Ej)t(16)
witha2/√
N≈1. Clearly, one obtains in this approx-
imation only neutrino oscillations, because the charged
lepton is in a pure mass eigenstate. Choosing further-
moreV=U(ν), we obtain the state normally considered
as initial state in pion decay,
|l+ν/an}bracketri}htπ≈3/summationdisplay
α=13/summationdisplay
j=1δ2αU(ν)
αj|l+
2νj/an}bracketri}ht=|l+
2νµ/an}bracketri}ht. (17)
The approximation a2≫a1which is widely used in text-
books is numerically well justified. However, its use ob-
scures the fact that even for the choice V=U(ν), i.e.
identifying mass and flavor eigenstates of the charged
leptons, the charged lepton is nevertheless produced in
a mixed state.
Let us now discuss different measurements of the ex-
act two-particle state, Eq. (13). Since we are only inter-
ested in flavor oscillations, we do not consider possible
momentum measurements of the two particles. Then,
a measurement of the state |l+ν(t)/an}bracketri}htπis complete if at
timetthe quantum numbers iorαof both the neutrino
and the charged lepton are determined. In the case that
only one quantum number is observed, the probability
P(l) of this measurement is obtained by summing over
the quantum number of the unobserved particle, symbol-
ically P(l) =/summationtext
l′P(l, l′).
To begin with, we recall the case normally treated in
the literature, namely that the neutrino flavor is detected
while the lepton is not observed. In a first try, we asso-
ciate the probability P(lk, να) =|/an}bracketle{tlkνα|l+ν(t)/an}bracketri}htπ|2to the
measurement of the lepton mass eigenstate kand the
neutrino flavor eigenstate αat time t. This would result
in
/an}bracketle{tlkνα|l+ν(t)/an}bracketri}htπ=ak√
N3/summationdisplay
l=1VklU(ν)∗
αle−i(Ek+El)t,(18)
i.e. in an amplitude which does not only depend on V
but also on the neutrino mixing matrix U(ν). However,
in practice one cannot observe the flavor of a neutrino
directly. Instead, the flavor of the neutrino is determined
looking at the mass eigenstates of the charged lepton
l′produced in a secondary charged current reaction, cf.
Fig. 1.
Therefore, we should calculate
/an}bracketle{tlkl′
m|ˆHCC(t)|l+ν(t)/an}bracketri}htπ=ak√
N3/summationdisplay
l=1VklV∗
mle−i(Ek+El)t,
(19)where the action of ˆHCCdestroys at time ta neutrino νβ
and creates a superposition of mass eigenstates of charged
leptons l′
m=U(ch)
mβlβ. Here, ˆHCCdenotes the second
quantized Hamiltonian of the usual charged-current in-
teraction. The corresponding probability to measure the
primary lepton lkfrom the pion decay and the secondary
lepton l′
mproduced by the neutrino is
P(lk, l′
m) =a2
k
N/braceleftBig3/summationdisplay
l=1|Vkl|2|Vml|2+ 23/summationdisplay
n>l|VklV∗
mlV∗
knVmn|
cos[(El−En)t+ξklmn]/bracerightBig
, (20)
where ξklmn= arg( VklV∗
mlV∗
knVmn). If both charged lep-
tons are observed, the probability (20) shows clearly an
oscillatory behavior.
In the case that only one of the two leptons is observed,
the result is completely different depending on if the pri-
marylkor the secondary l′
m(as indicator for the neutrino
flavor) is observed. In the first case, summing over m,
we obtain
P(lk) =/summationdisplay
mP(lk, l′
m) =a2
k
N(21)
using the unitarity of V, i.e./summationtext
kVikV∗
jk=δij. In the
second case, we cannot make use of these unitarity rela-
tions because the prefactors akdepend on the summation
index k. However, in the limit a2≫a1, the result sim-
plifies and we obtain the well-known neutrino oscillation
formula
P(l′
m) =/summationdisplay
kP(lk, l′
m)≈/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay
lV2lV∗
mlexp (−iElt)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
.(22)
Let us now comment shortly on the reason for this
asymmetry. A flavor oscillation experiment measures the
correlation between the flavor or mass quantum numbers
of two particles. In the case of pion decay, we can use our
knowledge about the initial state created in the decay as
replacement for an actual measurement of the primary
lepton. However, the kinematic of the decay does not
give us any useful information about the neutrino state:
all three flavor eigenstates are with equal probability pro-
duced. Therefore, the neutrino has to be measured via
its secondary lepton to obtain an oscillation pattern.
From this discussion, it should became clear that
the difference between charged leptons and neutrinos in
|l+ν/an}bracketri}htπis rather specific to the process considered: The
V–A structure of the weak current together with angular
momentum conservation forbid the decay of the spinless
pion into two massless spin 1/2 particles. Therefore, the
pion decay rate is proportional to the fermion masses m2.
Moreover, it is used that the neutrino masses can be ne-
glected compared to the masses of the charged leptons.
Therefore, the fact that we have no a-priori information
4about the neutrino but about the charged lepton is spe-
cific to pion decay.
Next, we want to discuss if it is possible to measure not
the flavor of a neutrino but its mass without measuring
the lepton. A possible way to do this is to use Cherenkov
or transition radiation of the neutrino [13]. In vacuum,
neutrinos can interact with real photons whose squared
four-momentum q2is zero only via their electromagnetic
dipole or transition moments. In a medium, the photon
acquires a more complicated dispersion relation ( q2/ne}ationslash= 0)
and, therefore, neutrinos can emit real photons without
decaying, νi→νi+γ. At least in principle, it is possible
to reconstruct the mass of the radiating neutrino from
the spectra of emitted photons.
The probability to find the neutrino in the mass eigen-
statelwithout measuring the charged lepton is
P(νl) =2/summationdisplay
k=1P(lk, νl) =1
N2/summationdisplay
k=1a2
k|Vkl|2≈a2
2
N|V2l|2.(23)
In this case, we have the interesting result that P(νl)
depends on the mixing matrix Vbut does not show an
oscillatory behavior. Compared to a flavor measurement
which introduces an additional summation/summationtext
jVjlin the
probability amplitude, the phase of the probability am-
plitude of a mass measurement is constant and, conse-
quently, no oscillation pattern can arise. On the other
hand, P(νl) depends on Vbecause of the factors ak.
Consequently, it is possible to extract information on the
mixing matrix Vmeasuring the mass eigenstate of the
neutrino.
Finally, we comment briefly on charged lepton–
neutrino states created in nuclear beta decay. Due to
the low nuclear energies involved, only the state l1is
produced and, therefore, the charged lepton is not only
approximately but exactly in a pure state.
IV.WDECAY AND LEPTON MIXING
We discuss now the evolution of the two-particle state
|lν/an}bracketri}htcreated by a decaying real Wboson. The Wbo-
son is a spin-1 particle and, therefore, can decay into
two massless fermions. Neglecting small corrections of
O(m2/m2
W), the state produced is equally populated for
all three generations. Thus, while the state
|l+ν/an}bracketri}htW=1√
33/summationdisplay
α=1|l+
ανα/an}bracketri}ht (24)
is produced in the SM without lepton mixing by a decay-
ingW+, the state
|l+ν/an}bracketri}htW=1√
33/summationdisplay
i,j=1Vij|l+
iνj/an}bracketri}ht (25)is created with mixing.
We can repeat now the discussion of different measure-
ments similar to the case of |l+ν/an}bracketri}htπ. The only change nec-
essary is the replacement of ak/√
Nby 1/√
3. Hence, the
probability to find the primary lepton in a mass eigen-
statekand the secondary lepton in l′
mbecomes
P(lk, l′
m) =1
3/braceleftBig3/summationdisplay
l=1|Vkl|2|Vml|2+ 23/summationdisplay
n>l|VklV∗
mlV∗
knVmn|
cos[(El−En)t+ξklmn]/bracerightBig
. (26)
In contrast to Eq. (20), the probability is now symmetric
inlkandl′
m. In particular, the oscillation pattern van-
ishes now in both cases as long as only one particle is ob-
served. Only when both the primary and the secondary
lepton are observed, an oscillation pattern according to
Eq. (26) is observed.
We note that the same observation was made in
Ref. [14]. There, the neutrino state produced in the
decay of a real Zwas examined. The authors of [14]
showed that also in this case neutrino flavor oscillations
can be observed, although the neutrinos are produced by
neutral-current interaction. Moreover, they showed that
it is necessary to measure both neutrinos in order to ob-
serve a oscillation pattern. Thus, their results are in line
with our findings presented above.
V. CONCLUSION
Flavor oscillations are observed by the detection of cor-
relations between two states. In an ideal experiment,
the composition of both states is measured. In experi-
ments which use nuclear beta decay to produce the ini-
tial charged lepton–neutrino state the energy available
is limited to nuclear energies. Only the state |l1νj/an}bracketri}htcan
be populated thus making a measurement of the charged
lepton obsolete.
Experiments in which pion decay create the initial
lepton–neutrino state, one exploits the known branching
ratios into the different states |liνj/an}bracketri}htas a substitute for
the measurement of the charged lepton. These branch-
ing ratios differ only for different charged lepton mass
eigenstates, but are the same for different neutrino states.
Therefore, the knowledge of the branching ratios “re-
places” only a measurement of the state of the charged
lepton, and the measurement of the neutrino state is nec-
essary to obtain information about lepton mixing. In
contrast, the lepton–neutrino states produced in the de-
cay of real WBosons are symmetrical in their branching
ratios. If experiments were carried out with such initial
states, both charged lepton and neutrino would be re-
quired to be measured in order to observe flavor oscilla-
tions. In summary, the specific nature of the initial state
5used in oscillation experiments explains the distinguishe d
rˆ ole of neutrinos compared to charged leptons.
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6π+
Source DetectorW+Vkl νll+
k
W+
n pl′
m
V∗
ml
FIG. 1. Production of a superposition of neutrino mass eigen states νlin pion decay and subsequent detection of the neutrino
flavour via the secondary lepton l′
m.
7 |
arXiv:physics/0102032v1 [physics.optics] 12 Feb 2001Two-photon absorption in potassium niobate
A. Ludlow, H. M. Nelson, and S. D. Bergeson
Department of Physics and Astronomy, Brigham Young Univers ity, Provo, UT 84602
February 17, 2014
Abstract
We report measurements of thermal self-locking of a Fabry-P erot cavity containing a potassium
niobate (KNbO 3) crystal. From the shape and width of the cavity transmissio n signals, we determine
absorption coefficients for our crystal at 846 nm [ α= 0.0034(22) m−1andβ= 3.2(5)×10−11m/W]
and 423 nm [13(2) m−1]. Our method is particularly well suited to bulk absorption measurements
where absorption is small compared to scattering. We also re port new measurements of the temperature
dependence of the index of refraction at 846 nm, and compare t o values in the literature.
1 Introduction
For cw laser second harmonic generation in a build-up cavity , thermal issues in the nonlinear crystal inevitably
come into play. This is particularly true at high laser power s. In typical applications, a Fabry-Perot build-
up cavity increases the fundamental laser intensity by one o r two orders of magnitude compared to the
single-pass intensity. The nonlinear crystal absorbs a sma ll fraction of the fundamental power, heats up,
and changes the cavity properties. Thermal lensing changes the cavity’s spatial mode. Nonlinear absorption
changes the cavity’s finesse. Thermal expansion and changes in the crystal’s index of refraction change the
cavity’s optical path length.
These effects are well known, and have been observed by severa l groups working in this field [1].
Thermally-induced changes in the build-up cavity sometime s limit the second-harmonic generation efficiency.
In many cases, they are nuisances that can be worked around. H owever, these changes can be used as a
sensitive diagnostic tool for accurately characterizing n onlinear crystal properties.
For gases and atomic vapors, measuring optical properties o f materials inside Fabry-Perot cavities has a
long history [2], particularly in relation to optical bista bility, saturable absorption, and index of refraction
measurements. Thermal and collisional properties of gases have also been measured in this way [3]. However,
no systematic analysis for solids has been carried out for Fa bry-Perot measurements.
In some ways, Fabry-Perot measurements are similar to laser calorimetry [4]. In those calorimetry
measurements, a laser passes through a thermally isolated a bsorbing crystal and heats it slightly. From the
temperature rise, laser power, thermal mass, and thermal co nductivity it is possible to determine an optical
absorption coefficient for the crystal. Comparing calorimet ry to our Fabry-Perot measurements, both are
photothermal methods. Both can have high sensitivities to s mall absorptions. In contrast to calorimetry,
our “thermometer” is the power-induced shift in the cavity r esonance frequency. Fabry-Perot measurements
can typically reach much higher intensities, making nonlin ear effects more apparent.
This paper presents a detailed study of the thermal response of KNbO 3inside a Fabry-Perot build-up
cavity. We use this response to determine a new value for the o ptical absorption coefficients in KNbO 3at
846 nm and 423 nm, and to measure the effects of blue-light-ind uced infrared absorption (BLIIRA). We also
measure the temperature-dependence of the index of refract ion for light polarized along the bandccrystal
axes. These measurements are compared to values from the lit erature.
2 Experimental Setup
Our experimental setup is typical for cw second-harmonic ge neration in an external build-up cavity (see
Figure 1). This laser system is designed for a calcium laser c ooling experiment at 423 nm. It produces
1typically 75 mW at 423 nm, measured outside the cavity.
The laser system consists of a single frequency extended cav ity diode laser at 846 nm amplified in a
single-pass through a tapered laser diode [5]. The extended cavity diode laser is frequency-stabilized relative
to a separate optical cavity using the Pound-Drever method [ 6] to provide both short and long-term stability.
After optical isolation, polarization correction, and mod e matching, we inject 250 mW into the Gaussian
mode of a four-mirror folded (bowtie) Fabry-Perot cavity wi th roughly 80% efficiency. Higher-order modes
in the cavity are less than 5% of the Gaussian mode. The input c oupler reflectivity is 96.6%, and the
other three mirror reflectivities are >99.5% in the infrared. The radius of curvature for the two curved
mirrors is 100 mm, and they are separated by a distance of 117 m m. The angle of incidence on the curved
mirrors is 6 .5o, minimizing the ellipticity of the beam waist inside the cry stal (41.1 and 40.7 µm in the
saggital and tangential planes, as determined from the cavi ty geometry). The round-trip cavity length is
618 mm. The KNbO 3crystal [7] is a-cut and antireflection-coated ( R <0.25%) at 846 nm. The non-critical
phase-matching temperature is −11.5oC, for light polarized along the b-axis. The crystal is mounted on a
Peltier-cooled copper block. An AD590 temperature sensor a nd a thermister embedded in the copper mount
monitor the crystal temperature. The entire bowtie cavity i s enclosed in a sealed aluminum box, filled with
dry oxygen at atmospheric pressure.
3 Transmission Lineshape Without Blue Light
When laser light incident on the bowtie cavity is polarized a long the c-axis, it is impossible to meet the
phase-matching condition for non-critical second harmoni c generation at λ= 846 nm, and no blue light is
generated. Previous measurements in KNbO 3have shown that even small amounts of blue light dramaticall y
change the infrared absorption properties [8, 9, 10]. We add ress this blue-light-induced-infrared-absorption
(BLIIRA) later in the paper. However, in this section we disc uss infrared absorption in the absence of blue
light,i.e.: when the incident laser beam is polarized along the c-axis.
One of the flat mirrors in the bowtie cavity is mounted on a piez oelectric crystal, and we use this crystal
to change the optical length of the cavity. A small amount of i nfrared light is transmitted through the cavity
mirrors, and we measure this light while sweeping the cavity length. When the piezo-mounted mirror passes
through the position for cavity resonance ( i.e.: when the roundtrip pathlength in the cavity is an integer
multiple of the laser wavelength), the power circulating in the cavity, and transmitted through the cavity
end mirror, increases. The transmission lineshape is Loren zian in the absence of disturbing effects:
P(ν) =P0
1 +F(ν−ν0)2, (1)
where ν0is the resonant frequency of the cavity, 2 /√
Fis the full width at half-maximum (FWHM), and P0
is the infra-red power at the peak of the Lorenzian.
The KNbO 3crystal inside the bowtie cavity absorbs a small amount of th e laser power and heats up.
This changes the crystal’s length and index of refraction, w hich in turn changes the optical path length of
the bowtie cavity. We model the overall effect by replacing ν0withν0+ηP/P 0.Accordingly, Equation 1
becomes
y=1
1 +F(ν−ηy)2. (2)
where y=P/P0is the normalized transmission, and the frequency νis measured relative to the unshifted
frequency ν0. Transmission lineshapes of this kind are typical of optica lly bi-stable devices.
The peak in the transmission curve occurs when y= 1 and ν=η. Figure 2 shows the transmission
lineshape for both low and high laser powers. On the same figur e we plot a fit of the data to Equation
2. The asymmetry in the transmission lineshape at high power s is due to thermal changes in the optical
pathlength. When sweeping the cavity from low to high freque ncies as in Figure 2, the power-dependent
cavity resonance is effectively “dragged along” from ν=ν0toν=nu0+ηas the cavity length changes.
This is known as “thermal self-locking” [3], and the value of ηis called the “self-locking range.”
For a given power incident on the cavity, we determine the sel f-locking range ηusing a nonlinear least-
squares fit of the measured transmission lineshape to Equati on 2. For the highest laser intensities, it appears
2that a better overall fit is obtained by replacing ηyin Equation 2 by a higher order polynomial. However,
the fit in these high intensity data is hampered because the fit function is not single-valued, as can be seen
in Figure 2. For these measurements, we determine the values ofηvisually by finding the frequency at the
peak of the transmission curves.
Our values for ηas a function of infrared power in the bowtie cavity are shown in Figure 3a. The error
bars indicate the 1 σstatistical spread in 5 measurements at each power setting. For linear absorption, the
cavity shift, η, should be a linear function of the circulating power. Howev er our measurements indicate a
quadratic dependence of the cavity shift on the circulating power, suggesting that the dominant absorption
mechanism is two-photon absorption in the infrared.
We convert ηinto a temperature rise inside the crystal by measuring the t emperature change necessary
to shift the cavity resonance frequency by one free spectral range. In these calibration measurements, we
attenuate the incident laser power to 10 µW to avoid laser-heating in the crystal. The cavity mirror po sitions
remain fixed, and the laser frequency is constant. Using the P eltier device, we change the temperature of
the crystal by several degrees while monitoring the cavity t ransmission. At T=−11.5oC, the resonant
frequency of the bowtie cavity shifts at a rate of 231 MHzoC−1for light polarized along the b-axis and 331
MHzoC−1for light polarized along the c-axis.
We calibrate our photodiode signal using a power meter, and m easure the efficiency of the optical path
to convert this IR signal to a power circulating inside the ca vity. The conversion from cavity shift to
temperature rise can also be derived from measurements of th e change in the index of refraction with
temperature. From the published values of dn/dT [11] and the geometry of our bowtie cavity, we calculated
what the temperature-dependent shift ought to be. The calcu lation agrees with our data for the b-axis.
However, we find a small difference between the calculation an d our data for the c-axis, as discussed later in
this paper.
Using our measurements, we convert ηinto ∆ T, the difference in temperature between the crystal axis
where the laser propagates and the crystal wall which is kept constant by our temperature control circuit.
Figure 3b shows a plot of the temperature rise as a function of the infrared power circulating inside the cavity.
It is relatively simple to convert this temperature rise int o a measure of the optical absorption coefficient of
KNbO 3by solving the heat transfer equation, which we do below.
4 Solving the Heat Transfer Equation
We solve the heat transfer equation in cylindrical coordina tes. The laser beam is in the TEM 00Gaussian
mode, with a 41 µm waist. The confocal parameter of the beam is 6.2 mm, and at th e face of the crystal, the
beam waist is only 7% larger than in the center of the crystal. So we approximate the laser beam as axially
symmetric and Gaussian with a 41 µm waist along the entire 10 mm length of the crystal. Our cryst al has
a square cross section, 3 mm on a side. However, because it is s o much larger than the laser beam waist, we
assume that the crystal is cylindrical also, with a 1.5 mm dia meter. With these assumptions, the equilibrium
temperature inside the crystal is determined by solving the heat equation in the form:
1
rd
dr/parenleftbigg
rdu
dr/parenrightbigg
=−G
K, (3)
where u(r) is the temperature as a function of radius, Gis the power density, and Kis the thermal conduc-
tivity. Following Sutherland [12], the absorbed power dens ity inside the crystal can be written as
G=αI+βI2(4)
where I= (2P/πa2)exp(−2r2/a2) is the laser intensity, Pis the laser power incident on the crystal, ais
the Gaussian beam waist, αis the linear optical absorption coefficient, and βis the two-photon absorption
(TPA) coefficient.
We can simplify Equation 3 by letting r=ay/√
2,A= (Pα/2πK), and B= (P2β/2π2a2K). With these
substitutions, Equation 3 integrates once to
du
dy=A
y/bracketleftbig
exp(−y2)−1/bracketrightbig
+B
y/bracketleftbig
exp(−2y2)−1/bracketrightbig
, (5)
3where the constant of integration is chosen to avoid a singul arity as y→0. The solution to this differential
equation is
u(y) =−A
2E1(y2)−A
2ln(y2)−B
2E1(2y2)−B
2ln(y2) +C (6)
where En(x) is the exponential integral [13], and Cis the constant of integration. This constant is determined
by requiring the temperature at the wall of the crystal to be Tc(i.e.:u(r=b) =Tc), which we control in
the experiment. With this constraint, we can evaluate the te mperature rise on axis inside the crystal. The
series expansion for the exponential integral has a logarit hmic term that exactly cancels the ln( y2) terms in
Equation 6.
∆T=u(0)−Tc (7)
=A
2/bracketleftbigg
ln/parenleftbigg2b2
a2/parenrightbigg
+γ/bracketrightbigg
+B
2/bracketleftbigg
ln/parenleftbigg2b2
a2/parenrightbigg
+γ+ ln(2)/bracketrightbigg
(8)
=α
4πK/bracketleftbigg
ln/parenleftbigg2b2
a2/parenrightbigg
+γ/bracketrightbigg
P+β
4π2a2K/bracketleftbigg
ln/parenleftbigg4b2
a2/parenrightbigg
+γ/bracketrightbigg
P2, (9)
where γ=.57721 . . .is Euler’s constant.
We can analyze the data in Figure 3 to extract the linear and tw o-photon absorption coefficients. A
weighted fit to the function T(P) =c1P+c2P2, gives c1= 5.7(36)×10−4oC W−1andc2= 1.09(12)×10−3oC
W−2. The number in parenthesis indicates the statistical uncer tainty from the weighted fit in the last digits.
Taking the thermal conductivity from [4] K= 4.0 W m−1oC−1,a= 41×10−6m,b= 1.5×10−3m, and
γ= 0.57721. We determine the one-photon (linear) absorption coe fficient to be α= 0.0034(21) m−1and the
two-photon absorption coefficient to be β= 3.16(35)×10−11m W−1. The uncertainties in these absorption
coefficients are only statistical. We have a systematic uncer tainty in the power measurements of 10%. Our
assumption of cylindrical symmetry in the crystal for solvi ng the heat equation also introduces an error. The
error is known, but related to this is the fact that our laser d oes not propagate exactly down the center of
the crystal. These errors show up in the ln( b2) term in Equation 9, and probably add ∼8% uncertainty to
the measurements. Our best numbers for the absorption coeffic ients are therefore α= 0.0034(22) m−1and
β= 3.2(5)×10−11m/W. These numbers do not reflect the uncertainty in the therm al conductivity, which
we do not know. To our knowledge, this is the first determinati on of the two-photon absorption coefficient
at any wavelength in KNbO 3.
Earlier work on the linear absorption coefficient for KNbO 3[4] found α= 0.001cm−1= 0.1m−1at 860
nm. This value is much higher than this work. It may be that som e of the absorption measured in previous
work was two-photon absorption. That work used laser calori metry to measure the absorption coefficient at
specific laser lines from 457 nm to 1064 nm. At 860 nm, they focu sed 900 mW into a 5 mm crystal, and it
is likely that the intensity was high enough for multiphoton absorption to be important.
Our method for determining the one and two-photon absorptio n coefficients is quite general. It is indepen-
dent of measurements of light scattering inside the crystal . It is also independent of reflection measurements
at the crystal faces. This method is particularly well suite d to measurements of optical absorption where
the absorption coefficients are comparable to or smaller than scattering and reflection coefficients.
5 Transmission Lineshape With Blue Light
When infrared light incident on the bowtie cavity is polariz ed along the b-axis of the KNbO 3crystal, it is
possible to meet the phase-matching conditions required fo r generating the second harmonic at 423 nm. The
presence of blue light in the crystal dramatically alters th e transmission lineshape. Not only is the blue
light itself absorbed by the crystal, it also significantly i ncreases the absorption of the infrared light. This is
called “blue-light-induced-infrared-absorption” (BLII RA), and it has been studied at length in the literature
[4, 8, 9, 10]. Previous work has found BLIIRA to be significant at blue light intensities down to 7 ×10−4
W/cm2. BLIIRA is minimized at longer wavelengths and at higher cry stal temperatures [10]. However, for
our calcium work, we require high powers at 423 nm.
4As before, we fix the laser frequency and scan the cavity lengt h through the resonance condition. Both
blue light at 423 nm and a small amount of infrared light at 846 nm exits the cavity. We separate these
wavelengths using dichroic mirrors, which transmit 95% in t he blue and reflect 99.5% in the infrared. To
make sure that no blue light reaches the infrared laser beam d etector, we use four of these mirrors in series
in the infrared beam path after the doubling cavity (see Figu re 1).
In these experiments, we do not independently control the in frared and blue beam intensities. Rather,
we optimize the crystal temperature to maximize the blue lig ht production for each infrared power setting
in the steady state. Losses inside the doubling cavity limit the maximum infrared power circulating inside
and therefore the maximum blue light produced.
The transmission lineshape for the cavity when blue light is present is shown in Figure 4a for our
highest IR powers. The temperature rise inside the crystal i s significant. At maximum power, as the cavity
approaches the resonance condition, only a small amount of b lue light is initially generated because the
crystal temperature is too low to meet the phase matching con dition for second harmonic generation exactly
(see Figure 4b). Closer to the cavity resonance condition, t he circulating power increases, the crystal heats
up, and more blue light is generated, which heats the crystal even more. This positive feedback continues,
and the cavity demonstrates thermal self-locking for up to 9 0 MHz, as shown in Figure 4a.
The blue light signal in Figure 4b has a shoulder on the low-fr equency side of the maximum. This is
a Maker fringe. As stated previously, we initially optimize the crystal temperature for optimum blue light
production in the steady state. At the highest intensities, the temperature shift ( ∼0.4oC) due to thermal
self-locking of the cavity is a few times the temperature pha se matching bandwidth.
Because of these complicated thermal conditions, and espec ially because the blue light production is not
constant across the transmission lineshape, we are relucta nt to fit the lineshape to any model for the highest
laser intensities. Instead, we find the self-locking range g raphically. It is the falling step at the far right hand
side of the infrared transmission peak shown in Figure 4. Thi s choice is valid as long as there is no significant
thermally-induced change in the cavity coupling efficiency. We can monitor these changes by measuring the
light reflected from the cavity. For the relatively modest in tensities in this study, the minimum reflected
light (which corresponds to the maximum light inside the cav ity) changes by only a few percent, and only
at the highest intensity measurements. The statistical sca tter in the data is larger than this, and we neglect
this small systematic error.
At lower laser intensities, where BLIIRA and other absorpti on processes are less severe, we use a least-
squares method similar to Section 3 to find the self-locking r angeη. The self-locking range is plotted as a
function of infrared laser power in Figure 5a. Note that exce pt for the two highest power measurements, the
frequency doubling process is not saturated (see Figure 4c) .
With blue light present in the crystal, the temperature rise inside the crystal has three sources: infrared
absorption (one and two photon), blue light absorption, and BLIIRA. We can easily subtract off the in-
frared contribution to the self-locking range using the dat a in Figure 3 as a look-up-table. The remaining
contribution to the self-locking range is due to temperatur e rise from blue light absorption and BLIIRA.
Because the blue light intensity and the temperature are not constant along the length of the axis of the
crystal, we must be careful about how we define the temperatur e rise and the blue light intensity. We will
define the absorption coefficient at 423 nm, δ, by the equation
∆P=/integraldisplay
VIbδdV=δPbL (10)
where Ibis the blue light intensity inside the crystal, Pbis the measured blue power, and Lis the length of the
crystal. In the non-depleted pump plane-wave approximatio n, the blue light intensity grows quadratically
with distance zin the crystal. With the definition from Equation 10, the blue light intensity can be written
asIb(r, z) = (12 Pbz2/πa2L2)exp(−4r2/a2), where ais the Gaussian waist for the infrared beam in the
crystal and we explicitly assume that the Gaussian waist for the blue beam is a/√
2.
The heat equation for blue light absorption and BLIIRA is wri tten as
1
r∂
∂r/parenleftbigg
r∂u
∂r/parenrightbigg
+∂2u
∂z2=−z2/parenleftBig
Ce−4r2/a2+De−6r2/a2/parenrightBig
. (11)
The first term on the right hand side of the equation is the blue light absorption term where C= 12Pbδ/(πa2L2K).
The second term on the right hand side of the equation is the BL IIRA term, where D= 24PbPirξ/(π2a4L2K),
5Piris the measured infrared power, assumed to be constant along the crystal length, and ξis the BLIIRA
absorption coefficient. As before, u(r, z) is the temperature inside the crystal and Kis the thermal conduc-
tivity.
The solution of this equation for r= 0 is
∆T(z) =3δ
4πK/bracketleftbigg
ln/parenleftbigg4b2
a2/parenrightbigg
+γ/bracketrightbiggz2
L2Pb+ξ
a2π2K/bracketleftbigg
ln/parenleftbigg6b2
a2/parenrightbigg
+γ/bracketrightbiggz2
L2PbPir, (12)
but we have to be careful about how we define the temperature ri se. In Equation 11, the radial derivatives
are much larger than the longitudinal derivatives, and the ∂2u/∂z2term can be treated perturbatively. This
approximation produces a temperature function that increa ses quadratically with zin the crystal.
We can write the temperature rise on axis as ∆ T(z) =Tmz2/L2, where Tmis the maximum temperature
rise in the crystal. We can calculate the change in the optica l pathlength inside the crystal with this
temperature profile. It is equal to the change in pathlength d ue to an equivalent temperature ∆ T′=Tm/3.
More expressly, we can write T(z) =Tmz2/L2= 3T′z2/L2, and rewrite Equation 12 as
T′=δ
4πK/bracketleftbigg
ln/parenleftbigg4b2
a2/parenrightbigg
+γ/bracketrightbigg
Pb+ξ
3a2π2K/bracketleftbigg
ln/parenleftbigg6b2
a2/parenrightbigg
+γ/bracketrightbigg
PbPir, (13)
Knowing this, we can convert the measured self-locking rang e of the cavity in the presence of blue light
into a temperature rise. The data is plotted in Figure 5. For t his data, the “temperature rise” is the average
temperature rise in the crystal, i.e.: T′. The “blue power” is the measured blue power, i.e.: the power at
the end of the crystal, Pb.
Again, we fit a function of the form
∆T=a1Pb+a2PbPir (14)
to the data in Figure 5b using a multivariable least-squares method and find a1= 2.41oC W−1and and
a2= 1.04oC W−2. Using equation 13, we determine the absorption coefficients δ= 13.3 m−1andξ=
2.2×10−8m/W. Apparently, there are no measurements in the literatur e of the KNbO 3absorption coefficient
at 423 nm. However, there are measurements close to this wave length [4], and a value of 13.3% per cm is
reasonable. For this determination of the absorption coeffic ient, we have the same kinds of uncertainties as
before, namely in the beam position in the crystal and in the p ower measurement. We estimate the error in
this measurement to be 15 %. Our final number for the absorptio n coefficient at 423 nm is δ= 13(2), where
the number in parenthesis is the uncertainty in the last digi t.
This uncertainty estimate is perhaps conservative. We have a second KNbO 3crystal. In a simple
transmission measurement at 423 nm, using the blue light aft er the frequency doubling cavity, we find the
linear absorption coefficient to be 13% per cm, in agreement wi th the work above.
We verified the validity of our perturbative approximation b y solving Equation 11 numerically. In the
numerical solution, the temperature profile is quadratic in zexcept at the very end of the crystal, where is
flattens out slightly to meet the boundary condition of no hea t flowing out the crystal face. With our values
ofδ,ξ,a,Pb, and Pir, the calculated temperature T′agrees with our data to about one percent.
It is interesting that the above treatment of BLIIRA does not distinguish between blue-light induced
infrared absorption and infrared light induced blue-light absorption. This model of BLIIRA only postulates
that there is a two-color two-photon absorption cross-sect ion. This treatment of two-color two-photon
absorption in KNbO 3is perhaps simplistic. However, it serves the purpose in thi s paper of demonstrating
that thermal self-locking of a Fabry-Perot cavity can be use d to sensitively characterize the linear and
nonlinear optical properties of an absorbing crystal.
6 New Measurements of dn/dT
From the calibration measurements that convert ηin Figures 3 and 5 to a temperature rise, we can extract
the temperature-dependent change in the index of refractio n for KNbO 3. When the crystal temperature
changes, the optical pathlength of the bowtie cavity also ch anges because both the crystal length and the
index of refraction depend on temperature:
6dL
dT=Lc/bracketleftbiggdnx
dT+αT(nx−n0)/bracketrightbigg
, (15)
whereLis the cavity optical pathlength, Lcis the crystal length (10 mm), αTis the coefficient of linear
expansion for KNbO 3, and nxandn0are the indices of refraction for the crystal and air, respec tively.
In a relatively simple measurement, we can determine dn/dT for the crystal across a wide range of
temperatures. We fix the laser frequency and the cavity mirro r positions. We attenuate the laser intensity
to 10µW to avoid thermal self-locking. The crystal temperature ca n be independently controlled because
it is mounted on a Peltier device. We measure the cavity trans mission while slowly sweeping the crystal
temperature over several degrees. At certain temperatures , the cavity optical pathlength meets the resonance
condition, and we measure a peak in the cavity transmission. The temperature between transmission peaks
in these measurements corresponds to changing the optical p athlength by one optical wavelenth, 846 nm.
We repeat these measurements over a wide temperature range t o determine dL/dTfrom -12oC to about 50
oC.
Using equation 15, we convert dL/dTtodn/dT . With nc= 2.13,nb= 2.28,αT= 5×10−6oC−1, and
dL/dT=λ/∆T, we determine dn/dT for several temperatures. Our values are shown in Figure 6 fo r light
polarized along the c-axis and b-axis, as labeled. Also shown in the figure is the data from Gho sh [11]. The
line from Ghosh is a fit to his measurements of the index of refr action between 0 and 140oC. We extrapolate
that fit to lower temperatures to compare with our data.
For light polarized along the c-axis, our measurements agree with those of Ghosh over our en tire tem-
perature range. For light polarized along the b-axis, in the range 5 to 50oC, which is roughly the range
for which our data overlaps Ghosh’s data, our measurements a lso agree. However, below ∼5oC, our data
departs slightly from Ghosh’s.
To check our data, we made an additional measurement. When th e polarization of the light incident
on the cavity is rotated slightly relative to the c-axis, the transmission lineshape has two features, one for
each polarization. Because the index of refraction is differ ent for the two polarizations ( bandc), the optical
pathlength is also different, and each polarization is trans mitted at a different cavity length. When we change
the temperature of the crystal, the two peaks in the transmis sion lineshape shift by different amounts in
different directions, as shown in Figure 7. We measured these shifts for a set of small temperature changes
near -11.5oC and determined the relative changes in the indices of refra ction, ( dnc/dT)/(dnb/dT). These
measurements confirm our low temperature data in Figure 6. Fo llowing the treatment of Ghosh, we fit our
dn/dT data for the b-axis to a second-order polynomial, and find
dnb
dT= 1.55809×10−9T2+ 4.06912×10−8T−33.76305×10−6(16)
where the units of dn/dT areoC−1, and Tis measured in Celcius.
7 Conclusion
We have demonstrated how thermal self-locking of a Fabry-Pe rot cavity can be used to determine optical
properties of nonlinear crystals. We have determined new va lues for the one- and two-photon absorption
coefficients in KNbO 3at 846 nm. We have also determined new values for the one photo n (linear) absorption
coefficient at 423 nm, and a BLIIRA absorption coefficient under specific circumstances. Finally, we present
new measurements of the temperature-dependent change in th e indices of refraction for the bandcaxes.
These measurements all derive from the thermal response of a n absorbing Fabry-Perot cavity to light
circulating in the cavity. The methods are general and can be widely applied to other systems, particularly
those in which the absorption coefficients are small compared to scattering and reflection coefficients.
8 Acknowledgements
We express our appreciation to Ross Spencer for his computat ional assistance. This work is supported in
part by a grant from the Research Corporation and from the Nat ional Science Foundation under Grant No.
PHY-9985027.
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[3] P. Dube, L.-S. Ma, J. Ye, P. Jungner, and J. L. Hall, “Therm ally induced self-locking of an optical cavity
by overtone absorption in acetylene gas,” J. Opt. Soc. Am. B 13, 2041-2054 (1996).
[4] L. E. Busse, L. Goldberg, M. R. Surette, and G. Mizell, “Ab sorption losses in MgO-doped and undoped
potassium niobate,” J. Appl. Phys. 751102 (1994).
[5] Our extended cavity laser is a Vortex laser from New Focus Corp. The tapered amplifier is a Model 8613
laser from SDL with the rear cavity optics removed, configure d as a single-pass amplifier.
[6] R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. F ord, A. J. Munley, and H. Ward, “Laser
phase and frequency stabilization using an optical resonat or,” Appl. Phys. B. 31, 97-105 (1983).
[7] Our crystal is from VLOC.
[8] L. Goldberg, L. E. Busse, and D. Mehuys, “High power conti nuous wave blue light generation in KNbO 3
using semiconductor smplifier seeded by a laser diode,” Appl . Phys. Lett. 63, 2327-2329 (1993).
[9] H. Mabuchi, E. S. Polzik, and H. J. Kimble, “Blue-light-i nduced infrared absorption in KNbO 3,” J. Opt.
Soc. Am. B 11, 2023-2029 (1994).
[10] L. Shiv, J. L. Sorensen, and E. S. Polzik, “Inhibited lig ht-induced absorption in KNbO 3,” Opt. Lett.
20, 2270-2272 (1995).
[11] G. Ghosh, “Dispersion of thermo-optic coefficients in a p otassium niobate nonlinear crystal,” Appl.
Phys. Lett. 65, 3311-3313 (1994).
[12] R. L. Sutherland. Handbook of Nonlinear Optics (Dekker, New York, 1996), p. 502
[13] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), p.
228
8ECDL Amplifier
crystalI I
DC1
Dλ/2
DPBS C2λ/4λ/2
BS2BS1
Figure 1: Schematic diagram of the experimental layout. ECD L = extended cavity diode laser, I = optical
isolator, λ/2 = half wave plate, λ/4 = quarter wave plate, C1 = frequency doubling “bowtie” cavi ty, C2
= stable optical cavity, D = photodiode detectors, PBS = pola rizing beam-splitter cube, BS1 = uncoated
quartz window, BS2 = dichroic beam splitter. The short heavy lines are mirrors.
−150150123 Transmitted infrared signal (mV)
Frequency (MHz)(a)
−150150204060 Transmitted infrared signal (mV)
Frequency (MHz)(b)
Figure 2: Infrared transmission lineshape for the bowtie ca vity in the absence of blue light. The points are
the measurements. The solid line is a fit to the data (see Equat ion 2). The infrared light is polarized along
thec-axis. (a) Low power. (b) High power. For high powers, the fit f unction is not single-valued above the
cavity resonant frequency.
9030600246
Infrared Signal (mV)η (MHz)(a)
0123401020
Circulating Power (W)Temperature Rise (10−3 oC)(b)
Figure 3: (a) Self-locking range, η, versus infrared detector signal. (b) The data from 3a with t emperature
and power calibrations. The solid line is a weighted fit to the data, T=c1P+c2P2.
050100(a)
(b)
Frequency (MHz)Signal (arb. units)
0120204060(c)
Circulating Power (W)Second Harmonic Power (mW)
Figure 4: (a) Cavity transmission lineshape in the infrared at 846 nm when generating the second harmonic
at 423 nm. This signal is taken from the highest intensities i n the experiment. Note that for low powers, the
FWHM is 5 MHz. (b) The second harmonic signal for the same sett ings as 4a. The shoulder to the left of
the signal is a Maker fringe (see text). (c) Peak second harmo nic power as a function of peak infrared power.
For all but the highest two power measurements, the peak blue power is given by Pblue= 36.2kW−1Pir.
10010200204060(a)
Infrared Signal (mV)η (MHz)
00.020.040.0600.10.2(b)
Blue Power (W)Temperature Rise (oC)
Figure 5: (a) Self-locking range, η, in the presence of blue light versus infrared detector sign al, excluding
the highest power data where the SHG process appears to be sat urating. (b) The data from 5a with the
temperature and power calibrations, and with the temperatu re rise from the infrared light subtracted off.
The solid line is a weighted fit to the data (see text).
−1001020304050−3003060
Temperature (oC)dn/dT (10−6 oC−1) c−axis
b−axis
Figure 6: The change in the index of refraction with temperat ure for light polarized along the bandc-
axes. The points are from this work. The solid line is from Gho sh [11]. The dashed line is a second order
polynomial fit to the data.
110 50 100 150→ b−axis
← c−axis
Frequency (MHz)
Figure 7: Transmitted lineshape as a function of cavity leng th for two temperatures. The upper trace has
been displaced for clarity. The polarization of the light in cident on the cavity is rotated a few degrees relative
to the c-axis. The larger feature is polarized along the c-axis, and the smaller feature is polarized along the
b-axis, as labeled. This shows the magnitude and direction of the change in the cavity resonance length with
temperature.
12 |
1SLAC–PUB–8142
May 1999
Planar Waveguide Hybrids for Very High Power RF*
C.D. Nantista, W.R. Fowkes, N.M. Kroll#, and S.G. Tantawi‡
Stanford Linear Accelerator Center, P.O. Box 4349, Stanford, CA 94309
Abstract
Two basic designs have been developed for waveguide hybrids, or 3-dB couplers, capable
of handling hundreds of megawatts at X-band. Coupling is provided by one or twoconnecting waveguides with h-plane junctions and matching elements. In the former case,the connecting waveguide supports two modes. Small apertures and field-enhancing e-bends are avoided to reduce the risk of rf breakdown. The h-plane symmetry also allowsthe use of over-moded rectangular waveguide in which the height has been increased toreduce field amplitudes without affecting the scattering matrix. The theory and designs arepresented, along with the results of prototype tests of functionality and power-handlingcapability. Such a device is integral to the rf pulse compression or power distributionsystem [2] of the Next Linear Collider (NLC) [1] for combining, splitting, and directingpower. This work was motivated by the observation of rf breakdown at power levelsabove 200 MW in conventional and modified magic-T's.
presented at the 1999 IEEE Particle Accelerator Conference
New York, New York, March 29-April 2, 1999
* Work supported by Department of Energy contract DE–AC03–76SF00515
and grant DE-FG03-93ER40695.
# Also University of California, San Diego, La Jolla, CA 92093.
‡ Also with the Communications and Electronics Department, Cairo University, Giza, Egypt.2I. INTRODUCTION
The design of the Next Linear Collider (NLC) [1] includes plans for powering the
high-gradient accelerator structures of the main linacs with 11.424 GHz X-band klystronsthrough a pulse compression or power distribution system [2]. In such a system pulsed rfwill need to be combined, split, or directed at peak power levels reaching 600 MW. Abasic component required is a waveguide hybrid, or 3-dB directional coupler, capable ofhandling very high power levels. Prototype rf systems have employed conventional,matched magic T's in WR90 (0.9"x0.4") waveguide. As power levels were increased,these proved inadequate and a modified design was developed, in which the matching postin the waveguide junction was replaced with a thick fin. While simulations showed thisdesign to have lower field strengths, it's reliability proved to still be inadequate. Thesemagic T's exhibited frequent rf breakdown at power levels above 200 MW, whichinspection showed to occur primarily at the mouth of the e-bend [3].
With this motivation, we have subsequently developed two novel planar hybrid
designs capable of reliably handling hundreds of megawatts of peak power at X-band(11.424 GHz). These each consist of four rectangular waveguide ports, which operate inthe TE
10 mode, connected through four or two h-plane T-junctions, yielding, respectively,
a two-rung ladder or an "H" geometry. In the latter case, the single connecting waveguidecarries two modes. Figure 1 illustrates the two design geometries.
Small apertures, slots, and field-enhancing e-bends are avoided to reduce the risk of
rf breakdown. Matching features maintain the translational symmetry of these cross-sections. Electric fields terminate only on the flat top and bottom surfaces. This h-planesymmetry also allows the use of over-moded rectangular waveguide in which the heighthas been increased to reduce field amplitudes without affecting the scattering matrix. Bothare quadrature hybrids (i.e. the coupled port fields are 90 ° out of phase), and directly3opposite port pairs are isolated. That is, the scattering matrices, with properly chosen,
symmetric reference planes and the indicated port numbering, are of the form
S=−
−
−
−
1
201 0
100
001
01 0i
i
i
i.
a) b)143
214 3
2
Figure 1. Schematic of the h-planar geometries of the a) two-
rung ladder and b) "H" hybrid designs. Power-flow arrowsindicate output ports for the indicated input port.
II. TWO-RUNG LADDER HYBRID
The two-rung ladder hybrid is basically a realization in rectangular waveguide of
R.H. Dicke's coaxial-line synthesis of a biplanar magic T [4], or of a modified microstripbranch-line hybrid. In transmission line theory, this device requires the distances betweenall adjacent junctions to be (n/2 plus) one quarter wavelength and the two connecting linesto have a characteristic admittance that is
2 times that of the main lines. The resulting
circuit can be shown to yield the desired scattering matrix.
In waveguide, whose width is not small compared to a wavelength, the two-
dimensional geometry necessitates matching elements in the T-junctions to adjust the4complex impedances. A mode matching code was used to determine the radius and
placement of a post that would yield a three-port junction with the desired scattering matrix,from which the hybrid ring circuit could then be constructed. The fields were expanded incylindrical harmonics about the post in the junction region and in normal modes in therectangular port regions. The full hybrid design was verified with the finite-element fieldsolver HFSS. The peak field amplitude was found to be 44.1 MV/m at 300 MW with awaveguide height of 0.900 inch.
III. "H" HYBRID
The "H" hybrid can be viewed as a variation of the above with the two connecting
waveguides collapsed into one in which two modes are utilized. Thus, in a transmissionline picture, the requirement of two connecting lines is not circumvented. The couplingmechanism is actually the same as that of the Riblet short-slot coupler [5], although thisgeometry provides separated ports and no sharp-edged wall interruptions.
The connecting guide is wide enough that both the TE
10 and the TE20 modes can
propagate, and these are excited with comparable amplitudes by the fields of a single port.They are excited with a relative phase such that their fields add constructively on the sidenearest the input port and destructively on the other side. If they were to slip in phase by
π
radians, the TE10 wave would enhance the opposite lobe of the TE20 wave, sending the
power out the farthest port. To get a 3-dB split, therefore, the total phase lengths for thesetwo modes through the connecting guide must differ by an odd multiple of
π/2.
The T-junctions in this design have been matched by shaping the walls with blunt,
triangular protrusions at the symmetry plane, rather than with free-standing posts. Theresult is essentially a side-wall coupler with the common wall removed and two back-to-back mitred 90 ° bends at either end. The connecting guide must be narrower than twice the
standard guide width in order to keep the TE
30 mode cut off. Simple mitred bends used in
a preliminary design therefore led to narrow ports (half the connecting guide width). To5accommodate standard-width ports and avoid the added length of width tapers, a small
vertical ridge was placed in each port to match into an effective asymmetric mitred bend.The TE
20 mode is thus matched independently at each junction. The width and length of
the connecting guide are adjusted to simultaneously meet the above phase length differencerequirement and cause the small TE
10 mismatch at the two junctions to cancel. HFSS was
used extensively in the design process to calculate scattering matrices. The peak field of thefinal design was found from simulation to be 39.5 MV/m at 300 MW with a waveguideheight of 0.900 inch.
IV. TESTS
A copper high-power vacuum-flanged prototype of each of these hybrids has been
built and tested. They were both made over-moded, the two-rung ladder in 0.9 inch squareguide and the "H" in double-height (0.9"x0.8") guide. This necessitated height tapers atthe ports for compatibility with our test setup and other WR90-based components. The topand bottom were tapered symmetrically with one-inch long half-cosine tapers. AnHP 8510C Network Analyzer was used to measure the scattering matrix parameters in thevicinity of the design frequency. The results over a 500 MHz span are presented inFigure 2.
The measured insertion losses, when corrected for the predicted loss of the flange
adaptors used and WR90 curved h-bends built onto two ports of the "H" hybrid toaccomodate a particular installation, give ~1.5% for the two–rung ladder hybrid and ~0.9%for the "H" hybrid. We define this loss as 1
212
312−+(| | | | )SS . It is dominated by ohmic
loss. For the two-rung ladder, the reflected signal and isolation at 11.424 GHz were bothabout -26 dB, accounting together for 0.49% misdirected power. For the "H" they were-33 dB and -37 dB, respectively, accounting for 0.07% misdirected power. The measuredcoupling at 11.424 GHz, corrected for loss (i.e. 10
312
212
312log[| | /(| | | | )]SSS + ), was
-3.19 dB for the former hybrid and -2.96 dB for the latter (the ideal being -3.01 dB).6Finally, Figure 2 shows the "H" hybrid to have a significantly broader bandwidth, as one
would expect from the more compact geometry.
The hybrids were later high-power tested in the pulse compression system of the
Accelerator Structure Test Area (ASTA) [3], where they were processed with pulsed rf topeak power levels exceeding 400 MW in 150 ns pulses and performed successfully withoutbreakdown problems or excessive X-ray production.
V. CONCLUSIONS
In response to the problem of rf breakdown in multi-hundred-megawatt X-band rf
systems being developed for a next generation linear collider, we have conceived andproduced two new types of rectangular waveguide hybrid, with relatively open interiorsand completely two-dimensional designs, perpendicular to the electric field lines. The latterfeature makes their circuit properties independent of height, allowing for their constructionin over-height waveguide to reduce fields. Prototypes of both designs performed similarlyand quite well. The "H"-shaped hybrid has the advantage of broader bandwidth and ismore compact. HFSS simulation suggests that it has peak fields lower by about 11% for agiven power flow and waveguide height. The absence of free-standing matching elementsmay also be an advantage with regard to cooling.
One goal in our component development program is to limit surface fields at
anticipated power levels to values below 40 MV/m in order to avoid rf breakdownproblems. For a power flow of 300 MW in one port, both hybrid designs meet orapproach this goal in square guide (0.9"x0.9"). By contrast, our original and modifiedmagic T's had peak fields of approximately 80 MV/m and 63 MV/m, respectively, at thispower level and could not be made over-moded in height.
For testing and for their intended use, we required standard, single moded ports on
our prototypes, for which the peak field at 300 MW is 49 MV/m. Smooth height taperswere incorporated at the ports to bring the peak fields in the interior of these devices, where7standing waves cause some enhancement, below the peak port field. With reference planes
taken just inside these tapers, the hybrids proper are thus over-moded. To take fulladvantage of these hybrid designs, one would not normally use single-moded ports, butremain over-moded, perhaps matching into a TE
11 mode in circular waveguide.
To comfortably handle 600 MW, the hybrid height would have to be increased to
1.75 inches. In such waveguide, taper design becomes non-trivial because the TE12 mode
can propagate. Mode conversion due to mechanical imperfections also becomes more of aconcern as a device becomes more over-moded. It may therefore be preferable to use aconfiguration which incorporates two hybrids to further increase power-hancdling capacity.
Figure 2. Scattering matrix elements for our two hybrid prototypes
measured over a frequency range of 500 MHz centered on the designfrequency of 11.424 GHz.8REFERENCES
[1] The NLC Design Group, Zeroth-Order Design Report for the Next Linear Collider,
LBNL-PUB 5424, SLAC Report 474, and UCRL-ID 124161, May 1996.
[2] S.G. Tantawi et al., "A Multi-Moded RF Delay Line Distribution System (MDLDS) for
the Next Linear Collider," to be published in Physical Review-Special Topics.
[3] A.E. Vlieks et al., "High Power RF Component Testing for the NLC," presented at the
19th International Linear Accelerator Conference (LINAC 98) Chicago, IL, August 23-28,1998, SLAC-PUB-7938 (1998).
[4] Montgomery, Dicke, and Purcell,
Principles of Microwave Circuits, Radiation Lab.
Series, 1948, p. 451.
[5] Henry J. Riblet, "The Short-Slot Hybrid Junction," Proceedings of the I.R.E.,
February 1952, p. 180. |
arXiv:physics/0102034 13 Feb 2001
1SLAC–PUB–8771
February 2001
A Compact, Planar, Eight-Port Waveguide Power Divide r/Combiner:
The Cross Potent Superhybrid*
Christopher D. Nantista and Sami G. Tantawi**
Stanford Linear Accelerator Center, Stanford University, Stanford CA 94309
Submitted to IEEE Microwave and Guided Wave Letters
* Work supported by Department of Energy contract DE– AC03–76SF00515 .
** Also with the Communications and Electronics Depart ment, Cairo University, Giza, Egypt. 2
Abstract-- In this letter, we present a novel four-way
divider/combiner in rectangular waveguide. The desi gn is
completely two-dimensional in the h-plane, with eig ht-fold
mirror symmetry, and is based on a recent four-port hybrid
design [6]. In combining mode, it can function as a phased
array with four inputs and four outputs. The plana r nature of
this design provides advantages, such as the freedo m to
increase the waveguide height beyond the over-modin g limit in
order to reduce field strengths. Along with its op en geometry,
this makes it ideal for high-power applications whe re rf break
down is a concern. Design criteria, field-solver s imulation
results, and prototype measurements are presented.
Index Terms -- waveguide, hybrid, phased array,
divider/combiner
I. INTRODUCTION
Designs for a next-generation 0.5–1 TeV center-of-m ass
electron/positron collider [1],[2] call for power c ombined
from each group of several 11.424 GHz 75 MW klystr ons
to power spacially separated sets of high-gradient
accelerator structures of the main linacs through a Delay
Line Distribution System (DLDS) [3],[4]. The rf is directed
appropriately through source phasing during differe nt time
bins, yielding a pulse compression effect, in which each
accelerator feed sees a peak power several times hi gher
than that of a single source for a fraction of the klystron
pulse width.
The basic requirement at the heart of this scheme i s a
phased array capable of directing pulsed rf from fo ur inputs
to four outputs (three delay lines and a local feed ) at peak
power levels reaching 600 MW. This can be accompli shed
with a set of hybrids and a pattern of discrete pha se changes
in the source drive signals. In prototype rf syste ms
employing WR90 magic T's under high-vacuum for
combining and directing power, rf breakdown proved to be
a problem at power levels much exceeding 200 MW [5] .
Autopsies revealed the problem to be between the ma tching
element and the mouth of the e-bend.
This work was supported by the Department of Energy under contract DE-
AC03-76SF00515.
The authors are with the Stanford Linear Accelerato r Center, Stanford, CA
94309.
Sami G. Tantawi is also with the Communications and Electronics
Department, Cairo University, Giza, Egypt.
This motivated the design of an alternative to the magic T, a
novel planar hybrid capable of reliably handling hu ndreds
of megawatts of peak power at X-band. Small apertu res,
slots, matching posts, and field-enhancing e-bends were
avoided to reduce the risk of rf breakdown. This n ew
hybrid, reported on in [6], consists of four rectan gular
waveguide ports, operated in the TE 10 mode, connected
through two h-plane T-junctions, yielding an "H" ge ometry.
The connecting waveguide carries two modes.
A further development led to a component, based on the
same design technique, which provides the desired f unction
of a set of four hybrids in a single compact device . This
eight-port combiner/splitter has been named the "cr oss
potent superhybrid", after the heraldic symbol its geometry
recalls. Each port is matched, isolated from thre e other
ports, and coupled equally to the four remaining po rts. As
the above-mentioned hybrid, or “magic H” serves as a sort
of stepping stone to the cross potent superhybrid, we shall
recount here the theory, design method and advantag es of
the former device.
II. MAGIC H HYBRID
The magic H is a quadrature hybrid (i.e. the couple d port
fields are 90 ° out of phase with each other) with directly
opposite port pairs isolated. It can be viewed as a variation
of the branchline coupler with the two connecting
transmission lines collapsed into one waveguide uti lizing
two modes. A closer comparison might be made with the
Riblet short-slot coupler [7], which employs the sa me
coupling mechanism, although the “H” geometry provi des
separated ports and no mismatch due to wall thickne ss.
A key feature of this hybrid is its planar design. Matching
features maintain the translational symmetry in the
direction of the electric fields, so that the field s terminate
only on the flat top and bottom surfaces. This H-p lane
symmetry allows the use of over-moded rectangular
waveguide, in which the height has been increased t o
reduce field amplitudes, without affecting the scat tering
matrix. Theoretically, the height of the device ca n thus be
arbitrarily increased to accommodate higher power.
A Compact, Planar, Eight-Port
Waveguide Power Divider/Combiner:
The Cross Potent Superhybrid
Christopher D. Nantista and Sami G. Tantawi, Member, IEEE 3The geometry of the magic H hybrid is illustrated i n
Figure 1. The connecting waveguide is wide enough to
accommodate both TE 10 and TE 20 mode propagation. The
fields of a single port excite these modes with equ al
amplitudes and a relative phase such that their fie lds add
constructively on the side nearest the input port a nd
destructively on the other side. As their guide
wavenumbers are different, their relative phase cha nges as
/G87/G75/G72/G92/G3/G83/G85/G82/G83/G68/G74/G68/G87/G72/G3/G68/G79/G82/G81/G74/G3/G87/G75/G72/G3/G74/G88/G76/G71/G72/G17/G3/G3/G36/G3/G81/G72/G87/G3/G83/G75/G68/G86/G72/G3/G86/G79/G76/G83/G3/G82/G73/G3 /G3
radians would cause the TE 10 wave to enhance the opposite
lobe of the TE 20 wave, sending the power out the farthest
port. To get a 3-dB split, therefore, the total ph ase lengths
/G73/G82/G85/G3/G87/G75/G72/G86/G72/G3/G87/G90/G82/G3/G80/G82/G71/G72/G86/G3/G80/G88/G86/G87/G3/G71/G76/G73/G73/G72/G85/G3/G69/G92/G3/G68/G81/G3/G82/G71/G71/G3/G80/G88/G79/G87/G76/G83/G79/G72/G3/G82/G73/G3 /G18/G21/G17
Figure 1. H-planar geometry of the magic H hybrid. Power-f low arrows
illustrate splitting function (reverse for combinin g).
The T-junctions are matched by shaping the walls wi th
blunt, triangular protrusions at the symmetry plane . The
result is essentially a side-wall coupler with the common
wall removed and two back-to-back mitred 90° bends at
either end. The connecting guide must be narrower than
twice the standard guide width in order to keep the TE30
mode cut off. With the ports constrained to half t he width
of the connecting guide, standard reference curves or tables
can be used to determine the dimension of the 45° m itres.
The TE 20 (odd) mode is matched independently at each
junction, as it sees an effective mitred bend. The TE10
(even) mode is matched for discrete lengths of the
connecting guide, at which the mismatches of the tw o
junctions cancel. The width of the connecting guid e is
adjusted until a reasonable matching length also yi elds the
required /G18/G21/G3/G83/G75/G68/G86/G72/G3/G79/G72/G81/G74/G87/G75/G3/G71/G76/G73/G73/G72/G85/G72/G81/G70/G72/G3/G87/G75/G85/G82/G88/G74/G75/G3/G87/G75/G72/G3/G75/G92/G69/G85/G76/G71/G3/G73/G82/G85/G3
the two modes. The code HFSS [8] was used extensiv ely in
the design process to calculate scattering matrices .
III. CROSS POTENT SUPERHYBRID
Often, a microwave circuit containing a configurati on of
multiple hybrids is required for a particular appli cation. As
mentioned above, the Delay Line Distribution System [3],
conceived as a means of efficiently delivering high -power
rf to the particle accelerator structures of a next -generation
linear collider, requires power from four sources t o be
combined and delivered sequentially (by means of ph ase
shifts) to four output waveguides. This can be don e with
four hybrids, two stages of two. One output port o f each of
the first two hybrids is connected to one input por t of each
of the second two, so that the combined circuit has four inputs and four outputs. The two sources feeding e ach first-
stage hybrid are phased so as to combine in one or the other
of its output ports (canceling in the fourth port). These
combined waves are directed to the same second-stag e
hybrid during a given time bin. The relative phase between
the source pairs provides another degree of freedom , so that
the power arriving at the second-stage hybrid can b e
combined and directed to one or the other of its ou tput
ports. This arrangement is illustrated schematical ly in
Figure 2.
Figure 2. Schematic of hybrid arrangement for combining and directing
four input signals to any of four outputs.
The same functionality can be achieved in a single eight-
port device based on the magic H. Imagine placing two
such such hybrids side-by-side and removing the com mon
wall, leaving only the diamond shaped post formed b y the
two triangular mitre protrusions. The two combined ports
formed are indistinguishable from the interior conn ecting
guides. That is, the junction around the diamond-s haped
post is four-fold symmetric. Adding mitred splits to these
double-width ports at the proper distance effective ly
completes a second pair of hybrids. The resulting device,
resembling a cross with cross beams at the extremit ies has
been christened the cross potent superhybrid. The
geometry is shown in Figure 3.
Figure 3. H-planar geometry of an eight-port cross potent s uperhybrid.
Power-flow arrows and an HP HFSS electric field plo t illustrate four-way
splitting function with power fed into port 1 (reve rse for combining).
4Note that the four ports at the ends of the horizon tal piece
couple to the four ports at the ends of the vertica l piece.
This corresponds to a slightly different orientatio n of the
hybrids in Figure 2 (albeit with the same connectio ns)
which circumvents the need to have one port connect ion
cross over another, thus allowing planar symmetry t o be
maintained. The four magic H’s which can be discer ned in
the cross potent superhybrid coalesce in such a way that the
connected ports disappear entirely into the interio rs of the
neighboring hybrids, elegantly yielding a device wh ich is
more compact than the sum of its parts.
The HP HFSS [9] electric field plot in Figure 3 ill ustrates
the power direction for a wave entering the lower l eft port.
The ideal scattering matrix, with properly chosen r eference
planes, can be written as follows:
.**
21 01 01000010 10
01 010000101001 0100001010
01 01000010 10
21
−−=
−−−−
−−− −− −−−−−− −− −−−
−−−−
=
H HH Hcp
iiii
iiiiiiii
iiii
SSSSS
Here SH is the scattering matrix of the simple hybrid.
A cold-test model of the cross potent superhybrid h as been
built. A waveguide height of 0.400” was used, and 90°
curved bends were incorporated to simultaneously tu rn the
ports outward and taper from the 0.721” port width to
0.900”. This allowed for easy testing with WR90
connectors and loads. Figure 4 shows the network a nalyzer
results. S 71, being by symmetry and reciprocity identical to
S31, obscures the latter. All isolations are better t han
−38.8 dB at the design frequency and remain below −20 dB
over a bandwidth of ~200 MHz. All couplings are wi thin
0.06 dB of an average of −6.07 dB, adjusted for the added
bends. Insertion loss is thus calculated to be ~0. 05 dB. This
device inherits all the advantages of the planar ma gic H
hybrid, making it suitable for over-height fabricat ion for
very high power applications.
IV. CONCLUSIONS
In response to the problem of rf breakdown in multi -
hundred-megawatt X-band rf systems being developed for a
next generation linear collider, we have conceived and
designed a rectangular waveguide, eight-port super hybrid,
capable of serving as a 4x4 phased array, with a re latively
open interior and a completely two-dimensional geom etry.
The latter feature makes its circuit properties ind ependent
of height, allowing for fabrication in over-height waveguide
to minimize field strengths.
Measurements of a prototype were very satisfactory and in
good agreement with simulation. Field plots from t he latter
suggest that the maximum field in our X-band design can
be limited to 50 MV/m at 600 MW with a waveguide height
of 1.35”, whereas the same power in a magic T would yield
surface fields exceeding 100 MV/m. A prototype of a
similar four-port hybrid performed quite well [5] a t high
power. The simplicity of both these devices makes them
attractive for low power use as well.
-40-30-20-100
11.224 11.324 11.424 11.524 11.624S11
S21
S31
S41
S51
S61
S71
S81scattering parameters in dB
frequency (GHz)
Figure 4. Measured amplitudes, in dB, of the first column s cattering
matrix elements (and by symmetry all others) for th e cross potent
superhybrid over a frequency range of 500 MHz centered on 11.424 GHz.
V. REFERENCES
[1] The NLC Design Group, Zeroth-Order Design Repor t for the Next
Linear Collider , LBNL-PUB 5424, SLAC Report 474, and UCRL-ID
124161, May 1996.
[2] The JLC Design Group, JLC Design Study , KEK-REPORT-97-1,
KEK, Tsukuba, Japan, April 1997.
[3] H. Mizuno and Y. Otake, "A New RF Power Distrib ution System for X
Band Linac Equivalent to an RF Pulse Compression Sc heme of Factor 2N,"
contributed to the 17th International Linac Conference (LINAC94),
Tsukuba, Japan, August 21 −26, 1994.
[4] S.G. Tantawi et al., "A Multi-Moded RF Delay Line Distribution
System for the Next Linear Collider," proceedings o f the 8th Workshop on
Advanced Accelerator Concepts, Baltimore, Maryland, July 5−11 1998.
[5] A.E. Vlieks et al., "High Power RF Component Testing for the NLC,"
presented at the 19th International Linear Accelera tor Conference (LINAC
98) Chicago, IL, August 23 −28, 1998, SLAC-PUB-7938 (1998).
[6] C.D. Nantista, et al., "Planar Waveguide Hybrids for Very High Power
RF," presented at the 1999 Particle Accelerator Con ference, New York,
NY, March 29 −April 2, 1999.
[7] Henry J. Riblet, "The Short-Slot Hybrid Junctio n," Proceedings of the
I.R.E., February 1952, p. 180.
[8] High Frequency Structure Simulator, Version A.0 4.01, copyright
1984−1995 Ansoft Corp., copyright 1990 −1995 Hewlett-Packard Co.
[9] HP High Frequency Structure Simulator, Version 5.3, copyright
1996−1998 Hewlett-Packard Co.
|
arXiv:physics/0102035v1 [physics.plasm-ph] 13 Feb 2001A walk in the parameter space of L–H transitions without
stepping on or through the cracks
BALL Rowena and DEWAR Robert L.
Department of Theoretical Physics and Plasma Research Labo ratory
Research School of Physical Sciences & Engineering
The Australian National University, Canberra ACT 0200 Aust ralia
e-mail: Rowena.Ball@anu.edu.au, Robert.Dewar@anu.edu. au
Abstract
A mathematically and physically sound three-
degree-of-freedom dynamical model that emu-
lates low- to high-confinement mode (L–H) tran-
sitions is elicited from a singularity theory cri-
tique of earlier fragile models. We construct a
smooth map of the parameter space that is con-
sistent both with the requirements of singular-
ity theory and with the physics of the process.
The model is found to contain two codimension
2 organizing centers and two Hopf bifurcations,
which underlie dynamical behavior that has been
observed around L–H transitions but not mir-
rored in previous models. The smooth traver-
sal of parameter space provided by this analysis
gives qualitative guidelines for controlling access
to H-mode and oscillatory r´ egimes.
I. INTRODUCTION
A unified, low-dimensional description of the
dynamics of L–H transitions [1] would be a valu-
able aid for the predictive design and control of
confinement states in fusion plasmas. In this
work we report significant progress made toward
this goal by developing the singularity theory ap-
proach to modeling L–H transitions that was in-
troduced in [2]. The results give new insights
into the role of energy exchange and dissipa-
tion in the onset, evanescence, and extinction of
discontinuous and oscillatory action in confined
plasmas.The title of this paper refers to the philoso-
phy of singularity theory [3] as applied to dy-
namical models: that paths through parame-
ter space should be smooth and continuous, and
that parameters should be independent and not
fewer than the codimension∗of the system.
Since 1988 [4] many efforts have been made
to derive unified low-dimensional dynamical
models that mimic L–H transitions and/or as-
sociated oscillatory behavior [5–21]. All of these
models have contributed to the current view in
which the coupled evolution of poloidal shear
flow and turbulence establishes a transport bar-
rier. However, as was shown in [2], the mod-
els often founder at singularities. Consequently,
much of the discussion in the literature concern-
ing the bifurcation properties of L–H transition
models is qualitatively wrong.
We examine the bifurcation structure of a
semi-empirical dynamical model for L–H tran-
sitions [8], and find it needs two major opera-
tions to give it mathematical consistency: (1)
a degenerate singularity is identified and un-
folded, (2) the dynamical state space is expanded
to three dimensions.
We then analyse the bifurcation structure of
the enhanced model obtained from these opera-
tions, the BD model, and find it consistent with
many known features of L–H transitions. In par-
ticular, this is the first model that can emulate
the onset and abatement of oscillations in H-
mode, and direct jumps to oscillatory H-mode
[22,23].
1II. BIFURCATION STRUCTURE OF
THE DLCT MODEL
This paradigmatic 2-dimensional model [8]
comprises an evolution equation for the turbu-
lence coupled with an equation for the flow shear
dynamics derived from the poloidal momentum
balance:
dN
dt=γN−αFN−βN2(1)
dF
dt=αFN−µF. (2)
Nis the normalized level of density fluctuations,
Fis the square of the averaged E×Bpoloidal
flow shear. The fluctuations grow linearly with
coefficient γand are damped quadratically with
coefficient β. The exchange coefficient αis re-
lated to the Reynolds stress, and the damping
rateµFis due to viscosity.†
Following the procedure outlined in [2] we
form the bifurcation function g=X(F,γ), and
identify the singular points where g=gF=
0. We find the unique physical singularity
(F,γ)T= (0, βµ/α ), which satisfies the addi-
tional defining conditions for a transcritical bi-
furcation:
gγ= 0, g FF∝negationslash= 0,detd2g <0, (3)
where det d2gis the Hessian matrix of second
partial derivatives with respect to Fandγ.
Evaluating (3) at Tgives gγ= 0, gFF=
−2α2/β, detd2g=−α2/β2. The bifurcation di-
agram showing the transcritical point Tis plot-
ted in Fig. 1a. (In this and subsequent diagrams
stable solutions are indicated by continuous lines
and unstable solutions by dashed lines.)
However, Fig. 1a does not represent the com-
plete bifurcation structure of the DLCT model
because of the following generic property of the
transcritical bifurcation: it is non-persistent to
an arbitrarily small perturbation. Since the
poloidal shear flow v′is symmetric under the
transformation v′→ −v′it is appropriate to
introduce the perturbation term ϕF1/2. (NotethatF∝v′2.) Thus, the modified DLCT model
is
dN
dt=γN−αFN−βN2(1)
dF
dt=αFN−µF+ϕF1/2. (4)
The perturbation term in Eq. 4 represents
a physically inevitable source of shear flow that
breaks the symmetry of the internal shear flow
generation and loss rates. The physics comes
from non-ambipolar ion orbit losses that produce
a driving torque for the shear flow. This con-
tribution to the total shear flow evolution can
be quite large [24], and in fact the early models
for L–H transitions relied exclusively on a large,
nonlinear ion orbit loss rate [4,5]. However, ion
orbit loss alone cannot explain turbulence sup-
pression. Here we treat this term as part of a
more complete picture of L–H transition dynam-
ics, and emphasize its symmetry-breaking na-
ture by assigning the simplest consistent form to
it while recognizing that ϕmay be a nonlinear
function ϕ(ζ), where ζmay include dynamical
variables and parameters.
Some bifurcation diagrams for increasing val-
ues of ϕare plotted in Fig. 1b–d. We see im-
mediately that solution of one problem causes
another: a nonzero perturbation term does in-
deed unfold the degenerate singularity T, but it
releases another degenerate singularity Ts.
Before proceeding with a treatment of the
new bifurcation Tswe highlight three important
issues:
1. Since ϕis inevitably nonzero in experiments,
no transition can occur at all in the vicinity of
T, neither first-order or second-order, contrary
to what is stated in [8].
2. Both NandFchange continuously with γin
the same direction. The fact that Tis a trans-
critical bifurcation tells us that only two param-
eters — ϕand any one of the other parameters
— are required to define the qualitative struc-
ture of the problem. The bifurcation diagram
withNas state variable is plotted in Fig. 2,
2which should be compared with Fig. 1c. As it
stands, the model therefore cannot emulate tur-
bulence stabilization by the shear flow, contrary
to what is stated in [8].
3. To ascertain whether the model can exhibit
periodic dynamics as stated in [8] we look for
a pair of purely complex conjugate eigenvalues.
For Eqs 1 and 4 the defining conditions for Hopf
bifurcations may be expressed as
g= trJ= 0,detJ >0,d
dγtrJ∝negationslash= 0,(5)
where Jis the Jacobian matrix. We find that
detJ <0 where the equalities in Eq. 5 are ful-
filled, therefore oscillatory dynamics arising from
Hopf bifurcations cannot occur.
This does not rule out the possible existence
of periodic behavior arising from rare and patho-
logical causes. According to Dulac’s criterion
[25] Eqs 1 and 4 possess noperiodic solutions
arising from anycause if there exists Dsuch that
the quantity
S=∂
∂N(DW) +∂
∂F(DY) (6)
never changes sign. Here W=W(N,F)≡
dN/dt ,Y=Y(N,F)≡dF/dt , and the Dulac
function D=D(N,F) is a real positive func-
tion. Choosing D= 1 we find that
S=α(N−F)−2Nβ+γ−µ+ϕF−1/2/2,
(7)
which clearly can switch sign. However, there
may exist a more exotic Dulac function that for-
bids a change of the sign of S. We have not found
oscillatory solutions numerically in this system.
Returning to the new singularity Tswe
find that it is also a transcritical bifurcation:
the conditions (3) evaluated at ( F,γ)Ts=
(ϕ2/µ2, αϕ2/µ2) yield gγ= 0, gFF=
−βµ6/(2α2ϕ4)−µ3/ϕ2, detd2g=−µ6/(4α2ϕ4).
Does the DLCT model therefore require a second
perturbation term, this time to Eq. 1, to unfold
Ts?We remark here that often there is more than
one universal unfolding for a given bifurcation
problem, and we turn to the physics to decide
which is physically consistent. For the pertur-
bation in Eq. 4 that unfolded Twe chose the
form ϕF1/2because it is physically inevitable
that the symmetry v′→ −v′be broken. How-
ever, there is no matching physics for a simi-
lar term in Eq. 1. Another possibility is that
Tsis spurious, created by an unwarranted col-
lapse of a larger state space. This idea leads to
a suggestion that issupported by the physics,
that another dynamical variable is intrinsic to
a low-dimensional description of L–H transition
dynamics.
III. INTRINSIC 3-DIMENSIONAL
DYNAMICS OF L–H TRANSITIONS
We introduce the the third dynamical vari-
able by assuming that γ=γ(P), where Pis
the pressure gradient, as have a number of other
authors [9,14,13,12,20]. Assuming the simplest
evolution of Pand that γ(P) =γP, we arrive at
the following augmented model, obtained purely
from dynamical and physical considerations:
εdP
dt=q−γPN (8)
dN
dt=γPN−αFN−βN2(9)
dF
dt=αFN−µF+ϕF1/2. (4)
In Eq. 8 qis the power input and εis a di-
mensionless parameter that regulates the contri-
bution of the pressure gradient dynamics to the
overall evolution. The dynamics is essentially 3-
dimensional with ε≈O(1), but for ε≪1 or
ε≫1 the system can evolve in two timescales:
1. The original “slow” time t. For ε→0,
εdP/dt ≈0 and P≈q/(γN). The system col-
lapses smoothly to
dN
dt=q−αFN−βN2(10)
dF
dt=αFN−µF+ϕF1/2. (4)
3The organizing center is the unique transcritical
bifurcation ( F,q,ϕ )T= (0, βµ2/α2,0), the spu-
riousTsis non-existent, and there are no Hopf
bifurcations. For ε≫1 we define δ≡1/εand
multiply Eq. 8 through by δ; taking the limit as
δ→0 gives dP/dt≈0, from which P=P0. We
recover the same form as Eqs 1 and 4,
dN
dt=γP0N−αFN−βN2(1′)
dF
dt=αFN−µF+ϕF1/2, (4)
along with the “good” bifurcation Tand the
“bad” bifurcation Ts— therefore we suggest
that this is a non-physical limit for ε.
2. In “fast” time τ≡ε/tand, recasting the
system accordingly, it can be seen that on this
timescale the dynamics becomes 1-dimensional
inPin both limits.
The organizing center of the bifurcation
problem obtained from Eqs 8, 9, and 4 is
the unique transcritical bifurcation ( F,q,ϕ )T=
(0, βµ2/α2,0),gFF=−α2/β, det d2g=
−α4/(4β2µ2), and the spurious singularity Tsis
non-existent. We now have the bones of an im-
proved dynamical model for L–H transitions, but
it still does not emulate the following character-
istics of L–H transitions: (a) Hysteresis: Since
there is no non-trivial point where gFF= 0 it
cannot model discontinuous transitions or clas-
sical hysteretic behavior. (b) Oscillations in H-
mode: These have not been found numerically.
In a 3 - dimensional dynamical system it is, of
course, very difficult to prove that oscillatory so-
lutions do notexist.
Evidently we need more nonlinearity or
higher order nonlinearity to produce enough
competitive interaction. To obtain multiple
solutions, at least, the bifurcation equation g
should map to the normal form for the pitchfork
bifurcation h=±x3±λx.
Several authors have taken the viscosity co-
efficient as a function of the pressure gradient,
but usually it is treated as a constant. In [12]
the viscosity was considered to be the sum ofneoclassical and anomalous or turbulent contri-
butions, both with separate power-law depen-
dences on the pressure gradient. We shall adopt
this bipartite form and in Eq. 4 take
µ=µ(P) =µneoPn+µanPm. (11)
Equations 8, 9 and 4 with (11) comprise the BD
model.
The values of the exponents nandmare not
precisely known empirically or from theory. In
[26]µanis given as having a P3/2dependence,
but is also subject to the additional influence of
aP-dependent curvature factor. In this work we
taken=−3/2 as in [12] and m= 5/2.
IV. BIFURCATION STRUCTURE OF
THE BD MODEL
The bifurcation problem obtained from the
BD model contains two codimension 2 organiz-
ing centers:
1. The defining conditions for the pitchfork,
g=gF=gFF=gq= 0, gFFF∝negationslash= 0, gFq∝negationslash= 0,
(12)
find this singularity occurring at ( F,q,β,ϕ )℘=
(0,8µ1/8
anµ7/8
neoγ/(77/8α),(73/8αγ)/(8µ5/8
anµ3/8
neo),0),
gFFF =−12(77/8)µ7/8
anµ1/8
neoγ/α,gFq=
2(7µan/µneo)1/4. The pitchfork ℘becomes a
transcritical bifurcation Tlaway from the criti-
cal value of β.
2. Another transcritical bifurcation Tuoccurs
at ( F,q,ϕ )Tu = (0 , P2γ/β,0),
gFF=−2Pγ3(7P5/2αγ−8µneoβ)/(P7/2αβγ2),
detd2g=−(−3P5/2αγ+ 8µneoβ)2/(4P7α2γ2).
TlandTuare annihilated at a second codi-
mension 2 bifurcation. The defining conditions
for this point are
g=gF=gq= det d2g= 0, gFF∝negationslash= 0, gFq∝negationslash= 0
(13)
At this point we find
(F,q,β,ϕ ) =
4(0,(8(71/8)µneoγ)/(3α(µneo/µan)1/8),
3(µneo/µan)5/8αγ/(8(75/8µneo)),0),
gFF =−64(7µan)5/8µ3/8
neoγ/(3α),gFq=
4(7µan/µneo)1/4.
In Fig 3a the partially perturbed bifurcation
diagram is plotted, showing the lower and up-
per transcritical bifurcations TlandTu. In Fig.
3b the fully perturbed, physical bifurcation dia-
gram is plotted, where ϕ >0. There are also two
Hopf bifurcations on the upper H-mode branch
in Fig. 3 linked by a branch of stable limit cycles.
The dotted lines mark the maximum and mini-
mum amplitude trace of the limit-cycle branch.
This reflects the passage through an oscillatory
r´ egime that is often observed in experiments.
Since it is a codimension 2 bifurcation prob-
lem, the qualitative structure is fully defined by
qand two auxiliary parameters. One of these
is obviously ϕ, the other may be any one of the
other parameters. We choose βbecause we are
interested in the effects of poor turbulence dis-
sipation (i.e. low β). Figure 4 illustrates how a
jump can occur directly to oscillatory states, a
phenomenon which is frequently observed.
Figure 5, to be compared with Fig. 3b, shows
that the BD model does indeed reflect shear flow
suppression of turbulence.
V. DISCUSSION AND CONCLUSIONS
A dynamical model that emulates much of
the typical behavior around L–H transitions has
been elicited from an earlier fragile model that
had serious flaws by considering the relationship
between bifurcation structure and the physics of
the process. Built in to this model are the follow-
ing major dynamical features of L–H transitions:
1. Discontinuous, hysteretic transitions, or
smooth changes with power input, depending on
the degree of turbulence dissipation β, or equiv-
alently, the viscosity.
2. Two Hopf bifurcations in H-mode. It is the
first model that can emulate the onset andabate-
ment of oscillatory behavior, and a transition di-rectly into oscillatory H-mode.
3. Turbulence suppression by the shear flow.
4. A maximum in the shear flow generated by
the turbulence, followed by a decrease as the
power input flowing to the turbulence is raised.
5. Turbulence generation from non-ambipolar
losses.
Finally, we note that the existence of two
codimension 2 bifurcations is suggestive: Should
there be an expansion of the system, perhaps ex-
pressing fluctuations of the magnetic field, that
creates (or annihilates) the two bifurcations at a
codimension 3 singularity? In other words, does
a more complete model contain an organizing
center of higher order? In singularity theory we
persevere in seeking higher order behavior: that
is how the relationship between a model and the
process it represents is tracked. This question is
currently under investigation.
5REFERENCES
∗Codimension is defined as the minimum
number of independent auxiliary parame-
ters required for a universal unfolding of the
most degenerate singularity, or organizing
center, in the system. A universal unfolding
captures all possible qualitative bifurcation
states of a given bifurcation problem.
†To avoid confusion in later discussion some
minor changes to the original notation have
been made. Also, for convenience a factor
of 2 is absorbed into the coefficients, that
arises because the dynamical variables are
squared quantities, implying that the phase
space is restricted to the first quadrant.
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Acknowledgment: This work is supported
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Fellowship.
6FIGURES
01
0 1 2F
γTa.
01
0 1 2F
γTsb.
01
0 1 2F
γTsc.
012
0 1 2F
γTsd.
FIG. 1. Bifurcation diagrams of the DLCT model showing how a p erturbation of the shear
flow unfolds Tbut introduces Ts.α= 1,β= 0.77,µ= 1. a. ϕ= 0, b. ϕ= 0.05, c. ϕ= 0.5,
d.ϕ= 1. The region F <0 is not within the phase space but is included to make the natu re ofT
clearer.
700.20.40.60.8
0 0.5 1 1.5 2N
γTs
FIG. 2. Same as Fig. 1(c) except with Nas the state variable. The region N < 0 is not
within the phase space but is included to make the nature of TSclearer.
8012
0.1 1 10 100a.
F
qTl Tu
012
0.1 1 10 100b.
F
q
FIG. 3. Bifurcation diagrams BD model. α= 2.4,β= 1, γ= 1, ε= 1, µneo= 1,
µan= 0.05,n=−1.5,m= 2.5. a. Bifurcation structure of the partially perturbed syst em,
withϕ= 0. b. ϕ= 0.05. For clarity the lower unstable branches are not plotted i n (b).
9012
0 0.5 1 1.5F
q
FIG. 4. Under extreme conditions, where the turbulence diss ipation rate is low relative to the
generation rate, the jump at the lower limit point can occur d irectly to an oscillatory state on the
H-mode branch. β= 0.1, other parameters as for Fig. 3.
100.10.5151015
0.1 1 10 100N
q
FIG. 5. Same as Fig. 3, except with Nas state variable.
11 |
Hot Points in Astrophysics
JINR, Dubna, Russia, August 22-26, 2000
HOT POINTS OF THE WAVE UNIVERSE CONCEPT:
NEW WORLD OF MEGAQUANTIZATION
Chechelnitsky A.M.
Joint Institute for Nuclear Research, Laboratory of Theoretical Physics, 141980 Dubna, Russia
ABSTRACT
We present the brief review of some central problems, anomalies, paradoxes of Astrophysics,
Cosmology and new approaches to its decisions, proposed by the Wave Universe Concept (WU
Concept) (see monography - Chechelnitsky /G3A. /G46. Extremum, Stability and Resonance in
Astrodynamics ..., etc. and consequent publications).
All these, anyhow, - Hot Points of modern science of Universe, about which Standard Model have
not authentic answers. The represented set of brief (justified) Novels reflects very perspective
directions of search.
Essential from thems – Headline (and Topics):
# New Phenomenon – Megaquantization: Observed Megaquantum Effects in Astronimical
Systems.
# Internal Structure of Celestial Bodies and Sun. Where is disposes the Convective Zone?
# Towards to Stars: Where the Heliopause will be found?
# Mystery of the Fine Structure Constant (FSC). Answer of the WU Concept: Theoretical
Representation of the Fine Structure Constant. FSC – As Micro and Mega Parameter of the Universe.
FSC, Orbits, Heliopause.
# What Quasars with Record Redshifts will be discovered in Future?
# Observed Universe: Monotonic Homogeneity and Anisotropy or Principal Hierarchy? Is the
(Large – Scale) Limit of Universe exist?
All answers, proposed by the Wave Universe Concept, in contrast with speculative schemes of
Standard Model, are effectively verified by experience, including, - and by justified in observations
prognoses.
It will be waited, that namely here it is possible the real break in the understanding of these
principal problems and enigmas, which Nature, infinite Universe offers (to us).
MODERN COSMOLOGY: HOT POINTS – CHALLENGE WITHOUT ANSWER.
The manifold, unexpected, astonishing information about the structure of the Universe, represented
by observations from Earth and Space, now is the real challenge to human Mind, to ability of the
modern theory (in particular, of the Standard Model of Cosmology) adequately to understand the
surrounding us World.
That challenge of observations and experiments, in many cases, remains without satisfactory
Answer.
For the noneffective theory, which can not make the justified predictions (evidently, namely these
are very essential attributes of the true science), every new significant phenomenon of observations
and experiments is a suddenness, surprise, reason for astonishment. But obstinantly noninterpreting
phenomena usually manifests a natural tendency to convert in anomalies, paradoxes, becomes Hot
Points of science.
CHALLENGE OF PROBLEMS.
[But to Many Even Existing Answers there are not Yet Questions
in the Modern Representations (in Standard Model)].
# Nothing or Something? Emptiness or Medium in Astrodynamics and Celestial Mechanics?
# In Search of Initial Principles.
What are Fundamental Laws, determining all variety of observed Micro and Mega Systems?
# What is commonness between Atom, Solar system, Galaxy?
Fundamental Isomorphism. Is it possible Unity and Universality of the dynamical description of
the observed Hierarchy of Nature objects (systems)?
# Why Quantization? Hidden (Latent) sense of Quantization.
Quants of Microworld. Quantization “in the Small”. Dynamical Genesis. Planck Constant /G84.
Quants of Megaworld. Megaquantization. Quantization “in the Large”. Megaquantum Effects.
Dynamical Genesis. Constant d− (of Sectorial Velocity). Quantizations – is Pattern of only Microworld
or of Megaworld and All Universe?
# Rhythms of Universe. Universal Phenomenon of periodicity. Why? Whence? From where?
Dynamical Genesis.
# Evolution. Whither is fly the “Arrow of Time”?
Direction of Evolution.
# Problems, Anomalies, Paradoxes of Modern Astrophysics, Cosmology (of Standard Model).
∗ Variable “Constant” of Hubble.
∗ Paradox of Missing Mass: Dark Matter – Phenomenon of Reality or only of the Theory.
∗ True of Microwave Background Radiation.
∗ Paradox of observed superlight velocities.
∗ Problem of the Redshift Quantization of far Qusars.
# What does the Universe look like on the very largest scales? How much Matter does it contain?
What are the biggest Things that exist?
Etc, Etc…
# History of Universe and Man: Problems of Genesis.
Genesis (Origin) – by Myths, by Bible (by Moses), by Vedas;
Genesis – by Modern Science: Big Bang, Birth – Ex Nihilo.
# Earth and Universe – Problems and Paradoxies of Age.
Etc, Etc…
The List of these Problems, Anomalies, Paradoxes – is nonterminating (Endless).
NEW PHENOMENON – MEGAQUANTIZATION:
Observed Megaquantum Effects in Astronimical Systems.
Wave Universe. Wave Astrodynamics.
The principle possibility of dynamical synthesis of the classical celestial mechanics and
astrophysics continual aspects, follows from the Wave Universe concept and wave concept of the
astrodynamics - the Megaquantum wave astrodynamics (Chechelnitsky, 1980 -1997) (see Fig.1).
Megawaves. One of the basic ideas of the Wave Universe concept is the assertion of the
existence of some waves in any megasystem (astronomical system) of Universe, in particular, in the
Solar system. These waves actual realise short-range interactions in the scale, compared with scale of
the system.
The Suggestion. Let us study the Solar system, - as the wave dynamic system (WDS)of
megaworld. There are megawaves (large astronomical scale waves with large lengths and periods),
which induce, propagate, and absorbe due to the cosmic medium. These megawaves are responsible
for the dynamic structure (and geometry) of the Solar system (the co - dimension principle
(Chechelnitsky, 1980)).
Micro-Mega Analogy.
When quan tum mechanics was developing, the main role in the comprehension of the microobject
dynamic structure belonged to the analogy of the structure between the atom and planetary system.
Apparently, the time has come to return debts. And at present this dynamic analogy works vise
versa: now astronomical (in particular, planetary) systems are studied in analogy - with atom systems
(Fig. 1).
Besides the investigation of Fundamental wave equations (Chechelnitsky, 1980, 1986), some
quantitive representations on megawaves properties (if these waves are considered as some analog
of the De Broglie waves for megaworld astronomical systems) can be received, in particular, from the
following relations of the Megaquantum wave astrodynamics:
v = d− ⋅k ε = d− ⋅Ω Δx⋅Δv ≥ (1/2)⋅d−
Here k - wave number, v - velocities, Ω - circular frequency,
ε - normalized (divided by m -mass) energy of megawaves (ε = /G3F/ /G46=(1/2)⋅v2),
d− = d/2π - Fundamental constant of normalized action (sectorial velocity, circulation) with the
dimension [cm2⋅s-1].
The quantization constant d− = d/2π in the microworld for the atom is determined by Planck's
constant /G84=h/2π and electron mass me
d− ∼ /G84/me = 1.158 cm2⋅s-1
The mentioned relations above represent the megaquantum analogs of the Bohr, Planck-Einstein,
Heisenberg relations well-known in the microworld quantum mechanics, accordingly. It should be
emphasized that, at least, in the frame of the Wave astrodynamics, the formal analog of Heisenberg
incertainty relation (Δx⋅Δv ≥ (1/2)d− - difractional incertainty relation) does not have such a wide
prohibitive sense, as in the Copenhagen interpretation of microworld quantum mechanics (Born, 1963,
Jammer, 1985).
Megaquantum World of the Solar System Distance Quantization.
Radial Quantization. The high-precision information on the geometry (in particular, on semi-major
axes) of planetary orbits, gives the possibility to receive some single-valued, intriguing facts. They
were found in the 70-ies using the quantum of (linear size) distance a∗ = a∗[1] = 8R/G7E (R/G7E - radius of
Sun) and caused the astonishment surprise showing a great set of integer numbers, which
characterized the dynamical structure and geo metry of the Solar system (Fig. 2).
It concerns differences of Δa∧ - normalized (divided by a∗) planetary distances a∧ = a/a∗ for orbits
ai, aj
Δa∧ i,j=a∧ i - a∧ j, (i, j = 1,2,3....),
as however, in many cases, and on the normalized distances a∧ i = ai /a∗ themselves.
The integerality phenomenon
Δa∧ = Integer [Semi-Integer],
as and the a∧ = Integer [Semi-integer],
has a deep ph ysical basis. Such a nonaccidental abundance of integer (semi-integer) numbers (in the
absence of anything ad hoc fitting parameters) shows the conceal, early unknown, but really existing
phenomenon of the Solar system wegawave structure (Chechelnitsky, 1984; 1992 b, c, Table 1) (see
Fig. 3).
Azimuthal Quantization. The effects of megaquantization (quantization "in the Large") - the
megaquantum effects, are not less interesting. They were discovered for pi = 2πai perimeters of
planetary orbits, normalized by a∗ quantum, i.e. for
p∧i = pi / a∗ = 2πai / a∗ = 2πa∧ i ,
Δp∧ i,j = p∧i - p∧j
Azimuthal Quantization p ∧, Δp∧ = Integer [Semi-Integer]
is a very characteristic pattern of the physically distinguished orbits and it occurs rather often
(Chechelnitsky, 1992 b,c, Table 1) (see Fig.3A).
Sectorial Velocity Quantization. In this case we speak about the observed effect of the
discreteness - the quantization of the dynamical value
LN = (K⋅a)1/2 = LN=1=1⋅N
of the sectorial velocity, normalized by
LN=1 = L∗/(2π)1/2, where L∗ = (K⋅a∗ )1/2.
Taking into account the interpretation of N quantum number, as N = LN/LN=1, we can talk about
integer (semiinteger) ability of the N quantum number.
For planetary orbits of Mercury (ME), Venus (V), Earch (E), Mars (MA), we have, in particular,
N=(2πa/a∗)1/2
N = 7.911; 11.050; 12.99; 15.969, close to integer
N = 8; 11; 13; 16, accordingly (Fig. 2) (see also Fig. 3).
At the definite conceptual supposition this effect may be connected with the well-known
quantization effect of kinetical momentum (K(m) = mva, where m - mass) or action.
In this aspect it has been known since the times Planck, Einstein, Bohr, De Broglie. Although, we
should stress, that in the quantum mechanics (of microworld) the quantization namely of the kinetical
momentum K(m) = mva was always discussed, but not of the sectorial velocity (or circulation) L =
K(m)/m= va.
Since Kepler's times (his second law) the notion and dynamical value of the sectorial velocity has
taken the importance place in the astrodynamics, in space sciences.
INTERNAL STRUCTURE OF CELESTIAL BODIES AND SUN.
Where is disposes the Convective Zone?
Results from Helioseismology, periodicities spectrum of the Sun and stars - can be adequa tely
understand and e ffectively interpreted only in the context of unified theoretical representations. That
possibility is permitted the Wave Universe concept (see monograph - Chechelnitsky A.M. Extremum,
Stability, Resonance in Astrodynamics ..., etc. and consequent publications), which comprehand the
Solar and any astronomical systems as the principle Wave dynamic systems (WDS). Investigations
with using of the theoretically calculated Fundamental Wave spectrum of (periods) those objects, its
Megaspectroscopy (Fig. 4) (including the Helioseismology - for the Sun, and Asteroseismology - for
the stars) lead to reconstruction of dynamics, physics, geometry of the Sun and stars (its internal and
external structure) (Fig. 5).
The Hierarchy set of TR∗[s] (s = …, -2, -1, 0, 1, 2, …) Transspheres of the Solar system with semi-
major axes
a∗[s] = χ2(s-1)a∗[1] = χ2(s-1)8R/G7E, s = …, -2, -1, 0, 1, 2, …,
it is clear, also contain the value of critical surface - TR∗[0] Transsphere
a∗[0] = χ-28R/G7E = 0.595R/G7E,
which lies inside the Sun and coincide with theoretical (geometrical) beginning of the Convective Zone
(with the bounda ry of External Nucleus) of the Sun:
Internal (Smallest) Nucleus of Sun … , a∗[-1] = χ-48R/G7E = 0.0442R/G7E,
External Nucleus of Sun (Beginning of Convective Zone) a∗[0] = χ-28R/G7E = 0.595R/G7E,
TR∗[1] Transsphere (for I Earth Group of Planets) a∗[1] = 8R/G7E = 0.0372193 AU,
TR∗[2] Transsphere (for II Jupiter Group of Planets) a∗[2] = χ28R/G7E = 0.500 AU,
TR∗[3] Transsphere (for Trans-Pluto Group of Objects) a∗[3] = χ48R/G7E = 6.727 AU,
TR∗[4] Transsphere – Heliopause a∗[4] = χ68R/G7E = 90.447 AU,
TR∗[5] Transsphere a ∗[5] = χ88R/G7E = 1216.016 AU,… ,
χ - Fundamental parameter of hierarchy - Chechelnitsky Number χ = 3.66(6).
TOWARDS TO STARS: WHERE THE HELIOPAUSE WILL BE FOUND?
Where the Solar System Ends?
Of course, such a problem is specially attracting and interesting, when spacecrafts Pioneer 10, 11
and Voyager 1,2 still did't achieve the intriguing, significant barrier of the Solar system - Heliopause
[Abstracts COSPAR, 1994; Belcher et al., 1993; Masek, 1996].
Some astrophysicists are strive to consider the Heliopause as objectively detectable external
boundary of the Solar system. But astronomers - observers will hardly agree with this conclusion.
Resulting from these and other, more fundamental ideas, it will be reasonable to suppose, that the
region beyond Heliopause also has very nontrivial properties. Just from the fact that for a long time it
has been considered as inexhaustable reservoure of comet bodies.
Near the Stars. Trans - Pluto Space. Heliopause Region
The prognosis, based upon the conceptions, besides the existence regions - G[1], that is occupied
by the space of I (Earth) group planets, and G[2], that is occupied by the space of II (Jupiter) group,
predicts the existens of at least, Trans - Pluto G[3] Shell and, probably, the following it G[4] Shell
[Chechelnitsky, 1992 a, b, c] (Fig. 6).
G[3] Shell. The dominant orbits system in this Trans-Pluto Shell with using of the (linear size)
distance quantum
a*[3] = K/G7E/( C*[3])2 = 6.7276 AU
in the main approach are as follow (a[s]=a*[s]N2/2π, K/G7E - gravitational parameter of Sun):
a[3] = 68.5(70); 90.44; 129.5; 181; 257; 274; 407(403); 495(500); 542(530) AU
G[4] Shell. In this far region of the Solar system with the usage of the (linear size) distance quantum
a*[4] = K/G7E/(C*[4]) 2 = 90.447 AU
the geometry of dominant orbits (semi-major axes values) seem as following
a[4] = 921; 1216; 1742; 2433; 3458; 3680; 5474; 6654; 7287 AU
Heliosphere Boundaries. Heliopause
According to the ideas of modern astrophysics, the interaction of the Sun with interstellar medium,
surrounding it, results in formation of the Solar system Heliosphere (that in some sence reminds the
well investigated Earth magnetosphere). Depending on the chosed parameters, it is believed that the
bow shock, after which the Heliosphere begins, can be located at the distance 30 - 50 AU or 75 - 200
AU.
There are reasons to expect, that the bow shock (and inside it - the Heliopause) may be discovered
at the heliocentric distance [Chechelnitsky, 1992 a,b,c]
a* = a*[4] = K/G7E/(C*[4]) 2 = 90.447 AU,
connected with the Solar system G[4] Shell (Fig. 6).
Trans - Pluto Celestial Bodies
The potentially possible Trans - Pluto celestial bodies most probably can be discovered on these
dominant orbits.
The experience of space researches of the Solar system and satellite systems of planets (not
depending on the succeses in far celestial bodies continuating search) show, that dominant orbits are
physically distinguished states in many other, fixed by observations, aspects (for instance, - in
measurements of energetic proton count intensity, etc.) [Chechelnitsky, 1992 a].
MYSTERY OF THE FINE STRUCTURE CONSTANT (FSC).
Microworld: Quantum Wave Mechanics and Fine Structure Constant
From all modern theories of microworld - quantum electrodynamics (QED) describes the dynamic
structure and the interaction of elementary particles (photons, electrons, muons) most exactly.
There is the fundamental parameter (coupling constant, interaction parameter), that lies in the basis
of that advanced and consistent theory - the Fine Structure Constant (FSC), [Born, 1963].
The theoretical representation of this constant is unknown up till now. "The Mysterious Number 137"
- so titled Max Born the famous paper of 1936 [Born, 1936].
The Fine Structure Constant (FSC) α = 2πe2/hc or nondimensional number α-1 ≈137 (where e -
electron charge, h - Planck constant, c - speed of the light) was introduced in the theoretical physics by
Arnold Sommerfeld in 1915 [Sommerfeld, 1973]. That is the fundamental parameter of the all atomic
spectroscopy. At present, only its experimental value is known (α-1 = 137.036).
Answer of the WU Concept:
Theoretical Representation of the Fine Structure Constant.
In the framework of Wave Universe concept may be naturally obtained the following surpriselly
simple analytical and numerical (closed) representation for the Fine Structure Constnt that is proved to
be correct by the logics of the consistent theory [Chechelnitsky, (1986) 1996]
α-1 = 239/4/2π = 137.0448088
FSC – As Micro and Mega Parameter of the Universe.
FSC, Orbits, Heliopause.
Fine Structure Constant is fundamental constant not only of microworld (atoms), but also – of
megaworld (astronomical systems) – one of the general nondimensional parameter of Universe.
Megaworld: Megaquantum Wave Astrodynamics and Astrophysics;
Earth Orbit and Heliopause
We shall cite only one fragment of the new knowledge [Chechelnitsky, 1996], that is spontaneously
connected with discussed theme - with wave structure, geometry and dynamics of Solar system.
The Assertion
∗ There is regular connection between planetary orbits arrangement and special critical surface of
Solar system - Heliopause location.
∗ This connection may be presented by using the Fine - Structure Constant, that is considered as
megaparameter of astronomical systems.
∗ In particular, when using the Earth orbit, the following most simple relation between the Keplerian
periods of Earth orbit TE =1a and of Heliopause T* is valid:
T* ≈ SαTE ≈ 861a, Sα = 2π/α ≈ 2π⋅137 ≈ 861
The appea ring from the above relation between semi-major axis of Earth orbit aE = 1 AU and of
Heliopause a* is like this:
a* ≈ Sα2/3 ⋅aE = 90.5 AU
These relations reflect the presence of spontaneous and close connection between Wave
astrodynamics (celestial mechanics) - geometry and dynamics of regular set of elite (dominante) Solar
system planetary orbits - and geometry and dynamics of Heliopause (of Solar system magnetosphere,
or of standing shock wave of Heliosphere), that is traditionally regarded as an object of astrophysics.
WHAT QUASARS WITH RECORD REDSHIFTS WILL BE DISCOVERED IN FUTURE?
Megaquantization in the Universe.
It is clear, Megaquantization (quantization “in the Large”), observed megaquantum effects are not
monopolic privelege of only Solar system.
Let us point the brief resume of research (prognosis), connected with problem of redshift
quantization of far objects of Universe – quasars (QSO) [Chechelnitsky, (1986) 1977]:
“Abstract: In the framework of the Wave Universe concept it is shown that the genesis of redshifts
can be connected with the intra-system (endogenou s) processes which take place in astronomical
systems. The existence of extremal redshift objects (quasars – QSO) with most probable
z = 3.513 (3.847); 4.677; 6.947 (7.4); 10.524; 14.7; 27.79; …
is predicted.”
Prognosis already had justified successively for extremal values of z redshifts
ztheory = 3.513, zobs = 3.53 (quasar OQ172)
ztheory = (3.847), zobs = 3.78 (quasar PKS2000-330)
ztheory = 4.677, zobs = 4.71 (Schmidt, Gunn, Schnaider, 1989)
zobs = 4.694 (4.672) (quasar BR1202-0725, Wampler et al., 1996)
At the present time, apparently, also the object Q2203+29 G73 with record value z of redshift
z=6.97 is discovered in special Astrophysical Observatory (SAO, Russia)
ztheory = 6.947, zobs = 6.97
(Q2203+29 G73, Dodonov et al., 2000).
The Quene – for objects with even more high redshifts z = 10.524; 14.7; …
Consequences of such successfully realizable prognosis, imperatives of observations not only are
unexpected for the Standard cosmology, but also, probably, its can stimulated the radical
reconsideration of many habitual representations, having become as freezen dogmas.
OBSERVED UNIVERSE:
MONOTONIC HOMOGENEITY AND ANISOTROPY OR PRINCIPAL HIERARCHY?
Is the (Large – Scale) Limit of Universe exist?
Invariablly justified representations of the Wave Universe Concept - WU Concept indicate a
principle incorrectness of expectations of Standard (Model) cosmology about homogeneity and
isotropy of the Universe.
It also is connected with observational data about apparent hierarchy of giant astronomical systems
(stellar systems, galaxies, clusters of galaxies, superclusters of galaxies, etc.), their megawave
structure, Megaquantization (quantization "in the Large"), non-homogeneity of microwave background
Space radiation, adequately interpreted (in frameworks of WU Concept) effects redshifts quantization
of quasars, etc.
The principle absence of a Limit of Hierarchy of Matter Levels asserts: "The Staircase of a Matter" -
is endless.
For orientation of the explorers, working with the observational data, in frameworks of WU Concept
the concrete characteristics of following (behind superclusters of galaxies) potentially possible
extremely large astronomical systems are calculated with using the Fundamental parameter of
Hierarchy – Chechelnitsky Number χ = 3.66(6).
The astronomical systems, belonging to the nearest hierarchy Levels of Solar-Like systems, are
characterized by external radiuses [a(k) = χk a(0), a(0) = 39.373 AU]
a(20) = 36.83, a(21) = 135, a(22) = 495, a(23) = 1815 Mpc.
It may be expected that in the Universe also exist and should show itself in observations (the Solar-
Like objects) – extremely large astronomical systems (ELAS), characterized by the external radiuses
(of peripherals)
a(26) = 89503, a(27) = 328177, a(28) = 1203318 Mpc.
FROM PARADOXES AND ANOMALIES – TO THE GREAT SYNTHESIS.
Foreseeing Future.
Many factors, - including megaquantum effects, manifestation of the FSC as megaparameter in the
science of giant astronomical systems – (Mega) Wave astrodynamics, in our opinion, are symptoms,
that in front of sciences about Universe new, (for many, possibly) surprisely representatives are
opening:
Sciences about Universe are found at the Way to the Great Synthesis [Chechelnitsky, 1996-1997],
in particular, of Substance and Medium.
REFERENCES
Abstracts 30th COSPAR Scientific Assembly, Hamburg, Germany, 11-21 July (1994).
Belcher J.W., Lazarus,A.J., McNutt, R.L. Jr.,and Gordon,G.S.Jr. Solar Wind Conditions in the Outer
Heliosphere and the Distance to the Termination Shock, J.Geophhys.Res.98, 15177-15183, (1993).
Born M., The Mysterious Number 137, Uspekhi Fis. Nauk, 16, N6, p. 697, (1936) (in Russian).
Born M., Atomic Physics, Blackie and San Limited, London - Glasgow, (1963).
Chechelnitsky A.M., Extremum, Stability, Resonance in Astrodynamics and Cosmonautics, M.,
Mashinostroyenie, 312 pp. (1980) (Monograph in Russian); (Library of Congress Control Number:
97121007 ; Name: Chechelnitskii A.M.).
Chechelnitsky A.M., Megaquantum Structure of the Halley Comet Orbit, Cometary Circular N317,
KGU - GAO USSR Academy of Science, pp. 2-3, (30 Dec. 1983b).
Chechelnitsky A.M., On the Quantization of the Solar System and Its State Structure of
Megawaves, Astronomical Circular of the AN SSSR, N1334, pp.1-4, (1984).
Chechelnitsky A.M. Astronomical Circular, NN 1410, 1411; (1985a).
Chechelnitsky A.M., The Shell Structure of Astronomical Systems. Astronomical Circular of the
USSR Academy of Science, N1410, pp.3-7, N1411, pp.3-7, (1985b).
Chechelnitsky A.M., Wave Structure, Quantization, Megaspectroscopy of the Solar System; In the
book: Spacecraft Dynamics and Space Research, M., Mashinostroyenie, pp.56-76 , (1986a) (in
Russian).
Chechelnitsky A.M., Wave Universe and the Possibility of Existence of Extremal Redshift Quasars.
(The original date of the promulgation - 30 November 1986), (1986b) (in Russian).
Chechelnitsky A.M., Uranus System, Solar System and Wave Astrodynamics; Prognosis of Theory
and Voyager - 2 Observations, Doklady AN SSSR, v. 303, N5, pp.1082-1088, (1988).
Chechelnitsky A.M., Neptune - Unexpected and Predicted: Prognosis of Theory and Voyager-2
Observations, Report (IAF-92-0009) to the World Space Congress, Washington, DC, (Aug.22-Sept.5),
AIAA Preprint (1992 a) .
Chechelnitsky A.M., Wave Structure of the Solar System, Report to the World Space Congress,
Washington, DC, (Aug.22-Sept.5), (1992 b).
Chechelnitsky A.M., Wave Structure of the Solar System, Tandem - Press, (1992c) (Monograph in
Russian).
Chechelnitsky A.M., Magnetospheres and Heliosphere - As Phenomena of Wave Astrodynamics,
Report to 30 COSPAR Assembly, Hamburg, 11-21 July (1994b)
Chechelnitsky A.M. Wave World of Universe and Life: Space-Time and Wave Dynamics of
Rhythms, Fields, Structure, Report to the XV Int.Congress of Biomathematics, Paris, September 7-9,
1995; Bio-Math (Bio - Mathematique & Bio - Theorique), Tome XXXIV, N134, pp.12-48, (1996).
Chechelnitsky A.M., Mystery of the Fine Structure Constant: Universal Constant of Micro and
Megaworld, Wave Genesis, Theoretical Representation, In the book: Proceedings of International
Conference "Systems Analysis on the Threshold of XXI Centure: Theory and Practice", Intellect
Publishing Hause, Moscow, (1996 - 1997).
Chechelnitsky A.M., On the Way to Great Synthesis of XXI Century: Wave Universe Concept, Solar
System, Rhythms Genesis, Quantization ″In the Large″, pp. 10-27; In book; Proceedings of
International Conference ″Systems Analysis on the Threshold of XXI Century: Theory and Practice″,
Moscow, 27-29 February 1996, v. 3, Intellect Publishing Hause, Moscow, (1997).
Chechelnitsky A.M., Wave Universe and Spectrum of Quasars Redshifts, Preprint E2-97-259, Lab.
Theor. Physics, Joint Institute for Nuclear Research, (1997).
Chechelnitsky A.M., Wave Astrodynamics Concept and It Consequences, In book: Search of
Mathematical Laws of Universe: Physical Ideas, Approaches, Concepts, Selected Proceedings of II
Siberian Conference on Mathematical Problem of Complex Systems Space–Time (PST – 98),
Novosibirsk, 19-21 June 1998 , Publishing House of Novosibirsk Mathematical Institute, p.74-91,
(1999) (In Russian)
Chechelnitsky A.M., Large - Scale Homogeneity or Principle Hierarchy of the Universe? Report
E1.5 – H0 – 0018 to COSPAR Scientific Assembly, Warsaw, 16-23 July 2000
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Report E2.3 – 0023 to COSPAR Scientific Assembly, Warsaw, 16-23 July 2000
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Quantization “in the Large”, Report H0.1 – 0039 to COSPAR Scientific Assembly, Warsaw, 16-23 July
2000
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Abundances at Redshifts Above z = 4, Astronomy and Astrophysics, v.316, p.33-42, (1996).
arXiv:physics/0102036v1 [physics.gen-ph] 13 Feb 2001? ?
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S
MICRO - MEGA ANALOGY
∇2Ψ +2
¯d2[E- U ]Ψ = 0FUNDAMENTAL WAVE EQUATIONASTRONOMICAL SYSTEM
SOLAR SYSTEMMEGASYSTEM
ATOMQUANTUM SYSTEMMICROSYSTEM
¯d- FUNDAMENTAL QUANTIZATION
CONSTANT [ cm2s−1] (¯d= d/2 π)
U=−K/a– Potential
K – Dynamical Parameter [ cm3s−2]
a( =r ) – Distance [ cm ]
E ∼Normalized Energy
(E ∼v2
2[ cm2s−2] )
CONSEQUENCES
OF THE
FUNDAMENTAL WAVE EQUATION¯d=¯de=¯h
me=1.158 cm2s−1
U=V
me, V=−e2
a
– Electric Potential
K = K e=e2
me
E=E
me, E – Energy
e – Electric Charge
¯h= Planck’s Constant
me– Electron Mass¯d∼1019cm2s−1=109km2s−1
U =−K⊙/a
K = K ⊙= 1.327 ·1011km3s−2
Gravitational
Parameter of the Sun
Relations of Wave
Astrodynamics
v=¯dK
E=¯dΩ
∆x∆v >1
2¯d
K – Wave Number
Ω – FrequencySCHR ˝ODINGER’S EQUATION
∇2Ψ +2me
¯h2[E - V]Ψ = 0
L =va= (Ka)1/2, LN=1– Constant
¯d∼LN=1(¯d=ξLN=1,ξ= Constant)
N=L/L N=1– Normalized Sectorial
Velocity – Quantum NumberL=L N=1N, N - IntegerVelocity ( Circulation ) L [ cm2s−1]Quantization of the SectorialRelations of Quantum
Mechanics
DE BROGLIE: P = ¯ hk
PLANCK-EINSTEIN: E=¯ hω
HEIZENBERG: ∆x∆p >1
2¯h
p=mv, k - Wave Number
ω- Frequency
BOHR’S STATE ORBITS ELITE ORBITS
PLANETARY
( DOMINANT )
ORBITS
Figure 1.arXiv:physics/0102036v1 [physics.gen-ph] 13 Feb 2001-
- - -
- - - -TRANSSPHERE
*MERCURY
0.387097676*
0.723335194VENUS
*
1112.9931.000007872
26.867992EARTH
*
16.038
1614.071
141.523749457
40.939766MARS SUN
*
10.400455 19.43441a[ AU ]
/hatwidea=a/a∗0.03721930ME V E MA TR∗ Sa∗=8R ⊙
9.0339 7.433 ∆/hatwidea=/hatwideai+1-/hatwideai
9 ∼7.5 ∆/hatwidea– INTEGER
8.084 11.050 N = ( 2 πa/a ∗)1/2
8 13INITIAL DATA
* The Astronomical Ephemeris DE19 JPL;
a-Semi-Major Axes of Planetary Orbits.
* The radius of the Sun
R⊙= 695992 km ( ≈696000 km )
*a=a∗= 8R ⊙= 5567928 km = 0.0372193 AU
Semi-Major Axes of the Transsphere
N –INTEGER
∆/hatwidea– INTEGER∆/hatwidea=/hatwideai+1−/hatwideai/hatwidea=a/a∗a[ AU ]
References
* Chechelnitsky A.M. Is the Solar System Quantized?, Knowle dge - is Power, 1983, N2, p.19
* Chechelnitsky A.M. Astronomical Circular of USSR Academy of Science, 1983, N1257, pp. 5-7
* Chechelnitsky A.M. Wave Structure, Quantization, Megasp ectroscopy of the Solar System;
In the book: Spacecraft Dynamics and Space Research. M., Mas hinostroyenie, 1986, pp.56-76.116.069255.85352 139.78381JJUPITER SATURN
259.033514.8866019.163718892UURANUS
293807.885709.522688738SANEPTUNE
*5.202655382* * *NE
30.0689404
1057.882439.373641353*MEGAQUANTUM EFFECTS IN THE SOLAR SYSTEM
QUANTIZATION ”IN THE LARGE”:
QUANTIZATION OF DISTANCIES
( Linear Values )
PPLUTO
259 116 250292.999 249.996
Figure 2.Fig. 3
MEGAQUANTUM EFFECTS IN THE SOLAR (PLANETARY) SYSTEM
RADIAL, AZIMUTHAL, SECTORIAL QUANTIZATION
(** - Close to Integer, * - Close to Semi-Integer)
Shells
Planetary
Groups Planets
and
Orbits Semi-Major Axes of the
Planetary Orbits
(Astronomical Ephemeris DE19 JPL) G[1] Shell
C∗ ∗[1] = 154.3864 km⋅ ⋅s-1
a∗ ∗[1] = 0.0372 AU = 8R G[2] Shell
C∗ ∗[2] = 42.1 km⋅ ⋅s-1
a∗ ∗[2] = 0.5 AU G[3] Shell
C∗ ∗[3] = 11.4 km⋅ ⋅s-1
a∗ ∗[3] = 6.727 AU
Orbits
(Planets) In-
dex a
[AU] a
[km] a/R
R =
=695992 km Radial
Quantization
a∧ ∧=a/a∗ ∗
Δ Δa∧ ∧=ai∧ ∧-aj∧ ∧ Azimuthal
Quantization
P∧ ∧=2π πa∧ ∧
Δ ΔP∧ ∧ Sectorial
Quantization
N=(2π πa∧ ∧)1/2
Δ ΔN
a
Δ Δa∧ ∧ P∧ ∧
Δ ΔP∧ ∧ N
Δ ΔN
a
Δ Δa∧ ∧
P∧ ∧
Δ ΔP∧ ∧
N N
Δ ΔN
I
(Earth)
Planetary
Group Mercury
Venus
Earth
Mars
Ceres ME
TR
VE
EA
MA
CE 0.3871
0.5004
0.7233
1.0000
1.5237
2.7675 57908887 .
74857428 .
108209375 .
149598917 .
227949605 .
414012105 . 83.
108.
155.
215.
328.
595.
10.400
** 3.044
* 13.444
** 5.990
* 19.434
* 17.434
26.868
** 14.072
** 40.940
* 33.417
74.357 65.348
19.126
* 84.474
122.110
168.816
* 88.416
257.232
** 209.964
467.196 ** 8.084
9.192
** 11.050
** 1.943
** 12.993
** 3.045
** 16.038
* 5.577
21.615 0.774
** 1.000
* 0.446
* 1.446
* 0.552
** 1.998
** 1.047
** 3.045
* 2.486
* 5.531 4.861
6.283
** 9.083
** 12.557
19.133
34.750 2.205
* 2.507
**3.014
* 3.544
4.374
* 1.521
5.895 0.058
0.074
0.108
0.149
0.226
* 0.411 0.362
* 0.467
0.676
**0.934
* 1.423
* 2.585 0.601
0.684
0.822
**0.966
1.193
1.608
II
(Jupiter)
Planetary
Group
Jupiter
Saturn
Uranus
Neptune
Pluto JP
TR
SA
UR
NE
PL 5.2027
6.7275
9.5227
19.1637
30.0689
39.3736 778306106 .
1006412639 .
1424573841 .
2866851394 .
4498249377 .
5890212828 . 1118.
1446.
2047.
4119.
6463.
8463. 139.781
** 40.968
180.752
75.102
255.854
** 259.033
514.887
** 292.999
807.886
** 249.997
1057.883 878.288
1135.697
* 1607.575
3535.128
** 5076.096
6646.872 29.636
33.700
** 40.095
56.878
71.247
* 81.528 10.397
** 3.047
* 13.444
** 19.030
38.297
** 60.091
78.686 65.327
19.147
* 84.474
** 35.098
* 119.572
**121.058
240.630
**136.932
* 377.362
494.397 ** 8.083
9.191
** 10.935
* 15.512
* 19.431
22.235 0.773
**1.000
0.415
* 1.415
* 1.434
2.849
* 4.470
5.853 4.859
6.283
8.891
** 9.004
17.898
** 28.083
36.773 2.204
* 2.507
* 0.475
** 2.982
4.231
** 1.068
5.299
** 6.064
arXiv:physics/0102036v1 [physics.gen-ph] 13 Feb 2001Mercury Venus Earth Mars Ceres Jupiter Saturn Uranus Neptune Pluto
Series Series Series Series Series Series Series Series Series Series
(Asteroids)
58,d6809
44,505
35,567
31,713
29,470
23,43
27,840
27,633
27,548
27,d277
184,d229
97,033
69,007
59,201
55,160
52,970
52,227
51,924
50,d970
205,d013
110,335
87,234
78,735
74,346
72,891
72,303
70,d467
238,d919
151,845
127,827
116,648
113,104
111,695
107,d373
416,d647
274,911
227,932
214,784
209,759
195,d016
808,d128 (2,a212)
503,233 (1,a377)
443,317 (1,a213)
422,431 (1,a156)
366,d613 (1,a00375)
3,a651
2,688
2,423
1,a837
10,a194
7,203
3,a697
24,a548
5,a801
7,a596
Figure 4. Fundamental Wave Spectrum of the Solar System ( Fra gment )
(τ– Wave Periods [ in d-days, a-years ] )Fig. 5
INTERNAL STRUCTURE OF CELESTIAL BODIES (Fragment: Sun, Earth, Moon)
Isomorphysm. Dominant Levels (States)
Sun (R=695992 km) Earth (Rmid=6371km) Moon (R = 1738 km)
Level Dominant a∧ ∧ = G[-2] Shell G[-1] Shell G[1] Shell G[2] Shell G[2] Shell G[3] Shell
Index (Planetary)
Values =a/a*
=N2/2π π Distance
Quantum a*[-2]
= = 0.00329 R/G7F Distance
Quantum a*[-1]
= = 0.044268
R/G7F Distance Quantum
a*[1]=16.735 km Distance Quantum
a*[2] = 225 km Distance Quantum
a*[2] = 2.765 km Distance Quantum
a*[3] = 37.174 km
N Radius
a=a*[-2] a∧ Radius
a=a*[-1] a∧ Radius
a=a*[1] a∧ Depth
N=R-a Radius
a=a*[2] a∧ Depth
N=R-a Radius
a=a*[2] a∧ Depth
N=R-a Radius
a=a*[3] a∧ Depth
N=R-a
[a/R/G7E] [a/R/G7E] [km] [a/R] [km] [km] [a/R] [km] [km] [a/R] [km] [km] [a/R] [km]
ME 8.083 10.398 0.0342 0.460 174 0.0272 6197 2430 0.366 4031 29 0.0165 1709 386 0.222 1352
TR 9.191 13.444 0.0442 0.595 225 0.0352 6146 3025 0.474 3346 37 0.0213 1701 500 0.287 1238
V 11.050 19.433 0.0639 0.860 325 0.051 6046 4372 0.685 1999 54 0.0309 1684 722 0.415 1016
1
E 12.993 26.868 0.0884 1.189 450 0.0705 5921 6045 0.947 326 74 0.0427 1664 999 0.574 739
1 0
(U) 15.512 38.296 0.126 641 0.100 5730 1.344 106 0.0609 1632 1424 0.819 314
MA 16.038 40.937 0.134 685 0.107 5686 113 0.065 1625 1522 0.875 216
1 0
(NE) 19.431 60.091 0.197 1005 0.157 5366 166 0.095 1572 1738 1.277
CE 21.614 74.351 0.244 1244 0.195 5127 205 0.118 1533
(P) 22.235 78.685 0.259 1317 0.206 5054 217 0.125 1521
C∗ ∗[0]=566.08 km⋅ ⋅s-1 C∗ ∗[1] = 154.3864 km⋅ ⋅s-1 C∗ ∗[2] = 42.1 km⋅ ⋅s-1 C∗ ∗[3] = 11.4 km⋅ ⋅s-1
a∗ ∗[0] = 0.595R
/G7E/G7E a∗ ∗[1] = 8R /G7E/G7E =0.0372 AU a∗ ∗[2] = 0.5 AU a∗ ∗[3] = 6.727 AU
Transitional
Transi- Region
tional Transitional Asteroids Transitional Transitional
Region Region Belt Regon Region
Sun Transsphere Mercury Venus Earth Mars Cerus Jupiter Saturn Uranus Neptune Pluto Heliopause
∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ • • • • • ← Comets
/G7B /G7B /G7B /G7B /G7B /G7B /G7B /G7B /G7B /G7B /G7B /G7B /G7B /G7B /G7B /G7B /G7B /G7B /G7B /G7B /G7B /G7B /G7B /G7B /G7B /G7B /G7B /G7B /G7B /G7B /G7B /G7B ’
Me V E (U) MA (NE) CE (P) Me V E (U) MA (NE) CE (P) Me V E (U) MA (NE) CE (P) Me V E (U) MA (NE) CE (P) ’
TR∗ ∗[1]=TR[0] TR∗ ∗[2]=TR[1] TR∗ ∗[3]=TR [2] TR∗ ∗[4]=TR [3]
Transspheres a∗ ∗[1] =8R /G7E/G7E a∗ ∗[2]=0.5 AU a∗ ∗[3]=6.72 AU a∗ ∗[4]=90.5 AU
Quantun
Numbers 8 11 15 .5 19.43 22 .23 8 11 15 .5 19 .43 22 .23 8 11 15 .5 19 .43 22 .23 8 11 15 .5 19 .43 22 .23
9.191 13 16 2 1.6 9.191 13 16 21 .6 9.191 13 16 21 .6 9.191 13 16 21 .6
Semi-Major
axes
0.0287 0.0537 0 .106 0 .166 0 .217 0.387 0.723 1 .425 2 .23 2 .928 5.20 2 9.72 19 .1 30 39 .4 70 130 257 403 530 AU
0.0372 0.0743 0 .113 0 .205 0.5 1.0 1 .523 2 .7675 6.72 13.44 20.5 37 .2 90.5 181 275 500 AU
Semi-Major
axes
6.17 11.15 22 .7 35 .7 46 .8
8 16 24 .3 44
Fig. 6 SHELL HIERARCHY OF THE SOLAR SYSTEM
ELITE (DOMINANT) ORBITS
G[0] Shell
(Intra-Mercurian)
Dominate Sound Velocity
Distance Quanta
G[1] Shell
(Earth) G[2] Shell
(Jupiter) G[3] Shell
(Trans-Pluto)
N
a[AU]
a[R /G7E/G7E] |
arXiv:physics/0102037v1 [physics.atm-clus] 13 Feb 2001Equilibrium sizes of jellium metal clusters in the
stabilized spin-polarized jellium model
M. Payami
February 2, 2008
Center for Theoretical Physics and Mathematics, Atomic Ene rgy Organization of Iran,
P. O. Box 11365-8486, Tehran, Iran
February 2, 2008
Abstract
We have used the stabilized spin-polarized jellium model to calculate the equilib-
rium sizes of metal clusters. Our self-consistent calculat ions in the local spin-density
approximation show that for an N-electron cluster, the equilibrium is achieved for
a configuration in which the difference in the numbers of up-sp in and down-spin
electrons is zero or unity, depending on the total number of e lectrons. That is, a
configuration in which the spins are maximally compensated. This maximum spin-
compensation results in both the alternation in the average distance between the
nearest neighbor ions and the odd-even alternations in the i onization energies of al-
kali metal clusters, in a good agreement with the molecular d ynamics findings and
the experiment. These suggest a realistic and more accurate method for calculating
the properties of metal clusters in the context of jellium mo del than previous jellium
model methods.
71.15.Hx, 71.15.Mb, 71.15.Nc
11 Introduction
The simplest model used in theoretical study of the properti es of simple metal clusters
is jellium model (JM) with spherical geometry. [1, 2, 3] In th is model, the ions are
replaced by a uniform positive charge background sphere of d ensity n= 3/4πr3
sand
radius R= (zN)1/3rswhere z,Nandrsare the valence of the atom, the number of
constituent atoms of the cluster and the bulk value of the Wig ner-Seitz (WS) radius of
the metal, respectively. This model can be useful only when t he pseudopotentials of
the ions do not significantly affect the electronic structure . But, it is well-known that
the JM has some drawbacks. [4, 5] Keeping the simplicity of th e JM, the stabilized
jellium model (SJM) of Perdew et al.[6] has lifted the essential deficiencies of the JM
and significantly improved the results of calculations. In t he SJM, the bulk metal (which
is spin-unpolarized in the absence of external magnetic fiel d) has been made stabilized
through the introduction of a pseudopotential and fixing its core radius by a value that
makes the pressure on the unpolarized bulk system to vanish. However, applying the SJM
in the framework of rigid jellium background [7] may be suita ble for closed-shell clusters
of which the spin polarization of the valence electrons vani shes. It is well-known that,
for example, the bond-length of a diatomic molecule depends on the relative orientations
of the valence electrons. Hence, considering a metal cluste r as a large molecule, or the
bulk metal as a huge molecule, one should take account of the v olume change due to spin
polarization. We therefore, expect that the spherical jell ium radius should be different for
anN-atom cluster with two different spin configurations. These f acts led us to consider
a stabilized jellium model in which the spin degrees of freed om be present. Stabilizing
the jellium system with non-zero spin polarization, ζ, for the valence electrons resulted
in the stabilized spin-polarized jellium model (SSPJM).[8 ] Here ζ= (n↑−n↓)/(n↑+n↓)
andn↑,n↓are the spin-up and spin-down electron densities, respecti vely. The SSPJM
can be applied to metal clusters in two different ways. The firs t method, which has been
used in Ref.[8], exploits the fact that the bulk metal expand s asζincreases. We call that
method as SSPJM1 throughout this paper. In that method, the j ellium sphere radius is
taken as R(ζ) = (zN)1/3¯rs(ζ). For ¯ rs(ζ) we had taken ¯ rs(ζ) = ¯rs(0) + ∆ rs(ζ) in which
¯rs(0) was the observed value of the bulk WS radius of the metal an d ∆rswas obtained
by the application of the local spin-density approximation (LSDA) to the infinite electron
gas system. However, in our phenomenological accounting of the volume change, the
core radius of the pseudo-potential for the electron-ion in teraction has been considered
as a parameter, which becomes polarization dependent as we f orce the pressure of the
polarized bulk system to vanish. Using that scheme, we had ca lculated the energies of
different metal clusters, both neutral and singly ionized, f or different spin configurations
2and had shown that instead of Hund’s first rule for the ground s tate, the maximum spin-
compensation (MSC) rule was governing.[8] The MSC property which originates here from
the polarization dependence of the core radius, leads to the odd-even alternations in the
ionization energies that had been observed experimentally in the alkali metal clusters.[9]
On the other hand, if one assumes a fixed, polarization-indep endent form for the electron-
ion interaction, the MSC property will be realized only for n on-spherical geometries of
the jellium background.
We have recently shown[10] that it is not always necessary fo r a finite spherical jellium
system to increase its size as the polarization, ζ, is increased. This can be explained by
considering the fact that for an open-shell cluster if one in creases the spin polarization
from the possible minimum value consistent with the Pauli ex clusion principle, one should
make a spin-flip in the last uncomplete shell. Because of high degeneracy for the spherical
geometry, this spin-flip in the last shell does not change app reciably the kinetic energy
contribution to the total energy but changes appreciably th e exchange-correlation energy
which in turn gives rise to a deeper effective potential that m akes the Kohn-Sham (KS)
[11] orbitals more localized and therefore a smaller size fo r the cluster. On the other
hand, although the SSPJM1 results in better ionization ener gies than the SJM [7] in that
it reproduces the odd-even alternation, it always predicts incorrect cluster sizes. That
is, in the SSPJM1, the equilibrium rsfor a cluster is taken to be greater or equal to the
bulk value of rs( see Fig. 3 of Ref. [8] ), so that it approaches the bulk value f rom the
above; whereas, the molecular dynamics (MD) results for the average distance between
the nearest neighbor atoms show that the equilibrium rsvalue of the neutral clusters are
less than the bulk value and it approaches the bulk value from the below [ see Fig 15(a)
of Ref. 13]. To incorporate this correct behavior into our SS PJM calculations, which is
the subject of this paper, we proceed parallel to the work of P erdew et al.[12] for the spin-
polarized case and call this method as SSPJM2. In the SSPJM2, for a given polarization,
we first obtain the value of the core radius of the pseudopoten tial that stabilizes the bulk
system, say rB
c, and then, using this value of rB
cin the energy functional of the cluster,
we change the radius of the jellium sphere until the minimum e nergy is achieved. Our
self-consistent calculations show that the absolute minim um energy corresponds to a spin
configuration with maximum compensation as in the SSPJM1 cas e. The equilibrium rs
values corresponding to these minima lie below the bulk valu e, reproducing the correct
behavior. These equilibrium rsvalues determine the equilibrium sizes of the clusters. If
we plot the equilibrium rsvalue as a function of the number of constituent atoms in an
alkali metal cluster, we see an alternating behavior, consi stent with the MD results.[13]
In this paper we have found the equilibrium properties of neu tral and singly ionized Cs,
Na and Al clusters of various sizes (2 ≤N≤42) using jellium with sharp boundaries. We
3have also repeated the SSPJM2 calculations using a jellium s phere with diffuse boundary.
For the sake of comparison, we have derived the results of the work by Perdew et al.[12]
which is denoted by SJM1. Comparing our SSPJM2 results with t hose of SSPJM1 show
that here, the average energies per electron and the ionizat ion energies remain more or
less the same but here, our SSPJM2 calculations show an impro vement over the SSPJM1
results for the equilibrium sizes of the clusters.
In section 2 the calculational schemes has been explained. S ection 3 is devoted to the
results of our calculations and finally, we conclude this wor k in section 4.
2 Calculational Scheme
In the context of the SSPJM, the average energy per valence el ectron in the bulk with
density parameter rsand polarization ζis given by[8]
ε(rs, ζ) =ts(rs, ζ) +εxc(rs, ζ) + ¯wR(rs, rc) +εM(rs), (1)
where tsandεxcare noninteracting kinetic energy and exchange-correlati on energies per
electron, respectively. ¯ wRis the average value (over the WS cell) of the repulsive part o f
the Ashcroft empty core[14] pseudopotential,
w(r) =−2z
r+wR, w R= +2z
rθ(rc−r), (2)
and is given by ¯ wR= 3r2
c/r3
s. In Eq.(2), zis the valence of the atom, and θ(x) is the
ordinary step function which assumes the value of unity for p ositive arguments, and zero
for negative values. The core radius, rc, will be fixed by setting the pressure of the bulk
system equal to zero at equilibrium density ¯ n(ζ) = 3/4π¯r3
s(ζ). In Eq.(1), εMis the average
Madelung energy, εM=−9z/5r0. Here, r0is the radius of the WS sphere, r0=z1/3rs,
and for monovalent metals z= 1, and for polyvalent metals we set z∗= 1 (for details
see Ref.[6]). All equations throughout this paper are expre ssed in Rydberg atomic units.
The bulk stability is achieved when rctakes a value that makes the pressure of the system
with a given ζto vanish at rs= ¯rs(ζ):
∂
∂rsε(rs, ζ, r c)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
rs=¯rs(ζ)= 0. (3)
The derivative is taken at fixed ζandrc. Solution of the above equation gives the bulk
value of rcas a function of ¯ rsandζ. Here, ¯ rs(ζ) is the equilibrium density parameter for
4the bulk system with given ζand is evaluated by
¯rs(ζ) = ¯rs(0) + ∆ rEG
s(ζ). (4)
Here, ¯ rs(0) takes the observed value for a metal and for Cs, Na, and Al i t takes the
values of 5.63, 3.99, and 2.07, respectively. In the second t erm of the right hand side,
the superscript “EG” refers to the electron gas, and ∆ rEG
s(ζ) is evaluated by setting the
pressure of the electron gas system equal to zero (see Eq.(19 ) of Ref.[8]). The solution of
Eq.(3) at equilibrium density gives the bulk value of the cor e radius:
rB
c(¯rs, ζ) =¯r3/2
s
3
−2ts(¯rs, ζ)−εx(¯rs, ζ) + ¯rs/parenleftBigg∂
∂¯rs/parenrightBigg
ζεc(¯rs, ζ)−εM(¯rs)
1/2
.(5)
Now, using rB
cin the SSPJM energy functional of a cluster [ Eq.(20) of Ref.[ 8]], we obtain
the SSPJM2 energy as
ESSPJM2 [n↑, n↓, n+] = EJM[n↑, n↓, n+] + (εM(rs) + ¯wR(rB
c, rs))/integraldisplay
drn+(r)
+/an}bracketle{tδv/an}bracketri}htWS(rB
c, rs)/integraldisplay
drΘ(r)[n(r)−n+(r)], (6)
where
EJM[n↑, n↓, n+] = Ts[n↑, n↓] +Exc[n↑, n↓]
+1
2/integraldisplay
drφ([n, n+];r)[n(r)−n+(r)] (7)
and
φ([n, n+];r) = 2/integraldisplay
dr′[n(r′)−n+(r′)]
|r−r′|. (8)
In Eq. (6), /an}bracketle{tδv/an}bracketri}htWSis the average of the difference potential over the Wigner-Se itz cell
and the difference potential, δv, is defined as the difference between the pseudopotential
of a lattice of ions and the electrostatic potential of the je llium background.
The first and second terms in the right hand side of Eq.(7) are t he non-interacting
kinetic energy and the exchange-correlation energy, and th e last term is the Coulomb
interaction energy of the system. In our spherical JM, we hav e
n+(r) =3
4πr3sθ(R−r) (9)
5in which R= (zN)1/3rsis the radius of the jellium sphere, and n(r) denotes the electron
density at point rin space.
Applying the SSPJM2 to an Nelectron cluster with N↑up-spin and N↓down-spin
electrons ( N=N↑+N↓) and polarization ζ= (N↑−N↓)/(N↑+N↓), the total energy
becomes a function of N,ζ,rs, and rB
cwhere rsis the density parameter of the jellium
background and rB
cis given by Eq.(5). The equilibrium density parameter, ¯ rs(N, ζ), for
a cluster is the solution of
∂
∂rsE(N, ζ, r s, rB
c)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
rs=¯rs(N,ζ)= 0. (10)
Here again, the derivative is taken at fixed values of N,ζ, andrB
c. For an N-electron
cluster, we have solved the KS equations[11] self-consiste ntly for various spin configura-
tions and rsvalues and obtained the absolute minimum-energy spin config uration and its
corresponding density parameter.
3 Results
We have applied the SSPJM2 to calculate the equilibrium ener gies and sizes of different
metal clusters. In our calculations for an Nelectron cluster, we have solved the KS
equations for all possible spin configurations 0 ≤ζ≤1 and obtained the minimum
values of energies and corresponding rsvalues of each configuration. The self-consistent
calculations for Cs, Na, and Al with 2 ≤N≤42 show that the absolute minimum-
energy configuration obeys the MSC rule and the equilibrium rsvalue for the cluster,
¯rs(N, ζ), is less than the bulk value because, for small clusters the ratio of surface to
volume energies become comparable and the surface tension c ompresses the cluster. This
effect is known as self-compression.[12] In Fig. 1 we have com pared the equilibrium rs
values of “generic clusters”, JM1 (see Ref. 10), with the SJM 1 results which reproduce
correct trends.[12] To clarify the concept of the “generic c luster”, suppose that one solves
the self-consistent KS-LSDA equations for a spherical simp le JM cluster with jellium
radius equal to R=N1/3rsand total number of electrons N. For a given N, these
calculations are performed for different rsvalues as well as different spin configurations
until the equilibrium rsvalue, ¯ rs(N, ζ), corresponding to the absolute minimum-energy
configuration is obtained ( See Fig. 4 of Ref.10 ). Since in the calculations one does not
use any specific parameter corresponding to a certain metal, the result does not simulate
any real cluster, and we call it an N-electron “generic cluster”. In the limit of N→ ∞, the
infinite generic cluster tends to the electron gas system for which ζ→0 and ¯ rs→4.18. As
6is seen from the figure, the equilibrium rsvalue for the generic cluster approaches the bulk
value, 4.18, from the above which does not simulate the corre ct behavior for a real metal
cluster. It is seen that for the generic clusters the equilib rium values are greater than the
value rs= 4.18 whereas the SJM1 predicts values that are smaller than the bulk value
for Na ( rs= 3.99) in agreement with the MD findings. This comparison clearl y shows
that simple JM gives wrong molecular bond lengths. We have pe rformed our calculations
based on Eq. (10) both for jellium with sharp boundary, SSPJM 2, and diffuse jellium,
dif-SSPJM2. For our diffuse jellium calculations we have use d the background density
with the radial dependence as[15]
n+(r) =/braceleftBigg
n{1−(R+t)e−R/t[sinh(r/t)]/r}, r≤R
n{1−((R+t)/2R)(1−e−2R/t)}Re(R−r)/t/r, r > R,(11)
where n= 3/4πr3
s,R=N1/3rs, and tis a parameter related to the surface thickness.
We have chosen t= 1 in all our diffuse jellium calculations and then have varie d the
value of rsuntil the minimum energy is achieved. Figure 2(a) compares t he equilibrium
rsvalues of neutral cesium clusters for different sizes. It is s een that in most cases (rather
large clusters for which ζ << 1 ), the SSPJM2 and the SJM1 predict the same values
for the equilibrium rs; whereas for N= 3, 5, and 7 the SSPJM2 predicts larger values.
These larger values give rise to an alternation in the plot. T he values obtained from the
dif-SSPJM2 lie below the values obtained from the SSPJM2 and the SJM1. Figure 2(b)
shows the results obtained for singly ionized cesium cluste rs. In this case, we see that
the alternations persist up to N=15 and have relatively large amplitudes. For N=3 i.e.,
singly ionized 4-atom cluster, the value obtained from the S SPJM2 has become larger
than the bulk value which is related to the rough evaluation o f ∆rs. In Fig. 3(a) we have
shown the same quantities for neutral Na clusters. The behav ior of SSPJM2 results is the
same as in Fig. 2(a) but, in the case of the dif-SSPJM2 the valu e forN=5 has become
nearly equal to that of N=6 and also, the value for N=3 is less than that of N=4 which
completely differs from the SSPJM2 results. Also, we could no t find any finite value for
N=2 case in the dif-SSPJM2. That is, as much as we decrease the i nputrsvalue, the
total energy correspondingly decreases. This means that th e surface tension dominates
the internal pressure and collapses the cluster. Of course, this is not the case in reality and
it is the consequence of the fact that here the surface thickn ess,t, has become comparable
to the cluster radius, R. Figure 3(b) compares the plots of average distance between the
nearest neighbors obtained from our SSPJM2 calculations an d the MD calculations of
R¨ othlisberger and Andreoni.[13] In order to estimate the a verage distance between the
nearest neighbor ions in the cluster, we have assumed a bccstructure as in the bulk of
7Na. Then the shortest distance between the ions, d, is related to the lattice constant, a,
through d=a√
3/2. But, in the bccstructure for Na, there are two electrons in a cell
and therefore, a= 2/radicalBig
π/3rs. Combining these two relations results in d=√π rs. The
valuers= 3.99 is appropriate for room temperature ( T= 300 K) which results in a value
ofd= 7.07 bohrs whereas, for T= 0Kthe appropriate value for rsis 3.93 which gives
rise to the value d= 6.96 bohrs. Therefore, our results should lie above the MD resu lts
[ see Fig 15(a) of Ref. 13] because, the MD calculations have b een performed for zero
temperature. In Fig. 3(c) we have shown the plots of equilibr iumrsvalues for singly
ionized sodium clusters as functions of number of electrons ,N. The behavior is more or
less the same as neutral one. Figure 4(a) compares the SSPJM2 results for neutral Al
clusters with the results obtained using the SJM1. Here, we h ave taken the effective value
ofz∗= 1. The diamonds and squares in the plot show the physical poi nts. The main
difference between our results and the SJM1 is in the size of th e jellium atom of Al. In
Fig. 4(b) we have compared the results for singly ionized Al c lusters. The results show
some differences for values of Naway from shell closings. Looking at the above-mentioned
figures, we note that in all the three cases of Cs, Na, and Al the SSPJM2 results in a larger
or equal values for the average distance than the SJM1, and in addition show alternations
for small clusters.
Finally, in Fig. 5(a) we have compared the plots of the total e nergies per electron in
the two schemes of the SSPJM2 and the SSPJM1 for Na clusters. I t is seen that in the
SSPJM2 the energies are relatively lower than those of the SS PJM1 for smaller clusters
but the same for larger ones. We have also calculated the ioni zation energies of the clusters
using the dif-SSPJM2 and and compared with the dif-SSPJM1 an d experimental values
in Fig. 5(b). Here also the odd-even alternations show up the mselves in the SSPJM2 as
well as in the dif-SSPJM2 results and the values obtained are more or less the same as in
the dif-SSPJM1 [see Fig. 7(c) of Ref. 8]. Therefore, the SSPJ M2 calculations for simple
metal clusters has improved the previous work, SSPJM1, in th at it not only reproduces
the odd-even alternations in the ionization energies, but a lso it gives correct behavior for
the equilibrium sizes of the clusters.
4 Summary and Conclusion
In this work we have performed the SSPJM calculations as in th e case of ab initio molecular
structure calculations. That is, we have firstly calculated the stabilizing core radius of
the pseudopotential, rB
c, for the bulk system with nonzero spin polarization. Then, u sing
this value in the energy functional of a cluster with given va lues of Nandζ, the energy
8becomes a function of the single variable rs, the density parameter of the uniform jellium
background. Minimizing this function with respect to rsgives us the equilibrium rs
value and energy of the cluster with that specified Nandζ. Our self-consistent KS-
LSDA calculations show that the equilibrium configuration i s one in which the spins are
maximally compensated as in our previous findings.[8] This m aximum spin compensation
gives rise to the odd-even alternations seen in the experime ntal ionization energy plot of
alkali metal clusters. Calculating the average distance be tween the nearest neighbors of
Na clusters, we find a good agreement between our SSPJM2 resul ts and those obtained
from MD calculations. We have therefore improved our previo us SSPJM1 results in that
the odd-even property is kept the same as before but here, the sizes of the smaller clusters
have been predicted correctly.
Acknowledgements
The author would like to thank John P. Perdew for reading the m anuscript and the
useful discussions on the subject. He also acknowledges Bah ram Payami for providing
computer facilities.
9References
[1] W. E. Ekardt, Phys. Rev. B 29, 1558 (1984).
[2] W. D. Knight, K. Clemenger, W. A. de Heer, W. A. Saunders, M . Y. Chou, and M.
L. Cohen, Phys. Rev. Lett. 52, 2141 (1984).
[3] M. Brack, Rev. Mod. Phys. 65, 677 (1993), and references therein.
[4] N. D. Lang and W. Kohn, Phys. Rev. B 1, 4555 (1970).
[5] N. W. Ashcroft and D. C. Langreth, Phys. Rev. 155, 682 (1967).
[6] J. P. Perdew, H. Q. Tran, and E. D. Smith, Phys. Rev. B 42, 11627 (1990).
[7] M. Brajczewska, C. Fiolhais, and J. P. Perdew, Int. J. Qua ntum Chem., Quantum
Chem. Symp. 27, 249 (1993). We have calculated the ionization energies and plotted
the resuls for Na clusters in Fig. 7(c) of Ref. [8].
[8] M. Payami and N. Nafari, J. Chem. Phys. 109, 5730 (1998).
[9] See Fig. 26 in W. A. de Heer, Rev. Mod. Phys. 65, 611 (1993), and references therein.
[10] M. Payami, J. Chem. Phys. 111, 8344 (1999).
[11] W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965).
[12] J. P. Perdew, M. Brajczewska, and C. Fiolhais, Solid Sta te Commun. 88, 795 (1993).
[13] U. R¨ othlisberger and W. Andereoni, J. Chem. Phys. 94, 8129 (1991).
[14] N. W. Ashcroft, Phys. Lett. 23, 48 (1966).
[15] A. Rubio, L. C. Balb´ as, and J. A. Alonso, Z. Phys. D 19, 93 (1991).
10Figure 1: The equilibrium rsvalues in atomic units as functions of the cluster size N.
The solid squares correspond to the “generic clusters” (JM1 ) defined in the text and the
large squares correspond to Na clusters using the method of R ef.12. The dashed and solid
lines correspond to the equilibrium rsvalue of the bulk (4.18) in simple JM and to the
bulk value of sodium (3.99), respectively.
Figure 2: (a) The equilibrium rsvalues in atomic units as functions of the cluster size for
cesium clusters obtained from using the three schems of SJM1 , SSPJM2, and dif-SSPJM2.
The dashed line corresponds to the bulk value of rsfor cesium (5.63). (b) Same as (a)
but for singly ionized cesium clusters.
Figure 3: (a) Same as Fig.2 for neutral Na clusters. The bulk v alue is 3.99. (b) The
average distance between nearest neighbors in atomic units for Na clusters. The squares
correspond to our findings through SSPJM2, appropriate for r oom temperature and the
diamonds correspond to the molecular dynamics results at ze ro temperature. The dotted
line correspond to the bulk value 7.07. (c) Same as (a) for sin gly ionized Na clusters.
Figure 4: (a) The equilibrium rsvalues in atomic units for neutral Al clusters ( z∗= 1,
rs= 2.07). The diamonds and squares show the physical points in SSP JM2 and SJM1
schemes, respectively. (b) Same as (a) for singly ionized Al clusters.
Figure 5: (a) The total energies per atom of Na clusters in ele ctron volts for the SSPJM2
and the SSPJM1. The SSPJM2 results are somewhat lower than th ose of the SSPJM1
for smaller clusters. (b) The ionization energies in electr on volts for Na clusters in the
dif-SSPJM2 and the dif-SSPJM1 are compared with the experim ental values.[9]
110 5 10 15 20 25 30 35 40 453.23.74.24.75.2rs(a.u.)
Number of Electrons NJM1SJM1 Bulk(JM) Bulk(Na)
Sodium"generic cluster"0 5 10 15 20 25 30 35 40 454.64.855.25.45.65.8rs(a.u.)
Number of Electrons NSJM1
SSPJM2
dif-SSPJM2
BulkCesium (neutral)0 5 10 15 20 25 30 35 40 4555.15.25.35.45.55.65.7rs(a.u.)
Number of Electrons NSJM1
SSPJM2
dif-SSPJM2
BulkCesium (singly ionized)0 5 10 15 20 25 30 35 40 452.72.93.13.33.53.73.94.1rs(a.u.)
Number of Electrons NSJM1
SSPJM2
dif-SSPJM2
BulkSodium (neutral)0 5 10 15 20 25 30 35 40 4555.566.577.5
Bulk
SSPJM2(T=300K)
MD (T=0K)
Number of atoms N<d>(a.u.)
Sodium0 5 10 15 20 25 30 35 40 452.72.93.13.33.53.73.94.1rs(a.u.)
Number of Electrons NSJM1
SSPJM2
dif-SSPJM2
BulkSodium (singly ionized)0 5 10 15 20 25 30 35 40 451.41.51.61.71.81.922.1rs(a.u.)
Number of Electrons NBulk
SJM1
SSPJM2Aluminum (neutral)0 5 10 15 20 25 30 35 40 451.61.71.81.922.1rs(a.u.)
Number of Electrons NBulk
SJM1
SSPJM2Aluminum (singly ionized)0 5 10 15 20 25 30 35 40 45-5.3 -5.5 -5.7 -5.9 -6.1
Number of Electrons NSSPJM2 SSPJM1E(N)/N (eV)_Sodium0 5 10 15 20 25 30 35 40 45234567
EXP.dif-SSPJM1 dif-SSPJM2
Number of Atoms NIonization Energy (eV)Sodium |
arXiv:physics/0102038v1 [physics.optics] 13 Feb 2001On the physical nature of the photon mixing processes
in nonlinear optics.
V.A.Kuz’menko
Troitsk Institute for Innovation and Fusion Research,
Troitsk, Moscow region, 142190, Russian Federation.
E-mail: kuzmenko@triniti.ru
Abstract
The physical nature of the photon mixing and photon echo pro-
cesses is discussed on the basis of inequality of the forward and back-
ward optical transitions.
PACS number: 42.50.-p
Wave (photon) mixing processes of different kind play an impo rtant role
in nonlinear optics. Interaction of light with substance ca n be described
both from the wave point of view and from that of the quantum in sight into
the nature of light. It has happen historically, that the so- called semiclas-
sical theory has got the greatest development [1,2]. The fac t of quantum
absorption and emission of light is recognized here, but all description of the
dynamics of optical transitions is performed from the pure w ave position of
the so-called rotating wave model.
The basic problems in this theory arise when one attempts to e xplain
the physical nature of the processes underlying the observa ble effects. Most
brightly, this difficulty manifests itself in the case of the p opulation transfer
effect at sweeping of resonance conditions [3]. The theory we ll describes the
dynamics of this effect, but fails to explain its physical nat ure. A similar
situation exists with a photon echo effect in the gas phase. Th e theory very
well explains the principle of a photon echo, and well descri bes its dynamics,
but can not explain the specific physical mechanism of the pro cess [1]. An
attempt to draw the Doppler effect to this explanation demons trates only
the inconsistency of the Bloch model in explanation of the pa rticular nature
of the photon echo.
In the framework of wave approximation, the reason of laser s timulated
wave mixing processes is generally believed to be due to the s o-called non-
linear susceptibility. If the corresponding factor of nonl inear susceptibility is
great enough, the wave mixing can occur. If this factor is equ al to zero, the
wave mixing is absent. Also, a restriction on the lowest orde r wave mixing
process exists, that is connected with the symmetry of the pr ocess. Namely,
1the factor of nonlinear susceptibility χ2is nonzero only in a medium without
center of inversion [4].
The purpose of the present report is to discuss an opportunit y of pho-
ton mixing explanation from the position of pure quantum app roximation. A
good theoretical basis exists for this approach, as the Dira c equation predicts
time invariance violation in electromagnetic interaction s [5]. For many years,
however, a generally accepted opinion exists, that electro magnetic interac-
tions proceed with T-invariance preservation [6]. Obvious ly, this is a mistake.
This point of view does not have any experimental proof. On th e contrary,
the opposite point of view (here we mean the T-invariance vio lation) has a
direct and complete experimental proof for the case of the ph oton interac-
tion with molecules [7]. The experiments show, that althoug h the integral
cross-sections of absorption and stimulated emission of ph otons by atoms and
molecules are, obviously, identical (the Einstein coefficie nts are equal), the
difference in spectral widths and cross-sections of the forw ard and backward
optical transitions can reach several orders of magnitude. The principle of
inequality of the forward and backward optical transitions is quite sufficient
for explanation of most nonlinear effects from the pure quant um point of
view without using any wave approximation.
In the quantum approach, the physical reason of efficient phot on mixing
is probable ultra high cross-section of the backward optica l transition to the
initial state. The ”initial state” concept must include, pr obably, not only
a set of quantum numbers, but also the orientation of molecul es in space
and the phase of vibration motion. Fig. 1a shows a general fou r photon
mixing scheme. Three laser beams, adjusted in the resonance with 0→1,
1→2 and 2 →3 transitions, interact with molecules. As a result of the
population transfer, directed superradiation on the 3 →0 transition appears,
which transfers molecules precisely to the initial state. A little more complex
scheme is submitted in Fig. 1b. Here the directed superradia tion arises at
first on the 2 →3 transition, for which level 3 is an initial state. Then
directed superradiation on the 3 →0 transition should appear, for which 0
level is an initial state. In this case we have a combined vari ant of four and
six photon mixing.
The selection rules can be deduced from the existence of the p hoton spin.
It is impossible to return a molecule precisely to the initia l quantum state
using odd number of photons. On the contrary, even number of p hotons
allows this to be done. Therefore, only four, six, and eight p hoton mixing
processes are possible in gases and liquids. In the solid sta te, the rotation of
molecules is absent. A crystal lattice allows to eliminate t he problem of the
photon’s spin. Therefore, in the crystals the three- and five - photon mixing
processes are also observed.
2What type of approximation (the wave or the quantum one) is mo re
acceptable for description of the photon (wave) mixing effec ts in nonlin-
ear optics? From our point of view, the quantum approximatio n is more
preferable, especially in the cases of using the short delay ed pulses of laser
radiation. Temporary delay between pulses badly coordinat es to the princi-
ples of nonlinear susceptibility and wave mixing. On the con trary, quantum
approximation well admits using the delayed laser pulses. I t only imposes
certain restrictions on the sequence of interactions of mol ecules with laser
pulses. Population of quantum levels must be transferred co nsistently and
the whole mixing process should return molecules to the init ial state. The de-
lay between laser pulses allows to study dynamics of the grou nd and various
excited states of molecules [8,9].
There are many experiments in the literature, which are rath er similar in
arrangement to that shown in Fig. 1, but simpler in implement ation. Some-
times these experiments are related to the concept of the pho ton echo. The
physical mechanism of the classical photon echo in the gas ph ase can be
shown to be a result of degenerative four photon mixing in a th ree-level sys-
tem, if one takes the photon’s spin into consideration. The p hoton’s spin can
play a role of the ”Maxwell’s demon” [1]. The basic scheme of t his process
is given in the Fig. 2. The photon (1) of the first laser pulse ex cites the
molecule and passes to it a rotation moment, connected with t he photon’s
spin. Now a process of rotation dephasing of molecules, conn ected with het-
erogeneity of rotational spectrum, begins. This process ha s been studied in
the beautiful experimental work [10]. (It is necessary to no te, that the probe
pulse in this work stimulated a backward Raman optical trans ition). The
inhomogeneity of the absorption spectrum is connected with the hyperfine
splitting due to higher order interactions like the centrif ugal distortion, the
electronic spin-spin splitting, and others. For typical ti me of the photon echo
in the gas phase ∼1µs[11] it makes the rotational dephasing of molecules
quite significant.
At time τthe second laser pulse starts. Two photons (2 and 3) are con-
nected with the second laser pulse (for the case of the two-pu lse echo). The
second photon deexcites molecule and also compensates the r otation moment
of the first photon. The absorption of the third photon is acco mpanied by
transfer of the rotation moment to the opposite direction. A fter that the
process of rotation rephasing begins, which is finished to th e time 2 τby di-
rected superradiation of the photon echo pulse (photon 4), w hich returns the
molecule precisely to the initial state. Existence of the so -called line wings
[12] in the absorption spectrum of polyatomic molecules (li keSF6andBCl 3)
allows to eliminate the problem of exact resonance of narrow laser radiation
with absorption lines of molecules. Thus, inhomogeneity of the absorption
3spectrum, connected with the Doppler effect, has no relation neither to the
rotation rephasing, nor to the photon echo effect. This simpl e quantum
model predicts also, that the main phase-match direction of the photon echo
superradiation ( ke=k1−k2+k2=k1) must be collinear with the beam of
the first pulse. On the contrast, the phase-match direction ( ke=−k1+2k2)
is impossible for the real photon echo, since the sign ”minus ” of vector k1
corresponds to stimulated emission of a photon.
If the second laser pulse split on two pulses, we obtain the so -called stim-
ulated photon echo variant [13,14]. Here a photon of the first laser pulse
excites molecules and gives a rise to the rotation dephasing process. The
second laser pulse transfers molecules to the ground state a nd stops the ro-
tation dephasing process. The delay between the second and t he third laser
pulses allows to study the dynamics of the ground state. The t hird laser pulse
again excites the molecules and initiates the process of the ir rotation rephas-
ing, which is finished by directed superradiation of a photon echo pulse. In
this case, the delay between the third laser pulse and the pho ton echo pulse
is equal to that between the first two laser pulses.
The experimental studies of the dynamics of molecules in the liquid phase
[15-18] are usually associated with stimulated photon echo . Such an associ-
ation for these experiments is not successful. Photon echo i s only a special
case of the four-photon mixing. The main reason of the photon echo is laser-
induced rephasing process of the rotation motion. If the dep hasing is not
so great, there is no need neither in the rotation rephasing, nor in the pho-
ton echo. In the general case, only rotational and vibration al alignment of
molecules is required. In the gas phase, the rotational alig nment of molecules
occurs by their free rotation [10]. In the liquid phase the li brational mo-
tion plays the role of rotation. Therefore the duration of a s uperradiation
pulse in four photon mixing in the liquids characterizes mai nly the period
of librational alignment of molecules (if the lifetime of ex cited molecules is
sufficiently long).
The so-called peak shift is measured in these works as one of t he most im-
portant experimental parameters. This peak shift is relate d to some abstract
correlation function [15]. However, the effect of peak shift can be given an
alternative pure physical explanation. When the third lase r pulse coincides
in time with the second one, the shift of pulses characterize s, probably, most
optimal conditions for population transfer in the system, w hich leads to ap-
pearance of the directed superradiation. Here we have a rath er general and
interesting example of the so-called effect of a counterintu itive sequence of
interactions of molecules with laser pulses. The most effect ive superradiation
(the population transfer) takes place when radiation of the first (in time) laser
pulse interacts with molecules exited by radiation of the se cond laser pulse
4(some overlapping of laser pulses should, certainly, take p lace). This effect
was described long ago in the works on the dynamics of populat ion transfer in
atoms and molecules in the gas phase under influence of nanose cond pulses
of laser radiation [19,20]. For the photon mixing process of this kind the
phase-matched direction for superradiation ( k=k2−k1+k2=2k2−k1)
can well be realized. It follows from the notes above, that th e maximal value
of the measured peak shift is determined mainly by the shape a nd width of
the used laser pulses.
The delay of the third pulse destroys optimum conditions for the popula-
tion transfer and results in the sharp and substantial reduc tion in intensity of
superradiation and measured value of the peak shift in liqui d [15]. Probably,
this is a consequence of librational and vibrational dephas ing. The temper-
ature dependence of the peak shift in solid samples is especi ally interesting
here [21]. May be this dependence characterizes some specifi c characteristics
of librational motion.
The optimal experimental conditions are, obviously, differ ent for residual
superradiation. A dephasing degree of molecules, which are prepared in the
ground state by the first two laser pulses, can be more substan tial. This
dephasing can be due to the fact, that the molecules stay diffe rent time in
the exited state. The inertia moments of molecules in the gro und and excited
states can be essentially different (especially in the case o f electronic excita-
tion). The least dephasing in the ground state occurs when th e molecules
spend minimal and equal time in the excited state. This condi tion can be
implemented, when the first and the second pulses practicall y coincide in
time.
So, from the given point of view the dependence of the peak shi ft on the
delay of the third laser pulse, and on the temperature, as wel l as the shape of
the superradiation pulse, characterize mainly the dynamic s of the libration
motion of molecules. Experiments, similar to work [10], but conducted in a
liquid phase, could also give important information on the d ynamics of libra-
tional alignment. The possible role of rotation rephasing a nd the existence
of a real stimulated photon echo in the liquid phase require f urther study
and discussion.
The principle of inequality of the forward and backward opti cal transitions
is suitable not only for explanation of the physical nature o f photon mixing
and photon echo effects. It allows to explain easily and in the natural way
such effects, as population transfer at sweeping the resonan ce conditions
[3], amplification without inversion [22], coherent popula tion trapping [23],
electromagnetically induced transparency [24], and other s.
Thus, two approaches can be considered for description of th e dynamics
of optical transitions, the wave approximation and the quan tum one. The
5wave approximation has a 50-year’s old history [2], advance d mathematical
tools, and some problems with physical interpretation. The quantum approx-
imation has a good theoretical base (the Dirac equation), si mple and clear
physical sense, and sufficient proofs. The quantum approxima tion requires
experimental study of the basic parameters of the backward o ptical transi-
tions and creation of the mathematical tools for descriptio n of the effects in
nonlinear optics.
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92, 5363 (1990).
[20] S.Schiemann, A.Kuhn, S.Steuerwald and K.Bergmann, Ph ys.Rev.Lett.
71, 3637 (1993).
[21] S.De Silvestri, A.M.Weiner, J.G.Fujimoto and E.P.Ipp en,
Chem.Phys.Lett. 112, 195 (1984).
[22] M.O.Scully, Quantum Opt. 6, 203 (1994).
[23] S.Brandt, A.Nagel, R.Wynands and D.Meschede, Phys. Re v. A56,
R1063 (1997).
[24] A.Kasapi, M.Jain, G.Y.Yin and S.E.Harris, Phys. Rev. L ett.74, 2447
(1995).
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arXiv:physics/0102039v1 [physics.pop-ph] 13 Feb 2001The Legality of Wind and Altitude Assisted Performances in
the Sprints
J. R. Mureika
Department of Physics
University of Toronto
Toronto, Ontario Canada M5S 1A7
Email: newt@palmtree.physics.utoronto.ca
Abstract
Based on a mathematical simulation which reproduces accura te split and velocity profiles for the 100
and 200 metre sprints, the magnitudes of altitude and mixed w ind/altitude-assisted performances
as compared to their sea-level equivalents are presented. I t is shown that altitude-assisted marks for
the 200 metre are significantly higher than for the 100 metre, suggesting that the “legality” of such
marks perhaps be reconsidered.
1 Introduction
According to IAAF regulations, sprint and jump performance s for which the measured
wind-speed exceeds +2.0 m /s are deemed illegal, and cannot be ratified for record
purposes (IAAF 1998). Similarly, performances which are ac hieved at altitudes ex-
ceeding 1000 metres above sea level are noted as “altitude-a ssisted”, but unlike their
wind-aided counterparts, these can and have qualified for re cord status. Indeed, the
1968 Olympics saw amazing World Records (WRs) set in the men’ s 100 m, 200 m,
and Long Jump, thanks in part to the lofty 2250 metre elevatio n of Mexico City.
Other examples of such overt assistance include Pietro Menn ea’s former 200 m WR
of 19.72 seconds, Marion Jones’ 1998 clocking of 10.65 s in Jo hannesburg, Obadele
Thompson’s wind- assisted 9.69 s, and Michael Johnson’s ear ly 2000-season marks of
19.71 s in the 200 m and World Best 30.85 s.
A search of the academic literature reveals a wealth of sourc es which discuss
the impact of wind and altitude assistance in the 100 metre sp rint. Based both on
statistical and theoretical models, the general consensus of most researchers is that
the maximum legal tail-wind of +2 .0 m/s yields roughly a 0.10-0.12 second advantage
over still conditions at low altitude. With no wind, every 10 00 m of elevation will
improve a performance by roughly 0.03-0.04 seconds, implyi ng that still conditions
in Mexico City have about a 0.07 s advantage over their sea-le vel equivalents. The
interested reader is directed to references (Davies 1980, D apena 1987, Dapena 2000,
Linthorne 1994a, Linthorne 1994b, Mureika 2001a) and citat ions therein for further
information.
Conversely, little attention has been paid to the equivalen t corrections in the 200
metres. There are multifold reasons why this is perhaps the c ase, the largest of which
being a lack of essential wind data for the first half of the rac e. The wind gauge is
1operated only after the first competitor has entered the stra ight, and without a second,
perpendicular wind gauge placed at the top of the curve, the a ctual conditions in the
first 100 metres remain a mystery. In a recent article (Mureik a 2001b), such effects
have been studied, accounting for variable wind effects on th e curve.
The model used in this investigation is a direct extension of that presented in
(Mureika 2001a), whose results are consistent with indepen dent investigations. The
underlying framework of the model is a part-mathematical, p art-physical force equa-
tion of the form
Fnet(t;v, w) =Fpropulsive (t)−Finternal(v;t)−Fdrag(v, w), (1)
Here, Fpropulsive (t) and Finternal(v;t) are functions of the “sprinter”, and are depen-
dent on the elapsed time tand athlete’s resulting velocity v(t). They are intended to
numerically represent both the forward driving of the sprin ter, as well as any internal
variables which govern the overall acceleration and speed ( e.g.flexibility, stride rate,
fast-twitch rate, and so forth). The drag term Fdrag(v, w;ρ) = 1/2ρ(H)Ad(v(t)−w)2
is an external, physical quantity, which is a function of the square of the sprinter’s rel-
ative velocity to the air, the athlete’s average drag area Ad (or frontal cross- sectional
area times the drag coefficient, normalized to mass), and the a tmospheric density
ρ=ρ(H) (dependent on the altitude Hof the venue). Fnetis twice-integrated with
respect to time to obtain the distance traveled as a function of time, d(t), and hence
for a suitable choice of input parameters, has been shown to e ffectively and realisti-
cally simulate the split/speed profiles of a 100 metre race.
Since the effects of cross-winds ( i.e.winds that are completely perpendicular
to the direction of motion) are assumed to be negligible, onl y the forward drag is
included. Hence, for the 200 metre simulations, only the com ponent of the wind in
the direction of motion are considered (which depends on the position of the athlete
through on curve). It is recognized that the influence of a str ong cross-wind will
undoubtedly affect the motion in some fashion, but since thes e effects are currently
unknown, they are left for future work.
This article is not designed to be an expository of the numeri cal model, but
rather a highlight of the results which address the “legalit y” of wind and altitude
assistance for the 200 m sprint, as compared to those in the 10 0 m. The interested
reader is referred to (Mureika 2001a, Mureika 2001b) for a co mplete mathematical
and methodological formulation.
2 Wind and Altitude Effects in the 100 m
Table 1 shows correction estimates for a 10.00 second 100 met re performance run
with 0-wind at sea level, obtained from the model discussed i n (Mureika 2000a).
Corrections are to be interpreted as ∆ t=tofficial −t0,0,i.e.the amount by which
the 0-wind, sea level performance t0,0is adjusted under the conditions. So, negative
2corrections mean faster official times (and vice versa for pos itive ∆ t). Note that a
legal-limit wind at high altitude will provide almost a 60% i ncrease over the same
conditions at sea-level. Only for extremely high elevation s does the magnitude of
the altitude assistance alone approach that of the wind, wit h the minimum altitude-
assisted correction being about 0.04 seconds. Only at altit udes at exceeding 2000 m
does the assistance begin to approach that provided by a low- altitude legal-limit wind.
For a +1 m /s wind at 2000 m, the theoretical assistance is equal to a lega l-limit, sea
level wind. These figures are in good agreement with those in ( Dapena 2000), with
showing only mild variations at higher altitudes and wind sp eeds.
w (m/s) 0 m 500 m 1000 m 1500 m 2000 m 2500 m
0.0 0.00 -0.02 -0.04 -0.05 -0.07 -0.08
+1.0 -0.05 -0.07 -0.08 -0.10 -0.11 -0.12
+2.0 -0.10 -0.11 -0.13 -0.14 -0.15 -0.16
0.0 0.00 -0.02 -0.04 -0.06 -0.07 -0.09
+1.0 -0.07 -0.08 -0.10 -0.11 -0.12 -0.14
+2.0 -0.12 -0.14 -0.15 -0.16 -0.17 -0.18
Table 1: Correction estimates (s) for 100 m at varying altitude (Top t hree rows or men, latter
three for women), as compared to 10.00 s (11.00 s) performanc e at sea level, 0-wind.
An easy to use “back-of-the-envelope” formula was presente d in (Mureika 2001a),
which can be used to quickly calculate the corresponding cor rections of Table 1. This
is
t0,0≃tw,H/bracketleftBig
1.03−0.03exp(−0.000125 ·H)(1−w·tw,H/100)2/bracketrightBig
, (2)
witht0,0, tw,H, w, and Hdefined as before. Thus, 100 metre sprint times may be
corrected to their 0-wind, sea level equivalents by inputti ng only the official time, the
wind gauge reading, and the altitude of the sporting venue. S ince Equation 2 is easily
programmable in most scientific calculators and portable co mputers, it may be used
track-side by coaches, officials and the media immediately fo llowing a race to gauge
its overall “quality”.
3 Wind and Altitude Effects in the 200 m
The story, however, is different for the longer sprint. Table s 2, 3 and exemplify
the degree of assistance which wind and altitude provide for World Class men and
women’s performances (20.00 s and 22.00 s). The estimates as sume a race run around
a curve of radius equivalent to about lane 4 of a standard IAAF track, implying the
distance run around the curve is 115.6 m, and 84.4 m on the stra ight. The model
Equations 1 are modified by the addition of an appropriate “da mping factor” to the
3propulsive forces (a function of the velocity and the lane’s radius). When coupled
with the effects of wind, the amplitude of the altitude correc tions escalates. While the
absolute value of the corrections may not be known at this poi nt, it is the magnitude
of these estimates to which this research note draws attenti on.
The data presented in Tables 2, 3 assume that the wind is entir ely in the direction
measured by the gauge. In this case, the athlete initially fa ces a head-wind out of the
blocks, which gradually subsides and increases to its maxim um value as the sprinter
rounds the bend. A straight wind of +2 .0 m/s adjusts the overall 200 m time by
−0.12 s for men ( −0.14 s for women), slightly more than the correction for the 100 m
under similar conditions. However, the difference between t he two race corrections
quickly grows for increasing wind-speed and altitude. In fa ct, the pure altitude effects
at the minimum 1000 m elevation are found to be equivalent to t hat provided by a
legal-limit wind in the 100 m. Furthermore, the combined win d and altitude effects
could become as high as 0.25-0.30 seconds for extreme elevat ions (H >2000 m).
w (m/s) 0 m 500 m 1000 m 1500 m 2000 m 2500 m
0.0 0.0 -0.05 -0.10 -0.15 -0.20 -0.24
+1.0 -0.06 -0.11 -0.16 -0.20 -0.24 -0.28
+2.0 -0.12 -0.16 -0.20 -0.25 -0.28 -0.32
0.0 0.0 -0.06 -0.11 -0.16 -0.21 -0.26
+1.0 -0.08 -0.16 -0.18 -0.23 -0.27 -0.31
+2.0 -0.14 -0.19 -0.23 -0.28 -0.32 -0.35
Table 2: Men’s and Women’s correction estimates (s) for 200 m at varyi ng altitudes, as compared
to 20.00 s (22.00 s) performance at sea level, 0-wind. The win d direction is assumed to be completely
in the direction of the gauge (100 metre straight).
Note that the 100 m splits do not significantly change for the w ind conditions
considered. Up to about 1000 m altitude, the head-wind equiv alent conditions in the
early part of the race actually serve to slow the splits from t heir 0-wind, sea level
equivalent. Even at high elevations, the splits are not sign ificantly affected, being
corrected by only -0.05 s at the most. The split corrections f or the 0-wind condition
are essentially identical to those for the 100 m, since the ad justments depend on the
velocity profile over the distance, and are not affected by the curve.
w (m/s) 0 m 500 m 1000 m 1500 m 2000 m 2500 m
0.0 0.00 -0.02 -0.03 -0.05 -0.06 -0.08
+1.0 +0.02 +0.00 -0.02 -0.04 -0.05 -0.07
+2.0 +0.03 +0.01 -0.01 -0.02 -0.04 -0.05
Table 3: Correction estimates (s) for 100 m splits of men’s (20.00 s) 2 00 m race. 0-wind split is
approximately 10.25 s including reaction.
4Using the corrections of Table 2, one can obtain “first-order ” adjustments of some
key 200 m performances. For example, Pietro Mennea’s WR of 19 .72 seconds run in
Mexico City with a +1 .8 m/s wind would be corrected by approximately 0.31 seconds,
yielding a 0-wind, sea-level equivalent of 20.03 s. This is e ssentially equivalent to his
low-altitude bests, e.g.(20.01 s; +0.0 m /s) in Rome (08 Aug 1980). Similarly, Michael
Johnson’s 19.71 s (+1 .8 m/s) in Pietersburg (approximately 1200 m) would roughly
adjust to a mid-19.9 s. This is also quite consistent with his low-altitude bests of 2000,
all of which clustered around 19.90 s. Tables4, 5 show the cor rected top-5 all-time
performances for men and women, as well as the re-ranked top- 5 performances.
It should be stressed that the calculations herein were perf ormed for an perfor-
mance around a curve of equivalent radius to lane 4 (and appro priate stagger). The
correction estimates for a straight wind will actually vary by several hundredths of a
second from lane 1 to 8. For a tail-wind, there will be minimal assistance provided
in lane 1, and maximal in lane 8. Thus, lane 4 is selected as the “standard” for
conversion.
tw,H(w) t0,0(s) Athlete Venue (altitude) Date
19.32 (+0.4) 19.38 Michael Johnson USA Atlanta, USA (350 m) 96/08/01
19.66 (+1.7) 19.79 Michael Johnson USA Atlanta, USA 96/06/23
19.68 (+0.4) 19.74 Frank Fredericks NAM Atlanta, USA 96/08/01
19.72 (+1.8) 20.03 Pietro Mennea ITA C. de Mexico, MEX (2250 m) 79/09/12
19.73 (-0.2) 19.73 Michael Marsh USA Barcelona, ESP (100 m) 92/08/05
19.75 (+1.5) 19.86 Carl Lewis USA Indianapolis, USA (200 m) 83/06/19
21.34 (+1.3) 21.45 Florence Griffith-Joyner USA Seoul, SKR (100 m) 88/09/29
21.56 (+1.7) 21.69 Florence Griffith-Joyner USA Seoul, SKR 88/09/29
21.62 (-0.6) 21.76 Marion Jones USA Johannesburg, SA (1800m) 98/09/11
21.64 (+0.8) 21.71 Merlene Ottey JAM Bruxelles, BEL (50 m) 91/09/13
21.71 (-0.8) 21.67 Heike Drechsler GER Stuttgart, GER (250 m) 86/08/29
21.72 (-0.1) 21.73 Gwen Torrence USA Barcelona, ESP 92/08/05
Table 4: Official top 5 all-time rankings for men and women, showing 0-w ind, sea-level equivalents
(t0,0). Best-per-athlete (excluding WR). Altitudes are assumed correct to within 50 m. Races are
assumed run in lane 4.
As previously mentioned, the lack of wind condition informa tion over the first
half of the 200 m race ultimately prevents completely accura te correction estimates.
For a wind w blowing at angle θto the straight, the gauge reads wcosθ. An angle
θ= 0 corresponds to a wind purely down the straight, with the va lue increasing in
the counterclockwise direction (such that θ <0 will provide a tail-wind assistance
around the bend, and θ >0 a head-wind). Preliminary results (see Mureika 2000b)
indicate that there is an extremely wide range of variations in corrections for 200
metre performances apparently run under the “same” wind con ditions (as read by
the gauge). In fact, for a raw 20.00 s race, the correction diff erential between effective
5t0,0(w) tw,H(s) Athlete Venue (altitude) Date
19.38 19.32 (+0.4) Michael Johnson USA Atlanta, USA 96/08/01
19.72 20.01 (-3.4) Michael Johnson USA Tokyo, JPN (0 m) 91/08/27
19.73 19.73 (-0.2) Michael Marsh USA Barcelona, ESP 92/08/05
19.74 19.68 (+0.4) Frank Fredericks NAM Atlanta, USA 96/08/01
19.75 19.80 (-0.9) Carl Lewis USA Los Angeles, USA (100 m) 84/08/08
19.83 19.61 (+4.0) Leroy Burrell USA College Station, USA (300 m) 90/05/19
21.45 21.34 (+1.3) F. Griffith-Joyner USA Seoul, SKR 88/09/29
21.62 21.66 (-1.0) Merlene Ottey JAM Zurich, SWI (400 m) 90/08/15
21.67 21.71 (-0.8) Heike Drechsler GER Stuttgart, GER 86/08/29
21.69 21.56 (+1.7) F. Griffith-Joyner USA Seoul, SKR 88/09/29
21.73 21.75 (-0.1) Juliet Cuthbert JAM Barcelona, ESP 92/08/05
21.73 21.72 (-0.1) Gwen Torrence USA Barcelona, ESP 92/08/05
Table 5: Corrected top 5 all-time rankings for men and women. ( t0,0). Best-per-athlete (excluding
WR). Altitudes are assumed correct to within 50 m. Races are a ssumed run in lane 4.
tail-winds and head-winds on the curve could be up to 0.3 s for lower altitudes. At
high altitudes, this differential could exceed 0.6 s!
In 1990, Leroy Burrell ran the fastest-ever wind-assisted 2 00 m, a startling time
of 19.61 s (+4.0 m /s) (see Table 5). Application of the straight-wind correcti ons
would give a 0-wind, sea-level time of 19.83 s, much faster th an his legal sea-level
best of 20.12 s ( −0.8 m/s, New Orleans; 20.06 s corrected). However, a wind in
excess of 5 m /s blowing at an angle of roughly -40 degrees would produce the proper
gauge reading, and assist the performance by up to 0.4 s, much more consistent with
Burrell’s previous marks. Also, if this race had been run in a higher lane than 4,
the correction would be larger (but still faster than his pre vious bests, if the wind is
assumed purely in the 100 m direction).
If these correction estimates are accurate, then the sugges tion is put forth to
the IAAF that the status of higher (but legal) wind and altitu de-assisted 200 metre
marks be reconsidered. The degree of variation from differin g wind angles would also
affect the IAAF Top Performance lists introduced this year, a s well as similar scoring
tables which account for accompanying wind speeds. Equival ently, these types of
corrections could be used to “rate” the quality sprints run u nder varying conditions.
A wind gauge placed at the top of the curve could help to shed li ght on these effects,
as well as assist in the evaluation and comparison of race per formances.
6Acknowledgements
This work was supported in part by grants from the Walter C. Su mner Foundation,
and the Natural Sciences and Engineering Research Council o f Canada (NSERC).
References
(IAAF 1998) Official 1998/1999 Handbook , International Amateur Athletics Federa-
tion, Monaco (1998)
(Davies 1980) C. T. M. Davies, “Effects of wind assistance and resistance on the
forward motion of a runner”, J. Appl. Physio. 48, 702-709 (1980)
( Dapena 1987) J. Dapena and M. Feltner, “Effects of wind and al titude on the times
of 100-meter sprint races”, Int. J. Sport Biomech. 3, 6-39 (1987)
(Dapena 2000) J. Dapena, in The Big Green Book , Track and Field News Press (2000)
(Linthorne 1994a) N. P. Linthorne, “The effect of wind on 100 m sprint times”, J.
App. Biomech. 10, 110-131 (1994)
(Linthorne 1994b) N. P. Linthorne, “Wind and altitude assis tance in the 100-m
sprint”, Proc. 8th Bienn. Conf. Can. Soc. Biomech. , W. Herzog, B. Nigg and T.
van den Bogert (Editors), 68-69 (1994)
(Mureika 2001a) J. R. Mureika, “A Realistic Quasi-physical Model of the 100 Metre
Dash”, to appear, Canadian Journal of Physics (2001)
(Mureika 2001b) J. R. Mureika, “Modeling Wind and Altitude E ffects in the 200
Metre Sprint”, in preparation (2001)
7 |
arXiv:physics/0102040v1 [physics.plasm-ph] 13 Feb 2001Coherent structures in a turbulent environment
F. Spineanu1,2, M. Vlad1,2
1Association Euratom-C.E.A. sur la Fusion, C.E.A.-Cadarac he,
F-13108 Saint-Paul-lez-Durance, France
2National Institute for Laser, Plasma and Radiation Physics ,
P.O.Box MG-36, Magurele, Bucharest, Romania
A systematic method is proposed for the determination of the statistical properties of a field
consisting of a coherent structure interacting with turbul ent linear waves. The explicit expression
of the generating functional of the correlations is obtaine d, performing the functional integration on
a neighbourhood in the function space around the soliton. Th e results show that the non-gaussian
fluctuations observed in the plasma edge can be explained by t he intermittent formation of nonlinear
coherent structures.
I. INTRODUCTION
In a recent work [1] it has been proposed a systematic analyti cal method for the investigation of the statistical
properties of a coherent structure interacting with turbul ent field. The method is here developed in detail and new
possible applications or developments arise.
The nonlinearity of the dynamical equations of fluids and pla sma is the determining factor in the behaviour of these
systems. The current manifestation is the generation, from almost all initial conditions, of turbulent states, with an
irregular aspect of fluctuations implying a wide range of spa ce and time scales. The fluctuations seem to be random
and a statistical characterization of the fluctuating fields is appropriate. However it is known both from theory and
experiment that the same fields can have, in particular situa tions, stable and regular forms which can be identified
as coherent structures, for example solitons and vortices. For most general conditions one should expect that these
aspects are both present and this requires to study the mixed state consisting of coherent structures and homogeneous
turbulence.
Numerical simulations of magnetohydrodynamics show that i n general cases a coherent structure emerges in a
turbulent plasma, it moves while deforming due to the intera ctions with the random fields around it and eventually
is destroyed. In plasma turbulence a coherent structure is b uild up by the inverse spectral cascade or by merging
and coalescence of small-scale structures [2], [3], [4]. Th e nonlinearity of the equations for the drift waves in a
non-uniform, magnetized plasma permits the formation of so litary waves in addition to the usual small-amplitude
dispersive modes. The convective nonlinearity (of the Pois son bracket type), can lead to low-frequency convective
structures in magnetized plasma [5], [6], [7], [8], [9]. The structures are not solitons in the strict sense but are very
robust. It is even possible that the state of plasma turbulen ce can be represented as a superposition of coherent vortex
structures (generated by a self-organization process) and weakly correlated turbulent fluctuations.
Naturally, the coherent structures influences the statisti cal properties of the fields (the correlations), in particul ar
the spectrum. In this context it is usual to say that the devia tion of the correlations of the fluctuating fields from the
gaussian statistics is associated with the presence of the c oherent structures and it is named intermittency . Numerical
simulations [10] of the 2-dimensional Navier-Stokes fluid t urbulence have shown coherent structures evolving from
random initial conditions and in general energy spectra ste eper thank−3have been atributed to intermittency (patchy,
spatial intermittent paterns). These coherent structures are long-lived and disappear only by coalescence, the latte r
being manifested as spatial intermittency. In these studie s it was underlined that the coherent structures have effects
which cannot be predicted by closure methods applied to mode -coupling hyerarchies of equations.
The difficulty of the analytical description consists in the a bsence from theory of well established technical methods
to investigate the plasma turbulence in the presence of cohe rent structures. While for the instability-induced turbu-
lence (via nonlinear mode-coupling) systematic renormali zation procedures have been developed, the problem of the
simultaneous presence of coherent structures and drift tur bulence has not received a comparable detailed description .
In the recently proposed method [1], the starting point is th e observation that the coherent structure and the drift
waves, although very different in form, are similar from a par ticular point of view: the first realizes the extremum
and the later is very close to the extremum of the action funct ional that describes the evolution of the plasma.
The analytical framework is developed such as to exploit thi s feature and is based on results from well established
theories: the functional statistical study of the properti es of the classical stochastic dynamical systems (in the Mar tin-
Siggia-Rose approach); the perturbed Inverse Scattering T ransform method, allowing to calculate the field of perturbe d
nonlinear coherent structures; the semi-classical approx imation in the study of the quantum particle motion in multip le
minima potentials.
1The dilute gas of plasma solitons has been studied by Meiss an d Horton [13] who assumed a probability density
function of the amplitudes characteristic of the Gibbs ense mble. We analyse the same nonlinear equation but take
into account the drift wave turbulence.
A brief discussion on the closure methods developed in the st udy of drift wave turbulence provides us the argumen-
tation for the need of a different approach (Section 2). The Se ction 3 contains a description of the general lines of
the method proposed. A more technical presentation of the ca lculation is given in the subsection 3.2. The particular
case of the drift wave equation is developed in detail in the S ection 4 and in the Section 5 the explicit expression of
the generating functional is used for the calculation of the correlation functions. The results and the conclusions are
presented in the last Section. Some details of calculations are given in the Appendix.
II. THE NONLINEAR DYNAMICAL EQUATIONS
We consider the plasma confined in a strong magnetic field and t he drift wave electric potential in the transversal
plane (x,y) whereycorresponds to the poloidal direction and xto the radial one in a tokamak. We shall work with
the radially symmetric Flierl-Petviashvili soliton equat ion [27] studied in Ref. [13] :
/parenleftbig
1−ρ2
s∇2
⊥/parenrightbig∂ϕ
∂t+vd∂ϕ
∂y−vdϕ∂ϕ
∂y= 0 (1)
whereρs=cs/Ωi,cs= (Te/mi)1/2and the potential is scaled as ϕ=Ln
LTeeΦ
Te. HereLnandLTare respectively the
gradient lengths of the density and temperature. The veloci ty is the diamagnetic velocity vd=ρscs
Ln. The condition
for the validity of this equation is: ( kxρs) (kρs)2≪ηeρs
Ln, whereηe=Ln
LTe.
The exact solution of the equation is
ϕs(y,t;y0,u) =−3/parenleftbiggu
vd−1/parenrightbigg
sech2/bracketleftbigg1
2ρs/parenleftBig
1−vd
u/parenrightBig1/2
(y−y0−ut)/bracketrightbigg
(2)
where the velocity is restricted to the intervals u>v doru<0. The function is represented in Fig.1. In the Ref.
[13] the radial extension of the solution is estimated as: (∆ x)2∼ρsLn. In our work we shall assume that uis very
close tovd,u>∼vd(i.e. the solitons have small amplitudes).
The nonlinear equations for the drift waves are known to gene rate as solutions irregular turbulent fields but also
exact coherent structures of the type (2), depending on the i nitial conditions. Typical statistical quantities are the
correlations, like: /an}b∇acketle{tϕ(x,y,t)ϕ(x′,y′,t′)/an}b∇acket∇i}ht ∼ |x−x′|ζ|t−t′|z, where for the homogeneous turbulence the exponents
ζandzare calculated by the theory of renormalization or by spectr al balance equations, using closure methods (
[20]). Various closure methods have been developed as pertu rbations around gaussianity and they are valid for small
deviation from the gaussian statistics [17]. We see intuiti vely that this approach cannot be extended to the descriptio n
of the coherent structures. This can also be seen in more anal ytical terms. A quantity which is unavoidable in the
calculation of the correlations is the average of the expone ntial of a functional of the fluctuating field, consider simpl y
/an}b∇acketle{texp (ϕ)/an}b∇acket∇i}ht(for example in the inverse of the Vlasov operator, using the Fourier transformation, the potential appears
in the formal expression of the trajectory, i.e. at the expon ent). This quantity can be written schematically as [17]:
/an}b∇acketle{texp(ϕ)/an}b∇acket∇i}ht= exp/bracketleftBigg/summationdisplay
n1
n!/an}b∇acketle{t/an}b∇acketle{tϕn/an}b∇acket∇i}ht/an}b∇acket∇i}ht/bracketrightBigg
(3)
where /an}b∇acketle{t/an}b∇acketle{tϕn/an}b∇acket∇i}ht/an}b∇acket∇i}htrepresents the cumulant of order n(i.e. the irreducible part of the correlation, after substr acting the
combinations of the lower order cumulants). For a gaussian s tatistics the first two cumulants are different of zero
(n= 1 : average and n= 2 : dispersion), all others are zero. Non-vanishing of the h igher order cumulants is the
signature of non-gaussian statistics. In the perturbative renormalization we assume slight deviation from gaussiani ty,
i.e. small absolute values of the next order cumulants (e.g. the kurtosis must be close to 3, the gaussian value) and
vanishing of the higher order cumulants. This assumption is obviously invalid in the case of coherent structures. The
field of a coherent structure has long range, persistent corr elations imposed by its regular geometry, which naturally
requires non-vanishing very large order cumulants ( i.e.many terms in the sum at the exponent in Eq.(3)) and excludes
any perturbative expansion.
In particular, the closure of the nonlinear equation for the two-point correlation (based on the retaining the directly
interacting triplet) can account for the small scale correl ations related to the space-dependent relative diffusion, i .e.
the clump effect ( [18], [19], [20]), but the spectrum obtaine d in this framework cannot account for the possible
existence of the coherent structures. This clearly suggest s that we must find a different approach.
2III. COHERENT STRUCTURES IN A TURBULENT BACKGROUND
A. The outline of the method
We present the basic lines of an approach which can provide a s tatistical description of the coherent structure in
a turbulent background. The physical origin of this approac h is the observation that the non-linear equation whose
solution is the coherent structure (the vortex soliton) als o has classical drift waves as solutions, in the case of very
weak nonlinearity. In a certain sense (which will become mor e clear further on), the vortex soliton and the drift waves
belong to the same family of dynamical configurations of the p lasma. Our approach, which is designed to put in
evidence and to exploit this property, consists of the follo wing steps.
We start by constructing the action functional of the system . The dynamical equation is the Euler-Lagrange
equation derived from the condition of extremum of this func tional and the exact solution is the vortex soliton (2).
By using the exponential of the action we construct the gener ating functional of the irreducible correlations of ϕ.
This functional contains all the information on the coheren t structure and the drift turbulence. The correlations
are obtained via functional differentiations. This require s the formal introduction of a perturbation of the system,
through the interaction with an external current. Througho ut the work, this perturbation will be considered a small
quantity and finally it will be taken zero.
The generating functional is by definition a functional inte gral over all possible configurations of the system and
this integral must be calculated explicitely. The simplest thing to do is to determine the configuration of the system
(with space and time dependence) which extremises the actio n, by equating the first functional variation of the action
with zero and solving this equation: this will give the vorte x soliton (modified due to the small interaction term).
Then one should replace this solution in the expression of th e action. This is the lowest approximation and it does
not contain anything related to the drift wave trubulence.
At this point we can benefit of the particular physics of the dr ift waves. The vortex soliton is the exact solution of
the fully nonlinear equation and is a localized potential pe rturbation with regular, cylindrical symmetric form. The
linear drift waves are harmonic potential perturbations pr opagating with constant velocity (the diamagnetic velocit y
in the case of the drift poloidal propagation in tokamak). Al though the drift waves have very different geometry they
are solutions of the same equation as the vortex, but for negl igible magnitude of the nonlinear term. The drift waves
do not exactly realize the extremum of the action functional , but obtain an action very close to this extremum. This
means that the drift waves and the vortex soliton are close in the function space in the sense of the measure defined
by the exponential of the action. In other terms the drift wav es are in a functional neighbourhood of the vortex (for
this measure). This suggests to perform the functional inte gral with better approximation, which means to perform
the integration over a functional neighbourhood of the vort ex solution. This will automatically include the drift wave s
in the generating functional of correlation which so will co ntain information on both the coherent structure and the
drift waves. The function space neighbourhood over which th e functional integration is extended is limited by the
measure (exponential of the action) which severly penalize s all configurations of the system which are far from the
solution realizing the extremum (i.e. the vortex soliton). As in any stationary phase method there are oscillations
which strongly suppress the contribution of the configurati ons which are far (in the sense of the measure) from the
soliton. In practice we shall expand the action in a function al Taylor series around the soliton solution and keep the
term with the second functional derivative.
In this perspective the drift waves appear as fluctuations ar ound the soliton solution. This is compatible with the
numerical simulations which show that the vortices are acco mpagned by a tail of drift waves. During the interaction
of the vortices linear drift waves are “radiated” [21]. On th e other hand, the analytical treatment of the perturbed
vortex solution by the perturbed Inverse Scattering Transf orm shows similar tail of perturbed field, following the
soliton. This strengthens our argument that integrating cl ose to the vortex means to include the drift waves in the
generating functional.
The functional integral can be performed exactly and we dete rmine the generating functional of the potential
correlations. We shall calculate the two-point correlatio n by performing double functional derivative at the externa l
current.
B. Expansion around a soliton
1. The action and the generating functional of the correlati ons
The analytical framework is similar to the model of quantum fl uctuations around the instanton solution in the
semi-classical calculation of the transition amplitude fo r the particle in a two-well potential (see reference [32]). Let
3us write formally the equation for a nonlinear plasma waves a s
/hatwideOϕ= 0 (4)
where the field ϕ(x,y,t) represents the “field” (coherent structure + drift waves) a nd the operator /hatwideOis the nonlinear
operator of the equation (1). This equation should be derive d from the condition of extremum of an action functional
which must reflect the statistical nature of our problem. The fieldϕobeys a purely deterministic equation, but the
randomness of the initial conditions generates a statistic al ensemble of realizations of the system evolutions (space -
time configurations). We shall follow the Martin-Siggia-Ro se method of constructing the action functional but in
the path-integral formalism, for which we give in the follow ing a very short description ( [22], [23], [34]). First, we
consider a formal extension from the statistical ensemble o f realizations of the system’s space-time configurations to
a larger space of functions which may include even non-physi cal configurations. Every function is discretized in space
and time, so it will be represented as a collection of varable sϕi, each attached to the corresponding space-time point
i. In this space of functions, the selection of the configurati ons which correspond to the physical ones (solutions of
the equation of motion) is performed through the identificat ion with Dirac delta-functions, in every space-time point
/productdisplay
iδ[ϕi−ϕ(xi,yi,ti)]δ/bracketleftBig
/hatwideOϕ/bracketrightBig
(5)
and integration over all possible functions ϕ,i.e.over the ensemble of independent variables ϕi. Using the Fourier
representation for every δfunction we get
/integraldisplay/productdisplay
idϕi/integraldisplay/productdisplay
idχiexp/bracketleftBig
iχi/hatwideOϕ(xi,yi,ti)/bracketrightBig
(6)
Going to the continuum limit, a new function appears, χ(x,y,t) which is similar to the Fourier conjugate of ϕ. The
generating functional of the correlation functions is
Z=/integraldisplay
D[ϕ(x,t)]D[χ(x,t)] exp/braceleftbigg
i/integraldisplay
dx′dt′χ(x′,t′)/hatwideOϕ(x′,t′)/bracerightbigg
(7)
where the functional measures have been introduced and x≡(x,y).
The random initial conditions ϕ0(y) can be included by a Dirac δfunctional: δ(ϕ(t0,y)−ϕ0(y)). As explained in
[1], instead of this exact treatment (accessible only numer ically) we exploit the particularity of our approach, i.e.the
connection between the functional integration and the deli mitation of the statistical ensemble: the way we perform
the functional integration is an implicit choice of the stat istical ensemble. We choose to build implicitely the statis tical
ensemble, collecting all configurations which have the same type of deformations (given in our formulas by /tildewideχJ). All
these configurations belong to the neighbourhood of the extr emum in function space and we take them into account,
by performing the integration over this space. In doing so we assume that the ensemble of perturbed configurations
induced by an “external” excitation ( Jbelow) of the system is the same as the statistical ensemble o f the system’s
configurations evolving from random initial conditions.
We must add to the expression in the integrand at the exponent ial a linear combination related to the interaction
of the fields ϕandχwith external currents JϕandJχ:
Z→ZJ=/integraldisplay
D[ϕ(x,t)]D[χ(x,t)] exp{iSJ} (8)
SJ≡/integraldisplay
dx′dt′/bracketleftBig
χ(x′,t′)/hatwideOϕ(x′,t′) +Jϕϕ+Jχχ/bracketrightBig
It is now possible to obtain correlations by functional diffe rentiation, for example
/an}b∇acketle{tϕ(x2,y2,t2)ϕ(x1,y1,t1)/an}b∇acket∇i}ht=1
ZJδ2ZJ
δJϕ(x2,y2,t2)δJϕ(x1,y1,t1)/vextendsingle/vextendsingle/vextendsingle/vextendsingle
J=0(9)
For the explicit calculation of the generating functional w e need the functions ϕandχwhich extremize the action
δSJ
δϕ= 0 (10)
δSJ
δχ= 0
42. Schema of calculation of the generating functional
In the absence of the current Jthe equations (10) have as solutions for ϕthe nonlinear solitons (vortices) [30], [31].
More generally, the basic solution of the KdV equation (on wh ich the Flierl-Petviashvili equation can be mapped) is
the periodic cnoidal function which becomes, when the modul us of the elliptic function is close to 1, the soliton. When
the distance between the centres of the solitons is much larg er than their spatial extension (dilute gas) the general
solution can be written as a superposition of individual sol itons, with different velocities and different positions [13 ].
For simplicity we shall consider in this work a single vortex soliton and in the last Section we shall comment on the
extension of the method to many solitons.
The position of the centre of the soliton rises the difficult pr oblem of the zero modes [32]. Except for a brief comment
about the relation of the zero modes with the gaussian functi onal integration (see below), we shall avoid this problem
and postpone the discussion of this topic to a future work.
In the presence of the external current J, the equations resulting from the extremization of the acti onSJbecome
inhomogeneous, and the solutions are perturbed solitons. This point is technically non-trivial and we shal l use the
results obtained by Karpman [29] who considered the Inverse Scattering Transform method applied to the perturbed
soliton equation. We find the approximate solution ϕJsandχJsof the inhomogeneous equations (i.e. including the
external current J). The result depends on the currents J, and this will permit us to perform functional differentiati ons
in order to calculate the correlation, as shown in Eq.(9). As a first step in obtaining the explicit form of ZJ, the
perturbed soliton solutions depending on Jmust be introduced in the expression of the action SJ. After that we
perform the expansion of the functions ϕandχaround the coherent solution,
ϕ=ϕJs+δϕ (11)
χ=χJs+δχ
This gives
ZJ= exp (iSJs)/integraldisplay
D[δϕ]D[δχ]
×exp
/integraldisplay
dx′dt′δχ(x′,t′)
δ2/hatwideO
δϕδχ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
ϕJs,χJs
δϕ(x′,t′)
or
ZJ= exp (iSJs)1
2nin(2π)n/2
detδ2/hatwideO
δϕδχ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
ϕJs,χJs
−1/2
(12)
since the integral is gaussian [24]. The determinant is calc ulated using the eigenvalues
δ2/hatwideO
δϕδχ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
ϕJs,χJs
ψn(x,t) =λnψn(x,t) (13)
and
det
δ2/hatwideO
δϕδχ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
ϕJs,χJs
=/productdisplay
nλn (14)
Since the action is invariant to the arbitrary position of th e centre of the soliton there are directions in the function
space where the fluctuations are not bounded and in particula r are not Gaussian. This requires the introduction
of a set of collective coordinates and after a change of varia bles the functional integrations along those particular
directions are replaced by usual integrations over the cole ctive variables, with inclusion of Jacobian factors. The
zero eigenvalues of the determinant (corresponding to the zero modes ) are excluded in this way. We shall avoid this
complicated problem and assume a given position for the cent re of the vortex.
5IV. APPLICATION TO THE VORTEX SOLUTION OF THE NONLINEAR DRIF T WAVE
A. The action functional
In order to adimensionalize the equation (1) we introduce th e space and time scales t→Ω−1tandy→ρsyand the
equation becomes
/parenleftbig
1− ∇2
⊥/parenrightbig∂ϕ
∂t+/parenleftbiggvd
Ωρs/parenrightbigg∂ϕ
∂y−/parenleftbiggvd
Ωρs/parenrightbigg
ϕ∂ϕ
∂y= 0 (15)
For simplicity of notation we keep the symbol vdfor the adimensional velocity/parenleftBig
vd
Ωρs/parenrightBig
. The equation does not change
of form but now all variables are adequately normalized and t heaction
S=/integraldisplay
dydtLϕ (16)
is also adimensional.
We have to calculate explicitely the scalar function χ. Based on the extended knowledge developed in field theory it
seems reasonable to assume that this function represents th e generalization of the functions which have the opposite
evolution compared to ϕ: ifϕevolves toward infinite time, then χcomes from infinite time toward the initial time.
Ifϕdiffuses then χanti-diffuses (see Ref. [33]). The general characteristics of this behaviour suggest to represent
χas the object with the opposite topology than ϕ. Ifϕhas a certain topological class, then χhas the opposite
topological class. If ϕis an instanton then χis an anti-instanton. In our case: if ϕis the vortex solution, then χmust
the “anti-vortex” solution, with everywhere opposite vort icity compared to ϕ. In our case of a single vortex, χmust
simply be a negative vortex.
In general terms, the direct ( i.e.the vortex + random drift waves) solution ϕarises from an initial perturbation
which evolving in time breaks into several distinct vortice s (solitons) and a tail of drift waves, as shown by the Inverse
Scattering Method. The functionally conjugated (“regress ive”) function χis att=∞a collection of vortices and
drift wave turbulence which evolving backward in time, towa rdt= 0, coalesce and build up into a single perturbation,
the same as the initial condition of ϕ. We can restrict our analysis to the time range where the two f unctions has
similar patterns (but opposite) which simply means to chose the time interval far from the initial and asymptotic
limits. As shown by analytical and numerical studies, the vo rtices (positive and negative) are robust patterns and the
time evolution simply consists of translations without dec ay. In conclusion we can take for the time range far from
the boundaries t= 0 andt=T:
χ=−ϕ (17)
To see this more clearly, we write down the action and then the Euler-Lagrange equations, with the current J
included.
SJ[χ,ϕ] =/integraldisplayL
0dy/integraldisplayT
0dtLJ,ϕ (18)
with the notation
LJ,ϕ=χ/bracketleftbigg/parenleftbig
1− ∇2
⊥/parenrightbig∂ϕ
∂t+vd∂ϕ
∂y−vdϕ∂ϕ
∂y/bracketrightbigg
+Jϕϕ+Jχχ (19)
When performing integrations by parts the boundary conditi ons of the two functions prevents us from taking the
integrals of exact differentials as vanishing, but this just produces terms which do not contribute to the determination
of the solution of extremum. We shall first change the Eq.(19) such as to obtain by functional extremization an
(Euler-Lagrange) equation for the function χ:
L(1)
J,ϕ=χ∂ϕ
∂t−/bracketleftbig
∇2
⊥χ/bracketrightbig∂ϕ
∂t+vdχ∂ϕ
∂y−vdχϕ∂ϕ
∂y+Jϕϕ+Jχχ (20)
Now we write the condition of extremum for the action functio nal and obtain the Euler-Lagrange equation
d
dtδL(1)
J,ϕ
δ/parenleftBig
∂ϕ
∂t/parenrightBig+d
dxδL(1)
J,ϕ
δ/parenleftBig
∂ϕ
∂x/parenrightBig+d
dyδL(1)
J,ϕ
δ/parenleftBig
∂ϕ
∂y/parenrightBig−δL(1)
J,ϕ
δϕ= 0 (21)
6This equation can be written
/parenleftbig
1− ∇2
⊥/parenrightbig∂χ
∂t+vd∂χ
∂y−vdϕ∂χ
∂y=Jϕ (22)
An equivalent form of the action is
SJ[χ,ϕ] =/integraldisplayL
0dy/integraldisplayT
0dtLJ,χ (23)
with
LJ,χ=−ϕ∂χ
∂t+/parenleftbig
∇2
⊥ϕ/parenrightbig∂χ
∂t−vdϕ∂χ
∂y+vdϕ2
2∂χ
∂y+Jϕϕ+Jχχ (24)
The equation Euler-Lagrange for the function χis obtained from the extremum condition on the functional Eq .(23)
d
dtδLJ,χ
δ/parenleftBig
∂χ
∂t/parenrightBig+d
dxδLJ,χ
δ/parenleftBig
∂χ
∂x/parenrightBig+d
dyδLJ,χ
δ/parenleftBig
∂χ
∂y/parenrightBig−δLJ,χ
δχ= 0 (25)
This equation reproduces the nonlinear vortex equation wit h an inhomogeneous term:
/parenleftbig
1− ∇2
⊥/parenrightbig∂ϕ
∂t+vd∂ϕ
∂y−vdϕ∂ϕ
∂y=−Jχ (26)
Comparing the homogeneous equations (22) (with Jϕ= 0) and (26) (with Jχ= 0) we see that
χ=−ϕ (27)
is indeed the solution of the homogeneous equation (22) i.e.the negative vortex is the solution for χ.
We must remember that the “external” currents are arbitrary and later, after functional differentation, they will
be taken zero. This allows us to start from the configurations given by the homogeneous equations and Eq.(27) and
study the small changes using perturbative methods develop ed in the framework of the Inverse Scattering Transform.
We will only use the current Jϕwhich will be denoted Jand already take Jχ= 0.
The final form of the action which will be used later in this wor k is
SJ[χ,ϕ] =/integraldisplayL
0dy/integraldisplayT
0dt/braceleftbigg
χ/parenleftbig
1− ∇2
⊥/parenrightbig∂ϕ
∂t+vdχ∂ϕ
∂y−vdχϕ∂ϕ
∂y+Jϕ/bracerightbigg
(28)
B. The condition of extremum of the action functional
The Euler - Lagrange equations for the two functions χandϕare obtained from the first functional derivative of
the actionSJ:δSJ/δχ= 0 andδSJ/δϕ= 0. The first equation (which is the original equation) has th e solution
(2). It does not depend on the current J(since the corresponding current Jχhas been taken zero). However, for
uniformity of notation we shall write ϕJs
ϕJs(x,y,t)≡ϕs(x,y,t) (29)
The second Euler-Lagrange equation is the equation for χ, with the inhomogeneous term given by the current J:
/parenleftbig
1− ∇2
⊥/parenrightbig∂χ
∂t+vd∂χ
∂y−vdϕ∂χ
∂y=J (30)
The solution is:
χJs(x,y,t) =−ϕs(x,y,t) +/tildewideχJ(x,y,t) (31)
where −ϕs(x,y,t) represents the “free” solution of the variational equatio n,i.e.the negative vortex (anti-soliton)
and/tildewideχJ(x,y,t) is the small modification induced by an inhomogeneous small term,J(x,y,t). Since the function
/tildewideχJ(x,y,t) is the perturbation of the negative-vortex solution we wil l use the equation (26) but with the opposite
current ( i.e.−Jinstead ofJ), as (30) requires.
7C. Second order functional expansion and the eigenvalue pro blem for the calculation of the Determinant
Now we shall expand the action SJ[ϕ] to second order around the saddle-point solution. Write
ϕ=ϕJs+δϕ (32)
χ=χJs+δχ
where the function ( δϕ,δχ ) is a small difference from the extremum solution.The expand ed form of the action will be
written:
SJ[χ,ϕ] =SJ[ϕJs,χJs] +1
2/parenleftBigg
δ2SJ
δϕδχ/vextendsingle/vextendsingle/vextendsingle/vextendsingle
ϕJs,χJs/parenrightBigg
δϕδχ
where obviously the absence of the linear term is due to the fa ct that (ϕJs,χJs) is the solution at the extremum and
SJ[ϕJs,χJs] =/integraldisplayL
0dy/integraldisplayT
0dt/bracketleftbigg
χJs∂ϕJs
∂t−/parenleftbig
∇2
⊥χJs/parenrightbig∂ϕJs
∂t(33)
+vdχJs∂ϕJs
∂y−vdχJsϕJs∂ϕJs
∂y+JϕJs
Few manipulations are necessary to make the second function al variation of SJsymmetric in δϕandδχ. Again
this will imply boundary terms, but these are now zero since t he variations δϕandδχvanishes at the limits of the
space-time domain, by definition. The transformations are s imply integrations by parts and give
1
2δχ/parenleftBigg
δ2SJ
δϕδχ/vextendsingle/vextendsingle/vextendsingle/vextendsingle
ϕJs,χJs/parenrightBigg
δϕ=1
2⌢
δϕ δχ
⌣/parenleftbigg
/hatwideγ−/hatwideα−/hatwideβ
/hatwideα−/hatwideβ 0/parenrightbigg/parenleftbigg
δϕ
δχ/parenrightbigg
(34)
where
/hatwideα=/parenleftbig
1− ∇2
⊥/parenrightbig∂
∂t+vd∂
∂y−vd/parenleftbigg∂ϕJs
∂y/parenrightbigg
(35)
/hatwideβ=1
2vd/parenleftbigg∂ϕJs
∂y/parenrightbigg
/hatwideγ=−2vdχJs∂
∂y
In the generating functional of the correlations, the expan sion gives, after performing the Gaussian integral:
ZJ= exp(iSJ)1
in(2π)n/2/bracketleftBigg
det/parenleftBigg
δ2SJ
δϕδχ/vextendsingle/vextendsingle/vextendsingle/vextendsingle
ϕJs,χJs/parenrightBigg/bracketrightBigg−1/2
(36)
As stated before, the det will be calculated as the product of the eigenvalues λn
det/parenleftBigg
δ2SJ
δϕδχ/vextendsingle/vextendsingle/vextendsingle/vextendsingle
ϕJs,χJs/parenrightBigg
=/productdisplay
nλn (37)
We must find the eigenvalues of the differential operator appe aring in Eq.(34):
/parenleftbigg
/hatwideγ−/hatwideα−/hatwideβ
/hatwideα−/hatwideβ 0/parenrightbigg/parenleftbigg
ψϕ
n
ψχ
n/parenrightbigg
=λn/parenleftbigg
ψϕ
n
ψχ
n/parenrightbigg
(38)
which gives the following equation
/bracketleftbigg
/hatwideγ−1
λn/parenleftBig
/hatwideα2+/hatwideβ/hatwideα−/hatwideα/hatwideβ−/hatwideβ2/parenrightBig/bracketrightbigg
ψϕ
n=λnψϕ
n (39)
8The functions δϕ(y,t) andδχ(y,t) represent the differences between the solutions at extremu m (solitons) and other
functions which are in a neighbourhood (in the function spac e) of the solitons. According to the discussion above,
the functions which are “close” to the solitons, for the Flie rl-Petviashvilli equation are drift waves. For this reason
the operator which represents the dispersion ( i.e.∇2
⊥) will be replaced with its simplest form , −k2
⊥for these waves,
withk⊥representing an average normalized wavenumber for the pure drift turbulence. However, the operator will be
retained when applied on the functions related to solitons, since these solutions owe their existence to the balance of
nonlinearity and dispersion.The following detailed expre ssions are obtained for the operators involved in this equat ion:
/hatwideα2=/bracketleftbigg/parenleftbig
1− ∇2
⊥/parenrightbig∂
∂t+vd∂
∂y−vd/parenleftbigg∂ϕJs
∂y/parenrightbigg/bracketrightbigg2
(40)
=/parenleftBig
1 +k2
⊥/parenrightBig2/parenleftbigg∂
∂t/parenrightbigg2
−vd/bracketleftbigg/parenleftbig
1− ∇2
⊥/parenrightbig∂ϕJs
∂t/bracketrightbigg∂
∂y
+vd(1−ϕJs)/parenleftBig
1 +k2
⊥/parenrightBig∂2
∂t∂y+vd(1−ϕJs)/parenleftBig
1 +k2
⊥/parenrightBig∂2
∂y∂t
−v2
d(1−ϕJs)/parenleftbigg∂ϕJs
∂y/parenrightbigg∂
∂y
+v2
d(1−ϕJs)2∂2
∂y2
/hatwideβ/hatwideα=1
2vd/parenleftbigg∂ϕJs
∂y/parenrightbigg/parenleftBig
1 +k2
⊥/parenrightBig∂
∂t+1
2v2
d/parenleftbigg∂ϕJs
∂y/parenrightbigg
(1−ϕJs)∂
∂y(41)
/hatwideα/hatwideβ=1
2vd/bracketleftbigg/parenleftbig
1− ∇2
⊥/parenrightbig∂2ϕJs
∂t∂y/bracketrightbigg
(42)
+1
2v2
d(1−ϕJs)/parenleftbigg∂2ϕJs
∂y2/parenrightbigg
+1
2v2
d(1−ϕJs)/parenleftbigg∂ϕJs
∂y/parenrightbigg∂
∂y
+1
2vd/parenleftbigg∂ϕJs
∂y/parenrightbigg/parenleftBig
1 +k2
⊥/parenrightBig∂
∂t
/hatwideβ2=1
4v2
d/parenleftbigg∂ϕJs
∂y/parenrightbigg2
(43)
The square brakets are used to underline that the differentia l operators are not acting outside and the only operation
is multiplication. We use the equation verified by ϕJsto make the following replacement
/bracketleftbigg/parenleftbig
1− ∇2
⊥/parenrightbig∂ϕJs
∂t/bracketrightbigg
=−vd(1−ϕJs)/parenleftbigg∂ϕJs
∂y/parenrightbigg
(44)
The equation becomes
−/braceleftbigg/parenleftBig
1 +k2
⊥/parenrightBig2∂2
∂t2+ 2/parenleftBig
1 +k2
⊥/parenrightBig
vd(1−ϕJs)∂2
∂y∂t(45)
+v2
d(1−ϕJs)2∂2
∂y2−3
4v2
d/parenleftbigg∂ϕJs
∂y/parenrightbigg2/bracerightBigg
ψϕ
n
+λn/parenleftbigg
−2vdχJs∂
∂y/parenrightbigg
ψϕ
n
=λ2
nψϕ
n
We now take into account the propagating nature of the drift w aves and make the change of variables t→t
andy→y−vdti.e.we change to the system of reference moving with the diamagne tic velocity. We simplify the
9equation assuming that the most important space-time varia tion is wave-like and replace∂
∂t=−vd∂
∂y. By this change
of variables the soliton will not be at rest in the new referen ce system, but it will move very slowly since we have
assumed that u>∼vd. We make another approximation by neglecting the slow motio n of the soliton. This restricts us
to the wavenumber spectrum but considerably simplifies the c alculations. The space variable which will be denoted
againymeasures the space from the fixed center of the soliton, in the moving system. The difference between the KdV
soliton, which is one-dimensional and depends exclusively onyand the vortex which is a two-dimensional structure
will be considered in the simplest form as described by the es timation of Meiss and Horton for the x- extension of
the vortex. For convenience we suppress the index nand replace ψϕ
nbyq.
/braceleftbigg/bracketleftBig/parenleftBig
1 +k2
⊥/parenrightBig
vd−vd(1−ϕJs)/bracketrightBig2∂2
∂y2(46)
+ (2λvdχJs)∂
∂y
+/bracketleftBigg
λ2−3
4v2
d/parenleftbigg∂ϕJs
∂y/parenrightbigg2/bracketrightBigg/bracerightBigg
q
= 0
We have a suggestive confirmation that the generating functi onZJ(via the action SJ) potentially contains con-
figurations of the system consisting of simple drift waves. A perturbation consisting of drift waves and propagating
with the diamagnetic velocity vdis an approximate solution of the original equation for smal l amplitude ( i.e.small
nonlinearity). Due to its particular structure, the Martin -Siggia-Rose action functional is exactly zero when calcul ated
with the exact solution, in the absence of any external curre ntJ. The action expanded to the second order then gives,
for no vortex ( ϕJs= 0,χJs= 0)
∂2
∂y2+/parenleftBigg
λ
k2
⊥vd/parenrightBigg2
q= 0 (47)
which implies periodic oscillations in the space variable ywith (recall that everything is adimensional)
λ=kyvd/parenleftBig
k2
⊥/parenrightBig
(48)
Returning to the equation (46) we write it in the following fo rm
/parenleftbigg∂2
∂y2+A∂
∂y+B/parenrightbigg
q= 0 (49)
where
A≡2λ
vdχJs/parenleftBig
k2
⊥+ϕJs/parenrightBig2(50)
B≡λ2
v2
d−3
4/parenleftBig
∂ϕJs
∂y/parenrightBig2
/parenleftBig
k2
⊥+ϕJs/parenrightBig2
Now we make the standard transformation of the unknown funct ion
q=wexp/parenleftbigg
−1
2/integraldisplayy
A(y′)dy′/parenrightbigg
(51)
and obtain
w′′+/parenleftbigg
B−A′
2−A2
4/parenrightbigg
w= 0 (52)
where prime means derivation with respect to y. After replacing the two extremum solutions ϕJsandχJsfrom
equations (29) and (31) this equation is written in the follo wing form, to exhibit the dependence on λ:
10w′′+/parenleftbig
λ2t1+λt2+t3/parenrightbig
w= 0 (53)
with the notations
t1(y)≡1
v2
dh2−ϕ2
s
h4+2
vdϕs
h4/tildewideχJ (54)
t2(y)≡ −1
vd/parenleftbigg∂ϕs
∂y/parenrightbigg2c−h
h3+2
vd/parenleftBig
∂ϕs
∂y/parenrightBig
h3/tildewideχJ−1
vd1
h2/parenleftbigg∂/tildewideχJ
∂y/parenrightbigg
(55)
t3(y)≡ −3
41
h2/parenleftbigg∂ϕs
∂y/parenrightbigg2
(56)
and
c≡k2
⊥ (57)
h=c+ϕs
The functions ti(y) are represented for i= 1,2 in Figure 2 and 3. The function
U(λ;y)≡λ2t1+λt2+t3 (58)
has singularities at the points where hvanishes. We introduce the notation yhfor the location of the singularities,
taking into account the symmetry around y= 0, the centre of the soliton
h(±yh) = 0. (59)
Since the soliton is very localized, the function Uhas very fast variations close to the singularities. The slo w variation
of the function U(λ;y) over most of the space interval ( −L/2,+L/2) becomes very fast due to the growth of the
absolute values of t1,t2andt3near±yh, on spatial intervals having an extension of the order of the spatial unit,
i.e.ρsin physical terms. Since the physical model leading to our or iginal equation cannot accurately describe the
physical processes at such scales, we shall adopt the simple st approximation of U, assuming that it reaches infinite
absolute value at points which are located whithin a distanc e ofρsof the actual positions of the singularities, ±yh. We
have checked that the exact position of the assumed infinite value ofUhas no significant impact on the final results,
which can be explained by observing that t1,2,3will be integrated on. The total space interval is now divide d into
three domains: ( −L/2,−yh) (external left), ( −yh,yh) (internal) and ( yh,L/2) (external right). Here “internal” and
“external” refer to the region approximatly occupied by the soliton. The form of the function Uimposes the function
wto vanish at the limits of these domains. In a more general per spective, the fact that wbehaves independently on
each domain has a consequence with statistical mechanics in terpretation: the generating functional (similar to any
partition function) is obtained by integrating over the ful l space of the system’s physical configurations and behaves
multiplicatively for any splitting of the whole function sp ace into disjoint subspaces. In particular the functional
integration over the space of functions δϕandδχactually consists of three functional integrations over th e disjoint
function subspaces corresponding to the three spatial doma ins. The fact that our physical model is restricted to
spatial scales larger than ρsnecessarly has an impact on the maximum number of eigenvalue sλnthat should be
retained in the infinite product giving the determinant, but we shall not need to use this limitation.
For absolute values of the parameter λgreater than unity (which will be confirmed a posteriori , by the expressions
(62) and (71) below), the three terms in the expression of Uhave very different contributions. The terms t3is
practically negligible, and the term with t1is always much greater than t2in absolute value. In the following we
consider separately the three domains.
On the “external left” domain, the function t1is positive. If we fix at zero the amplitude and the phase of wat the
limit−L/2 the condition that the solution vanishes at −yhgives, forλreal,
/integraldisplay−yh
−L/2dy′/parenleftbig
λ2t1+λt2+t3/parenrightbig1/2= 2πn (60)
In the integrand, the first term is factorized and, taking int o account the relative magnitude of the terms, we expand
the square root and obtain
11λl
nα1+β1+γ1
λln= 2πn (61)
i.e.
λl
n=2πn
α1/parenleftbigg
1−β1/(2π)
n/parenrightbigg
(62)
where
α1=/integraldisplay−yh
−L/2dy′/radicalbig
t1(y′) (63)
β1=/integraldisplay−yh
−L/2dy′t2(y′)/radicalbig
t1(y′)(64)
γ1=/integraldisplay−yh
−L/2dy′t3(y′)/radicalbig
t1(y′)(65)
andγ1has been neglected. We note that β1is positive.
On the “external right” domain the function t1is positive but t2is negative. The condition on the phase is
/integraldisplayL/2
yhdy′/parenleftbig
λ2t1+λt2+t3/parenrightbig1/2= 2πn′(66)
and introduce similar notations
α2=/integraldisplayL/2
yhdy′/radicalbig
t1(y′) =α1 (67)
β2=/integraldisplayL/2
yhdy′t2(y′)
2/radicalbig
t1(y′)=−β1 (68)
γ2=/integraldisplayL/2
yhdy′t3(y′)
2/radicalbig
t1(y′)(69)
The equation then becomes
λr
n′α2+β2+γ2
λr
n′= 2πn′(70)
or
λr
n′=2πn′
α2/parenleftbigg
1 +β1/(2π)
n′/parenrightbigg
(71)
The infinite product of eigenvalues gives, for the “external ” region [28]:
/productdisplay
nλl
n/productdisplay
n′λr
n′=/productdisplay
n/parenleftbigg2πn
α1/parenrightbigg2/productdisplay
n/parenleftBigg
1−β2
1/(2π)2
n2/parenrightBigg
(72)
=sin (β1/2)
β1/2/productdisplay
n/parenleftbigg2πn
α1/parenrightbigg2
In the “internal” region, the function t1is negative. The relations between the magnitudes of the abs olute values
of the functions t1,t2andt3are preserved. Then λwill be complex. Due to the anti-symmetry of the function t2we
can suppose that the unknown function wtakes zero value at y= 0. We introduce the notations
12αc=/integraldisplayyh
0dy′/radicalbig
−t1(y′) (73)
βc=/integraldisplayyh
0dy′t2(y′)
2/radicalbig
−t1(y′)(74)
γc=/integraldisplayyh
0dy′t3(y′)
2/radicalbig
−t1(y′)(75)
which are realnumbers. The condition
λi
nαc+βc+γc
λin= 2πin (76)
gives (after neglecting γc) for the complex number λi
n:
λi
n=α−1
c(2πn)/parenleftBigg
1 +β2
c
(2π)2n2/parenrightBigg1/2
exp/bracketleftbigg
−iarctan/parenleftbigg2πn
βc/parenrightbigg/bracketrightbigg
(77)
The infinite product of these eigenvalues is
/productdisplay
nλi
n=/productdisplay
nα−1
c(2πn)exp/bracketleftbigg
−iarctan/parenleftbigg2πn
βc/parenrightbigg/bracketrightbigg/productdisplay
n/parenleftBigg
1 +β2
c/(2π)2
n2/parenrightBigg1/2
(78)
The number βcis smaller than unity and for large nthe argument of the exponential will be more and more close to
−iπ/2. We make the approximation that the exponential can be repl aced with −i. Then we obtain
/productdisplay
nλi
n=/bracketleftbiggsinh (βc/2)
βc/2/bracketrightbigg1/2/productdisplay
n(−i)2πn
αc(79)
On the “external” regions the functions t1,t2are not symmetrical around the centre y= 0 since the perturbed
soliton develops a “tail” which is not symmetrical. However we take this perturbation to be small and assume the
same absolute value for the function β1on both external domains.
We remark that we remain with two quantities in which all the f unctional depencence on the current Jis packed:
for “exterior” β1(hereafter denoted σ) and for “interior” βc(hereafter denoted β).
ZJ= exp (iSJ)/parenleftBigg/productdisplay
n(2π)
i/parenrightBigg/bracketleftBigg
det/parenleftBigg
δ2SJ
δϕδχ/vextendsingle/vextendsingle/vextendsingle/vextendsingle
ϕJs,χJs/parenrightBigg/bracketrightBigg−1/2
(80)
=const exp(iSJ)/bracketleftbiggβ/2
sinh (β/2)/bracketrightbigg1/4/bracketleftbiggσ/2
sin (σ/2)/bracketrightbigg1/2
where
const =/productdisplay
n/parenleftbigg(−i)αc
2πn/parenrightbigg1/2α1
n(81)
will disappear after the normalizations required by the cal culation of the correlations (see below).
V. CALCULATION OF THE CORRELATIONS
The two-point correlation can be obtained by a double functi onal differentiation at the external current J.
/an}b∇acketle{tϕ(y2)ϕ(y1)/an}b∇acket∇i}ht=Z−1
Jδ2ZJ
iδJ(y2)iδJ(y1)/vextendsingle/vextendsingle/vextendsingle/vextendsingle
J=0
13The main achivement of this approach is that it provides the e xplicit expression of the generating functional. We
introduce the notations
A=A[J]≡/bracketleftbiggβ/2
sinh (β/2)/bracketrightbigg1/4
(82)
B=B[J]≡/bracketleftbiggσ/2
sin (σ/2)/bracketrightbigg1/2
(83)
and drop the factor const; actually the latter depends on α1andαcand thus on the current Jand contributes to the
functional derivatives. However, taking a formal limit Nto the number of factors in (81) we find that the functional
derivatives of α1andαcgive additive terms which vanish in the limit N→ ∞. Then we drop const since it disappears
after dividing to ZJand taking J≡0. In this way (80) becomes
ZJ= exp (iSJ)AB (84)
We calculate the functional derivatives.
δZJ
iδJ(y1)=/bracketleftbiggδSJ
δJ(y1)+1
AδA
iδJ(y1)+1
BδB
iδJ(y1)/bracketrightbigg
exp (iSJ)AB (85)
We will also need the functional derivative at J(y2), with a similar expression.The second derivative:
Z−1
Jδ2ZJ
iδJ(y2)iδJ(y1)/vextendsingle/vextendsingle/vextendsingle/vextendsingle
J=0=δSJ
δJ(y2)δSJ
δJ(y1)+δ2SJ
iδJ(y2)δJ(y1)(86)
+1
AδA
iδJ(y2)δSJ
δJ(y1)+1
BδB
iδJ(y2)δSJ
δJ(y1)
+1
AδA
iδJ(y1)δSJ
δJ(y2)+1
BδB
iδJ(y1)δSJ
δJ(y2)
+1
AδA
iδJ(y1)1
BδB
iδJ(y2)+1
AδA
iδJ(y2)1
BδB
iδJ(y1)
+1
Aδ2A
iδJ(y2)iδJ(y1)+1
Bδ2B
iδJ(y2)iδJ(y1)
The detailed expressions of these terms are given in the Appe ndix. The terms are calculated numerically using the
detailed expressions of ϕs,∂ϕs
∂y,/tildewideχJand∂/tildewideχJ
∂y.
The first term reproduces the self-correlation of the solito n and represents the connection with the results of Ref.
[13], with our particular simplifications: single soliton a nd fixed (non-random) position of its centre. As can easily
be seen, the first order functional derivatives of SJto the current Jreduce to the function ϕscalculated in the
corresponding points. The term with the double functional d erivative of the action represents the contribution to the
self-correlation of the soliton due to a statistical ensemb le of initial conditions, without drift waves. All mixed ter ms
(i.e.containing both the action and one of the factors AorB) represent interaction between the perturbed soliton
and the drift waves. The terms containing exclusively the fa ctorsAand/orBrefers to the drift waves in the presence
of the perturbed soliton.
VI. DISCUSSION AND CONCLUSIONS
The formulas obtained by functional differentiation of the g enerating functional are complicated and a numerical
calculation is necessary. We chose a particular value of the soliton velocity (which also fixes its amplitude): u= 1.725vd
and let the variables y1andy2sample the one-dimensional volume of length L= 0.2m. The physical parameters are
chosen such that ρs≈10−3mandvd≈571m/s. We recall that there are two particular symmetry limitatio ns of our
calculation. (1) The soliton centre is assumed fixed (at y= 0) , especially for avoiding the complicated problem of
thezero modes . (2) Due to the asymmetry of the perturbed soliton tail the te rms which results from the functional
differentiation are also asymmetric. These are only limitat ions of our calculation and in no way reflect the reality of a
isotropic motion of many solitons in a real turbulent plasma . In order to see to what extent our result can be useful
14for understatnding the (much more complicated) real situat ion we will symmetrize these terms in the unique mode
which is accessible to our one-dimensional calculation, i. e. take into account the mixing of perturbed solitons moving
in the two directions on the line.
The amplitude of the modifications of the soliton depends on a parameter which is the average time of interaction
with the perturbation. This average time is comparable with the time required to cross Lat a speed of vdand is
limited since the growth of the perturbation cannot exceed t he soliton itself.
The figures are conventional representations of functions of two varia bles(y1,y2); they do not correspond to a
two-dimensional geometry. For this reason it is not expecte d to have circular symmetry. The contributions to the
correlation from the last two factors in Eq.(86) have amplit udes similar or less by a factor of few units, compared to
the pure soliton. The factors coming from “internal” part ar e peaked and localized on the soliton extension while the
“external” part gives terms oscillating on ( y1,y2). In wavenumber space, there are contributions to both low- kand
high-kregions. The spectrum of an unperturbed soliton is smooth an d monotonously decreasing from the peak value
atk=0. Fig.5 shows much more structure. In the low- kpart there are many local peaks, an effective manifestation
of the periodic character of the terms (as shown by (72) ). Thi s arises from the discrete nature of the eigenvalues,
which is induced by the second order differential operator an d the vanishing of the eigenmodes at the positions of the
singularities ≈ ±yh. The singularities are generated by the vanishing of the nor m of the operator /hatwideα, which makes
ambiguous the assumption of propagating wave character, ∂t=−vd∂y. The large- kpart mainly reflects the structure
of the small-scale shape perturbation of the soliton, commi ng fromβ-related terms. Fig.6 is an ( k,ω) spectrum
obtained from ω−ku= 0 and repeating the calculations for various soliton veloc itiesumax>u >v d. Although we
cannot afford high umaxsince the expressions of t1,2,3(y) depend on the assumption u>∼vd, we remark local peaks
in contrast to the “pure soliton” result of Ref. [13].
For simplicity we have assumed a single soliton. However the calculation can be readily extended to the multi-
soliton case, considering instead of (29) and (31) sums over many individual soliton solutions with different velocitie s
and positions of the centres. These sums replace the functio nsϕJsandχJsin the expressions of the operators /hatwideα,/hatwideβ
and/hatwideγ. If the velocities are all greater but not too different of vdthe change of variables to the referential moving
withvd(described in the paragraph below Eq.(45) ) will leave a very slow time variation which eventually may be
treated perturbatively. Many solitons will also generate m any singularities arising from the vanishing of the functio n
h, and this will factorize the space of functions and correspo ndingly the generating functional. It will become however
possible to consider random positions and random velocitie s and average them with distribution functions for the
Gibbs ensemble, like in [13]. This is very simple with the firs t term of (86), which should be compared directly with
Ref. [13], but technically very difficult with the terms invol ving functional derivatives of Aand/orB.
The first results suggests that the non-gaussianity at the pl asma edge can be explained by the presence of coherent
structures. The contribution of avalanches to the deviatio n from the gaussian statistics cannot be excluded but, as
shown for self-organized systems [36], they have a scaling w hich should be easy recognized, at least in frequency
domain.
In conclusion we have developed an approach which allows us t o calculate the statistical properties of a coherent
structure in a turbulent background. Compared to the standa rd renormalization, this approach is at the opposite limit
in what concerns the relation “coherent structure / wave tur bulence”, highlightning the coherent structure. However
it offers comparatively greater possibilities for the exten sion of this studies to the more realistic problem of cascadi ng
wave turbulence mixed with rising and decaying coherent str uctures.
Acknowledgments . The authors are indebt to J.H. Misguich and R. Balescu for ma ny stimulating and enlightening
discussions. F. S. and M.V. gratefully acknowledge the supp ort and hospitality of the D´ epartement de Recherche sur
la Fusion Control´ ee , Cadarache, France.
This work has been partly supported by the NATO Linkage Grant CRG.LG 971484.
APPENDIX A: EXPLICIT EXPRESSIONS FOR THE FUNCTIONAL DERIVA TIVES
We shall first concentrate on the derivatives of the two facto rsAandB.
δB
δJ(y1)=δ
δJ(y1)/bracketleftbiggσ/2
sin (σ/2)/bracketrightbigg1/2
(A1)
=1
4/bracketleftbigg1
sin (σ/2)+σ
2cos(σ/2)
sin2(σ/2)/bracketrightbigg/bracketleftbiggσ/2
sin (σ/2)/bracketrightbigg−1/2/parenleftbiggδσ
δJ(y1)/parenrightbigg
and
15δ2B
δJ(y2)δJ(y1)=δ2
δJ(y2)δJ(y1)/bracketleftbiggσ/2
sin (σ/2)/bracketrightbigg1/2
(A2)
=/braceleftBigg
−1
8/bracketleftbiggσ/2
sin (σ/2)/bracketrightbigg1/21 + cos2(σ/2)
sin2(σ/2)
−1
16/bracketleftbiggσ/2
sin (σ/2)/bracketrightbigg−3/2/bracketleftbigg1
sin (σ/2)+σ
2cos(σ/2)
sin2(σ/2)/bracketrightbigg2/bracerightBigg/parenleftbiggδσ
δJ(y2)/parenrightbigg/parenleftbiggδσ
δJ(y1)/parenrightbigg
+1
4/bracketleftbiggσ/2
sin (σ/2)/bracketrightbigg−1/2/bracketleftbigg1
sin(σ/2)+σ
2cos(σ/2)
sin2(σ/2)/bracketrightbigg /parenleftbiggδ2σ
δJ(y2)δJ(y1)/parenrightbigg
For the exterior domains,
σ=σ0+/tildewideσJ1+/tildewideσJ2 (A3)
with
σ0=1
2/integraldisplay−yh
−L/2dy′/bracketleftBigg
−/parenleftbigg∂ϕs
∂y/parenrightbigg2c
h−1
(h2−ϕ2s)1/2/bracketrightBigg
(A4)
/tildewideσJ1=1
2/integraldisplay−yh
−L/2dy′/bracketleftBigg/parenleftbigg∂ϕs
∂y/parenrightbigg1
h(h2−ϕ2s)1/2/parenleftbigg
2−ϕs(2c−h)
h2−ϕ2s/parenrightbigg
/tildewideχext
J/bracketrightBigg
(A5)
/tildewideσJ2=1
2/integraldisplay−yh
−L/2dy′/bracketleftBigg
−1
(h2−ϕ2s)1/2/parenleftbigg∂/tildewideχext
J
∂y/parenrightbigg/bracketrightBigg
(A6)
We have the following connected expressions:
δσ
δJ(y1)=δ/tildewideσJ1
δJ(y1)+δ/tildewideσJ2
δJ(y1)(A7)
δ/tildewideσJ1
δJ(y1)=1
2/integraldisplay−yh
−L/2dy′/parenleftbigg∂ϕs
∂y/parenrightbigg1
h(h2−ϕ2s)1/2/parenleftbigg
2−ϕs(2c−h)
h2−ϕ2s/parenrightbigg/parenleftbiggδ/tildewideχext
J
δJ(y1)/parenrightbigg
(A8)
δ/tildewideσJ2
δJ(y1)=1
2/integraldisplay−yh
−L/2dy′(−1)
(h2−ϕ2s)1/2δ
δJ(y1)/parenleftbigg∂/tildewideχext
J
∂y/parenrightbigg
(A9)
and:
δ2σ
δJ(y2)δJ(y1)=δ2/tildewideσJ1
δJ(y2)δJ(y1)+δ2/tildewideσJ2
δJ(y2)δJ(y1)(A10)
δ2/tildewideσJ1
δJ(y2)δJ(y1)=1
2/integraldisplay−yh
−L/2dy′/parenleftbigg∂ϕs
∂y/parenrightbigg1
h(h2−ϕ2s)1/2/parenleftbigg
2−ϕs(2c−h)
h2−ϕ2s/parenrightbigg/parenleftbiggδ2/tildewideχext
J
δJ(y2)δJ(y1)/parenrightbigg
(A11)
δ2/tildewideσJ2
δJ(y2)δJ(y1)=1
2/integraldisplay−yh
−L/2dy′(−1)
(h2−ϕ2s)1/2δ2
δJ(y2)δJ(y1)/parenleftbigg∂/tildewideχext
J
∂y/parenrightbigg
(A12)
For the “interior” region, the derivatives of A, (which are strightforward) will require the calculation o f the deriva-
tives ofβ.
16β=1
2/integraldisplayyh
0dy−1
vd/parenleftBig
∂ϕs
∂y/parenrightBig
2c−h
h3+2
vd1
h3/parenleftBig
∂ϕs
∂y/parenrightBig
/tildewideχint
J−1
vd1
h2d/tildewideχint
J
dy
/parenleftBig
1
v2
dϕ2s−h2
h4/parenrightBig1/2/parenleftBig
1−2ϕs
ϕ2s−h2/tildewideχint
J/parenrightBig1/2
The function /tildewideχint
Jand its derivative are present in the expression of β:
β=β0+/tildewideβJ1+/tildewideβJ2
β0=1
2/integraldisplayyh
0dy/bracketleftBigg
−/parenleftbigg∂ϕs
∂y/parenrightbigg2c−h
h(ϕ2s−h2)1/2/bracketrightBigg
/tildewideβJ1=1
2/integraldisplayyh
0dy/bracketleftBigg/parenleftbigg∂ϕs
∂y/parenrightbigg1
h(ϕ2s−h2)1/2/parenleftbigg
2−ϕs(2c−h)
ϕ2s−h2/parenrightbigg
/tildewideχint
J/bracketrightBigg
/tildewideβJ2=1
2/integraldisplayyh
0dy/bracketleftBigg
−1
(ϕ2s−h2)1/2/parenleftbiggd/tildewideχint
J
dy/parenrightbigg/bracketrightBigg
and the derivatives at Jare easily calculated, as for σ.
The formulas above need to specify the expression of the func tions/tildewideχext
J,∂/tildewideχext
J
∂yand of their functional derivatives.
We use the results of the analysis carried out by Karpman.
17[1] F. Spineanu and M. Vlad, Phys. Rev. Letter 84,4854 (2000)
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[14] J.D. Meiss and W. Horton, Phys.Fluids 26990 (1983).
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1984), Vol. 2, Chap. 5.5, p. 183.
[18] J.H. Misguich and R.Balescu, Plasma Phys. 24,284 (1982).
[19] W. Y. Zhang and R. Balescu, Plasma Phys. Control. Fusion 29, 993 (1987); Plasma Phys. Control. Fusion 29, 1019 (1987).
[20] P. W. Terry and P.H. Diamond, Phys. Fluids 28, 1419 (1985).
[21] W. Horton, J. Liu, J.D.Meiss and J.E. Sedlak, Phys.Flui ds29, 1004, (1986).
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[24] D. J. Amit, Field Theory, the Renormalization Group and Critical Pheno mena, Singapore: World Scientific, (1984).
[25] J. A. Krommes, Phys. Rev. E 53, 4865 (1996).
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1.431.1.
[29] V. I. Karpman, Physica Scripta 20, 462, (1979).
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18Figure Captions
Fig.1 The form of the soliton ϕs(y) for the velocity u= 1.725vd.
Fig.2 The function t1(y) foru= 1.725vd.
Fig.3 The function t2(y) of the Eq.(53) for the same u.
Fig.4 The perturbation to the correlation in physical space.
Fig.5 Contour plot of the spectrum of the vortex perturbed by the tu rbulent drift waves.
Fig.6 The contour plot of the frequency-wavenumber spectrum, wit hω−ku= 0.
19−60 −40 −20 0 20 40 6000.10.20.30.40.50.60.70.8
space y/ρs−φs(u=1.725 vd;y)
FIG. 1. Fig.1 Variation of the form of the soliton ϕs(y) with the velocity, u.
−50−40−30−20−10 01020304050−10−505x 109
space y normalized to ρsthe term t1(y)1.2784 x 1091.2784 x 109
−3.144 x 104
FIG. 2. Fig.2 The function t1(y) for a particular soliton velocity, u= 1.725vd.
−50−40−30−20−10 01020304050−1−0.8−0.6−0.4−0.200.20.40.60.81x 106
space y normalized to ρsthe term t2(y)
FIG. 3. Fig.3 The function t2(y) of the Eq.(53) for the same u.
20FIG. 4. Fig.4 The perturbation to the correlation in physical space.
−1 −0.5 0 0.5 1−1−0.500.51
k1ρs k2ρs
FIG. 5. Fig.5 Contour plot of the spectrum of the vortex perturbed by the tu rbulent drift waves.
0 0.5 1 1.5 2 2.5012345678910x 106
k ρsω (sec−1)
FIG. 6. Fig.6 The contour plot of the frequency-wavenumber spectrum, wit hω−ku= 0.
21 |
arXiv:physics/0102041v1 [physics.comp-ph] 13 Feb 2001Quasilinearization Approach to Nonlinear Problems in Phys ics
V. B. Mandelzweig1∗and F. Tabakin2†
1Racah Institute of Physics, Hebrew University, Jerusalem 9 1904, Israel
2Department of Physics and Astronomy, University of Pittsbu rgh, Pittsburgh, PA 15260,USA
Abstract
The general conditions under which the quadratic, uniform a nd monotonic
convergence in the quasilinearization method could be prov ed are formulated
and elaborated. The method, whose mathematical basis in phy sics was dis-
cussed recently by one of the present authors (VBM), approxi mates the so-
lution of a nonlinear differential equation by treating the n onlinear terms as
a perturbation about the linear ones, and unlike perturbati on theories is not
based on the existence of some kind of a small parameter.
It is shown that the quasilinearization method gives excell ent results when
applied to difficult nonlinear differential equations in phys ics, such as the Bla-
sius, Duffing, Lane-Emden and Thomas-Fermi equations. The fir st few quasi-
linear iterations already provide extremely accurate and n umerically stable
answers.
PACS numbers: 02.30.Mv, 04.25.Nx, 11.15.Tk
∗Electronic mail: victor@helium.fiz.huji.ac.il
†Electronic mail: tabakin@pitt.edu
1I. INTRODUCTION
In a series of recent papers, [1,2] the possibility of applyi ng a very powerful approxima-
tion technique called the quasilinearization method (QLM) to physical problems has been
discussed. The QLM is designed to confront the nonlinear asp ects of physical processes. The
method, whose iterations are constructed to yield rapid con vergence and often monotonicity,
was introduced years ago by Bellman and Kalaba [3,4] to solve individual or systems of non-
linear ordinary and partial differential equations. Modern developments and applications of
the QLM to different fields are given in a monograph [5].
However, the QLM was never systematically studied or extens ively applied in physics,
although references to it can be found in well known monograp hs [6,7] dealing with the
variable phase approach to potential scattering, as well as in a few scattered research papers
[8–11]. The reason for the sparse use of the QLM in Physics is t hat the convergence of
the method has been proven only under rather restrictive con ditions [3,4], which generally
are not fulfilled in physical applications. Recently, thoug h, it was shown [1] by one of the
present authors (VBM) that a different proof of the convergen ce can be provided which we
will generalize and elaborate here so that the applicabilit y of the method is extended to
incorporate realistic physical conditions of forces define d on infinite intervals with possible
singularities at certain points.
In the first paper of the series [1], the quasilinearization a pproach was applied to the
nonlinear Calogero equation in a variable phase approach to quantum mechanics and the
results were compared with those of perturbation theory and the exact solutions. It was
found analytically and by examples that the n-th approximation of the QLM exactly sums
2n−1 terms of perturbation theory. In addition, a similar numbe r of terms is reproduced
approximately. The number of the exactly reproduced pertur bation terms thus doubles with
each subsequent QLM approximation, and reaches, for exampl e, 127 terms in the 6-th QLM
approximation, 8191 terms in the 12-th QLM approximation, a nd so on.
The computational approach in the work [1] was mostly analyt ical, and therefore one
2was able to compute only two to three QLM iterations, mainly f or power potentials. Only in
the case of the 1 /r2potential, could the calculation of QLM iterations be done a nalytically
for any n.
The goal of the next work [2] was, by dropping the restriction of analytical computation,
to calculate higher iterations as well as to extend the analy sis to non-power potentials, in
order to better assess the applicability of the method and of its numerical stability and the
convergence pattern of the QLM iterations. It was shown that the first few iterations already
provide very accurate and numerically stable answers for sm all and intermediate values of
the coupling constant and that the number of iterations nece ssary to reach a given precision
only moderately increases for larger values of the coupling . The method provided accurate
and stable answers for any coupling strengths, including fo r super singular potentials for
which each term of the perturbation theory diverges and the p erturbation expansion does
not exist even for a very small coupling.
The quasilinearization approach is applicable to a general nonlinear ordinary or partial n-
th order differential equation in N-dimensional space. In this paper, we consider the case of
nonlinear ordinary differential equations in one variable w hich, unlike the nonlinear Calogero
equation [6] considered in references [1,2], contain not on ly quadratic nonlinear terms but
various other forms of nonlinearity and not only a first, but a lso higher derivatives. Namely,
we apply it to a panopoly of well-known and difficult nonlinear ordinary first, second and
third order differential equations and show that again with j ust a small number of iterations
one can obtained fast convergent and uniformly excellent an d stable numerical results.
The paper is arranged as follows: in the second chapter we pre sent the main features of
the quasilinearization approach, while in the third chapte r we consider, as a warm-up exer-
cise, a simple first-order differential equation with a nonli nearn-th power term and compare
its exact analytic solution with the perturbation theory an d with the QLM iterations in
order to demonstrate the main features of the quasilineariz ation approach. In the next four
chapters, we apply our method to four nonlinear ordinary sec ond and third order differential
equations, namely to the Lane-Emden, Thomas-Fermi, Duffing, and Blasius equations, re-
3spectively. The results, convergence patterns, numerical stability, advantages of the method
and its possible future applications are discussed in the fin al chapter.
II. THE QUASILINEARIZATION METHOD (QLM)
The aim of the QLM [1,3–5] is to solve a nonlinear n-th order ordinary or partial dif-
ferential equation in Ndimensions as a limit of a sequence of linear differential equ ations.
This goal is easily understandable since there is no useful t echnique for obtaining the general
solution of a nonlinear equation in terms of a finite set of par ticular solutions, in contast
to a linear equation which can often be solved analytically o r numerically in a convenient
fashion using superposition. In addition, the QL sequence s hould be constructed to assure
quadratic convergence and, if possible, monotonicity.
As we have mentioned in the Introduction, we will follow here the derivation outlined in
ref. [1], which is not based, unlike the derivations in refs. [3,4], on a smallness of the interval
and on the boundness of the nonlinear term and its functional derivatives, the conditions
which usually are not fillfilled in physical applications.
For simplicity, we limit our discussion to nonlinear ordina ry differential equation in one
variable on the interval [0 , b],which could be infinite:
/suppress L(n)u(x) =f(u(x), u(1)(x), ....u(n−1)(x), x), (2.1)
withnboundary conditions
gk(u(0), u(1)(0), ...., u(n−1)(0)) = 0 , k= 1, ...l (2.2)
and
gk(u(b), u(1)(b), ...., u(n−1)(b)) = 0 , k=l+ 1, ..., n. (2.3)
HereL(n)is linear n-th order ordinary differential operator and fandg1, g2, ....., g nare
nonlinear functions of u(x) and its n−1 derivatives u(s)(x), s= 1, ...n−1. The more general
case of partial differential equations in N-dimensional spa ce could be considered in exactly
4the same fashion by changing the definition of L(n)to be a linear n-th order differential
operator in partial derivatives and xto be an N-dimensional coordinate array.
The QLM prescription [1,3,4] determines the r+ 1-th iterative approximation ur+1(x) to
the solution of Eq. (2.1) as a solution of the linear different ial equation
L(n)ur+1(x) =f(ur(x), u(1)
r(x), ....., u(n−1)
r(x), x)
+n−1/summationdisplay
s=0(u(s)
r+1(x)−u(s)
r(x))fu(s)(ur(x), u(1)
r(x), ....., u(n−1)
r(x), x), (2.4)
where u(0)
r(x) =ur(x), with linearized two-point boundary conditions
n−1/summationdisplay
s=0(u(s)
r+1(0)−u(s)
r(0))gku(s)(ur(0), u(1)
r(0), ....., u(n−1)
r(0),0) = 0 , k= 1, ...l (2.5)
and
n−1/summationdisplay
s=0(u(s)
r+1(b)−u(s)
r(b))gku(s)(ur(b), u(1)
r(b), ....., u(n−1)
r(b), b) = 0, k=l+ 1, ..., n. (2.6)
Here the functions fu(s)=∂f/∂u(s)andgku(s)=∂gk/∂u(s), s = 0,1, ..., n −
1 are functional derivatives of the functionals f(u(x), u(1)(x), ....u(n−1)(x), x) and
gk(u(x), u(1)(x), ....u(n−1)(x), x), respectively.1.
The zeroth approximation u0(x) is chosen from mathematical or physical considerations.
To prove that the above procedure yields a quadratic and ofte n monotonic convergence
to the solution of Eq. 2.1 with the boundary conditions 2.2 an d 2.3, we follow reference [1]
and consider a differential equation for the difference δur+1(x)≡ur+1(x)−ur(x) between
two subsequent iterations:
1For example, in case of a simple nonlinear boundary conditio nu′(b)u(b) =cwhere c is a con-
stant, one has g(r)≡g(u(r),u′(r),r) =u′(r)u(r) so that gu=u′(r) and gu′=u(r). The lin-
earized boundary condition 2.6 has a form ( ur+1(b)−ur(b))u′
r(b) + (u′
r+1(b)−u′
r(b))u(b) = 0
or (ur+1(b)ur(b))′= (ur(b)ur(b))′so the nonlinear boundary condition for the initial guess
u0(b)u′
0(b) =cwill be propogated to the linear boundary condition for the n ext iterations.
5L(n)δur+1(x) = [f(ur(x), u(1)
r(x), ....., u(n−1)
r(x), x)−f(ur−1(x), u(1)
r−1(x), ....., u(n−1)
r−1(x), x)]
+n−1/summationdisplay
s=0[δu(s)
r+1(x)fu(s)(ur(x), u(1)
r(x), ....., u(n−1)
r(x), x)
−δu(s)
r(x)fu(s)u(r−1(x), u(1)
r−1(x), ....., u(n−1)
r−1(x), x)]. (2.7)
The boundary conditions are similarly given by the differenc e of Eqs. 2.5 and 2.6 for two
subsequent iterations:
n−1/summationdisplay
s=0[δu(s)
r+1(0)gku(s)(ur(0), u(1)
r(0), ....., u(n−1)
r(0),0)
−δu(s)
r(0)gku(s)(ur−1(0), u(1)
r−1(0), ....., u(n−1)
r−1(0),0)] = 0 ,
k= 1, ...l (2.8)
and
n−1/summationdisplay
s=0[δu(s)
r+1(b)gku(s)(ur(b), u(1)
r(b), ....., u(n−1)
r(b), b)
−δu(s)
r(b)gku(s)(ur−1(b), u(1)
r−1(b), ....., u(n−1)
r−1(b), b)] = 0,
k=l+ 1, ...n. (2.9)
In view of the mean value theorem [12]
f(ur(x), u(1)
r(x), ....., u(n−1)
r(x), x)−f(ur−1(x), u(1)
r−1(x), ....., u(n−1)
r−1(x), x) =
n−1/summationdisplay
s=0δu(s)
r(x)fu(s)(ur−1(x), u(1)
r−1(x), ....., u(n−1)
r−1(x), x) +
1
2n−1/summationdisplay
s,t=0δu(s)
r(x)δu(t)
r(x)fu(s)u(t)(¯ur−1(x),¯u(1)
r−1(x), ....., ¯u(n−1)
r−1(x), x), (2.10)
where ¯ u(s)
r−1(x) lies between u(s)
r(x) and u(s)
r−1(x). Now Eq. 2.7 can be written as
L(n)δur+1(x)−n−1/summationdisplay
s=0δu(s)
r+1(x)fu(s)(ur(x), u(1)
r(x), ....., u(n−1)
r(x), x) =
1
2n−1/summationdisplay
s,t=0δu(s)
r(x)δu(t)
r(x)fu(s)u(t)(¯ur−1(x),¯u(1)
r−1(x), ....., ¯u(n−1)
r−1(x), x). (2.11)
Denoting G(n)
r(x, y) as the Greens function, which is the inverse of the followin g differential
operator and incorporates linearized boundary conditions 2.5 and 2.6,
6˜L(n)=L(n)−n−1/summationdisplay
s=0fu(s)(ur(x), u(1)
r(x), ....., u(n−1)
r(x), x)ds
dxs, (2.12)
one can express the solution for the difference function δur+1as
δur+1(x) =
1
2/integraldisplayb
0G(n)
r(x, y)n−1/summationdisplay
s,t=0δu(s)
r(y)δu(t)
r(y)fu(s)u(t)(¯ur−1(y),¯u(1)
r−1(y), ....., ¯u(n−1)
r−1(y), y)dy. (2.13)
The functions δu(s)
r(y)δu(t)
r(y) could be taken outside of the sign of the integral at some poi nt
y= ¯xbelonging to the interval, so one obtains
δur+1(x) =1
2n−1/summationdisplay
s,t=0δu(s)
r(¯x)δu(t)
r(¯x)Mst(x). (2.14)
where Msr(x) equals
Mst(x) =/integraldisplayb
0G(n)
r(x, y)fu(s)u(t)(¯ur−1(y),¯u(1)
r−1(y), ....., ¯u(n−1)
r−1(y), y)dy (2.15)
IfMst(x) is a strictly positive (negative) matrix for all xin the interval, then δur+1(x)
will be positive (negative), and the monotonic convergence from below (above) results.
Obviously, from Eq. 2.13 follows
|δur+1(x)| ≤kr(x)||δur||2(2.16)
where kris given by
kr(x) =1
2/integraldisplayb
0|G(n)
r(x, y)|n−1/summationdisplay
s,t=0|fu(s)u(t)(¯ur−1(y),¯u(1)
r−1(y), ....., ¯u(n−1)
r−1(y), y)|dy (2.17)
and||δur||is a maximal value of any of |δ¯u(s)
r|on the interval (0,b).
Since Eq. 2.16 is correct for any xon the interval (0,b), it is correct also for some x= ¯x
where |δur+1(x)|reaches its maximum value ||δur+1||. One therefore has
||δur+1||| ≤kr(¯ x)||δur||2(2.18)
Assuming the boundness of the integrand in expression 2.17, that is the existence of the
bounding function F(x) such that integrand at x= ¯xand at any yis less or equal to F(y),
one finally has
7||δur+1||| ≤k||δur||2, (2.19)
where
k=/integraldisplayb
0F(x)dx. (2.20)
The linearized boundary conditions 2.5 and 2.6 are obtained from exact boundary con-
ditions 2.2 and 2.3 by using the mean value theorem Eq. 2.10 an d neglecting the quadratic
terms, so that the error in using linearized boundary condit ions vis-a-vis the exact ones
is quadratic in the difference between the exact and lineariz ed solutions. The maximum
difference between boundary conditions 2.5 and 2.6 correspo nding to two subsequent quasi-
linear iterations is therefore quadratic in ||δur||. In view of this result and of Eq. 2.19, the
difference between the subsequent iterative solutions of Eq .2.4 with boundary conditions 2.5
and 2.6 decreases quadratically with each iteration. In a si milar way, one can show [1] that
the difference ∆ ur+1(x) =u(x)−ur(x) between the exact solution and the r-th iteration is
decreasing quadratically as well:
||∆ur+1||| ≤k||∆ur||2. (2.21)
A simple induction of Eq. 2.19 shows [4] that δun+1(x) for an arbitrary l < r satisfies
the inequality
/bardblδur+1/bardbl ≤(k/bardblδul+1/bardbl)2r−l/k, (2.22)
or, for l= 0, we can relate the n+ 1 th order result to the 1st iterate by
/bardblδun+1/bardbl ≤((k/bardblδu1/bardbl)2n/k. (2.23)
The convergence depends therefore on the quantity q1=k/bardblu1−u0/bardbl, where, as we have
mentioned earlier, the zeroth iteration u0(x) is chosen from physical and mathematical
considerations. Usually it is advantageous (see discussio n below) that u0(x) would satisfy
at least one of the boundary conditions. From Eq. (2.22) it fo llows, however, that for
convergence it is sufficient that just one of the quantities qm=k/bardblδum/bardblis small enough.
8Consequently, one can always hope [4] that even if the first co nvergent coefficient q1is
large, a well chosen initial approximation u0results in the smallness of at least one of the
convergence coefficients qm, m > 1, which then enables a rapid convergence of the iteration
series for r > m . It is important to stress that in view of the quadratic conve rgence of the
QLM method, the difference ||∆ur+1||between the exact solution and the QLM iteration
always converges to zero if the difference δur+1(x) between two subsequent QLM iterations
becomes infinitesimally small.
Indeed, if δur(x) is close to zero, it means, since δur+1(x) = ∆ ur(x)−∆ur+1(x) that
∆ur(x) = ∆ ur+1(x) orQr=Qr+1where Qr=k||∆ur||. When one assumes the possi-
bility that QrandQr+1could be not small, one could conclude that the iteration pro cess
“stagnates”, which means convergence to the wrong answer or no convergence at all.
However, such a conclusion is wrong since Eq. 2.21, which can be written as Qr+1≤
Q2
r, forQr≤1 (this last inequality, starting from some r is a necessary c ondition of the
convergence) could be not satisfied unless both ||Qr+1||and||Qr||equal to zero. This proves
that stagnation of the iteration process is impossible and c onvergence of ||δur+1||to zero
automatically leads to convergence of the QLM iteration seq uence to the exact solution.
Hence the QLM assures not only convergence,i but also conver gence to the correct solution.
Another corollary of this iteration process is that if the so lution and its derivatives are
continuous functions of x, the convergence of the QLM in the whole region will follow.
Indeed, even if the zero iteration u0(x) is chosen not to satisfy the boundary conditions,
the next iteration u1(x), being a solution of a linear equation with linearized boun dary
conditions 2.5 and 2.6, will automatically satisfy the exac t boundary conditions 2.2 and 2.3,
at least up to the second order in difference δu1at the boundaries. This means that the
difference between the exact and first QLM iterations at some i ntervals near the boundaries
will be small, so that the QLM iterations in this interval wou ld converge. Because the
subsequent values of kδum(x),m>2became much smaller for this interval, in view of
assumed continuity of the solution and its derivatives thes e differences will also be small at
the neighboring intervals. The subsequent iterations will extend the convergence to the next
9neighboring intervals and so on, until the convergence in th e whole region will be reached.
The predicted trend is therefore that the QLM yields rapid co nvergence starting at the
regions where the boundary conditions are imposed and then s preading from there to all
other regions.
An additional important corollary is that, in view of Eq. 2.2 2, once the quasilinear
iteration sequence starts to converge, it will continue to d o so, unlike the perturbation
expansion, which is often given by an asymptotic series and t herefore converges only up to
a certain order and diverges thereafter.
Based on this summary of the QLM, one can deduce the following important features of
the quasilinearization method:
i) The method approximates the solution of nonlinear differe ntial equations by treating
the nonlinear terms as a perturbation about the linear ones, and is not based, unlike
perturbation theories, on the existence of some kind of smal l parameter.
ii) The iterations converge uniformly and quadratically to the exact solution. In case of
matrix Mstin Eq. 2.15 being strictly positive (negative) for all xin the interval, the
convergence is also monotonic from below (above).
iii) For rapid convergence it suffices that an initial guess fo r the zeroth iteration is suffi-
ciently good to ensure the smallness of just one of the quanti tiesqr=k/bardblur+1−ur/bardbl. If
the solution and its derivatives are continuous, convergen ce follows from the fact that
starting from the first iteration, all QLM iterations automa tically satisfy the quasilin-
earized boundary conditions 2.5 and 2.6. The convergence is extremely fast: if, for
example, q1is of the order of1
3, only 4 iterations are necessary to reach the accuracy
of 8 digits, since (1
3)2nis of the order of (1
10)2n−1.
iv) Convergence of ||δur+1||to zero automatically leads to convergence of the QLM itera-
tion sequence to the exact solution.
10v) Once the quasilinear iteration sequence at some interval starts to converge, it will
always continue to do so. Unlike an asymptotic perturbation series, the quasilin-
earization method yeild the required precision once a succe ssful initial guess generates
convergence after a few steps.
III. ANALYTICALLY SOLVABLE EXAMPLE: COMPARISON OF
QUASILINEARIZATION APPROACH WITH EXACT SOLUTION AND WITH
PERTURBATION THEORY.
In order to investigate the applicability of the quasilinea rization method and its conver-
gence and numerical stability, let us start from a simple exa mple of an analytically solvable
nonlinear ordinary differential equation suggested in ref. [13]:
u′(r) =−g un(r), u(0) = 1 , (3.1)
where the boundary condition at r= 0 is also given and′means differentiation in variable
r. The exact solution to this problem is
u(r) = (1 + ( n−1)g r)−1
n−1 (3.2)
Since
(1 +x)q=∞/summationdisplay
0Γ(q+ 1)
m!Γ(q+ 1−m)xm, (3.3)
the expansion of the solution 3.2 in powers of gis given by
u(r) =∞/summationdisplay
0Γ(n−2
n−1)(g(n−1))m
m!Γ(n−2
n−1−m)rm(3.4)
The convergence radius of the series 3.4 is R= 1/(g(n−1)),which is inversely proportional
to the extent n−1 of the nonlinearity and to the value gof the perturbation parameter.
Now consider the quasilinearization approach to this equat ion, taking, for example, g= 1
andn= 6.Here we consider Eq. 3.1 with a rather strong degree of nonlin earity. In this
case, one can expect the convergence of the perturbation exp ansion only up to r≤R=1
5.
11The QLM procedure in the case where nonlinear term depends on ly on the solution itself
and not on its derivatives reduces to setting u′
k+1(r) =f(uk)+(uk+1(r)−uk(r))fu(uk). Here
f=−g un(r) while its functional derivative fuequels to −g nun−1(r). The quasilinearized
equation 2.4 for the ( k+ 1)-th iteration for this case has therefore the following f orm:
u′
k+1(r) +ng un−1
k(r)uk+1(r) = (n−1)g un
k(r), u k+1(0) = 1 , (3.5)
where uk(r) is a previous iteration which is considered to be a known fun ction. Let us choose
as a zero iteration u0(r)≡1 which satisfies the boundary condition u0(0) = 1.
The results of our QLM calculations with Eq. 3.5 are presente d in Fig. 1 which displays
the exact solution for the case of n= 6 and g= 1,together with the first four QLM
iterations. Convergence to the exact solution in Fig. 1 is mo notonic from above as it should
be as discussed in Section II and in Refs. [1,3,4] due to fact t hat the second functional
derivative −n(n−1)un−2(x) of the left-hand side of Eq.3.1 for even nis strictly negative.
The convergence starts at the boundary, exactly as expected from the discussion in section
II, and expands with each iteration to larger values of the va riable r. The difference between
the exact solution and the sixth QLM iteration for all r in the range between zero and five
where our calculations were performed is less than 10−6.Note that the QLM yields a solution
beyond the convergence radius limit on the series solution o f 1/5.
A. Lane-Emden equation
The Lane-Emden equation
y′′(r) +2
ry′(r) +yn(r) = 0, y (0) = 1 , y′(0) = 0 (3.6)
is a nonlinear second order differential equation which aris es in the study of stellar structure.
It describes the equilibrium density distribution in a self -gravitating sphere of polytropic
isothermal gas. The parameter ncorresponds to a particular choice for an equation of
state with its physically interesting range being 0 ≤n≤5. The equation also appears in
12other contexts, e.g., in case of radiatively cooling, self g ravitating gas clouds, in the mean-
field treatment of a phase transition in critical absorption or in the modeling of clusters of
galaxies. The equation can be solved analytically for the sp ecial cases n= 0,1 and 5 .For
other values of n, power series approximations as well as nonperturbative ap proaches have
been developed (see, for example, [13,14] and references th erein). Setting y=u
rtransforms
the equation to a more convenient form without a first derivat ive:
u′′(r) +un(r)
rn−1= 0, u(0) = 0 , u′(0) = 1 . (3.7)
Let us consider this nonlinear equation for the physically i nteresting and analytically non-
solvable case of n= 4. The quasilinearized form of equation 3.7 is
u′′
k+1(r) +nun−1
k(r)
rn−1uk+1(r) =n−1
rn−1un
k(r), uk+1(0) = 0 , u′
k+1(0) = 1 . (3.8)
The simplest initial guess, satisfying the boundary condit ions will be u0(r) =r.Comparison
of the quasilinear solutions corresponding to the first five i terations with the numerically
computed exact solution are given in Fig. 2. The figure shows t hat the convergence to
the exact solution is very fast. It starts, as in the example o f the previous section, at the
left boundary and spreads with each iteration to larger valu es ofras expected from the
discussion in section II. The difference between the exact so lution and the eighth QLM
iteration for all rin the range between zero and ten, where our calculations wer e performed,
is less than 10−11.
IV. THOMAS-FERMI EQUATION
The Thomas-Fermi equation [15,16]
√x u′′(x) =u3
2(x), u (0) = 1 , u(∞) = 0, (4.1)
is an equation for the electron density around the nucleus of the atom. The left hand
side of the above equation equals zero for u <0.The Thomas-Fermi equation is also very
useful for calculating form-factors and for obtaining effec tive potentials which can be used
13as initial trial potentials in self-consistent field calcul ations.. It is also applicable to the
study of nucleons in nuclei and electrons in metal. It is long known (see [17] and references
therein) that solution of this equation is very sensitive to a value of the first derivative at
zero which insures smooth and monotonic decay from u(0) = 1 to u(∞) = 0 as demanded
by boundary conditions. Finding the value of u′(0) accurately is a tedious procedure requir-
ing a considerable computer time. By contrast, the computat ion is much simpler for the
quasilinearized version of this equation. The QLM procedur e in this case reduces to setting
u′′
k+1(r) =f(uk) + (uk+1(r)−uk(r))fu(uk), where f=u3/2(r)√xand the functional derivative
isfu= (3/2)u1/2√x, so that the QLM equation has a form:
√x u′′
k+1(x)−3
2u1
2
k(x)uk+1(x) =−1
2u3
2
k(x), uk+1(0) = 1 , uk+1(∞) = 0, (4.2)
which is easily solved by specifying directly the boundary c ondition at infinity without
searching first for the proper value of the first derivative. T he initial guess, satisfying the
boundary condition at zero was chosen to be u0(x)≡1. The results of QLM calculations
with Eq. 4.2 are presented in Fig. 3 which displays the exact s olution together with the
first four QLM iterations. The convergence starts at the boun daries, exactly as expected
from the discussion in section II, and expands with each iter ation to a wider range of values
of the variable x. The difference between the exact solution and the eighth QLM iteration
for all xin the range between zero and forty where our calculations we re performed is less
than 10−7.
V. CLASSICAL ANHARMONIC OSCILLATOR
The classical anharmonic oscillator satisfies the nonlinea r second-order equation
¨u(t) +u(t) +g u3(t) = 0 (5.1)
commonly referred to as the Duffing equation. In our example, w e impose the following
boundary conditions at zero t
14u(0) = 1 ,˙u(0) = 0 . (5.2)
The solution oscillates strongly and thus is more difficult to approximate. It is, for example,
well known [13] that the usual perturbative solution is vali d only for times tsmall compared
with1
g, so that for larger gthe perturbative solution is adequate only on a small time
interval. In contrast, the quasilinearization approach gi ves solution in the whole region also
for large g-values.
The quasilinearized equation is
¨uk+1(t) + (1 + 3 g u2
k(t))uk+1(t)−2g u3
k(t) = 0, u k+1(0) = 1 ,˙uk+1(0) = 0 . (5.3)
The results of QLM calculations with Eq. 5.3 for g= 3 are presented in Figs. 4 and 5.
Fig. 4 displays the exact solution together with the QLM solu tions for the first, second and
fourth iterations while Fig. 5 shows comparison of exact sol ution with sixth, seventh and
eighth QLM iterations. Again, the convergence starts at the left boundary as expected from
the discussion in section II, and expands with each iteratio n to larger values of the variable
t. The difference between the exact solution and the eleventh Q LM iteration for all t in the
range between zero and seven where our calculations were per formed is less than 10−10.
VI. BLASIUS EQUATION
The Blasius equation [18]
u′′′(x) +u′′(x)u(x) = 0, u(0) = u′(0) = 0 , u′(∞) = 1 (6.1)
is a third order nonlinear differential equation which descr ibes the velocity profile of the fluid
in the boundary layer which forms when fluid flows along a flat pl ate. The Blasius equation
is similar to the Thomas-Fermi equation in that it has a two-p oint boundary condition.
However, it differs from the Thomas-Fermi case in that it is of higher order and also contains
a second derivative term times u(x).Therefore, Eq. 6.1 is even more difficult to solve. The
QLM procedure in this case is given by u′′′
k+1(x) =f(uk, u′′
k) + (uk+1−uk)fu(uk, u′′
k) +
15(u′′
k+1−u′′
k)fu′′(uk, u′′
k), where f(u, u′) =−u′′u,fu(u, u′) =−u′′andfu′′(u, u′) =−u. The
quasilinearized version of the Blasius equation thus has a f orm
u′′′
k+1(x) +uk(x)u′′
k+1(x) +uk+1(x)u′′
k(x)−uk(x)u′′
k(x) = 0,
uk+1(0) = u′
k+1(0) = 0 , u′
k+1(∞) = 1. (6.2)
The initial guess, satisfying the boundary condition for th e derivative at zero was chosen
to beu0(x)≡1. The results of QLM calculations with Eq. 6.2 are presented in Fig. 6 which
displays the exact solution together with the first QLM itera tion. The convergence starts
at the left boundary as follows from the discussion in sectio n II, and expands with each
iteration to larger values of the variable x. The difference between the exact solution and
the fifth QLM iteration for all xin the range between zero and ten where our calculations
were performed is less than 10−11.
VII. CONCLUSION
Summing up, we formulated here the conditions under which th e quadratic, uniform and
often monotonic convergence of the quasilinearization met hod are valid.
We have followed here the derivation outlined in ref. [1], wh ich is not based, unlike the
derivations in refs. [3,4], on a smallness of the interval an d on the boundness of the nonlinear
term and its functional derivatives, the conditions which u sually are not fulfilled in physical
applications.
In order to analyze and highlight the power and features of th e quasilinearization method
(QLM), in this work we have also made numerical computations on different ordinary second
and third order nonlinear differential equations, applied i n physics, such as the Blasius,
Duffing, Lane-Emden and Thomas-Fermi equations and have comp ared the results obtained
by the quasilinearization method with the exact solutions. Although all our examples deal
only with linear boundary conditions, the nonlinear bounda ry conditions can be handled
readily after their quasilinearization as explained in Sec tion II.
16Our conclusions are as follows:
The QLM treats nonlinear terms as a perturbation about the li near ones [1,3,4] and is
not based on the existence of some kind of small parameter. As a result, as we see in all our
examples, the QLM is able to handle large values of the coupli ng constant and any degree
of the nonlinearity, unlike perturbation theory. Thus the Q LM provides extremely accurate
and numerically stable answers for a wide range of nonlinear physics problems.
In view of all this, since most equations of physics, from cla ssical mechanics to quantum
field theory, are either not linear or could be transformed in to a nonlinear form, the quasi-
linearization method appears to be extremely useful and in m any cases more advantageous
than the perturbation theory or its different modifications, like expansion in inverse powers
of the coupling constant, the 1 /Nexpansion, etc.
ACKNOWLEDGMENTS
The research was supported in part by the U.S. National Scien ce Foundation PHY-
9970775 (FT) and by the Israeli Science Foundation founded b y the Israeli Academy of
Sciences and Humanities (VBM).
17REFERENCES
[1] V. B. Mandelzweig, J. Math. Phys. 40, 6266 (1999).
[2] R. Krivec and V. B. Mandelzweig, Computer Physics Commun ications, 2001, accepted
for publication.
[3] R. Kalaba, J. Math. Mech. 8, 519 (1959).
[4] R. E. Bellman and R. E. Kalaba, Quasilinearization and Nonlinear Boundary-Value
Problems , Elsevier Publishing Company, New York,1965.
[5] V. Lakshmikantham and A. S. Vatsala, Generalized Quasilinearization for Nonlinear
Problems , MATHEMATICS AND ITS APPLICATIONS, Volume 440, Kluwer Aca-
demic Publishers, Dordrecht,1998.
[6] F. Calogero, Variable Phase Approach to Potential Scattering , Academic Press, New
York,1965.
[7] V. V. Babikov, Metod Fazovyh Funkcii v Kvantovoi Mehanike (Variable Metho d of Phase
Functions in Quantum Mechanics) , Nauka, Moscow 1968; V. V. Babikov, Sov. Phys.
Uspekhi 10, 271 (1967).
[8] A. A. Adrianov, M. I. Ioffe and F. Cannata, Modern Phys. Let t.11, 1417 (1996).
[9] M. Jameel, J. Physics A: Math. Gen. 21, 1719 (1988).
[10] K. Raghunathan and R. Vasudevan, J. Physics A: Math. Gen .20, 839 (1987).
[11] M. A. Hooshyar and M. Razavy, Nuovo Cimento B75, 65 (1983).
[12] V. Volterra Theory of Functionals , Blackie and Son, London,1931.
[13] C. M. Bender, K. A. Milton, C. C. Pinsky, L. M. Simmons, Jr ., J. Math. Phys 30, 1447
(1989).
[14] H. Goenner and P. Havas, J. Math. Phys 41, 7029 (2000).
18[15] L. H. Thomas, Proc. Cambrige. Phil. 23, 542 (1927).
[16] E. Fermi, Z. Physik. 48, 73 (1928).
[17] Hans A. Bethe, Roman W. Jackiw , Intermediate Quantum Mechanics , W. A. Ben-
jamin Inc., New York, 1968.
[18] H. Schlichting , Boundary Layer Theory , McGrow-Hill , New York, 1978.
19FIGURE CAPTIONS
FIG. 1. Convergence of QLM iterations for the analytic examp le of section III and
comparison with the exact solution. Thin solid, dot-dashed , short-dashed and dotted curves
correspond to the first, second, third and fourth QLM iterati on respectively, while the thick
solid curve displays the exact solution. The convergence is monotonic from above as it
should be according to the discussion in the text. The differe nce between the exact solution
and the sixth QLM iteration for all r in the figure is less than 1 0−6.
FIG. 2. Convergence of QLM iterations for the Lane-Emden equ ation and comparison
with the numerically obtained exact solution. Thin solid, d ot-dashed, short-dashed, long-
dashed and dotted curves correspond to the first, second, thi rd, fourth and fifth QLM
iteration, respectively, while the thick solid curve displ ays the exact solution. The difference
between the exact solution and the eighth QLM iteration for a ll r in the figure is less than
10−11
FIG. 3. Convergence of QLM iterations for the Thomas-Fermi e quation and comparison
with the numerically obtained exact solution. Thin solid, d ot-dashed, short-dashed and
dotted curves correspond to the first, second, third and four th QLM iteration, respectively,
while the thick solid curve displays the exact solution. The difference between the exact
solution and the eighth QLM iteration for all x in the figure is less than 10−7.
FIG. 4. Convergence of the first few QLM iterations for the Duffi ng equation and compar-
ison with the numerically obtained exact solution. The dott ed curves on three consecutive
graphs correspond to the first, second and fourth QLM iterati on respectively, while the solid
curve displays the exact solution.
FIG. 5. Convergence of the higher QLM iterations for the Duffin g equation and compari-
son with the numerically obtained exact solution. The dotte d curves on the three consecutive
graphs correspond to the sixth, seventh and eighth QLM itera tion respectively, while the
20solid curve displays the exact solution. The difference betw een the exact solution and the
eighth QLM iteration for all t in the figure is less than 10−10.
FIG. 6. Comparison of the first QLM iteration for the Blasius e quation with the numer-
ically obtained exact solution. The difference between the e xact solution and the fifth QLM
iteration for all x in the figure is less than 10−10.
21FIGURES
1 3 5r0.70.9u
FIG. 1. Convergence of QLM iterations for the analytic examp le of section III and comparison
with the exact solution. Thin solid, dot-dashed, short-das hed and dotted curves correspond to
the first, second, third and fourth QLM iteration respective ly, while the thick solid curve displays
the exact solution. The convergence is monotonic from above as it should be according to the
discussion in the text. The difference between the exact solu tion and the sixth QLM iteration for
all r in the figure is less than 10−6.
2 4 6 810r246u
FIG. 2. Convergence of QLM iterations for the Lane-Emden equ ation and comparison with the
numerically obtained exact solution. Thin solid, dot-dash ed, short-dashed, long-dashed and dotted
curves correspond to the first, second, third, fourth and fift h QLM iteration, respectively, while
the thick solid curve displays the exact solution. The differ ence between the exact solution and
the eighth QLM iteration for all r in the figure is less than 10−11.
2210 20 30 40x0.20.40.60.81u
FIG. 3. Convergence of QLM iterations for the Thomas-Fermi e quation and comparison with
the numerically obtained exact solution. Thin solid, dot-d ashed, short-dashed and dotted curves
correspond to the first, second, third and fourth QLM iterati on, respectively, while the thick solid
curve displays the exact solution. The difference between th e exact solution and the eighth QLM
iteration for all x in the figure is less than 10−7.
231 3 5 7t
-22u1 3 5 7t
-22u1 3 5 7t
-22u
FIG. 4. Convergence of the first few QLM iterations for the Duffi ng equation and comparison
with the numerically obtained exact solution. The dotted cu rves on three consecutive graphs
correspond to the first, second and fourth QLM iteration resp ectively, while the solid curve displays
the exact solution.
241 3 5 7t
-11u1 3 5 7t
-11u1 3 5 7t
-11u
FIG. 5. Convergence of the higher QLM iterations for the Duffin g equation and comparison
with the numerically obtained exact solution. The dotted cu rves on the three consecutive graphs
correspond to the sixth, seventh and eighth QLM iteration re spectively, while the solid curve
displays the exact solution. The difference between the exac t solution and the eighth QLM iteration
for all t in the figure is less than 10−10.
254 8 12x2610u
FIG. 6. Comparison of the first QLM iteration for the Blasius e quation with the numerically
obtained exact solution. The difference between the exact so lution and the fifth QLM iteration for
all x in the figure is less than 10−10.
26 |
Physical Interpretation of the 26 dimensions of Bosonic String Theory
February 2001
Frank D. (Tony) Smith, Jr.
tsmith@innerx.net
http://www.innerx.net/personal/tsmith/Rzeta.html
Abstract:
The 26 dimensions of Closed Unoriented Bosonic String Theory
are interpreted as the 26 dimensions of
the traceless Jordan algebra J3(O)o of 3x3 Octonionic matrices,
with each of the 3 Octonionic dimenisons of J3(O)o
having the following physical interpretation:
4-dimensional physical spacetime plus 4-dimensional internal symmetry space;
8 first-generation fermion particles;
8 first-generation fermion anti-particles.
This interpretation is consistent with interpreting the strings
as World Lines of the Worlds of Many-Worlds Quantum Theory
and the 26 dimensions as the degrees of freedom of
the Worlds of the Many-Worlds.
Details about the material mentioned on the above chart can beseen on these web pages:
Clifford algebras -http://www.innerx.net/personal/tsmith/clfpq.html
Discrete -http://www.innerx.net/personal/tsmith/Sets2Quarks2.html#sub2
Real -http://www.innerx.net/personal/tsmith/clfpq.html#whatclifspin
Octonions -http://www.innerx.net/personal/tsmith/3x3OctCnf.html
Jordan algebras -http://www.innerx.net/personal/tsmith/Jordan.html
Lie algebras -http://www.innerx.net/personal/tsmith/Lie.html
Internal Symmetry Spaee -http://www.innerx.net/personal/tsmith/See.html
Segal Conformal theory -http://www.innerx.net/personal/tsmith/SegalConf.html
MacDowell-Mansouri gravity -http://www.innerx.net/personal/tsmith/cnfGrHg.html
Standard Model Weyl groups -http://www.innerx.net/personal/tsmith/Sets2Quarks4a.html#WEYLdimredGB
Fermions -http://www.innerx.net/personal/tsmith/Sets2Quarks9.html#sub13
HyperDiamond lattices - -http://www.innerx.net/personal/tsmith/HDFCmodel.html
Generalized Feynman Checkerboards -http://www.innerx.net/personal/tsmith/Fynckb.html13/2/01 9:36 PM 26 Dimensions of Bosonic Strings
Page 1 of 6 file:///iPurple/26dimBosonStrings/String26.html The following sections of this paper are about:
MacroSpace of Many-Worlds
Unoriented Closed Bosonic Strings
An M-theory of the full 27-dimensional Jordan algebraJ3(O)
Some descriptions of a few relevant terms
The 26-dimensional traceless subalgebra J3(O)o is thefoundation for a representation of
the 26-dim Theory of Unoriented Closed Bosonic Strings as the , MacroSpace of Many-Worlds
since each World of the can be seen as a 1-Timelike-dimensional String ofSpacelike States, like a World Line or World String, and MacroSpace ofMany-Worlds
since the is represented by corresponding to the
complexification of the27-dimensional J3(O) and by related to the same , and of Many-Worlds MacroSpace geometrically E7/ E6xU(1) with 54 real dimensions and 27 complexdimensions
Jordan algebra algebraically structure 27-dimensional Jordanalgebra J3(O)
since the E6 of the can be represented in terms of 3 copies ofthe 26-dimensional traceless subalgebra J3(O)o of the 27-
dimensional J3(O) by using the of 78-dimensional E6 over 52-dimensional F4 and thestructure of based on the 26-
dimensional representation of .Lie algebra D4-D5-E6-E7-E8VoDou Physics model
Jordan algebra fibrationE6 / F4 F4 as doubled J3(O)o
F4
Unoriented Closed Bosonic Strings:
Michio Kaku, in his books, Introduction to Superstrings andM-Theory (2nd ed) (Springer-Verlag 1999) and Strings, ConformalFields, and M-Theory (2nd ed) (Springer-Verlag
2000) diagrams theUnoriented Closed Bosonic String spectrum:
Joseph Polchinski, in his books String Theory vols. I and II(Cambridge 1998), says: "... [In] the simplest case of 26 flatdimensions ... the closed bosonic string ... theory has the
maximal26-dimensional Poincare invariance ... [and] ... is theunique theory with this symmetry ... It is possible to have aconsistent theory with only closed strings ...
, with Guv representing the graviton [and] ... PHIthe dilaton ... [and also] ... the tachyon ... [forthe] [are]...":massless
spectra Closed unoriented bosonic string
Guv, as to which Green,Schwartz, and Witten, in their book Superstring Theory, vol. 1, p.181 (Cambridge 1986) say the long-wavelength
limit of theinteractions of the massless modes of the bosonic closed string... [which] ... can be put in the formmassless spin-2 Gravitons
INTEGRAL d^26 x sqrt(g) R
.. .[of 26-dimensional general relativistic EinsteinGravitation] ... by absorbing a suitable power of exp(-PHI) inthe definition of the [26-dimensional MacroSpace]space-time
metric g_uv ...";
PHI, as to which Joseph Polchinski says"... The massless dilaton appears in the tree-level spectrum ofevery string theory, but not in nature: it would
mediate along-range scalar force of roughly gravitational strength.Measurements of the gravitational force at laboratory and greaterscales restrict any force with a range
greater than a fewmillimeters ( corresponding to a mass of order of 10^(-4) eV ) tobe several orders of magnitude weaker than gravity, ruling out amassless dilaton. ...". In
the , Dilatons could through and through the
and the and related ; andscalar Dilatons
D4-D5-E6-E7-E8VoDou Physics model get an effectively realmass dimensionalreduction of spacetime X-scalarHiggs field
of SU(5) GUT ElectroWeakSU(2)xU(1) Higgs scalar field conformalstructures
with , as to which JosephPolchinski says "... the negative mass-squared means that theno-string 'vacuum' is actually unstable ... whether the
bosonicstring has any stable vacuum ... the answer is not known. ...". Inthe interpretation of Closed Unoriented Bosonic String Theory asthe MacroSpace of the Many-
Worlds of World Strings, theinstability of a no-string vacuum is natural, because:Tachyons imaginary mass
if MacroSpace had no World Strings, or just one WorldString, the other possible World Strings would automatically becreated, so that any MacroSpace would be
"full" of "all"possible World Strings.
What about the size/scale of each of the 26 dimensions ofClosed Unoriented Bosonic String Theory ?
Represent the size/scale of each dimension as a radius R, with R =infinity representing a flat large-scale dimension. Let Lpl denotethe Planck length, the size of the lattice spacing in
the version of the . Joseph Polchinski says "... as R ->infinity winding states become infinitely massive, while the
compactmomenta go over to a continuous spectrum. ... at the opposite limit R-> 0 ... the states with compact momentum become infinitelymassive, but the spectrum of winding
states ... approaches acontinuum ... it does not cost much energy to wrap a string around asmall circle. Thus as thr radius goes to zero the spectrum againseems to approach that
of a noncompact dimension. ... In fact, theR-> 0 and R-> infinity limits are physically identical. Thespectrum is invariant under ...[HyperDiamondLattice D4-D5-E6-E7-E8VoDou Physics model
R -> R' = (Lpl)^2 / R
]... This equivalence is known as T-duality. ... The space ofinequivalent theories is the half-line [ R Lpl ]. We could take instead the range [ 0 R Lpl ] but it is more natural to
think in terms of the larger ofthe two equivalent radii ... in particular questions of locality areclearer in the larger-R picture. Thus [from the larger-R point ofview], there is no radius
smaller than the self-dual radius [Rself-dual = Lpl ]. ...". T-duality structures are is similar to .> <<
Planck Pivot Vortex structures13/2/01 9:36 PM 26 Dimensions of Bosonic Strings
Page 2 of 6 file:///iPurple/26dimBosonStrings/String26.html
Consider a ( ) world-lineString of one World of the MacroSpace of Many-Worlds and itsinteractions with another ( )world-line World String, from the point of view of
one point of the( ) World String, seen so close-upthat you don't see in the diagram that the( ) and( ) World Strings are both reallyclosed strings when seen at very
large scale:purple gold
purple purple gold
From the given point (diagram origin) of the( ) World String: purple
travel along the( ) MacroSpace light-cones tointeract with the intersection points of those( ) light-cones with the( ) World String; massless spin-2 Gravitons red red gold
travelwithin the ( ) MacroSpacelight-cone time-like interior to interact with the intersectionregion of the ( ) light-
conetime-like interior region with the( ) World String; andscalar Dilatons, with effectively real mass, yellow yellow
gold
travel within the( ) MacroSpace light-conespace-like exterior to interact with the intersection points ofthe ( ) light-cone space-
likeexterior region with the ( ) WorldString.Tachyons, with imaginary mass, cyan cyan
gold
The Gravitation of the massless spin-2 Gravitons of MacroSpaceis equivalent to the Gravitation of our physical SpaceTime, thusjustifying the Hameroff/Penrose
idea:
Superposition Separationis the separation/displacement of a mass separated from itssuperposed self. The picture is spacetime geometry separating fromitself .
, Gravitation from nearby World Strings might account for atleast some Dark Matter that isindirectly observed in our World String an ideasimilar to one described (in the
context of a superstring model that is in many ways very different from the D4-D5-E6-E7-E8VoDou Physics model ) by Nima Arkani-Hamed,Savas Dimopoulos, Gia Dvali, Nemanja Kaloper in their paper
ManyfoldUniverse, hep-ph/9911386 , and also in anarticle by the first three authors in the August 2000 issue of ScientificAmerican .
Bosonic Unoriented Closed String Theory describes the structure of andis related through (1+1) conformal structures to the
. For a nice introductorydiscussion of the mathematics of Bosonic Closed Strings, see and andother relevant works of
.Bohm's SuperImplicate Order MacroSpace LargeN
limit of the AN Lie Algebras Week 126 Week 127 JohnBaez
,such as: Branching among the Worlds of 27-dim M-Theory may bedescribable in termsof Singularities
simple (classified precisely by the Coxeter groups Ak, Dk, E6, E7, E8); singularities
unimodal ( asingle infinite three-suffix series and ); and singularities 14"exceptional" one-parameter families
bimodal ( 8infinite series and ). singularities 14exceptional two-parameter families
An M-theory of the full 27-dimensional Jordan algebra J3(O)
that could be to has beendiscussed in some recent (1997 and later) papers: S-dual BosonicString theory representing MacroSpace on 26-dim J3(O)o
Discussing both open and closed bosonic strings, Soo-Jong Rey, inhis paper ,Heterotic M(atrix) Strings and Their Interactions, says: hep-th/970415813/2/01 9:36 PM 26 Dimensions of Bosonic Strings
Page 3 of 6 file:///iPurple/26dimBosonStrings/String26.html"... We would like to conclude with a highly speculativeremark on apossible . It is well-known that
... The regularizedone-loop effective action of d-dimensional Yang-Mills theory ...For d=26, the gauge
kinetic term does not receive radiativecorrection at all ... We expect that this non-renormalizationremains the same even after dimensional reductions. ... one may
wonder if it is possible to construct ... for bosonic string as well despite theabsence of supersymmetry and BPS states.M(atrix) theory description of bosonicstrings bosonic Yang-
Mills theory intwenty-six dimensions is rather special
M(atrix)string theory
The bosonic strings also have D-brane extended solitons ...whose tension scales as 1 / gB for weak string coupling gB<< 1. Given the observation that the leading
order stringeffective action of and antisymmetrictensor field , let us make an assumption that
the 27-th `quantum'dimension decompactifies as the string coupling gB becomes large.For D0-brane, the dilaton exchange force may be interpreted as the27-th
diagonal component of d = 27 metric. Gravi-photon issuppressed by compactifying 27-th direction on an rather than on a circle. Likewise,
its mass may be interpreted as 27-th Kaluza-Klein momentum of amassless excitation in d = 27. In the infinite boost limit, thelight-front view of a bosonic string is that
infinitely manyD0-branes are threaded densely on the bosonic string. ...".graviton, dilaton may be derived from an Einstein gravity in d =27
orbifold[ such as S1 / Z2 ]
Gary T. Horowitz and Leonard Susskind, in their paper ,Bosonic M Theory, say: hep-th/0012037
"... The possibility that was ... discussed ...[ by Soo-Jong Reyin his paper ]... in the context of a
proposed matrix string formulation.... We conjecture that there exists a strong coupling limit ofbosonic string theory which is related to the 26 dimensionaltheory in
the same way that is related to superstring theory. More precisely, webelieve that
. The line intervalbecomes infinite in the strong
coupling limit, and this mayprovide a stable ground state of the theory. ...the bosonic string has a 27dimensional origin hep-th/9704158
11 dimensional Mtheory bosonic string theory is the compactification on aline
interval of a 27 dimensional theory whose low energy limitcontains gravity and a three-form potential
we ... argue that the tachyon instability may be removed inthis limit. ... The main clue motivating our guess comes from theexistence of the dilaton and its connection to
the couplingconstant. ... Evidently, as in , the dilaton enters the action just as it would if itrepresented the compactification scale of a Kaluza Klein
theory.We propose to take this seriously and try to interpret . We will refer to this theory
as . ...IIA stringtheory
bosonicstring theory as a compactification of a 27 dimensionaltheory
bosonic Mtheory
Closed bosonic string theory does not have a massless vector.This means it cannot be a compactification on an S1 . ...Accordingly, we propose that
. ... closed bosonic string theory is a compactification of27 dimensional bosonic M theory on [an orbifold ] S1 / Z2
In the bosonic case, since there are no fermions or chiralbosons, there are no anomalies to cancel. So there are no extradegrees of freedom living at the fixed points.
... the weaklycoupled string theory is the limit in which the compactificationlength scale becomes much smaller than the 27 dimensional Plancklength and the strong
coupling limit is the decompactificationlimit. The 27 dimensional theory should contain membranes but nostrings, and would not have a dilaton or variable coupling
strength. The usual bosonic string corresponds to a membranestretched across the compactification interval. ...
... In order to reproduce the known spectrum ofweakly coupled bosonic string theory, bosonic M theory will haveto contain an
additional field besides the 27 dimensionalgravitational field, namely a three-form potential CFT. Let usconsider
.the lowenergy limit of bosonic M theory ... is a
gravity theory in 27dimensions
the various massless fields that would survive in
theweak coupling limit
First of all, there would be the . As usual, general covariance in 26 dimensionswould insure that it remains massless. 26 dimensionalgraviton
The component of the 27 dimensional gravitational fieldg27;27 is a . It isof course the . No symmetry protects
the mass ofthe dilaton. In fact we know that at the one loop level adilaton potential is generated that lifts the dilatonic atdirection. Why the mass vanishes in the
weak coupling limit isnot clear.scalar in the 26 dimensional theory dilaton
Massless vectors have no reason to exist since there is notranslation symmetry of the compactification space. This isobvious if we think of this space [
] as a line interval.the
orbifold S1 / Z2
...[ with respect to ]... Even if27 dimensional flat space, M27, is a stable vacuum, one mightask what is the "ground state" of the theory at finite string
coupling, or finite compactification size? Tachyon condensationis not likely to lead back to M27, and there is probably nostable minimum of the tachyon
potential in 26 dimensions ...Instead, we believe . It is an old ideathat
quantum gravity may have an essentially topological phasewith no metric. We have argued that the tachyon instability isrelated to nucleation of "bubbles of
nothing" which iscertainly reminiscent of zero metric.tachyons
tachyon condensation may lead to anexotic state with zero metric guv = 0
... As an aside, we note that there is also a brane solution of26 dimensional bosonic string theory which has both electric andmagnetic charge associated with the
three-form H. It is a 21-branewith fundamental strings lying in it and smeared over theremaining 20 directions. Dimensionally reducing to six dimensionsby
compactifying on a small T 20 , one recovers the usual selfdual black string in six dimensions. ...
... We have proposed that a . One recovers the usual
bosonic string bycompactifying on andshrinking its size to zero. In particular, a Planck tension2-brane stretched along the compact direction has the right
tension to be a fundamental string. This picture offers aplausible explanation of the tachyon instability and suggests thatuncompactified 27 dimensional flat space may
be stable. A definiteprediction of this theory is , which should be its holographic dualfor AdS4 x S23
boundary conditions. ... if there does not exist a2+1 CFT with SO(24) global symmetry, bosonic M theory would bedisproven.bosonic version of M theoryexists, which is a 27 dimensional theory with 2-branes and21-branes
S1 / Z2
the existence of a 2+1 CFT withSO(24) global symmetry
... ? ... webelieve the limit of
bosonic M theory compactified on a circle asthe radius R --> 0 is the same as the limit R --> infinity,i.e., the uncompactified 27 dimensional theory. If we compactify
bosonic M theory on S1 x ( S1 / Z2 ), and take the second factorvery small, this is a consequence of the usual T-duality of thebosonic string. More generally, it
appears to be the onlypossibility with the right massless spectrum. ...".What kind of theory do we get if we compactify bosonic Mtheory on a circle instead of [ theorbifold S1 / Z2 ] a line interval
,such as: Branching among the Worlds of 27-dim M-Theory may bedescribable in termsof Singularities
simple (classified precisely by the Coxeter groups Ak, Dk, E6, E7, E8); singularities
unimodal ( asingle infinite three-suffix series and ); and singularities 14"exceptional" one-parameter families
bimodal ( 8infinite series and ). singularities 14exceptional two-parameter families
13/2/01 9:36 PM 26 Dimensions of Bosonic Strings
Page 4 of 6 file:///iPurple/26dimBosonStrings/String26.htmlHere are some descriptions of a few relevant terms:
Michio Kaku, in his book Introduction to Strings and M-Theory(second edition, Springer 1999), says:
"... the ... the fields can either be chiral or not. Closed strings are, bydefinition, periodic in sigma, which yields the following normal
mode expansion:closed [super] string ( Type II )
S1a(s,t) = Sum( n = -infinity; n = + infinity ) San exp( -2 i n( t - s ) ) ,
S2a(s,t) = Sum( n = -infinity; n = + infinity ) S'an exp( -2 i n( t + s ) ) .
If these two fields have different chiralities, then they arecalled . ... this
. ...Type IIA represents the N = 2, D =10-dimensional reduction of ordinary N = 1, D = 11
supergravity
... there exists a new 11-dimensional theory, called , containing 11-dimensional supergravity as itslow-energy limit, which reduces to Type IIA [super] string
theory (with Kaluza-Klein modes) when compactified on a circle.... the strong coupling limit of 10-dimensional Type IIAsuperstring theory is equivalent to the weak
coupling limit of anew 11-dimensional theory [ M-theory ], whose low-energylimit is given by 11-dimensional supergravity. ... Usingperturbation theory around weak
coupling in 10-dimensional TypeIIA superstring theory, we would never see 11-dimensional physics,which belongs to the strong coupling region of the theory. ...M-
theory is much richer in its structure than string theory. InM-theory, there is a three-form field Amnp, which can couple to anextended object. We recall that in
electrodynamics, a pointparticle acts as the source of a vector field Au. In[open] string theory, the [open] string acts asthe source for a tensor field Buv. Likewise, in
M-theory, amembrane is the source for Amnp. ...M-theory
... Ironically, 11-dimensional supergravity was previouslyrejected as a physical theory because:
(a) it was probably nonrenormalizable (i.e., there exists acounterterm at the seventh loop level);
(b) it does not possess chiral fields when compactified onmanifolds; and
(c) it could not reproduce the Standard Model, because itcould only yield SO(8) when compactified down to fourdimensions.
Now we can veiw 11-dimensional supergravity in an entirely newlight, as the low-energy sector of a new 11-dimensional theory,called M-theory, which suffers from
none of these problems. Thequestion of renormalizability is answered because the fullM-theory apparently has higher terms in the curvature tensor whichrender the
theory finite. The question of chirality is solvedbecause ... M-theory gives us chirality when we compactify on aspace which is not a manifold (such as [
] line segments). And the problem thatSO(8) is too small to accommodate the Standard Model is solvedwhen we analyze the theory nonperturbatively,
where we find E8 xE8 symmetry emerging when we compactify on [ ] line segments. ...".orbifoldssuch
as S1 / Z2
orbifoldssuch as S1 / Z2
Note that theD4-D5-E6-E7-E8 VoDou Physics Model solves the problems of11-dimensional supergravity in differentways , but uses many similarmathematical structures
and techniques.
Michio Kaku, in his book Strings, Conformal Fields and M-Theory(second edition, Springer 2000), says:
"... ... TypeIIA [super] string theory is S dual to a new, D = 11theory called M-theory, whose lowest-order term is given by D = 11
supergravity. ...S: M-theory on S1 <---> IIA
... ...[11-dimensional ]... M-theory, when compactified on a linesegment [S1 / Z2 ], is dual to the ... [ E8 x E8heterotic ]...
string ...".S: M-theory on S1 / Z2 <---> E8 x E8
Lisa Randall and Raman Sundrum, in their paper ,say: hep-ph/9905221
"... we work on the space . We take therange of PHI to be from -pi to pi; however the metic is completelyspecified by the values in the range 0 PHI pi.
The fixed points at PHI = 0, pi...[may]... be taken as the locations of ... branes ...".S1 / Z2 < <
orbifold
Note that S1 / Z2 can have two different interpretations.
says:
"... Z_2 acts in various ways on the circle.
Let's think of the circle as the subset
{(x,y): x^2 + y^2 = 1} of R^2.
Z_2 can act on it like this:
(x,y) |-> (-x,-y)
and then S^1/Z_2 =
which is a manifold, in fact a circle.
...
Z_2 also can act on the circle like this:
(x,y) |-> (-x,y)
and then S^1/Z_2 is an orbifold,
in fact a closed interval. ...".
The physical interpretations of RP1 in
the
as Time of SpaceTime and
as representation space for Neutrino-type
(only one helicity state) Fermions
might be viewed as having some
of the characteristics of a orbifold line interval. John Baez
RP1 [Real Projective 1-space]
D4-D5-E6-E7-E8 VoDou Physics model
Joseph Polchinski, in his book String Theory (volume 1, Cambridge1998), says:
"... orbifold
1. ... , where H is a group ofdiscrete symmetries of a manifold M. ; a coset space M / H The coset is singularat the fixed points of H
2. ... the CFT or string theory produced by the gauging ofa discrete world-sheet symmetry goup H. If the elements of Hare spacetime symmetries, the result is
a theory of stringspropagating on the coset space M / H . A non-Abelian orbifoldis one whose point group is non-Abelian. An asymmetric orbifoldis one 13/2/01 9:36 PM 26 Dimensions of Bosonic Strings
Page 5 of 6 file:///iPurple/26dimBosonStrings/String26.htmlwhere H does not have a spacetime interpretation andwhich in general acts differently on the right-movers andleft-movers of the string;
3. ... to produce such a CFT or string theory by gauging H; this is synonymous with the second definitioin of twist.
... ... a duality under which the couplingconstant of a quantum theory changes nontrivially, including thecase of weak-strong duality. ... In compactified
theories, theterm S-duality is limited to those dualities that leave the radiiinvariant, up to an overall coupling-dependent rescaling ...S-duality
... ... a duality in string theory, usually ina toroidally compactified theory, that leaves the couplingconstant invariant up to a radius-dependent rescaling and
therefore holds at each order of string perturbation theory. Mostnotable is R --> a' / R duality, which relates string theoriescompactified on large and small tori by
interchanging winding andKaluza-Klein states. ...T-duality
... ... any of the dualities of a stringtheory ... This includes the S-dualities and T-dualities, but incontrast to these includes also transformations that mix the
radiiand couplings. ...".U-duality
13/2/01 9:36 PM 26 Dimensions of Bosonic Strings
Page 6 of 6 file:///iPurple/26dimBosonStrings/String26.html |
arXiv:physics/0102043v1 [physics.bio-ph] 14 Feb 2001Statistical Approach to Gene Evolution
Sujay Chattopadhyay∗, William A. Kanner and Jayprokas Chakrabarti
Department of Theoretical Physics, Indian Association for the Cultivation of
Science, Calcutta 700 032, INDIA.
Abstract
The evolution in coding DNA sequences brings new flexibility and freedom to the codon
words, even as the underlying nucleotides get significantly ordered. These curious contra-rules
of gene organisation are observed from the distribution of w ords and the second moments of the
nucleotide letters. These statistical data give us the phys ics behind the classification of bacteria.
PACS numbers: 87.10.+e, 87.15.-v, 05.40.+j
1Over the years the statistical approach to genes has become p rominent. The hidden Markov
models are used in the alignment routines of biological sequ ences [1]. For the secondary structures of
the sequences stochastic context-free and context-sensit ive grammars [2] are applied [3]. The recent
discovery of the fractal inverse power-law correlations [4 ] in these biological chains have led to ideas
that statistically these sequences have features of music a nd languages [5-7]. Languages evolve with
time. The vocabulary increases; the rules that dominate get progressively optimised so the order
and information content is more. The purpose of this work is t o track the statistical basis of the
evolution in the coding DNA sequences (CDS).
The CDS are multiple of 3-tuples, the codons. The nucleotide s adenine (A), cytosine (C), guanine
(G) and thymine (T) taken in groups of three work to build the a mino acid chains called proteins.
The word-structure of CDS is, therefore, well known. We want to study evolution in terms of these
words, their distributions and the moments.
It is known that any prose does not carry all the ingredients o f evolution of languages. Similarly
the CDS of any gene does not have all the salient features that accompany change. The genes that
are present in the whole range of organisms, from the lowest b acteria to the highest mammals, and
therefore connected to fundamental life processes are norm ally considered to be best suited to function
as evolutionary markers. With this in view we choose glycera ldehyde-3-phosphate dehydrogenase
(GAPDH) CDS for its ubiquitous presence in all living beings . The enzyme it codes for catalyses
one of the crucial energy-producing steps of glycolysis, th e common pathway for both aerobic and
2anaerobic respiration.
Distribution of words is studied for languages. The frequen cy of words is plotted against the
rank. Here the total number of occurrences of a particular wo rd is termed its frequency. The word
most frequent has rank=1, the next most has rank=2, and so on. For natural languages, the plot
gives the Zipf [8] behaviour:
fN=f1
N(1)
where Nstands for the rank and f1andfNare the frequencies of words of rank 1 and Nrespectively.
The Zipf-type approach to the study of DNA has brought method s of statistical linguistics into DNA
analysis [6]. The generalized Zipf distribution of n-tuple s has provided hints that the DNA sequences
may have some structural features common to languages. In th is work we confine ourselves to the
distribution of 3-tuples, the codons, in the CDS. The words, therefore, are non-overlapping and on
the single reading frame.
The frequency-vs-rank plot of the codon words show that thes e distributions, given the frequency
of rank 1 and the length of the sequence, are almost completel y defined through the universal
exponential functional form [9]:
fn=f1.e−β(N−1)(2)
The parameter, called β, is determined by the ratio
β≈f1
L(3)
3βmeasures the frequency of rank 1 per unit length of the sequen ce. The exponential form (2) is
to be compared to the usual Boltzmann distribution. The rank of the word is akin to energy; βis
analogous to inverse temperature. The relationship (3) tha tβis frequency of rank 1 per unit length
is supported well from data [9]. The analogy between word dis tributions and the classical Boltzmann
concepts goes deeper. A decrease in β, from (3), implies frequency of rank 1 per unit length goes
down. In that case the vocabulary clearly increases. More wo rds are used, thereby more states are
accessed. For the GAPDH CDS we find the evolution is driving it to higher temperatures; into more
freedom for words, into more randomisation. βevolves monotonically.
Underneath, however, there runs a curious counterflow. Supp ose we look into the nucleotides
that constitute the sequence, once again in windows of size 3 and in the same reading frame. First,
we ask how much order there is in the sequence. To find out we stu dy the second moments of the
letters A, C, G and T. These second moments, by themselves, do not produce any pattern. The
GAPDH CDS has about 1000 bases. For each organism the proport ions of A, C, G and T in the
GAPDH CDS are different. This strand-bias, interestingly, m asks a remarkable underlying trend.
To get there the strand-bias has to be eliminated. The order i n the sequence, we assume, is its
deviation from the random. We define the quantity X, a measure of this deviation, as follows:
X=Second Moment of the Base Distribution in GAPDH CDS
Second Moment of the Base Distribution in the random sequenc e with identical strand bias
4Normalised as above, the effect of the strand bias is unmasked .Xvalues of GAPDH change mono-
tonically with evolution. The data tells us there is an incre ase in persistence amongst the letters (in
windows of size 3) with evolution in the CDS [10].
The evolution in the GAPDH CDS is then the result of these two c ontra trends: while words
acquire greater uniformisation, the underlying letters ha ve more order. The monotonic behaviours
ofβandXwith evolution give us the physics behind the biological cla ssification of bacteria.
Methods
Word Distributions
For the codons it is known [9] the exponentials give somewhat better fits over the usual power laws.
The exponential form, equation (2), is characterized by the parameter β. The quantity has some
universal features in that it is almost completely determin ed by f1and the length of the CDS. The
relationship [9]
β=f1−1
L+1
2.(f1−1)2
L2(4)
is known to fit observations on diverse genes. For the bacteri al GAPDH CDS the results of βare
given in Table 2.
Moments
Consider the 4-dimensional walk model [11,12] such that A, C , G and T correspond to unit steps, in
5the positive direction, along XA,XC,XGandXTaxes. After n-steps if the co-ordinate of the walker
is (nA,nC,nG,nT), then, clearly,
n=nA+nC+nG+nT (5)
andni(i≡A,C,G,T), is the number of nucleotide of type iin the sequence just walked.
If the sequence has nbases, and niis the number of base of type i, the strand bias of the sequence
is the proportion of niinn, defined as
pi=ni
n(6)
The probability distribution for the single step in this 4-d walk is
P1(x) =/summationdisplay
ipiδ(xi−1) (7)
where δis the usual δ-function of Dirac. The characteristic function of the step is the Fourier
transform of equation (7),
P1(k) =/summationdisplay
ipieiki(8)
The characteristic function of lsteps
Pl(k) = [P1(k)]l(9)
The second moments (i.e. the average values) of distributio ns may be obtained taking derivatives
ofPl(k) with respect to k. Thus for the random sequence (indicated by the subscript r) with the
6strand bias (6), we get the average values:
< n2
i>r=l[(l−1)p2
i+pi] (10)
< ninj>r=l(l−1)(pi.pj) ( i/negationslash=j) (11)
We are interested in codons, therefore, the window size lin equations (10) and (11) is chosen to
be 3. For the actual sequences we calculate < n2
i>seqand< ninj>seq. The quantities
XD=< n2
i>seq
< n2
i>r(12)
[where D ≡AA,CC,GG,TT]
and
XOD=< ninj>seq
< ninj>r(i/negationslash=j) (13)
[where OD ≡AC,AG,AT,CG,CT,GT]
measure the deviation of the diagonal and off-diagonal secon d moments of the sequence to those
of the random sequence of identical strand bias respectivel y. Finally, we come up with an over-all
averaged index, X, given by
X=/summationtextXD+/summationtextXOD
10(14)
ThisXprovides a measure of the order in the sequences.
7Observations and Results
To set the basis for what we discuss later, we begin by recordi ng the βand the Xvalues of higher
organisms, the eukaryotes (Table 1). We confine our discussi on of the eukaryotes to three broad
categories: fungi, invertebrates and vertebrates. It is kn own [13] from fossil records the oldest fungi
came about 900 million years (Myr) before present (bp). The o ldest fungal species, identified with
certainty, are from the Ordovician period, i.e., some 500 My r bp. The fossil records of invertebrates
suggest this group came about the same time as the fungi. The v ertebrates came later, about 400
Myr bp, in late Ordovician and Silurian period.
Let us look at the βand the Xvalues of these eukaryotic groups. Fungi has the highest, fo llowed
by invertebrates, while for the vertebrates the βand the Xreach minima. We conclude the βand
theXdecrease with evolution. The data further suggest fungi and invertebrates came about the
same time and underwent parallel evolution, while the repre sentives of the vertebrate group came
later in the evolutionary line-up.
Having set the basis, let us now look at 14 bacterial species f rom three groups: cyanobacteria,
proteobacteria (that includes vast majority of gram-negat ive bacteria), and the Bacillus /Clostridium
group, a type of gram-positive bacteria. Table 2 summarises theβand the Xvalues of these samples.
These bacterial groups arose during the Precambrian period of geological time-scale, but there are
several schools of thought regarding their specific times of origin within this period.
We approach the bacterial GAPDH CDS with two differing statis tical measures, the βand the
8X. Interestingly, both give us almost identical trends (Figs . 1 and 2). Lactobacillus delbrueckii ,
a member of the Bacillus /Clostridium group, has the highest βandXvalues (Table 2). There
is then a large measure of overlap between the Bacillus /Clostridium group and the proteobacteria
(Figs. 1 and 2). The extent of overlap of the βvalues is somewhat more than that of the X. The
cyanobacterial samples have the minimum values of the βand the X. There is no overlap between the
cyanobacterial values of the βand the Xwith the Bacillus /Clostridium group. The overlap between
the proteobacteria and the cyanobacteria is small. Only one proteobacterial sample, Brucella abortus
has greater βvalue than the cyanobacterial member, Synechocystis sp. (strain PCC 6803).
The averages of the βor the Xhas the maximum value in the Bacillus /Clostridium group,
followed by the proteobacteria, while the cyanobacteria sa mples have the lowest values. In line with
our observations on the eukaryotes, we propose (Figs. 1 and 2 ) that the Bacillus /Clostridium group
originated some time before the proteobacterial species, b ut later both groups evolved in parallel.
The cyanobacterial samples are of recent origin compared to these groups. The trends in the βand
theXgive us identical patterns that segregate the bacterial spe cies into groups. Amusingly, the
results seem to be in agreement with what is accepted so far re garding the phylogenetic relationships
among these three groups [14]. Our study of the GAPDH CDS, its word distributions, and the
moments gives us the physics underlying evolution.
The decrease in βwith evolution for the GAPDH CDS tells us that evolution is ta king the gene
progressively towards higher temperatures. The βvalue, we recall, is the frequency of rank one per
9unit length. Lowering of the βimplies less dominance of the maximum weight. In consequenc e,
the other words enjoy greater freedom, the vocabulary incre ases and more states are accessed. In a
sense the evolution in the GAPDH CDS mirrors Boltzmannian st atistics. Even though the GAPDH
CDS has evolved in a complex evolutionary regime in contact w ith environment, the Boltzmannian
behaviour is useful. For instance, it allows us to define the w ord-entropy of the CDS. That gives us
a measure of the information content of the words in biologic al chain.
At the level of the nucleotide letters A, C, G and T, the order i s measured by the quantity X.
As we look into the diagonal averages XD, (12), we find it increases with evolution. For the window
of size 3, this growing diagonal moment implies a rising pers istent correlation. In consequence, the
off-diagonal averages XOD, (13), go down, decreasing antipersistence. Looked at from the letters, the
sequences become less uniform and deviate more from the rand om sequence of identical strand bias.
The order, or the information, in the arrangement of letters shows a rising trend with evolution.
Does any CDS that is an evolutionary marker evolve in ways sim ilar to the GAPDH? We have
worked with the CDS of some other glycotic enzymes, such as ph osphoglycerate kinase, and found
they behave similarly. Other evolutionary markers such as t he ribulose-1,5-bisphosphate carboxy-
lase/oxygenase enzyme large segment (rbcL) show similar be haviour. We use these data for biological
subclassification. The CDS for ribosomal RNA is another clas s of sequence that is being investigated.
It does not code for protein, but for RNA, and has periods othe r than 3. The 3 period does exist,
but is not predominant.
10Sequence modeling has recently become important. The fract al correlations in the sequences led
to the expansion-modification system [15]. Later came the in sertion models [16]. Here the differences
in the CDS and non-coding sequences were observed and the non -coding sequences modeled. The
unifying models of copying-mistake-maps [17] modeled both the coding and the non-coding parts.
In these models the statistical features of the non-coding s equences have received emphasis. The
evolutionary features of the GAPDH CDS isolates the statist ical aspects that underlie evolution in
coding sequences. The statistics of the word distributions and the subtle cross current of the second
moments, we hope, will lead further in these efforts.
Acknowledgments
S.C. thanks Professor Anjali Mookerjee for many discussion s. W.A.K. is supported by the John
Fulbright foundation in the laboratory of J.C.
11∗Electronic address: tpsc@mahendra.iacs.res.in
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14Figure Legends
Figure 1. The average βvalues for the GAPDH CDS from three bacterial groups (see Tab le 3).
The error bars indicate the standard deviation from the aver age values.
Figure 2. The average Xvalues for the GAPDH CDS from three bacterial groups (see Tab le 3). The
error bars indicate the standard deviation from the average values.
15Table 1: The average βandXvalues of GAPDH CDS for eukaryotic groups, along with
the range of deviations in the respective groups.
Group β X
Vertebrates 0.05398 ( ±0.00414) 0.99698 ( ±0.004)
Invertebrates 0.07503 ( ±0.01067) 1.00235 ( ±0.00261)
Fungi 0.07742 ( ±0.00389) 1.00705 ( ±0.00175)
16Table 2: Theβand the Xvalues of the GAPDH CDS from the bacterial species that
have been used in our study (source: GenBank and EMBL databas es).
Organism Accession No. Group β X
Bacillus megaterium M87647 Bacillus /Clostridium 0.07662 1.01185
Bacillus subtilis X13011 Bacillus /Clostridium 0.07431 1.00912
Clostridium pasteurianum X72219 Bacillus /Clostridium 0.07837 1.00483
Lactobacillus delbrueckii AJ000339 Bacillus /Clostridium 0.08529 1.01861
Lactococcus lactis L36907 Bacillus /Clostridium 0.06038 1.00245
Pseudomonas aeruginosa M74256 Proteobacteria 0.08166 1.00338
Escherichia coli X02662 Proteobacteria 0.08366 1.00447
Brucella abortus AF095338 Proteobacteria 0.05713 1.00604
Zymomonas mobilis M18802 Proteobacteria 0.07721 1.00457
17Organism Accession No. Group β X
Rhodobacter sphaeroides M68914 Proteobacteria 0.06539 1.00564
Xanthobacter flavus U33064 Proteobacteria 0.06839 1.00086
Anabaena variabilis L07498 Cyanobacteria 0.04547 1.00073
Synechococcus PCC 7942 X91236 Cyanobacteria 0.05025 0.99988
Synechocystis PCC 6803 X83564 Cyanobacteria 0.06043 0.99101
18Table 3: The average βandXvalues of the GAPDH CDS for the three bacterial groups,
along with the range of deviations in the respective groups.
Group β X
Bacillus /Clostridium 0.07499 ( ±0.00914) 1.00937 ( ±0.00633)
Proteobacteria 0.07224 ( ±0.01033) 1.00416 ( ±0.00187)
Cyanobacteria 0.05205 ( ±0.00764) 0.99721 ( ±0.00538)
190 1 2 30.0400.0450.0500.0550.0600.0650.0700.0750.0800.085
Figure 11 - Bacillus / Clostridium group
2 - Proteobacteria
3 - Cyanobacteriaβ
Bacterial Groups0 1 2 30.9900.9920.9940.9960.9981.0001.0021.0041.0061.0081.0101.0121.0141.016
Figure 21 - Bacillus / Clostridium group
2 - Proteobacteria
3 - CyanobacteriaX
Bacterial Groups |
arXiv:physics/0102044 14 Feb 2001
1Optimality of the genetic code with respect to protein stability
and amino acid frequencies
Dimitri Gilis 1, Serge Massar 2, Nicolas Cerf 3, and Marianne Rooman 1,2
1. Biomolecular Engineering, Université Libre de Bruxelles, CP 165/64, 50 av. F. D. Roosevelt, 1050
Bruxelles, Belgium
2. Service de Physique Théorique, Université Libre de Bruxelles, CP225, bld. du Triomphe, 1050
Bruxelles, Belgium
3. Ecole Polytechnique, Université Libre de Bruxelles, CP 165/56, 50 av. F. D. Roosevelt, 1050
Bruxelles, Belgium
Classification: Biological Sciences, Evolution
Corresponding author: D. Gilis, dgilis@ulb.ac.be, tel +32-2-650 2067, fax +32-2-650 3606
Manuscript information: 31 pages, 3 tables, 2 figures, supplementary material
Character counts: 45884 characters (including tables and figures)2Abstract
How robust is the natural genetic code with respect to mistranslation errors? It has long been
known that the genetic code is very efficient in limiting the effect of point mutation. A misread codon
will commonly code either for the same amino acid or for a similar one in terms of its biochemical
properties, so the structure and function of the coded protein remain relatively unaltered. Previous
studies have attempted to address this question more quantitatively, namely by statistically estimating
the fraction of randomly generated codes that do better than the genetic code regarding its overall
robustness. In this paper, we extend these results by investigating the role of amino acid frequencies in
the optimality of the genetic code. When measuring the relative fitness of the natural code with respect
to a random code, it is indeed natural to assume that a translation error affecting a frequent amino acid is
less favorable than that of a rare one, at equal mutation cost (being measured, e.g., as the change in
hydrophobicity). We find that taking the amino acid frequency into account accordingly decreases the
fraction of random codes that beat the natural code, making the latter comparatively even more robust.
This effect is particularly pronounced when more refined measures of the amino acid substitution cost
are used than hydrophobicity. To show this, we devise a new cost function by evaluating with computer
experiments the change in folding free energy caused by all possible single-site mutations in a set of
known protein structures. With this cost function, we estimate that of the order of one random code out
of 100 millions is more fit than the natural code when taking amino acid frequencies into account. The
genetic code seems therefore structured so as to minimize the consequences of translation errors on the
3D structure and stability of proteins.3Introduction
One of the tantalizing questions raised by molecular biology is whether the basic structures of life
as we know them arose through a Darwinian evolutionary process, and if so, what were the evolutionary
pressures acting on them? One such structure which could have changed through evolution is the genetic
code. The genetic code is almost universal throughout life, with very minor variations in mitochondria,
and any trace of the different stages of a possible evolutionary process, if any, has disappeared.
Nevertheless the idea that the genetic code could have evolved to its present form has been repeatedly
suggested in the literature (1). For instance, it has been proposed that early codes were simpler in that
they coded for only a few amino acids, and that the number of amino acids coded in the genetic code
increased as the code evolved (2-5). Several hypotheses have been put forward to explain the evolution
of the genetic code to its present form, and to find out what is the genetic code optimized for (4, 6-15).
One possible scenario is that the genetic code evolved so as to minimize the consequence of errors
during transcription and translation (7, 8, 10, 11, 16). To test this hypothesis, some authors tried to
estimate the percentage of achievement of the natural code by quantifying the cost of single-base
changes (17-19).
More recently, Haig and Hurst (20) and Freeland and Hurst (21) improved the latter approach by
comparing the natural code with random codes. To this end, they defined a fitness function Φ that
measures the efficiency of the code to limit the consequences of transcription and translation errors. This
function Φ supposedly evolved towards a minimum through evolution. To measure how close the
natural code is to the actual minimum of Φ, they generated random genetic codes, and computed the
fraction of those that are better - i.e., have a smaller value of Φ - than the natural code. They found that
only a very small fraction of the random codes are better than the natural code, and concluded that the
natural code is therefore optimal in that it minimizes the effect of translation and transcription errors.* These results have already been presented at the Conference Frontiers of Life, Blois, France
(June 2000)._________________________
4Haig and Hurst (20) tested several fitness functions Φ based on different physico-chemical
parameters, and found that single-base changes in the natural code had the smallest average effect when
using, as a cost measure, the change in polarity or hydropathy between the corresponding amino acids.
These parameters, although not unique, are clearly biologically relevant as they are related to
hydrophobicity, a property known to play an important role in protein conformation. Changing, through
a transcription or a translation error, a non-polar amino acid into a polar one at some strategic position in
the sequence of a protein can have dramatic consequences on its conformation. Using these parameters,
and assuming that all point mutations occur with the same frequency, Haig and Hurst (20) found that the
fraction of random codes that beat the natural code is of the order of 10 -4.
It has been shown experimentally that individual translation errors occur more frequently at the
first and third codon positions than at the second (8, 22, 23), and that there are transition/transversion
biases (24-27). Taking this into account, Freeland and Hurst (21) proposed a modified fitness function Φ
which models more accurately the probability of translation errors. They found that with this improved
modeling, the fraction of random genetic codes that are better than the natural one decreases from 10-4
to 10-6. They retrieved from their calculations a well known property of the genetic code: single-base
substitutions in the first and third codon position are strongly conservative with respect to changes in
polarity (8, 28).
In the present paper*, we highlight the importance of another parameter in the optimization of the
genetic code, namely the frequency with which different amino acids occur in proteins. This frequency
differs from protein to protein, and even from species to species, but there is a general pattern that
prevails (Table 1). For instance Leu is the most common amino acid, and Trp the rarest. In Fig. 1, we 5have plotted the number of codons coding for the same amino acid (synonyms) versus the amino acid
frequency. The clear correlation between these two quantities, first noted by King and Jukes (30), led us
to suspect that the amino acid frequency is an important parameter in the optimization of the genetic
code, which should also be taken into account in the fitness function Φ. Our calculations indeed confirm
that the genetic code is even more optimal with respect to translation errors if the amino acid frequencies
of Table 1 are properly incorporated in Φ.
In addition, we bring further improvements to the fitness function Φ by using other quantities than
polarity to measure the roles of the different amino acids in protein conformation and stability. It should
be stressed that the biological relevance of the parameters used in Φ is crucial in the estimation of the
relative robustness of the natural code. Indeed, one can always construct an artificial fitness function Φ
such that the natural biological structure apparently lies at its minimum. Clearly, the hydrophobicity
parameters used by Haig and Hurst (20) are biologically motivated, but we would like to do better by
refining our cost measure. In particular, we devise a mutation matrix describing the average cost of
single amino acid substitutions in protein stability, obtained by computer experiments. This mutation
matrix combines many different physico-chemical properties of the amino acids. For instance, it takes
into account that mutating Cys into any other amino acid is always very costly since this can break a
disulfide bond. Such an effect would not be apparent if only a single property, say hydrophobicity, was
taken into account. We show that, with a fitness function Φ depending on this mutation matrix and the
amino acid frequencies, only about 1 out of 108 randomly generated codes are better than the natural
code. This suggests that the genetic code is even better optimized to limit translation errors than was
previously thought. After completion of this paper, we became aware that Freeland et al. (29) have
recently improved the fitness function in a different way by using as a cost measure an amino acid
substitution matrix derived from protein sequence alignements. They found that the relative fitness of
the genetic code increases with this more realistic cost function. It is reassuring that this conclusion is
reached using both our mutation matrix and Freeland et al.’s substitution matrix, although the derivation 6of the two cost functions use very different starting points (protein 3D structures in one case, sequence
similarity in the other).
Fitness of the genetic code with respect to translation errors
Consider the natural genetic code. It is built out of 64 codons, each consisting of 3 consecutive
DNA bases (A, G, C, T) or RNA bases (A, G, C, U). These 64 codons are divided into 21 sets of
synonyms, which code each for one of the 20 natural amino acids or correspond to a stop signal; hence,
to each codon c, an amino acid (or stop signal) a is assigned through a function a(c). Consider now an
error during transcription from DNA to RNA or during translation from RNA to protein, in which codon
c is mistaken for codon c'. This error thus results in amino acid a(c) being replaced by amino acid
a'=a(c'). The associated cost is estimated by a function g(a,a'), which measures the difference between
the amino acids a and a' with respect to their physico-chemical properties or their role in (de)stabilizing
protein structures; when a or a' corresponds to a stop codon, we set g(a,a')=0. Different cost functions g
will be discussed in the next section. Following Freeland and Hurst (21), the fitness Φ of a code is
measured by the average of the cost g over all codons c and all single-base errors c→c' :
ΦFH=1
64∑64
∑64
p( )c′|cg
c′=1 c=1( )a(c),a(c′) , (1)
where p(c'|c) is the probability to misread codon c as codon c'. If one focuses on transcription errors
only, as Haig and Hurst (20), then all p(c'|c)'s must be taken equal. But here we consider translation
errors, as Freeland and Hurst (21), and hence p(c'|c) changes according to whether c and c' differ in the
first, second or third base, and lead to a transition or a transversion. A transition is the substitution of a
purine (A, G) into another purine, or a pyrimidine (C, U/T) into another pyrimidine, whereas a
tranversion interchanges purines and pyrimidines. Based on experimental data indicating that transitions 7are more common than transversions (24-27), and that errors on the third base are more frequent than
errors on the first base, which are themselves more frequent than errors on the second base (8, 22, 23),
Freeland and Hurst (21) have chosen the following values of p(c'|c) , which we also use here:
p(c′|c)=1/Nifc andc′differinthe3rdbaseonly,
p(c′|c)=1/Nifc andc′differinthe1stbaseonlyandcause atransition,
p(c′|c)=0.5/Nifc andc′differinthe1stbaseonlyandcause atransversion,
p(c′|c)=0.5/Nifc andc′differinthe2stbaseonlyandcause atransition,
p(c′|c)=0.1/Nifc andc′differinthe2stbaseonlyandcause atransversion,
p(c′|c)=0ifc andc′differbymorethan 1 base,
where N is a normalization factor ensuring that Σc' p(c'|c)=1.
Incorporating amino acid frequencies in the fitness function
Let us now come back to the correlations between the number of codons coding for an amino acid
and the frequency of this amino acid (see Fig. 1). King and Jukes (30), who first noted this correlation,
suggested that most of the amino acids in the genomes have arisen by random mutations which do not
affect the properties and function of the proteins. As a consequence, the number of synonymous codons
determines the frequency of amino acids. The fitness function ΦFH is in accordance with this point of
view, since each codon is given equal weight in this function.
An alternative interpretation, assuming a very different chain of causality, is that the amino acid
frequencies are fixed by their physico-chemical properties. For instance, Trp would be a rare amino acid
because its specific properties are seldom needed in proteins or because it is difficult to synthesize. The
correlation between the amino acid frequencies and number of synonymous codons (Fig. 1) would then
be interpreted as being due to an adjustment of the natural genetic code to the frequency of the amino 8acids. The conclusions which are reached using these two opposite interpretations are discussed in the
final section of this paper.
A codon error substituting a rare amino acid into another has obviously less consequence, at least
on the average, than an error affecting a frequent amino acid. The frequencies with which the different
amino acids occur in proteins, which are approximately universal in all organisms (Table 1), are only
imperfectly taken into account in the fitness function ΦFH given by eq. (1), because of the imperfect
correlation between amino acid frequency and number of synonymous codons (Fig. 1). In order to
properly account for this effect, we propose a modified fitness function Φfaa:
Φfaa=∑64p( )a(c)
n( )c∑64
p( )c′|cg
c′=1 c=1( )a(c),a(c′) , (2)
where p(a) is the relative frequency of amino acid a, and n(c) is the number of codons in the block to
which c belongs. In other words, n(c) is the number of synonyms coding for the amino acid a(c) that c
codes for. Note that eq. (2) supposes that there is no codon bias, i.e., the different synonyms of a given
amino acid appear with the same frequency.
In order to measure the effect of the amino acid frequency on the value of the fitness function
Φfaa, we define, for the sake of comparison, another fitness function Φequif where all the amino acids
are supposed equifrequent, i.e. p(a)=1/20:
Φequif=1
20∑64
1
n( )c∑64
p( )c′|cg
c′=1 c=1( )a(c),a(c′) . (3)9Cost of substituting an amino acid into another
The function g(a,a') in eqs (1) and (2) measures the cost - as far as protein stability and structure is
concerned - of substituting amino acid a by a’. This cost depends on several physico-chemical and
energetic factors. Hydrophobic interactions are known to constitute the dominating energetic
contribution to protein stability. Hence, a natural choice for g consists of taking the squared difference
in hydrophobicity h of the amino acids a and a':
ghydro(a,a′)=( )h(a)−h(a′)2. (4)
There exist various hydrophobicity scales for amino acids. We have tested two of them. The first is the
polarity scale defined by Woese et al. (31), which is the one that was used by Haig and Hurst (20) and
Freeland and Hurst (21). In the second scale, h(a) is the average solvent accessibility of amino acid a
derived from a set of 141 well resolved and refined protein structures with low sequence identity (see
Appendix) ; solvent accessibilities are computed using SurVol (32). We denote the associated cost
functions as gpol and gaccess, respectively.
Although hydrophobic forces dominate in proteins, other types of interactions also contribute to
protein stability. We therefore also attempted to devise a better cost function g(a,a'), measuring more
accurately the difference between amino acids a and a'. This new function is inspired by recent
computations of the change in free energy of a protein when a single amino acid is mutated (33-35). It is
obtained by mutating in silico, in all proteins of the aforementioned set of 141 protein structures, and at
all positions, the wild type amino acids into the 19 other possible ones, and evaluating the resulting
changes in folding free energy with mean force potentials derived from the same structure dataset. The
matrix elements M(a,a') are obtained as the average of all the computed folding free energy changes
which correspond to a substitution a→a'. Details on the procedure and the value of the matrix elements
M(a,a') are given in the Appendix. This matrix is taken as a cost function:10gmutate(a,a′)=M(a,a′). (5)
As a last cost function, we consider the "blosum62" substitution matrix (36, 37), one of the most
commonly used matrices in the context of protein sequence alignment :
gblosum(a,a′)=blosum62(a,a′). (6)
This matrix is computed from the frequency of amino acid substitutions in families of evolutionary
related proteins. However, it reflects not only the similarity between amino acids with respect to their
physico-chemical and energetic properties, but also the facility with which one amino acid is mutated
into another and thus their proximity in the genetic code. Strictly speaking, it should therefore not be
used to estimate the fitness of the genetic code; we only use it here as a reference. This potential
problem might also affect the substitution matrix used by Freeland et al. (29), but probably to a smaller
extent as their matrix was derived from highly diverged protein sequences.
Results: the genetic code versus random codes
To evaluate the robustness of the natural genetic code with respect to translation errors, we
computed the fitness functions ΦFH, Φequif and Φfaa using eqs (1-3) for the natural genetic code, and
compared it to the corresponding fitnesses of random codes. The random codes are obtained by
maintaining the codon block structure of the natural genetic code, where each block corresponds to
synonyms coding for the same amino acid (or stop signal). When generating a random code, the stop
signal is kept assigned to the same block as in the natural genetic code, whereas the different amino
acids are randomly interchanged among the 20 remaining blocks. Thus, each random code is simply
specified by a different function a(c) in eqs (1-3). This is the procedure previously used by Haig and
Hurst (20) and Freeland and Hurst (21).
Thus, in a first stage, we computed the fitness functions Φequif and Φfaa for the natural genetic
code and for 108 randomly generated codes, using the three cost functions gpol, gaccess and gmutate. 11We then calculated the fraction f of random codes whose value of Φ is lower than that of the natural
code. This fraction is supposedly a good estimate of the relative merit of the natural genetic code
comparatively to other codes. The results are given in Table II. It appears that, for all cost functions g,
this fraction f is between 10 and 100 times smaller for Φfaa than for Φequif. This indicates that the
natural code appears to be better optimized with respect to translation errors if the amino acid
frequencies are taken into account.
In order to investigate this further, we have analyzed which of the cost functions gpol, gaccess or
gmutate the genetic code appears to be best optimized for. For this purpose, we compared the fraction f
of better codes for each of the cost functions using the fitness function Φfaa (cf. Table II). For the
hydrophobicity functions gpol and gaccess, the result is roughly the same: f is about 1-8 in a million.
The relative statistical error on this value is of the order of N-1/2, where N is the number of random
codes better than the natural one that were found in our sample of 108 random codes; thus, N is about
100-800, and the error is insignificant. For the mutational cost function gmutate, we did not find any
random code better than the natural one among the 108 random codes. Then, to estimate the fraction f
without having to generate a larger ensemble, we used the following procedure. We computed, from the
values of Φfaa for the 108 random codes, the probability function π(Φfaa) to have a given value of
Φfaa. We fitted log(π(Φfaa)) to a polynomial of fourth degree, and extrapolated this curve down to the
value of Φfaa for the natural code. This provides an estimate of the fraction f of random codes that have
a lower Φfaa value. Note that this estimate is essentially insensitive to the degree of the polynomial. We
found that using gmutate as a cost function this fraction is of the order of 1 in 10 8.
This result shows that the natural genetic code appears even more optimal if the cost function
gmutate is used than if hydrophobicity-based cost functions are considered. Since gmutate has been
computed from protein stability changes effected by point mutations, we may conclude that evolution
has optimized the genetic code in such a way as to limit the effect of translation errors on the 3D 12structure and stability of the coded proteins. Note that the improvement brought by the choice of
gmutate results from the fact that it probably better accounts for the cost of a mutation than a mere
difference of hydrophobicity; for example, Gly, Pro, and Cys have close neighbors in hydrophobicity,
while the cost of their mutation as accounted for by gmutate is high. This is due to their special role in
determining protein structure: Gly and Pro can adopt backbone torsion angles essentially inaccessible to
other amino acids, and Cys can form disulfide bonds.
For completeness, we have added in Table II the values of the fraction f of random codes with a
lower Φfaa value than the natural one, using the cost function gblosum. With this function, f
(extrapolated as above) is about three times smaller than with gmutate. This was expected as the blosum
matrix (36, 37) is computed from amino acid substitutions in families of evolutionary related proteins,
which are more frequent between amino acids that are closer in the genetic code. Using gblosum can
therefore be considered as superimposing some information on the proximity of amino acids in the
genetic code to the desired measure of their similarity in preserving protein structure. So, it is not
surprising that gblosum does better in minimizing Φfaa than gmutate, which only includes information
related to protein structure. In contrast with what happens with Φfaa, the fraction of random codes
having a lower Φequif value than the natural code is larger when using gblosum than with gmutate.
Thus, if all amino acids are assumed to be equifrequent then the apparent merit of gblosum disappears.
This means that, besides informations about proximity, gblosum also incorporates information about the
amino acid frequencies. For these reasons, gblosum is probably an intrinsically bad cost measure for our
purposes here.
Finally, we have attempted to check the significance of our main result that the natural code is
better optimized if amino acid frequencies are taken into account. To this end, we have computed the
fraction f of random codes that beat the natural one for random choices of the amino acid frequencies ,
distinct from the natural frequencies p(a). We have generated 102 sets of random p(a)’s, and, for each of 13them, estimated the fraction f (out of a sample of 106 random codes). The percentage of random amino
acid frequency sets that result in a lower fraction f than the natural frequencies is shown in Table III. We
find that a random assignment of the amino acid frequencies does not decrease f in the great majority (at
least 94 %) of the cases, and this tendency persists for all cost functions g. Thus, the probability that the
decrease of f, observed in Table II, when passing from Φequif to Φfaa, was due to chance is quite
limited. We may therefore conclude that the genetic code is optimized so as to take into account the
natural amino acid frequencies.
For comparison, we have also included in Table II the results based on the fitness function ΦFH. It
can be argued that this function takes in part, but imperfectly, the amino acid frequencies into account.
Indeed, for this fitness function each codon is assigned the same weight, which corresponds to each
amino acid being assigned a frequency proportional to the number of synonyms n(a) coding for it. In the
case of the natural genetic code, this frequency corresponds approximatively to the amino acid
frequency since there is a correlation between n(a) and p(a), as shown in Fig. 1. But for random codes,
where the amino acids are randomly interchanged between the codon blocks, this correspondence breaks
down. Thus, the way in which ΦFH takes amino acid frequencies into account depends on the code
considered. This explains why the fraction f of random codes better than the natural one is sometimes
smaller or sometimes larger using ΦFH instead of Φequif. Note, however, that f is always larger for
ΦFH than for Φfaa, indicating again the importance of the amino acid frequencies in the optimality of
the genetic code.
Discussion and conclusion
Our results confirm and specify those of Freeland and Hurst (21): the genetic code seems
structured so as to minimize the consequences of translation errors on the 3D structure and stability of
the coded proteins. We have shown that, using the cost function gmutate, which best reflects the roles of
various amino acids in protein structures, and taking amino acid frequencies into account, about 1 out of 14108 random codes does better than the natural code. However, we have to keep in mind that there exist
20!≈2•1018 possible codes preserving the codon block structure, which means that we can expect about
1010 better codes overall. Moreover, if the codon block structure is not preserved (14), the number of
possible codes is larger by orders of magnitude, and therefore the number of codes better than the
natural one will certainly be much larger.
So, we can assert from our analysis that the genetic code has been optimized through evolution up
to a certain point, even though it is probably not fully optimal at least with respect to the parameters
considered here. However, our analysis does not give us information about the mechanism of this
evolution since there is unfortunately no trace left of evolution of the code or amino acid frequencies in
early times. For instance, we do not know whether the relative frequency of occurrence of amino acids
in proteins adapted so as to increase the optimality of the genetic code with respect to translation errors,
or, on the contrary, whether the genetic code evolved to take into account pre-existing amino acid
frequencies. We can, however, argue that if the amino acid frequencies adapted to the genetic code, as
assumed by King and Jukes (30), a discrepancy in amino acid composition between frequently and
unfrequently expressed genes might be detectable today (unless the period during which evolution took
place was long enough for this discrepancy to vanish). If, alternatively, the genetic code adapted to the
amino acid frequencies, and thus if these frequencies acted as an evolutionary pressure, one can imagine
two scenarios. Either the code optimized to take into account the pre-biotic frequencies of the amino
acids that became involved in it, or it optimized for the amino acid frequencies of already formed
proteins (or of a subset of them) that were important for life and maybe linked to the code’s control.
Perhaps can we assume, more realistically, that the genetic code and amino acid frequencies coevolved
during some evolutionary period, thereby approaching an optimal code/amino acid relation.
More generally, the parameters that acted as evolutionary pressure on the genetic code probably
included all the mechanisms that code and maintain the genetic information, and were not just restricted
to the frequency of amino acids and the preservation of protein structure. For example, the genetic code 15is obviously related to the translation apparatus, composed of the ribosomes and transfer RNA, whose
action we described schematically here by the probabilities p(c'|c) to misread codon c as c'. This
apparatus was certainly less reliable at the beginning of evolution. All these mechanisms probably
evolved together with the genetic code during the early stages of life.
The evolution of the code came to an end at an early stage of life development, as reflected by its
universality among all organisms. This probably arose because even small modifications in the code
would entail loss of functionality of previously expressed genes. Moreover, the advent of more
sophisticated transcription/translation control mechanisms, which involve huge protein systems, could
have decreased the evolutionary pressure on the genetic code. Even though the present data on the
genetic code are insufficient to discriminate between evolution scenarios, our analysis enables us to put
some constraints on the situation at the time when the code evolution was frozen. In particular, it appears
that the frequencies of the amino acids that were used in proteins synthesized at that time were similar to
the present frequencies. We do not know what determines the present amino acid frequencies, but,
presumably, they are due at least in part to their physico-chemical properties. For instance, the
hydrophobic to hydrophylic ratio is intrinsically related to the globular structure of proteins and certainly
contributes to the pressure on amino acid frequencies. Also, amino acid that are easily synthesized may
be used more often. Thus, we can assert that some of the pressures that determine the present amino acid
frequencies were already present at the time when the code took its definitive form. In addition, the
increased optimality of the genetic code with respect to gmutate implies that the 3D structure of proteins
probably played an equally important role in fixing the structure of the code. Since the 3D structure of a
protein essentially determines its function, this suggests, more generally, that the protein function acted
as a main evolutionary pressure on the code structure. Consequently, at the time when the genetic code
took its present form, primitive life was presumably synthesizing complex proteins already. This
provides a tentative picture of primitive life at that time: the translation apparatus was similar to the 16present one, and organisms where made of complex proteins whose amino acid frequencies was
comparable to the present ones.
Acknowledgments
We are grateful to Jacques Reisse for fruitful discussions and to a referee for useful comments.
DG benefits from a "FIRST-université" grant of the Walloon Region. SM and MR are, respectively,
research associate and senior research associate of the Belgian National Fund for Scientific Research.17Appendix: Derivation of the mutation matrix
The derivation is based on a dataset of 141 high-resolution protein structures determined by X-ray
crystallography listed in Wintjens et al. (38). In order to avoid biases, these 141 proteins are chosen so as
to either present less than 20% sequence identity or to present less than 25% sequence identity and no
structure similarity.
The protein main chains are described by their heavy atoms, and each side chain is represented by
a pseudo-atom Cµ. For a given amino acid type, the Cµ has a well-defined position relative to the main
chain, corresponding to the geometric average of all heavy side chain atoms of this type in the dataset
(39); for glycine, the Cµ pseudo-atom is positioned on the Cα. Side chain degrees of freedom are thus
neglected.
Each residue, at each position of each of the 141 proteins, is mutated in turn into the 19 non-wild
type amino acids. The mutations are performed by keeping the main chain structure unchanged, and
substituting the Cµ of the mutated amino acid by that of the mutant amino acid. For each of these
mutations, the change in folding free energy is evaluated using database derived potentials. For each
substitution of amino acid a into a', the average of all computed changes in folding free energy, at all
protein positions, is computed and defined as minus the matrix element M(a,a'). We then symmetrize M
by setting M(a,a')=[M(a,a')+M(a',a)]/2 and only consider the lower half of M (a≤a'). This procedure
does not define the diagonal elements of M. Based on the principle that the structural role of a given
amino acid is fulfilled by no other amino acid better than by itself, we assign to all the diagonal element
the same maximum value: M(a,a)= Max[M(a',a")]+1. Then, to simplify M without modifying its
structure, we center it around its mean value :18M(a,a′)→M(a,a′)−<M>with<M> =1
210∑M(a,a′)
a′≤a
Finally, we multiply all matrix elements M(a,a') by 2 and replace them by the closest integer. The
resulting half matrix is given in Fig. 2. For more details, see the supplementary material. 19References
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Ala 7.85
Arg 5.33
Asp 5.37
Asn 4.55
Cys 1.88
Glu 5.83
Gln 3.77
Gly 7.35
His 2.35
Ile 5.80
Leu 9.43
Lys 5.88
Met 2.28
Phe 4.07
Pro 4.56
Ser 6.04
Thr 6.17
Trp 1.31
Tyr 3.27
Val 6.92
Table I: The mean frequency p(a) with which the different amino acids a appear in the genomes of many
different organisms, derived from the Swiss-Prot database (40) (see
http://cbrg.inf.ethz.ch/ServerBooklet/section2_11.html).23 f ΦFH Φequif Φfaa
gpol2.7•10-6 1.9•10-5 1.5•10-6
gaccess3.3•10-5 6.2•10-5 8.4•10-6
gmutate1.8•10-4 1•10-6 1•10-8 *
gblosum1.5•10-3 1.1•10-4 3•10-9 *
Table II: Fraction f of random codes that have a lower value of the fitness function (ΦFH, Φequif, or
Φfaa) than the natural code, using each of the four cost functions gpol, gaccess, gmutate and
gblosum. The values with ’*’ have been obtained by extrapolation as explained in text.24%
gpol 6
gaccess 4
gmutate <1
gblosum <1
Table III: Percentage of the sets of random amino-acid frequency assignments for which the fraction f of
random codes that beat the natural code is lower than the corresponding fraction computed with
the natural frequencies p(a)’s. This percentage is estimated using the four cost functions gpol,
gaccess, gmutate and gblosum. In the case of gmutate and gblosum we are only able to give an
upper bound, because our sample of 100 random frequencies and 106 random codes is too small.
For these cost functions we found respectively 4 and 2 random frequencies for which none of the
random codes was better than the natural code. The extrapolation method used for the natural
frequencies showed that for none of these 6 frequency sets, the fraction f was smaller than the
corresponding fraction for the natural amino acid frequencies. Given the approximate character of
this extrapolation, and the limited size of our sample, we give an upper bound of 1% for the
fraction of random amino acid frequencies whose f is smaller than for the natural frequencies. 25Figure 1: The relative frequency p(a) (in %) of the amino acids a, taken from Table I, as a function of
the number of synonyms n(a) that code for them. The linear regression line is indicated; the
correlation coefficient is equal to 0.71.26
A
A+7 C
C -3 +7 D
D 0 -4 +7 E
E 0 -5 +2 +7 F
F 0 -2 -2 -3 +7 G
G -2 -4 -1 -2 -3 +7 H
H +1 -2 +1 +1 -1 0 +7 I
I -2 -3 -4 -4 0 -4 -2 +7 K
K +1 -5 +2 +3 -2 -1 +1 -3 +7 L
L -1 -3 -3 -3 0 -4 -2 0 -2 +7 M
M +1 -3 -1 -1 0 -2 0 0 -1 +1 +7 N
N 0 -3 +2 +2 -2 0 +2 -3 +2 -2 -1 +7 P
P -3 -6 -1 -2-4 -4-2 -5 -1 -5 -4 -1 +7 Q
Q +1 -4 +2 +3 -1 -1 +2 -2 +3 -1 0 +2 -2 +7 R
R +1 -4 +2 +2 -2 -1 +1 -3 +3 -2 0 +2 -2 +2 +7 S
S +1 -2 +2 +1 -1 0 +2 -2 +2 -1 0 +2 -1 +2 +2 +7 T
T 0 -2 +1 0 0 -1 +2 -1 +1 -1 0 +1 -1 +1 +1 +2 +7 V
V -1 -3 -3 -4 0 -4 -2 +1 -3 0 0-3 -4 -2 -2 -1 0 +7 W
W +1 -2 0 -1 0 -2 +1 -1 0 -1 +1 0 -3 0 0 +1 0 -1 +7 Y
Y 0 -1 -1 -2 +1 -1 +1 -1 -1 -1 0 0 -3 0 -1 +1 +1 0 +1 +7
Figure 2: Mutation matrix M27Supplementary material: details of the derivation of the mutation matrix
Database derived potentials
The potentials we use to evaluate the protein conformations are derived from observed frequencies
of sequence and structure patterns in the aforementioned dataset of 141 proteins. We consider two types
of potentials, called torsion (1, 2) and C µ-Cµ (3) potentials.
Torsion potentials describe only local interactions along the sequence. They take into account the
propensities of single residues and residue pairs to be associated with a (ϕ, ψ, ω) backbone torsion angle
domain. Seven (ϕ, ψ, ω) domains are considered, defined in Rooman et al. (1). We use two variants of
the torsion potential, called torsionshort-range and torsionmiddle-range . Both are computed from
propensities of a (ϕ, ψ, ω) domain ti, at position i along the sequence, or pairs of domains (ti, tj), at
positions i and j, to be associated with an amino acid ak at position k. But we have k-1 ≤ i,j ≤ k+1 for the
torsionshort-range potential and k-8 ≤ i,j ≤ k+8 for the torsionmiddle-range potential. The folding free
energy ∆G(S,C) of a sequence S in the conformation C computed from these propensities is expressed as
(4, 5):
∆Gtorsion(S,C)=-kT∑N
1
ζk
i,j,k=1lnP(ak,ti,tj)
P(ti,tj)P(ak)
where P are normalized frequencies, N is the number of residues in the sequence S, k is the Boltzmann
constant and T is a conformational temperature taken to be room temperature (6). The normalization
factor ζk ensures that the contribution to ∆G(S,C) of each residue in the window [k-1, k+1] for the
torsionshort-range potential or [k-8, k+8] for the torsionmiddle-range potential is counted once. It is
equal to the window width, except near the chain ends .28The Cµ-Cµ potentials are distance potentials dominated by non-local, hydrophobic interactions.
They are based on propensities of pairs of amino acids (ai,aj) at position i and j along the sequence to be
separated by a spatial distance dij, calculated between the pseudo atoms Cµ. We consider two variants of
Cµ-Cµ potentials. The first one, called Cµ-Cµlong_range potential, describes purely non-local
interactions along the sequence, and only takes into account residues separated by at least 15 residues
along the sequence, i.e. j≥i+16. The second one, simply called Cµ-Cµ potential, though dominated by
non-local interactions, possesses a local interaction component. The non-local component is obtained by
considering together the frequencies of all residues separated by seven sequence positions and more,
thus with j≥i+8. The local component is obtained by computing separately the frequencies of residues
separated by one to six positions along the sequence, for i+1<j<i+8. Consecutive residues along the
sequence are not considered. The folding free energies are expressed as:
∆GCµ−Cµ(S,C)=-kT∑N
lnPj-i(ai,aj,dij)
Pj−i(ai,aj)Pj-i(dij)
i<j
with j≥i+16 and the normalized frequencies Pj-i independent of j-i for the Cµ-Cµlong_range potential,
and i+1<j and the normalized frequencies Pj-i independent of j-i for j≥i+8 for the Cµ-Cµ potential. The
discretisation of the spatial distances dij is performed by dividing the distances between 3 and 8 Å into
25 bins of 0.2 Å width and merging the distances greater than 8 Å. To increase the reliability of the
statistics, these bins are smoothed by combining the counts in each bin with those of the 10 flanking bins
at each side. The predominance of the central bin is preserved by weighting the counts from each
flanking bin by a factor 1/n, where n is the position relative to the central bin; n is equal to 1 for the two
closest bins and to 10 for the two most distant bins.
The so-defined folding free energies are reliable for common amino acids and structure motifs, but
not for less common ones. To correct for the sparse data, we substitute the sequence-specific frequencies 29P(c,s), where s denotes a sequence pattern and c a structure motif, which appear in the two above
equations defining the torsion and Cµ-Cµ folding free energies, by a linear combination of these
frequencies and the product of the separate frequencies of s and c, denoted P(s) and P(c) respectively
(7).
P(c,s)→1
σ+ms[σP(c)P(s)+msP(c,s)]
where ms is the number of occurrences of the sequence pattern s in the dataset, and σ a parameter. This
expression ensures that the sequence-specific contribution dominates for seldom sequence patterns and
tends to zero for frequent ones. This behavior is modulated by the parameter σ, which we consider here
equal to 50.
Evaluation of folding free energy changes
To estimate the stability changes caused by a single-site mutation, we compute the folding free
energy changes as:
∆∆G(Sm,Cm;Sw,Cw)=∆G(Sm,Cm)−∆G(Sw,Cw)
where Cm and Cw are the mutant and wild-type conformations and Sm and Sw the mutant and wild-type
sequences, respectively. With this convention, ∆∆G is positive when the mutation is destabilizing, and
negative when it is stabilizing. The conformations Cm and Cw of the mutant and wild-type protein are
assumed to be nearly identical. More precisely, the backbone conformations are taken as identical and
only the position of the Cµ pseudo-atom, which is amino acid dependent, is different in the mutant and
wild-type structures.
The folding free energies of the wild-type and mutant proteins are computed with linear
combinations of the torsion and Cµ-Cµ potentials described in the previous section. Previous analyses
(8-10) have shown that the combination that gives the best evaluation of the ∆∆G’s depends on the 30solvent accessibility A of the mutated residue; A is defined as the solvent accessible surface in the
protein structure, computed by SurVol (11), times 100 and divided by its solvent accessible surface in an
extended tripeptide Gly-X-Gly (12). These analyses have revealed that the mutations can be divided in
three subsets. When the mutated residue is at the surface, with a solvent accessibility A equal to or larger
than 50%, the optimal folding free energy changes has been shown to be equal to:
∆∆GA≥50%=1.14×∆∆Gtorsionshort_range+0.27
When the mutated residue is half buried, half exposed to the solvent, with a solvent accessibility
comprised between 20 and 40%, the optimal folding free energy is:
∆∆G20<A≤40%=1.39×∆∆Gtorsionshort_range+0.97×∆∆GCµCµ+0.21
Finally, when the mutated residue is totally buried in the protein core, with a solvent accessibility less
than or equal to 20%, the optimal folding free energy is:
∆∆GA≤20%=1.44×∆∆Gtorsionmiddle_range+1.70×∆∆GCµCµ
long−range+1.44
When the mutated residue has a solvent accessibility comprised between 40 and 50%, we do not
evaluate its folding free energy. We have indeed observed that in this case, the solvent accessibility of
the mutated residue is not a good measure to guide the choice of the optimal potential.
References of the supplementary material
1. Rooman, M.J., Kocher, J.-P.A. & Wodak, S.J. (1991) J. Mol. Biol. 221, 961-979.
2. Rooman, M.J., Kocher, J.-P.A. & Wodak, S.J. (1992) Biochemistry 31, 10226-10238.
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10. Gilis, D. & Rooman, M. (1999) Theor. Chem. Acc. 101, 46-50.
11. Alard, P. (1991) PhD thesis, Université Libre de Bruxelles
12. Rose, G.D., Geselowitz, A.R., Lesser, G.J., Lee, R.H. & Zehfus, M.H. (1985) Science 229, 834-838. |
arXiv:physics/0102045v1 [physics.atom-ph] 14 Feb 2001Some exact analytical results and a semi-empirical formula for
single electron ionization induced by ultrarelativistic h eavy ions
A. J. Baltz
Physics Department, Brookhaven National Laboratory, Upto n, New York 11973
(June 8, 2011)
Abstract
The delta function gauge of the electromagnetic potential a llows semiclassi-
cal formulas to be obtained for the probability of exciting a single electron
out of the ground state in an ultrarelativistic heavy ion rea ction. Exact for-
mulas have been obtained in the limits of zero impact paramet er and large,
perturbative, impact parameter. The perturbative impact p arameter result
can be exploited to obtain a semi-empirical cross section fo rmula of the form
σ=Alnγ+Bfor single electron ionization. AandBcan be evaluated for any
combination of target and projectile, and the resulting sim ple formula is good
at all ultrarelativistic energies. The analytical form of AandBelucidates
a result previously found in numerical calculations: scale d ionization cross
sections decrease with increasing charge of the nucleus bei ng ionized. The
cross section values obtained from the present formula are i n good agreement
with recent CERN SPS data from a Pb beam on various nuclear tar gets.
PACS: 34.90.+q, 25.75.-q
I. INTRODUCTION
In a recent work [1] ionization cross sections were calculal ated for a number of represen-
tative cases of collisions involving ultrarelativistic Pb , Zr, Ca, Ne and H ions. The method
1of calculation (on a computer) involved an exact semiclassi cal solution of the Dirac equation
in the ultrarelativistic limit [2]. A single electron was ta ken to be bound to one nucleus with
the other nucleus completely stripped. The probability tha t the electron would be ionized
in the collision was calculated as a function of impact param eter, and cross sections were
then constructed by the usual integration of the probabilit ies over the impact parameter.
The results of the probability calculations were used to con struct cross sections for various
ion-ion collision combinations in the form
σ=Alnγ+B (1)
where AandBare constants for a given ion-ion pair and γ(= 1/√
1−v2) is the relativistic
factor one of the ions seen from the rest frame of the other.
In Section II of this paper analytic results are derived for t he probability that a single
ground state electron will be excited in an ultrarelativist ic heavy ion reaction. Exact semi-
classical formulas are presented for the limits of zero impa ct parameter and perturbational
impact parameters. In Section III the perturbational impac t parameter analytical form is
used as a basis to construct semi-empirical formulas for AandB. These formulas reproduce
the previous numerical results for single particle ionizat ion, and they illuminate the system-
atic behavior of AandBwith changing target and projectile ion species. Ionizatio n cross
sections calculated with Eq.(1) are then compared with data .
II. IMPACT PARAMETER DEPENDENT PROBABILITIES
If one works in the appropriate gauge [3], then the Coulomb po tential produced by an
ultrarelativistic particle (such as a heavy ion) in uniform motion can be expressed in the
following form [4]
V(ρ, z, t) =−αZ1(1−αz)δ(z−t) ln(b−ρ)2
b2. (2)
bis the impact parameter, perpendicular to the z–axis along which the ion travels, ρ,z, and
tare the coordinates of the potential relative to a fixed targe t (or ion), αzis the Dirac matrix,
2αthe fine structure constant, and Z1andvthe charge and velocity of the moving ion. This
is the physically relevant ultrarelativistic potential si nce it was obtained by ignoring terms
in (b−ρ)/γ2[4] [3]. Its multipole expansion is
V(ρ, z, t) =αZ1(1−αz)δ(z−t)
/braceleftbigg
−lnρ2
b2ρ > b
+/summationdisplay
m>02 cosmφ
m
×/bracketleftbigg/parenleftbiggρ
b/parenrightbiggm
ρ < b
+/parenleftbiggb
ρ/parenrightbiggm/bracketrightbigg/bracerightbigg
. ρ > b (3)
Forb >> ρ
V(ρ, z, t) =δ(z−t)αZ1(1−αz)2ρ
bcosφ. (4)
As will be shown in Section III, when bbecomes large enough that expression Eq.(4) is inac-
curate for use in calculating a probability, we match onto a W eizsacker-Williams expression
which is valid for large b. Note that the b2in the denominator of the logarithm in Eq.(2) is
removable by a gauge transformation, and we retain the optio n of keeping or removing it as
convenient.
It was shown in Ref. [2] that the δfunction allows the Dirac equation to be solved exactly
at the point of interaction, z=t. Exact amplitudes then take the form
aj
f(t=∞) =δfj+/integraldisplay∞
−∞dtei(Ef−Ej)t/angbracketleftφf|δ(z−t)(1−αz)
×(e−iαZ1ln(b−ρ)2−1)|φj/angbracketright (5)
where jis the initial state and fthe final state. This amplitude is in the same form as the
perturbation theory amplitude, but with an effective potent ial to represent all the higher
order effects exactly,
V(ρ, z, t) =−iδ(z−t)(1−αz)(e−iαZ1ln(b−ρ)2−1), (6)
3in place of the potential of Eq.(2).
Since an exact solution must be unitary, the ionization prob ability (the sum of probabil-
ities of excitation from the single bound electron to partic ular continuum states) is equal to
the deficit of the final bound state electron population
/summationdisplay
ionP(b) = 1−/summationdisplay
boundP(b) (7)
The sum of bound state probabilities includes the probabili ty that the electron remains in
the ground state plus the sum of probabilities that it ends up in an excited bound state.
From Eq.(5) one may obtain in simple form the exact survival p robability of an initial state
Pj(b) =|/angbracketleftφj|(1−αz)e−iαZ1ln (b−ρ)2|φj/angbracketright|2. (8)
By symmetry the αzterm falls out and we are left with
Pj(b) =|/angbracketleftφj|e−iαZ1ln (b−ρ)2|φj/angbracketright|2. (9)
The ground state wave function φjis the usual K shell Dirac spinor [5]
φj=
g(r)χµ
κ
if(r)χµ
−κ
(10)
with upper and lower components wave functions gandf
g(r) =N/radicalig
1 +γ2rγ2−1e−αZ2r
f(r) =−N/radicalig
1−γ2rγ2−1e−αZ2r(11)
where Z2, is the charge of the nucleus that the electron is bound to, γ2=/radicalig
1−α2Z2
2, and
N2=(2αZ2)2γ2+1
2Γ(γ2+ 1). (12)
Let us first consider b= 0. We have
Pj(b= 0) = |/angbracketleftφj|e−2iαZ1lnρ|φj/angbracketright|2=|/angbracketleftφj|e−2iαZ1(lnr+ln(sin θ))|φj/angbracketright|2. (13)
Putting in the explicit form of the upper and lower component s for the K shell lowest bound
state Dirac wave function and carrying out the integration w e have
4Pj(b= 0) =π
4/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleΓ(2γ2+ 1−2iαZ1)Γ(1−iαZ1)
Γ(2γ2+ 1)Γ(3
2−iαZ1)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
, (14)
or
Pj(b= 0) =παZ 1ctnh(παZ 1)
(1 + 4 α2Z2
1)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleΓ(2γ2+ 1−2iαZ1)
Γ(2γ2+ 1)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
. (15)
It is interesting to compare this result with a previous calc ulation of the probability of
ionization in “close collisions” by Bertulani and Baur [6]. For a one electron atom they find
Pion(b < λ c/αZ 2) = 1.8α2Z2
1, (16)
where λc= ¯h/m ecis the electron Compton wavelength. If we take the low Z1limit of our
expression Eq.(15) and then subtract it from one we obtain
Pion(b= 0) = (π2
3−1)α2Z2
1= 2.29α2Z2
1 (17)
However our expression Eq.(15) only gives the flux lost from t he initial state; some of
that flux goes into excited bound states and is not ionized. Fr om our previous numerical
calculations we find that the actual ionization probabiliti es obtained either by summing up
final continuum states or else by subtracting all the final bou nd states from unity were 76%
– 80% respectively of the flux lost from the initial state. Thu s if we multiply the constant
in Eq.(17) by such a percentage we are in remarkable agreemen t with Bertulani and Baur
for the perturbative limit.
Now let us consider the case of b >> ρ . From Eq.(4) and Eq.(9) we have
Pj(b) =|/angbracketleftφj|e−2iαZ1cos(φ)(ρ/b)|φj/angbracketright|2. (18)
Expanding the exponential up to ρ2/b2we have
Pj(b) =|/angbracketleftφj|1−2iαZ1cos(φ)ρ
b−2α2Z2
1cos2(φ)ρ2
b2|φj/angbracketright|2(19)
The term in cos( φ) vanishes by symmetry, and integrating, we obtain
Pj(b) = 1−2Z2
1
Z2
2(1 + 3 γ2+ 2γ2
2)
3λ2
c
b2(20)
5by ignoring the term in 1 /b4.
Both limits, Eq.(15) for b= 0 and Eq.(20) for b >> ρ , are relativistically correct and
thus correct for all Z1andZ2since exact Dirac wave functions were used.
III. A SEMI-EMPIRICAL FORMULA FOR SINGLE ELECTRON IONIZATI ON
It is well known that the cross section for ionization of any p air of projectile and target
species can be expressed as a sum of a constant term and a term g oing as the log of the
relativistic γof the beam as seen in the target rest frame [6] [7] [1]
σion=Alnγ+B. (21)
The cross section of this form is constructed from an impact p arameter integral
σion= 2π/integraldisplay
P(b)ionb db (22)
where P(b) is the probability of ionization at a given impact paramete r. If all the flux lost
from the initial state went into the continuum then Eq.(20) w ould provide the ionization
probability at moderately large b
Pion(b) = 2Z2
1
Z2
2(1 + 3 γ2+ 2γ2
2)
3λ2
c
b2. (23)
We will take this form as a physical basis to build a semi-empi rical formula for ionization.
In any case we need to integrate the probability up to a natura l energy cutoff. In order to
do this we match the delta function solution Eq.(23) at some m oderately large bonto the
known Weizsacker-Williams probability for larger bby noting that if bω << γ then
K2
1(ωb
γ) =γ2
ω2b2, (24)
and we can rewrite Eq.(24) in the Weizsacker-Williams form f or large b
Pion(b) = 2Z2
1
Z2
2(1 + 3 γ2+ 2γ2
2)
3λ2
cω2
γ2K2
1(ωb
γ). (25)
To perform the large bcutoff recall that to high degree of accuracy
6ω2
γ2/integraldisplay∞
b0K2
1(ωb
γ)b db= ln(0.681γ
ωb0) = ln γ+ ln(0.681
ωb0). (26)
We immediately obtain the following expression for A
A=4πλ2
c
3Z2
1
Z2
2(1 + 3 γ2+ 2γ2
2), (27)
where λ2
c, the square of the electron Compton wave length, is 1491 barn s. However, as it turns
out, uniformly for all species of heavy ion reactions, at per turbational impact parameters a
little over 70% of the flux lost from the initial state goes int o excited bound states and does
not contribute to ionization. But since the ratio of flux goin g into continuum states to the
total flux lost is so uniform we can use a fit to previously publi shed numerical results [1] to
obtain a semi-analytical form for A:
A= (0.2869)4πλ2
c
3Z2
1
Z2
2(1 + 3 γ2+ 2γ2
2), (28)
or in barns
A= 1792Z2
1
Z2
2(1 + 3 γ2+ 2γ2
2). (29)
Now one can use the second term in Eq.(26) to obtain a provisio nal expression for B
B=Aln(0.681
ωb0). (30)
Obviously we need to evaluate ωand to discuss b0.ωcan be taken as the minimum ionization
energy, 1 −γ2, times a constant a little larger than one. One next observes that if Pion(b)
varies as 1 /b2the impact parameter integral has to be cut off on the low side a t some value b0
to avoid divergence. In fact the 1 /b2dependence continues down to the surface of the atom
where other terms evident in Eq.(3) begin to contribute. The atomic size is just the electron
Compton wave length divided by αZ2. In this region Pion(b) first rises faster than 1 /b2and
then levels off to approach a constant change with batb= 0 [1]. One could try to add a
low impact paramenter contribution to Abased on Eq.(15) to our provisional form Eq.(30),
but that turns out to unduly complicate things without impro ving the phenomenology. Our
approach will be to set b0to an empirical constant divided by αZ2
7Eq.(30) now takes the form
B=Aln(CαZ 2
1−γ2). (31)
Putting in two analytical fine tuning factors and fitting the r emaining constant to the nu-
merical results of Ref. [1] we obtain a semi-analytical form forB:
B=Aγ1/10
1(1−α2Z1Z2)1/4ln(2.37αZ2
1−γ2). (32)
Table I expands a corresponding table from Ref. [1] by adding cross sections of symmetric
ion-ion pairs calculated with the formulas for AandB. There is good agreement between the
formula values for the cross sections (first rows) and the num erical cross sections calculated
by subtracting the bound state probabilities from unity (se cond rows) or calculated by
summing continuum electron final states (third rows). For bo thAandBthe agreement is
also good with the Anholt and Becker calculations [7] in the l iterature for the lighter ion
species. However with increasing mass of the ions the pertur bative energy dependent term
Adecreases in the formula calculations and in our previous nu merical calculations, whereas
it increases in the Anholt and Becker calculations. The grea test discrepancy is for Pb + Pb,
with Anholt and Becker being about 60% higher. The reason tha t the Ashould decrease
with increasing mass (actually Z) of the ions is explained by the
(1 + 3 γ2+ 2γ2
2) = 3−2α2Z2
2+ 3/radicalig
1−α2Z2
2 (33)
factor in the formula for A(and thereby Balso). As we noted before, perhaps the discrepancy
between our Adecreasing with Zand the Anholt and Becker Aincreasing with Zis due
to the fact that Anholt and Becker use approximate relativis tic bound state wave functions
and the present calculations utilize exact Dirac wave funct ions for the bound states. For
the term B(which has the non-perturbative component) the agreement i s relatively good
between all the calculations.
In the perturbative limit (small Z1, Z2) the cross section formula goes over to
σ= (0.2869)8 πλ2
cZ2
1
Z2
2ln2.37γ
αZ2= 7.21λ2
cZ2
1
Z2
2ln2.37γ
αZ2. (34)
8By way of comparison, Bertulani and Baur [6] using the equiva lent photon method and
taking the contribution of b≥λc/αZ 2found
σ= 4.9λ2
cZ2
1
Z2
2ln2γ
αZ2(35)
for this case of ionization of a single electron.
Table II shows results of the calculation of B(multiplied by Z2
2/Z2
1) for a number of
representative non-symmetric ion-ion pairs. (Since Ais perturbative, scaling as Z2
1, its
value can be taken from Table I for the various pairs here.) On ce again there is good
agreement between the formula values for the cross sections (first rows) and the numerical
cross sections calculated by subtracting the bound state pr obabilities from unity (second
rows) or calculated by summing continuum electron final stat es (third rows). The only
notable disagreement is with Anholt and Becker for Pb target s.
The availabilty of the present semi-empirical formula faci litates a comparison with avail-
able CERN SPS data. Calculations with the formula are in cons iderably better agreement
with the data of Krause et al. [8] for a Pb beam on various targe ts than are the Anholt and
Becker numbers with or without screening. Note that in this c ase the role of target and
beam are reversed. It is the single electron Pb ion in the beam that is ionized by the various
nuclei in the fixed targets. The formula numbers do not includ e screening, which should in
principle be included for a fixed target case. However, one mi ght infer from the Anholt and
Becker calculations that the effect of screening is smaller t han the error induced by using
an approximate rather than proper relativistic wave functi on for the electron bound in Pb.
Note that the formula has not been fit to experimental data. It is compared with ex-
perimental data. The “empirical” aspect of this formula ref ers to adjusting the formula to
previous numerical calculations of Ref. [1]
At RHIC the relativistic γof one ion seen in the rest frame of the other is 23,000, and
of course there is no screening, so the present formula shoul d be completely applicable. The
present formula predicts a single electron ionization cros s section of 101 kilobarns for Au +
Au at RHIC. The corresponding cross section from Anholt and B ecker is 150 kilobarns.
9IV. ACKNOWLEDGMENTS
After this work was completed, a paper by Voitkiv, M¨ uller an d Gr¨ un [9], which includes
screening in ionization of relativistic projectiles, was b rought to my attention. I would like
to thank Carlos Bertulani for pointing out this paper to me an d for reading the present
manuscript. This manuscript has been authored under Contra ct No. DE-AC02-98CH10886
with the U. S. Department of Energy.
10REFERENCES
[1] A. J. Baltz, Phys. Rev. A 61, 042701 (2000).
[2] A. J. Baltz, Phys. Rev. Lett. 78, 1231 (1997).
[3] A. J. Baltz, M. J. Rhoades-Brown, and J. Weneser, Phys. Re v.A 44, 5568 (1991).
[4] A. J. Baltz, Phys. Rev. A 52, 4970 (1995).
[5] M. E. Rose, Relativistic Electron Theory (Wiley, New York, 1961).
[6] Carlos A. Bertulani and Gerhard Baur, Physics Reports 163, 301 (1988).
[7] R. Anholt and U. Becker, Phys. Rev. A 36, 4628 (1987).
[8] H. F. Krause, C. R. Vane, S. Datz, P. Grafstr¨ om, H. Knudse n, C. Scheidenberger, and
R. H. Schuch, Phys. Rev. Lett. 80, 1190 (1998).
[9] A. B. Voitkiv, C. M¨ uller, and N. Gr¨ un, Phys. Rev. A 62, 062701 (2000).
11TABLES
TABLE I. Calculated Ionization Cross Sections Expressed in the Form Alnγ+B(in barns)
Pb + Pb Zr + Zr Ca + Ca Ne + Ne H + H
A Formula 8400 10,212 10,618 10,718 10,752
1−/summationtext
bnde−8680 10,240 10,620 10,730 10,770
/summationtext
conte−8450 9970 10,340 10,440 10,480
Anholt & Becker [7] 13,800 11,600 10,800 10,600 10,540
B Formula 14,133 27,375 36,623 44,638 69,629
1−/summationtext
bnde−14,190 28,450 38,010 46,080 71,090
/summationtext
conte−12,920 27,110 36,530 44,430 68,780
Anholt & Becker 13,000 27,800 37,400 45,400 70,000
12TABLE II. Calculated values of the scaled quantity ( Z2
2/Z2
1)Bfor non-symmetric combinations
of colliding particles. The second nucleus ( Z2) is taken to be the one with the single electron to
be ionized. Since Anholt and Becker cross sections without s creening are completely perturba-
tive, their values of of B also can be taken from Table IV, and a re repeated here for convenient
comparison.
H + Ne H + Ca Ca + H H + Zr H + Pb Pb + H
Formula 44,716 36,890 69,462 28,226 16,487 66,539
1−/summationtext
bnde−46,150 38,270 70,820 29,440 17,090 67,550
/summationtext
conte−44,490 36,790 68,520 28,070 15,680 65,330
Anholt & Becker [7] 45,400 37,400 70,000 27,800 13,000 70,000
Pb +Ne Ne + Pb Pb + Ca Ca + Pb Pb + Zr Zr + Pb
Formula 42,308 16,313 34,503 16,097 25,751 15,592
1−/summationtext
bnde−42,560 17,030 34,720 16,870 26,010 16,250
/summationtext
conte−41,000 15,690 33,330 15,530 24,730 14,930
Anholt & Becker 45,400 13,000 37,400 13,000 27,800 13,000
13TABLE III. Cross sections for the ionization of a 160 GeV/A on e electron Pb projectile ( Z2)
by various fixed nuclear targets ( Z1). Unlike in Table II, here the appropriate ( Z2
1/Z2
2) factor has
been included. Cross sections are given in kilobarns to matc h the format of the CERN SPS data.
Target Z1 Formula SPS Data Anholt & Becker Anholt & Becker
(with screening) (no screening)
Be 4 0.14 0.14 0.24 0.20
C 6 0.32 0.31 0.49 0.45
Al 13 1.5 1.3 2.0 2.1
Cu 29 7.4 6.9 9.0 10.5
Sn 50 22 15 25 31
Au 79 53 42 60 78
14 |
Submitted to Phys. Rev. A 23 January 2001
Ablation of solids by femtosecond lasers:
ablation mechanism and ablation thresholds for metals and dielectrics
E. G. Gamaly*1 A. V. Rode1, V. T. Tikhonchuk2, and B. Luther-Davies1
1Research School of Physical Science and Engineering,
Australian National University, Canberra, ACT 0200 Australia
2P.N. Lebedev Physical Institute, Moscow, Russia
ABSTRACT
The mechanism of ablation of solids by intense femtosecond laser pulses is described in
an explicit analytical form. It is shown that at high intensities when the ionization of the targetmaterial is complete before the end of the pulse, the ablation mechanism is the same for bothmetals and dielectrics. The physics of this new ablation regime involves ion acceleration in theelectrostatic field caused by charge separation created by energetic electrons escaping from thetarget. The formulae for ablation thresholds and ablation rates for metals and dielectrics,combining the laser and target parameters, are derived and compared to experimental data. Thecalculated dependence of the ablation thresholds on the pulse duration is in agreement with theexperimental data in a femtosecond range, and it is linked to the dependence for nanosecond
pulses.
PACS: 79.20.Ds; 32.80.Rm; 52.38.Mf
* e-mail: gam110@rsphysse.anu.edu.au ; ph.: ++61-2-6125-0171; fax: ++61-2-6125-07322
I. INTRODUCTION:
THE ULTRA SHORT PULSE LASER-MATTER INTERACTION MODE
The rapid development of femtosecond lasers over the last decade has opened up a wide
range of new applications in industry, material science, and medicine. One important physicaleffect is material removal or laser ablation by femtosecond pulses which can be used for thedeposition of thin films; the creation of new materials; for micro-machining; and, in the arts, forpicture restoration and cleaning. Femtosecond laser ablation has the important advantage in suchapplications compared with ablation using nanosecond pulses because there is little or nocollateral damage due to shock waves and heat conduction produced in the material beingprocessed. In order to choose the optimal laser and target parameter it is useful to have simplescaling relations, which predict the ablation condition for an arbitrary material. In this paper wepresent an analytical description of the ablation mechanism and derive appropriate analyticalformulae.
In order to remove an atom from a solid by the means of a laser pulse one should deliver
energy in excess of the binding energy of that atom. Thus, to ablate the same amount of materialwith a short pulse one should apply a larger laser intensity approximately in inverse proportion tothe pulse duration. For example, laser ablation with 100 fs pulses requires an intensity in a range~ 10
13 – 1014 W/cm2 [1], while 30-100 ns pulse ablates the same material at the intensities ~ 108 –
109 W/cm2 [2]. At intensities above 1013 – 1014 W/cm2 ionization of practically any target
material takes place early in the laser pulse time. For example, if an intense, 1013 – 1014 W/cm2,
femtosecond pulse interacts with a dielectric, almost full single ionization of the target occurs atthe beginning of the laser pulse. Following ionization, the laser energy is absorbed by freeelectrons due to inverse Brehmstrahlung and resonance absorption mechanisms and does notdepend on the initial state of the target. Consequently, the interaction with both metals anddielectrics proceeds in a similar way which contrasts to the situation when a long pulse is where3
ablation of metals occurs at relatively low intensity compared with that for a transparent
dielectric whose absorption is negligibly small.
Another distinctive feature of the ultra short interaction mode is that the energy transfer
time from the electrons to ions by Coulomb collisions is significantly longer (picoseconds) thatthe laser pulse duration (t
p ~ 100 fs). Therefore, the conventional hydrodynamics motion does
not occur during the femtosecond interaction time.
There are two forces which are responsible for momentum transfer from the laser field
and the energetic electrons to the ions in the absorption zone: one is due to the electric field ofcharge separation and another is the ponderomotive force. The charge separation occurs if theenergy absorbed by the electrons exceeds the Fermi energy, which is approximately a sum of thebinding energy and work function, so the electrons can escape from the target. The electric fieldof charge separation pulls the ions out of the target. At the same time, the ponderomotive forceof the laser field in the skin layer pushes electrons deeper into the target. Correspondingly itcreates a mechanism for ion acceleration into the target. Below we demonstrate that the formermechanism dominates the ablation process for sub-picosecond laser pulses at an intensity of 10
13
– 1014 W/cm2. This mechanism of material ablation by femtosecond laser pulses is quite
different from the thermal ablation by long pulses.
Femtosecond ablation is also sensitive to the temporal and spatial dependence of the
intensity of the laser pulse. The Chirped Pulse Amplification (CPA) technique commonly usedfor short pulse generation [3] can produce a main (short) pulse accompanied by a nanosecondpre-pulse or pedestal that can be intense enough itself to ablate the target. Therefore, animportant condition for the practical realization of the pure femtosecond interaction mode shouldbe that the intensity in any pre-pulse has to be lower than the thresholds for ablation or ionizationin the nanosecond regime. There are fortunately several methods for achieving high pulsecontrast (nonlinear absorbers, conversion to second harmonic, etc.) [4,5].4
A rather simple and straightforward analytical model can describe the ultra short pulse
mode of laser-matter interaction. The main features of this model were developed more than 10years ago in connection with the ultra short and super intense laser-matter interaction [5]. In whatfollows this model is modified and applied to the problems of the laser ablation at relativelymoderate intensities near the ablation threshold for solids. The absorption coefficient, ionizationand ablation rates, and ablation thresholds for both metals and dielectrics are expressed in termsof the laser and target parameters by explicit formulae. The comparison to the long-pulseinteraction mode and to the experimental data is presented and discussed.
2. LASER FIELD PENETRATION INTO A TARGET: SKIN-EFFECT
The femtosecond laser pulse interacts with a solid target with a density remaining
constant during the laser pulse (the density profile remains step-like). The laser electromagneticfield in the target (metal or plasma) can be found as a solution to Maxwell equations coupled tothe material equations. The cases considered below fall in framework of the normal skin-effect[5,6] where the laser electric field E(x) decays exponentially with the depth into the target:
Ex Ex
ls( ) ( )exp=−
0 ; (1)
here ls is the field penetration (or absorption) length (skin-depth); the target surface corresponds
to x = 0, and Eq.(1) is valid for x > 0. The absorption length in general is expressed as follows
[6]:
lc
ks=ω(2)
where k is the imaginary part of the refractive index, N = ε1/2 = n + ik (εε ε=′+′′i is the
dielectric function and ω is the laser frequency). We take the dielectric function in the Drude
form for the further calculations:
εω
ωω ν=−+12
pe
effi ()(3)5
where 21 2)/ 4(e e pe mneπω= is the electron plasma frequency, and νeff is an effective collision
frequency of electrons with a lattice (ions). In the case of a high collision rate νωeff>> and thus
′′>>′εε one can reduce Eq.(2) to the conventional skin depth expression for the high-conducting
metals [6]:
lc
kc
s
peeff=≈
ωων
ω212/
. (4)
The main difference between these formulae and those ones for the conventional low-
intensity case resides in the fact that the real and imaginary parts of the dielectric permittivity,and thus, the plasma frequency and the effective collision frequency, all are intensity and time-dependent. The finding of these dependencies is the subject of next sections.
3. ABSORPTION MECHANISMS AND ABSORPTION COEFFICIENT
The light absorption mechanisms in solids are the following [7]:
1. intraband absorption, and contribution of free charge carriers in metals and
semiconductors;
2. interband transitions and molecular excitations;
3. absorption by collective excitations (excitons, phonons);
4. absorption due to the impurities and defects.
At high intensities ~ 10
13 – 1014 W/cm2, the electron oscillation energy in the laser electric field
is a few electron-volts, which is of the order of magnitude of the ionization potential.Futhermore, at intensities above 10
14 W/cm2 the ionization time for a dielectric is just a few
femtoseconds, typically much shorter than the pulse duration (~100 fs). The electrons producedby ionization then in dielectrics dominate the absorption in the same way as the free carriers inmetals, and the characteristics of the laser-matter interaction become independent of the initialstate of the target. As a result the first mechanism becomes of a major importance for both metalsand dielectrics. In the presence of free electrons, inverse Bremsstrahlung and resonant6
absorption (for p-polarized light at oblique incidence) become the dominant absorption
mechanisms.
However, one should not oversimplify the picture. The electron interaction with the
lattice (the electron-phonon interaction) and the change in the electron effective mass might besignificant in dielectrics and even in some metals [11]. The number density of conductivityelectrons in metal changes during the pulse. We should also note that in many cases the real partof the dielectric function is comparable to the imaginary part. Then the skin-effect solution (forexample, the simple formula Eq.4) should be replaced by a more rigorous approach. We usebelow the Fresnel formulae [8] with the Drude-like dielectric function for the absorptioncoefficient calculations taking into account that the density of the target during the pulse remainsunchanged. Then the conventional formulae for the reflection R and absorption A coefficients
are the following [8]:
Rnk
nkAR =−()+
+()+=−1
1122
22; . (5)
In the limit of low absorption A << 1, which holds for the high conductivity perfect metals, one
finds a simple expression the absorption coefficient (cf. Appendix A [3]):
Al
cs≈2ω. (6)
It should be noted that the skin-depth is the function of the laser intensity and time. This function
we shall find below.
4. INTENSITY THRESHOLD FOR IONISATION OF DIELECTRICS
The dielectric function in dielectrics at low intensity of the electric field is characterized
by the large real and small imaginary parts. The imaginary part increases mainly due toionization. There are two major mechanisms of ionization in the laser field: ionization byelectron impact (avalanche ionization); and the multiphoton ionization. The time dependence of7
the number density of free electrons ne stripped off the atoms by these processes is defined by the
rate equation [1,9]:
dn
dtnw nwe
e imp a mpi=+ (7)
here na is the density of neutral atoms, wimp is the time independent probability (in s-1) for the
ionization by electron impact, and wmpi is the probability for the multiphoton ionization [5,9].
For the case of single ionization it is convenient to present the probabilities wimp and wmpi, in the
form:
wJ2
imposc
i2
eff
2
eff2≈
+
εων
ων; (8)
w2Jmpiosc
i≈
ωεnphnph
32/; (9)
here, εosc is the electron quiver energy in the laser field, nJph i=hω/ is the number of photons
necessary for atom ionization by the multiphoton process, Ji, is the ionization potential, and νeff is
the effective collision frequency. It must be emphasised that the effective collision frequency in
Eq. (8) accounts for the inelastic collisions leading to the energy gain by the electrons. Ingeneral, it differs from the effective collision frequency in Eq. (3) and in the Sections below,which accounts for the momentum exchange due to elastic collisions.
One can see from the Eqs. (8) and (9) that the relative role of the impact and multiphoton
ionization depends dramatically on the relation between the electron quiver energy and the
ionization potential. If
εosc > Ji then wmpi > w imp, and the multiphoton ionization dominates for
any relationship between the frequency of the incident light and the efficient collision frequency.
By presenting the oscillation energy in a scaling form:
εαλ µoscIm eV
W/cm2 []=+()[][]() 93 1
102
142. (10)
where α accounts for the laser polarization ( α = 1 for the circular and α = 1 for the linear8
polarisation), it is evident that the multiphoton ionization dominates in the laser-interaction at
intensities above 1014 W/cm2 (for the 100-fs pulse duration this condition corresponds to the laser
fluence of 10 J/cm2.)
The general solution to Eq. (7) with the initial condition ne(t = 0) = n 0 is the following:
nI t nnw
wwt wtea mpi
impimp imp , , exp expλ()=+ −− ()[]
() 0 1 (11)
It is in a good agreement with the direct numerical solution to the full set of kinetic equations [1].
Electron impact ionization is the main ionization mechanism in the long (nanosecond) pulse
regime. Then εosc << Ji and ω << νeff, and one can neglect the second term in Eq. (11) and the
number of free electrons exponentially increases with the product of wimp and the pulse duration:
nn w te imp p~ exp0 ×{} . Therefore in the long-pulse regime the ionization threshold depends on
the laser fluence F = I×tp. In the case of high intensities multiphoton ionization dominates and
the number of free electrons increases linearly with time, nn w te a mpi~ . In this case the ionization
time could be shorter than the pulse duration and the ionization threshold depends on the laser
intensity and laser wavelength. It is conventional to suggest that the ionization threshold (orbreakdown threshold) be achieved when the electron number density reaches the critical densitycorresponding to the incident laser wavelength [7]. The ionization threshold for the majority of
materials lies at intensities in between (10
13 - 1014) W/cm2 (λ ~ 1 µm) with a strong nonlinear
dependence on intensity. For example, for a silica target at the intensity 2 ×1013 W/cm2 avalanche
ionisation dominates, and the first ionisation energy is not reached by the end of 100-fs pulse at
1064 nm. At 1014 W/cm2 multiphoton ionisation dominates and the full ionisation is completed
after 20 fs. It should be also mentioned that the ionisation threshold decreases with the increasein the photon energy.9
5. ELECTRON COLLISION FREQUENCY AND ENERGY TRANSFER FROM
ELECTRONS TO IONS
When the ionisation is completed, the plasma formed in the skin-layer of the target has a
free-electron density comparable to the ion density of about 1023 cm-3. In order to meet the
ablation conditions the average electron energy should increase up to the Fermi energy εF, i.e. up
to several eV. The electron-electron equilibration time is of the order of magnitude of the
reciprocal electron plasma frequency, i.e. ~ ωpe−1 ~ 10-2 fs that is much shorter than the pulse
duration. Therefore the electron energy distribution is close to equilibrium and follows the laser
intensity evolution in time adiabatically adjusting to any changes. The electron gas is non-idealin the high-density conditions: the energy of Coulomb interaction is comparable to the electronkinetic energy and there are only few electrons in the Debye sphere. There are no reliableanalytical expressions for the effective electron collision frequency in this energy-densitydomain. There are interpolation formulae for some materials in [10]. Physically sound estimatescould also be made.
The effective electron-ion collision frequency could be estimated by approaching the
Fermi energy from two limiting cases: from the low and from the high temperature limits. In thelow-temperature limit the electron-phonon collisions dominate. The electron-phonon collisionfrequency increases in direct proportion with the temperature for T above the Debye temperature
T
D. In the opposite high-temperature limit the effective frequency of electron-ion Coulomb
collisions decreases with the electron temperature. Thus, the effective collision frequency hasmaximum at the electron temperature approaching Fermi energy. The electron-phonon collisionfrequency (or, the probability for an electron to emit or to absorb a phonon) one can estimate atthe temperature T
D << T << TF as the following [7,10]:
νep hei
Dm
MJT
T−≈
12/
h(12)
Taking, for example Ji = 7.7 eV (first ionization potential for copper), TD(Cu) ~ 300 K, and MCu10
= 63.54 a.m.u. one obtains νeff ~ 9×1015 s-1. This estimate is very close to the effective frequency
at the room temperature 8.6 ×1015 s-1 extracted from the conductivity measurement [11].
At high temperatures TeF>>ε the effective electron-ion collision frequency could be
estimated by using the model for an ideal plasma at solid state density. The collision is
considered as a 90-degree deflection of an electron path due to the Coulomb interaction with theion, and the collision frequency is the frequency of the momentum exchange. According to [10]:
νeie
eVnZ
T≈×−31 06
32ln/Λ . (13)
For example, from Eq.13 the electron-ion collision frequency in Copper at the electron
temperature coinciding with the Fermi energy ( ne = 0.845×1023 cm-3, ωp = 1.64×1016 s-1, TeV ~ 7.7
eV, lnΛ ~ 2) is νei = 2.38×1016 s-1. This value is about twice higher than that estimated from the
low temperature case, and almost coincides with the plasma frequency, νeff ~ ωpe = 2.39×1016 s-1.
It seems reasonable to assume that νeff ≈ ωpe for the further estimates of the ablation threshold, as
it has been suggested in [1]. The value of νeff can be corrected by experimental measurements of
the skin depth (ablation depth). Some more advanced models and interpolations for the effective
collision frequency were derived in [10].
Thus, in the ablation conditions νei >> ω. Therefore the electron mean free path is much
smaller than the skin depth. That is, the condition for the normal skin effect is valid.
The electron-ion energy transfer time in a dense plasma can be expressed through the
collision frequency (13) as follows:
τνei
eeiM
m≈−1(14)
The estimation for copper yields the ion heating time τei = 4.6×10-12 s, which is in agreement
with the values suggested by many authors [1,5,13]. A similar estimate for silica gives 6.4 ×10-12
s. Therefore, for the sub-picosecond pulses ( tp ~ 100 fs) the ions remain cold during the laser11
pulse interaction with both metals and dielectrics.
6. ELECTRON HEATING IN THE SKIN LAYER
In the previous Section we have demonstrated that electrons have no time to transfer the
energy to the ions during the laser pulse τei > tp. That means that the target density remains
unchanged during the laser pulse. The electrons also cannot transport the energy out of the skin
layer because the heat conductivity time is much longer than the pulse duration. It is easy to seethat the electron heat conduction time t
heat (the time for the electron temperature smoothing
across the skin-layer ls) is also much longer than the pulse duration. Indeed, the estimates for this
time with the help of conventional thermal diffusion [6] give:
tll v
heatse e≈=2
3 κκ; ;
here κ is coefficient for thermal diffusion, le and ve are the electron mean free path and velocity
correspondingly. Using Copper as an example yields ls = 67.4 nm, κ ~1 cm2/s, and the electron
heat conduction time is in the order of tens of picoseconds.
The energy conservation law takes a simple form of the equation for the change in the
electron energy Te due to absorption in a skin layer [5]:
cTnT
tQ
xQA Ix
lee ee
s()∂
∂=−∂
∂=−
; exp02; (15)
here Q is the absorbed energy flux in the skin layer, A = I/I0 is the absorption coefficient, I0 =
cE2/4π is the incident laser intensity, ne and ce are the number density and the specific heat of the
conductivity electrons. In a simple model of the ideal Fermi gas the electron specific heat
increases with electron temperature from the low-temperature level ce = π2Te/2εF for Te << εF
[11] up to the maximum value of ce ~ 3/2 for the conventional ideal gas at high temperature Te >
εF. The specific heat could also be found as a tabulated function corrected on the experimental
measurements, which are usually evidencing the deviations from the simple model of the ideal12
Fermi gas [11]. The absorption coefficient and the skin depth are the known functions of the
incident laser frequency ω, the number density of the conductivity electrons ne, (or, plasma
frequency ωpe), the effective collision frequency including electron-ion and electron-phonon
collisions νeff, the angle of the incidence, and polarisation of the laser beam [5]. In fact, both
material parameters ωpe and νeff, are temperature dependent. Therefore, Eq. (15) describes the
skin effect interaction with the time-dependent target parameters. In order to obtain convenient
scaling relations for the ablation rate we use, as a first approximation, the conventional skineffect approach with time-independent characteristics and with the specific heat of the ideal gas.
Such an approximation is applicable because at the ablation threshold T
e ≈ εF. Thus, the time
integration of the Eq.(15) yields time and space dependencies of the electron energy in the skin
layer:
TAI t
lnx
lTe
se seF =−
≈4
320exp ; ε. (16)
This approach is well justified for metals because the temperature dependent skin-depth and
absorption coefficient enter into the above formula as a ratio A/ls, which is a weak function of
temperature. Indeed, in the low-absorption case (A<<1) for the highly conductive perfect metals,the absorption coefficient expresses by (6), and the ratio A/l
s is almost constant:
A
lcs≈2ω. (17)
In the high absorption case this ratio changes weakly being of the same order of magnitude with
the correction factor of ~ 1.3 (see Appendix A). The number density of the conductivityelectrons is also almost constant during the interaction time.
The relationship Eq.(16) represents an appropriate scaling law for the electron
temperature in the skin layer. The experimental data correlate well with the prediction of theEq.(16). For example, the estimate of the skin depth in a Copper target irradiated by a
Ti:sapphire laser (
λ = 780 nm, ω = 2.4×1015 s-1; νeff ≈ ωpe = 1.639×1016 s-1, ne = 0.845×102313
cm-3) gives with the help of Eq.(4) ls = 67.4 nm. The maximum electron temperature at the
surface of the Copper target under the fluence AI0 tp = 1 J/cm2 reaches Te = 7,5 eV, which is close
to the Fermi energy for Copper.
7. ABLATION MECHANISM:
IONS PULLED OUT OF THE TARGET BY ENERGETIC ELECTRONS
It has been shown in the preceding section that the free electrons in the skin layer can
gain the energy exceeding the threshold energy required to leave a solid target during the pulsetime. The energetic electrons escape the solid and create a strong electric field due to chargeseparation with the parent ions. The magnitude of this field depends directly on the electron
kinetic energy
εe ~ (Te – εεsc) (εesc is the work function) and on the gradient of the electron density
along the normal to the target surface (assuming one dimensional expansion) [6,14]:
Et
en
zaee=−()∂
∂ε ln. (18)
A ponderomotive force of the electric field in the target is another force applied to the
ions during the laser pulse [15]:
Fe
mcIpf
e=− ∇22
2π
ω.
However, for the solid density plasma and at the intensities of ~1014 W/cm2 it is significantly
smaller than the electrostatic force eE a.
The field Eq.(18) pulls the ions out of the solid target if the electron energy is larger than
the binding energy, εb, of ions in the lattice. The maximum energy of ions dragged from the
target reaches: εi(t) ≈ Zεe(t) ≈ (Te – εesc – εb). The time necessary to accelerate and ablate ions
could be estimated with the help of the equation for the change of ion momentum:
dp
dteEi
a≈ . (19)14
The characteristic scale length for the expanding electron cloud is the Debye length lD ~ v e/ωpe,
where ve = [(Te – εesc)/me]1/2 is the electron thermal velocity. Thus, the ion acceleration time, i.e.
the time required for the ion to acquire the energy of εe could be found with the help of (19) as
the following:
tl
vm
mT
TaccD
ip ei
ee esc
e esc b=≈
−
−−
21
21
2
ωε
εε. (20)
Below the ablation threshold, when Te ~ εesc + εb the acceleration time is much longer than the
pulse duration. However, when the laser fluence exceeds the ablation threshold this time is
comparable and even shorter than the pulse duration. For example, for Copper at F = 1 J/cm2
this time is less than 40 fs. This means that for high intensities (fluences) well over the ablation
threshold the equation (15) for the electron temperature should include the energy losses for ionheating. This effect of electrostatic acceleration of ions is well known from the studies of theplasma expansion [6,14] and ultrashort intense laser-matter interaction [5].
A. Ablation threshold for metals
According to Eq. (20), the minimum energy that electron needs to escape the solid equals
to the work function. In order to drag ion out of the target the electron must have an additionalenergy equal to or larger than the ion binding energy. Hence, the ablation threshold for metalscan be defined as the following condition: the electron energy must reach, in a surface layer d <<
l
s by the end of the laser pulse, the value equal to the sum of the atomic binding energy and the
work function. Using the Eq.(16) for the electron temperature we obtain the energy condition forthe ablation threshold:
εεεe b escp
seAI t
ln=+ =4
30. (21)
The threshold laser fluence for ablation of metals is then defined as the following:15
FI tln
Athm
p b escse≡= +() 03
4εε . (22)
We assume that the number density of the conductivity electrons is unchanged during the
laser-matter interaction process. After insertion (17) into (22) the approximate formula for theablation threshold takes the following form:
FI tcn n
thm
p b esce
b esce≡≈ +() ≡+() 03
83
82εεωεελ
π. (23)
The formula (23) predicts that the threshold fluence is proportional to the laser
wavelength: Fth ~ λ. We demonstrate below that this relation agrees well with the experimental
data.
B. Ablation threshold for dielectrics
The ablation mechanism for the ionized dielectrics is similar to that for metals. However,
there are several distinctive differences. First, an additional energy is needed to create the freecarriers, i.e. to transfer the electron from the valence band to the conductivity band. Therefore,the energy equal to the ionization potential J
I, should be delivered to the valence electrons.
Second, the number density of free electrons depends on the laser intensity and time during theinteraction process as has been shown in Section IV. However, if the intensity during the pulseexceeds the ionization threshold then the first ionization is completed before the end of the pulse,and the number density of free electrons saturates at the level n
e ~ n a, where na is the number
density of atoms in the target. Then the threshold fluence for ablation of dielectrics, taking intoaccount the above corrections, is defined as the following:
FJln
Athd
bise=+()3
4ε (24)
Therefore, as a general rule, the ablation threshold for dielectric in the ultra short laser-matter
interaction regime must be higher than that for the metals, assuming that all the atoms in theinteraction zone are at least singly ionized. Because the absorption in the ionized dielectric also16
occurs in a skin layer, one can use the relation ls/A ≈ λ/4π for the estimates and the scaling
relations (see Appendix A).
Another feature of the defined above ablation thresholds (21) and (24) is that they do not
depend explicitly on the pulse duration and intensity. However it is just a first orderapproximation. A certain, though weak, dependence is hidden in the absorption coefficient and inthe number density of free electrons.
8. COMPARISON TO THE LONG PULSE REGIME
It is instructive to compare the above defined ablation threshold to that for the long laser
pulses. This also helps in considering a general picture of the ablation process in a whole rangeof laser pulse duration.
The ultra short pulse laser-matter interaction mode corresponds to conditions when the
electron-to-ion energy transfer time and the heat conduction time exceed significantly the pulse
duration,
τei ~ t heat >> tp. Then the absorbed energy is going into the electron thermal energy, and
the ions remain cold εion << εe, making the conventional thermal expansion inhibited. However,
as it was shown above, if the laser intensity is high enough, the electrons can gain the energy in
excess of the Fermi energy and escape from the target. The electromagnetic field of the chargeseparation created by the escaped electrons pulls the ions out of the target. Hence, the extremenon-equilibrium regime of material ablation takes place. This regime occurs at the laser pulseduration t
p < 200 fs and at the intensities above 1013-1014 W/cm2. The escaped electrons
accelerate the ions by the electrostatic field of charge separation.
An intermediate regime takes place at the laser pulse duration 0.5 ps < tp < 100 ps and at
the intensities less than 1011 W/cm2, when τei ~ t heat ~ tp, and Te ~ T i. The most appropriate
description of the heating and expansion processes in this regime is given by the conventional
two temperature approach [16].17
At the longer laser pulse duration tp > 10 ps the heat conduction and hydrodynamic
motion dominate the ablation process, tp >> {τei; theat}. The electrons and the lattice (the ions)
are in equilibrium early in the beginning of the laser pulse Te ~ T i. Hence, the limiting case of
thermal expansion (thermal ablation) is suitable for the description of the long-pulse ablationmode. The ablation threshold for this case is defined by condition that the absorbed laser energyAI
0tp, is fully converted into the energy of broken bonds in a layer with the thickness of the heat
diffusion depth lheat ~ (κtp)1/2 during the laser pulse [16]:
AI t t npp b a01
2≅()κε . (25)
The well-known tp1/2 time dependence for the ablation fluence immediately follows from this
equation:
Ftn
Athpb a≈()κε12/
. (26)
Equations (23), (24), and (26) represent two limits of the short- and the long-pulse laser
ablation with a clear demonstration of the underlying physics. The difference in the ablationmechanisms for the thermal long pulse regime and the non-equilibrium short pulse mode is two-fold.
Firstly, the laser energy absorption mechanisms are different. The intensity for the long
pulse interaction is in the range 10
8-109 W/cm2 with the pulse duration change from nanoseconds
to picoseconds. The ionization is negligible, and the dielectrics are almost transparent up to UV-range. The absorption is weak, and it occurs due to the interband transitions, defects andexcitations. At the opposite limit of the femtosecond laser-matter interaction the intensity is inexcess of 10
13 W/cm2 and any dielectric is almost fully ionized in the interaction zone.
Therefore, the absorption due to the inverse Bremmstrahlung and the resonance absorptionmechanisms on free carriers dominates the interaction, and the absorption coefficient amounts toseveral tens percent.18
Secondly, the electron-to-lattice energy exchange time in a long-pulse ablation mode is of
several orders of magnitude shorter than the pulse duration. By this reason the electrons and ionsare in equilibrium, and ablation has a conventional character of thermal expansion. By contrast,for the short pulse interaction the electron-to-ion energy exchange time, as well as the heatconduction time, is much larger than the pulse duration, and the ions remain cold. Electrons cangain energy from the laser field in excess of the Fermi energy, and escape the target. The electricfield of a charge separation pulls ions out of the target thus creating an efficient non-equilibriummechanism of ablation.
9. ABLATION DEPTH AND EVAPORATION RATE
The depth of a crater x = d
ev, drilled by the ultra short laser with the fluence near the
ablation threshold F = I 0t > F th (23) is of the order of the skin depth. According to Eq. (15), it
increases logarithmically with the fluence:
dlF
Fevs
th=2ln (27)
due to the exponential decrease of the incident electric field and electron temperature in the target
material. Equation (27) coincides apparently with that from [17]. However, one should notedifference in definitions of the threshold fluence and skin depth in this paper from that in [17].The skin depth calculated above for the laser interaction with copper target of 74 nm qualitativelycomplies with the ablation depth fitting to the experimental value of 80 nm [17].
The average evaporation rate, which is the number of particles evaporated per unit area
per second, can be estimated for the ultra short pulse regime from (27) as the following:
nvdn
tshortev a
p()= . (28)
One can see a very weak logarithmic dependence on the laser intensity (or, fluence). For dev ≈ ls
≈ 70 nm, na ≈ 1023 cm-3, and tp ~100 fs, one gets the characteristic evaporation rate for the short19
pulse regime of ~7 ×1030 1/cm2 s. The evaporation rate for the long pulse regime depends only on
the laser intensity [2]:
nvI
longa
b()≈ε. (29)
Taking Ia ~ 109 W/cm2 and εb ~ 4 eV [2], the characteristic ablation rate for the long pulse
regime of ~ 3 ×1027 1/cm2s is about 2 ×103 times lower.
The number of particles evaporated per short pulse dev×na×Sfoc (Sfoc is the focal spot area)
is of several orders of magnitude lower than that for a long pulse. This effect eliminates the
major problem in the pulsed laser deposition of the thin films, which is formation of droplets andparticulates on the deposited film. The effect has been experimentally observed with 60 pspulses and 76 MHz repetition rate by producing diamond-like carbon films with the rms surfaceroughness on the atomic level [2].
One also can introduce the number of particles evaporated per Joule of absorbed laser
energy as a characteristic of ablation efficiency. One can easily estimate using Eqs.(28) and (29)that this characteristic is comparable for both the short-pulse and the long-pulse regimes.
10. COMPARISON TO THE EXPERIMENTAL DATA
Let us now to compare the above formulae to the different experimental data. Where it is
available we present the full span of pulse durations from femtosecond to nanosecond range forablation of metals and dielectrics.
A. Metals
Let us apply Eq.(23) for calculation of the ablation threshold for Copper and Gold targets
ablated by 780-nm laser. The Copper parameters are: density 8.96 g/cm
3, binding energy, e.g.
heat of evaporation per atom εb = 3.125 eV/atom, εesc= 4.65 eV/atom, na = 0.845×1023 cm-3.
The calculated threshold Fth ~ 0.51 J/cm2 is in agreement with the experimental figure 0.5-0.6
J/cm2 [17], though the absorption coefficient was not specified in [17]. For the long pulse20
ablation taking into account thermal diffusivity of Copper 1.14 cm2/s Eq. (26) predicts Fth =
0.045[J/cm2]×(tp [ps])1/2.
For a gold target ( εb = 3.37 eV/atom, εesc = 5.1 eV, ne = 5.9×1022 cm-3) evaporated by
laser wavelength 1053 nm the ablation threshold from Eq. (23) is Fth = 0.5 J/cm2. That figure
should be compared to the experimental value of 0.45 ± 0.1 J/cm2 [15]. For the long pulse
ablation assuming the constant absorption coefficient of A = 0.74 (see Appendix A) one finds
from Eq.(26) Fth = 0.049[J/cm2]×(tp [ps])1/2. The experimental points [15] and the calculated
curve are presented in Fig.1.
1 1 1 1 1 1 1 11
0.1110
0.01 0.1 1 10 100 1000 10000Threshold fluence, J/cm2
time, psAu-theory
1Au-experiment
Fig.1. Threshold laser fluence for ablation of gold target versus laser pulse duration. The
experimental error is ±0.5 J/cm2 [15].
B. Silica
An estimate for the ablation threshold for silica from Eq.(26) ( ne ~1023 cm-3,εb+Ji ≈ 12
eV [24]) by a laser with λ = 1053 nm ( ω = 1.79×1015 s-1; ls/A ~ 83.8 nm) gives Fth = 2.4 J/cm2,21
which is in a qualitative agreement with the experimental figures ~2 J/cm2 [1]. Formula (26) also
predicts the correct wavelength dependence of the threshold: Fth = 1.8 J/cm2 for λ = 800 nm ( ls/A
~ 63.7 nm) and Fth = 1.2 J/cm2 for λ = 526 nm (cf. Fig. 2). The experimental threshold fluences
for the 100 fs laser pulse [1] are: 2 – 2.5 J/cm2 (λ = 1053 nm), ~ 2 J/cm2 (λ = 800 nm), and 1.2
– 1.5 J/cm2 (λ = 526 nm).
111
012345
200 400 600 800 1000 1200Threshold fluence, J/cm2
Wavelength, nmTheory
1Experiment
Fig.2. Threshold fluence for laser ablation of fused silica target as a function of the laser
wavelength for 100 fs pulses. The experimental points are from the Ref. [1].
Using the following parameters for the fused silica at wavelength of 800 nm ( κ = 0.0087
cm2/s, εb = 3.7 eV/atom; na = 0.7×1023 cm-3; and A ~ 3×10-3) one obtains a good agreement with
the experimental data collected in [1] for the laser pulse duration from 10 ps to 1 ns. The long
pulse regime Eq.(26) holds: Fth = 1.29[J/cm2]×(tp [ps])1/2 (see Fig. 3).22
J JJJ JJJ J JJJJJJJJJJJ J JJJ JJJ JJJJJJ JJJ JJJJ
HHH HH HHH H HHHH
11050
0.01 0.1 1 10 100 1000Threshold fluence, J/cm2
time, psSilica; 1053 nm
Silica; 800 nm
Silica; 526 nm
J1053 nm exp
H825 nm exp
Fig. 3. Threshold laser fluence for ablation of fused silica target vs laser pulse duration. The
experimental error is ±15% [1].
The ablation threshold of 4.9 J/cm2 for a fused silica with the laser tp = 5 fs, λ = 780 nm,
intensity ~1015 W/cm2 has been reported in [21]. This value is three times higher than that of [1]
and from the prediction of Eq.(26). However, the method of the threshold observation, theabsorption coefficient, as well as the pre-pulse to main pulse contrast ratio were not specified in[21].
In the Ref. [22] the crater depth of 120 nm was drilled in a BK7 glass by a 100-fs 620-nm
laser at the intensity 1.5 ×10
14 W/cm2. Assuming that the skin depth in the BK7 glass target is the
same 84 nm as in the fused silica, the Eq.(29) for the ablation depth predicts the threshold value
of 0.9 J/cm2. This is in a reasonable agreement with the measured in [22] Fth = 1.4 J/cm2.
It should be noted that the definition of the ablation threshold implies that at the threshold
condition at least a mono-atomic layer x << l s, of the target material should be removed.
Therefore, the most reliable experimental data for the ablation threshold are those obtained by the23
extrapolation of the experimental dependence of the ablated depth vs the laser fluence to the
‘zero’ depth. As one can see from above comparison, the experimental data on the ablationthreshold determined this way are in excellent agreement with the formulae in this paper. Itshould be particularly emphasised that there were no any fitting coefficients in the calculationspresented here.
11. DISCUSSION AND CONCLUSIONS
We described here a new regime of material ablation in the ultra short laser-matter
interaction mode. The regime is characterised by the laser intensity in a range ~ 10
13 – 1014
W/cm2 and the pulse duration shorter than the plasma expansion time, the heat conduction time,
and the electron-to-ion energy transfer time. The interaction at such conditions results inionisation of practically any target material. The interaction with the metals and dielectricsproceeds in a similar way in contrast to the conventional, long pulse interaction mode. Thephysics of this new regime of ablation consists in the ion acceleration in the electrostatic fieldcreated by hot electrons escaping from the target. We derived the explicit analytical formulae forthe ablation threshold, the electron temperature in the skin layer, and the ablation rates for metalsand dielectrics in terms of laser and target parameters. These formulae do not contain any fittingparameters and agree well with the available experimental data. In this new regime the thresholdfluence is almost independent on the pulse duration, and the material evaporation rate is muchhigher than in the long pulse interaction regime.
An important condition for the ultra short pulse interaction mode in the real experiments
is the high contrast ratio of the pulse: the target surface should not be ionized, damaged orablated during the pre-pulse action. For the nanosecond-scale pre-pulse and the 100-fs mainpulse the intensity contrast ratio must be of the order of ~ 10
6. The ultra short laser ablation can
do a variety of fine jobs without any collateral damage to the rest of a target: cutting and drillingholes with a high precision, ablating all available materials with the ablation rate of several24
orders of magnitude faster than that with nanosecond lasers. The application of the ultra short
lasers with high repetition rate for film deposition allows totally eliminate the problem ofdroplets and particulates on the deposited film. The theoretical background developed in thispaper for laser ablation allows the appropriate laser parameters to be chosen for any givenmaterial and the laser-target interaction process to be optimized.
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th edition)
12. A. M. Malvezzi, N. Bloembergen, C. Y. Huang, Phys. Rev. Lett . 57, 146 – 149 (1986).
13. E. G. Gamaly, Phys. Fluids B5, 944 - 949 (1993).25
14. V. Yu. Bychenkov, V. T. Tikhonchuk and S. V. Tolokonnikov, JETP 88, 1137 - 1142 (1999).
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Science 109/110, 15 - 19 (1997).
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Moscow, 1970 (in Russian).
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24. R. B. Sosman, The Phases of Silica , (Rutgers University Press, New Bruswick, 1965).26
APPENDIX A:
ABSORPTION COEFFICIENT AND SKIN DEPTH NEAR THE ABLATION THRESHOLD
1. Metals: νω ωei pe~>>
In these conditions the refraction coefficient expresses as the following:
Nn i kn i n kpe=+≈ +() ≈=
1212
; /ω
ω; (A1)
the Fresnel absorption coefficient reads:
ARnnpe pe=− ≈ − =
−
121 812212 12
ω
ωω
ω//
; (A2)
and correspondingly the skin-depth takes the form:
lc
kcs
pe=≈
ωω ω212/
. (A3)
The ratio of ls/A that enters into the ablation threshold, expresses as the follows:
l
Acs
pe pe≈−
=−
−−
212412121121
ωω
ωλ
πω
ω//
. (A4)
Correction in the brackets for ablation of Copper ablation at 780 nm ( ω = 2.415×1015 s-1; ωpe =
1.64×1016 s-1) comprises 1.37. For a Gold target ablation at 1064 nm ( ω = 1.79×1015 s-1; ωpe =
1.876×1016 s-1) it amounts to 1.28. For the short wavelength such as KrF-laser or higher
harmonics of Nd laser one should use the general formulae for the absorption coefficient and the
skin-length.
2. Dielectrics : νωωei pe~~
Repeating the above procedure for dielectrics one obtains R ~ 0.05, A ~ 0.95, and
l
As≈3
2λ
π. (A5)27
APPENDIX B: IONIZATION OF SILICA
The ionization potential of Si is Ji = 8.15 eV. For Nd:YAG laser ( λ = 1064 nm) at the
intensity 2 ×1013 W/cm2 the probability for the ionization by electron impact is wimp = 1013 s-1, for
the multiphoton ionization is wmpi = 5×10-4 s-1, and the number density of created free electrons in
100 fs is ne ~ 107. At the intensity 1014 W/cm2 wimp = 1013 s-1; wmpi = 5×1014 s-1; and the number
density of free electrons reaches the solid density ne ~ na ~1023 cm-3 in 20 fs – this is the time
required for full first ionization. |
arXiv:physics/0102048v1 [physics.bio-ph] 16 Feb 2001Self-Organizing Approach for Finding Borders
of DNA Coding Regions
Fang Wu1and Wei-Mou Zheng2
1Department of Physics, Peking University, Beijing 100871, China
2Institute of Theoretical Physics, Academia Sinica, Beijin g 100080, China
Abstract
A self-organizing approach is proposed for gene finding base d on the model of codon usage for coding
regions and positional preference for noncoding regions. T he symmetry between the direct and reverse
coding regions is adopted for reducing the number of paramet ers. Without requiring prior training,
parameters are estimated by iteration. By employing the win dow sliding technique and likelihood ratio,
a very accurate segmentation is obtained.
PACS number(s): 87.15.Cc, 87.14.Gg, 87.10.+e
The data of raw DNA sequences is increasing at a phenomenal pa ce, providing a rich source of data
to study. As a consequence, we now face the tremendous challe nge of extracting information from the
formidable volume of DNA sequence data. Computational meth ods for reliably detecting protein-coding
regions are becoming more and more important.
Genome annotation by statistical methods is based on variou s statistical models of genomic sequences
[1, 2], one of the most popular being the inhomogeneous, thre e-period Markov chain model for protein-coding
regions with an ordinary Markov model for noncoding regions . The independent random chain model can
be included in this category by regarding it as a Markov chain of order 0. The codon usage model is the
independent random chain model of non-overlapping triplet s, and corresponds to an inhomogeneous Markov
model of order 2. Signals in a short segment are usually burie d in large fluctuations. With well chosen
parameters statistical models work as a noise filter to pick o ut the signals.
Methods based on local inhomogeneity, e.g. position asymme try or periodicity of period 3, suffer fluc-
tuations. Most of the current computer methods for locating genes require some prior knowledge of the
sequence’s statistical properties such as the codon usage o r positional preference [3, 4, 5]. That is, a sizable
training set is necessary for estimating good parameters of the model in use [6, 7]. Strongly biased by the
training, such models have little power to discover surpris ing or atypical features. Thus, it is desirable to
decipher the genomic information in an objective way. Audic and Claverie [8] have proposed a method which
does not require learning of species-specific features from an arbitrary training set for predicting protein-
coding regions. They use an ab initio iterative Markov modeling procedure to automatically part ition genome
sequences into direct coding, reverse coding, and noncodin g segments. This is an expectation-maximization
(EM) algorithm, which is useful in modeling with hidden vari ables, and is performed in two steps of ex-
pectation and maximization [9, 10, 11]. Such a self-organiz ing or adaptive approach uses all the available
unannotated genomic data for its calibration.
Before introducing the model we use and describing the techn ical details, we explain the EM algorithm
with a simple pedagogic model which assumes that a DNA sequen ce written in four letters {a, c, g, t }is
generated by independent tosses of two four-sided dice. An a nnotation maps the DNA sequence site-to-site
to a two-letter sequence of the alphabet {C, N}(Cfor coding and Nfor noncoding). Two sets {pa, pc, pg, pt}
and{qa, qc, qg, qt}of positional nucleotide probabilities are associated wit h the two dice CandN, respectively.
The total probability for the given DNA sequence S=s1s2. . .to be seen under the model is the partition
or likelihood function
Z=/summationdisplay
HP(S|Hα) =/summationdisplay
H/productdisplay
iP(si|hα
i), (1)
1where the summation is over all the possible “annotations” H={Hα}withHα=hα
1hα
2. . .,hα
i∈ {N, C},
andP(s|C) =ps,P(s|N) =qs. The unknown two sets of probabilities can be determined by m aximizing
the likelihood Z. From Bayesian statistics
P(Hα|S) =P(S|Hα)P(Hα)/summationtext
HP(S|Hα)P(Hα), (2)
with prior P(Hα) assumed, the most possible Hαcan then be selected as the inferred annotation. As we
know, coding regions are organized in blocks. The first simpl ification is the window coarse-graining. The
sequence Sis divided into nonoverlapping window segments of constant length w, and each whole window
is entirely assigned to either NorC. Conducting Bayesian analysis for window Wjand accepting uniform
prior, we have P(h|Wj)∝P(Wj|h). The second simplification is to introduce “temperature” τ(as in the
simulated annealing), replace P(Wj|hj) with [ P(Wj|hj)]1/τand take the limit τ→0. In this way we keep
only a single term, i.e. the greatest one, in the summation fo rZ. Window Wis inferred to belong to either
NorCdepending on whether P(W|N) orP(W|C) is larger. The likelihood maximization is then equivalent
to estimating nucleotide probabilities with frequencies i n two window classes inferred from the pre-assumed
{ps}and{qs}. Consistency requires that the estimate probabilities mus t be equal to {ps}and{qs}. This
“fixed point” can be found by iteration. As an example, we use t he first 99 ×5 051 = 500 049 nucleotides of
the complete genome of E. coli as the input data. Statistical significance requires that the window size cannot
be too small, while a large window size would give poor resolu tion in discriminating different regions. The
window size is chosen to be w= 99. We assign the 5 051 fixed nonoverlapping windows to the tw o subsets of
NandCin either a periodic or a random way. We estimate psandqsfrom the counts of different nucleotides
in each subset. The likelihood functions for each window are then calculated using the estimated psandqs,
and the assignment of the windows to CorNis updated according to which of P(W|C) and P(W|N) is
larger. This ends one iteration. The process of iteration co nverges to a single fixed point of precision 10−4
around step 28 for different initializations with a final wind ow assignment to NandCalso given. The final
psandqsare{0.219,0.270,0.289,0.222}and{0.279,0.213,0.227,0.281}. The qsestimated from the complete
genome are {0.285,0.214,0.218,0.283}, which are rather close to the corresponding convergent val ues.
More realistic models take the three phases in the coding reg ions and the opposite ordering of the direct
and reverse coding regions into account. Such models adopt 7 subsets: one for noncoding (N), three for direct
coding (C 1, C2, C3) and three for reverse coding (C 4, C5, C6). The subscript iin C iindicates the phase 0, 1
or 2 accordng to i(mod 3). From the genomic data statistics, we may assume that there is symmetry between
the direct and reverse coding regions, which means that a rev erse coding sequence is indistinguishable from
a direct coding sequence if we make the exchanges a↔t,c↔gand reverse the order. For the model
based on the positional preference of codons, instead of 7 se ts of positional nucleotide probabilities, we need
only 4 sets. The reduction of the total number of parameters b y the symmetry consideration improves the
statistics. The procedure of iteration is similar to that fo r the last model, the only difference being that now
we have to estimate 4 sets of probabilities and calculate 7 li kelihood functions for a window.
We use a better model based on the codon usage. We now need a set of 64 probabilities for coding
regions. For noncoding segments, 4 positional nucleotide p robabilities are used just as before. To simplify
the programming, we move the windows with a phase-shift othe r than zero by one or two nucleotides to clear
the phase-shift, although we can calculate the marginal dis tribution probabilities for uni- and bi-nucleotides.
For example, we replace the window W=sisi+1. . .s i+w−1marked as C 2withW′=si+1si+2. . . s i+w. (The
alternative way is to consider a cyclic transformation.) Ou r further discussions are all based on this model.
It is observed that the iteration also quickly converges to a fixed point. Contrary to the two-sets model
where coding and noncoding are symmetric, and extra knowled ge is required to relate one set to coding and
the other to noncoding, we can now distinctly distinguish co ding from noncoding regions, even with their
phases fixed. Direct and reverse coding sets are symmetric in the model. However, the fact that stop codons
taa,tagandtgaare rare can be used to remove the symmetry between direct and reverse coding. That is, if
the convergent probabilities for taa,tagandtgaare all significantly small in comparison with the other 61,
sets C 1, C2and C 3then do indeed correspond to direct coding. (Otherwise, tho se oftta,ctaandtcawould
be small instead.)
We employ the sliding window technique to improve the resolu tion as follows. We shift each window
2by 3 nucleotides, initiate the window assignment with the co nvergent probabilities just obtained, and then
find new assignments for the shifted windows by iteration. We repeat the shifting process 32 times to cover
the window width. This ends with 33 assignments for triplets , except for a few sites at the two ends. By a
majority vote we can obtain a triplet assignment of the whole sequence.
Recently, an entropic segmentation method that uses the Jen sen-Shannon measure for sequences of a
12-letter alphabet has been proposed to find borders between coding and noncoding regions [12]. Their best
result was obtained on the genome of the bacterium Rickettsia prowazekii . We test our approach with the
same genome data. To inspect the accuracy we obtain the “true ” assignment of sites based on the known
annotation as follows. If a nucleotide is in a noncoding regi on, it belongs to N. If it is in a coding (or
reverse coding) region and the site-index of the beginning n ucleotide plus 1 is congruent to imodulo 3, the
nucleotide under consideration will belong to C 1+i(or C 4+i). For overlapping coding zones we may keep
two alternative assignments. We define three rates of accura cyR2,R3andR7:R2only discriminates coding
from noncoding segments while R7covers full discrimination of the 7 sets, and R3ignores the phases. For
the total N= 1 111 523 nucleotides, we obtain R2= 91.7%,R3= 89.8% and R7= 89.7%. (The rates
without window sliding are R2= 89.1%,R3= 84.8% and R7= 84.4%.)
For finding block borders, to eliminate illusary fluctuation s we accept only the assignments with the 33
identical samplings, and regard others as undetermined. Wh en two adjacent identified blocks are of the
same assignment we join the two together with the sites betwe en into a single zone of the same assignment.
Otherwise, we take the middle site of the intervening undete rmined zone as the border, and assign the two
sides according to their corresponding flank blocks. We can d o the job better by means of the likelihood
ratio. Suppose that the left block is assigned to l, and the right to r. A point min the intervening zone
divides the zone into two segments LmandRm. The likelihood ratio is defined as
Γm=P(Lm|l)P(Rm|r)
P(Lm|r)P(Rm|l). (3)
The maximal Γ mplaces the border at m. This segmentation finally gives the accuracy rates R2= 93.3%,
R3= 92.8% and R7= 92.7%.
In Ref. [12] the quantity quantifying the coincidence betwe en borders inferred from the segmentation and
those from the known annotation is defined by
D=1
2N
/summationdisplay
imin
j|bi−cj|+/summationdisplay
jmin
i|bi−cj|
, (4)
where {bi}is the set of all borders between coding and noncoding region s, and {cj}is the set of all cuts
produced by the segmentation. We use an even harsher quantit yDby interpreting {bi}and{cj}as the
borders of all coding zones. That is, we include borders of ea ch overlapping coding zone. The total number
of “CDS” in the annotation is 834, one of which has two joint zo nes. We obtain 1 −D= 87.7%, compared
with∼80% of Ref. [12]. In Fig. 1 we show a comparison of the inferred segmantation with the known
coding regions. In the section from 475 500 to 497 500 there ar e two overlaps (one for direct, and the other
for reverse coding regions), and the shortest gap separatin g adjacent coding regions is just 1 nucleotide (at
486 215). They do not escape detection. As mentioned in [12], there are two very close coding regions in
the same phase (538 197 : 539 879 and 539 937 : 540 887). The resu lt from the majority vote is shown in
Fig. 2 for the section. We see indeed a peak of the counts for se t N between the two coding regions. The
highest count for N is 32, and so is ignored in our strategy. Th ere is indeed plenty of room for improving
this approach. A larger width w= 123 gives higher accuracy rates: R2= 93.6%,R3= 93.3%,R7= 93.0%
and 1−D= 88.2%. When we consider only the triplets with all 33 assignment s identical in window sliding
the rates are R2= 98.7%,R3= 98.6% and R7= 98.6%. In the above we avoid setting up an arbitary
cut-off threshold. If a threshold of 17 counts is used to deter mine the segments whose central parts have
33 identical samplings, for w= 99 we predict a total of 1 001 351 (90 .1%) sites with accuracies R2= 97.4%
andR3= 95.4%. The accuracy rate for noncoding regions is 96.5%, much hi gher than that of Ref. [8]. It
is important and feasible to integrate biological signals i nto our algorithm. We expect our algorithm, with
certain modifications, should work well for other species, t oo.
3This work was supported in part by the Special Funds for Major National Basic Research
Projects, the National Natural Science Foundation of China and Research Project 248 of Beijing.
References
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[3] R. Grantham, C. Gautier, M. Gouy, M. Jacobzone, and R. Mer cier, Nucleic Acids Res. 9,0 R43 (1981).
[4] J. W. Fickett, Nucleic Acids Res. 10, 5303 (1982).
[5] S. Karlin and J. Mrazek, J. Mol. Biol. 262, 459 (1996).
[6] M. Borodovsky and J. D. McIninch, Comput. Chem. 17, 123 (1993).
[7] M. Borodovsky, J. D. McIninch, E. V. Koonin, K. E. Rudd, C. Medigue, and A. Danchin, Nucleic Acids
Res.23, 3554 (1995).
[8] S. Audic and J.-M. Claverie, Proc. Natl. Acad. Sci. USA, 95, 10026 (1998).
[9] P. Baldi, Bioinformatics 16, 367 (2000); P. Baldi and S. Brunak, Bioinformatics: The Mechine Learning
Approach (The MIT Press, Cambridge, Ma., 1998).
[10] C. E. Lawrence and A. A. Reilly, Proteins 7, 41 (1990).
[11] L. R. Cardon and G. D. Stormo, J. Mol. Biol. 223, 159 (1992).
[12] P. Bernaola-Galv´ an, I. Grosse, P. Carpena, J.L. Olive r, R. Rom´ an-Rold´ an, and H.E. Stanley, Phys.
Rev. Lett. 85, 1342 (2000).
Figure 1: Comparison between the inferred segmentation (do tted lines) and the known coding regions of
Rickettsia (shaded areas).
Figure 2: Counts of majority assignment in the section conta ining two very close coding regions (shaded
areas) in the same phase. A peak corresponding to noncoding a ssignment is clearly seen.
4segment borders codingregions
Position (kbp)480 485 490 495Position(kbp)539.8 539.9 540.0 540.130
20Counts
coding phase 2 noncoding |
arXiv:physics/0102049v1 [physics.comp-ph] 16 Feb 2001Implementation of analytical Hartree-Fock gradients for p eriodic
systems
K. Doll
CLRC, Daresbury Laboratory, Daresbury, Warrington, WA4 4A D, UK
Institut f¨ ur Mathematische Physik, TU Braunschweig, Mend elssohnstraße 3, D-38106
Braunschweig
Abstract
We describe the implementation of analytical Hartree-Fock gradients for pe-
riodic systems in the code CRYSTAL, emphasizing the technic al aspects of
this task. The code is now capable of calculating analytical derivatives with
respect to nuclear coordinates for systems periodic in 0, 1, 2 and 3 dimensions
(i.e. molecules, polymers, slabs and solids). Both closed- shell restricted and
unrestricted Hartree-Fock gradients have been implemente d. A comparison
with numerical derivatives shows that the forces are highly accurate.
Typeset using REVT EX
1I. INTRODUCTION
The determination of equilibrium structure is one of the mos t important targets in elec-
tronic structure calculations. In surface science especia lly, theoretical calculations of surface
structures are of high importance to explain and support exp erimental results. Therefore,
a fast structural optimization is an important issue in mode rn electronic structure codes.
Finding minima in energy surfaces is substantially simplifi ed by the availability of analytical
gradients. As a rule of thumb, availability of analytical gr adients improves the efficiency by
a factor of order NwithNbeing the number of parameters to be optimized. UK’s Collab-
orative Computational Project 3 has therefore supported th e implementation of analytical
gradients in the electronic structure code CRYSTAL1–4. This implementation will also be
valuable for future projects which require analytical grad ients as a prerequisite. Another
advantage of having analytical gradients is that higher der ivatives can be obtained with less
numerical noise (e.g. the 2nd derivative has less numerical noise when only one numerical
differentiation is necessary).
CRYSTAL is capable of performing Hartree-Fock and density- functional calculations for
systems with any periodicity (i.e. molecules, polymers, sl abs and solids). The periodicity
is ”cleanly” implemented in the sense that, for example, a sl ab is considered as an object
periodic in two dimensions and is notrepeated in the third dimension with one slab being
separated from the others by vacuum layers. The code is based on Gaussian type orbitals
and the technology is therefore in many parts similar to that of molecular quantum chemistry
codes. As the density-functional part of the code relies in b ig parts on the Hartree-Fock
part, the strategy of the project was to implement Hartree-F ock gradients first.
The implementation of Hartree-Fock gradients for multicen ter basis sets was pioneered by
Pulay5; the theory had already been derived earlier independently6. Meanwhile, analytical
gradients have been implemented in many molecular codes, an d several review articles have
appeared (see, e.g., references 7–13).
Substantial work has also been done in the case of one-dimens ional periodicity: Hartree-
2Fock gradients with respect to nuclear coordinates and with respect to the lattice vector have
already been implemented in codes periodic in one dimension14–16. Moreover, correlated
calculations based on the MP2 scheme17,18and MP2 gradients15have been coded. Also,
density functional gradients have been implemented19,20. Even second derivatives at the
Hartree-Fock level have meanwhile been coded21.
The implementation of Hartree-Fock gradients with respect to nuclear coordinates in
CRYSTAL is to the best of our knowledge the first implementati on for the case of 2- and
3-dimensional periodicity. The aim of this article is to des cribe the implementation of the
gradients in the code, with an emphasis on the technical aspe cts. Therefore, the article is
supposed to complement our first article on the purely theore tical aspects22. An attempt of
a detailed description is made; however, as the whole code is undergoing constant changes, it
can not be too detailed. For example, it did not seem advisabl e to give any variable names
because they have already undergone major changes after the code moved to Fortran 90
with the possibility of longer variable names.
The article is structured as follows: In section II, we give a brief introduction to Gaussian
and Hermite Gaussian type basis functions. The definition of the density matrix is given in
section III. The individual integrals, their derivatives, and details of the implementation are
discussed in section IV. Formulas for total energy and gradi ent are given in section V. The
structure of the gradient code is explained in section VI, fo llowed by examples in section
VII and the conclusion.
II. BASIS FUNCTIONS
Two sets of basis functions are relevant for CRYSTAL: firstly , unnormalized spherical
Gaussian type functions, in a polar coordinate system chara cterized by the set of variables
(|/vector r|,ϑ,ϕ), and centered at /vectorA. They are defined as
S(α,/vector r−/vectorA,n,l,m ) =|/vector r−/vectorA|2n+lP|m|
l(cosϑ) exp(imϕ) exp( −α|/vector r−/vectorA|2) (1)
3with P|m|
lbeing the associated Legendre function. CRYSTAL uses real s pherical Gaussian
type functions defined as
R(α,/vector r−/vectorA,n,l, 0) =S(α,/vector r−/vectorA,n,l, 0)
R(α,/vector r−/vectorA,n,l, |m|) = ReS(α,/vector r−/vectorA,n,l, |m|)
R(α,/vector r−/vectorA,n,l, −|m|) = ImS(α,/vector r−/vectorA,n,l, |m|)
This is in the following denoted as φµ(α,/vector r−/vectorAµ,n,l,m ) =NµR(α,/vector r−/vectorAµ,n,l,m ), with
the normalization Nµ.µis an index enumerating the basis functions in the reference cell
(e.g. the primitive unit cell). In fact, CRYSTAL uses only ba sis functions with quantum
numbern= 0 and angular momentum l=0,1 or 2 (i.e. s,pordfunctions).
The exponents are defined by the user of the code. A huge amount of basis sets for
molecular calculations is available in the literature and o n the world wide web; also for
periodic systems a large number of basis sets has been publis hed. Molecular basis sets can,
with a little effort, be adopted for solid state calculations . High exponents which are used to
describe core electrons do not have to be adjusted, but expon ents with low values (e.g. less
than 1a−2
0, witha0being the Bohr radius) should be reoptimized for the corresp onding solid.
Very diffuse exponents should be omitted because they cause l inear dependence problems
in periodic systems.
A second type of basis functions, which CRYSTAL uses interna lly to evaluate the inte-
grals, is the Hermite Gaussian type function (HGTF) which is defined as:
Λ(γ,/vector r−/vectorA,t,u,v ) =/parenleftbigg∂
∂Ax/parenrightbiggt/parenleftbigg∂
∂Ay/parenrightbiggu/parenleftbigg∂
∂Az/parenrightbiggv
exp(−γ|/vector r−/vectorA|2) (2)
CRYSTAL uses the McMurchie-Davidson algorithm to evaluate the integrals. The basic
idea of this algorithm is to map the product of two spherical G aussian type functions on
two centers onto a set of Hermite Gaussian type functions at o ne center.
S(˜α,/vector r−/vectorB,˜n,˜l,˜m)S(α,/vector r−/vectorA,n,l,m ) =
/summationdisplay
t,u,vE(˜n,˜l,˜m,n,l,m,t,u,v )Λ(γ,/vector r−/vectorP,t,u,v ) (3)
4withγ=α+ ˜αand/vectorP=α/vectorA+˜α/vectorB
α+˜α.
The starting point E(0,0,0,0,0,0,0,0,0) = exp( −α˜α
α+˜α|/vectorB−/vectorA|2) is derived from the
Gaussian product rule23:
exp(−α|/vector r−/vectorA|2) exp( −˜α|/vector r−/vectorB|2) = exp/parenleftbigg
−α˜α
α+ ˜α|/vectorB−/vectorA|2/parenrightbigg
exp/parenleftbigg
−(α+ ˜α)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vector r−α/vectorA+ ˜α/vectorB
α+ ˜α/vextendsingle/vextendsingle/vextendsingle/vextendsingle2/parenrightbigg
(4)
As indicated in section IV, all the integrals can be expresse d with the help of the
coefficients E(˜n,˜l,˜m,n,l,m,t,u,v )24–27. These coefficients are generated by recursion
relations24,25. They are zero for the case t+u+v >2n+ 2˜n+l+˜land for all nega-
tive values of t,uorv. CRYSTAL uses only basis functions with n= 0. Therefore, there are
(l+˜l+1)(l+˜l+2)(l+˜l+3)
3!coefficients E(0,˜l,˜m,0,l,m,t,u,v ) for fixed values of l,m,˜l,˜m. As the
maximum angular quantum number is l= 2, this results in 25 possible combinations of m
and ˜m. Therefore, the maximum number of coefficients is 25 ×35 = 875. These coefficients
are pre-programmed in the subroutine DFAC3. Pre-programmi ng is the fastest possible way
of evaluating these coefficients which is important because t his is one of the key issues of the
integral calculation. On the other hand, the code has become inflexible as no E-coefficients
are available for higher quantum numbers.
Derivatives of Gaussian type functions are again Gaussian t ype functions. Therefore,
the evaluation of gradients is closely related to the evalua tion of integrals. In a similar way
as all the integrals can be expressed with the help of coefficie ntsE, all the derivatives of
the integrals can be expressed with the help of coefficients fo r the gradients, GA
x,GA
y,GA
z.
TheseG-coefficients can be obtained with recursion relations deriv ed by Saunders4,22. The
recursions are similar to the ones for the E-coefficients. However, as the existing subroutine
DFAC3 cannot compute the G-coefficients, the recursions were newly coded. This has in
addition the advantage that, by small modifications of the ne w subroutines, E-coefficients
for higher quantum numbers than l=˜l= 2 can now be computed by recursion. There are
three sets of G-coefficients because of the three spatial directions. The G-coefficients are
zero for the case t+u+v >2n+ 2˜n+l+˜l+ 1 and for all negative values of t,uorv.
This means that for a maximum quantum number of l= 2, there are 3 ×5×5×56 = 4200
5coefficients. Three other sets of G-coefficients are necessary because of the second center.
However, the sets on the second center are closely related to the sets on the first center and
can be derived from them in an efficient way28,4,22.
III. DENSITY MATRIX
After solving the Hartree-Fock equations29, the crystalline orbitals are linear combina-
tions of Bloch functions
Ψi(/vector r,/vectork) =/summationdisplay
µaµi(/vectork)ψµ(/vector r,/vectork) (5)
which are expanded in terms of real spherical Gaussian type f unctions
ψµ(/vector r,/vectork) =Nµ/summationdisplay
/vector gR(α,/vector r−/vectorAµ−/vector g,n,l,m )ei/vectork/vector g(6)
The sum over /vector gis over all direct lattice vectors.
In the case of closed shell, spin-restricted Hartree-Fock, the spin-free density matrix in
reciprocal space is defined as
Pµν(/vectork) = 2/summationdisplay
iaµi(/vectork)a∗
νi(/vectork)Θ(ǫF−ǫi(/vectork)) (7)
with the Fermi energy ǫFand the Heaviside function Θ; iis an index enumerating the
eigenvalues.
In the case of unrestricted Hartree-Fock (UHF)30, we use the notation
Ψ↑
i(/vector r,/vectork) =/summationdisplay
µa↑
µi(/vectork)ψµ(/vector r,/vectork) (8)
and
Ψ↓
i(/vector r,/vectork) =/summationdisplay
µa↓
µi(/vectork)ψµ(/vector r,/vectork) (9)
for the crystalline orbitals with up and down spin, respecti vely. We define the density
matrices
6P↑
µν(/vectork) =/summationdisplay
ia↑
µi(/vectork)a∗↑
νi(/vectork)Θ(ǫF−ǫ↑
i(/vectork)) (10)
for up spin and
P↓
µν(/vectork) =/summationdisplay
ia↓
µi(/vectork)a∗↓
νi(/vectork)Θ(ǫF−ǫ↓
i(/vectork)) (11)
for down spin. In the following, Pµνrefers to the sum P↑
µν+P↓
µνin the UHF case.
The density matrices in real space Pµ/vector0ν/vector g,P↑
µ/vector0ν/vector g,P↓
µ/vector0ν/vector gare obtained by Fourier transfor-
mation.
IV. INTEGRALS AND THEIR DERIVATIVES
The calculation of the integrals is fundamental to all quant um chemistry programs.
CRYSTAL uses two integral packages: a package derived from G AUSSIAN7031is the default
for calculations when only sandspshells are used; alternatively Saunders’ ATMOL Gaussian
integral package can be used and it must be used for cases when pordfunctions are involved.
The implementation of gradients has been done with routines based on the ATMOL package.
This is not a restriction, and it is possible to use routines b ased on GAUSSIAN70 for the
integrals and routines based on ATMOL for the gradients.
The calculation of the integrals is essentially controlled from MONMAD and MONIRR
for one-electron integrals and from SHELLC or SHELLX for the bielectronic integrals.
SHELLC is used in the case of non-direct SCF, i.e. when the int egrals are written to
disk and read in each cycle. SHELLX is the direct version when the integrals are computed
in each cycle without storing them on disk. The direct mode is the preferred one when the
integral file is too big or when input/output to disk is too slo w. The gradients are com-
puted only once after the last iteration, when convergence i s achieved. Therefore, a direct
implementation of gradients has been done.
One of the bottlenecks of the CRYSTAL code is the restriction to a highest quantum
number of l= 2, i.e. the code can only cope with s,p,spanddfunctions, but not with
basis functions with higher angular momentum. Introducing gradients, however, is similar
7to increasing the quantum number from dtoffor the corresponding basis function. This
means that many subroutines had to be extended to higher quan tum numbers, and array
dimensions in the whole code had to be adjusted.
A. One-electron integrals
In this section we summarize the appearing types of integral s and the corresponding gra-
dients. We restrict the description to the x-component of th e gradient; y- and z-component
can be obtained in similar way. Note that the integrals depen d on the dimension because
of the Ewald scheme used. Therefore, there are four different routines for the one-electron
integrals for the case of 0,1,2 and 3-dimensional periodici ty: CJAT0, CJAT1, CJAT2 and
CJAT3. Similarly, four gradient routines have been develop ed which have been given the
preliminary names CJAT0G, CJAT1G, CJAT2G and CJAT3G. These routines calculate all
the one-electron integrals except for the multipolar integ rals which are computed in POLIPA
(with the corresponding gradient routine OLIPAG).
1. Overlap integral
The basic integral is the overlap integral:
Sµ /vector g1ν /vector g2=/integraldisplay
φµ(˜α,/vector r−/vectorAµ−/vector g1,˜n,˜l,˜m)φν(α,/vector r−/vectorAν−/vector g2,n,l,m )d3r =
/integraldisplay/summationdisplay
t,u,vE(˜n,˜l,˜m,n,l,m,t,u,v )Λ(γ,/vector r−/vectorP,t,u,v )d3r = (12)
E(˜n,˜l,˜m,n,l,m, 0,0,0)/parenleftBiggπ
γ/parenrightBigg3
2
The x-component of the gradient with respect to center Aµis obtained as
∂
∂Aµ,xSµ /vector g1ν /vector g2= (13)
∂
∂Aµ,x/integraldisplay
φµ(˜α,/vector r−/vectorAµ−/vector g1,˜n,˜l,˜m)φν(α,/vector r−/vectorAν−/vector g2,n,l,m )d3r =
∂
∂Aµ,x/integraldisplay/summationdisplay
t,u,vE(˜n,˜l,˜m,n,l,m,t,u,v )Λ(γ,/vector r−/vectorP,t,u,v )d3r =
8/integraldisplay/summationdisplay
t,u,vGAµ
x(˜n,˜l,˜m,n,l,m,t,u,v )Λ(γ,/vector r−/vectorP,t,u,v )d3r =
GAµ
x(˜n,˜l,˜m,n,l,m, 0,0,0)/parenleftBiggπ
γ/parenrightBigg3
2
(14)
Equation 14 thus
defines the coefficients GAµx; similarly the coefficients GAµy,GAµz,GAνx,GAνy,GAνzcan be de-
fined.
In the following, we use the identity
Sµ /vector g1ν /vector g2=Sµ/vector0ν(/vector g2−/vector g1)=Sµ/vector0ν/vector g.
2. Kinetic energy integrals
In equation 15, the expression for the kinetic energy integr als for the case of spherical
Gaussian type functions is reiterated25:
Tµ/vector0ν/vector g=
/integraldisplay
φµ(˜α,/vector r−/vectorAµ,˜n,˜l,˜m)/parenleftbigg
−1
2∆/vector r/parenrightbigg
φν(α,/vector r−/vectorAν−/vector g,n,l,m )d3r =
−n(2n+ 2l+ 1)/integraldisplay
φµ(˜α,/vector r−/vectorAµ,˜n,˜l,˜m)φν(α,/vector r−/vectorAν−/vector g,n−1,l,m)d3r +
α(4n+ 2l+ 3)/integraldisplay
φµ(˜α,/vector r−/vectorAµ,˜n,˜l,˜m)φν(α,/vector r−/vectorAν−/vector g,n,l,m )d3r−
2α2/integraldisplay
φµ(˜α,/vector r−/vectorAµ,˜n,˜l,˜m)φν(α,/vector r−/vectorAν−/vector g,n+ 1,l,m)d3r =
−n(2n+ 2l+ 1)/integraldisplay/summationdisplay
t,u,vE(˜n,˜l,˜m,n−1,l,m,t,u,v )Λ(γ,/vector r−/vectorP,t,u,v )d3r +
α(4n+ 2l+ 3)/integraldisplay/summationdisplay
t,u,vE(˜n,˜l,˜m,n,l,m,t,u,v )Λ(γ,/vector r−/vectorP,t,u,v )d3r−
2α2/integraldisplay/summationdisplay
t,u,vE(˜n,˜l,˜m,n+ 1,l,m,t,u,v )Λ(γ,/vector r−/vectorP,t,u,v )d3r (15)
The x-component of the gradient is therefore:
∂
∂Aµ,xTµ/vector0ν/vector g=
−n(2n+ 2l+ 1)/integraldisplay/summationdisplay
t,u,vGAµ
x(˜n,˜l,˜m,n−1,l,m,t,u,v )Λ(γ,/vector r−/vectorP,t,u,v )d3r +
9α(4n+ 2l+ 3)/integraldisplay/summationdisplay
t,u,vGAµ
x(˜n,˜l,˜m,n,l,m,t,u,v )Λ(γ,/vector r−/vectorP,t,u,v )d3r−
2α2/integraldisplay/summationdisplay
t,u,vGAµ
x(˜n,˜l,˜m,n+ 1,l,m,t,u,v )Λ(γ,/vector r−/vectorP,t,u,v )d3r (16)
As CRYSTAL uses spherical Gaussian type functions with n= 0, this reduces to
∂
∂Aµ,xTµ/vector0ν/vector g=
/parenleftBiggπ
γ/parenrightBigg3
2
α(4n+ 2l+ 3)GAµ
x(0,˜l,˜m,0,l,m, 0,0,0)−
2/parenleftBiggπ
γ/parenrightBigg3
2
α2GAµ
x(0,˜l,˜m,1,l,m, 0,0,0) (17)
Explicit differentiation with respect to the other center /vectorAνis more difficult because the
kinetic energy operator applies to that center. However, th e differentiation can easily be
avoided by applying translational invariance:
∂
∂Aµ,xTµ/vector0ν/vector g=−∂
∂Aν,xTµ/vector0ν/vector g (18)
3. Nuclear attraction integrals
The nuclear attraction integrals are defined as
Nµ/vector0ν/vector g=−/summationdisplay
aZa/integraldisplay
φµ(˜α,/vector r−/vectorAµ,˜n,˜l,˜m)A(/vector r−/vectorAa)φν(α,/vector r−/vectorAν−/vector g,n,l,m )d3r =
−/summationdisplay
aZa/integraldisplay/summationdisplay
t,u,vE(˜n,˜l,˜m,n,l,m,t,u,v )Λ(γ,/vector r−/vectorP,t,u,v )A(/vector r−/vectorAa)d3r (19)
whereAis the Coulomb potential function in the molecular case, the Euler-MacLaurin
potential function for systems periodic in one dimension32, Parry’s potential function33for
systems periodic in two dimensions, and the Ewald potential function for systems periodic
in three dimensions34,35,26. The summation with respect to aruns over all nuclei of the
primitive unit cell.
The x-component of the partial derivative with respect to th e centerAµ,xis obtained as:
10∂
∂Aµ,xNµ/vector0ν/vector g=
−/summationdisplay
aZa/integraldisplay/summationdisplay
t,u,vGAµ
x(˜n,˜l,˜m,n,l,m,t,u,v )Λ(γ,/vector r−/vectorP,t,u,v )A(/vector r−/vectorAa)d3r (20)
In the same way, the partial derivative with respect to Aν,xis obtained. The partial
derivative with respect to the set of third centers /vectorAais obtained by translational invariance:
for each center /vectorAa, there is a derivative with value
−∂
∂/vectorAµ−∂
∂/vectorAν.
4. Multipolar integrals
The electronic charge density is expressed with a lattice ba sis as:
ρ(/vector r) =−/summationdisplay
/vector g,µ,νPν/vector gµ/vector0φµ(˜α,/vector r−/vectorAµ,˜n,˜l,˜m)φν(α,/vector r−/vectorAν−/vector g,n,l,m ) (21)
Then, the Ewald potential due to this charge density is given by:
Φew(ρ;/vector r) =/integraldisplay
A(/vector r−/vector r′)ρ(/vector r′)d3r′(22)
The Ewald energy of the electons (i.e. the Ewald energy of the electrons in the primitive
unit cell with all the electrons) is obtained as
E=1
2/integraldisplay /integraldisplay
ρ(/vector r)A(/vector r−/vector r′)ρ(/vector r′)d3rd3r′(23)
For efficiency reasons, the calculation of the Ewald potentia l is done approximatively.
A multipolar expansion up to an order Lis performed for the charge distribution in the
long range. Therefore, the electrons do not feel the Ewald po tential created by the correct
charge distribution, but the Ewald potential created by the multipolar moments. It is thus
necessary to compute the multipolar moments of the charge di stribution which are defined
as
ηm
l(ρc;/vectorAc) =/integraldisplay
ρc(/vector r)Xm
l(/vector r−/vectorAc)d3r (24)
withXm
lbeing regular solid harmonics26and the charge ρc(/vector r) defined as
11ρc(/vector r) =−/summationdisplay
/vector g,µ∈c,νPν/vector gµ/vector0φµ(˜α,/vector r−/vectorAµ,˜n,˜l,˜m)φν(α,/vector r−/vectorAν−/vector g,n,l,m ) =
−/summationdisplay
/vector g,µ∈c,νPν/vector gµ/vector0/summationdisplay
t,u,vE(˜n,˜l,˜m,n,l,m,t,u,v )Λ(γ,/vector r−/vectorP,t,u,v ) (25)
cis an index for the shell. The total electronic charge ρ(/vector r) is thus obtained by summing
over all shells c:
ρ(/vector r) =/summationdisplay
cρc(/vector r) (26)
In CRYSTAL, the multipole is located at center /vectorAµand therefore it is convenient to take
the derivative with respect to center /vectorAνand obtain the derivative with respect to /vectorAµby
translational invariance. The expression computed for the gradients is thus
−/summationdisplay
/vector g,µ∈c,νPν/vector gµ/vector0/integraldisplay∂
∂Aν,x/parenleftBig
φµ(˜α,/vector r−/vectorAµ,˜n,˜l,˜m)φν(α,/vector r−/vectorAν−/vector g,n,l,m )Xm
l(/vector r−/vectorAµ)/parenrightBig
d3r =
−/summationdisplay
/vector g,µ∈c,νPν/vector gµ/vector0/integraldisplay/summationdisplay
t,u,vGAν
x(˜n,˜l,˜m,n,l,m,t,u,v )Λ(γ,/vector r−/vectorP,t,u,v )Xm
l(/vector r−/vectorAµ)d3r (27)
5. Field integrals
If the electronic charge distribution is approximated with an expansion up to the maxi-
mum quantum number L, the Ewald potential of this model charge distribution is ob tained
as
Φew(ρmodel;/vector r) =/summationdisplay
cΦew(ρmodel
c;/vector r) =/summationdisplay
cL/summationdisplay
l=0l/summationdisplay
m=−lηm
l(ρc;/vectorAc)Zm
l(ˆ/vectorAc)A(/vector r−/vectorAc) (28)
withZm
l(ˆ/vectorAc) being the spherical gradient operator in a renormalized fo rm26. The model
charge distribution is expressed as
ρmodel
c(/vector r) =L/summationdisplay
l=0l/summationdisplay
m=−lηm
l(ρc;/vectorAc)δm
l(/vectorAc,/vector r) (29)
and
δm
l(/vectorAc,/vector r) = limα→∞Zm
l(ˆ/vectorAc)Λ(α,/vector r−/vectorAc,0,0,0) (30)
12The integral of the electronic charge distribution and the E wald potential function is
required which gives rise to the field integrals which are defi ned as follows:
Mm
lµ/vector0ν/vector gc=
Zm
l(ˆ/vectorAc)/integraldisplay
φµ(˜α,/vector r−/vectorAµ,˜n,˜l,˜m)φν(α,/vector r−/vectorAν−/vector g,n,l,m )/bracketleftbigg
A(/vector r−/vectorAc)−pen/summationdisplay
/vector n1
|/vector r−/vectorAc−/vector n|/bracketrightbigg
d3r =
Zm
l(ˆ/vectorAc)/integraldisplay/summationdisplay
t,u,vE(˜n,˜l,˜m,n,l,m,t,u,v )Λ(γ,/vector r−/vectorP,t,u,v )/bracketleftbigg
A(/vector r−/vectorAc)−pen/summationdisplay
/vector n1
|/vector r−/vectorAc−/vector n|/bracketrightbigg
d3r (31)
The term/bracketleftbigg
A(/vector r−/vectorAc)−/summationtextpen
/vector n1
|/vector r−/vectorAc−/vector n|/bracketrightbigg
instead ofA(/vector r−/vectorAc) appears because the multipolar
approximation is only done for the charge distribution in th e long range. The penetration
depthpenis a certain threshold for which the integrals are evaluated exactly26,3.
For the gradients, the derivative with respect to all the cen ters is needed. The partial
derivative with respect to Aµ,xis obtained as
∂
∂Aµ,xMm
lµ/vector0ν/vector gc=
Zm
l(ˆ/vectorAc)/integraldisplay/summationdisplay
t,u,vGAµ
x(˜n,˜l,˜m,n,l,m,t,u,v )Λ(γ,/vector r−/vectorP,t,u,v )/bracketleftbigg
A(/vector r−/vectorAc)−pen/summationdisplay
/vector n1
|/vector r−/vectorAc−/vector n|/bracketrightbigg
d3r (32)
In similar way, the partial derivative with respect to cente r/vectorAνis computed. Finally, the
partial derivatives with respect to the centers /vectorAcare obtained from translational invariance.
6. Spheropole
This term arises because the charge distribution is approxi mated by a model charge
distribution in the long range26:
Φew(ρc;/vector r) = Φew(ρmodel
c;/vector r) + Φew(ρc−ρmodel
c;/vector r) = Φew(ρmodel
c;/vector r) + Φcoul(ρc−ρmodel
c;/vector r) +Qc(33)
The calculation of the Coulomb potential Φcoul(ρc−ρmodel
c;/vector r) is restricted to contributions
from those charges inside the penetration depth pen. The use of the Coulomb potential
Φcoul(ρc−ρmodel
c;/vector r) instead of the Ewald potential Φew(ρc−ρmodel
c;/vector r) is correct, if ρc−ρmodel
c
is of zero charge, dipole, quadrupole and spherical second m oment35. However, this condition
13leads to a correction in the three-dimensional case36,35,26: although the difference ρc−ρmodel
c
has zero charge, dipole and quadrupole moment, it has in gene ral a non-zero spherical second
momentQc. Therefore, the potential must be shifted by Qdefined as:
Q=/summationdisplay
cQc=/summationdisplay
c2π
3V/integraldisplay
(ρc(/vector r)−ρmodel
c(/vector r))|/vector r|2d3r (34)
Three types of contributions are obtained26: zero, first and second order HGTFs. They
have to be combined with the corresponding E-coefficient. For the zeroth order, a contri-
bution of
E(˜n,˜l,˜m,n,l,m, 0,0,0)/parenleftbigg
3
2γ+/parenleftBig/vectorAµ−/vectorP/parenrightBig2/parenrightbigg
is computed. The derivative is therefore
∂
∂Aµ,x/parenleftBigg
E(˜n,˜l,˜m,n,l,m, 0,0,0)/parenleftBigg3
2γ+ (/vectorAµ−/vectorP)2/parenrightBigg/parenrightBigg
(35)
To obtain the derivative∂
∂Aµ,xE(˜n,˜l,˜m,n,l,m, 0,0,0), we use the identity
∂
∂Aµ,x
/summationdisplay
t,u,vE(˜n,˜l,˜m,n,l,m,t,u,v )Λ(γ,/vector r−/vectorP,t,u,v )
=
/summationdisplay
t,u,vE(˜n,˜l,˜m,n,l,m,t,u,v )˜α
γΛ(γ,/vector r−/vectorP,t+ 1,u,v) +
/summationdisplay
t,u,vΛ(γ,/vector r−/vectorP,t,u,v )∂
∂Aµ,xE(˜n,˜l,˜m,n,l,m,t,u,v ) =
/summationdisplay
t,u,vGAµ
x(˜n,˜l,˜m,n,l,m,t,u,v )Λ(γ,/vector r−/vectorP,t,u,v ) (36)
which gives
∂
∂Aµ,xE(˜n,˜l,˜m,n,l,m,t,u,v ) =GAµ
x(˜n,˜l,˜m,n,l,m,t,u,v )−˜α
γE(˜n,˜l,˜m,n,l,m,t −1,u,v) (37)
A similar operation is necessary for the components with E(˜n,˜l,˜m,n,l,m, 1,0,0),
E(˜n,˜l,˜m,n,l,m, 0,1,0) andE(˜n,˜l,˜m,n,l,m, 0,0,1) (first order HGTFs) which are mul-
tiplied with prefactors 2( Px−Aµ,x), 2(Py−Aµ,y) and 2(Pz−Aµ,z), respectively. Fi-
nally, derivatives of the products of E(˜n,˜l,˜m,n,l,m, 2,0,0),E(˜n,˜l,˜m,n,l,m, 0,2,0) and
E(˜n,˜l,˜m,n,l,m, 0,0,2) (second order HGTFs) with 2 are required.
14B. Bielectronic integrals
We define a bielectronic integral as
Bµ/vector0ν/vector gτ/vector nσ/vector n +/vectorh=
/integraldisplayφµ(α1,/vector r−/vectorAµ,n1,l1,m1)φν(α2,/vector r−/vectorAν−/vector g,n2,l2,m2)
|/vector r−/vector r′|
φτ(α3,/vector r′−/vectorAτ−/vector n,n3,l3,m3)φσ(α4,/vector r′−/vectorAσ−/vector n−/vectorh,n4,l4,m4)d3r d3r′=
/summationdisplay
t,u,vE(n1,l1,m1,n2,l2,m2,t,u,v )/summationdisplay
t′,u′,v′E(n3,l3,m3,n4,l4,m4,t′,u′,v′)[t,u,v |1
|/vector r−/vector r′||t′,u′,v′] (38)
The expression [ t,u,v |1
|/vector r−/vector r′||t′,u′,v′] is defined as24,25
[t,u,v |1
|/vector r−/vector r′||t′,u′,v′] =
/integraldisplay /integraldisplay
Λ(γ,/vector r−/vectorP,t,u,v )1
|/vector r−/vector r′|Λ(γ′,/vector r′−/vectorP′,t,′u′,v′)d3r d3r′(39)
The partial derivative with respect to Aµ,xis obtained as
∂
∂Aµ,xBµ/vector0ν/vector gτ/vector nσ/vector n +/vectorh=
/summationdisplay
t,u,vGAµ
x(n1,l1,m1,n2,l2,m2,t,u,v )/summationdisplay
t′,u′,v′E(n3,l3,m3,n4,l4,m4,t′,u′,v′)[t,u,v |1
|/vector r−/vector r′||t′,u′,v′] (40)
Similarly, gradients with respect to the other centers are o btained. One of the gradients
can be obtained by translational invariance if the other thr ee gradients have been computed.
In the context of periodic systems, it is necessary to perfor m summations over the lattice
vectors/vector g,/vectorh,/vector n. We define a Coulomb integral as follows
Cµ/vector0ν/vector gτ/vector0σ/vectorh=pen/summationdisplay
/vector nBµ/vector0ν/vector gτ/vector nσ/vector n +/vectorh(41)
Similarly, we define an exchange integral as follows:
Xµ/vector0ν/vector gτ/vector0σ/vectorh=/summationdisplay
/vector nBµ/vector0τ/vector nν/vector gσ/vector n +/vectorh(42)
15V. TOTAL ENERGY AND GRADIENT
A. Total energy
The correct summation of the Coulomb energy is the most sever e problem of the total
energy calculation. The individual contributions to the Co ulomb energy, such as for example
the nuclear-nuclear interaction, are divergent for period ic systems. Thus, a scheme based
on the Ewald method is used to sum the individual contributio ns26. The total energy is
then expressed as the sum of kinetic energy Ekin, the Ewald energies of the nuclear-nuclear
repulsionENN, nuclear-electron attraction Ecoul−nuc, electron-electron repulsion Ecoul−el,
and finally the exchange energy Eexch−el.
Etotal=Ekinetic+ENN+Ecoul−nuc+Ecoul−el+Eexch−el=
=/summationdisplay
/vector g,µ,νPν/vector gµ/vector0Tµ/vector0ν/vector g+ENN
−/summationdisplay
/vector g,µ,νPν/vector gµ/vector0/summationdisplay
aZa/integraldisplay
φµ(˜α,/vector r−/vectorAµ,˜n,˜l,˜m)φν(α,/vector r−/vectorAν−/vector g,n,l,m )A(/vector r−/vectorAa)d3r
+1
2/summationdisplay
/vector g,µ,νPν/vector gµ/vector0/parenleftbigg
−QSµ/vector0ν/vector g+/summationdisplay
/vectorh,τ,σPσ/vectorhτ/vector0Cµ/vector0ν/vector gτ/vector0σ/vectorh−/summationdisplay
cL/summationdisplay
l=0l/summationdisplay
m=−lηm
l(ρc;/vectorAc)Mm
lµ/vector0ν/vector gc/parenrightbigg
−1
2/summationdisplay
/vector g,µ,νP↑
ν/vector gµ/vector0/summationdisplay
/vectorh,τ,σP↑
σ/vectorhτ/vector0Xµ/vector0ν/vector gτ/vector0σ/vectorh−1
2/summationdisplay
/vector g,µ,νP↓
ν/vector gµ/vector0/summationdisplay
/vectorh,τ,σP↓
σ/vectorhτ/vector0Xµ/vector0ν/vector gτ/vector0σ/vectorh(43)
B. Gradient of the total energy
The force with respect to the position of the nuclei can be cal culated similarly to the
molecular case6,5. The derivatives of all the integrals are necessary, and the derivative of the
density matrix is expressed with the help of the energy-weig hted density matrix. The full
force is obtained as:
/vectorFAi=−∂Etotal
∂/vectorAi=
−/summationdisplay
/vector g,µ,νPν/vector gµ/vector0∂Tµ/vector0ν/vector g
∂/vectorAi−∂ENN
∂/vectorAi
16+/summationdisplay
/vector g,µ,νPν/vector gµ/vector0/summationdisplay
aZa∂
∂/vectorAi/bracketleftbigg/integraldisplay
φµ(α2,/vector r−/vectorAµ,n2,l2,m2)φν(α1,/vector r−/vectorAν−/vector g,n1,l1,m1)A(/vector r−/vectorAa)d3r/bracketrightbigg
−1
2/summationdisplay
/vector g,µ,νPν/vector gµ/vector0/braceleftbigg
−Sµ/vector0ν/vector g2π
3V/summationdisplay
c/summationdisplay
/vectorh,σ,τ∈cPσ/vectorhτ/vector0
∂
∂/vectorAi/integraldisplay/bracketleftbigg
−φτ(α2,/vector r−/vectorAτ,n2,l2,m2)φσ(α1,/vector r−/vectorAσ−/vectorh,n1,l1,m1)
+L/summationdisplay
l=0l/summationdisplay
m=−l/integraldisplay
φτ(α2,/vector r′−/vectorAτ,n2,l2,m2)φσ(α1,/vector r′−/vectorAσ−/vectorh,n1,l1,m1)Xm
l(/vector r′−/vectorAc)d3r′δm
l(/vectorAc,/vector r)/bracketrightbigg
r2d3r
+/summationdisplay
τ,σPσ/vectorhτ/vector0∂Cµ/vector0ν/vector gτ/vector0σ/vectorh
∂/vectorAi
+/summationdisplay
cL/summationdisplay
l=0l/summationdisplay
m=−l/summationdisplay
/vectorh,τ∈c,σPσ/vectorhτ/vector0
∂
∂/vectorAi/bracketleftbigg/integraldisplay
φτ(α2,/vector r−/vectorAτ,n2,l2,m2)φσ(α1,/vector r−/vectorAσ−/vectorh,n1,l1,m1)Xm
l(/vector r−/vectorAc)d3rMm
lµ/vector0ν/vector gc/bracketrightbigg/bracerightbigg
+1
2/summationdisplay
/vector g,µ,νP↑
ν/vector gµ/vector0/summationdisplay
/vectorh,τ,σP↑
σ/vectorhτ/vector0∂Xµ/vector0ν/vector gτ/vector0σ/vectorh
∂/vectorAi+1
2/summationdisplay
/vector g,µ,νP↓
ν/vector gµ/vector0/summationdisplay
/vectorh,τ,σP↓
σ/vectorhτ/vector0∂Xµ/vector0ν/vector gτ/vector0σ/vectorh
∂/vectorAi
+/summationdisplay
/vector g,µ,ν∂Sµ/vector0ν/vector g
∂/vectorAi/integraldisplay
BZexp(i/vectork/vector g)/summationdisplay
j{a↑
νj(/vectork)a∗↑
µj(/vectork)(ǫ↑
j(/vectork) +Q)Θ(ǫF−ǫ↑
j(/vectork)−Q)
+a↓
νj(/vectork)a∗↓
µj(/vectork)(ǫ↓
j(/vectork) +Q)Θ(ǫF−ǫ↓
j(/vectork)−Q)}d3k (44)
The last addend is the energy weighted density matrix; the in tegral is over the first
Brillouin zone.
VI. STRUCTURE OF THE GRADIENT CODE
The present structure of the gradient code is indicated in fig ure 1. The first step is to
compute the gradient of the Ewald energy of the nuclei in subr outine GRAMAD (the Ewald
energy is computed in ENEMAD). The control module TOTGRA the n first calls routines to
compute the gradient of the bielectronic integrals (labele d with SHELLX ∇as these routines
will change their structure). The subroutine SHELLX ∇calls subroutines which explicitly
compute the derivatives of Coulomb and exchange integrals, and multiplies the gradients a
first time with the density matrix. Back in TOTGRA again, the s econd multiplication with
the density matrix is performed. The next step is to compute t he derivatives of the multi-
17poles (MONIRG) and to compute the energy weighted density ma trix (PDIGEW). Then,
the gradients of the one-electron integrals are computed (C JAT0G, CJAT1G, CJAT2G or
CJAT3G, depending on the dimension). The field integrals and their gradients are now
multiplied with the multipolar integrals and their gradien ts, and a multiplication with the
density matrix is performed. This concludes the calculatio n of the gradients.
The structure has been simplified to focus on the most importa nt parts. In addition, as
already mentioned, the code will undergo changes during the optimization process so that
a too detailed description seems to be unadvised.
18FIGURES
FIG. 1. The present structure of the gradient code. The left c olumn describes the purpose
of the routines, the middle column gives the names of the corr esponding routines, and the right
column gives the name of the routines in the energy code. One a rrow indicates that the routine is
a subroutine, two arrows indicate that it is a subroutine cal led from a subroutine.
19n uclear/-n uclear repulsiongradien ts
GRAMAD ENEMADgradien t con trol mo duleTOTGRA
TOTENY /+ MONMADgradien t of bielectronic in/-tegrals/, / rst m ultiplica/-tion with densit y matrix
/& SHELLX rSHELLXgradien t of Coulom b andexc hange in tegrals
/& /& VIC/5J r /, VIC/5K r /,VIC/5L r
VIC/5J/, VIC/5K/, VIC/5Lsecond m ultiplication ofgradien ts of bielectronicin tegrals with densit y ma/-trixCalculation of m ultip olesand their gradien ts/, m ulti/-plication with densit y ma/-trix
/& MONIR G MONIRR/+QGAMMAm ultip olar gradien ts /& /& OLIP A G POLIP Aenergy w eigh ted densit ymatrix
/& PDIGEW deriv ed from PDIGgradien ts of one/-electronin tegrals
/& CJA T/0G/, CJA T/1G/,CJA T/2G/, CJA T/3G
CJA T/0/, CJA T/1/,CJA T/2/, CJA T/3m ultiplication of / eld in/-tegrals and their gradien tswith m ultip olar in tegralsand their gradien ts/, m ulti/-plication with densit y ma/-trixprin ting of forces
20VII. EXAMPLES
In tables I, II and III, we give examples of the accuracy of the gradients. First, in table
I, a chain of NiO molecules is considered, with ferromagneti c ordering (all the Ni spins up)
and with antiferromagnetic ordering (nearest Ni spins are a ntiparallel). The oxygen atoms
are moved by 0.01 ˚A from their equilibrium positions which results in a non-va nishing
force. The agreement between numerical and analytical grad ient is better than 0.0001Eh
a0.
As we discussed in our first article22, the agreement can be improved by using stricter
”ITOL”-parameters (these are parameters which control the accuracy of the evaluation of
the integrals3). Indeed, when increasing these parameters, the agreement further improves
up to an error of less than 10−5Eh
a0.
In table II, a LiF layer with a lattice constant of 5 ˚A is considered with one atom being
displaced from its equilibrium position. The forces agree t o 2×10−5Eh
a0when default ITOL
parameters (6, 6, 6, 6, 12) are used.
Finally, in table III, a three-dimensional, ferromagnetic ally polarized NiO solid is con-
sidered. When displacing the oxygen ions, the forces agree t o better than 2 ×10−5Eh
a0.
As a whole, the accuracy is certainly very high and can furthe r be improved by applying
stricter cutoff (ITOL) parameters.
VIII. CONCLUSION
In this article, we described the implementation of analyti cal gradients in the code CRYS-
TAL. In its present form, the code is capable of computing hig hly accurate Hartree-Fock
gradients for systems with 0,1,2 and 3-dimensional periodi city. Both closed-shell restricted
Hartree-Fock as well as unrestricted Hartree-Fock calcula tions can be performed.
A first step of improving the efficiency of the code has been comp leted with the coding of
gradients for the bipolar expansion, and a further enhancem ent of the efficiency will be one
of the future directions. Of highest importance is the imple mentation of symmetry which
21will lead to high saving factors37. Other targets are the implementation of gradients with
respect to the lattice vector, an extension to metallic syst ems38, and the implementation of
density functional gradients.
IX. ACKNOWLEDGMENTS
The author would like to thank CCP3 and Prof. N. M. Harrison fo r their interest and
support of this work (EPSRC grant GR/K90661), Mr. V. R. Saund ers for many helpful
discussions, and Prof. R. Dovesi and the Turin group for help ful discussions and hospitality.
22REFERENCES
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1988).
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Int. J. Quantum Chem. 17, 517 (1980).
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Zicovich-Wilson, crystal 98 User’s Manual, Theoretical Chemistry Group, University
of Torino (1998).
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Molecules to Crystals (in preparation)
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(1958).
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(Plenum, New York, 1977).
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10T. Helgaker and P. Jørgensen, in Methods in Computational Physics , edited by S. Wilson
and G. H. F. Diercksen, 353 (Plenum, New York, 1992).
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2313H. B. Schlegel, Theor. Chim. Acta 103, 294 (2000).
14H. Teramae, T. Yamabe, C. Satoko and A. Imamura, Chem. Phys. L ett.101, 149 (1983);
H. Teramae, T. Yamabe and A. Imamura, J. Chem. Phys. 81, 3564 (1984).
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(1998); S. Hirata and S. Iwata, J. Phys. Chem. A 102, 8426 (1998).
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(1992).
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25TABLES
TABLE I. Ferromagnetic (FM) and antiferromagnetic (AFM) Ni O chain (i.e. a chain with
alternating nickel and oxygen atoms). The distance between two oxygen atoms is chosen as 5 ˚A.
The force is computed numerically and analytically with the oxygen atoms being displaced. A
[5s4p2d] basis set was used for nickel, and a [4 s3p] basis set for oxygen.
magnetic ITOL displacement analytical derivative numeric al derivative
ordering parameter of oxygen (x-component) (x-component)
in˚A Eh/a0 Eh/a0
FM 6 6 6 6 12 0.01 0.001274 0.001188
FM 8 8 8 8 14 0.01 0.001246 0.001249
AFM 6 6 6 6 12 0.01 0.001276 0.001191
AFM 8 8 8 8 14 0.01 0.001250 0.001252
TABLE II. Forces on the atoms of a LiF layer when one of the atom s is displaced from its
equilibrium position. A [4 s3p] basis set was used for the fluorine atom and a [2 s1p] basis set for
the lithium atom. Default ITOL parameters were used.
atom analytical derivative numerical derivative
(x-component) (x-component)
F at (0.5 ˚A, 0˚A, 0˚A) 0.001379 0.001400
Li at (2.5 ˚A, 0˚A, 0˚A) -0.020731 -0.020726
F at (2.5 ˚A, 2.5 ˚A, 0˚A) 0.010384 0.010376
Li at (0 ˚A, 2.5 ˚A, 0˚A) 0.008969 0.008950
26TABLE III. Ferromagnetic NiO in an fcc structure at a lattice constant of 4.2654 ˚A. We com-
pare numerical and analytical derivatives when moving the o xygen ion parallel to the x-direction.
Default ITOL parameters were used, the basis sets are the sam e as in table I.
displacement of oxygen analytical derivative numerical de rivative
(x-component) (x-component)
in˚A Eh/a0 Eh/a0
0.01 0.001499 0.001485
0.02 0.002939 0.002925
0.03 0.004387 0.004378
0.04 0.005857 0.005847
0.05 0.007352 0.007346
27 |
arXiv:physics/0102050v1 [physics.optics] 16 Feb 2001The coherent radiation of the ordered or crystalline electr on beam is inves-
tigated. For the first time it is shown that crystallization c onditions for charged
particles beams differs from those of plasma. Crystallizati on conditions for
charged relativistic bunches are found and shown that this c onditions satisfies
in some existing linacs.The coherence undulator radiation of crystalline beams
in the XUV region are considered.Necessary conditions of co herency are found.
The influence of deviations from mean distance on the radiati on coherency is
considered.
1Coherence Radiation of the Crystalline Beam
L.A.Gevorgian,R.V. Tumanian
13 feb 2001
1 Introduction
The importance of tunable and powerful sources of the cohere nt radiation on
XUV wavelenghts is cause in recent years few projects of FEL [ 1, 2]. This
FEL’s operating in the SASE mode, i.e. starting from noise in the initial
electron beam longitudinal density distribution. This evo lution determines the
undulatory length needed to reach saturation practically v ery long (about
hundred meters). The needed power of laser radiation is poss ible to obtain in
other way. In last two decades very important experimental [ 9, 10]and
theoretical [ ?] results are achieved in investigations of the ordered bunc hes or
crystalline beams.The possibility of such new and interest ing state of matter is
become real in the storage rings of charged particles and as w e show below in
linear accelerators with high density beams. The case of rea l storage ring
lattice was considered in [15]. The crystalline or ordered b eams have of course
many interesting and important properties and application s.Few properties
and one very important application is considered in this pap er. It is clear that
particles of dense bunches are arranged in the certain order s.In these ordered
bunches particles are complete transverse planes,which ar e same spacing in the
longitudinal direction.The conditions and properties of s uch ordering are
considering in the next section of this paper.Such bunches a re radiate coherent
at the wavelengths integer times smaller than inter plane di stances, as shown
in the section 3.This radiation calls super radiant regime o f FEL radiation [ ?]
or coherent spontaneous emission CSE.As shown in above refe rences it is
possible in two cases.First,for short bunches and second fo r modulated
bunches. The super radiant regime because of self bunching o f the beam in the
FEL is considered by Bonifacio [6, 7]. In difference from abov e references
where is considered the coherency of long wavelength radiat ion (the
wavelength much more than mean distance between bunch elect rons), in this
report is considered the coherent radiation of the bunch whe n the radiation
wavelength is about or less than distance between particles . In this case it is
important the discreteness of beam and correlations betwee n beam particles
positions,because of strong Coulomb interaction of the bea m particles.
22 Ordered or Crystalline beams
The requirement of bunch uniformity assumes that electrons of bunch with
density n are replaced on the same mean distance ¯ a=n−1/3from each
other.Such replacement is possible only when each three par ticles are compose
equilateral triangle as result of strong correlation betwe en particles
positions.This is connected with approximately hexagonal structure of the
disordered medium [8] at most probable.Each particle in suc h medium have 14
nearest neighbors,but numerical calculations show that th e mean number of
nearest neighbors is about 15 with mean deviation 10 percent s. For medium
with long distance inter particle interaction such as beam o r bunch the
triangle apices are the equilibrium points of the particles positions.The
particles which are not replaced in his equilibrium points a re oscillate around
his equilibrium point with frequency Ω = Nωp,where ωp=/radicalbig
4πe2n/mc
plasma frequency ,and amplitude equal to deviation from equ ilibrium point.It
is well known that properties of any medium are depend on dime nsion less
parameter Γ,which is the ratio of the depth of the interparti cle potential well
and medium temperature. For neutral and one component plasm a (OCP)
because of Debay screening inter particle potential is abou t pure two particle
Coulomb potential e2/¯a[14].The numerical Molecular Dynamics (MD)
calculations show that for Γ ≥172 OCP is crystallize with body centered cube
(bcc) lattice [13].Detailed MD calculations [11] for finite full charged Coulomb
systems with number of particles about thousand was show tha t in this case
crystallization take place at low values of Γ,but this resul t is not explained
theoretically.We show in this report that for full charged C oulomb systems the
potential well of each particle is about N0(number of particles in the one
transverse plane) times deeper than pure two particle Coulo mb potential.
After expanding of the full force between particles with cha rge e and fixed
distance R moving along longitudinal z direction with veloc ityv=βc
Fz=e2
R2(1−β2)cosθ
(1−β2(sinθ)2)3/2(1)
F⊥=e2
R2(1−β2)sinθ
(1−β2sin2θ)3/2(2)
where θ-angle between R and z directions, around equilibrium point s of the
bcc lattice,one can find that potential wells in the longitud inal and transverse
directions are
U/bardbl=U0N0
4γ2sl, sl=/summationdisplay1
n3(3)
U⊥=U0λl
λtγ
4st, st=/summationdisplay n1
(n2
1+n2
2)2(4)
where U0=e2/λlbare Coulomb potential, N0-number of particles on each
transverse plane.The sum in the slis carrying out over number of transverse
3planes,and the sum in the stover particles of one plane.Here is assumed that
self mean space charge forces of the bunch are compensated by external
restoring forces [11, 12]. Such deepening of the potential w ell in the bunches in
comparison with known potential well in the OCP is mean that c rystallization
or ordering of bunch is possible at comparable higher temper atures than of
OCP.Notice, that transverse ordering is more difficult than l ongitudinal
ordering, because of small tranverse well.So,we can neglec t the transverse
potential and consider the bunch as consisting of transvers e planes with
spacing, but particles on this planes are replaced randomly . Particles of the
bunch in this longitudinal potential well is oscillate with frequency
Ω2=U
2mλ2
where λ-mean interparticle distance in correspondence direction , m is effective
mass of bunch particle equal to mγ3for linacs and mγ3/(1−αγ2) for cyclic
accelerators. The case of real storage ring lattice was cons idered in [15]. If the
crystallization time which equal few oscillation periods lc=c/Ω much less
than period of betatron oscillations of the accelerator the bunch becomes
ordered.The electron bunch with λabout 1 nm, and γ= 104(SLAC)have lcis
about 5m and shorter than betatron oscillation wavelength. The calculations
show that beam of the SLAC in the Final Focus have Γ about few hu ndreds
which is means that bunch may be crystallized.
3 Coherence Radiation of the Bunch
Radiation of the ensemble of N electrons moving along identi cal trajectories
but with arbitrary spatial displacement may be written in th e form [3]
I=iNF (5)
where i-intensity of single electron radiation,N-number o f the electrons in the
bunch,F-factor of coherency of the bunch and may be written i n the follow
form [3]
F=1
N/summationdisplay
ei/vectork /vector rj/summationdisplay
e−i/vectork /vector rj
The position of the j-th electron /vector rjmay be written as a sum of transverse and
longitudinal parts /vector rj=/vectorrjtr+zj. For longitudinal waves the transverse part of
the F is equal to N2
r. This is right because of transverse coherency of the
radiation [3]. For Nzplanes with Nrelectrons on each plane the coherency
factor is may be written in the following form
F=N2
r
N/summationdisplay
eikj˜a/summationdisplay
e−ikj˜a
4where/summationtext-means sum over planes in longitudinal direction. It is not d ifficult
to obtain that
F=N2
rsin2Nzx
sin2x
and when the value x=k˜ais approach to nπ(resonance condition) the
coherency factor is becomes N2. This consideration is true for bunch with step
function for longitudinal density distribution (for homog eneous beam). Now
consider the influence of fluctuations or deviations of the pa rticles positions
from hexagonal model above on the coherence short wave lengt h radiation of
the bunch.Let us assume that particle is on the mean distance , but not in the
z-direction. In this case the particle position is fluctuate d on the value ˜ aθ2/2,
and if xθ2/4≪1 we may neglect this fluctuation.Notice, that in important
practical cases this condition is satisfied. It is not satisfi ed only for very short
radiation when x≫1. The second source of fluctuations from our model is
possible when the particle is replaced on the z-direction bu t not in the
distance a. In this case F multiplied by the factor e−k2b2, where b- the
dispersion of the fluctuation. It is clear that if b/λ≪1, this fluctuations not
disturb the radiation coherency.Executed numerical calcu lations show that
fluctuations about ten times smaller than inter particle dis tance a. This show
that radiation of the wavelengths few times shorter than int erparticle distance
is availably too.Notice,that such full coherent radiation is much powerful than
any other radiation regime in the same conditions.
4 Conclusions
So,in this paper are find the crystallization or ordering con ditions of dense
bunches or beams,which may be satisfied much easier than thos e for OCP.This
very exotic and interesting station of the matter can be obta ined at existing
linac beams.This possibility is very important for attaini ng crystalline state of
the beams as well as for many other applications of new state o f matter.We
show that ordered bunches can radiate coherently ,i.e. much powerful than
spontaneous one or SASE. In dependence of beam energy and den sity this
radiation may have wavelength at XUV region also.
References
[1] Pellegrini,C. et. al., Nucl.Inst. and Methods, 341(199 4)326
[2] J. Rossbach,Nucl.Instr.and Methods,A 393(1997)86
[3] Korkhmazian,N.A. et.al.,Zh.Tekh.Fiz.47(1977)1583,
Gevorgian,L.A.,Zhevago,N.K.,Sov.Phys.Dokl.,27(1982) 946, in Russian
[4] Y.Pinhasi,A.Gover,Nucl.Instr. and Methods, A393(199 7)343
5[5] A.Gover et. al.,Phys.Rev.Lett.,72(1994)1192
[6] R.Bonifacio,B.W.J.McNeil,P.Pierini,Phys.Rev.A40( 1989)4467
[7] R.Bonifacio,C.Maroli,N.Piovella,Optics Comm.,68(1 988)369
[8] J.M.Ziman,’Models of disorder’,Cambridge University Press,1979
[9] E.N.Dementiev,N.S.Dikansky,A.S.Medvedko,V.V.Park homchuk, and
D.V.Pestrikov,Zh.Tekh.Fiz.50(1980)1717[Sov.Phys.Tec h.Phys.25(1980)1001].
[10] A.V.Aleksandrov et al.,Europhys.lett.18(1992)151; M.Steck et al.,Electron
Cooling at ESR,Workshop on Crystalline Beams and Related
Issues,Erice,Sicily,11-21 nov.1995,eds. D.M. Maletic an d
A.V.Ruggiero(World Scientific).
[11] J.P.Schiffer and P.Kienle,Z.Phys.A321(1985)181; A.R ahman and
J.P.Schiffer,Phys.Rev.Lett.57(1986)1133;
J.P.Schiffer,Phys.Rev.Lett.61(1988)1843
[12] A.G.Ruggiero,Proceedings of the PAC93,p.3530
[13] W.L.Slattery,G.D.Dooley,andH.E.deWitt,Phys.rev. A21(1980)2087
[14] S.Ichimaru,H.Iyetomi,and S.Tanaka,Phys.Rep.149(1 987)93
[15] X.-P.Li,A.M.Sessler,and J.Wei,in Proceedings of the EPAC’04,p.1379.
J.Wei,H.Okamoto,and A.M.Sessler,Phys.Rev.Lett.,80(19 98)2606.
6 |
arXiv:physics/0102051v1 [physics.soc-ph] 16 Feb 2001String Theory: An Evaluation
Peter Woit
Department of Mathematics, Columbia University
woit@math.columbia.edu
February 2, 2008
For nearly seventeen years now most speculative and mathema tical work
in particle theory has centered around the idea of replacing quantum field
theory with something that used to be known as “Superstring T heory”, but
now goes under the name “M-Theory”. We’ve been told that “str ing theory
is a part of twenty-first-century physics that fell by chance into the twentieth
century”, so this year the time has perhaps finally come to beg in to evaluate
the success or failure of this new way of thinking about parti cle physics. This
article will attempt to do so from the perspective of a quantu m field theorist
now working in the mathematical community.
The theory has been spectacularly successful on one front, t hat of public
relations. Best-selling books and web sites are devoted to e xplaining the
subject to the widest possible audience. The NSF is funding a series of NOVA
programs on string theory, and the ITP at Santa Barbara is org anizing a
conference to train high school teachers in string theory so that they can teach
it to their students. The newspaper of record informs us that “Physicists
Finally Find a Way to Test Superstring Theory” (NYT 4/4/00).
The strongest scientific argument in favor of string theory i s that it ap-
pears to contain a theory of gravity embedded within it. It is not often
mentioned that this is not yet a consistent quantum theory of gravity. All
that exists at the moment is a divergent series that is conjec tured to be an
asymptotic perturbation series for some as yet undefined non -perturbative
string theory (the terms in the series are conjectured to be fi nite, unlike the
situation in the standard quantization of general relativi ty). String theorists
actually consider the divergence of this series to be a virtu e, since otherwise
they would have an infinity (one for each compactification of s ix dimensions)
1of consistent theories of gravity on their hands, with no pri nciple for choosing
amongst them.
String theory has lead to many striking new mathematical res ults. The
concept of “mirror symmetry” has been very fruitful in algeb raic geometry,
and conformal field theory has opened up a new, fascinating an d very deep
area of mathematics. Unfortunately the mathematically int eresting parts of
string theory have been pretty much orthogonal to those part s that attempt
to connect with the real world.
The experimental situation is best described with Pauli’s p hrase “it’s
not even wrong”. No one has managed to extract any sort of expe rimental
prediction out of the theory other than that the cosmologica l constant should
probably be at least 55 orders of magnitude larger than exper imental bounds.
String theory not only makes no predictions about physical p henomena at
experimentally accessible energies, it makes no predictio ns whatsoever. Even
if someone were to figure out tomorrow how to build an accelera tor capable
of reaching Planck-scale energies, string theorists would be able to do no
better than give qualitative guesses about what such a machi ne might see.
This situation leads one to question whether string theory r eally is a scientific
theory at all. At the moment it’s a theory that cannot be falsi fied by any
conceivable experimental result. It’s not even clear that t here is any possible
theoretical development that would falsify the theory.
String theorists often attempt to make an aesthetic argumen t, a claim
that the theory is strikingly “elegant” or “beautiful”. Sin ce there is no well-
defined theory, it’s hard to know what to make of these claims, and one is
reminded of another quote from Pauli. Annoyed by Heisenberg ’s claims that
modulo some details he had a wonderful unified theory (he didn ’t), Pauli sent
his friends a postcard containing a blank rectangle and the t ext “This is to
show the world I can paint like Titian. Only technical detail s are missing.”
Since no one knows what “M-theory” is, its beauty is that of Pa uli’s painting.
Even if a consistent M-theory can be found, it may very well be a theory of
great complexity and ugliness.
From a mathematician’s point of view, the idea that M-theory will replace
the Standard Model with something aesthetically more impre ssive is rather
suspicious. Two of the most important concepts of the Standa rd Model
are that of a gauge field and that of the Dirac operator. Gauge fi elds are
identical with connections, perhaps the most important obj ects in the modern
formulation of geometry. Thinking seriously about the infin ite dimensional
space of all connections has been a very fruitful idea that ma thematicians
2have picked up from physicists. The importance of the Dirac o perator is well
known to physicists, what is less well known is that it is of si milar importance
in mathematics where it plays the role of “fundamental class ” in K-theory.
This is reflected in the central role the Dirac operator plays in the Atiyah-
Singer index theorem, one of the great achievements of twent ieth century
mathematics.
To the extent that the conceptual structure of string theory is understood,
the Dirac operator and gauge fields are not fundamental, but a re artifacts
of the low energy limit. The Standard Model is dramatically m ore “elegant”
and “beautiful” than string theory in that its crucial conce pts are among
the deepest and most powerful in modern mathematics. String theorists are
asking mathematicians to believe in the existence of some wo nderful new
mathematics completely unknown to them involving concepts deeper than
that of a connection or a Dirac operator. This may be the case, and one
must take this argument seriously when it is made by a Fields m edalist, but
without experimental evidence or a serious proposal for wha t M-theory is,
the argument is unconvincing.
Given the lack of experimental or aesthetic motivation, why do so many
particle theorists work on string theory? Sheldon Glashow d escribes string
theory as “the only game in town”, but this begs the question. Why is it the
only game in town?
During much of the twentieth century there were times when th eoreti-
cal particle physics was conducted quite successfully in a s omewhat faddish
manner; there was often only one game in town. Experimentali sts regu-
larly discovered new unexpected phenomena, each time leadi ng to a flurry
of theoretical activity and sometimes to Nobel prizes for th ose quickest to
correctly understand the significance of the new data. Since the discovery
of the J/Ψ in November 1974, there have been no solid experimental res ults
that disagree with the Standard Model (except perhaps recen t indications of
neutrino masses). It is likely that this situation will cont inue at least until
2006 when experiments at the LHC at CERN are scheduled to begi n. To
a large extent particle theory research has continued to be c onducted in a
faddish manner for the past quarter century, but now with lit tle success.
Graduate students, post-docs and untenured junior faculty interested in
physics beyond the Standard Model are under tremendous pres sures in a bru-
tal job market to work on the latest fad in string theory, espe cially if they are
interested in speculative and mathematical research. For t hem, the idea of
starting to work on an untested new idea that may very well fai l looks a lot
3like a quick route to professional suicide. Many physics res earchers do not
believe in string theory but work on it anyway. They are often intimidated
intellectually by the fact that some leading string theoris ts are undeniably
geniuses, and professionally by the desire to have a job, get grants, go to
conferences and generally have an intellectual community i n which to partic-
ipate.
What can be done? Even granting that string theory is an idea t hat
deserves to be pursued, how can theorists be encouraged to tr y and find
more promising alternatives? Here are some modest proposal s, aimed at
encouraging researchers to strike out in new directions:
1. Until such time as a testable prediction (or even a consist ent compelling
definition) emerges from string theory, theorists should pu blicly acknowledge
the problems theoretical particle physics is facing, and sh ould cease and
desist from activities designed to sell string theory to imp ressionable youths,
popular science reporters and funding agencies.
2. Senior theorists doing string theory should seriously re evaluate their
research programs, consider working on less popular ideas a nd encourage
their graduate students and post-docs to do the same.
3. Instead of trying to hire post-docs and junior faculty wor king on
the latest string theory fad, theory groups should try and id entify young
researchers who are working on original ideas and hire them t o long enough
term positions that they have a chance of making some progres s.
4. Funding agencies should stop supporting theorists who pr opose to
continue working on the same ideas as everyone. They should a lso question
whether it is a good idea to fund a large number of conferences and work-
shops on the latest string theory fad. Research funds should be targeted at
providing incentives for people to try something new and amb itious, even if
it may take many years of work with a sizable risk of ending up w ith nothing.
Particle theorists should be exploring a wide range of alter natives to string
theory, and looking for inspiration wherever it can potenti ally be found. The
common centrality of gauge fields and the Dirac operator in th e Standard
Model and in mathematics is perhaps a clue that any fundament al physi-
cal model should directly incorporate them. Another powerf ul and unifying
idea shared by physics and mathematics is that of a group repr esentation.
Some of the most beautiful mathematics to emerge from string theory in-
volves the study of (projective) representations of the gro up of conformal
transformations and of one-dimensional gauge groups (“loo p groups”). This
work is essentially identical with the study of two dimensio nal quantum field
4theory. The analogous questions in four dimensions are terr a incognita, and
one of many potentially promising areas particle theorists could look to for
inspiration.
During the 1960’s and early 1970’s, quantum field theory appe ared to be
doomed and string theory played a leading role as a theory of t he strong
interactions. Could it be that just as string theory was wron g then, it is
wrong now, and in much the same way: perhaps the correct quant um theory
of gravity is some form of asymptotically free gauge theory? As long as the
best young minds of the field are encouraged to ignore quantum field theory
and pursue the so far fruitless search for M-theory, we may ne ver know.
5 |
arXiv:physics/0102052v1 [physics.ed-ph] 16 Feb 2001A simple solvable model of body motion in a
one-dimensional resistive medium
M. I. Molina∗
Facultad de Ciencias, Departamento de F´ ısica, Universida d de Chile
Casilla 653, Las Palmeras 3425, Santiago, Chile.
Abstract
We introduce and solve in closed form a simple model of a macro scopic
body propagating in a one-dimensional resistive medium at t emperature T.
The assumption of completely inelastic collisions between the body and the
particles composing the medium leads to a resistive force th at is opposite
and proportional to the square of the body’s velocity.
Key words: air drag, collisions
PACS: 45.20.Dd , 45.50.Tn
∗mmolina@abello.dic.uchile.cl
1The topic of macroscopic bodies moving through resistive me dia, such as
air or viscous fluids, gives rise to one aspect that students o f Introductory
Physics courses find rather mysterious: The origin of the ‘fo rce law’ that
describes the effective force on the moving body as it propaga tes through
the resistive medium. The student is usually told that the eff ective force on
the body is either proportional to the speed of the body or to t he square of
the body’s speed, according to whether the body has, or does n ot have, a
small cross–sectional area, or whether it is moving at low or high speeds1.
Under further questioning, the instructor might tell the st udent that these
‘laws’ are based on ‘experimental observations’ which are d ifficult to obtain
analytically. There are basic models, however, that show in a simple manner
how the energy and momentum exchange between the moving body and the
particles composing the resistive medium lead to some of the se ‘force laws’.
In this article, we present an extremely simplified model tha t leads to a very
well–known ‘force law’: a resistive force that is opposite a nd proportional to
the square of the body’s velocity: F=−γ V2.
Consider a body, represented by a heavy point ‘particle’ of m assM0and
initial speed V0, that propagates inside a one–dimensional medium composed
of identical point particles of mass m, with m≪M0which are in thermal
equilibrium at temperature T(Fig.1). We will consider here the ‘short’ time
scale where the body does not have enough time to reach therma l equilibrium
with the surrounding medium. The ‘brownian motion’ case whe re the body
is in thermal equilibrium with the medium have been nicely di scussed by de
Grooth3. Let us denote by vj, the velocity of the jth medium particle. Since
2the medium is one-dimensional the particles can be labelled unambiguously.
For instance, the particles to the right of the body could be l abelled by odd
values of j, while the ones to the left, by even jvalues. Because of thermal
equilibrium the {vj}are random quantities whose values are taken from a
gaussian distribution of width proportional to the medium t emperature T.
We will assume for simplicity that the body undergoes comple tely inelastic
collisions with the medium particles.
After the first collision we have, because of momentum conser vation,
M0V0+mv1= (M0+m)V1
i.e., the speed of the body after its first collision is
V1=/parenleftbiggM0
M0+m/parenrightbigg
V0+/parenleftbiggm
M0+m/parenrightbigg
v1,
where v1denotes the velocity of the medium particle with which the bo dy
collides first(this particle could come from the left or right of M). Some time
afterwards, the body (now with mass M0+m) will suffer a second inelastic
collision from which will emerge with velocity:
V2=/parenleftbiggM0
M0+ 2m/parenrightbigg
V0+/parenleftbiggm
M0+ 2m/parenrightbigg
(v1+v2).
where v2is velocity of the medium particle who suffers the second collision
withM0, and so on. After nof these collisions, the speed of the body will be
Vn=/parenleftbiggM0
M0+nm/parenrightbigg
V0+/parenleftbiggm
M0+nm/parenrightbiggn/summationdisplay
j=1vj,
where we remind the reader that the {vj}are random with /an}bracketle{tvj/an}bracketri}ht= 0,/an}bracketle{tv2
j/an}bracketri}ht=
kT/m and/an}bracketle{t.../an}bracketri}htdenotes a thermal average. This implies,
/an}bracketle{tVn/an}bracketri}ht=/parenleftbiggM0
M0+nm/parenrightbigg
V0. (1)
3On the other hand,
/an}bracketle{tV2
n/an}bracketri}ht=M2
0V2
0
(M0+nm)2+2mM 0V0
(M0+nm)2/angbracketleftBigg/summationdisplay
ivi/angbracketrightBigg
+m2
(M0+nm)2/angbracketleftBigg/summationdisplay
i,jvivj/angbracketrightBigg
=/an}bracketle{tVn/an}bracketri}ht2+nmkT
(M0+nm)2
=/parenleftBigg
1−kT
M0V2
0/parenrightBigg
/an}bracketle{tVn/an}bracketri}ht2+kT
M0V0/an}bracketle{tVn/an}bracketri}ht. (2)
We note that, as the number of collisions tends to infinity (i. e., after a ‘long’
time), /an}bracketle{tV2
n/an}bracketri}ht →(kT/M 0V0)/an}bracketle{tVn/an}bracketri}ht= (kT/M (n)), where M(n) =M0+nmis the
effective body mass after ncollisions. This is nothing else but equipartition:
M(n)/an}bracketle{tV2
n/an}bracketri}ht →kT=m/an}bracketle{tv2/an}bracketri}ht, where vis the velocity of a medium particle.
If we now assume that ρ, the density of medium particles per unit length
is constant, then we can express nasn=ρ xwhere xis the distance travelled
by the body between its first and n-thcollision. We are assuming here, as
in hydrodynamics, that an element of length ∆ xwhile ‘small’ will contain
a great number of medium particles. By re-expressing nin terms of xin
(1), we can express the average velocity of the body after it h as travelled a
distance xas
/an}bracketle{tV(x)/an}bracketri}ht=/parenleftBiggM0
M0+ρ m x/parenrightBigg
V0, (3)
and the average of the velocity squared as
/an}bracketle{tV(x)2/an}bracketri}ht=/parenleftBigg
1−kT
M0V2
0/parenrightBigg
/an}bracketle{tV(x)/an}bracketri}ht2+kT
M0V0/an}bracketle{tV(x)/an}bracketri}ht. (4)
The average velocity decreases monotonically with distanc e. Its explicit time
dependence can be found from (3): dx/dt =M0V0/(M0+ρmx), which can
be integrated to give
X(t)
X0=−1 +/radicalBig
1 + 2( t/t0) (5)
4where X0≡M0/(ρm) and t0≡X0/V0constitute natural length and time
scales. Finally, after replacing (5) into (3), or by direct d ifferentiation of (5),
one obtains
/an}bracketle{tV(t)/an}bracketri}ht
V0=1/radicalBig
1 + 2 ( t/t0)(6)
and
M(t)
M0= 1 +/parenleftbiggρm
M0/parenrightbigg
X(t) =/radicalBig
1 + (2 t/t0) (7)
is the effective body mass as a function of time. In Fig.2 we sho wM(t),X(t)
andV(t), all of which diverge at long times.
Average resistive force . As the body propagates, it is being hit from front and
back by medium particles which stick completely to it after c olliding. This
accretion process is rather akin to the opposite process that occurs in the
propulsion of a rocket engine: instead of expelling matter o ur body absorbs
it. One process is the time-reversal of the other. The averag e effective force
on the body can be directly computed from /an}bracketle{tF/an}bracketri}ht=M(t)d/an}bracketle{tV(t)/an}bracketri}ht/dt. From
Eqs.(6) and (7) one obtains:
/an}bracketle{tF/an}bracketri}ht=−M0V0
t01
1 + 2( t/t0)(8)
which can be recast as
/an}bracketle{tF/an}bracketri}ht=−γ V(t)2(9)
withγ≡ρm.
Another way to compute /an}bracketle{tF/an}bracketri}htis to start from conservation of momentum
during an infinitesimal collision, M(x)V(x)+dM(x)v= (M(x)+dM(x))(V(x)+
dV(x)). This implies that the instantaneous force on the body is
MdV
dt=−/parenleftBiggdM
dt/parenrightBigg
(V−v) (10)
5where vis random. The average force on the moving body will then be
/an}bracketle{tF/an}bracketri}ht=−/parenleftBiggdM
dt/parenrightBigg
/an}bracketle{tV/an}bracketri}ht=−/parenleftBiggdM
dx/parenrightBigg
/an}bracketle{tV/an}bracketri}ht2. (11)
Since M(x) =M0+ρmx, we now have /an}bracketle{tF/an}bracketri}ht=−γ/an}bracketle{tV/an}bracketri}ht2, with γ≡ρmas
before.
Stopping power . The stopping power /an}bracketle{tS/an}bracketri}htof a medium is defined by the
average energy per unit length, lost by a projectile while tr aversing a resistive
medium:
/an}bracketle{tS/an}bracketri}ht=/angbracketleftBiggdE
dx/angbracketrightBigg
=d
dx/braceleftbigg1
2M(x)/an}bracketle{tV(x)2/an}bracketri}ht/bracerightbigg
. (12)
From Eqs.(4), (3) and the relations ( d/dx)/an}bracketle{tV(x)/an}bracketri}ht=−(ρm/M 0V0)/an}bracketle{tV(x)/an}bracketri}ht2
andM(x) =M0V0//an}bracketle{tV(x)/an}bracketri}ht, we have
/an}bracketle{tS/an}bracketri}ht=−ρm
2/parenleftBigg
1−kT
M0V2
0/parenrightBigg
/an}bracketle{tV(x)/an}bracketri}ht2
=−ρm
2/parenleftBigg
1−kT
M0V2
0/parenrightBigg/parenleftBiggM0
M0+ρmx/parenrightBigg2
V2
0
=−ρm
2M0(2E−kT)2
(M0V2
0−kT). (13)
Figure 2 shows /an}bracketle{tS/an}bracketri}htas a function of distance traversed inside the medium for
several temperatures. Note that, since we are assuming the b ody’s initial
kinetic energy to be higher than the average thermal energy, the body will
always lose energy to the medium, on average. This energy los s becomes
smaller and smaller as the body traverses the medium. Only af ter an infinite
amount of time, or distance travelled, will the body’s avera ge energy loss
reach zero, where a thermalization process will occur.
In summary, we have introduced and solved in closed form the d ynamics of
a simple model of a body moving through a resistive medium. We find that
6the effective resistive force is opposite and proportional t o the square of the
body’s speed2.
7References
1Raymond A. Serway, Physics for Scientists and Engineers with modern
physics , 2nd. ed. (Saunders College Publishing, Philadelphia, 198 6), pp.
115–118.
2A related model, where the collisions between the body and th e medium
particles is assumed to be completely elastic, leads to the s ame force law
(but a different γ) and is reported in: M. I. Molina, “Body Motion in a One-
Dimensional Resistive Medium”, M.I. Molina, Am. J. of Phys. 66, 973–975
(1998).
3Bart G. de Grooth, “A Simple model for Brownian motion leadin g to the
Langevin equation”, Am. J. Phys. 67, pp. 1248–1252.
8Figure Captions
FIG 1: Macroscopic body of mass M0propagating inside a one-dimensional
resistive medium composed by identical particles of mass m << M 0in ther-
mal equilibrium at temperature T.
FIG 2: Effective body mass, average velocity and distance tra velled as
a function of time, for body moving through our resistive med ium (S0≡
ρmV2
0/2).
FIG 3: Stopping power of the one-dimensional resistive medi um as a function
of the distance traversed by the body, for several medium tem peratures.
9Mmm m mmV0
Fig.1t/t00.00.51.01.52.00.00.51.01.52.0
V(t)/V0X(t)/X0M(t)/Mx/x00.00.40.81.21.62.0<S>/S0
-1.0-0.8-0.6-0.4-0.20.0
τ = 0τ = 0.5τ = 0.8τ = 1
FIG. 3 |
arXiv:physics/0102053v1 [physics.ed-ph] 16 Feb 2001The Attractive Nonlinear Delta-Function
Potential
M. I. Molina and C. A. Bustamante
Facultad de Ciencias, Departamento de F´ ısica, Universida d de Chile
Casilla 653, Las Palmeras 3425, Santiago, Chile.
1Abstract
We solve the continuous one-dimensional Schr¨ odinger equa tion for the case
of an inverted nonlinear delta–function potential located at the origin, ob-
taining the bound state in closed form as a function of the non linear expo-
nent. The bound state probability profile decays exponentia lly away from
the origin, with a profile width that increases monotonicall y with the non-
linear exponent, becoming an almost completely extended st ate when this
approaches two. At an exponent value of two, the bound state s uffers a
discontinuous change to a delta–like profile. Further incre ase of the expo-
nent increases again the width of the probability profile, al though the bound
state is proven to be stable only for exponents below two. The transmission
of plane waves across the nonlinear delta potential increas es monotonically
with the nonlinearity exponent and is insensitive to the sig n of its opacity.
2The delta-function potential δ(x−x0) has become a familiar sight in the
landscape of most elementary courses on quantum mechanics, where it serves
to illustrate the basic techniques in simple form. As a physi cal model, it has
been used to represent a localized potential whose energy sc ale is greater than
any other in the problem at hand and whose spatial extension i s smaller
than other relevant length scales of the problem. Arrays of d elta-function
potentials have been used to illustrate Bloch’s theorem in s olid state physics
and also in optics, where in the scalar approximation, wave p ropagation in a
periodic medium resembles the dynamics of an electron in a cr ystal lattice. It
is well known that the single “inverted” delta-function pot ential −Ωδ(x−x0)
possesses one exponentially localized bound state for all v alues of the opacity
parameter Ω. Its existence and stability has been tested aga inst the effects
of different boundary conditions1and symmetry-breaking perturbations2. In
addition, the inverted delta potential has been used as a sem i-permeable
barrier to examine resonance phenomena in scattering theor y3, among others.
Other interesting applications of the delta-function pote ntial concept are
found in Ref.4.
In this work we examine the problem of finding the bound state a nd the
transmission coefficient of plane waves across an inverted nonlinear delta–
function potential, described by the so-called Nonlinear S chr¨ odinger (NLS)
equation:
−¯h2
2mφ′′(x)−¯h2
2mΩδ(x)|φ(x)|αφ(x) =E φ(x), (1)
where Ω>0 is the opacity coefficient and αis the nonlinearity exponent.
3Forα= 0 we recover the familiar problem of the linear “inverted” d elta-
potential which possesses an exponentially decaying bound state profile for
any opacity strength: φ(x) =/radicalBig
Ω/2 exp( −(Ω/2)|x|). The rate of decay in
space is determined by the localization length 2 /Ω which increases (decreases)
as the opacity decreases (increases).
It might seem odd at first to see a nonlinear–looking Schr¨ odinger equation like
(1), since we know that quantum mechanics is linear. Therefo re, all physical
systems should be described by coupled sets of linear equati ons. However
we oftentimes can only concentrate on a few “relevant” degre es of freedom,
making suitable approximations to deal with the rest. At tim es, the price to
pay for this reduction is the appearance of nonlinear evolut ion equations for
the variables of interest, such as Eq. (1).
For instance, in atomic physics, a well-known approximatio n when dealing
with multi-electron atoms is the self-consistent field appr oximation (Hartree-
Fock). In this case each electron is described by a single-pa rticle wave func-
tion that solves a Schr¨ odinger-like equation. The potenti al appearing in this
equation is that generated by the average motion of all the ot her electrons,
and so depends on their single-particle wave functions. Thi s results in a set of
nonlinear eigenvalue equations5. A more recent application of the mean-field
ideas to a weakly interacting Bose condensate can be found in Ref.[6].
Forα= 2, Eq.(1) could model the problem of an electron propagatin g in a
one–dimensional linear medium which contains a vibrationa l “impurity” at
the origin that can couple strongly to the electron. In the ap proximation,
where one considers the vibrations completely “enslaved” t o the electron, one
4obtains Eq.(1) as the effective equation for the electron.
A closely related equation, given by the discrete version of (1) is known as
the discrete nonlinear Schr¨ odinger (DNLS) equation. It wa s introduced in
its time–independent form, in the late fifties by Holstein in his studies of
the polaron problem7in condensed matter physics. The DNLS equation
was derived in a fully time–dependent form by Davydov in his s tudies of
energy transfer in proteins and other biological materials8. In the continuum
limit the time-dependent DNLS equation reduces to the time- dependent NLS
equation, which supports soliton solutions. Therefore, a s oliton–based energy
transport appears as an attractive candidate mechanism for energy transport
in biomolecules. A recent review of the status of Davydov’s p roposal can be
found in Ref.[9]. The time-dependent DNLS equation can also be viewed
as the evolution equation for a Hamiltonian system of classi cal anharmonic
oscillators10.
An important application of the continuous model (1) is that of a wave
propagating in a one–dimensional linear medium which conta ins a narrow
strip of nonlinear (general Kerr-type) material11. This nonlinear strip is as-
sumed to be much smaller than the typical wavelength. In fact , periodic and
quasiperiodic arrays of nonlinear strips have been conside red by a straight-
forward generalization of Eq.(1) in order to model wave prop agation in some
nonlinear superlattices12.
Bound State ( E=E(b)<0):Our system consists of a single, infinitely
localized potential well in a continuous infinite line and th erefore, lacks any
natural length scale. If the delta potential were confined be tween two infinite
5walls, the distance between the walls would provide a length scale. If, instead
of a continuous line, the potential were defined on a discrete lattice, its lattice
constant would define a natural length scale. Also, if instea d of one delta
potential, we had at least two of them, their mutual distance would constitute
a natural length scale for the system.
In our case we have none of these. Thus, Ω serves only to define t he unit of
distance (as it does in the linear case). It is possible to get rid of Ω formally
as follows: From Eq.(1) we see that Ω must have units of [dista nce](α/2)−1,
which suggests the definition of a dimensionless distance uasu=x/L, with
L≡Ω2/(α−2). In terms of uandφ(u)≡(1/√
L)φ(x), Eq.(1) can be recast
in a dimensionless form:
φ′′(u)−k2φ(u) =−δ(u)|φ(u)|αφ(u), (2)
withk2=−2mL2E(b)/¯h2andφ′′(u) = (d2/du2)φ(u). The opacity Ω has now
disappeared from view since it only determines the unit of di stance. We try:
φ(u) =/braceleftBigg
Aexp(k u)u<0
Bexp(−k u)u>0(3)
Using the continuity of φ(u) and the discontinuity of φ′(u) atu= 0, one ob-
tainsA=Bandk= (1/2)|A|α. Finally, use of the normalization condition
1 =/integraltext∞
−∞|φ(u)|2du, leads to
φ(u) =/parenleftbigg1
2/parenrightbigg1/(2−α)
exp/bracketleftBigg
−/parenleftbigg1
2/parenrightbigg2/(2−α)
|u|/bracketrightBigg
(4)
with a dimensionless bound state energy
E(b)=−/parenleftbigg1
2/parenrightbigg4/(2−α)
. (5)
6As in the linear ( α= 0) case, the bound state profile is exponentially de-
creasing away from the delta potential with localization le ngth 22/(2−α). As
αincreases from zero, the probability profile widens and the b ound state
energy decreases in magnitude. At α= 2−, the state is completely extended
all over the real axis and the bound state energy is vanishing ly small. At
α= 2+, the bound state becomes infinitely localized, with a delta– like prob-
ability profile and with an infinite bound state energy. Furth er increase in
the nonlinear exponent leads to a widening of the probabilit y profile and to
a corresponding reduction in the magnitude of the bound stat e energy.
Figure 1 shows the amplitude, or the inverse square probabil ity profile width,
of the bound state as a function of the nonlinearity exponent . However, at
this point, an important observation is in order. The total energy of the
system does not coincide with the bound state energy. In orde r to see this, we
must consider the full time-dependent nonlinear Schr¨ odin ger equation that
gives rise to Eq. (2). By using τ≡t/Tas a dimensionless time variable,
withT≡(¯h/2mL2)−1, we have
id ψ(u,τ)
d τ=−d2ψ(u,τ)
d τ2−δ(u)|ψ(u,τ)|αψ(u,τ). (6)
In other words, we have i(d/dτ)ψ(u,τ) =H ψ(u,τ), where the Hamiltonian
operator can be decomposed as H=H0+ONL, whereH0=p2, withp=
i(d/du) andONL=−δ(u)|ψ(u,τ)|αas the nonlinear part. We see that H
depends on time explicitly, through the time dependence of ONL:
∂H
∂τ=∂ONL
∂τ(7)
7This implies that /angbracketleftH/angbracketrightis no longer a constant of the motion:
d/angbracketleftH/angbracketright
dτ=i/angbracketleft[H,H]/angbracketright+/angbracketleftBigg∂H
∂τ/angbracketrightBigg
=/angbracketleftBigg∂H
∂τ/angbracketrightBigg
/negationslash= 0. (8)
For the nonlinear part, we have
d/angbracketleftONL/angbracketright
dτ=i/angbracketleft[H,O NL]/angbracketright+/angbracketleftBigg∂ONL
∂τ/angbracketrightBigg
. (9)
But,/angbracketleftBigg∂ONL
∂τ/angbracketrightBigg
=/integraldisplay
du|ψ(u,τ)|2(∂ONL/∂τ). (10)
By expressing ONLin terms of ψ(u,τ) and using Eq. (6), we can recast (10)
as/angbracketleftBigg∂ONL
∂t/angbracketrightBigg
=iα
2/angbracketleft[H,O NL]/angbracketright, (11)
which means
d/angbracketleftONL/angbracketright
dτ=i/parenleftbigg
1 +α
2/parenrightbigg
/angbracketleft[H,O NL]/angbracketright. (12)
By comparing Eqs. (11) and (12), we conclude
/angbracketleftBigg∂ONL
∂τ/angbracketrightBigg
=/parenleftbiggα
α+ 2/parenrightbiggd
dτ/angbracketleftONL/angbracketright. (13)
Finally, by inserting this back into Eq.(7), Eq.(8) becomes :
d
dτ/angbracketleftH/angbracketright=/parenleftbiggα
α+ 2/parenrightbiggd
dτ/angbracketleftONL/angbracketright, (14)
which implies
d
dτ/angbracketleftbigg
H−/parenleftbiggα
α+ 2/parenrightbigg
ONL/angbracketrightbigg
= 0. (15)
Therefore, the true energy operator for our system is Ht≡H−(α/α+2)ONL.
For a stationary-state, ψ(u,τ) =φ(u) exp( −iE(b)t/¯h), the total dimensionless
8energy is then
Et=E(b)−/parenleftbiggα
α+ 2/parenrightbigg
(−|φ(0)|α+2) =−/parenleftbigg1
2/parenrightbigg4/(2−α)/parenleftbigg2−α
2 +α/parenrightbigg
. (16)
Thus, forα <2 the total energy is negative and the eigenstate is a stable
localized state. On the contrary, when α>2, the total energy is positive and
the eigenstate is localized but possibly unstable, which me ans that any weak
‘perturbation’ could make it disappear into the continuum. This explains the
‘stable’ and ‘unstable’ labelling in Fig.1. Only for α= 0, i.e., the linear case,
both the total energy and the energy eigenvalue coincide. Fi gure 2 shows
some probability profiles for several different values of the nonlinear exponent
that give rise to true (stable) bound states. This distincti on between the
eigenenergy and the total energy must always be kept in mind w hen dealing
with effectively nonlinear systems.
Transmission of plane waves ( E >0):We now cast Eq.(1) as
ψ′′(x) +k2ψ(x) =−Ωδ(x)|ψ(x)|αψ(x) (17)
wherek2= 2mE/¯h2is the electron wavevector. Unlike the bound state
problem, we now have 1 /kas a natural length scale and can therefore con-
sider Ω as a bona fide opacity coefficient. The problem looks similar to the
usual single delta-barrier problem, with the exception of t he nonlinear term
|ψ|αthat modulates the strength of the barrier opacity, dependi ng on how
much electronic probability is sitting on the barrier. We wi ll examine the
dependence of the transmission coefficient on Ω and α.
9Since we are interested in plane wave transmission, we set
ψ(x) =/braceleftBigg
R0exp(ikx) +Rexp(−ikx)x<0
Texp(ikx) x>0(18)
From the continuity of ψ(x) and discontinuity of ψ′(x) atx= 0, we obtain
T=R0+R (19)
ikT=ik(R0−R)−Ω|T|αT. (20)
From here, one obtains T= 2R0/(2−(iΩ/k)|T|α). Defining the trans-
mission coefficient as t≡ |T|2/|R0|2, we obtain the following equation for the
transmission coefficient:
t=4
4 + (Ω∗/k)2tα(21)
where Ω∗≡Ω|R0|αis the “effective” opacity. We note that (21) is invari-
ant under a sign change in Ω. In other words, both the “upright ” and the
“inverted” delta potentials possess identical transmissi vities.
For arbitrary α, Eq. (21) is a nonlinear equation for tand must be solved
numerically. There are, however, four exactly solvable cas es, three of which
can be described shortly:
1.α= 0 (linear case): From (21) we immediately obtain the well–k nown
result
t=1
1 + (Ω/2k)2(22)
2.α= 1: Now Eq. (1) can be recast as the quadratic equation (Ω∗/2k)2t2+
t−1 = 0, with physical solution
t= 2/parenleftBiggk
Ω∗/parenrightBigg2
−1 +/radicalBigg
1 +/parenleftbiggΩ∗
k/parenrightbigg2
(23)
103.α= 2: Now we deal with a cubic equation for t: (Ω∗/2k)2t3+t−1 = 0.
Its physical solution is
t= (2/9)1/3(1/|Ω∗|)A(k,Ω∗)−(32/3)1/3(k2/|Ω∗|)A(k,Ω∗)−1/3(24)
whereA(k,Ω∗) = 9k2|Ω∗|+/radicalBig
3(16k6+ 27k4Ω∗2)
The caseα= 3 is exactly solvable in principle, but it leads to a cumbers ome
expression for tthat is not particularly illuminating.
If we recast the general equation for tast(1 + (Ω∗/2k)2tα) = 1, a bit
of simple analysis will convince the reader that the left han d side is always a
monotonically increasing function of tforα >0. Therefore, there is always
only one solution in the interval 0 ≤t≤1. Figure 3 shows the transmission
coefficienttas a function of k/Ω∗and several different nonlinearity exponents
α. Unlike the bound state calculation, there is no restrictio n here on the
magnitude of the nonlinear exponent α. For all wavevectors, the transmission
increases with increasing αand does not display any special behavior at
α= 2. The increase of twithαcan be easily understood with the help of
Eq.(21): For any α >0,tα<1 sincetis less than unity. Thus, Ω∗tα<Ω∗
which means that the total “nonlinear” opacity is always sma ller than the
“linear” one, hence a higher transmission.
Summary. In this work we have calculated the bound state correspond-
ing to a single “inverted” nonlinear delta-function potent ial, with opacity Ω
and nonlinearity exponent α. Following the usual methods of elementary
quantum mechanics, we arrived at a closed form expression fo r the bound
state characterized by an exponentially–decreasing proba bility profile, with
11a localization length that decreases with increasing α. The most significant
feature of this solution is the existence of a critical αvalue, namely 2, beyond
which the total energy (not the eigenenergy) of the bound sta te becomes pos-
itive, making the state unstable against a collapse into the continuum. The
transmission of plane waves across the nonlinear delta pote ntial is invariant
under a sign change in opacity, and increases monotonically with an increas-
ing nonlinearity exponent. The transmission is always high er than in the
linear case, for a nonzero exponent.
Finally, it is important to remark that because of the nonlin ear nature of
Eqs. (1) and (6), it is no longer possible to superpose statio nary states in
order to find the time evolution of a given initial state. A sta tionary state
solution of Eq.(6) is now only a particular solution whose re lation to the
solution of the time-dependent problem is unclear. Other fe atures that arise
in similar ‘nonlinear’ quantum mechanical problems includ e the fact that
eigenstates are no longer guaranteed to be orthogonal to eac h other. Also,
the number of eigenstates is no longer constant, but depends on nonlinearity.
Thus, ‘nonlinear’ quantum mechanics is considerably more c hallenging than
the linear one, although the reader should be aware that, as w as mentioned
at the beginning of this Note, nonlinearity in quantum mecha nics is the
consequence of some underlying assumption about the system .
12ACKNOWLEDGMENTS
This work was supported in part by FONDECYT grants 1990960 (M .I.M),
2980033 (C.A.B) and 4990004 (C.A.B). The authors are gratef ul to J. R¨ ossler
for illuminating discussions.
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(1985).
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(1988).
4 See for instance, Daniel A. Atkinson, “An exact treatment o f the Dirac
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Roy, “Regularization and renormalization in scattering fr om Dirac delta
potentials”, Am. J. Phys. 66, pp. 1101–1109 (1998).
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1968).
6 H. J. Davies and C. S. Adams, “Mean–field model of a weakly int eract-
ing Bose condensate in a harmonic potential”, Phys. Rev. A 55, pp.
2527–2530 (1997).
7 T. Holstein, “Studies of Polaron Motion. Part I. The molecu lar–crystal
model”, Ann. Phys. (N.Y.) 8, 325–342 (1959).
8 A. S. Davydov, Theory of molecular excitons (Plenum Press, New York,
1971); A. S. Davydov, Biology and Quantum Mechanics (Pergamon,
Oxford, 1982).
9 Peter L. Christiansen and Alwyn C. Scott, Davydov’s Soliton Revisited
(Plenum, New York, 1990).
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trapping Equation”, Physica D 16, 318–338 (1985).
11 P. Yeh, Optical Waves in Layered Media , New York: Wiley, 1988.
12 D. Hennig, G.P. Tsironis, M. Molina and H. Gabriel, “A Nonl inear
Quasiperiodic Kronig-Penney Model”,Phys. Lett. A 190, pp. 259–
263 (1994); D. Hennig, H. Gabriel, G.P. Tsironis and M. Molin a,
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Molina and G.P. Tsironis, “Wave Propagation in a Nonlinear Q uasiperi-
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15Captions List
Fig.1 : Normalized bound state amplitude at the origin as a function of the
nonlinearity exponent. The wavefunction changes disconti nuously at α= 2,
becoming unstable for α>2.
Fig.2 : Bound state probability profile for the “inverted” nonlinea r delta-
function potential, for several nonlinearity exponents αthat give rise to a
stable bound state.
Fig.3 : Transmission coefficient of plane waves across the nonlinear delta-
function potential versus wavevector, for several differen t nonlinearity expo-
nent values. The transmission is the same for the “upright” a nd “inverted”
delta potentials.
16α0 1 2 3 4φ(0)/(1/2)1/2
01234
STABLE
UNSTABLE
FIG. 1u-2 -1 0 1 2PROBABILITY
0.00.10.20.30.40.50.6
α = 1.5α = 1.0α = 0.5α = 0
FIG. 2k/Ω*0 1 2 3 4TRANSMISSION
0.00.20.40.60.81.0
α = 0α = 1α = 2α = 3α = 4
FIG. 3 |
EXACT ZERO-ENERGY SOLUTION
FOR A NEW FAMILY OF ANHARMONIC POTENTIALS
Yoël Lana-Renault
Departamento de Física Teórica
Universidad de Zaragoza. 50009 Zaragoza. Spain
e-mail: yoel@kepler.unizar.es
Abstract. An explicit, oscillatory solution of the type r = [ro + Ao·sin(ω·t + φo)]a
is found for the zero-energy one-dimensional motion of a particle under a specific
family of anharmonic potentials.
Key words: Anharmonic Potentials. Oscillatory Motion.
I. The Potential
In classical mechanics there are very few non-trivial potentials that admit an
explicit analytical solution. Our aim in this paper is to present a new family of
one-dimensional anharmonic potentials represented by the function:
()1V()r .....1
2ma2ω2r21 ..2ror1
a . ro2Ao2r2
a
for which one is able to find an analytical solution.
In (1), r is the position of a point-like particle of mass m; ro and Ao are
two independent parameters, fulfilling ro > Ao . As usual, ω will represent the
angular frequency of the oscillatory solutions; -a- is a free non-null parameter,
and we will study the equation of the oscillation of zero-energy around the
minimum of the potential.
1 For a large enough r, V(r) is always confining. If a < 0 , the third term within the
brackets is the leader. If a > 0 , then, the first term in the bracket produces a
-parabolic- confinement. Our study will be devoted to the analysis of the oscillations
generated around the potential minimum, which has at its right the confining wall
just mentioned.
At the origin, r = 0 , V(r) can exhibit different behaviours, but they are of
no interest here.
Beyond the origin, V(r) has, in general, a minimum, at r = R1 , and a maximum,
at r = R2 . These values are
()2R1rororo2 . ..4a( ) a1Ao2
.2aa
()3R2rororo2 . ..4a( ) a1Ao2
.2aa
Note that R1 always exists. R2 , on the contrary, is not real for 0 < a < 1 .
The oscillatory motion is confined to the region where V(r) < E . Thus, for
oscillations of E = 0 , imposing the condition V(r) = 0 we find:
If>a0,RMroAoaand RmroAoa()4
If<a0,RMroAoaand RmroAoa()5
where RM and Rm are the maximum and minimum limiting distances, respectively.
In all cases, R2 < Rm < R1 < RM .
II. Several Examples
In the five cases depicted in Sections II and III, we will assume that m = 1 ,
ω = 10 , ro = 2 and Ao = 1 .
The purpose of these drawings is to give an idea of the several types of geometries
included in Eq. (1). In all cases the continuous line represents the potential, whilst the
dotted straight line represents the E = 0 level.
2 Let us consider first a = -1.1 (a < 0) . This is illustrated in Fig. 1 . Some of the
results are:
=R20.188 , =Rm0.299 , =R10.779 , =RM1
Here = VR20.574 and = VR110.37
1510505
0 0.2 0.4 0.6 0.8 1 1.2V()r
E()r
rFig. 1
In the second example, a = -0.1 (a < 0) . The result appears in Fig. 2.
Here =R20.732 , =Rm0.896 , =R10.963 , =RM1
and = VR20.222 , = VR10.154
0.200.20.40.6
0 0.2 0.4 0.6 0.8 1 1.2V()r
E()r
rFig. 2
3 In the third example, a = 0.2 (0 < a < 1) . This is plotted in Fig. 3 . In this case,
there is no maximum because R2 in (3) is complex. Besides, we observe that the limit
of V(0 +) is infinite.
Here =Rm1 , =R11.096 , =RM1.246
and = VR10.792
20246
0.9 1 1.1 1.2 1.3V()r
E()r
rFig. 3
In the fourth example, a = 1.6 (a > 1) . The values are (see Fig. 4):
=R20.33 , =Rm1, =R13.66 , =RM5.8
and = VR269.644 , = VR1317.541
4002000200400600
0 2 4 6V()r
E()r
rFig. 4
4III. Harmonic Behaviour
We arrive at the harmonic limit, when we adopt a = 1 . This conduct is illustrated
in Fig. 5 . The value of the other parameters is identical to that of the other figures.
The form of the harmonic potential, VH(r) , is
VH()r...1
2mω2rro2Ao2
and the values are:
R2rororo2 . ..4a( ) a1Ao2
.2aa
0 , =Rm1
R1rororo2 . ..4a( ) a1Ao2
.2aa
ro2, =RM3
Here the limit of V(0 +) = VH(0) = 150 and = VR150
50050100150200
0 1 2 3 4V()r
E()r
rFig. 5
5
IV. Oscillatory SolutionIV. Oscillatory Solution
To find the general solution of the oscillatory motion, of energy E = 0 , in the
interval between Rm and RM , we proceed, in the usual way [1], departing from the
energy equation:
tto= d
rtor()t
r1
.2
m( ) 0V()r=I ()6
The function I , can be worked out easily by performing the following change of
variable
r=ro.Aosen().ωsa()7
In terms of s , -I- adopts the form:
I= .1
.aωd
stos()t
s...aωAocos().ωs
.ro.Aosen().ωsaa1.Aocos().ωs
ro.Aosen().ωs=
= d
stos()t
s1=s()tsto()8
Thus, coming back to the r variable and taking for convenience to = 0 , we easily
find
r=ro.Aosen .ωtφoa()9
The phase, φo , is given by:
φo=arsenr()01
aro
Ao()10
Adopting φo = 0 , we have:
r()0=roa()11
6V. Final Comments
We have presented a new parametric family of anharmonic potentials for which one
is able to obtain closed analytical solutions for the trajectories of zero-energy.
The degree of anharmonicity is expressed by the departure of the exponent -a-
from the value 1 . Several illustrative examples have been provided.
Acknowledgements
The author wishes to express his gratitude to Dr. Amalio F. Pacheco, who provided
invaluable help in the writing of the manuscript.
REFERENCES
[1] See, for example, T. B. Kibble, "Classical Mechanics". Mc. Graw Hill (1966)
7 |
arXiv:physics/0102055v1 [physics.flu-dyn] 16 Feb 2001Mathematics of structure-function equations of all orders
Reginald J. Hill
National Oceanic and Atmospheric Administration, Environ mental Technology
Laboratory, Boulder CO 80305-3328, USA
(February 2, 2008)
Exact equations are derived that relate velocity structure functions of arbitrary order with other
statistics. “Exact” means that no approximation is used exc ept that the Navier-Stokes equation and
incompressibility condition are assumed to be accurate. Th e exact equations are used to determine
the structure-function equations of all orders for locally homogeneous but anisotropic turbulence as
well as for the locally isotropic case. These equations can b e used for investigating the approach to
local homogeneity and to local isotropy as well as the balanc e of the equations and identification of
scaling ranges.
I. INTRODUCTION
Full mathematical exposition on the topic of structure-fun ction equations is given here. A brief summary
of results derived here will appear in the Journal of Fluid Me chanics in the paper “Equations relating structure
functions of all orders.” The two sections below are sufficien tly similar to that paper so as to guide the reader to
the relevant mathematical details, much of which resides in the appendices herein. The two sections below contain
more mathematical detail than does that paper. The derivati on of the structure function equations of all orders
produces substantial mathematical detail. This is true for reduction of the viscous term and for the term involving
the pressure gradient when deriving the exact equations. Ap plying isotropic formulas for structure functions of
arbitrary order requires the invention of new notation and m uch use of combinatorial analysis. The divergence
and Laplacian operating on isotropic formulas necessarily appear in the equations; evaluation of which requires the
derivation of many identities. Finally, matrix-based algo rithms are invented such that the isotropic formulas for the
divergence and Laplacian of isotropic tensors of any order c an be generated by computer.
There is some difference in notation between the paper and thi s document. In the paper, a component
of a structure function is denoted by D[N1,N2,N3], whereas here it is denoted by the more complicated notation
D[N:N1,N2,N3]. The reason for the more complicated notation here is to avoi d ambiguity at several places in the mathe-
matics. In the paper, the components of the tensor/braceleftbig
W[N−2P](r)δ[2P]/bracerightbig
are denoted by/braceleftbig
W[N−2P](r)δ[2P]/bracerightbig
[N1,N2,N3].
Here, there is no symbolic distinction between the tensor/braceleftbig
W[N−2P](r)δ[2P]/bracerightbig
and its components. The distinction
is implied by the context.
II. EXACT TWO-POINT EQUATIONS
The Navier-Stokes equation for velocity component ui(x, t) and the incompressibility condition are
∂tui(x, t) +un(x, t)∂xnui(x, t) =−∂xip(x, t) +ν∂xn∂xnui(x, t) , and ∂xnun(x, t) = 0, (1)
where p(x, t) is the pressure divided by the density (density is constant ),νis kinematic viscosity, and ∂denotes partial
differentiation with respect to its subscript variable. Sum mation is implied by repeated Roman indexes. Consider
another point x′such that x′andxare independent variables. For brevity, let ui=ui(x, t),u′
i=ui(x′, t), etc.
Require that xandx′have no relative motion. Then ∂xiu′
j= 0,∂x′
iuj= 0, etc., and ∂tis performed with both held
xandx′fixed. Subtracting (1) at x′from (1) at xand using the aforementioned properties gives
∂tvi+un∂xnvi+u′
n∂x′nvi=−Pi+ν/parenleftbig
∂xn∂xnvi+∂x′n∂x′nvi/parenrightbig
, (2)
where vi≡ui−u′
i,Pi≡/parenleftig
∂xip−∂x′
ip′/parenrightig
. (3)
Change independent variables from xandx′to the sum and difference independent variables:
X≡(x+x′)/2 and r≡x−x′,and define r≡ |r|. (4)
The relationship between the partial derivatives is
1∂xi=∂ri+1
2∂Xi,∂x′
i=−∂ri+1
2∂Xi,∂Xi=∂xi+∂x′
i,∂ri=1
2/parenleftig
∂xi−∂x′
i/parenrightig
. (5)
The change of variables organizes the equations in a reveali ng way because of the following properties. In the case of
homogeneous turbulence, ∂Xioperating on a statistic produces zero because that derivat ive is the rate of change with
respect to the place where the measurement is performed. Con sider a term in an equation composed of ∂Xioperating
on a statistic. For locally homogeneous turbulence, that te rm becomes negligible as ris decreased relative to the
integral scale. For the homogeneous and locally homogeneou s cases, the statistical equations retain their dependence
onr, which is the displacement vector of two points of measureme nt. Using (5), (2) becomes
∂tvi+Un∂Xnvi+vn∂rnvi=−Pi+ν/parenleftbig
∂xn∂xnvi+∂x′n∂x′nvi/parenrightbig
, (6)
where Un≡(ui+u′
i)/2. (7)
Now multiply (6) by the product vjvk· · ·vl, which contains N−1 factors of velocity difference, each factor
having a distinct index. Sum the Nsuch equations as required to produce symmetry under interc hange of each pair
of indexes, excluding the summation index n. French braces, i.e., {◦}, denote the sum of all terms of a given type
that produce symmetry under interchange of each pair of inde xes. The differentiation chain rule gives
{vjvk· · ·vl∂tvi}=∂t(vjvk· · ·vlvi), (8)
{vjvk· · ·vlUn∂Xnvi}=Un∂Xn(vjvk· · ·vlvi) =∂Xn(Unvjvk· · ·vlvi), (9)
{vjvk· · ·vlvn∂rnvi}=vn∂rn(vjvk· · ·vlvi) =∂rn(vnvjvk· · ·vlvi). (10)
The right-most expressions in (9) and (10) follow from the in compressibility property obtained from (5) and the fact
that∂xiu′
j= 0,∂x′
iuj= 0, namely
∂XnUn= 0, ∂Xnvn= 0, ∂rnUn= 0, ∂rnvn= 0. (11)
The viscous term in (6) produces ν/braceleftbig
vjvk· · ·vl/parenleftbig
∂xn∂xnvi+∂x′n∂x′nvi/parenrightbig/bracerightbig
; this expression is treated in Appendix A.
These results give
∂t(vj· · ·vi) +∂Xn(Unvj· · ·vi) +∂rn(vnvj· · ·vi) =
− {vj· · ·vlPi}+ 2ν/bracketleftbigg/parenleftbigg
∂rn∂rn+1
4∂Xn∂Xn/parenrightbigg
(vj· · ·vi)− {vk· · ·vleij}/bracketrightbigg
, (12)
where eij≡(∂xnui)(∂xnuj) +/parenleftbig
∂x′nu′
i/parenrightbig/parenleftbig
∂x′nu′
j/parenrightbig
= (∂xnvi) (∂xnvj) +/parenleftbig
∂x′nvi/parenrightbig/parenleftbig
∂x′nvj/parenrightbig
. (13)
The quantity {vj· · ·vlPi}can be expressed differently on the basis that (5) allows Pito be written as Pi=
∂Xi(p−p′). The derivation is in Appendix B; the alternative formula i s
{vjvk· · ·vlPi}={∂Xi[vjvk· · ·vl(p−p′)]} −(N−1)(p−p′)/braceleftbig/parenleftbig
sij−s′
ij/parenrightbig
vk· · ·vl/bracerightbig
, (14)
where the rate of strain tensor sijis defined by
sij≡/parenleftbig
∂xiuj+∂xjui/parenrightbig
/2. (15)
III. AVERAGED EQUATIONS
Consider the ensemble average because it commutes with temp oral and spatial derivatives. The above notation
of explicit indexes is burdensome. Because the tensors are s ymmetric, it suffices to show only the number of indexes.
Define the following statistical tensors, which are symmetr ic under interchange of any pair of indexes, excluding the
summation index nin the definition of F[N+1]:
D[N]≡ /an}b∇acketle{tvj· · ·vi/an}b∇acket∇i}ht,F[N+1]≡ /an}b∇acketle{tUnvj· · ·vi/an}b∇acket∇i}ht,T[N]≡ /an}b∇acketle{t{vj· · ·vlPi}/an}b∇acket∇i}ht,E[N]≡ /an}b∇acketle{t{vk· · ·vleij}/an}b∇acket∇i}ht, (16)
where angle brackets /an}b∇acketle{t/an}b∇acket∇i}htdenote the ensemble average, and the subscripts NandN+ 1 within square brackets denote
the number of indexes. The argument list ( X,r, t) is understood for each tensor. The left-hand sides of each
definition in (16) are in implicit-index notation for which o nly the number of indexes is given; the right-hand sides in
(16) are in explicit-index notation. The ensemble average o f (12) is
∂tD[N]+∇X·F[N+1]+∇r·D[N+1]=−T[N]+ 2ν/bracketleftbigg/parenleftbigg
∇2
r+1
4∇2
X/parenrightbigg
D[N]−E[N]/bracketrightbigg
, (17)
where, ∇X·F[N+1]≡∂Xn/an}b∇acketle{tUnvj· · ·vi/an}b∇acket∇i}ht,∇r·D[N+1]≡∂rn/an}b∇acketle{tvnvj· · ·vi/an}b∇acket∇i}ht,∇2
r≡∂rn∂rn,∇2
X≡∂Xn∂Xn. The notations
∇X·,∇2
X,∇r·, and ∇2
rare the divergence and Laplacian operators in X-space and r-space, respectively.
2A. HOMOGENEOUS AND LOCALLY HOMOGENEOUS TURBULENCE
Consider homogeneous turbulence and locally homogeneous t urbulence; the latter applies for small rand large
Reynolds number. The variation of the statistics with the lo cation of measurement or of evaluation is neglected
for these cases. That location being X,the result of ∇X·operating on a statistic is neglected. Thus the terms
∇X·F[N+1]and1
4∇2
XD[N]are neglected in (17); then (17) becomes
∂tD[N]+∇r·D[N+1]=−T[N]+ 2ν/bracketleftbig
∇2
rD[N]−E[N]/bracketrightbig
. (18)
Because the X-dependence is neglected, the argument list ( r, t) is understood for each tensor. The ensemble av-
erage of (14) contains /an}b∇acketle{t∂Xi[{vjvk· · ·vl(p−p′)}]/an}b∇acket∇i}ht, which can be written as the sum of N−1 statistics of the form
/an}b∇acketle{t{vjvk· · ·vl(p−p′)}/an}b∇acket∇i}htoperated upon by the X-space gradient. Since such X-space derivative terms are neglected,
(14) gives the alternative that
T[N]=−(N−1)/angbracketleftbig
(p−p′)/braceleftbig/parenleftbig
sij−s′
ij/parenrightbig
vk· · ·vl/bracerightbig/angbracketrightbig
. (19)
Locally homogeneous turbulence is also locally stationary such that the term ∂tD[N]in (18) may be neglected.
However, ∂tD[N]is not necessarily negligible for homogeneous turbulence.
B. ISOTROPIC AND LOCALLY ISOTROPIC TURBULENCE
Consider isotropic turbulence and locally isotropic turbu lence; the latter applies for small rand large Reynolds
number. The tensors D[N],T[N], andE[N]in (16) obey the isotropic formula. The Kronecker delta δij= 1 if
i=jandδij= 0 if i/ne}ationslash=j. Let δ[2P]denote the product of PKronecker deltas having 2 Pdistinct indexes, and let
W[N](r) denote the product of Nfactorsri
reach with a distinct index; the argument ris omitted when clarity does
not suffer. Because each tensor in (16) is symmetric under int erchange of any two indexes, their isotropic formulas
are particularly simple. Each formula is a the sum of M+ 1 terms where
M=N/2 ifNis even, and M= (N−1)/2 ifNis odd. (20)
Each term is the product of a distinct scalar function with a W[N]and a δ[2P]. From one term to the next, a pair
of indexes is transferred from a W[N]to aδ[2P]; examples are given in (65-67) of Appendix E. For the tensor D[N],
denote the Pth scalar function by DN,P(r, t). Thus the scalar functions belonging to the isotropic form ulas for T[N],
E[N], andD[N+1]are denoted by TN,P(r, t),EN,P(r, t), and DN+1,P(r, t), respectively. The scalar functions depend
on the magnitude of the spacing rrather than on the vector spacing r. The isotropic formula for D[N]is
D[N](r, t) =M/summationdisplay
P=0DN,P(r, t)/braceleftbig
W[N−2P](r)δ[2P]/bracerightbig
, (21)
and the isotropic formulas for T[N]andE[N]have the analogous notation. Recall that {◦}denotes the sum of all
terms of a given type that produce symmetry under interchang e of each pair of indexes. Henceforth, the argument
list (r, t) will be deleted.
A special Cartesian coordinate system is typically used bec ause it simplifies the isotropic formula. This
coordinate system has the positive 1-axis parallel to the di rection of r, and the 2- and 3-axes are therefore perpendicular
tor. Let N1,N2, and N3be the number of indexes of a component of D[N]that are 1, 2, and 3, respectively; such
thatN=N1+N2+N3. Because of symmetry, the order of indexes is immaterial suc h that a component of D[N]
can be identified by N1,N2, and N3. Thus, denote a component of D[N]byD[N:N1,N2,N3], which is a function
ofrandt. The projection of (21) using N1,N2, and N3unit vectors in the directions of the 1-, 2-, and 3-axes,
respectively, results in the component D[N:N1,N2,N3]on the left-hand side of (21), and numerical values of the pro jection
of/braceleftbig
W[N−2P](r)δ[2P]/bracerightbig
appear on the right-hand side. Henceforth the word ”project ion” will be omitted for brevity.
Those values of the coefficients/braceleftbig
W[N−2P](r)δ[2P]/bracerightbig
in (21) are needed; the values obtained for the special coord inate
system are determined in Appendix C; they are, from (44-45),
if 2P < N 2+N3then/braceleftbig
W[N−2P]δ[2P]/bracerightbig
= 0; otherwise, (22)
/braceleftbig
W[N−2P]δ[2P]/bracerightbig
=N1!N2!N3!//bracketleftbigg
(N−2P)!2P/parenleftbiggN2
2/parenrightbigg
!/parenleftbiggN3
2/parenrightbigg
!/parenleftbigg
P−N2
2−N3
2/parenrightbigg
!/bracketrightbigg
. (23)
3By applying (21) and (22-23) for all combinations of indexes , one can determine which components D[N:N1,N2,N3]
are zero and which are nonzero, identify M+ 1 linearly independent equations that determine the DN,Pin terms
ofM+ 1 of the D[N:N1,N2,N3], and find algebraic relationships between the remaining non zeroD[N:N1,N2,N3]. The
derivations are in Appendix D; a summary follows.
A component D[N:N1,N2,N3]is nonzero only if both N2andN3are even and when N1is odd if Nis odd and
when N1is even if Nis even. Thereby, ( M+ 1)(M+ 2)/2 components are nonzero. There are 3Ncomponents of
D[N]; thus the other 3N−(M+ 1)(M+ 2)/2 components are zero.
There exists exactly ( M+ 1)M/2 kinematic relationships among the nonzero components of D[N]. For each
of the M+ 1 cases of N1, these relationships are expressed by the proportionality
D[N:N1,2L,0]:D[N:N1,2L−2,2]:D[N:N1,2L−4,4]:· · ·:D[N:N1,0,2L]=
[(2L)!0!/L!0!] : [(2 L−2)!2!/(L−1)!1!] : [(2 L−4)!4!/(L−2)!2!] : · · ·: [0! (2 L)!/0!L!]. (24)
Previously, only one such kinematic relationship was known (Millionshtchikov 1941). For N= 4, (24) gives D[4:0,4,0]:
D[4:0,2,2]:D[4:0,0,4]= 12 : 4 : 12. In explicit-index notation this can be written as D2222= 3D2233=D3333, which
was discovered by Millionshtchikov (1941). Now, all such re lationships are known.
There remain M+ 1 linearly independent nonzero components of D[N]. This must be so because there are
M+ 1 terms in (21), and the M+ 1 scalar functions DN,Ptherein must be related to M+ 1 components. Consider
theM+ 1 linearly independent equations that determine the DN,Pin terms of M+ 1 of the D[N:N1,N2,N3]. For
simplicity, the chosen components can all have N3= 0, i.e., the choice of linearly independent components can be
D[N:N,0,0],D[N:N−2,2,0],D[N:N−4,4,0],· · ·,D[N:N−2M,2M,0]. As described above, projections of (21) result in the
chosen components on the left-hand side and algebraic equat ions on the right-hand side. These equations can be
expressed in matrix form and solved by matrix inversion meth ods; the result is given in (88) of Appendix F. Given
experimental or DNS data or a theoretical formula for the cho sen components, the solution in (88) determines the
functions DN,Pin (21); then (21) completely specifies the tensor D[N]. The matrix algorithm is an efficient means
of determining isotropic expressions for the terms ∇r·D[N+1]and∇2
rD[N]in (18). Those algorithms are given in
Appendix F. From the example for N= 2 in Appendix F, use of the matrix algorithm and the isotropi c formulas in
(18) gives the two scalar equations
∂tD11+/parenleftbigg
∂r+2
r/parenrightbigg
D111−4
rD122=−T11+ 2ν/bracketleftbigg/parenleftbigg
∂2
r+2
r∂r−4
r2/parenrightbigg
D11+4
r2D22−E11/bracketrightbigg
= 2ν/bracketleftbigg
∂2
rD11+2
r∂rD11+4
r2(D22−D11)/bracketrightbigg
−4ε/3, (25)
∂tD22+/parenleftbigg
∂r+4
r/parenrightbigg
D122=−T22+ 2ν/bracketleftbigg2
r2D11+/parenleftbigg
∂2
r+2
r∂r−2
r2/parenrightbigg
D22−E22/bracketrightbigg
= 2ν/bracketleftbigg
∂2
rD22+2
r∂rD22−2
r2(D22−D11)/bracketrightbigg
−4ε/3, (26)
where use was made of the fact (Hill, 1997) that local isotrop y gives T11=T22= 0 and 2 νE11= 2νE22= 4ε/3 where
εis the average energy dissipation rate per unit mass of fluid. Now, (25-26) are the same as equations (43-44) of
Hill (1997), and Hill (1997) shows how these equations lead t o Kolmogorov’s equation and his 4/5 law. From the
example for N= 3 in Appendix F,
∂tD111+/parenleftbigg
∂r+2
r/parenrightbigg
D1111−6
rD1122=−T111+ 2ν[C−E111], (27)
∂tD122+/parenleftbigg
∂r+4
r/parenrightbigg
D1122−4
3rD2222=−T122+ 2ν[B−E122], (28)
where
C≡/parenleftbigg
−4
r2+4
r∂r+∂2
r/parenrightbigg
D111,andB≡1
6/parenleftbigg4
r2−4
r∂r+ 5∂2
r+r∂3
r/parenrightbigg
D111. (29)
The incompressibility condition, D122=1
6(D111+r∂rD111), was substituted in (113) to obtain (29). The matrix
algorithm is checked by the fact that (27-29) are the same as g iven by Hill and Boratav (2001).
The equations for N= 4 are
4∂tD1111+/parenleftbigg
∂r+2
r/parenrightbigg
D11111 −8
rD11122=
=−T1111+ 2ν/bracketleftbigg/parenleftbigg
∂2
r+2
r∂r−8
r2/parenrightbigg
D1111+14
r2D1122+10
3r2D2222/bracketrightbigg
−2νE1111, (30)
∂tD1122+/parenleftbigg
∂r+4
r/parenrightbigg
D11 122 −8
3rD12222=
=−T1122+ 2ν/bracketleftbigg2
r2D1111+/parenleftbigg
−52
3r2+∂2
r+2
r∂r/parenrightbigg
D1122+34
9r2D2222/bracketrightbigg
−2νE1122, (31)
∂tD2222+/parenleftbigg
∂r+6
r/parenrightbigg
D12222=−T2222+ 2ν/bracketleftbigg2
r2D1122+/parenleftbigg
−2
3r2+∂2
r+2
r∂r/parenrightbigg
D2222/bracketrightbigg
−2νE2222. (32)
Since these equations have a repetitive structure, it suffice s to give the divergence and Laplacian terms. For N= 5
to 7 these are, respectively:
/parenleftbig
∂r+2
r/parenrightbig
D111111 −10
rD111122 /parenleftbig
∂r+4
r/parenrightbig
D111122 −4
rD112 222 /parenleftbig
∂r+6
r/parenrightbig
D112222 −6
5rD222222
,2ν
/parenleftbig
∂2
r+2
r∂r−10
r2/parenrightbig
D11 111 −14
r2D11122+54
r2D12222
2
r2D11 111+/parenleftbig
−154
5r2+∂2
r+2
r∂r/parenrightbig
D11122+94
5r2D12222
6
5r2D11122+/parenleftbig
−16
5r2+∂2
r+2
r∂r/parenrightbig
D12222
/parenleftbig
∂r+2
r/parenrightbig
D1111111 −12
rD1111 122 /parenleftbig
∂r+4
r/parenrightbig
D1111122 −16
3rD1112 222 /parenleftbig
∂r+6
r/parenrightbig
D1112222 −12
5rD1222 222 /parenleftbig
∂r+8
r/parenrightbig
D1222222
,
2ν
/parenleftbig
∂2
r+2
r∂r−12
r2/parenrightbig
D111111 −108
r2D111122 +920
3r2D112222 −416
15r2D222222
2
r2D111111 +/parenleftbig
−242
5r2+∂2
r+2
r∂r/parenrightbig
D111122 +824
15r2D112 222 −248
75r2D222222
4
5r2D111122 +/parenleftbig
−112
15r2+∂2
r+2
r∂r/parenrightbig
D112222 +4
3r2D222222
2
3r2D112222 +/parenleftbig
−2
15r2+∂2
r+2
r∂r/parenrightbig
D222222
;
/parenleftbig
∂r+2
r/parenrightbig
D11111 111 −14
rD11111 122 /parenleftbig
∂r+4
r/parenrightbig
D11111 122 −20
3rD11112 222 /parenleftbig
∂r+6
r/parenrightbig
D11112 222 −18
5rD11222 222 /parenleftbig
∂r+8
r/parenrightbig
D11222 222 −8
7rD22 222222
,
2ν
/parenleftbig
∂2
r+2
r∂r−14
r2/parenrightbig
D1111111 −316
r2D1111 122 +3376
3r2D1112222 −1376
5r2D1222222
2
r2D1111 111 +/parenleftbig
−1472
21r2+∂2
r+2
r∂r/parenrightbig
D1111122 +7808
63r2D1112222 −304
15r2D1222 222
4
7r2D1111122 +/parenleftbig
−206
15r2+∂2
r+2
r∂r/parenrightbig
D1112 222 +1132
175r2D1222222
2
7r2D1112 222 +/parenleftbig
−76
35r2+∂2
r+2
r∂r/parenrightbig
D1222 222
.
Acknowledgement 1 The author thanks the organizers of the Hydrodynamics Turbu lence Program held at the In-
stitute for Theoretical Physics, University of California at Santa Barbara, whereby this research was supported in par t
by the National Science Foundation under grant number PHY94 -07194.
IV. REFERENCES
Abramowitz, M. and I. A. Stegun 1964 Handbook of Mathematical Functions with Formulas, Graphs, and Math-
ematical Tables, National Bureau of Standards Applied Math ematics Series 55 , U. S. Government Printing Office,
Washington DC.
Hill, R. J. 1997 Applicability of Kolmogorov’s and Monin’s e quations of turbulence. J. Fluid Mech. 353, 67.
Hill, R. J. and O. N. Boratav 2001 Next-order structure-func tion equations. Phys. Fluids 13, 276.
Millionshtchikov, M. D. 1941 On the theory of homogeneous is otropic turbulence. Dokl. Akad. Nauk. SSSR 32,
611.
5V. APPENDIX A: THE VISCOUS TERM
The quantity {vjvk· · ·vl∂xn∂xnvi}requires special attention. Consider the repeated applica tion of the identity
∂xn∂xn(fg) =f∂xn∂xng+g∂xn∂xnf+ 2 (∂xnf)(∂xng) (33)
to the quantity
∂xn∂xn(vjvkvm· · ·vi) (34)
forNfactors of velocity difference in ( vjvkvm· · ·vi). For the first application of (33) let f=vjand let gbe the
remaining factors ( vkvm· · ·vi); this gives
∂xn∂xn(vjvkvm· · ·vi) =vj∂xn∂xn(vkvm· ·vi) + (vkvm· ·vi)∂xn∂xnvj+ 2 [∂xnvj] [∂xn(vkvm· ·vi)]. (35)
From the differentiation chain rule, ∂xn(vkvm· ·vi) is the sum of N−1 terms of the form ( vm· ·vp∂xnvi). Thus,
the right-most term in (35) is N−1 terms of the form 2 vm· · ·vp(∂xnvi)(∂xnvj) each term containing Nfactors;
two of those factors are distinguished by being derivatives of velocity differences. The second application of (33) is
performed on vj∂xn∂xn(vkvm· ·vi) in (35), for which purpose f=vkandg= (vm· ·vi); this gives
vj∂xn∂xn(vkvm· · ·vi) =vjvk∂xn∂xn(vm· ·vi) +vj(vm· ·vi)∂xn∂xnvk+ 2vj[∂xnvk] [∂xn(vm· ·vi)].
The right-most term gives N−2 terms of the form 2 vjvm· · ·vp(∂xnvi)(∂xnvk) each term containing Nfactors.
There are N−1 steps to complete reduction of the formula. The number of te rms of the form 2 vjvm· · ·
vp(∂xnvi)(∂xnvk) is (N−1) from the first step, ( N−2) from the second step, etc. such that the total number of ter ms
is (N−1)+(N−2)+···+(N−(N−1)) = N(N−1)/2. Now, N(N−1)/2 =/parenleftbigN
2/parenrightbig
is the binomial coefficient equal
to the number of ways of choosing two indexes from a set of Nindexes; the quantities ( ∂xnvi) and ( ∂xnvj) in 2vm·
· ·vp(∂xnvi)(∂xnvj) contain the chosen two indexes iandj. The/parenleftbigN
2/parenrightbig
terms constitute 2 {vm· · ·vp(∂xnvi)(∂xnvj)}.
Because two factors of the form ( vjvm· ·vi)∂xn∂xnvkappear in the last step, the total number of terms of the form
(vj· · ·vn)∂xn∂xnviisN. Not surprisingly, these Nterms constitute {(vj· · ·vn)∂xn∂xnvi}, and N=/parenleftbigN
1/parenrightbig
is the
binomial coefficient equal to the number of ways of choosing on e index from a set of Nindexes, the quantity ∂xn∂xnvi
contains the chosen one index i.
That is, for any N
∂xn∂xn(vj· · ·vi) ={(vj· · ·vl)∂xn∂xnvi}+ 2{vk· · ·vl(∂xnvi)(∂xnvj)}. (36)
The left-hand side is symmetric under interchange of any pai r of indexes (not including nbecause summation is
implied over n), and the French brackets make the right-hand side likewise symmetric.
Use of (36) within the viscous term ν/braceleftbig
vjvk· · ·vl/parenleftbig
∂xn∂xnvi+∂x′
n∂x′
nvi/parenrightbig/bracerightbig
that arrises from (6), gives
/braceleftbig
vjvk· · ·vl/parenleftbig
∂xn∂xnvi+∂x′
n∂x′
nvi/parenrightbig/bracerightbig
=/parenleftbig
∂xn∂xn+∂x′
n∂x′
n/parenrightbig
(vj· · ·vi)
−2/braceleftbig
vk· · ·vl/bracketleftbig
(∂xnvi)(∂xnvj) +/parenleftbig
∂x′nvi/parenrightbig/parenleftbig
∂x′nvj/parenrightbig/bracketrightbig/bracerightbig
, (37)
where the right-most term in (36) has been subtracted from bo th sides of (36) to obtain (37). Note that
(∂xnui)(∂xnuj) = (∂xnvi)(∂xnvj) and/parenleftbig
∂x′nu′
i/parenrightbig/parenleftbig
∂x′nu′
j/parenrightbig
=/parenleftbig
∂x′nvi/parenrightbig/parenleftbig
∂x′nvj/parenrightbig
, and that use of (5) gives
/parenleftbig
∂xn∂xn+∂x′n∂x′n/parenrightbig
= 2/parenleftbigg
∂rn∂rn+1
4∂Xn∂Xn/parenrightbigg
.
Then, (37) can be written as
/braceleftbig
vjvk· · ·vl/parenleftbig
∂xn∂xnvi+∂x′n∂x′nvi/parenrightbig/bracerightbig
= 2/parenleftbigg
∂rn∂rn+1
4∂Xn∂Xn/parenrightbigg
(vj· · ·vi)
−2/braceleftbig
vk· · ·vl/bracketleftbig
(∂xnui)(∂xnuj) +/parenleftbig
∂x′
nu′
i/parenrightbig/parenleftbig
∂x′
nu′
j/parenrightbig/bracketrightbig/bracerightbig
. (38)
6VI. APPENDIX B: DERIVATION OF (14)
The purpose of this appendix is to derive (14). Since (5) allo wsPito be written as Pi=∂Xi(p−p′), the
differentiation chain rule gives
vjvk· · ·vlPi=vjvk· · ·vl∂Xi(p−p′) =∂Xi[vjvk· · ·vl(p−p′)]−(p−p′){(∂Xivj)vk· · ·vl}/i/, (39)
where the notation {◦}/i/denotes the sum of all terms of a given type that produce symme try under interchange of
each pair of indexes with the index iexcluded. Recall that the product vjvk· · ·vlconsists of N−1 factors. Sum
theNequations of type (39) such that the sum is even under interch ange of all pairs of indexes; then
{vjvk· · ·vlPi}={∂Xi[vjvk· · ·vl(p−p′)]} −(N−1)(p−p′){(∂Xivj)vk· · ·vl}, (40)
where use was made of the fact that the N−1 terms in the sum {(∂Xivj)vk· · ·vl}/i/each give the same result,
namely, {(∂Xivj)vk· · ·vl}. From (5), ∂Xivj=∂xiuj−∂x′
ju′
i; such that the definition of strain rate (15) gives
/parenleftbig
∂Xivj+∂Xjvi/parenrightbig
/2 =sij−s′
ij. (41)
Use of (41) gives {(∂Xivj)vk· · ·vl}=/parenleftbig
{(∂Xivj)vk· · ·vl}+/braceleftbig/parenleftbig
∂Xjvi/parenrightbig
vk· · ·vl/bracerightbig/parenrightbig
/2 =/braceleftbig/parenleftbig
sij−s′
ij/parenrightbig
vk· · ·vl/bracerightbig
, substi-
tution of which into (40) gives
{vjvk· · ·vlPi}={∂Xi[vjvk· · ·vl(p−p′)]} −(N−1)(p−p′)/braceleftbig/parenleftbig
sij−s′
ij/parenrightbig
vk· · ·vl/bracerightbig
. (42)
Alternatively, consider that (5) allows Pito be written as Pi= 2∂ri(p+p′). Analogous to (40), it follows that
{vjvk· · ·vlPi}= 2{∂ri[vjvk· · ·vl(p+p′)]} −(N−1)(p+p′){(2∂rivj)vk· · ·vl}.
The 2 ∂rivjcan be replaced by, 2 ∂rivj = ∂xiuj+∂x′
ju′
i, such that
{(2∂rivj)vk· · ·vl}=/parenleftbig
{(2∂rivj)vk· · ·vl}+/braceleftbig/parenleftbig
2∂rjvi/parenrightbig
vk· · ·vl/bracerightbig/parenrightbig
/2 =/braceleftbig/parenleftbig
sij+s′
ij/parenrightbig
vk· · ·vl/bracerightbig
. Then, analogous to
(42), it follows that
{vjvk· · ·vlPi}= 2{∂ri[vjvk· · ·vl(p+p′)]} −(N−1)(p+p′)/braceleftbig/parenleftbig
sij+s′
ij/parenrightbig
vk· · ·vl/bracerightbig
.
VII. APPENDIX C: THE COEFFICIENT IN (21)
The purpose of this appendix is to obtain a formula for evalua tion of the coefficient/braceleftbig
W[N−2P]δ[2P]/bracerightbig
in the
special Cartesian coordinate system. In this coordinate sy stem,ri
r=δ1isuch that W[N]is the product of NKronecker
deltas of the form δ1i. Consider setting N1of the indexes equal to 1, an even number N2to 2, and an even number N3
to 3 such that N=N1+N2+N3. Then/braceleftbig
W[N−2P]δ[2P]/bracerightbig
becomes a sum of zeros and ones. What is that sum? If
all indexes are set to 1, i.e., N1=N, then all terms in the sum/braceleftbig
W[N−2P]δ[2P]/bracerightbig
are unity such that/braceleftbig
W[N−2P]δ[2P]/bracerightbig
equals its number of terms; from (49) in Appendix E, that numb er is/parenleftbigN
N−2P/parenrightbig
(2P−1)!!. The notation/parenleftbigN
N−2P/parenrightbig
is a
binomial coefficient. To interpret (2 P−1)!!, recall that q!!≡q(q−2)(q−4)· · ·qL, where qLis 2 or 1 for qeven or
odd, respectively, and ( −1)!!≡1. Now consider setting two indexes to 2, thus N1=N−2, and N2= 2. Name the
two indexes i= 2 and j= 2. The only term in/braceleftbig
W[N−2P]δ[2P]/bracerightbig
that is nonzero is that which has iandjtogether
in a single Kronecker delta, δij, within δ[2P]. For P= 0 there is no such δij, in which case/braceleftbig
W[N]δ[0]/bracerightbig
= 0. For
P≥1, there is one such δij, which is set to 1 and it multiplies the quantity/braceleftbig
W[N−2P]δ[2(P−1)]/bracerightbig
; since this quantity
is evaluated with all 1s, it is equal to its number of terms, na mely/parenleftbig(N−2P)+2(P−1)
(N−2P)/parenrightbig
(2 (P−1)−1)!!. Now consider
setting four indexes to 2; thus N1=N−4, and N2= 4. Name the four indexes i= 2,j= 2,k= 2,l= 2. The only
terms in/braceleftbig
W[N−2P]δ[2P]/bracerightbig
that are nonzero are those that have factors δijδkl,δikδjl, orδilδjkwithin δ[2P]. For P≤1
there is no such pair of Kronecker deltas such that/braceleftbig
W[N]δ[0]/bracerightbig
= 0 and/braceleftbig
W[N−2]δ[2]/bracerightbig
= 0. For P≥2, there are the
above 3 = ( N2−1)!! nonzero factors and each multiplies the quantity/braceleftbig
W[N−2P]δ[2(P−2)]/bracerightbig
; since this latter quantity
is subsequently evaluated with all 1s, it is equal to its numb er of terms, namely/parenleftbig(N−2P)+2(P−2)
(N−2P)/parenrightbig
(2 (P−2)−1)!!.
Continuation of this study reveals the pattern that when/braceleftbig
W[N−2P]δ[2P]/bracerightbig
is evaluated with N22s and N11s such
thatN=N1+N2then
7if 2P < N 2then/braceleftbig
W[N−2P]δ[2P]/bracerightbig
= 0,otherwise
/braceleftbig
W[N−2P]δ[2P]/bracerightbig
= (N2−1)!!/parenleftbigg(N−2P) + 2/parenleftbig
P−N2
2/parenrightbig
(N−2P)/parenrightbigg/parenleftbigg
2/parenleftbigg
P−N2
2/parenrightbigg
−1/parenrightbigg
!!.
= (N2−1)!!/bracketleftigg
N1!/parenleftbig
N1−2/parenleftbig
P−N2
2/parenrightbig/parenrightbig
!/parenleftbig
2/parenleftbig
P−N2
2/parenrightbig/parenrightbig
!/bracketrightigg/parenleftbigg
2/parenleftbigg
P−N2
2/parenrightbigg
−1/parenrightbigg
!!
If one ceases increasing the number of 2s and commences incre asing the number of 3s in pairs
such that N=N1+N2+N3, then ( N2−1)!!/parenleftbig(N−2P)+2(P−N2
2)
(N−2P)/parenrightbig/parenleftbig
2/parenleftbig
P−N2
2/parenrightbig
−1/parenrightbig
!! is replaced by
(N2−1)!! (N3−1)!!/parenleftbig(N−2P)+2(P−N2
2−N3
2)
(N−2P)/parenrightbig/parenleftbig
2/parenleftbig
P−N2
2−N3
2/parenrightbig
−1/parenrightbig
!!. Of course, the binomial coefficient can be ex-
pressed as follows:/parenleftbig(N−2P)+2(P−N2
2−N3
2)
(N−2P)/parenrightbig
= (N−N2−N3)!/[(N−2P)! (2P−N2−N3)!]; also, ( N−N2−N3)! =
N1!. The double factorial can be eliminated by means of the foll owing identities:
(2Q−1)!!/(2Q)! = 1/(2Q)!! = 1 //parenleftbig
2QQ!/parenrightbig
. (43)
That is, (2 P−N2−N3−1)!!/(2P−N2−N3)! = 1 /(2P−N2−N3)!! = 1 //bracketleftig
2(P−N2
2−N3
2)/parenleftbig
P−N2
2−N3
2/parenrightbig
!/bracketrightig
; also,
(N2−1)!! = N2!//bracketleftbig
2N2/2(N2/2)!/bracketrightbig
. Finally,
if 2P < N 2+N3then/braceleftbig
W[N−2P]δ[2P]/bracerightbig
= 0,otherwise (44)
/braceleftbig
W[N−2P]δ[2P]/bracerightbig
= (N2−1)!! (N3−1)!!N1!//bracketleftbigg
(N−2P)!2(P−N2
2−N3
2)/parenleftbigg
P−N2
2−N3
2/parenrightbigg
!/bracketrightbigg
=N1!N2!N3!//bracketleftbigg
(N−2P)!2P/parenleftbiggN2
2/parenrightbigg
!/parenleftbiggN3
2/parenrightbigg
!/parenleftbigg
P−N2
2−N3
2/parenrightbigg
!/bracketrightbigg
. (45)
VIII. APPENDIX D: PROPERTIES OF ISOTROPIC SYMMETRIC TENSOR S
The purpose of this appendix is to determine (1) which compon ents of a symmetric, isotropic tensor are
zero; (2) how many components are zero and how many are nonzer o; and (3) the relationships between the nonzero
components.
Consider which components D[N:N1,N2,N3]are nonzero and which are zero. In (21), W[N−2P]vanishes if any
of its indexes is 2 or 3, δ[2P]vanishes unless it contains an even number of indexes equal t o 2 and a likewise even
number of 3s (and of 1s). Thus, a component D[N:N1,N2,N3]is nonzero only if both N2andN3are even. Because
N=N1+N2+N3,D[N:N1,N2,N3]is nonzero only when N1is odd if Nis odd and only when N1is even if Nis
even. The values of N1that can give nonzero values of D[N:N1,N2,N3]areN,N−2,· · ·, 0 or 1; i.e., M+ 1 cases
ofN1. Given N1, the values of [ N2, N3] that give nonzero values of D[N:N1,N2,N3]are [N−N1,0], [N−N1−2,2],
···,[0, N−N1]; i.e., ( N−N1+ 2)/2 cases (note that N−N1=N2+N3is necessarily even). Counting the number of
cases of [ N2, N3] asN1varies from Nto 0 or 1, (i.e., substituting N1=N, then N1=N−2,···into (N−N1+ 2)/2
and adding the resultant numbers) shows that there are 1 + 2 + 3 +· · ·+ (M+ 1) = ( M+ 1)(M+ 2)/2 components
D[N:N1,N2,N3]that are nonzero. Since there are 3Ncomponents of D[N], the remaining 3N−(M+ 1)(M+ 2)/2
components are zero. Since there are M+ 1 linearly independent components (that are related to the DN,P), there
are (M+ 1)(M+ 2)/2−(M+ 1) = M(M+ 1)/2 relationships among the nonzero components D[N:N1,N2,N3].For
instance, interchange of the values of N2andN3produces components that are equal.
Consider the M+ 1 linearly independent equations that determine the DN,Pin terms of M+ 1 of the
D[N:N1,N2,N3].DN,0is related by (21) to only the component D[N:N,0,0]because (22) shows that the coefficient
ofDN,0, namely/braceleftbig
W[N]δ[0]/bracerightbig
, vanishes unless N1=N. That is, if all indexes in (21) are 1, then D[N:N,0,0]appears
on the left-hand side and the DN,Pfor all Pappear in the equation. This equation is essential for deter mining DN,0
and is called “the equation for DN,0”; similar terminology “the equation for DN,P” is used below. With the equation
forDN,0in hand, consider DN,1.DN,1is related by (21) to D[N:N−2,2,0]orD[N:N−2,0,2]. When D[N:N−2,2,0]or
D[N:N−2,0,2]is on the left-hand side of (21) the equation for DN,1results because the coefficients/braceleftbig
W[N−2P]δ[2P]/bracerightbig
of
DN,PforP≥1 do not vanish, but the coefficient of DN,0does vanish. Now consider an equation for DN,2.DN,2
is related by (21) to D[N:N−4,4,0]orD[N:N−4,0,4]orD[N:N−4,2,2]; these components also involve DN,PforP≥3 but
not for P≤1. This procedure repeats until the last equation is produce d; only DN,Mappears in the last equation.
8IfNis even then DN,Mis related to D[N:0,N2,N3]withN2andN3equal to any positive even numbers such that
N=N2+N3. IfNis odd then DN,Mis related to D[N:1,N2,N3]withN2andN3equal to any positive even numbers
such that N= 1+ N2+N3. This procedure results in a set of M+1 linearly independent equations that can be solved
to obtain the DN,Pin terms of the D[N:N1,N2,N3]. Note that M+ 1 components must be chosen for use in the M+ 1
equations. For instance, from the above example, D[N:N,0,0]must be used; either D[N:N−2,2,0]orD[N:N−2,0,2]must
be chosen, and one of D[N:N−4,4,0]orD[N:N−4,0,4]orD[N:N−4,2,2]must be chosen, etc. For simplicity, the chosen
components can all have N3= 0, i.e., the choice can be D[N:N,0,0],D[N:N−2,2,0],D[N:N−4,4,0],· · ·,D[N:N−2M,2M,0].
The above procedure also reveals algebraic relationships b etween the nonzero D[N:N1,N2,N3]. The equation
forDN,1can be expressed in terms of either D[N:N−2,2,0]orD[N:N−2,0,2]; the left-hand side is the same in either
case because the coefficients (23) are the same; hence D[N:N−2,2,0]=D[N:N−2,0,2]. The equation for DN,2can be
expressed in terms of D[N:N−4,2,2]orD[N:N−4,4,0]orD[N:N−4,0,4]such that (23) gives D[N:N−4,4,0]=D[N:N−4,0,4];
but what is the relationship of D[N:N−4,2,2]toD[N:N−4,4,0]andD[N:N−4,0,4]? When D[N:N−4,0,4]orD[N:N−4,4,0]is
on the left-hand side of (21), the nonzero coefficients are, fr om (23), ( N−4)!4!0! //bracketleftbig
(N−2P)!2P2!0! (P−2−0)!/bracketrightbig
, but
when D[N:N−4,2,2]is on the left-hand side, the nonzero coefficients are ( N−4)!2!2! //bracketleftbig
(N−2P)!2P1!1! (P−1−1)!/bracketrightbig
.
The ratio of these coefficients is 3. Since this ratio is indepe ndent of P, the entire right-hand side of (21) is three
times greater when D[N:N−4,0,4]is on the left-hand side as compared to when D[N:N−4,2,2]is on the left-hand side.
Therefore, the proportionality D[N:N−4,0,4]:D[N:N−4,2,2](and also D[N:N−4,4,0]:D[N:N−4,2,2]) is 3 : 1. In general,
for given NandN1, and hence given N2+N3=N−N1, and another choice of N2andN3, call them N′
2andN′
3,
such that N′
2+N′
3=N2+N3=N−N1, the proportionality obtained from (23) is D[N:N1,N2,N3]:D[N:N1,N′
2,N′
3]=
[N2!N3!/(N2/2)! (N3/2)!] : [ N′
2!N′
3!/(N′
2/2)! (N′
3/2)!]. Parameterized in terms of an integer Lsuch that N=N1+2L,
for given N1, the proportionalities are D[N:N1,2L,0]:D[N:N1,2L−2,2]:D[N:N1,2L−4,4]:···:D[N:N1,0,2L]= [(2L)!0!/L!0!] :
[(2L−2)!2!/(L−1)!1!] : [(2 L−4)!4!/(L−2)!2!] : · · ·: [0! (2 L)!/0!L!]. This constitutes ( N−N1)/2 relationships
among the nonzero components D[N:N1,N2,N3]for given N1. Substituting the M+ 1 cases of N1(i.e.,N1=N,
N1=N−1,· · ·) into ( N−N1)/2 the number of relationships thus identified among the compo nents of D[N]is
0+1+2+ ···+M=M(M+ 1)/2. In the paragraph above, it was determined that the total nu mber of relationships
among the nonzero components of D[N]isM(M+ 1)/2. Consequently, all such relationships have now been found .
IX. APPENDIX E: DERIVATIVES OF ISOTROPIC TENSORS
The objective of this appendix is to develop succinct notati on for isotropic tensors and their derivatives with
specific attention to their first-order divergence and their Laplacian. Those derivatives appear in (18). A derivation
of those derivatives operating on an isotropic tensor that i s symmetric under interchange of any pair of indexes is
given.
First, notation is developed: δ[2P]is the product of PKronecker deltas having 2 Pdistinct indexes. For
example, δ[6]=δijδklδmn, where δij= 1 if i=jandδij= 0 if i/ne}ationslash=j.W[N]is the product of Nfactorsri
reach with
an index distinct from the other indexes. For example, W[4]=ri
rrj
rrk
rrl
r. For convenience, define
δ[0]≡1, δ[−2]≡0,W[0]≡1,W[−1]≡0,W[−2]≡0. (46)
The plural of δisδs and that of WisWs. It is understood that products of Ws (e.g., W[N]W[K]) and of
δs (e.g., δ[2N]δ[2K]) and of Ws with δs (e.g., W[N]δ[2K]) have all distinct indexes. Then, the Ws factor, e.g.,
W[4]=W[1]W[3]=W[2]W[2]=W[1]W[1]W[2], etc., and the δs likewise factor. The operation “contraction” means
to set two indexes equal and sum over their range of values; th e summation convention over repeated Roman indexes
is used, e.g., δii≡3/summationtext
i=1δii=δ11+δ22+δ33= 3. A contraction of two indexes of δ[2P]produces either 3 δ[2(P−1)]or
δ[2(P−1)]depending on whether the two indexes are on the same Kronecke r delta or different ones, respectively. The
contraction of W[N]on two indexes produces W[N−1]becauseri
rri
r=r2
r2= 1. Consider the contraction ofri
rwith
W[N]δ[2P]. If the index iis inW[N]then the contractionri
rW[N]δ[2P]isW[N−1]δ[2P]becauseri
rri
r= 1. If the index
iis inδ[2P]then the contractionri
rW[N]δ[2P]isW[N+1]δ[2(P−1)]becauseri
rδij=rj
r. The notation /a∇∇owdblbothv◦/a∇∇owdblbothvN
jmeans the
sum of Nterms where each term contains a distinct index j, and the index jis interchanged with all implied indexes,
butjis not interchanged with any explicit index. For example,
/arrowdbltp/arrowdblbtW[2]δij/arrowdbltp/arrowdblbt3
j=rk
rrl
rδij+rk
rrj
rδil+rj
rrl
rδik, (47)
and/arrowdbltp/arrowvertexdbl/arrowdblbtW[1]δ[2]ri
rrj
r/arrowdbltp/arrowvertexdbl/arrowdblbt4
j=rk
rδnmri
rrj
r+rj
rδnmri
rrk
r+rk
rδjmri
rrn
r+rk
rδnjri
rrm
r, (48)
9where, iis an explicit index and is therefore not interchanged with j.
French braces (i.e., {◦}) means: add all such distinct terms required to make the tens or symmetric under
interchange of any pair of indexes. For example,
/braceleftbig
W[2]δ[2]/bracerightbig
=/braceleftigri
rrj
rδkl/bracerightig
=ri
rrj
rδkl+ri
rrl
rδkj+rl
rrj
rδki+rk
rrl
rδij+rk
rrj
rδil+ri
rrk
rδjl.
Note that terms that are necessarily equal do not appear; i.e ., sinceri
rrj
rδklappears, neitherrj
rri
rδklnorri
rrj
rδlk
appear. Because of the commutative law of addition, {◦}commutes with addition; e.g.,/braceleftbig
W[N]/bracerightbig
+/braceleftbig
W[Q]δ[2P]/bracerightbig
=/braceleftbig
W[N]+W[Q]δ[2P]/bracerightbig
. Because of the distributive law of multiplication, multip lication by a scalar function commutes
with the {◦}notation; i.e., A(r)/braceleftbig
W[N]δ[2P]/bracerightbig
=/braceleftbig
A(r)W[N]δ[2P]/bracerightbig
.
The number of terms in various sums, {◦}, is required repeatedly:/braceleftbig
δ[2P]/bracerightbig
has (2 P−1)!! =
(2P−1)(2P−3)(2P−5)· · ·(1) terms. Since W[N]δ[2P]has 2P+Nindexes,/braceleftbig
W[N]δ[2P]/bracerightbig
has/parenleftbig2P+N
N/parenrightbig
(2P−1)!!
terms, where the binomial coefficient/parenleftbig2P+N
N/parenrightbig
is the number of ways of selecting the Nindexes in W[N]from the
total 2 P+Nindexes. If iis an index in/braceleftbig
W[N]δ[2P]/bracerightbig
, then/braceleftbig
W[N]δ[2P]/bracerightbig
has/parenleftbig2P+N−1
N/parenrightbig
(2P−1)!! terms in which i
appears in a Kronecker delta because there are Nindexes to select for W[N]from the remaining 2 P+N−1 indexes.
Similarly,/braceleftbig
W[N]δ[2P]/bracerightbig
has/parenleftbig2P+N−1
N−1/parenrightbig
(2P−1)!! terms in which iappears in a factor ( ri/r) because there are N−1
indexes remaining to select for W[N]from the remaining 2 P+N−1 indexes. Note that/braceleftbig
W[N]/bracerightbig
has only 1 term.
Hence,/braceleftbig
W[N]/bracerightbig
=W[N]. In summary,
/braceleftbig
W[N]δ[2P]/bracerightbig
has/parenleftbigg2P+N
N/parenrightbigg
(2P−1)!! terms; (49)
/braceleftbig
W[N]δ[2P]/bracerightbig
has/parenleftbigg2P+N−1
N/parenrightbigg
(2P−1)!! terms with iinδ[2P]; (50)
/braceleftbig
W[N]δ[2P]/bracerightbig
has/parenleftbigg2P+N−1
N−1/parenrightbigg
(2P−1)!! terms with iinW[N]. (51)
The sum of the number of terms in (50-51), namely/parenleftbig2P+N−1
N/parenrightbig
(2P−1)!! +/parenleftbig2P+N−1
N−1/parenrightbig
(2P−1)!! =
(2P+N−1)!/bracketleftig
2P
N!(2P)!+N
N!(2P)!/bracketrightig
(2P−1)!! =/parenleftbig2P+N
N/parenrightbig
(2P−1)!!, agrees with the total number of terms in (49).
Now, rules for differentiation of symmetric, isotropic tens ors are developed. Note the identity
∂ri(rj/r) =/bracketleftbig
δij−/parenleftbig
rirj/r2/parenrightbig/bracketrightbig
/r=/parenleftbig
δ[2]−W[2]/parenrightbig
/r, (52)
from which it follows that
∂ri(ri/r) = (3 −1)/r= 2/r, and ri∂ri(rj/r) =/bracketleftbig
rj−/parenleftbig
r2rj/r2/parenrightbig/bracketrightbig
/r= 0. (53)
The latter formula greatly simplifies the divergence of W[N]because ∂rioperating on W[N]vanishes when it operates
on any factor other than the factorri
rwithin W[N].
The divergence and gradient of W[N]are needed. If iis an index in W[N]then the divergence of W[N]is denoted
by∇r·W[N]=∂riW[N]. Application of (53) gives ∂riW[1]=∂ri(ri/r) = 2/r, and ∂riW[N]=W[N−1]∂ri(ri/r) =
2
rW[N−1]. The gradient of W[N]is denoted by ∂riW[N]where iis not an index in W[N]. From the differ-
entiation chain rule, ∂riW[N]is the sum of Nterms, each of which has the form W[N−1]∂ri(rj/r). Therefore,
by use of (52), ∂riW[N]=/arrowdbltp/arrowdblbtW[N−1]/bracketleftbig
δij−/parenleftbig
rirj/r2/parenrightbig/bracketrightbig
/r/arrowdbltp/arrowdblbtN
j=1
r/arrowdbltp/arrowdblbtW[N−1]δij/arrowdbltp/arrowdblbtN
j−N
rW[N+1], where use was made
of/arrowdbltp/arrowdblbtW[N−1]/bracketleftbig
−/parenleftbig
rirj/r2/parenrightbig/bracketrightbig
/r/arrowdbltp/arrowdblbtN
j= [−(ri/r)/r]/arrowdbltp/arrowdblbtW[N−1](rj/r)/arrowdbltp/arrowdblbtN
j=−(ri/r)/r/parenleftbig
NW[N]/parenrightbig
=−N
rW[N+1], because
/arrowdbltp/arrowdblbtW[N−1](rj/r)/arrowdbltp/arrowdblbtN
jis the sum of Nidentical terms each equal to W[N]. In summary,
Ifiis inW[N]then ∇r·W[N]=∂riW[N]=2
rW[N−1]. (54)
Ifiis not in W[N]then∂riW[N]=1
r/arrowdbltp/arrowdblbtW[N−1]δij/arrowdbltp/arrowdblbtN
j−N
rW[N+1], (55)
Consider the divergence of W[N]δ[2P].If the index iis inW[N]then, from (54), ∂ri/parenleftbig
W[N]δ[2P]/parenrightbig
=
δ[2P]W[N|1]=2
rδ[2P]W[N−1]. If the index iis inδ[2P], then (given that index kis not in W[N])∂ri/parenleftbig
W[N]δ[2P]/parenrightbig
=
10δ[2(P−1)]δik∂riW[N]=δ[2(P−1)]∂rkW[N]=δ[2(P−1)]1
r/arrowdbltp/arrowdblbtW[N−1]δkj/arrowdbltp/arrowdblbtN
j−δ[2(P−1)]N
rW[N+1], where the last expression
follows from (55). In summary,
Ifiis inW[N]then∂ri/parenleftbig
W[N]δ[2P]/parenrightbig
=2
rδ[2P]W[N−1], (56)
Ifiis inδ[2P]then∂ri/parenleftbig
W[N]δ[2P]/parenrightbig
=δ[2(P−1)]1
r/arrowdbltp/arrowdblbtW[N−1]δkj/arrowdbltp/arrowdblbtN
j−δ[2(P−1)]N
rW[N+1]. (57)
The above results allow evaluation of the divergence ∂ri/braceleftbig
W[N]δ[2P]/bracerightbig
≡ ∇r·/braceleftbig
W[N]δ[2P]/bracerightbig
. It follows from use
of (56) and the distributive law of multiplication and the fa ct that the number of terms in/braceleftbig
W[N]δ[2P]/bracerightbig
in which i
appears in the factorri
ris the same as the number of terms in/braceleftbig
W[N−1]δ[2P]/bracerightbig
[see (49-51)], that for those terms in
which iis inW[N], the divergence of/braceleftbig
W[N]δ[2P]/bracerightbig
yields2
r/braceleftbig
W[N−1]δ[2P]/bracerightbig
. Similar use of (57) gives that for those
terms in which iis inδ[2P], the divergence of/braceleftbig
W[N]δ[2P]/bracerightbig
yields2P
r/braceleftbig
W[N−1]δ[2P]/bracerightbig
−N(N+1)
r/braceleftbig
W[N+1]δ[2(P−1)]/bracerightbig
.
Thus,
∇r·/braceleftbig
W[N]δ[2P]/bracerightbig
=2
r/braceleftbig
W[N−1]δ[2P]/bracerightbig
+/bracketleftbigg2P
r/braceleftbig
W[N−1]δ[2P]/bracerightbig
−N(N+ 1)
r/braceleftbig
W[N+1]δ[2(P−1)]/bracerightbig/bracketrightbigg
=2
r(P+ 1)/braceleftbig
W[N−1]δ[2P]/bracerightbig
−N(N+ 1)
r/braceleftbig
W[N+1]δ[2(P−1)]/bracerightbig
. (58)
Because of the definitions in (46), (58) remains valid if Nis 0 or 1 or if Pis 0 or 1.
Derivation of the formula for the divergence of an isotropic tensor requires evaluation of the contraction/braceleftbig
W[N]δ[2P]/bracerightbigri
r. From (50), in/braceleftbig
W[N]δ[2P]/bracerightbig
there are/parenleftbig2P+N−1
N/parenrightbig
(2P−1)!! occurrences of the index iwithin δ[2P]and
each gives the contraction δijri
r=rj
r, which decreases Pby unity and increases Nby unity thereby producing several/braceleftbig
W[N+1]δ[2(P−1)]/bracerightbig
. From (49), there are/parenleftbig2(P−1)+(N+1)
(N+1)/parenrightbig
(2 (P−1)−1)!! terms in a/braceleftbig
W[N+1]δ[2(P−1)]/bracerightbig
; thus the
number of/braceleftbig
W[N+1]δ[2(P−1)]/bracerightbig
so produced is/bracketleftig/parenleftbig2P+N−1
N/parenrightbig
(2P−1)!!/bracketrightig
//bracketleftig/parenleftbig2(P−1)+(N+1)
(N+1)/parenrightbig
(2 (P−1)−1)!!/bracketrightig
= (N+ 1).
From (51), the contraction/braceleftbig
W[N]δ[2P]/bracerightbigri
rcontains/parenleftbig2P+N−1
N−1/parenrightbig
(2P−1)!! terms in which iappears within W[N]and
each results in the contractionri
rri
r= 1, which decreases Nby unity. The number of/braceleftbig
W[N−1]δ[2P]/bracerightbig
so produced is/bracketleftig/parenleftbig2P+N−1
N−1/parenrightbig
(2P−1)!!/bracketrightig
//bracketleftig/parenleftbig2P+(N−1)
(N−1)/parenrightbig
(2P−1)!!/bracketrightig
= 1 because/braceleftbig
W[N−1]δ[2P]/bracerightbig
has/parenleftbig2P+(N−1)
(N−1)/parenrightbig
(2P−1)!! terms, which
is also the number of terms given in (51). Thus,
contraction on i:/braceleftbig
W[N]δ[2P]/bracerightbigri
r= (N+ 1)/braceleftbig
W[N+1]δ[2(P−1)]/bracerightbig
+/braceleftbig
W[N−1]δ[2P]/bracerightbig
. (59)
The general isotropic formula for a tensor A[N](r) of order Nthat is symmetric under interchange of any pair
of indexes is
A[N](r) =A0(r)/braceleftbig
W[N]/bracerightbig
+A1(r)/braceleftbig
W[N−2]δ[2]/bracerightbig
+A2(r)/braceleftbig
W[N−4]δ[4]/bracerightbig
+· · ·+Tlast
=M/summationdisplay
P=0AP(r)/braceleftbig
W[N−2P]δ[2P]/bracerightbig
, (60)
where the A0(r),A1(r), etc., are scalar functions of r, and Tlastis the last term. Note that for brevity in this
appendix, the subscript Nhas been omitted from AN,0(r),AN,1(r), etc. If Nis even, then Tlast=AN/2(r)/braceleftbig
δ[N]/bracerightbig
andM=N/2. If Nis odd, then Tlast=A(N−1)/2(r)/braceleftbig
W[1]δ[N−1]/bracerightbig
andM= (N−1)/2.
All of the foregoing has set the stage for efficient derivation of a formula for the divergence ∇r·A[N](r). Also
needed is the fact that the gradient of a scalar function of r≡√ririis∂riA(r) = (∂rir)∂rA(r) =ri
r∂rA(r) =
W[1]∂rA(r). Consider the divergence of a term in (60). By use of the diffe rentiation chain rule, and substitution of
(59) and (58),
∇r·/bracketleftbig
AP(r)/braceleftbig
W[N−2P]δ[2P]/bracerightbig/bracketrightbig
=
/braceleftbig
W[N−2P]δ[2P]/bracerightbigri
r∂rAP(r) +AP(r)∇r·/braceleftbig
W[N−2P]δ[2P]/bracerightbig
=/bracketleftbig
(N−2P+ 1)/braceleftbig
W[N−2P+1]δ[2(P−1)]/bracerightbig
+/braceleftbig
W[N−2P−1]δ[2P]/bracerightbig/bracketrightbig
∂rAP(r)
+AP(r)/bracketleftbigg2
r(P+ 1)/braceleftbig
W[N−2P−1]δ[2P]/bracerightbig/bracketrightbigg
11−AP(r)/bracketleftbigg(N−2P)(N−2P+ 1)
r/braceleftbig
W[N−2P+1]δ[2(P−1)]/bracerightbig/bracketrightbigg
=BN,P(r)/braceleftbig
W[N−2P+1]δ[2(P−1)]/bracerightbig
+CP(r)/braceleftbig
W[N−2P−1]δ[2P]/bracerightbig
, (61)
where BN,P(r) and CP(r) are defined by the following operators, OB(N, P) and OC(P), operating on AP(r):
BN,P(r)≡OB(N, P)AP(r),where OB(N, P)≡(N−2P+ 1)/bracketleftbigg
∂r−N−2P
r/bracketrightbigg
, (62)
CP(r)≡OC(P)AP(r) where OC(P)≡/bracketleftbigg
∂r+2
r(P+ 1)/bracketrightbigg
. (63)
Thereby, the divergence of (60) is
∇r·A[N](r) =M/summationdisplay
P=0BN,P(r)/braceleftbig
W[N−2P+1]δ[2(P−1)]/bracerightbig
+M/summationdisplay
P=0CP(r)/braceleftbig
W[N−2P−1]δ[2P]/bracerightbig
, (64)
where (61) and (46) were used.
Now, (64) can be checked by comparison with the divergence pe rformed on the explicit-index formulas for
symmetric, isotropic tensors of rank 1 to 4. The lowest-orde r tensor for which the divergence is defined is a vector
(i.e.,N= 1), in which case (64) gives
∂riAi(r) =∂rA0(r) +2
rA0(r),
which is easily verified by evaluating the divergence of a iso tropic vector, namely ∂ri/bracketleftbig
A0(r)ri
r/bracketrightbig
. Expressed with
explicit indexes as well as in the implicit-index form of (60 ), isotropic tensors of rank 2 to 4 that are symmetric under
interchange of any pair of indexes are:
Aij(r) =A0(r)ri
rrj
r+A1(r)δij=A0(r)W[2]+A1(r)δ[2]. (65)
Aijk(r) =A0(r)ri
rrj
rrk
r+A1(r)/parenleftigri
rδjk+rj
rδik+rk
rδij/parenrightig
=A0(r)W[3]+A1(r)/braceleftbig
W[1]δ[2]/bracerightbig
. (66)
Aijkl(r) =A0(r)rirjrkrl
r4+A1(r)/parenleftigrirj
r2δkl+rirk
r2δjl+rjrk
r2δil+rirl
r2δjk+rjrl
r2δik+rkrl
r2δij/parenrightig
+A2(r) (δijδkl+δikδjl+δjkδil)
=A0(r)W[4]+A1(r)/braceleftbig
W[2]δ[2]/bracerightbig
+A2(r)/braceleftbig
δ[4]/bracerightbig
. (67)
One can see the brevity of the implicit-index formula as the r ank of the tensor increases. The first-order divergences
obtained by differentiating the above explicit-index formu las as well as from (64) are:
∇r·A[2](r) =/bracketleftbigg/parenleftbigg
∂r+2
r/parenrightbigg
A0(r) +∂rA1(r)/bracketrightbiggrj
r(68)
= [B2,1(r) +C0(r)]/braceleftbig
W[1]δ[0]/bracerightbig
. (69)
∇r·A[3](r) =/bracketleftbigg/parenleftbigg
∂r+2
r/parenrightbigg
A0(r) +/parenleftbigg
2∂r−2
r/parenrightbigg
A1(r)/bracketrightbiggrjrk
r2+/bracketleftbigg/parenleftbigg
∂r+4
r/parenrightbigg
A1(r)/bracketrightbigg
δjk (70)
= [B3,1(r) +C0(r)]/braceleftbig
W[2]δ[0]/bracerightbig
+C1(r)/braceleftbig
W[0]δ[2]/bracerightbig
. (71)
∇r·A[4](r) =/bracketleftbigg/parenleftbigg
∂r+2
r/parenrightbigg
A0(r) +/parenleftbigg
3∂r−6
r/parenrightbigg
A1(r)/bracketrightbiggrjrkrl
r3
+/bracketleftbigg/parenleftbigg
∂r+4
r/parenrightbigg
A1(r) +∂rA2(r)/bracketrightbigg/parenleftigri
rδjk+rj
rδik+rk
rδij/parenrightig
(72)
= [B4,1(r) +C0(r)]/braceleftbig
W[3]δ[0]/bracerightbig
+ (B4,2(r) +C1(r))/braceleftbig
W[1]δ[2]/bracerightbig
. (73)
12In the implicit-index formulas in (69, 71, 73), terms from (6 4) that are zero because of (46) have been omitted.
Equation (64) has been checked by using the implicit-index f ormulas in (69, 71, 73) to obtain the explicit-index
formulas in (68, 70, 72), respectively.
The Laplacian of a symmetric, isotropic tensor is also neede d for the term 2 ν∇2
rD[N]in (18). Application of
(36) to ∇2W[N−2P]and use of (52-53) gives
∇2W[N−2P]=/braceleftig
W[N−2P−1]∂rn∂rnrj
r/bracerightig
+ 2/braceleftig
W[N−2P−2]/parenleftig
∂rnrk
r/parenrightig/parenleftig
∂rnrj
r/parenrightig/bracerightig
. (74)
Now,/braceleftbig
W[N−2P−1]∂rn∂rnrj
r/bracerightbig
is/parenleftbigN−2P
1/parenrightbig
terms, each one is of the form W[N−2P−1]∂rn∂rnrj
r=W[N−2P−1]/parenleftbig−2
r2rj
r/parenrightbig
=
−2
r2W[N−2P]. Also,/braceleftbig
W[N−2P−2]/parenleftbig
∂rnrk
r/parenrightbig/parenleftbig
∂rnrj
r/parenrightbig/bracerightbig
is/parenleftbigN−2P
2/parenrightbig
terms, each one is of the form
W[N−2P−2]/parenleftbig
∂rnrk
r/parenrightbig/parenleftbig
∂rnrj
r/parenrightbig
=W[N−2P−2]1
r2/parenleftbig
δkj−rkrj
r2/parenrightbig
=1
r2W[N−2P−2]δij−1
r2W[N−2P]. From (49),/braceleftbig
W[N−2P−2]δ[2]/bracerightbig
has/parenleftbigN−2P
2/parenrightbig
terms; thus, (74) gives
∇2W[N−2P]=2
r2/braceleftbig
W[N−2P−2]δ[2]/bracerightbig
−2
r2/bracketleftbigg/parenleftbiggN−2P
2/parenrightbigg
+/parenleftbiggN−2P
1/parenrightbigg/bracketrightbigg
W[N−2P]. (75)
The binomial coefficients prevent a nonzero term in (75) when W[N−2P−1]orW[N−2P−2]vanish in (74) as required
by definition (46) provided that we define
/parenleftbiggN−2P
1/parenrightbigg
≡0 ifN−2P <1,and/parenleftbiggN−2P
2/parenrightbigg
≡0 ifN−2P <2. (76)
Of course, (76) is consistent with 1 /K! = 0 for K <0 (Abramowitz and Stegun, 1964, equation 6.1.7). Given (76) ,
we can define, for brevity
SN−2P≡2/parenleftbiggN−2P
2/parenrightbigg
+ 2/parenleftbiggN−2P
1/parenrightbigg
. (77)
Now (75) and (77) give
∇2W[N−2P]=2
r2/braceleftbig
W[N−2P−2]δ[2]/bracerightbig
−SN−2P
r2W[N−2P]. (78)
Now, use of (78) gives
∇2/braceleftbig
W[N−2P]δ[2P]/bracerightbig
=/braceleftbig/bracketleftbig
∇2W[N−2P]/bracketrightbig
δ[2P]/bracerightbig
=/braceleftbigg/bracketleftbigg2
r2/braceleftbig
W[N−2P−2]δ[2]/bracerightbig
−SN−2P
r2W[N−2P]/bracketrightbigg
δ[2P]/bracerightbigg
=2
r2/braceleftbig/braceleftbig
W[N−2P−2]δ[2]/bracerightbig
δ[2P]/bracerightbig
−SN−2P
r2/braceleftbig
W[N−2P]δ[2P]/bracerightbig
(79)
=R(N, P)
r2/braceleftbig
W[N−2(P+1)]δ[2(P+1)]/bracerightbig
−SN−2P
r2/braceleftbig
W[N−2P]δ[2P]/bracerightbig
(80)
Noting the appearance of W[N−2P−2]in (79) and recalling that W[N−2P−2]= 0 if N−2P−2<0, the coefficient
R(N, P) is defined by
R(N, P)≡0 ifN−2P−2<0, (81)
otherwise, R(N, P)≡2/parenleftbiggN−2P
2/parenrightbigg
//bracketleftbigg/parenleftbiggN
2P+ 2/parenrightbigg
(2P+ 1)!!/bracketrightbigg
. (82)
The coefficient/parenleftbigN−2P
2/parenrightbig
//bracketleftig/parenleftbigN
2P+2/parenrightbig
(2P+ 1)!!/bracketrightig
is the number of terms in/braceleftbig
W[N−2P−2]δ[2]/bracerightbig
divided by the number in/braceleftbig
W[N−2(P+1)]δ[2(P+1)]/bracerightbig
as obtained from (49). Because of (43), (81-82) can be simpli fied to
R(N, P)≡/bracketleftbig
2P+1(N−2P)! (P+ 1)!/N!/bracketrightbig
Θ (N−2P−2), (83)
where Θ ( x) = 1 for x≥0 and Θ ( x) = 0 for x <0.
13The Laplacian of the product of two functions fandgis (33). When applied to (60), the case f=AP(r)
andg=/braceleftbig
W[N−2P]δ[2P]/bracerightbig
are needed. Recall that ∂riA(r) =ri
r∂rA(r). The last term in (33) vanishes as follows:
(∂rif)(∂rig) =/bracketleftbig1
r∂rAP(r)/bracketrightbig
ri∂ri/braceleftbig
W[N−2P]δ[2P]/bracerightbig
=/bracketleftbig1
r∂rA(r)/bracketrightbig/braceleftbig/bracketleftbig
ri∂riW[N−2P]/bracketrightbig
δ[2P]/bracerightbig
= 0; this vanishes because
(53) shows that ri∂riW[N−2P]= 0. Then (80) used in (33) combined with ∇2A(r) =/parenleftbig
∂2
r+2
r∂r/parenrightbig
A(r) give
∇2/bracketleftbig
AP(r)/braceleftbig
W[N−2P]δ[2P]/bracerightbig/bracketrightbig
=/bracketleftbigg/parenleftbigg
∂2
r+2
r∂r−SN−2P
r2/parenrightbigg
AP(r)/bracketrightbigg/braceleftbig
W[N−2P]δ[2P]/bracerightbig
+AP(r)R(N, P)
r2/braceleftbig
W[N−2(P+1)]δ[2(P+1)]/bracerightbig
. (84)
The Laplacian operation on (60) is simply the sum,M/summationtext
P=0, of terms (84).
X. APPENDIX F: MATRIX ALGORITHMS
For computations, it is useful to write (21) as a matrix equat ion. Let the column index be J≡P+ 1, and
the row index be I≡(N2/2) + 1, such that both JandIrange from 1 to M+ 1 in (21). Use N3= 0 in (22-23) to
define the following matrix elements
MN(I, J) = 0, for J < I, i.e., MN(I, J) = 0 below the main diagonal; (85)
MN(I, J) = (N−2I+ 2)! (2 I−2)!//bracketleftbig
(N−2 (J−1))!2J−1(I−1)! (J−I)!/bracketrightbig
, forJ≥I. (86)
The chosen M+1 linearly independent components of D[N]are arranged in a column vector having D[N:N−2I+2,2I−2,0]
as in its I-th row, and the M+ 1 scalar functions DN,Pare likewise arranged in a column vector having DN,I−1in
itsI-th row. Then (21) is written as the matrix equation
D[N:N,0,0]
D[N:N−2,2,0]
...
D[N:N−2M,2M,0]
=
MN(1,1)MN(1,2)· · · MN(1, M+ 1)
0 MN(2,2)· · · MN(2, M+ 1)
............
0 0 · · ·MN(M+ 1, M+ 1)
DN,0
DN,1
...
DN,M
. (87)
Denote a matrix having matrix elements A(I, J) byA(I, J)Then (87) and its solution are (respectively)
D[N:N−2J+2,2J−2,0]=MN(I, J)DN,I−1,andDN,I−1=MN(I, J)−1D[N:N−2J+2,2J−2,0], (88)
where MN(I, J)−1is the inverse of MN(I, J). The determinant of MN(I, J)is the product of its diagonal elements;
from (86) that product is nonzero, hence MN(I, J)−1exists. This inverse matrix is to be calculated numerically . In
effect, evaluation of the components D[N:N−2J+2,2J−2,0]by means of experimental data or DNS data and use of the
solution in (88) produces the DN,Pfor use in (21) to completely specify D[N].
A matrix algorithm is useful for determining the isotropic f ormula for the first-order divergence ∇r·D[N+1].
By replacing NbyN+ 1 and the symbol AbyDin the divergence formula (64), we have
∇r·D[N+1]=M′/summationdisplay
P=0/braceleftbig
W[N−2(P−1)]δ[2(P−1)]/bracerightbig
OB(N+ 1, P)DN+1,P
+M′/summationdisplay
P=0/braceleftbig
W[N−2P]δ[2P]/bracerightbig
OC(P)DN+1,P, (89)
OB(N+ 1, P)≡((N+ 1)−2P+ 1)/parenleftbigg
∂r−(N+ 1)−2P
r/parenrightbigg
, (90)
OC(P)≡/bracketleftbigg
∂r+2 (P+ 1)
r/bracketrightbigg
, (91)
M′=N/2 ifNis even, and M′= 1 + ( N−1)/2 ifNis odd. (92)
14The differential operators, i.e., ∂r≡∂/∂r, in (90-91) are obtained from (62-63), and (92) is obtained b y replacing
NbyN+ 1 in (20) and simplifying and rearranging the terms. Compar ison of (92) with (20) shows that if Nis
even then M′=M; thus the matrix representation of/braceleftbig
W[N−2P]δ[2P]/bracerightbig
within (89) is the same as in (87), which
representation was abbreviated by MN(I, J)above. On the other hand, if Nis odd, then M′=M+ 1, and the
last column of the matrix representation of/braceleftbig
W[N−2P]δ[2P]/bracerightbig
within (89) corresponds to P=M′= 1 + ( N−1)/2, in
which case/braceleftbig
W[N−2P]δ[2P]/bracerightbig
contains W[−1]= 0 such that the last column of the matrix is zero. Thus, the ma trix
representation of/braceleftbig
W[N−2P]δ[2P]/bracerightbig
within (89) is
M∗
N(I, J)=
0
MN(I, J)...
0
ifNis odd; (93)
M∗
N(I, J)=MN(I, J)ifNis even. (94)
In addition to the coefficient/braceleftbig
W[N−2P]δ[2P]/bracerightbig
, (89) contains the coefficient/braceleftbig
W[N−2(P−1)]δ[2(P−1)]/bracerightbig
. From
the matrix representation of/braceleftbig
W[N−2P]δ[2P]/bracerightbig
, namely (85-86), the matrix representation of/braceleftbig
W[N−2(P−1)]δ[2(P−1)]/bracerightbig
is (recall that J≡P+ 1)
M′
N(I, J) = 0, for J−1< I, i.e., M′
N(I, J) = 0 on and below the main diagonal; (95)
M′
N(I, J) = (N−2I+ 2)! (2 I−2)!//bracketleftbig
(N−2 (J−2))!2J−2(I−1)! (J−1−I)!/bracketrightbig
, forJ−1≥I. (96)
The matrix having these elements is denoted by M′
N(I, J). Because of (92), if Nis odd, then the matrix M′
N(I, J)
contains the matrix MN(I, J)shifted to the right by one column and a first column of zeros is included; that is,
M′
N(I, J)=
0
...MN(I, J−1)
0
ifNis odd. (97)
Because of (92), the same is true if Nis even except that the right-most column of MN(I, J)is discarded. Thus,
M′
N(I, J)=
0
...MN(I, J−1)
0
ifNis even (discard the right-most column). (98)
Define operator matrices that are of dimension M′+1 by M′+ 1, that have zeros off of the diagonal, and that
have the operators (90-91) on the diagonals. Thus, recall th atJ≡P+ 1, and that ∂r≡∂/∂r, and define matrix
elements
B(I, J)≡δIJ(N−2J+ 4)/parenleftbigg
∂r−N−2J+ 3
r/parenrightbigg
andC(I, J)≡δIJ/parenleftbigg
∂r+2J
r/parenrightbigg
. (99)
The matrices corresponding to OB(N+ 1, P) andOC(P) in (90-91) are denoted by B(I, J), andC(I, J), respectively.
Let the components of ∇r·D[N+1]be denoted by/parenleftbig
∇r·D[N+1]/parenrightbig
[N:N1,N2,N3], which denotes the fact that
∇r·D[N+1]is a tensor of order N. In matrix notation, (89) gives
/parenleftbig
∇r·D[N+1]/parenrightbig
[N:N,0,0] /parenleftbig
∇r·D[N+1]/parenrightbig
[N:N−2,2,0]
.../parenleftbig
∇r·D[N+1]/parenrightbig
[N:N−2M,2M,0]
=/bracketleftig
M′
N(I, J)B(I, J)+M∗
N(I, J)C(I, J)/bracketrightig
DN+1,0
DN+1,1
...
DN+1,M′
. (100)
When applied to D[N+1], the solution of (87) is DN+1,I−1=MN+1(I, J)−1D[N+1:N+1−2J+2,2J−2,0], substitution
of which into (100) gives
15
/parenleftbig
∇r·D[N+1]/parenrightbig
[N:N,0,0] /parenleftbig
∇r·D[N+1]/parenrightbig
[N:N−2,2,0]
.../parenleftbig
∇r·D[N+1]/parenrightbig
[N:N−2M,2M,0]
=Y(I, J)
D[N+1:N+1,0,0]
D[N+1:N+1−2,2,0]
...
D[N+1:N+1−2M′,2M′,0]
, (101)
where, Y(I, J)≡/bracketleftig
M′
N(I, J)B(I, J)+M∗
N(I, J)C(I, J)/bracketrightig
MN+1(I, J)−1. (102)
We see that Y(I, J)is the operator matrix that operates on the column matrix rep resentation of D[N+1]to produce
∇r·D[N+1]; this is true for any completely symmetric isotropic tensor , not just true for D[N+1].
It is helpful to illustrate this algorithm for N= 2 and N= 3. Two examples are needed because the algorithm
differs for even Nas compared to odd N. For N= 2, (102) is
Y(I, J)=
/parenleftbigg
0M2(1,1)
0 0/parenrightbigg/parenleftbigg
B(1,1) 0
0B(2,2)/parenrightbigg
+
+/parenleftbigg
M2(1,1)M2(1,2)
0 M2(2,2)/parenrightbigg/parenleftbigg
C(1,1) 0
0C(2,2)/parenrightbigg
·/parenleftbigg
M3(1,1)M3(1,2)
0 M3(2,2)/parenrightbigg−1
(103)
=/parenleftbigg
∂r+2
r−4
r
0∂r+4
r/parenrightbigg
. (104)
Computer evaluation of (103) produced (104). Consequently , (101) is
/parenleftigg/parenleftbig
∇r·D[3]/parenrightbig
[2:2,0,0] /parenleftbig
∇r·D[3]/parenrightbig
[2:0,2,0]/parenrightigg
=/parenleftbigg
∂r+2
r−4
r
0∂r+4
r/parenrightbigg/parenleftbigg
D[3:3,0,0]
D[3:1,2,0]/parenrightbigg
=/parenleftbigg/parenleftbig
∂r+2
r/parenrightbig
D111−4
rD122 /parenleftbig
∂r+4
r/parenrightbig
D122/parenrightbigg
, (105)
where explicit-index notation is given at far right by use of D[3:3,0,0]≡D111andD[3:1,2,0]≡D122.
ForN= 3,Y(I, J)from (102) is
/parenleftbigg
0M3(1,1)M3(1,2)
0 0 M3(2,2)/parenrightbigg
B(1,1) 0 0
0B(2,2) 0
0 0 B(3,3)
+
/parenleftbigg
M3(1,1)M3(1,2) 0
0 M3(2,2) 0/parenrightbigg
C(1,1) 0 0
0C(2,2) 0
0 0 C(3,3)
·
M4(1,1)M4(1,2)M4(1,3)
0 M4(2,2)M4(2,3)
0 0 M4(3,3)
−1
=/parenleftbigg
∂r+2
r−6
r0
0∂r+4
r−4
3r/parenrightbigg
As with (104), the matrix was evaluated using a computer prog ram. Consequently, (101) is
/parenleftigg/parenleftbig
∇r·D[4]/parenrightbig
[3:3,0,0] /parenleftbig
∇r·D[4]/parenrightbig
[3:1,2,0]/parenrightigg
=/parenleftbigg
∂r+2
r−6
r0
0∂r+4
r−4
3r/parenrightbigg
D[4:4,0,0]
D[4:2,2,0]
D[4:0,4,0]
(106)
=/parenleftbigg /parenleftbig
∂r+2
r/parenrightbig
D1111−6
rD1122 /parenleftbig
∂r+4
r/parenrightbig
D1122−4
3rD2222/parenrightbigg
, (107)
where explicit-index notation is used in (107).
A matrix algorithm is also needed for the Laplacian of a symme tric tensor. Performing the Laplacian of (60)
and use of (84) gives
∇2
rD[N](r) =M/summationdisplay
P=0/parenleftigg/braceleftbig
W[N−2P]δ[2P]/bracerightbig/parenleftig
∂2
r+2
r∂r−SN−2P
r2/parenrightig
DN,P
+/braceleftbig
W[N−2(P+1)]δ[2(P+1)]/bracerightbigR(N,P)
r2DN,P/parenrightigg
. (108)
It is necessary to recall the definitions (77) and (83). The ma trix representation of/braceleftbig
W[N−2P]δ[2P]/bracerightbig
within (108)
is the same as in (85-86), namely MN(I, J). The matrix representation of/braceleftbig
W[N−2(P+1)]δ[2(P+1)]/bracerightbig
within (108) is
obtained from (85-86) by replacing JbyJ+ 1, i.e.,
16M#
N(I, J) = 0, for J+ 1< I, (109)
M#
N(I, J) = (N−2I+ 2)! (2 I−2)!//bracketleftbig
(N−2J)!2J(I−1)! (J+ 1−I)!/bracketrightbig
, forJ+ 1≥I. (110)
This is just the square matrix that appears in (87) except tha t the left-most column in (87) is discarded and the
matrix is then shifted leftward by one column and the right-m ost column is zeros. Those zeros appear because in
the right-most column J=P+ 1 = M+ 1 such that M#
N(I, M+ 1) contains the factor 1 /(N−2 (M+ 1))! which is
1/(−2)! = 0 if Nis even and is 1 /(−1)! = 0 if Nis odd(see Abramowitz and Stegun, 1964, equation 6.1.7). Th us,
M#
N(I, J)=
0
MN(I, J+ 1)...
0
(discard the left-most column).
Define two operator matrices that are zero off the main diagona l and contain/parenleftig
∂2
r+2
r∂r−SN−2P
r2/parenrightig
andR(N,P)
r2
on the main diagonal; i.e., their matrix elements are E(I, J) = δIJ/parenleftig
∂2
r+2
r∂r−SN−2(J−1)
r2/parenrightig
andF(I, J) =
δIJR(N, J−1)/r2. Analogous to the derivation of (101), the matrix represent ation of (108) is
/parenleftbig
∇2
rD[N]/parenrightbig
[N:N,0,0] /parenleftbig
∇2
rD[N]/parenrightbig
[N:N−2,2,0]
.../parenleftbig
∇2
rD[N]/parenrightbig
[N:N−2M,2M,0]
=X(I, J)
D[N:N,0,0]
D[N:N−2,2,0]
...
D[N:N−2M,2M,0]
, (111)
where, X(I, J)≡/bracketleftbigg
MN(I, J)E(I, J)+M#
N(I, J)F(I, J)/bracketrightbigg
MN(I, J)−1.
For both N= 2 and N= 3, the matrix representation of X(I, J)is
X(I, J)≡
/parenleftbigg
MN(1,1)MN(1,2)
0 MN(2,2)/parenrightbigg/parenleftbigg
E(1,1) 0
0E(2,2)/parenrightbigg
+/parenleftbigg
MN(1,2) 0
MN(2,2) 0/parenrightbigg/parenleftbigg
F(1,1) 0
0F(2,2)/parenrightbigg
/parenleftbigg
MN(1,1)MN(1,2)
0 MN(2,2)/parenrightbigg−1
ForN= 2 (111) is
/parenleftigg/parenleftbig
∇2
rD[2]/parenrightbig
[2:2,0,0] /parenleftbig
∇2
rD[2]/parenrightbig
[2:0,2,0]/parenrightigg
=X(I, J)/parenleftbigg
D[2:2,0,0]
D[2:0,2,0]/parenrightbigg
=/parenleftbigg/parenleftbig
∇2
rD[2]/parenrightbig
11/parenleftbig
∇2
rD[2]/parenrightbig
22/parenrightbigg
=
/parenleftbigg
∂2
r+2
r∂r−4
r24
r2
2
r2 ∂2
r+2
r∂r−2
r2/parenrightbigg/parenleftbigg
D11
D22/parenrightbigg
=/parenleftbigg/parenleftbig
∂2
r+2
r∂r−4
r2/parenrightbig
D11+4
r2D22
2
r2D11+/parenleftbig
∂2
r+2
r∂r−2
r2/parenrightbig
D22/parenrightbigg
, (112)
where the matrix was evaluated using a computer program. For N= 3 the matrix algorithm is
/parenleftbigg/parenleftbig
∇2
rD[3]/parenrightbig
111/parenleftbig
∇2
rD[3]/parenrightbig
122/parenrightbigg
=X(I, J)/parenleftbigg
D111
D122/parenrightbigg
=/parenleftbigg
∂2
r+2
r∂r−6
r212
r2
2
r2 −8
r2+∂2
r+2
r∂r/parenrightbigg/parenleftbigg
D111
D122/parenrightbigg
=/parenleftbigg /parenleftbig
∂2
r+2
r∂r−6
r2/parenrightbig
D111+12
r2D122
2
r2D111+/parenleftbig
−8
r2+∂2
r+2
r∂r/parenrightbig
D122/parenrightbigg
. (113)
17 |
arXiv:physics/0102056v1 [physics.flu-dyn] 16 Feb 2001Alternatives to Rλ-scaling of Small-Scale Turbulence Statistics
Reginald J. Hilla
National Oceanic and Atmospheric Administration, Environ mental Technology
Laboratory, 325 Broadway, Boulder CO 80305-3328
(PACS 47.27.Gs, 47.27.Jv)
atel:3034976565, fax:3034976181, Reginald.J.Hill@noaa. gov
(February 15, 2014)
Traditionally, trends of universal turbulence statistics are presented versus Rλ, which is the
Reynolds number based on Taylor’s scale λand the root-mean-squared (rms) velocity urms.λ
andurms, and hence Rλ, do not have the attribute of universality. The ratio of rms fl uid-particle
acceleration to rms viscous acceleration, Ra, and the ratio of rms pressure-gradient acceleration to
rms viscous acceleration, R∇p, are alternatives to Rλthat have the advantage of being determined
by the small scales of turbulence. These ratios have the foll owing attributes: Rais a Reynolds
number, RaandR∇pare dimensionless, are composed of statistics of the small s cales of turbulence,
can be evaluated with single-wire hot-wire anemometry, and likeRλ, can be partially evaluated by
means of flow similarity. Experimental data have shown R∇pandRawithRλon the abscissa;
those graphs show the nonuniversal behavior of Rλfor a variety of flows.
I. INTRODUCTION
Reynolds [1] sought, from the Navier-Stokes equation, “the dependence of the character of motion on a relation
between the dimensional properties and the external circum stances of motion.” Assuming that the motion depends
on a single velocity scale Uand length scale c, Reynolds found that the accelerations are of two distinct t ypes
and thereby deduced that the relevant solution of the Navier -Stokes equation “would show the birth of eddies to
depend on some definite value of cρU/µ ,” [1] where ρis the mass density of the fluid and µis the coefficient of
viscosity. Reynolds performed exhaustive experiments tha t demonstrated his deduction, as well as experiments on
the stabilization of fluctuating flow. [1] He discovered the s udden onset of flow instability [1]. The Navier-Stokes
equation is a=∂u/∂t+u·∇u=−∇p+ν∇2u, where pis pressure divided by ρ,ν=µ/ρis kinematic viscosity, uis the
velocity vector, and ais the acceleration. Batchelor [2] discussed (in his Sec. 4. 7) the interpretation of the Reynolds
number as a measure of arelative to the viscous term ν∇2u. He noted that the balance of the Navier-Stokes equation
can also be parameterized in terms of the relative magnitude s of∇pandν∇2u. The latter parameterization does
not technically lead to a Reynolds number, but it will be show n that the two parameterizations become equivalent at
large Reynolds numbers.
To paraphrase Nelkin’s [3] description of Reynolds number s caling: if two turbulent flows have the same
geometry and the same Reynolds number, then their statistic s, when appropriately scaled, should be equal. A
statistic of the small scales of turbulence is an average of q uantities that contain only products of differences, such as
two-point velocity difference or derivatives of velocity. U niversality of the small scales of turbulence is the hypothe sis
that statistics of the small scales, when appropriately sca led, should become equal as Reynolds number increases [3]
[4]; that is, the flow geometry becomes negligible in the limi t that the Reynolds number is infinite. Discovering the
appropriate scaling that results in universality is the top ic of a vast amount of research [3] [4] and will not be pursued
here. The relevance of universality to real turbulent flows i s discussed by Nelkin [3] and Sreenivasan and Antonia [4].
The Reynolds number based on the root-mean-square (rms) of t he longitudinal-velocity component urms≡
/angbracketleftbig
u2
1/angbracketrightbig1/2and Taylor’s length scale λisRλ≡urmsλ/ν, where νis kinematic viscosity, and λ≡urms//angbracketleftBig
(∂u1/∂x1)2/angbracketrightBig1/2
;
angle brackets denote an average. Here, u1andx1are the components of velocity and spatial coordinate in the
direction of the 1-axis; λis the spacing (i.e., the distance between two points) at whi ch the two-term Taylor-series
expansion of the two-point correlation of u1equals one-half of its value at the origin. [5] Thus, λandurmsare natural
scales for presenting measurements of the velocity correla tion function for isotropic turbulence. For decades, Rλhas
been used as the abscissa for presenting statistics that are believed to be universal aspects of small-scale turbulence
(such as velocity derivative statistics normalized by powe rs of/angbracketleftBig
(∂u1/∂x1)2/angbracketrightBig
). The observed trends as Rλincreases
are an often-sought quantification of scaling universality .Rλhas the advantage of being easily measured because
it requires only measurement of u1(which yields ∂u1/∂x1by means of Taylor’s hypothesis); that measurement can
be obtained with a single hot-wire anemometer. Alternative ly, flow similarity can be used to estimate the energy
dissipation rate ε, and by substituting the local-isotropy relationship that ε= 15ν/angbracketleftBig
(∂u1/∂x1)2/angbracketrightBig
,Rλcan be obtained
1fromRλ=u2
rms/(εν/15)1/2. Because Rλdepends on urms, it depends on large-scale geometry of the flow. Nelkin
[3] discussed the nonuniversal attributes of Rλ. As a result of the nonuniversality of Rλ, statistics of the small scales,
e.g., normalized derivative moments, when graphed with Rλon the abscissa, can have different curves corresponding
to dissimilar flows. One example is the derivative moments sh own in Fig. 6 by Belin et al. [6], which shows a
distinctly different trend in the limited range of Rλ= 700 to 1000 for the flow between counter-rotating blades as
compared to data from wind tunnels and the atmospheric surfa ce layer; the latter data is summarized by Sreenivasan
and Antonia (1997). Another example is Fig. 2 (curves c, g, f o nly) of Gotoh and Rogallo [7], which shows that DNS
of differing flows produces different curves for normalized ac celeration variance when Rλis the abscissa. Because of
the nonuniversality of Rλthere is no requirement that the curves lie upon one another.
II. ALTERNATIVES
In addition to graphing such statistics with Rλon the abscissa, it would seem advantageous to use a quantity
on the abscissa that is solely a property of the small scales o f turbulence. That advantage has long been recognized.
[8] [9] [10] Here, we seek a small-scale quantity that has the meaning of a Reynolds number, and it must be measurable
with an instrument no more complex than a single-wire hot-wi re anemometer. Consider the two ratios:
R∇p≡ /angb∇acketleft∇p· ∇p/angb∇acket∇ight1/2//angbracketleftbig
ν2/parenleftbig
∇2u/parenrightbig
·/parenleftbig
∇2u/parenrightbig/angbracketrightbig1/2andRa≡ /angb∇acketlefta·a/angb∇acket∇ight1/2//angbracketleftbig
ν2/parenleftbig
∇2u/parenrightbig
·/parenleftbig
∇2u/parenrightbig/angbracketrightbig1/2. (1)
Because of the intended application to statistical charact eristics of the small scales, it is appropriate to simplify
these ratios on the basis of local isotropy. Indeed, local is otropy is a precondition for universality. [3] [4] On
this basis, /angb∇acketlefta·a/angb∇acket∇ight=/angb∇acketleft∇p· ∇p/angb∇acket∇ight+ν2/angbracketleftbig/parenleftbig
∇2u/parenrightbig
·/parenleftbig
∇2u/parenrightbig/angbracketrightbig
[11] [12]; in which case Ra=/radicalBig
1 +R2
∇pandR∇p=/radicalbig
R2a−1.
In high Reynolds number turbulence, /angb∇acketleft∇p· ∇p/angb∇acket∇ight ≫ν2/angbracketleftbig/parenleftbig
∇2u/parenrightbig
·/parenleftbig
∇2u/parenrightbig/angbracketrightbig
[11]. Although this has been known for a
long time [12] [13] [14], the old estimates based on the joint Gaussian assumption greatly underestimated /angb∇acketleft∇p· ∇p/angb∇acket∇ight.
[15] [11] Because /angb∇acketleft∇p· ∇p/angb∇acket∇ight ≫ν2/angbracketleftbig/parenleftbig
∇2u/parenrightbig
·/parenleftbig
∇2u/parenrightbig/angbracketrightbig
at high Reynolds numbers, Ra=/radicalBig
1 +R2
∇pbecomes Ra≃R∇p.
Furthermore, on the basis of local isotropy and for all Reyno lds numbers, ν2/angbracketleftbig/parenleftbig
∇2u/parenrightbig
·/parenleftbig
∇2u/parenrightbig/angbracketrightbig
=−35ν/angbracketleftBig
(∂u1/∂x1)3/angbracketrightBig
/2
[14] [11], and /angb∇acketleft∇p· ∇p/angb∇acket∇ight= 4∞/integraltext
0r−3[D1111(r) +Dαααα(r)−6D11ββ(r)]dr[15], where D1111(r),Dαααα(r), and
D11ββ(r) are components of the fourth-order velocity structure-fu nction tensor, which is defined by Dijkl(r)≡/angbracketleftbig
(ui−u′
i)/parenleftbig
uj−u′
j/parenrightbig
(uk−u′
k)(ul−u′
l)/angbracketrightbig
, where uiandu′
iare velocity components at spatial points separated by the
vector r, andr≡ |r|; the 1-axis is parallel to the separation vector r;αandβdenote the Cartesian axes perpendicular
to the 1-axis. Thus, αandβare 2 or 3; equally valid options under local isotropy are α=βorα/negationslash=β.
There is
enough cancellation between the positive and negative part s of the integrand, i.e., between r−3[D1111(r) +Dαααα(r)]
and−r−36D11ββ(r), to make evaluation of the integral∞/integraltext
0r−3[D1111(r) +Dαααα(r)−6D11ββ(r)]drdifficult by
means of experimental or DNS data [15] [16] [17]. Hill and Wil czak [15] gave strong arguments that the ratio
Hχ≡∞/integraltext
0r−3[D1111(r) +Dαααα(r)−6D11ββ(r)]dr/∞/integraltext
0r−3D1111(r)dris a universal constant at high Reynolds
numbers. Subsequent studies of the inertial-range exponen ts of structure functions cast some doubt on the uni-
versality of Hχ, but recent research shows equal exponents for the fourth-o rder structure-function components for
an asymptotic inertial range; these developments are revie wed in the Appendix. Universality of Hχis equivalent
to the assertion that /angb∇acketleft∇p· ∇p/angb∇acket∇ightscales with∞/integraltext
0r−3D1111(r)drat high Reynolds numbers. Independent of Reynolds
number, /angb∇acketleft∇p· ∇p/angb∇acket∇ightdoes scale with∞/integraltext
0r−3[D1111(r) +Dαααα(r)−6D11ββ(r)]drfor locally isotropic turbulence in
the sense that ascales with bis proven by a= 4b. Hill and Wilczak [15] pointed out that the utility of de-
termining Hχis that the pressure-gradient variance can then be measured with a single-wire hot-wire anemome-
ter by means of /angb∇acketleft∇p· ∇p/angb∇acket∇ight= 4Hχ∞/integraltext
0r−3D1111(r)dr. Using DNS data, the preferable evaluation of Hχis via
Hχ=/angb∇acketleft∇p· ∇p/angb∇acket∇ight//bracketleftbigg
4∞/integraltext
0r−3D1111(r)dr/bracketrightbigg
so as to avoid the statistical uncertainty caused by the canc ellations within
the integrand. Vedula and Yeung [18] evaluated Hχusing DNS data and obtained a small variation from Hχ≃0.55
2atRλ= 20 to an approach to a constant value of Hχ≃0.65 at their highest Rλ(namely 230). Evaluations at yet
higher Rλwould be welcome. The above relationships give
R∇p=
4Hχ∞/integraldisplay
0r−3D1111(r)dr
1/2
//bracketleftBig
−35ν/angbracketleftBig
(∂u1/∂x1)3/angbracketrightBig
/2/bracketrightBig1/2
(1a)
Since the value of Hχvaries little from 0 .65 even for low Reynolds numbers, it is pragmatic to standard izeR∇pby
replacing the numerical factor [4 Hχ]1/2/[35/2]1/2by 0.4. Then,
R∇p= 0.4
∞/integraldisplay
0r−3D1111(r)dr
1/2
//vextendsingle/vextendsingle/vextendsingleν/angbracketleftBig
(∂u1/∂x1)3/angbracketrightBig/vextendsingle/vextendsingle/vextendsingle1/2
andRa=/radicalBig
1 +R2
∇p. (1b)
These ratios have the desired properties. They are dimensio nless; Rais a Reynolds number in Batchelor’s
aforementioned interpretation [2] and R∇pis closely related to Ra; they can be evaluated with single-wire hot-wire
anemometry; they are composed of statistics of the small sca les of turbulence. The integral∞/integraltext
0r−3D1111(r)drdoes
not require as great a spatial resolution as does measuremen t of/angbracketleftBig
(∂u1/∂x1)2/angbracketrightBig
because this integral is dominated by
dissipation-range r-values with little contribution from the viscous range. Th us,/angbracketleftBig
(∂u1/∂x1)3/angbracketrightBig
is the most challenging
measurement in (1b). For many flow geometries, both R∇pandRaincrease as Rλincreases. It is appropriate to
change perspective: The nonuniversal abscissa Rλtypically, but not necessarily, increases as the universal abscissas
R∇pandRaincrease.
Sreenivasan and Antonia [4] compiled existing derivative- moment data as functions of Rλ. They show that
the skewness/angbracketleftBig
(∂u1/∂x1)3/angbracketrightBig
//angbracketleftBig
(∂u1/∂x1)2/angbracketrightBig3/2
is about −0.3 atRλ= 2, decreases to a nearly constant value of −0.5
over the range Rλ= 10 to 103, and thereafter decreases to about −1 atRλ= 2×104. Assigning the skewness the
value −0.5 gives/vextendsingle/vextendsingle/vextendsingleν/angbracketleftBig
(∂u1/∂x1)3/angbracketrightBig/vextendsingle/vextendsingle/vextendsingle≃0.5ν/angbracketleftBig
(∂u1/∂x1)2/angbracketrightBig3/2
= 0.5 (ε/15)3/2ν−1/2, where ε= 15ν/angbracketleftBig
(∂u1/∂x1)2/angbracketrightBig
was
substituted. Substituting this approximation into (1b) gi ves an estimate of R∇pdenoted by Rappr
∇p; i.e.,
Rappr
∇p≡4.3
∞/integraldisplay
0r−3D1111(r)dr//parenleftBig
ε3/2ν−1/2/parenrightBig
1/2
andRappr
a=/radicalbigg
1 +/parenleftBig
Rappr
∇p/parenrightBig2
. (2)
Thus, like Rλ,R∇pandRacan be estimated using flow similarity estimates of εcombined with a measured time series
ofu1even if the spatial resolution of u1is not sufficient to calculate εor/angbracketleftBig
(∂u1/∂x1)3/angbracketrightBig
. The resulting approximations
toR∇pandRaareRappr
∇pandRappr
agiven in (2).
III. RELATIONSHIP TO RECENT DATA
In their Fig. 1, Vedula and Yeung [18] show a ratio that they ca llζ, which equals R2
∇p, as well as a quantity
a(I)
0, which is 20/parenleftBig
Rappr
∇p/parenrightBig2
, both graphed versus Rλ. Similarly, Gotoh and Rogallo [7] show F∇p= 3a(I)
0= 60/parenleftBig
Rappr
∇p/parenrightBig2
versus Rλin their Figs. 1 and 2. Figure 2 of Vedula and Yeung [18] and Fig . 12 of Voth et al. [19] show
/angb∇acketlefta·a/angb∇acket∇ight//parenleftbig
3ε3/2ν−1/2/parenrightbig
and/angbracketleftbig
a2
1/angbracketrightbig
//parenleftbig
ε3/2ν−1/2/parenrightbig
, respectively, with Rλon the abscissa. The corrected data of Voth
et al. is given in Fig. 4.13 of Voth [19] and Fig. 4 of LaPorta et al. [21]. The above analysis shows
that/angbracketleftbig
a2
1/angbracketrightbig
//parenleftbig
ε3/2ν−1/2/parenrightbig
=/angb∇acketlefta·a/angb∇acket∇ight//parenleftbig
3ε3/2ν−1/2/parenrightbig
=R2
a/vextendsingle/vextendsingle/vextendsingle/vextendsingle/angbracketleftBig
(∂u1/∂x1)3/angbracketrightBig
//angbracketleftBig
(∂u1/∂x1)2/angbracketrightBig3/2/vextendsingle/vextendsingle/vextendsingle/vextendsingle//bracketleftbig
3 (2/35)153/2/bracketrightbig
≃R2
a/20.
Therefore, the above-mentioned graphs show R2
∇pandR2
aasRλvaries; this is exact in the case of ζ≡R2
∇p, but it is
approximate for the other quantities on the basis of neglect of the variation of the skewness. Reversing the role of
ordinates and abscissas in their graphs, the graphs show the nonuniversal behavior of Rλfor a variety of flows as the
universal Reynolds number Ravaries.
3IV. SUMMARY
As defined in (1), Rais a Reynolds number; it is a measure of the ratio of rms accele ration to rms viscous
acceleration; it is composed of statistics of the small scal es of turbulence; it can be used as a universal abscissa for
judging the universality of turbulence statistics. Throug h its relationship (1b) to R∇p, it can be evaluated with
single-wire hot-wire anemometry. Although R∇pis not strictly a Reynolds number, it can also be used as a univ ersal
abscissa. At high enough Reynolds numbers R∇p≃Ra. If the Reynolds-number variation of the velocity-derivat ive
skewness is neglected, then R∇pandRamay be approximated on the basis of flow similarity estimates of energy
dissipation rate εcombined with a measured time series of u1even if the spatial resolution of u1is not sufficient to
calculate εor skewness. Those approximations of R∇pandRaare denoted by Rappr
∇pandRappr
aand are given in (2).
It is recommended that R∇pandRabe used in preference to Rappr
∇pandRappr
awhenever possible.
In Sec. 3, it is shown that data from several experiments have graphed R∇p,Rappr
∇p, and RawithRλon the
abscissa. Those graphs show the nonuniversal behavior of Rλfor a variety of flows as the universal Reynolds number
Ravaries.
Models of the small-scale statistics of turbulence should b e expressed in terms of universal attributes instead of
in terms of Rλ. For example, in Table II of Belin et al. [6], the model by Pullin and Saffman [22] is in good agreement
with data when judged in terms of power laws between derivati ve moments, but it is in relatively poor agreement
with data when judged in terms of power laws between normaliz ed derivative moments and Rλ. The latter can be
specific to the flow geometry.
V. APPENDIX
A number of experiments and DNS have shown different scaling e xponents for longitudinal versus transverse
velocity differences [23] [24] [25] [26] [27] [28] [29] [30] [ 31] [32]; whereas other experiments suggest equal scaling
exponents [33] [34] [35]. The differing exponents have been l inked to observed differences in scaling exponents
of enstrophy and dissipation [23] [24] [25], differences whi ch must, according to Nelkin [36], disappear in the high-
Reynolds-number limit. Others [32] [37] observe differing e xponents which they attribute to departures from isotropy,
and they suggest that the difference disappears at very large Reynolds numbers, and they find that differences in
scaling of enstrophy and dissipation are insufficient to acco unt for different scaling exponents of longitudinal versus
transverse velocity differences. Recently, Kerr et al. [38] used DNS of isotropic turbulence at the highest attaine d
Rλto show that equality of longitudinal and transverse scalin g exponents in fourth-order structure functions requires
a more restrictive definition of the extent of the inertial ra nge than has previously been used and that Rλmust be
at least 390. Kerr et al [38] then quantitatively explain the previous observations of different scaling exponents on
the basis that Rλwas too small or that the range chosen for evaluation of the ex ponents was too extensive. With
fourth-order scaling exponents returning to equal values a t sufficiently large Rλ, the reasons given by Hill and Wilczak
[15] that Hχis a constant at very large Rλare again strong. The empirical evidence by Vedula and Yeung [18] for
the value of Hχis strong as well.
Acknowledgement 1 The author thanks the organizers of the Hydrodynamics Turbu lence Program held at the Insti-
tute for Theoretical Physics, UCSB, whereby this research w as supported in part by the National Science Foundation
under grant number PHY94-07194. This work was partially sup ported by ONR Contract No. N00014-96-F-0011.
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Cornell U. (2000) (available at: http://milou.msc.cornel l.edu).
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turbulence,”(accepted for publication in Nature, xxx.lan l.gov/physics/0011017)
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turbulence,” Phys. Rev. Lett. 79, 2253 (1997).
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1400 (1997).
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(1995).
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5 |
arXiv:physics/0102057v1 [physics.gen-ph] 19 Feb 2001LARGE SCALE QUANTIZATION
ARBAB I. ARBAB1
Department of Physics, Faculty of Applied Sciences, Omdurm an Ahlia University, P.O. Box
786, Omdurman, SUDAN
Abstract
We have investigated the implications of Quantum Mechanics to macroscopic scale. Evaporation of
Black Holes and evolution of pulsars may be one of the consequ ences of this conjecture. The two
equationsGM=Rc2andGM2= ¯hcwhereRandMare the radius and the mass of the universe,
are governing the evolution of the universe throughout its e ntire cosmic expansion, provided that the
appropriate Planck constant is chosen. The existence of ver y large values of physical quantities are
found to be due to cosmic quantization.
PACS(numbers) 03.65, 98.80
Keywords: Cosmology, Black holes, quantum mechanics, vacuum energy d ensity, ideas and mod-
els
1. INTRODUCTION
The general theory of relativity (GTR), which attributes th e structure of space-time to the
gravitating mass, is incompatible with the theory of quantu m mechanics (QM). While quantum
mechanics is a linear theory the GTR is highly nonlinear so th at the two theories are dissim-
ilar. Attempts to linearize the GTR do no give all manifestat ion of the theory. Initially the
intention of the development of GTR was to describe the large scale structure of the universe,
i.e., solar and galactic scales, whereas QM deals with the mi croscopic scales. The unification of
electromagnetic, weak and strong was successful and gave a l ot of hints to further unification
with the hitherto unachievable gravitational interaction . Particle physicists are ambitious to get
the unification of all interactions at an energy scale of 1019GeV!
However, these disparate worlds can meet in some cases. In th ese situations, phenomena in
atomic physics appear to be written in the language of gravit y and Einstein’s GTR. By knowing
1arbab64@hotmail.com
1the working of one subject, we can thus make educated guesses about the other.
One therefore should not look for a paradigm in which gravity is manifested as a force, but
we should treat gravity as a background (framework) on which other interactions occur. Thus
gravity provides only the shape of the space-time membrane o n which other interactions are
carried. We therefore, expect that gravitational effects ar e always present and manifested in the
way space-time affects our physical phenomena. We remark tha t one should treat gravity not as
an independent interaction but rather as a framework in whic h all interactions take place and
consider only the strong, weak and electromagnetic interac tions to be independent. We therefore
expect to observe quantum effects at microscopic as well as ma croscopic scale depending on
the dimension of the system under consideration. This would imply that quantum effect at
cosmic scale to be similar to those at microscopic scale, the y only differ in the magnitude of
the quantization. This is plausible since the masses and dis tances are enormous for macroscopic
system.
After making these arrangement we would end up with a cosmic P lanck constant having a huge
value for macroscopic scales. The relative errors in determ ining the physical properties of the
macroscopic system would have the same value as for microsco pic one. One would also expect
that the same laws governing the microscopic scale to be stil l functioning but with scaled ones.
This might require some physical quantities to evolve in ord er to satisfy these laws.
Based on Tiftt’s [1] experimental data on galactic red-shif t DerSarkissian [2] suggested a
cosmic quantum mechanics (CQM) characterized by the Planck constant [3]
hg= 10102h= 1068J.s (1)
wherehis the ordinary Planck constant of the quantum mechanics. Th e quantum state of a
galaxy is described by a wave function ψ(x,t) so that galaxies become the elementary particles
of CQM. For a gravitational system and according to Einstein equivalence principle a body with
observable inertial mass ( mi) has observable active mass ( ma) and hence an observable gravita-
tional field. The observability of the this field requires tha t the total energy (self-gravitational
energy) stored in it is measurable and, therefore, greater t hanh/2t(according to the uncertainty
principle ∆ E∆t>h/ 2 and ∆t∼t,∆E∼E). For a spherical body of mass Mand radius Rwe
have
GM2
R>h/2t. (2)
But we know, from general relativity, thatGM
Rc2≤1 so thatMc2>h/2t.
In QM a body is observable only if its self-gravitational ene rgyEGis greater than h/2t(EG>
h/2t). All bodies in QM obey this inequality. By supposing that th e CQM obey the same rules
as QM except that the corresponding Planck constant is differ ent we can write,
EG>hc/2t (3)
which is obeyed by all cosmic bodies. For galaxies this formu la gives
GM2
R>hc/2t. (4)
This implies that, for t= 1017sec,R= 1019m, and
hc= 1068J.s, (5)
2M >1040kg. This represents a constraint for galaxies obeying CQM (h owever all galaxies obey
this). A system with spatial dimension R, massM, spinSand a positive energy density has
R>S
Mc[4]. This together with eq.(4), yield the inequality
M3>Shc
2Gct(6)
that holds for cosmic bodies obeying CQM with intrinsic spin angular momentum. The quanti-
zation of spin requires that S=nh(nis an integer). Hence,
M3>h2
c
Gct(7)
valid for spinning cosmic bodies. By considering a cluster m ade by many galaxies with quantized
spin Ruffini and Bonazzola [5] have shown that there exists no e quilibrium configuration for a
cluster of more that N∼(MP
M)Agalaxies of mass Min their ground state, where MPis the
cosmic Planck mass
MP= (hcc
G)1/2∼1043kg, (8)
whereA= 2 for bosons, A= 3 for fermions. Thus CQM applies well to galaxies. Is there o ther
CQM that applies at planetary, solar or universal scale with different quantum of action ( hc)?
Caldirola et al[6] suggested in a framework of unified theory of strong and gr avitational
interaction that a quantum of action for the universe is give n by
hc=MRc (9)
whereM,Rare the mass and the radius of the universe. Equation (9) give s
hc= 1087J.s (10)
withM= 1053kg andR= 1026m [6].
A possible connection between handhcfor the universe exists whereh
r3∼hu
R3whereris the size
of a typical hadron and huregarded as nonvanishing total angular momentum of the univ erse
due to torsion field [7]. Massa [3] enlarged the above relatio n to become
h
r3∼hu
R3∼hg
R3
g(11)
whereRgis a typical radius of a galaxy ( Rg∼1019m). It was suggested by Caldirola et al[6]
that “these forms are special case of a more general form stil l unknown”. In fact, the CQM for
planetary scale is introduced by Pierucci [8] in which
hc= 10−13J.s (12)
which explains the Titius-Bode law and other regularities o f the solar system. Massa considered
the possibility of the growth of hcwith cosmic time ( t). He, according to the Large Number
hypothesis (LNH) [9], suggested that for a flat universe hc∼t5/2, and he concluded that the
CQM is incompatible with this LNH. Carneiro [10] investigat ed the extension of scale invariance
to quantum behavior. The price that paid was the scaling of Pl anck constant ( h) leading to
3quantization of large structure that treated till now as cla ssical systems. He used the LNH in
his theory and concluded that the angular momentum for a rota ting universe is of the order of
magnitude of 1087J.s.
So far there is yet no evidence that the universe is rotating b ut if it does it should do that
with this value! However, Kuhne [11] claimed an observation of a rotation of the universe. We
remark that an order of magnitude for the universe angular mo mentum is within the limit for the
global rotation obtained from the cosmic microwave backgro und radiation (CMBR) anisotropy
obtained by Kogut et al[12]. Carneiro [10] suggested that “one of the possible expl anation of
the large scale quantization could be based on an evolutiona l point of view: the quantum nature
of the universe during its initial times has molded its-appa rently quantized nowadays large scale
structure”.
We also note that Nottale obtained, on a different ground, a sc ale Planck constant of order
1042J.s for a QM model for the solar system [13].
Carneiro [10] applied the Bohr quantization condition to th e motion of Earth around the Sun
and found that this is consistent provided that Planck const ant has the value 1042J.s. Muradian
[14] found an angular momentum of stars around 1042J.s which is close to Kerr limit of a rotating
black hole with a mass of 1030kg.
In this work we try to explain the origin of large numbers occu rring in nature by appealing to this
conjecture. The hierarchical nature of mass, cosmological constant, entropy, temperature and
angular momentum are found to be one of the manifestation of t his conjecture. The evolution
and characteristics of astrophysical objects like, black h oles and binary pulsars are investigated
in the framework of this conjecture.
We present our model in sect.2 and discuss its implication in a framework in which Gand
Λ vary with cosmic time in sec.3. In sect.4 we apply our conjec ture to Black Holes and Binary
Pulsars. In sec.4 we constrain the maximal acceleration and force appearing in the universe,
according to our model. We wind up our paper with a conclusion .
2. THE MODEL
The usual definition of Planck mass ( mP) can be inverted to give
h=Gm2
P
c. (13)
In a similar fashion, we suggest a cosmic Planck constant ( hc) with a cosmic Planck mass MP
as
hc=GM2
P
c. (14)
Another possibility, from a dimensional point of view, is
hc=c3
GΛ, (15)
where Λ is the cosmological constant. By supposing that the e nergy density of the vacuum
is caused by the gravitational interaction of the neighbori ng particles with mass m, Lima and
Carvalho [15] obtained
Λ =G2m6
¯h4. (16)
4from which one can write
hc=G1/2M3/2
P
Λ1/4. (17)
Hence, we have three different forms of cosmic Planck constan t to be tested with observation.
2. A MODEL WITH VARIABLE GAND Λ
We have shown in an earlier work that [16] G∼t2and Λ ∼t−2,m∼t−1in the early uni-
verse.
Therefore, all three forms reduce to the ordinary Planck con stant in the early universe (radi-
ation dominated era). Thus one has hc=hin the early universe. Hence CQM and QM are
equal in the early universe. However, in the matter dominate universe [16] G∼t,Λ∼t−2and
m= constant. Therefore hc∼t. This asserts that the Planck constant evolves with time and
that why it has different values for different systems. Thus QM applies to macroscopic system in
the same way as in microscopic system except Planck constant is replaced by the cosmic Planck
constant.
Equation (15) shows that the quantum of action for the univer se is related to the vacuum
energy density (Λ) of the universe, so that one obtains
ΛP
Λ0=hc
h= 10120. (18)
This gives a present value for Λ (i.e., Λ 0) a value of 10−52m−2as expected from present obser-
vations. Equations (15) and (17) yield
Λ =c4
G2M2
P, (19)
as a cosmological law governing the evolution of the univers e. This implies that in the early
universe [16] ( G∼t2,m∼t−1) Λ∼t−2. Similarly in the matter dominated phase ( G∼t,m=
const.) Λ∼t−2. Hence Λ ∼t−2during both phases. To reproduce the result in eqs. (1), (5) a nd
(10) one requires Mu
P= 1053kg,Mg
P= 1043kg andMs
P= 1030kg, representing the corresponding
Planck masses at universal, galactic and solar scales, resp ectively. We remark that cosmic Planck
constant for the planetary system is 1026kg. This represents the average mass of the planets in
the solar system. Hence,
hc=GM2
P
c= 1034J.s. (20)
This value is different from the Planck constant for the plane tary system quoted in eq.(12). Thus
eqs.(14), (15) and (17) provide a bridge connecting macrosc opic and microscopic phenomena
through a simple formula. It is found that the universe satis fies the equation
GM=Rc2. (21)
This is an statement of the equality of the rest mass energy of the universe to its gravitational
energy (i.e., Mc2=GM2
R). We observe that eq.(9) is same as eq.(14), if we use the Mach relation
(eq.(21)) [25].
5We have shown in an earlier work [16] that eq.(21) gives R∼t, so that eq.(21) yields
M≤c3t
G. (22)
which is similar to eq.(7). This inequality is obtained by [1 7] by different approaches. One may
define a maximal mass for a bound system to be gravitational st able with
Mmax.=c3t
G. (23)
So that the above equation becomes
M≤Mmax.. (24)
This formula is obtainable from Friedman cosmology (3 H2= 8πGρ) withR∼t,H∼t−1giving
M=c3t
G.
If we consider the stars to be the atoms of the universe we will observe that the universe is a
typical one solar mole , consisting of 1023stars(suns).
We now turn to calculate the Compton wavelength of the univer se, i.e.,λU=hc
Mc= 1026m,
which is of the same order of magnitude of the present radius o f the universe. Thus the use of
CQM for the present universe is also logical and plausible.
3. APPLICATION OF CQM TO BLACK HOLES
Consider a spinning ( S) black hole (or neutron star) with frequency ω. We have
S=Iω, (25)
withI=MR2,MandRare the mass and radius of the object. For an object with a grav itational
radiusR=2GM
c2and spinS∼hc, eq.(23) yields
ω=c3
GM, (26)
Pulsars are believed to be rotating neutron stars and a newbo rn pulsar formed in supernova
may be rotating with a frequency of 104s−1emitting a gravitational radiation with a rate of
1048Js−1.For a pulsar of a mass M=M⊙one gets, from CQM, a period of 10−4sec. One of the
most promising source is the pulsar NP0532 in the Crab nebula [18]. This pulsar is observed to
emit pulses of electromagnetic radiation, at optical, X-ra y and radio frequencies with a period
of 33 msec. Thus our model, though based on rough estimates, i s in a good agreement with
observations.
A black hole emitting radiation as a black body with a tempera tureTgiven by
kBT= ¯hω. (27)
Using eqs.(25) and (26) one gets
T=¯hc3
GMk B(28)
6in comparison with Hawking [19] formula obtained from QM tre atment for a non-rotating black
hole, viz.,
T=¯hc3
4πGMk B. (29)
A rotating galaxy of mass 1043kg would have a frequency of 10−8s−1or a period of 10 years.
We can compare this value with the presently observed rotati on of galaxies. The time for the
evaporation of the black hole can be estimated from the uncer tainty relation
∆E∆t∼hc (30)
with ∆E=Mc2and ∆t=τ. This upon using eq.(14) becomes
τ=GM
c3. (31)
SinceGM2
c¯h= 1, one can write eq.(30) as
τ=GM
c3.GM2
c¯h=G2M3
c4¯h. (32)
Which is obtained by Hawking [19] from a quantum mechanical t reatment. Hence, we may write
for the evaporation of Black holes the formula
τ=G2M3
c4hc(33)
as a CQM analogue.
We see that a black hole of one Planck mass mPevaporates during Planck time (10−43s). A
galactic rotating black mass evaporates during a time of 1 ye ar while a solar rotating black hole
evaporates during a time of µsec. A rotating black hole of size of the universe evaporates dur ing
a time of 1010years.
The entropy of a black hole is given by [19]
S=GM2
c¯hkB (34)
which upon using eq.(14) yields
S=¯hc
¯hkB= (kB
¯h)¯hc (35)
This entropy is independent of the mass of the object in quest ion as long as ¯ hcdescribes that
object. We see that for a black hole formed in the early univer se ¯hc= ¯h, and therefore irrespective
of its mass the black hole will have one unit of entropy, i.e., S=kB. Black holes forming during
solar and galactic time will have entropy that is multiple of ¯hc. We thus conclude that the
entropy of black holes is quantized. A galactic mass black ho le will have an entropy of 10102kB,
while a solar mass black hole will have an entropy of 1076kB. However, a black hole of the mass
of the universe has an entropy of 10120kB. We observe that in the early universe we have
m∝t−1andT∝t−1∝R−1(36)
7so that if these relations hold throughout the cosmic expans ion, one would obtain the relation
m=mP(tP
t) (37)
wheremPis the Planck mass at Planck time and Tis the temperature. We note that De Sabbata
and Sivaram [20] relate the temperature ( T) to curvature ( κ) and showed that T∝√κ, but the
time (t) scales ast∝1√κ. For a maximal curvature κmax.=c3
¯hG, which implies
G∝t2,and T ∝t−1. (38)
Comparison with eq.(15) immediately yields
κmax.= Λ. (39)
Thus one may connect the smallness of the present value of the cosmological constant to the
flatness of our present universe.
One would obtain a minimal mass at the present time given by
mmin.= 10−5(10−43
1017) = 10−65g. (40)
Eq.(35) implies that the temperature is given by
T=TP(tP
t), (41)
whereTPis the temperature at Planck time. Similarly, if this relati on is retained till now (for
some kind of interaction) one would obtain a minimal tempera ture at the present time
Tmin.= 1032(10−43
1017) = 1032×10−60= 10−29K, (42)
whereTP= 1032K. We remark that De Sabbata and Sivaram obtained a similar va lue by
considering a time-temperature uncertainty relation (∆ t∆T= ¯h/kB) and relate the maximum
time to Hubble time. They suggested that they could obtain su ch a value by considering a
black hole of a mass of the universe using the formula outline d in eq.(27). Or by considering
the maximal possible entropy of 10120kBwhich would imply this minimal temperature. They
also found a similar value and noted that this minimal temper ature corresponds to the quantum
fluctuations of cosmological torsion background.
Massa [21] has obtain a similar value and relates this to the m ass of graviton. He argued that
in an expanding universe this mass depends on time. Rosati [2 2] found the quantum field today
has typically a mass of the order of 10−66g. Larionov [23] attributed a similar mass term to an
effective mass associated with the vacuum energy density (or Λ). He assigned this extremely
low value of this effective mass to a quantum with wavelength e qual to the present radius of
the universe. One may also add to this conjecture the cosmolo gical constant problem (rooted
in the enormous value, i.e., Λ P= 10120Λ0) as due to the cosmic quantization, as is evident from
eq.(18)!
According to Massa assertion the graviton has a mass; this wo uld mean that the range of the
8gravitational interaction is not infinite but limited by thi s mass scale. Hence, the maximum
possible interaction distance between any two gravitating objects has to be at a maximum
distance of 1026m. A similar assertion would also hold for electromagnetic i nteraction if it
turned out that a photon is not massless!
We have so far shown that the two formulae (eqs.(14) and (21))
GM=c2Rand GM2= ¯ hcc (43)
hold throughout the cosmic evolution of the universe, provi ded we consider the CQM to be a
valid principle.
The Planck energy density is defined as
ρP=c5
G2¯h(44)
which represents the maximum density of the universe at Plan ck time. We now employ the
CQM and evaluate the present maximum energy density of the un iverse, i.e.,
ρ0
P= 5.4×10−28gcm−3. (45)
Present observations set a limit on the present energy densi ty as
10−30gcm−3<ρ0<10−29gcm−3. (46)
We have thus obtained a constraint on the density of the unive rse viz.,ρ0≤ρ0
P. It is evident
that the Planck energy density of the universe indeed evolve s with time.
4. MAXIMAL ACCELERATION AND FORCE
The self-gravitational force of a system of mass Mand radius Ris given by
F=GM2
R2. (47)
Using eq.(41)
F=c4
G, (48)
for the universe. This force is clearly independent of the ma ss of the object into consideration. It
is thus a universal force, and since it depends on Ginversely it defines a maximal self-gravitational
force. There corresponds to this maximal force a maximal acc eleration (Fmax.=Mamax.) defined
by
amax=c4
GM. (49)
If we consider a variable gravitational constant as suggest ed in [16], one gets a minimal gravita-
tional force in the universe during the nuclear (or hadronic ) epoch as
Fmin.= 10−2N , (50)
since during nuclear (or hadronic) epoch the gravitational constant was GN= 1040G0. This
smallness of this force may account for the fact that quarks a re asymptotically free inside hadrons,
9according to the theory of quantum chromodynamics (QCD). A m aximal force in the universe
during the present or Planck time is
Fmax.= 1044N . (51)
We remark that the factorc4
8πGappearing in the Einstein’s equation may be interpreted as t he
force per area required to give space-time unit curvature, t hat is 1043Nm−2for a curvature of
1m−2. Space-time is therefore an extremely stiff medium. De Sabba ta and Siviram noted that
there exists a maximal acceleration given by
amax.=c7/2
G1/2¯h1/2, (52)
originated as quantum effect due to torsion. Their formula re duces to our formula upon using
eq.(14) into (47). Therefore, we have, according to CQM
amax.=c7/2
G1/2h1/2
c. (53)
We now turn to calculate the maximal acceleration at a univer sal scale, according to CQM.
amax.= 10−9ms−2. (54)
However, De Sabbata obtained a value of 10−10ms−2on different grounds that agrees with [24].
Note that the maximal acceleration at Planck time was amax= 1051ms−2.
5. CONCLUSION
We have extended the implication of quantum mechanics form m icroscopic scale to include
macroscopic scale. This extension resulted in a lot of inter esting consequences concerning the
evolution and characteristics of black holes and binary pul sars. We have found that quantities
like entropy, cosmological constant and time are quantized for macroscopic scales. Limiting val-
ues for temperature, entropy, angular momentum (¯ hc), force, acceleration are obtained due to
this conjecture. The universe is found to have an energy dens ity less that Planckian density at
the present time. We have also shown that we have different Pla nckian parameters for the uni-
verse for different time. The relation GM2= ¯hcwhich was found to apply in the early universe
is still valid during other phases, provided that we replace the ordinary Planck constant with
the cosmic Planck constant, whose value depends on the prope rties of the macroscopic entity.
Acknowledgements
I would like to thank the Sudanese Physicists Association (S PA) for providing financial support
and the research grants number OAU-2000/2001/PHYS-32 from Omdurman Ahlia University.
I would also like to thank Dr. H.M. Widatallah, Dr. O.I. Eid an d Dr. O.F.Osman for their
enlightening discussion.
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11 |
arXiv:physics/0102058v1 [physics.flu-dyn] 20 Feb 2001Velocity Field Distributions Due to Ideal Line Vortices
Thomas S. Levi and David C. Montgomery
Department of Physics and Astronomy
Dartmouth College, Hanover, NH 03755-3528
(February 2, 2008)
Abstract
We evaluate numerically the velocity field distributions pr oduced by a
bounded, two-dimensional fluid model consisting of a collec tion of parallel
ideal line vortices. We sample at many spatial points inside a rigid circular
boundary. We focus on “nearest neighbor” contributions tha t result from
vortices that fall (randomly) very close to the spatial poin ts where the veloc-
ity is being sampled. We confirm that these events lead to a non -Gaussian
high-velocity “tail” on an otherwise Gaussian distributio n function for the
Eulerian velocity field. We also investigate the behavior of distributions that
do not have equilibrium mean-field probability distributio ns that are uniform
inside the circle, but instead correspond to both higher and lower mean-field
energies than those associated with the uniform vorticity d istribution. We
find substantial differences between these and the uniform ca se.
Typeset using REVT EX
1I. INTRODUCTION
Study of the hydrodynamics of ideal line vortices goes back a t least as far as Helmholtz in
the 19th century, and was developed in the 20th by Lin [1] and O nsager [2], who first made
the dynamical system an object of statistical mechanical in quiry. The system appeared in
plasma physics when Taylor and McNamara [3,4] calculated th e Bohm-like coefficients of
self-diffusion for a strongly-magnetized, two-dimensiona l, electrostatic guiding-center plasma
model, a system whose mathematical description becomes ide ntical with that of the ideal line
vortex system under appropriate substitutions; the fact th at these diffusion coefficients were
inversely proportional to the first power of the magnetic fiel d, even in thermal equilibrium,
was startling.
The system is one for which interesting statistical-mechan ical and fluid-mechanical ques-
tions can be asked, but must be asked with care, for two reason s. First, viscous effects have
never been fully included in the model, although some forms o f Navier-Stokesian behavior
have on occasion been observed for it. Secondly, no classica l, extensive “thermodynamic
limit” exists for the system in the conventional sense, and t he partition function, even for
the case in which there is no overall net vorticity, does not i n general exist in the infinite
volume limit [5]. None of the standard machinery of equilibr ium statistical mechanics can
be trusted completely without re-examination.
One question that can be asked, motivated in part by various p robability distribution
function measurements for turbulent fluid velocities that h ave been made in recent years,
concerns the distribution of the velocity field at a fixed poin t in space, one at which no vortex
necessarily resides. The field in question is one that is prod uced by all the vortices. This is a
close analogue of the question of the probability distribut ion of the vector gravitational field
due to a large collection of point masses, a question address ed in detail by Chandrasekhar
in 1943 [6]. Under the assumption that the point masses in thr ee dimensions are uniformly
distributed and uncorrelated, the resulting Holtsmark dis tribution has many non-standard
properties, including the divergence of some of its low-ord er moments: a consequence of the
long range of the inverse-square force field and the fact that point masses (or charges) each
have an infinite “self energy” that reflects itself in the tota l force field when the single-particle
contributions are combined additively.
In a recent interesting paper [7], Kuvshinov and Schep consi dered the statistics of the
velocity field of a large but finite number of ideal line vortic es inside a circular boundary
(see also the paper of Chukbar [8], which is of some importanc e). They assumed uniformly
distributed and uncorrelated line vortices of a single sign of vorticity. They noted that
the Holtsmark-style treatment carried out by Chandrasekha r for the three-dimensional case
contained a divergent integral in two dimensions, and so was not immediately applicable.
They then performed repeated numerical measurements of the two-dimensional (2D) velocity
field, near the center of the circular boundary, that resulte d from uncorrelated random
distributions of large numbers of vortices, thrown at each t rial into the circular boundary
without correlation and without any mean density variation .
The most interesting result of Kuvshinov and Schep was thus a n “experimentally” de-
termined probability distribution for the velocity which s eemed to split naturally into two
parts: a Gaussian distribution for the lower velocities and high-energy “tails” for the larger
velocities that fell off approximately as the third power of t he fluctuating velocity. (Here,
2“fluctuating” velocities are interpreted to mean those with the mean-field rigid rotation as-
sociated with the uniform vorticity density distribution s ubtracted out.) They hypothesized
that the approximate inverse third-power dependence of the tail was a consequence of occa-
sional “near neighbor” contributions, in which one vortex f ound itself very close to the point
where the velocity field was being sampled, and generalized a three-dimensional “nearest-
neighbor” algebraic argument of Chandrasekhar’s [6] to acc ount for this high-velocity power
law contribution. In a rather different continuum model, som ething not totally dissimilar
had previously been reported by Jimenez [9].
We have in this paper repeated certain features of Kuvshinov and Schep’s numerical
experiment, and have attempted to modify and amplify it in a v ariety of ways. (1) We have
inserted an ideal, perfectly-reflecting wall boundary at th e radius of the confining circle by
changing the Green’s function to one that, by the method of im ages, guarantees the vanishing
of all radial velocities at the boundary [10], rather than us ing the inverse logarithmic Green’s
function appropriate to the unbounded region. (2) We have, u pon finding the non-Gaussian
high-velocity tails in the probability distribution funct ion, implemented a program that
searches numerically for near neighbor contributions to th e locally measured velocity field,
and when it finds one, deletes its contribution to the local ve locity field. We find that as
a consequence, the high-velocity tails disappear, thus rei nforcing the conjecture of Ref. [7].
(3) We study the velocity field away from the origin, to determ ine how representative of the
entire spatial volume the velocity field sampled at the cente r is. (4) Finally, we allow the
mean vorticity density with which the vortex particles are d istributed to vary, and rather
than placing them randomly with a spatially-uniform mean-fi eld distribution, we weight their
locations with a probability distribution function that de pends exponentially upon a mean-
field stream function and has a temperature that can be positi ve or negative [10,11]. The
equilibrium statistical mechanics of the ideal line vortex system has undergone considerable
development since it was introduced (e.g., [10–13], and ref erences therein) and we take
advantage of results which we will not go into full detail des cribing here. We note only that
the pairwise, additive Coulomb potentials, summed over all the pairs in the system, are an
ideal invariant dynamically which can take on virtually any value and which determines the
single-time thermal-equilibrium probability distributi ons of all particles. Only one value of
this energy is represented by the uniform distribution. We fi nd significant differences in the
velocity field statistics that result from total mean energi es that are significantly higher or
lower than those associated with the unfiorm (rigidly-rotat ing) mean-field distribution.
In Sec. II, we describe the comptutational procedure and the results for the uniform
mean-field vorticity density distribution for points near t he center of the circle, with an
emphasis on non-Gaussian, high-velocity “tails” that appe ar in the probability distribution
function for the velocity. In Sec. III, we introduce a cutoff b elow which “near neighbor”
contributions to the velocity field are locally removed, and derive an analytic expression
for the contribution of very near neighbors to the local velo city field distribution. Sec.
IV discusses the statistics of the velocity field for the unif orm density distribution away
from the center of the container and near the boundary. Sec. V is devoted to the case in
which the mean number density of vortices is not uniform, but rather follows from a self-
consistent, mean-field theory which permits the study of hig h and low energy states, relative
to the uniform density state. Sec. VI presents the results fo r the non-uniform mean-field
distributions. Sec. VII summarizes the results and indicat es possible future directions for
3further investigations.
II. GENERAL PROCEDURE
In a point vortex model, where each vortex has strength κjthe flow is two-dimensional
in the (x,y) plane, has only xandycomponents, and is given by
v(r) =/summationdisplay
jκj∇ ×(G(r,rj)ez) (2.1)
Hereezis the unit vector pointing perpendicular to the plane of the spatial variation of the
fluid,Gis the Green’s function that relates the vorticity to the str eam function, and the
sum is over all (two-dimensional) vortex positions rj. Thus, we see that the velocity at a
given point is due to all the vortices not at that point. For a t wo-dimensional fluid, in a
rigid, circular container of radius R, the boundary condition is that the normal component
ofvgo to zero at the wall. The appropriate Green’s function to ch oose is [10]
G(r,r′) =1
2πln(|r−r′|)−1
2πln/parenleftBig/vextendsingle/vextendsingle/vextendsingler−R2
r′2r′/vextendsingle/vextendsingle/vextendsingler′
R/parenrightBig
(2.2)
Where here we have replaced rjwithr′. Using Eq. (2.1) we get
vr=κ
2π/parenleftbiggR2r′sinθ12
r2r′2+R4−2R2rr′cosθ12−r′sinθ12
r2+r′2−2rr′cosθ12/parenrightbigg
(2.3a)
vθ=κ
2π/parenleftbigg
−rr′2−R2r′cosθ12
r2r′2+R4−2R2rr′cosθ12+r−r′cosθ12
r2+r′2−2rr′cosθ12/parenrightbigg
(2.3b)
Wherevrandvθrepresent the randθcomponents of velocity due to one point vortex of
strengthκ, andθ12is the angle between the radii to the point where the velocity is measured
and the position of the vortex. For each component the terms w ithRrepresent the terms
that are a result of the finite boundary.
All quantities will be expressed throughout in terms of dime nsionless variables appro-
priate to the model. Since the Euler dynamics contain no visc osity, all quantities in the
dynamics before non-dimensionalization contain only comb inations of lengths and times, or
equivalently, velocities and times, so units are not of grea t significance. For a convenient
basic unit of length, we may take the mean nearest-neighbor s eparation in a uniform vortic-
ity benchmark case divided by π1/2and for the basic unit of velocity, the speed with which
an isolated vortex of strength 2 πwill rotate the fluid in which it is imbedded at unit length
distance from the vortex.
The general procedure we use is to place a large number, N, of vortices of strength
κ= 2πinto a circular region of radius Rusing a random number generator and study the
statistics of the resulting velocity field. Specifically, we examine the probability distribution
for the scalar fluctuating velocity |w|=w, where w=v−V, andVis the mean-field
velocity. Let f(w)dwbe the probability that the velocity is in the area element (i n velocity
space)dwcentered at w. We are here assuming that the distribution is isotropic in v elocity
4space, which is confirmed by our numerics everywhere except i n a very thin layer near
the radial boundary. We wish to switch to a one-dimensional i ntegral, which is done by
lettingF(w)dw= 2πwf(w)dw. The resulting distribution F(w) is normalized such that/integraltext∞
oF(w)dw= 1. Our graphs contain a numerically obtained F(w). The procedure for
obtaining this Fis to first run a series of trials, each trial representing a se t of random
choices for the vortex positions inside the circle. For the u niform vorticity density case,
we have run 3000 trials. Then, we record a velocity value at ea ch point sampled in the
circle. Here we have sampled at 50 points separated by unifor m intervals from r= 0 to
r= 399, where R= 400 and N= 1.6×105. We then bin the velocities using a histogram
with uniform spacing between bins. This procedure gives us a n unnormalized probability
distribution for f. To get from this step to the actual Fplotted requires two steps: (1) We
first multiply each bin value by the wat the center of its bin. (2) We normalize the result
using a trapezoidal numerical integration, so that, numeri cally/integraltextF(w)dw= 1. It is easiest
to see the probability distribution’s behavior on a natural log plot, so we plot ln( F(w)/w)
versusw2. The error bars are one standard deviation of the mean in leng th above and below;
namely, we calculate the standard deviation of ln( F/w) and then divide by the square root
of the number of actual events that fall into that histogram b in. We present two graphs
for each point sampled in the uniform case: (1) A graph that in cludes all numerical events.
(2) A graph with the “nearest-neighbor” events subtracted o ut. The subtraction procedure
is defined relatively simply and somewhat arbitrarily. At ea ch point sampled, the program
records the distance to all of the vortices placed in the regi on. If the distance dis such that
d <0.65 then that event is deleted from the distribution for that p oint only. That is, if
there is a nearest neighbor event recorded at r= 200 for example, its removal will notaffect
the resulting distribution at any other point. The resultin g distribution can be thought of
as the probability distribution if there were never any “nea rest-neighbor” events. In each
plot, the solid line is a best-fit Gaussian given by [7]:
F(w) =w
w2exp(−w2
2w2) (2.4)
Wherewis a measure of the average velocity and is numerically deter mined for a best fit.
The dashed line represents an analytical expression for nea r neighbor contributions in the
bounded case which will be calculated below.
III. NEAREST NEIGHBORS
Here we follow the general procedure of Chandrasekhar [6], b ut carry it out in two
dimensions and for a general mean-field vorticity density n(r) to get an analytical expression
for nearest neighbor events. Let Fn(r′)dr′represent the probability that that the nearest
neighbor lies between r′andr′+dr′. This probability must be equal to the probability that
no neighbors are interior to r′times the probability that a particles does exist in the circ ular
shell between r′andr′+dr′. ThusFn(r′) must satisfy [6]
Fn(r′) =/parenleftBig
1−/integraldisplayr′
0Fn(r)dr/parenrightBig
2πr′n(r′) (3.1)
5wherer′is the distance to the nearest neighbor. Differentiating bot h sides, we get a differ-
ential equation for Fn
d
dr′/parenleftbiggFn(r′)
2πr′n(r′)/parenrightbigg
=−2πr′n(r′)Fn(r′)
2πr′n(r′)(3.2)
This equation is not hard to solve; its solution is
Fn(r′) = 2πr′n(r′)Cexp/parenleftBig
−2π/integraldisplayr′
0n(r)rdr/parenrightBig
(3.3)
Where C is a normalization constant such that/integraltextR
0Fn(r′)dr′= 1. In general, C∼1
1−e−N,
and sinceN≫1,C∼=1. In particular, for n=constant and smallr′, we get
Fn(r′) = 2πr′nexp(−πnr′2)∼=2πr′n (3.4)
Usingw=κr′
2π/parenleftBig
1
r′2−1
R2/parenrightBig
which is exact at the origin ( r= 0), and a good approximation at
points not at the origin, we get
Fn(w) = 2πr′(w)ndr′
dw(3.5)
ThisFn(w) will be plotted as a dashed line when exhibiting the measure d velocity distribu-
tion vs.w.
IV. RESULTS FOR UNIFORM VORTICITY DENSITY CASE
Figs. 1 and 2 display results for the numerically determined velocity distribution for the
uniform mean-field vorticity density runs, a total of 3000 tr ials. Fig. 1 shows results of
sampling at r= 0, and Fig. 2 at r= 399, quite close to the wall. At intermediate points,
the results are quite similar to those at r= 0.
In Figs. 1a and 1b, the solid line represents the Gaussian, Eq . (2.4), with the same mean-
square velocity fluctuation. The dashed line represents the nearest neighbor contribution,
as predicted by Eq. (3.5). The “experimentally” determined points are shown with their
associated error bars, estimated as described in Sec. II. Fi g. 1a shows the results for the
raw data, with no “nearest neighbor” events removed. Fig. 1b (the lower figure) shows the
results of deleting the nearest neighbor events. The reason no data points appear above
w2of about 85 is that all the computed points above that value co ntain nearest neighbor
events. A similar set of statements applies to Figs. 2a,b, wh ich are for the radius r= 399.
In both cases, it appears that the high-velocity events are r easonably well predicted by Eq.
(3.5). In both cases, the Gaussian (2.4) is clearly a good app roximation only for the lower
values ofw.
Fig. 3 shows the distribution of the numerically-obtained m agnitude of the radial com-
ponent of velocity as a function of r. The intent is to assess the effect of the rigid boundary
atr= 400, the location of the wall. It will be seen that the decrea se of the radial velocities
is significant only within a relatively thin boundary layer n ear the wall. If the vortex dy-
namics were allowed to evolve in time, it is expected that the boundary layer would persist,
6but might acquire dimensions not necessarily the same as obs erved for the purely random
distribution.
Summarizing, we conclude that for the case in which the unifo rm mean-field vorticity
density applies, there are indeed non-Gaussian tails prese nt in the probability distributions,
and we confirm the conjecture of Kuvshinov and Schep that they may be explained as the
result of nearest-neighbor contributions. Only near the ra dial boundary does its presence
result in any significant departure from the statistics obse rved in the interior, for this case.
V. NON-UNIFORM MEAN-FIELD VORTICITIES: “MOST PROBABLE”
DISTRIBUTIONS
Up to this point, we have considered only the case of the unifo rm probability distribution
for vortices. However, a much wider variety of thermal equil ibrium states is possible for ideal
line vortices, considered as a dynamical system ( [2,4,5,10 –16], and references therein). The
Hamiltonian or energy of the system is equivalent to the Coul omb energies of the pairs of
interacting line vortices, summed over all the pairs, and is a constant of the motion for
these boundary conditions. More extensive investigations have been carried out for the
two-species case than for the present one-species case, but one species may equally well be
considered. The preceding results do not apply to any value o f the energy expectation (which
is determined by the initial conditions chosen when the syst em is considered dynamically)
except the one associated with the completely uniform mean- field distribution. For either
higher or lower energies, the thermal equilibrium, mean-fie ld, one-body distribution is not
spatially uniform. It is concentrated toward r= 0 for higher energies, and around the rim
for lower ones. In this Section, we provide an expression for the probability distribution for
these higher and lower energy cases, referring to the rather extensive cited literature for the
formalism and justification ( [10–16], and references there in).
We find the mean fields from solving the one-species analogue o f the “sinh-Poisson”
equation,
∇2ψ=−ω=−e−α−βψ(5.1)
whereψis the “most probable” stream function, and ωis its associated mean-field vorticity
distribution. In the present case, it will be assumed that th e relevant solutions are symmetric
with respect to rotations about r= 0.
Eq. (5.1) is to be solved subject to the constrainst that E=1
2/integraltext(∇ψ)2d2xand Ω =
−/integraltext∇2ψd2x, where Eis the mean-field energy, and Ω is the total vorticity. If we as sumeψ
is a function of radius only, Eq. (5.1) becomes simply1
rd
drrdψ
dr=−ω=−e−α−βψ, which is
sometimes called Liouville’s equation and has been widely s tudied (e.g., [17]).
We may solve the equation for ψby writingω=c1/(1 +c2r2)2. Taking the Laplacian of
the natural logarithm, we get
1
rd
drrdψ
dr=8c2
β(1 +c2r2)2=−ω=−c1
(1 +c2r2)2(5.2)
The equality demands that c1=−8c2/β. Inserting the expression into the constraint equa-
tions, we find that
7Ω =−8π
βc2R2
1 +c2R2(5.3a)
E=8π
β2/bracketleftbigg
ln(1 +c2R2)−c2R2
1 +c2R2/bracketrightbigg
(5.3b)
The goal is to solve Eqs. (5.3a) and (5.3b) for the constants c2andβ. The result is
E
Ω2=1
8π(1 +c2R2)2
(c2R2)2/bracketleftbigg
ln(1 +c2R2)−c2R2
1 +c2R2/bracketrightbigg
(5.4)
which must be solved numerically for c2in terms of Ω and E. The result is β=−8π
Ωc2R2
1+c2R2
andω=Ω
πR21+c2R2
(1+c2r2)2, wherec2is given by Eq. (5.4). We have now expressed the mean-field
vorticity directly in terms of energy and vorticity. It foll ows that when placing vortices
“randomly” into the circular region for numerical trials, w e should weight their placements
by a probability distribution that wil lead to the correct ωin the mean-field limit. That is,
p(r,θ)rdr=r
πR21 +c2R2
(1 +c2r2)2dr (5.5)
Here, the radial probability density pis normalized such that/integraltextp(r,θ)rdrdθ = 1. The
spatially uniform case treated previously corresponds to t he casec2→0, in which case we
getE0= Ω2/8π. The nearest neighbor formula must be modified to
Fn(w) = 2r′(w)N
R21 +c2R2
(1 +c2r′2(w))2dr′
dwexp/parenleftbigg
−(1 +c2R2)N
R2r′2(w)
1 +c2r′2(w)/parenrightbigg
(5.6)
VI. RESULTS FOR NON-UNIFORM TRIALS
As might be expected, noticeable differences occur when the m ean-field vorticity is a
function of radius. First, the mean azimuthal velocity no lo nger corresponds to a rigid
rotation, and the fluctuating velocity must be referred to it locally. Qualitatively, it might
be expected that the higher energy trials will produce more n earest-neighbor events, at
constant mean density over the whole circle, and hence a more intense velocity fluctuation
spectrum, and the opposite for the lower energy cases. That s eems to be what happens.
We conducted two runs of 1790 trials each, with N= 1.6×105andR= 400, as before.
One of the sets of trials corresponded to mean-field energy E= 4E0and the other set to
E=E0/4. Fig. 4 shows the mean probability distribution, Eq. (5.5) , evaluated for the two
cases. Consistently with Ampere’s law and the remarks above , more (less) vorticity must
be crowded toward the origin for the higher (lower) energies . We should bear in mind that
associated with each individual line vortex, there is an infi nite positive self-energy. This is not
included in what we are calling the “mean-field energy,” whic h is a sum of potential energies
between pairs only. Nevertheless, choosing mean-field ener gies above that of the uniform
distribution greatly enchances the ability of a given numbe r of line vortices to strengthen the
high-velocity tails: crowding the vortices together produ ces more opportunities for nearest
8neighbor events in the regions of enhanced mean-field vortic ity. Also, where there is a
high probability density, we may expect a large value of the a verage velocity that is not
attributable to nearest neighbor events.
Fig. 5 displays the vorticity probability distribution at r= 40.7 for the E= 4E0case;
this is inside the region of high radial probability density . Note the very large value of w
and the associated large values of w2. The probability of finding a vortex near this point
is so high, in fact, that every single trial contained at leas t one nearest-neighbor event,
so the corresponding graph with nearest neighbor events del eted has no data points in it,
according to our previously-chosen criterion. We also obse rve that the nearest-neighbor
formula (broken line) and the Gaussian (solid line) are not f ar apart for this case.
Figs. 6a,b are also for the high-energy case, but sample the v elocity field at r= 114, an
intermediate value. Here we observe, as in the uniform vorti city density case, a noticeable
high-velocity tail attributable to the nearest-neighbor e vents which disappears when those
events are deleted. The much lower value of w= 3.3 is close to what was seen in the uniform
vorticity case, and far lower than in Figs. 5a,b. Not only the mean-fields, but the statistics
of the fluctuations, are now strongly position-dependent. T his point is made even more
strongly by looking at the velocity distribution at r= 399, near the wall (Figs. 7a,b). Here,
where the probability distribution is very low, there is lit tle velocity fluctuation ( w= 0.35).
Here, the nearest-neighbor calculation is of severely limi ted applicability. The Gaussian is
still present, as is the high-velocity tail, but the high-ve locity tail does not disappear when
the nearest neighbor events are deleted. The nearest neighb or formula derivation takes no
account of the proximity of the wall, effectively assuming a r otational symmetry about the
point of observation which is not even approximately fulfill ed near the wall. The boundary
condition begins to make itself strongly felt in this case, a nd it is not obvious how to include
it in any theory.
Turning now to the second set of trials, with E=E0/4, we consider the case where the
probability is concentrated near the walls. We present the r esults of sampling at the radius
r= 147 (Figs. 8a,b). This is again an intermediate regime wher e the results are not greatly
different from the uniform mean-vorticity case. Closer to th e wall, the locally larger values
ofpagain diminish the differences between this case and the unif ormωcase.
In summary, there are some strong qualitative similarities between the uniform and non-
uniform mean field vorticity cases: the division into Gaussi an plus high-velocity tail is usually
applicable. One principal quantitative difference is that t he fluctuation level becomes more
intense for the high-energy cases in those regions where the vorticity is concentrated. The
mean velocity can also go up, and the mean field also becomes mo re intense. The overall
fluctuation level goes up dramatically with mean-field energ y. Though we do not have a
theory for how fast it should go up, we can see from Fig. 9 that i t is considerably faster than
linear. Fig. 9 shows the mean field energy, normalized to the u niform mean-vorticity values,
as a function of mean-field energy, for the three values of mea n-field energy considered.
Adding points to this graph is an expensive and time-consumi ng activity, but would seem
to be a worthwhile undertaking. The significantly noisier hi gh-energy states for the system
is something that will be characteristic of the ideal line vo rtex model but not for continuum
models of a fluid.
9VII. CLOSING REMARKS
We have investigated numerically the statistics of the Eule rian velocity field in two di-
mensional flows generated by a large number of ideal, paralle l, line vortices inside an axisym-
metric rigid boundary. This is a dynamical system the statis tical mechanics of which have
been interesting to investigate in their own right, and whic h also seem to have implications,
not fully elucidated, for two-dimensional viscous continu um flows [14–16]. By considering
the numerical effects of “near neighbors” and their contribu tions to the velocity fields at
fixed spatial points, we have to a considerable degree confirm ed the hypothesis of Kuvshi-
nov and Schep [7] that the observed non-Gaussian, approxima tely third power “tails” in the
velocity field distribution are due to these near neighbor ev ents. These tails coexist with a
“bulk” Gaussian distribution at lower velocities.
The phenomenon of non-Gaussian high-velocity tails in meas urement and computation
of three-dimensional continuum fluid turbulence has been ob served before (e.g., Vincent
and Meneguzzi [18]; see also Jimenez [9]). In computations, also simultaneously visible have
been concentrated vortex configurations that have variousl y been called “tubes,” “worms”
or “spaghetti,” since they are longer by a considerable amon t in one dimension than they are
in the other two. Accounting for these configurations has bee n an important problem. It is
difficult not to imagine that the one might be responsible for t he other. That is, we suggest
that the non-Gaussian tails are a signature of physically pr oximate strong, tubular vortices
which are enough like “line” vortices that they account for t he tails in three dimensions in
the manner observed here in pure two-dimensional form.
A second part of the investigation has been motivated by the r ecognition that pairwise
interaction energies, summed over all the pairs of an assemb ly of identical line vortices,
provides a finite integral of the motion that can be set at any v alue, and determines as
much about the thermal equilibria that are possible as energ y usually does for conservative
statistical-mechanical systems. The non-uniform mean-fie ld distribution which results can
impact the microscopic fluctuation distribution for a fixed n umber of vortices by creating
more (and therefore noisier) regions where “near neighbors ” reside. Such an effect will
undoubtedly enhance transport properties, such as the coeffi cient of self diffusion [3,4],
because of the larger random velocities which result.
It would be of interest to follow up these investigations wit h dynamical computations,
in which an assembly of line vortices was moved around by its s elf-consistent velocity field,
with an eye toward measuring two-time statistical correlat ions of Eulerian velocity fields,
diffusion and decay rates. Measured coefficients of self-diffu sion may be determined numer-
ically, and may be found to depend fundamentally on the mean- field energy and consequent
temperature that characterize a vortex equilibrium and not to be representable by any “uni-
versal” formula. Much earlier computations and theories fo r ideal line vortex dynamics [9–11]
showed unexpected late implications for Navier-Stokes flui d turbulence in two dimensions
[14,15]. Standard “homogeneous turbulence” theories were shown to be very poor predictors
for the late-time states of turbulent fluids in two dimension s, once this step was taken. We
may speculate that the present considerations, which exten d Holtsmark statistics beyond the
spatially uniform case, might substantially revise, for ex ample, the magnitudes of transport
coefficients that are often assigned to such diverse systems a s galaxies or globular clusters
[6] and dilute magnetized plasmas [3,4].
10ACKNOWLEDGMENTS
One of us (T.S.L.) was supported under a Waterhouse Research Grant from Dartmouth
College. The other (D.C.M.) would like to express gratitude for hospitality in the Fluid
Dynamics Laboratory at the Eindhoven Technical University in the Netherlands, where
part of this work was carried out.
11REFERENCES
[1] C.C. Lin, “On the motion of vortices in two dimensions,” ( University of Toronto Press,
Toronto, 1943).
[2] L. Onsager, Nuovo Cimento Suppl. 6, 279 (1949).
[3] J.B. Taylor and B. McNamara, Phys. Fluids 14, 1492 (1971).
[4] D. Montgomery, in “Plasma Physics: Les Houches 1972”, ed . by C. de Witt and J.
Peyraud (Gordon & Breach, New York 1975) pp. 427-535.
[5] e.g. M.K.H. Kiessling, Commun. Pure & Appl. Math. 46, 27 (1993), or L.J. Campbell
and K. O’Neil, J. Stat. Phys. 65, 495 (1991).
[6] S. Chandrasekhar, Rev. Mod. Phys. 15, 1 (1943), Chaps. IV and VII.
[7] B.N. Kuvshinov and T.J. Schep, Phys. Rev. Lett. 84, 650 (2000).
[8] K.V. Chukbar, Plasma Phys. Rep. 25, 77 (1999). Russian Ref: Fisika Plazmy 25, 83
(1999).
[9] J. Jimenez, J. Fluid Mech. 313, 223 (1996).
[10] Y.B. Pointin and T.S. Lundgren, Phys. Fluids 19, 1459 (1976).
[11] G.R. Joyce and D. Montgomery, J. Plasma Phys. 10, 107 (1973).
[12] D. Montgomery and G.R. Joyce, Phys. of Fluids 17, 1139 (1974).
[13] A.C. Ting, H.H. Chen, and Y.C. Lee, Physica D 26, 37 (1987).
[14] W.H. Matthaeus, W.T. Stribling, S. Oughton, D. Martine z, and D. Montgomery, Phys-
ica D51, 531 (1991).
[15] D. Montgomery, W.H. Matthaeus, W.T. Stribling, S. Ough ton, and D. Martinez, Phys.
Fluids A 4, 3 (1992).
[16] B.N. Kuvshinov and T.J. Schep, Phys. Fluids 12, 3282 (2000).
[17] D. Montgomery, L. Turner and G. Vahala, J. Plasma Phys. 21, 239 (1979).
[18] A. Vincent and M. Meneguzzi, J. Fluid Mech. 225, 1 (1991).
12FIGURES
−15−10−50ln(F/w)
0100200300400500600−15−10−50
w2ln(F/w)(a)
(b)
FIG. 1. Plot of ln( F/w) vs.w2atr= 0 for the uniform case. The upper graph (a) contains
nearest neighbor events. The lower graph (b) has nearest nei ghbor events deleted. The solid line
represents a best fit Gaussian ( w= 3.5). The dashed line is the analytical expression for the
nearest neighbor effects.
13−15−10−50ln(F/w)
0100200300400500600−15−10−50
w2ln(F/w)(a)
(b)
FIG. 2. Plot of ln( F/w) vs.w2atr= 399 for the uniform case. The upper graph (a) contains
nearest neighbor events. The lower graph (b) has nearest nei ghbor events deleted. The solid line
represents a best fit Gaussian ( w= 3.0). The dashed line is the analytical expression for the
nearest neighbor effects.
140 100 200 300 40000.511.522.53
r<vr>
FIG. 3. Plot of < vr>vs.r. Notice the sharp drop towards zero near the wall at r=R= 400.
This is evidence of a relatively thin boundary layer near the wall.
150 100 200 300 40000.511.522.533.5x 10−3
rr p(r,θ)
FIG. 4. Plot showing rp(r,θ) vs.r. The solid line is the case where E= 4E0. The dashed line
is the case where E=E0/4.
16010002000300040005000−20−15−10−505
w2ln(F/w)
FIG. 5. Plot of ln( F/w) vs.w2atr= 40.7 for the E= 4E0case. Here, every point has a nearest
neighbor event recorded and thus, the corresponding graph w ith nearest neighbor events deleted
contains no points. The solid line represents a best fit Gauss ian (w= 17). The dashed line is the
analytical expression for the nearest neighbor effects.
17−15−10−50ln(F/w)
0 100 200 300 400−15−10−50
w2ln(F/w)(a)
(b)
FIG. 6. Plot of ln( F/w) vs.w2atr= 114 for the E= 4E0case. The upper graph (a) contains
nearest neighbor events. The lower graph (b) has nearest nei ghbor events deleted. The solid line
represents a best fit Gaussian ( w= 3.3). The dashed line is the analytical expression for the
nearest neighbor effects.
18−15−10−505ln(F/w)
012345−15−10−505
w2ln(F/w)(a)
(b)
FIG. 7. Plot of ln( F/w) vs.w2atr= 399 for the E= 4E0case. The upper graph (a) contains
nearest neighbor events. The lower graph (b) has nearest nei ghbor events deleted. The solid line
represents a best fit Gaussian ( w= 0.35). The dashed line is the analytical expression for the
nearest neighbor effects.
19−15−10−505ln(F/w)
050100150200250−15−10−505
w2ln(F/w)(a)
(b)
FIG. 8. Plot of ln( F/w) vs.w2atr= 147 for the E=E0/4 case. The upper graph (a) contains
nearest neighbor events. The lower graph (b) has nearest nei ghbor events deleted. The solid line
represents a best fit Gaussian ( w= 2.3). The dashed line is the analytical expression for the
nearest neighbor effects.
200 1 2 3 40246810
ε / ε0w2/w20
FIG. 9. Average of w2/w2
0for every point sampled plotted as a function of E/E0where w2
0= 23.3
is the value at E0.
21 |
arXiv:physics/0102059v1 [physics.bio-ph] 20 Feb 2001The coherent dynamics of photoexcited green fluorescent pro teins
Riccardo A. G. Cinelli, Valentina Tozzini, Vittorio Pelleg rini, and Fabio Beltram
Scuola Normale Superiore and Istituto Nazionale per la Fisi ca della Materia, Piazza dei Cavalieri
7, I-56126 Pisa, Italy
Giulio Cerullo, Margherita Zavelani-Rossi, and Sandro De S ilvestri
Istituto Nazionale per la Fisica della Materia and Centro di Elettronica Quantistica e
Strumentazione Elettronica, Dipartimento di Fisica, Poli tecnico di Milano, Piazza Leonardo da
Vinci 32, I-20133 Milano, Italy
Mudit Tyagi and Mauro Giacca
Molecular Medicine Laboratory, International Centre for G enetic Engineering and Biotechnology,
Padriciano 99, I-34012 Trieste, Italy
Abstract
The coherent dynamics of vibronic wave packets in the green fl uorescent pro-
tein is reported. At room temperature the non-stationary dy namics following
impulsive photoexcitation displays an oscillating optica l transmissivity pat-
tern with components at 67 fs (497 cm−1) and 59 fs (593 cm−1). Our results
are complemented by ab initio calculations of the vibrational spectrum of
the chromophore. This analysis shows the interplay between the dynamics of
the aminoacidic structure and the electronic excitation in the primary optical
events of green fluorescent proteins.
PACS numbers: 87.15.He, 87.15.Aa, 87.14.Ee, 78.47.+p
1The green fluorescent protein (GFP) of the Aequorea victoria jellyfish has emerged in
recent years as a unique fluorescent label in several biologi cal studies [1]. GFP is a large
intrinsically-fluorescent protein (238 amino acids) chara cterized by a cylinder-shaped three-
dimensional structure with a diameter of 24 ˚A and a height of 42 ˚A [2]. The chromophore,
located at the center of the cylinder, is a photoexcitable gr een-light emitter autocatalytically
generated by the post-translational modification of a 3-ami no-acid sequence (Ser65-Tyr66-
Gly67) [3]. It consists of the hydroxybenzyl side chain of Ty r66 (phenolic ring) and the
imidazolidinone ring formed by cyclization of the tripepti de (heterocyclic ring). The spectral
characteristics of its emission and absorption bands have r eceived a tremendous amount of
attention [2–5] in an effort to understand GFP photophysics d own to the single-molecule
level [6] and design mutants with optical properties tailor ed to specific needs. To date there
exists a large set of GFP mutants with absorption and emissio n bands ranging from the violet
to the red part of the spectrum. Recently time-dependent ana lyses have shed light on internal
photoconversion mechanisms [7,8]. However, a definitive mi croscopic model accounting for
the optical properties of the chromophore and its blinking a nd photobleaching dynamics
[5–7] is still missing, partly due to a lack of precise inform ation on the electronic states
involved and to the limited effort in theoretical modelling. Only in recent years, in fact, a
few electronic-structure calculations of the GFP chromoph ore in different protonation states
have been performed [9,10].
Few-optical-cycle laser pulses allow ultrafast spectrosc opy with unprecedented temporal
resolution and are opening new avenues in the study of optica l properties of molecules
[11]. Creation of non-stationary vibronic wave packets and their observation have already
provided new insights into the microscopic mechanisms resp onsible for the optical activity
of few simple physical and biophysical systems [12–15]. Obs ervation of coherent phenomena
in a protein, however, still remains a challenge because of f ast dephasing times. There exist
only few reports on coherent dynamics in proteic systems who se functionality, however, is
determined by non-proteic cofactors [14,15].
In this Letter we demonstrate that GFP ultrafast response af ter femtosecond laser ex-
2citation is dominated by the coherent dynamics of single-el ectron vibronic wave packets,
created in both the ground and excited states of the protein c hromophore [16]. The coher-
ent dynamics manifests as an oscillatory modulation of the d ifferential optical transmissivity.
These results provide direct measurements of the chromopho re collective vibrations during
the optical process. In order to elucidate this dynamics, we also report the first ab initio
calculations of the vibrational properties of the GFP chrom ophore based on the density
functional theory (DFT).
The time-resolved technique here reported employed a pump- probe experimental scheme
[13] with identical pump and probe pulses, resonant with GFP absorption. Sub-10 fs pulses
centered around 500 nm were generated using a non-collinear optical parametric amplifier
in the visible, pumped by the second harmonic of a Ti:sapphir e laser [17]. The pulses had
energies of about 40 nJ and spectral width allowing analysis in a broad wavelength range
(490-550 nm). Samples consisted of GFP solutions at a concen tration of 350 µM in volumes
of 40 µl kept in a 0.5 mm-thick cuvette. Additionally, room-temper ature steady-state ab-
sorption and emission spectra were measured. In particular , absorption was measured in a
300µl GFP sample at a concentration of 70 µM in a 1 cm-long cuvette. For these experi-
ments we selected a representative GFP mutant, the enhanced GFP (EGFP), exhibiting a
single absorption peak around 490 nm (associated to the anio nic form of the chromophore
[18]) and improved brightness after blue-light excitation with respect to wild-type GFP [5].
Figure 1 shows the steady-state absorption (dotted line) an d emission (solid line) spectra
of EGFP together with a schematic configuration-coordinate diagram for the anionic state.
The absorption peak at 490 nm corresponds to the vertical ele ctronic transition labeled as
A. The emission lineshape displays two bands associated to t ransitions (B and C in the
diagram) from the bottom of the excited state band into the fir st two vibrational levels of
the ground state. The lower panel of Fig.1 shows the waveleng th dependence of the average
EGFP differential transmissivity (∆T/T) in the very first pic oseconds following impulsive
excitation. The observed positive values at the absorption and emission maxima can be
assigned to ground-state absorption bleaching and stimula ted emission from the excited
3state. In the inset of the lower panel of Fig.1, a typical time trace with low sampling
frequency of ∆T/T (at 500 nm) is reported. The signal remains constant during the first
tens of picoseconds in agreement with expected values of fluo rescence lifetime ( ≈3.3 ns [8]).
Superimposed to the long-lasting value of the EGFP transmis sivity we detected sub-
picosecond oscillations during the first two picoseconds fo llowing excitation. In Fig.2 (left
side) a representative time-dependent ∆T/T at 530 nm is repo rted. Similar modulations
were observed at other wavelengths. This oscillation patte rn is a direct evidence of coher-
ent dynamics of electronic wave packets in the GFP chromopho re and yields two distinct
frequencies at 497 cm−1(period 67 fs) and 593 cm−1(period 59 fs) with dephasing time
of about 1 ps. The two peaks in the power spectrum shown in the i nset of the figure are
clearly distinguishable and were found at unaltered positi on for the wavelengths studied.
The measured frequencies correspond to the energy spacing b etween consecutive vibronic
levels of the chromophore in the ground and excited states. I n our case, for pump-pulse du-
ration significantly shorter than the oscillation period, w e expect the oscillatory signal to be
dominated by excited-state dynamics, since the wave packet does not move greatly from the
Franck-Condon region during excitation [19]. The ground-s tate vibrational frequency can
also be roughly extracted from the energy spacing between ba nds B and C in the emission
lineshape shown in Fig.1. This procedure gives a value of ≈660 cm−1.
However, in order to unambiguously establish that the domin ant mode (at 497 cm−1)
corresponds to the GFP dynamics in the excited state, we cons idered the spectral depen-
dence of the amplitude and phase of the oscillatory pattern [ 15]. The coherent oscillation
of a vibronic wave packet in the excited state, in fact, cause s periodic wavelength shifts in
stimulated emission. This should yield periodic changes in the intensity of the stimulated
emission with maxima where the slope of the emission linesha pe is large (at 500 and 530
nm, see Fig.1) and minima at the peak position and wings of the emission spectrum. In ad-
dition, the oscillatory pattern at wavelengths correspond ing to the two sides of the emission
peak must be out of phase. Similar arguments relate the vibro nic wave packet in the ground
state and absorption modulation to the absorption lineshap e. The right side of Fig.2 shows
4the amplitude (upper panel) and the relative phase (lower pa nel) of the measured trans-
missivity oscillations as a function of wavelength. This an alysis reveals weak oscillations at
wavelengths near the emission peak and at the wings, while la rger on the steep sides. A
strong phase change (approximately a phase inversion) cent ered around the wavelength of
the emission maximum is observed. These data confirm that the origin of the dominant
modulation at 497 cm−1is the vibrational dynamics in the excited state. We associa te the
other frequency to oscillations in the ground state in light of the results of the steady-state
emission lineshape analysis and of our model-chromophore c alculations (see below).
On the basis of these arguments, we are now able to elucidate t he vibrational pattern
driven by the optical excitation process and identify the co upling mechanism of the vibra-
tional motion to the electronic excitation. To this end we pe rformed electronic, structural,
and vibrational calculations of the isolated EGFP chromoph ore in the anionic state within
a DFT-based ab initio molecular dynamics approach [20]. The electronic structur e was
calculated by using a local density exchange and correlatio n functional with Becke and
Perdew gradient corrections [21]. Soft first-principle pse udopotential [22] were used for the
interactions between valence electrons and inner cores wit h a 25 Ryd energy cutoff for the
plane-wave basis set. Simulations with 0.15 fs time-step we re performed in a 15 ˚A cubic
box, large enough to prevent interactions with the periodic images.
In the top part of Fig.3 we show the HOMO (Highest Occupied Mol ecular Orbital) and
LUMO (Lowest Unoccupied Molecular Orbital) of the chromoph ore. A charge transfer from
the phenolic ring to the heterocyclic ring and a redistribut ion of the charge within each
ring following electronic excitation is observed. This cha rge transfer is responsible for an
increase of the proton affinity of the Tyr66 heterocyclic-rin g nitrogen and may be linked to
the blinking dynamics [9,23].
The charge redistribution also causes changes in the streng ths of some of the molec-
ular bonds owing to the difference in bonding character betwe en the ground and excited
electronic states. This stimulates the dynamics seen in our experiments. The ground-state
vibrational spectrum was calculated at T=300 K from the Four ier-transform of the veloc-
5ity autocorrelation function on a ≈1.5 ps trajectory of Car-Parrinello molecular dynamics.
The resulting spectrum is shown in the lower part of Fig.3 (da shed line). The vibrational
frequencies in the region above 1000 cm−1agree within 5% with recent Raman data on the
EGFP chromophore [24] and correspond to (localized or colle ctive) stretching modes [23].
However, in thermally-equilibrated conditions not all the modes associated to the optical
excitation have a significant spectral strength. We note tha t these modes are very unlikely
to be observed in steady-state resonant Raman experiments o wing to fluorescence or sample
degradation [24] but are accessible within our experimenta l approach. In order to specif-
ically enhance these modes, we performed runs in appropriat e non-thermally-equilibrated
conditions reproducing the atomic displacements induced b y the electronic excitation [23].
The corresponding spectrum is shown in Fig.3 (solid line). A s expected, this simulation al-
lows us to better identify the high-frequency streching mod es in the range 1000-1650 cm−1.
Remarkably, the same simulation also yields vibrations bel ow 800 cm−1corresponding to an-
gular deformations of the chromophore. This fact provides e vidence of intramolecular mode
coupling among vibrations in two different and characterist ic frequency ranges [25]. The
inset of Fig.3 reports an enlarged view in the low-frequency region of the solid-line spectrum
shown in the main panel. The arrow indicates the experimenta lly-measured ground-state
frequency (593 cm−1). At frequencies close to this one, we found two modes at 575 a nd 615
cm−1. The identification of the specific one corresponding to the e xperiment is beyond our
accuracy. However they both correspond to collective vibra tions involving angular in-plane
deformation of the rings (mainly the phenolic one) and of the bridge between them. We can
therefore draw unambiguous conclusions on the relevant mic roscopic processes involved.
The steps leading to the coupling of these low-frequency mod es to the electronic pho-
toexcitation can now be easily understood: in structures wi th aπ-bonding system, the high-
frequency stretching modes are usually directly coupled to the electronic excitation, since
the primary effect of the induced electronic-density change is to shorten the single bonds and
lengthen the double bonds (for example in retinals [26]). Th e double-ring structure of the
GFP chromophore, however, introduces a strong geometric co nstraint that allows efficient
6coupling to the low-frequency angular modes responsible fo r the observed coherent dynam-
ics. This process highlights the intramolecular coupling p athways and nuclear dynamics
following photoexcitation and is a peculiar photophysical property of the GFP family.
In conclusion, we presented the coherent dynamics of single -electron vibronic wave pack-
ets following ultrafast excitation in EGFP. The analysis of coherent oscillations provided
the vibrational frequencies of both the ground and excited s tates of the EGFP chromophore
and allowed to evaluate the wave-packet dephasing time. The collective vibration excited
during the optical process and its coupling mechanism to the electronic excitation have been
identified by ab initio calculations.
Acknowledgements . One of us (V.T.) wishes to thank F. Buda for making available
the code for Molecular Dynamics.
7FIGURES
FIG. 1. Upper panel: Enhanced green fluorescent protein (EGF P) absorption (dotted line) and
fluorescence (solid line) spectra at room temperature. Lett ers indicate transitions between excited
and ground electronic states as depicted in the schematic co nfiguration-coordinate model. Lower
panel: Wavelength dependence of the average EGFP differenti al transmissivity in the very first
picoseconds after impulsive excitation (line is a guide to e ye). Typical error bars and positions of
the absorption and emission peaks are shown. The inset shows the differential transmissivity at
500 nm (acquired with low sampling frequency) as a function o f delay between pump and probe
pulses.
FIG. 2. Left side: Room-temperature enhanced green fluoresc ent protein (EGFP) differential
transmissivity at 530 nm as a function of delay between pump a nd probe pulses. The inset shows
the Fourier power spectrum of the data. Right side: Spectral dependence of the amplitude (upper
panel) and relative phase (lower panel) of the measured tran smissivity oscillations before damping
(lines are guides to eye and typical error bars are shown).
FIG. 3. Upper side: Simulated chromophore (C 10O2N2H−
7). The bonds with the protein back-
bone are cut at the level of the heterocyclic ring and saturat ed with hydrogen atoms. HOMO
(Highest Occupied Molecular Orbital) and LUMO (Lowest Unoc cupied Molecular Orbital) are
represented as isocharge surfaces. Carbon, hydrogen, and n itrogen are shown in black, white,
and grey, respectively. Lower side: Calculated vibrationa l spectra of the chromophore in ther-
mally-equilibrated (dashed line) and non-thermally-equi librated (solid line) conditions (see text).
Inset: Enlarged view of non-thermally-equilibrated vibra tional spectrum. The arrow corresponds
to the experimentally-measured ground-state frequency (5 93 cm−1).
8REFERENCES
[1] M. Chalfie et al., Science 263, 802 (1994); Green Fluorescent Proteins , edited by K. F.
Sullivan and S. A. Kay (Academic Press, San Diego, 1999); A. M arcello et al., Proc.
Natl. Acad. Sci. USA, submitted.
[2] M. Orm¨ o et al., Science 273, 1392 (1996).
[3] R. Heim, D. C. Prasher, and R. Y. Tsien, Proc. Natl. Acad. S ci. USA 91, 12501 (1994).
[4] R. Heim, A. B. Cubitt, and R. Y. Tsien, Nature 373, 663 (1995).
[5] G. H. Patterson et al., Biophys. J. 73, 2782 (1997).
[6] D. W. Pierce, N. Hom-Booher, and R. D. Vale, Nature 388, 338 (1997); R. M. Dickson
et al., Nature 388, 355 (1997); R. A. G. Cinelli et al., Photochem. Photobiol. 71, 771
(2000).
[7] P. Schwille et al., Proc. Natl. Acad. Sci. USA 97, 151 (2000).
[8] M. Chattoraj et al., Proc. Natl. Acad. Sci. USA 93, 8362 (1996).
[9] A. A. Voityuk, M. E. Michel-Beyerle, and N. R¨ osch, Chem. Phys.231, 13 (1998).
[10] W. Weber et al., Proc. Natl. Acad. Sci. USA 96, 6177 (1999).
[11] M. J. Rosker, F. W. Wise, and C. L. Tang, Phys. Rev. Lett. 57, 321 (1986).
[12] T. Tokizaki et al., Phys. Rev. Lett. 67, 2701 (1991); J. Feldmann et al., Phys. Rev.
Lett.70, 3027 (1993); T. Kobayashi et al., Chem. Phys. Lett. 321, 385 (2000).
[13] M. Nisoli et al., Phys. Rev. Lett. 77, 3463 (1996).
[14] L. Zhu et al., Phys. Rev. Lett. 72, 301 (1994); M. H. Vos et al., Proc. Natl. Acad. Sci.
USA91, 12701 (1994); U. Liebl et al., Nature 401, 181 (1999).
[15] Q. Wang et al., Science 266, 422 (1994).
9[16] The term “single-electron wave packet” is used to indic ate that only one electron is
photoexcited in the GFP molecules and that the following dyn amics of each molecule
is independent of the others.
[17] G. Cerullo, M. Nisoli, and S. De Silvestri, Appl. Phys. L ett.71, 3616 (1997).
[18] K. Brejc et al., Proc. Natl. Acad. Sci. USA 94, 2306 (1997).
[19] S. Mukamel, Principles of Nonlinear Optical Spectrosocpy (Oxford University Press,
New York, 1995).
[20] R. Car and M. Parrinello, Phys. Rev. Lett. 55, 2471 (1985).
[21] J. P. Perdew and A. Zunger, Phys. Rev. B 23, 5048 (1981); A. D. Becke, Phys. Rev. A
38, 3098 (1988); J. P. Perdew, Phys. Rev. B 33, 8811 (1986).
[22] D. Vanderbilt, Phys. Rev. B 43, 7892 (1990).
[23] V. Tozzini and R. Nifos` ı, J. Phys. Chem. B, submitted.
[24] A. F. Bell et al., Biochemistry 39, 4423 (2000).
[25] K. Moritsugu, O. Miyashita, and A. Kidera, Phys. Rev. Le tt.85, 3970 (2000).
[26] B. Curry et al., inAdvances in Infrared and Raman Spectroscopy , vol. 12, edited by R.
J. H. Clark and R. E. Hester (Heyden, London, 1985).
10350 400 450 500 550 600 65012345
BAC C
BB
AAEGFP
70 µM
Fluorescence (A. U.) Absorbance (x 10-2)
Wavelength (nm)
500 525 550 5752468
∆T/T (x 10-3)
Wavelength (nm)10 20 302468
λ = 500 nm
∆T/T (x 10-3)
Delay time (ps)0.5 1.0 1.5 2.02345
λ = 530 nm
∆T/T (x 10-3)
Delay time (ps)10 20 30Frequency (cm-1)
Power (A. U.)
Frequency (THz)250 500 750 1000
593 cm-1497 cm-1
500 520 540-1000100
Phase (degree)
Wavelength (nm)0.40.81.2
∆T/T (x 10-3) HOMO ONNO
LUMO |
arXiv:physics/0102060v1 [physics.plasm-ph] 20 Feb 2001Novel Mechanism for Single Bubble Sonoluminescence
Boris P. Lavrov
Faculty of Physics, St.-Petersburg State University, 1989 04, Russia
(February 2, 2008)
Careful re-examination of typical experimental data made i t possible to show that the UV
continua observed in multi-bubble (MBSL) and single-bubbl e (SBSL) sonoluminescence spectra
have the same physical nature - radiative dissociation of el ectronically excited H∗
2(a3Σ+
g) molecules
[and probably hydrides of heavy rare gases like ArH∗(A2Σ)] due to spontaneous transitions between
bound and repulsive electronic states. The proposed mechan ism is able to explain all available
spectroscopic observations without any exotic hypothesis but in terms usual for plasma spectroscopy.
42.50.Fx, 42.65.Re, 43.25.+y, 47.40.Nm
Sonoluminescence (SL) is one of most exciting features
of acoustic cavitation - the formation and non-linear os-
cillation of gas bubbles generated in liquids by ultrasound
[1]. Bubble collapses lead to enormous local pressures
(about 1000 atm and more) and temperatures, erosion
of hard materials, chemical reactions and light emission
(SL). Discovered in early 1930s the SL observed in ca-
vitation clouds is called multi-bubble sonoluminescence.
MBSL spectra consist of a continuum of unknown origin,
atomic lines and molecular bands of species connected
with host liquids [1]. Atomic and molecular emissions
give effective temperatures ≈3000–5000 K [2].
Suddenly the situation in this traditional branch of
non-linear acoustics has been changed dramatically after
the discovery [4] of single bubble sonoluminescence - light
emission of a single, stable, oscillating bubble trapped
by acoustic levitation inside an ultrasound resonator [4].
Such eccentric change in the performance of SL experi-
ment provided unique opportunity to study the dynamics
of oscillating bubble, obviously masked in MBSL. It was
shown [5] that every cycle consists of: 1) relatively slow
growth of bubble radius up to ≈50µm, 2) extremely fast
implosive collapse to ≈0.5µm, 3) damping and wait-
ing for the expansion phase of sound wave in so-called
“dead mode” with constant radius ≈5µm, correspond-
ing to atmospheric pressure. SL appears as light flashes
at the moment of the collapse. Further studies (see
bibl. in [1,3,5–7]) have shown: 1) water being friendli-
est liquid for SBSL; 2) clockwise regularity of the flashes
with≈50 ps stability; 3) abnormally short pulse dura-
tion (≈50–350 ps); 4) featureless spectrum with the in-
tensity increasing to UV (see Fig. 1), which was fitted as
a “tail of a blackbody spectrum” with abnormally high
temperatures T= 25000 −100000 K [3,5,6]; 5) absence
of characteristic emissions associated with host liquid; 6 )
stimulating influence of Ar. Those peculiarities are often
presented and discussed as mysteries and unknowns of
SBSL (see i.e. [5,7]) not only in respected scientific jour-
nals, but also in popular ones and newspapers as well.
Spectra of MBSL and SBSL measured “under simi-
lar experimental conditions” [8] are shown in Fig. 2 (ac-
tually the conditions are sufficiently different, see be-
1low). Such observations are often interpreted as evidence
that “these two phenomena are fundamentally different”
(“sonochemistry” and “sonophysics”) [6]. Unbelievably
high temperatures “observed” in SBSL experiments were
so exciting that the mechanism of highly spherical im-
ploding shock wave in SBSL was even considered as “the
key to reaching temperatures and densities sufficient to
realize the fusion of these hydrogen nuclei to yield he-
lium and neutrons” [5]. The stir stimulated tremendous
growth of speculations about the nature of SBSL. Many
mechanisms have been proposed so far from traditional
ones (quasi-adiabatic heating, chemiluminescence, elec-
tric breakdown, shock wave and others) up to such ex-
otic as Schwinger’s Dynamical Casimir Effect. Although
most popular hypothesis is the Bremsstrahlung radiation
of electrons in dense plasma (see i.e. [1,7,9]), the real na-
ture of SBSL is still an open question [1].
The main goal of the present work is to propose and
consider electronically excited H∗
2(a3Σ+
g) molecules [and
possibly hydrides of rare gases like ArH∗(A2Σ)] as light
emitters responsible for continua observed in both MBSL
and SBSL experiments with hydrogen-containing liquids.
This provides new sight on the well known “mysteries and
unknowns” of the SBSL phenomenon.
Perhaps the most simple and natural explanation of
the continuum radiation observed in SL spectra (never
even considered previously) is spontaneous emission of
hydrogen dissociation continuum appearing due to the
a3Σ+
g, v, J→b3Σ+
utransitions ( v, J- vibrational and ro-
tational quantum numbers) between lowest triplet bound
(upper) and repulsive (lower) states (Fig. 3). It is a well-
known spectral feature of hydrogen-containing plasmas
widely used in UV spectroscopy (see bibl. in [10]). The
shape of the continuum is determined by the distribu-
tion of population density among vibro-rotational levels
of the upper a3Σ+
gelectronic state [11]. It is not directly
connected with any translational temperature but is de-
termined by a dynamic balance between excitation and
deactivation of the a3Σ+
g, v, Jlevels.
In low-pressure gas discharges, the spectral intensity
distribution of the continuum may be calculated not only
in relative [11], but even in absolute scale [12] being
in rather good accordance with experimental data. In
the H 2+Ar mixtures the Ar∗(4s)→H2(a3Σ+
g) excitation
transfer from long living (metastable and resonant) levels
of Ar [13] may play important role as well as the forma-
tion of excited ArH∗excimer molecules and A2Σ→X2Σ
spontaneous emission due to bound-repulsive transitions
[10,14]. The ArH continuum overlaps that of H 2being
located in the same wavelength range. The energy of He∗
and Ne∗metastables is too high to participate in an ex-
citation transfer leading to populating of the a3Σ+
gstate.
However, Kr∗and Xe∗can do the job in three-body col-
lisions with two 1S hydrogen atoms.
Correct calculation of the continuum shape in plasma
with temperature higher than 1000 K is impossible just
2because the transition probabilities are available only fo r
rotation-less molecule [10]. On the other hand, it needs
development of certain model of microscopic excitation-
deactivation processes and certain values of plasma pa-
rameters determined by macroscopic dynamics of collaps-
ing bubble. Two main de-populating processes are for
sure: spontaneous emission and radiationless collisional
quenching [15]. But there is a great variety of competitive
volume and surface processes responsible for the gener-
ation of H∗
2(a3Σ+
g) excited molecules (electron impact,
electron-ion and ion-ion recombination, associative thre e-
body collisions, photo and/or collision-induced fragmen-
tation of water, etc.). Nevertheless, very rough estima-
tions can be made by neglecting the rotational structure
ofa3Σ+
g, vlevels in two simple cases: 1) thermodynamic
equilibrium (TDE) populations of a3Σ+
g, vlevels relative
to ground X1Σ+
g, v=0 vibronic state; 2) direct electron
impact excitation and spontaneous decay of the levels
[10]. The results are shown in Fig. 4. One may see
that calculated spectra are in accordance with experi-
mental observations at least qualitatively. They have
“featureless structure” with the intensity rising to UV
cutoff λ≈250 nm. In the range of observation, they
may be fitted as “blackbody spectrum” with “enormous
temperatures”.
Experimental data are also not free from criticism:
1) Intensity calibration should take into account re-
absorption in plasma and transparency of plasma-liquid
boundary neglected in [3,8].
2) Determination of the background level is not so sim-
ple, because light scattered inside a flask and a spectrom-
eter overlaps dark signal of a detector. The huge differ-
ence between backgrounds of SBSL and MBSL curves is
caused by use of two different optical systems in [8]. The
emission of MBSL was focused on the entrance slit of the
spectrometer by a lens. In the case of SBSL, no collec-
tion lens was used. The entrance slit was placed close to
the side of the levitation cell being 2.25 cm from the bub-
ble. It means that all light coming from within 2 πsolid
angle (much bigger than the instrumental aperture and
that for radiation directly coming from the bubble) was
able to enter the spectrograph and partly be detected as
a background.
3) The separation of the continuum intensity from the
total signal of a detector is also a rather delicate and
ambiguous deal. For example the peculiarities on the
SBSL curve of Fig. 2 may be interpreted as some ad-
ditional emissions at λ≈310−360 nm (OH bands)
andλ≈400−500 nm (NaH bands or H 2continuum
in the second order of the grating), or as absorptions at
λ <320 nm (OH) and λ≈350−420 nm (NaH).
4) Proper normalization of experimental curves is nec-
essary for a comparison of their shapes.
Experimental data of [3,8] have been treated taking
into account what is written above. Results of such re-
calculations are presented in Fig. 4 & 5. One may see
3that the results of two independent SBSL experiments
[3,8] are in good agreement as well as intensity distribu-
tions obtained by MBSL [8] and SBSL [3,8].
Taking into account experimental errors and the un-
certainties in the data processing we have to come to the
following conclusion. After proper treating the experi-
mental data show: 1) The continua emitted by SBSL and
MBSL (Fig. 5) have identical spectral intensity distribu-
tion, therefore they may have the same nature; 2) Mea-
sured spectral intensity distributions and those roughly
calculated for the a3Σ+
g, v, J→b3Σ+
uspontaneous emis-
sion of H 2molecule (Fig. 4) are in semi-quantitative
agreement good enough to propose H∗
2(a3Σ+
g) molecules
to be responsible for the continuum emission. “Enor-
mous temperatures” of SBSL reported so far have no
physical meaning being the result of incorrect fitting (un-
proper treated experimental data were approximated by
un-proper analytical expression – Planck formula).
There is actually a great difference between MBSL and
SBSL experiments even if they are carried out with the
same chemical solutions: 1) The amplitude of sound wave
in MBSL (10 atm) is about one order of magnitude bigger
than that used in SBSL ( ≈1.3 atm). Therefore in MBSL
the action of ultrasound should be much more power-
ful and destructive for bubbles. The widely distributed
opinion that SBSL is a stronger phenomenon is based
only on the “observation” of “enormous temperatures”,
not more. 2) MBSL experiments are made with 100%
air saturation of a solution, while SBSL experiments are
performed with degassed water.
Thus, the bubbles have qualitatively different gas con-
tents in those two types of SL experiments. MBSL bub-
bles are mainly air-filled with small amount of water va-
por. Dissociation of N 2, O2, H2O during the collapse
leads to formation of very aggressive species (like HN 3,
HNO 3, N2O2, N2O3, etc.) which disappear by chemical
reactions with water boundary. In the expansion phase
a bubble (if it would be able to survive!) is again filled
with air due to 100% air saturation. New bubbles are
definitely generated as air-filled.
An absolutely other situation should occur in the case
of SBSL when the action of acoustic waves is much more
gentle and water is degassed. The SBSL bubble can ac-
cumulate not only Ar (1% in air) [16], but molecular
hydrogen as well. The hydrogen molecule in its ground
state has almost the same electronic structure as that
of He atom - its united atom analogue (two 1s elec-
trons with anti-parallel spins). Thus, H 2itself has low
chemical activity in great contrast with hydrogen atom.
The solubility of H 2in water is much smaller than that
of radicals made from N, O and H atoms. Therefore,
a stable-oscillating bubble in SBSL mode actually con-
sists of a H 2+Ar gas mixture with periodically changing
amount of water vapor (increasing during the expansion
and decreasing in the collapse). These additional H 2O
molecules disappear in the collapse and serve as an en-
4gine (and fuel) for the transformation of the translational
energy of collapsing liquid-gas boundary into the energy
of light emission. This mechanism explains: 1) Why the
light flash appears only at the first collapse but not at
the second one in the series of damping oscillations in
spite of almost the same compression [5]; 2) Why the av-
erage radius of a bubble generally increasing with a rise
in acoustic amplitude suddenly shrinks when the onset of
SL is reached [5]; 3) Great rise of SBSL with a decrease
of water temperature [3].
Clockwise regularity of SBSL flashes should not be so
surprising and does not need unusual mechanisms be-
cause the experimental setup used in SBSL experiments
is essentially a resonant system. The stability of this
regularity means that after each collapse in the “dead
mode” the bubble contents returns to the initial one -
98% of (H 2+Ar) mixture and 2% of H 2O.
The abnormally short duration of SBSL light flashes
may be explained by extremely high rise of both the
rate of excitation and the rate of collisional quenching
of excited states. Thus the conditions suitable for spon-
taneous emission may be realized only in rather limited
period of time. The situation is obviously different for
different excited species. For some of them the favorable
conditions could not be achieved at all (this explains also
the existence of upper threshold of SBSL). The quenching
may lead to a dissociation of molecules and to emission
of vacuum UV radiation being out of the range of obser-
vation. The dim luminosity cloud surrounding a hot spot
most probably is the fluorescence induced by L αatomic
line and/or Lyman and Werner bands of H 2. The esti-
mation of the characteristic time of H∗
2(a3Σ+
g) collisional
quenching with cross sections from [15] gives ≈1 ps.
The positive influence of heavy inert gases Ar, Kr, Xe
(in contrast to He, Ne [17]) may be connected with the
excitation transfer from their metastables and with for-
mation of excited hydrides like ArH∗(A2Σ). Absence
of characteristic emissions of Na∗and OH∗in SBSL is
caused by better evacuation and/or quenching of upper
states in hydrogen-dominated contents of the bubble.
It is common practice in plasma spectroscopy that an
investigator should find proper answers for three ques-
tions: 1) Who is the emitter of the emission? 2) What
are main processes of excitation and deactivation of the
upper state of the transition? 3) How the population
density of the upper level(s) can be related to plasma
parameters? Only when all three are answered the emis-
sion may be used for plasma diagnostics (spectroscopic
determination of temperatures, particle densities etc.).
From such point of view any speculations around abnor-
mally high temperatures “observed” in SBSL and “the
opportunity” to make one more “cold fusion” are mean-
ingless.
5ACKNOWLEDGMENTS
My gratitude to M. K¨ aning for useful advices and help.
[1] S. Suslick et al., Phil. Trans. R. Soc. Lond. A 357, 335
(1999).
[2] W. B. McNamara III, J. T. Didenko, and K. S. Suslick,
Nature 401, 772 (1999).
[3] R. Hiller, S. J. Putterman, and B. P. Barber,
Phys. Rev. Lett. 69, 1182 (1992).
[4] D. F. Gaitan and L. A. Crum, in Frontiers in Nonlinear
Acoustics (Elsevier, N. Y., 1990), pp. 459–463.
[5] S. J. Putterman, Scientific American 33 (Feb. 1995).
[6] L. A. Crum, J. Acoust. Soc. Am. 95, 559 (1994).
[7] B. P. Barber et al., Phys. Lett. 281, 66 (1997).
[8] T. J. Matula et al., Phys. Rev. Lett. 75, 2602 (1995).
[9] K. Yasui, Phys. Rev. E 60, 1754 (1999).
[10] B. P. Lavrov, A. S. Melnikov, M. K¨ aning, and J. R¨ opcke,
Phys. Rev. E 59, 3526 (1999).
[11] B. P. Lavrov and V. P. Prosikhin, Opt. Spectrosc. 58,
317 (1985).
[12] B. P. Lavrov and V. P. Prosikhin, Opt. Spectrosc. 64,
298 (1988).
[13] B. P. Lavrov and A. S. Melnikov, Opt. Spectrosc. 85, 666
(1998).
[14] C. R. Lishawa, J. W. Feldstein, T. N. Stewart, and E. E.
Muschlitz, J. Chem. Phys. 83, 133 (1985).
[15] J. Bretagne, J. Godart, and V. Puech, J. Phys. B:
At. Mol. Phys. 14, 761 (1981).
[16] D. Lohse et al., Phys. Rev. Lett. 78, 1359 (1997).
[17] Y. T. Didenko, W. B. McNamara III, and K. S. Suslick,
Phys. Rev. Lett. 84, 777 (2000).
6FIG. 1. Spectral density of SBSL measured in [3] with two
different light sources used for the absolute intensity cali bra-
tion: the Deuterium lamp (dotted line) and the quartz tung-
sten halogen (QTH) lamp (points with error bars). The solid
line represents blackbody spectrum for T=25000 K. The hor-
izontal lines are background levels used in present work.
FIG. 2. Spectra of SBSL and MBSL obtained in [8] (solid
lines). Dashed lines were used in present work as the conti-
nuum intensities with backgrounds shown by horizontal line s.
7FIG. 3. Grotrian diagram of the ground and lowest triplet
excited electronic states of H 2molecule and Ar atom. Ar-
rows indicate a3Σ+
g, v→b3Σ+
uspontaneous emission transi-
tions. Double arrows show some processes of populating of
thea3Σ+
g, vvibronic states: direct excitation of atoms and
molecule and Ar∗→H2excitation transfer.
FIG. 4. The comparison of recalculated experimental SL
continuum spectra [3,8] with those calculated for the TDE
conditions with T= 4000, 5000 and 6000 K (dash-dot lines).
Solid line corresponds to the case of direct electron impact
excitation and spontaneous decay of a3Σ+
g, vlevels of H 2[10].
8FIG. 5. Relative continuum intensity obtained from SBSL
and MBSL spectra [3,8] (Fig. 1, 2) after the subtraction of
the background and renormalization for unity at λ= 350 nm.
9 |
ON SUPERLUMINAL PROPAGATION AN
D INFORMATION VELOCITY
Akhila Ra
man
Berkeley, CA.
Email: akhila_r
aman@yahoo.com
Fe
b 19,2001.
Abstract:
This paper examines s
ome of the recent experiments o
n superluminal propagation.
The
meaning of information velocity
from the perspective of a digi
tal communication
system is anal
yzed. It is shown that practic
al digital communication system
s use
bandlimited signals to tra
nsmit information consisting of
random bits which cannot
be pre
dicted by looking at previously
transmitted bits and that info
rmation is not
limited to discon
tinuous step-functions alone. T
hese random bits are pulse-shap
ed by
using bandlimited pulse (e
.g. gaussian pulse), in a causa
l fashion (meaning waveform
duri
ng any given bit is obtained by
convolution of present and pas
t bits with the causal
pulse, bu
t cannot have information about
future bits!). If these pulse-
shaped random bits
were not cons
idered as information, if someh
ow the future random bits could
be
predicted by, say looking at
the shape of the first bit, t
hen there would be no need for
e
laborate communication systems
like cellular systems and data
modems! We could
transmit the fi
rst bit waveform, turn off the
transmitter and expect the rece
iver to
correctly detect all fut
ure random bits, which clearly
is impossible! It is shown that
it is
possible to achieve infor
mation velocity greater than sp
eed of light, in media with
zero
dispersion and positive index
of refraction less than unity i
n the frequency range of
interes
t, and of sufficient length to
make the time gained during med
ium transmission in
comparison t
o vaccum transmission, more th
an the duration of several bits
. It is shown
that while signal
causality is preserved from the
perspective of an LTI system,
Einstein
causality is not preser
ved and hence what this means t
o relativistic causality violat
ions is
analyzed. It is further
shown that in cases where the
index of refraction or the grou
p
index is negative, the observe
d negative shift of the peak of
the output pulse with respect
t
o the input peak is merely due
to the fact that the pulse was
predictable.
1. Introduction:
Le
t us consider the case of a Lin
ear Time Invariant(LTI) system
which is characterized
by its ca
usal impulse response h(t). The
term causality is used through
out this paper to
indicate signa
l and system causality, which r
equires that a signal g(t) or s
ystem h(t) is
zero for t<0. Rela
tivistic or Einstein causality,
where specified is qualified b
y the
adjective "relativistic" o
r "Einstein".
Let p(t) be a caus
al gaussian input pulse to this
system. Gaussian pulse is chos
en because
it is both time-limit
ed and bandwidth-limited to a h
igh degree that it can be appro
ximated
to zero outside the rang
e of interest with negligible d
istortion in either domain. For
example, in the GSM cellular sy
stems
8
, a time-limited gaussian pul
se is used to shape
the random b
its to produce the transmit wav
eform X(t) which is a random pr
ocess with
gaussian PSD and is f
iltered in the receiver using a
causal filter in the frequency
range of
interest, say
0
ff≤
,and this process introduces n
egligible distortion in the wav
eform.Any distortion introduced can
be mitigated either by equaliza
tion or by simply boosting
up th
e transmit power if appropriate
, to achieve a specified bit er
ror rate(BER). Thus
information
is still communicated to the re
ceiver with a specified delay w
hich includes
two components:
1.
Processing delay: this include
s pulse shaping, filtering, buf
fering delays
2. Medium delay: T
his includes the delay encounte
red by each frequency component
of
the transmit waveform X(t) i
n
0
ff≤
.If the medium delay is consta
nt in
0
ff≤
, it is
linear-phase, and the r
eceived waveform Y(t) is a dela
yed version of X(t). The mediu
m
delay in
0
ff≤
will be the dominant component
of the delay of X(t). The medi
um delay
outside
0
ff≤
will have a negligible effec
t on X(t) because of the fact t
hat the power
spectral density(P
SD) of X(t) which is gaussian h
as negligible values in
0
ff>
.
Thus we see that the frequenc
y response of the medium in
0
ff>
has practically no
effect on t
he delay of X(t). If the medium
is not linear-phase, which is
usually the case in
dispersive w
ireless channels, then it intro
duces varying delay(dispersion)
for each
frequency component of
X(t) in
0
ff≤
, thus distorting X(t) . Agai
n, this distortion can
be mitiga
ted as mentioned before.
The Fou
rier Transform notations used i
n this paper are as follows
1
:
dfftjefGtg∫
∞
∞
−
=
π
2)()(
dtftjetgfG ∫
∞
∞
−
−=
π
2)()(
This is emphasized because som
e papers on superluminal propag
ation
2
have used an
alternate notatio
n with "f" replaced by "-f". He
nce the notation used in this p
aper has to
be borne in mind.
2.
Superluminal propagation analys
is in an LTI system:
Let ∞≤≤∞−
=
−
t
t
etg
2
)(
π
(1)
be a gaussian signal whose
standard deviation
π σ
2/1=
t
Let G(f) be the Fourier Transf
orm of g(t) which is also gauss
ian.
G(f)= F[ g(t) ] =
∞≤≤∞−
−
f
f
e
2
π
(2)
And itsstandard deviation
π σ
2/1=
f
For
0
tt>
where
t
t
σ
3
0
=
,
)(tg
drops to less than 39 dB of g
(0)=1 and can be
approximated to
zero. Similarly, for
0
ff>
where
f
f
σ
3
0
=
, G(f) can be approximated
to z
ero. This is justified because
more than 99.995% of the signal
energy is contained inthe range
0
tt≤
and
0
ff≤
respectively in either domain
! Thus we see that the gaussian
signal is both time-limited and
band-limited to a high degree.
Note that the choice of
t
t
σ
3
0
=
is arbitrary and is assumed on
ly as an example, it can be any
multiple of
t
σ
.
A time-limited gaussian baseb
and pulse p(t) is formed by tru
ncating g(t) for
0
tt>
, and
time-shifting it by t
0
.
=
−
=
)()(
0
ttgtp
2
0
)(tte−
−
π
for
0
20tt
≤
≤
and is zero elsewhere. (3)
I
ts Fourier Transform is given b
y
00 0
)))2(sin2()(()]([)( zftctfGtpFfP
⊗
=
=
(4)
where
02
0
ftez
j
π
−
= is a linear phase term
Where
⊗
denotes convolution.
Though P(
f) appears complicated,
)(fP
is very nearly equal to
)(fG
for
0
ff≤
,
and contains more than 99.995
% of total signal energy in tha
t frequency range.
P(f)= G(f) *
z
0
for
0
ff≤
; z
0
is a linear phase term, does
not affect magnitude.
=
0 for
0
ff>
since
)(fP
< 39 db of G(0)=1 (5)
Le
t us consider the simple case o
f a binary phase-shift-keying(B
PSK) system. The
transmitted bas
eband signal X(t) is formed by
convolving the random binary im
pulse
train representing the sym
bols with the gaussian pulse p(
t). We shall analyze BPSK in th
e
baseband only, using the equiv
alence of bandpass and lowpass
systems, since we can
always shi
ft it to any desired frequency
range by multiplying the lowpas
s signal by the
carrier frequenc
y f
c
>f
0
. In general, number of symbol
s M=2
b
, where b = number of bits.
For
BPSK, one symbol equals one bi
t. Throughout this paper, we wi
ll use the term
"symbol" to mean
random binary information for
the case of BPSK.
∑ ∑
−
=−
=
− =− ⊗=
1
01
0
)( )( )()(
N
ns nN
ns n
nTtpa nTtatptX
δ
(6)
where T
s
=symbol duration. Choose
t s
tT
σ
3
0
=
=
a
n
= binary i.i.d. random variab
le; takes the value
1
±
; 0
=
n
a for n<0.
N = number
of symbols in a given informat
ion stream.
X(t) is shown in the
figure below. It is very impor
tant to note that, though the w
aveform
appears "smooth", new ra
ndom information is present in
the peak of every symbol. There
is no way to predict with certa
inty the value of X(t) during a
given symbol duration,
based on
the values of X(t) during the
past symbols. In other words, d
uring a given
symbol interval, X
(t) carries information about p
resent and past bits, but carri
es no
information about the futu
re bits. This is precisely due
to the fact that the systemimplemented in Eq.(6) is a cau
sal system, where the output at
a given instant does not
depend
on future input!
Note that each
division along the X axis deno
tes the symbol number.
X(t) is a
random process. Its autocorrel
ation function is given by the
autocorrelation of
the gaussian
pulse p(t). The power spectral
density(PSD) of X(t) is obtaine
d by the
Fourier Transform of i
ts autocorrelation function
1
.
s X
TfPfS /)()(
2
=
(7)
Thus we see that the spectr
al content of transmit waveform
X(t) has the same
characteristi
cs as that of the underlying ga
ussian pulse p(t) and contains
99.995% of the
power within the
range
0
ff≤
.
Now, let us pass X(t) through
a medium which is an LTI syste
m with impulse response
h(t).The
frequency response of the syst
em is H(f). We know that the ou
tput PSD is given
by
2
)()()( fHfSfS
X Y
=
s Y
TfHfPfS /)()()(
2 2
=
(8)
Let us consider the case of
a completely inverted medium o
f length L whose refractive
inde
x is given by
3
:
21
222
1)(
−−−=
γωωωωω
in
rp
(9)
where
γ
is phenomenological linewidth,
r
ω
is the resonance frequency of
the medium,
and
p
ω
is the effective plasma freque
ncy. Let us consider the typica
l situation in which
r p
ω
ω
γ
<<
<<
are obeyed. For
r
ω
ω
<<
,
1)0()(
<
=
nn
ω
and is dispersionless near DC
.
The gain of the medium is clos
e to unity near DC.
-1.5
-1
-0.5
0
0.5
1
1.5
0
5
10
15
20
25
30
35
X(t)
time/Symbol durationBPSK signal shaped by a gaussian pulse
line 1Let us input an eigen function
ftjetE
i
π
2)(=to this medium. The refra
ctive index of the
medium is n(f
). Hence transmission velocity
of this eigen function is v(f)=
c/n(f). If the
medium length is
L, transmission delay experien
ced by this eigen function is
cL
fnfvLfT
d
/)()(/)(
=
=
. The output of the medium is
given by
))((2)(fTtfjetE
d
o
−=
π
. Given that the eigen value o
f the system is its frequency
re
sponse, eigen value is obtained
as
)(2)(/)()(ffTjetEtEfH
d
i o
π
−= =
.
cLfnfjecLfnfecLfnfjefH/)](Re[2*/)]
(Im[2]/)([2)(
π
π
π
−
−
=
−
=
(10)
In particular, n(f) is con
stant and equals n(0) for
0
ff≤
, Re[n(f)]=n(0), Im[n(f)]=0
0
/)0(2*1)( ffcLfnjefH ≤
−
=
π
(11)
The impulse response of th
e medium h(t) is the inverse Fo
urier Transform of H(f) .The
rec
eiver input waveform is the ran
dom process Y(t) obtained by co
nvolving X(t) with
h(t). In an L
TI system, we can interchange t
he order of convolution.
)()()('
);()()(')12.(....);........(' )( )(')()( )()()()](
)([)()()(
1
01
01
01
0
fHfPfPthtptpEq nTtpa nTtatptYnTtatht
pthnTtatpthtXtY
sN
nnN
ns nN
ns nN
ns n
= ⊗=− =− ⊗=− ⊗⊗=⊗− ⊗=⊗=
∑ ∑∑ ∑
−
=−
=−
=−
=
δδ δ
Given that p(t) contains 99.99
5% of the energy in
0
ff≤
, from Eq.(5), we note that
0
;/)0(2)()(' ffcLfnjefPfP ≤
−
=
π
(13)
= 0
0
ff>
; irrespective of non-zero H(f
) in this range
Hence ;/)0();()(' c
Lntttptp
mm
=
−
=
represents the delay encounte
red by p(t) in the
medium. Subst
ituting p'(t) in Eq.(12), we ha
ve,
)() ( )(
1
0mN
ns m n
ttXnTttpatY −=−− =
∑
−
=
(14)
Thus we see that the recei
ver input Y(t) is a time-shifte
d version of the transmitter ou
tput
X(t), with negligible disto
rtion. The only criterion for t
he choice of the medium is that
the
medium should have zero dis
persion, unity gain and a refra
ctive index less than unity and
positive, in the desired freque
ncy range
0
ff≤
. If we modulate X(t) with a
carrier
frequency f
c
, then the medium should have
above-mentioned properties in t
he range
0 0
fffff
c c
+≤≤−
.Comparing this case with trans
mission through a vaccum medium
where vaccum delay
t
v
=L/c, the receiver input for t
his case is
);()(
v v
ttXtY
−
=
(15)
We can see clearly that t
m
< t
v,
since n(0)<1. The time gained
by transmission through
inverted
medium, in comparison with va
ccum, is given by
cLnt
g
/))0(1(
−
=
(16)
In general, if we modulate
X(t) with a carrier f
c
,
cLfnt
c g
/))(1(
−
=
.
If t
g
>T
s
, then we have achieved an inf
ormation velocity faster than l
ight(FTL), in
comparison with va
ccum, between points B and C in
Fig.2. Because during a given
s
ymbol duration in the receiver,
the received symbol does not c
ontain information about
future
symbols. If t
g
<T
s
, it could be explained by the
pulse correlation within the s
ymbol.
Decoded
Input bits
X(t) ` Y(t) Output bit
s
A B C D
Fig.2. Block diagr
am of BPSK communication system
In order to compute the bit err
or rate(BER) of this communicat
ion system, we must
include a re
ceive baseband filter h
r
(t) to filter out out-of-band
signal and noise
components. Thi
s filter should be chosen such
that it has unity gain and appr
oximately
linear phase in the ra
nge
0
ff≤
and very good attenuation out
side the range. This filter
intr
oduces a delay T
f
. The output of the receive fi
lter is approximately equal to
);()(
f
ttYtZ
−
=
(17)
Thus we see that the outpu
t Z(t) is an undistorted, delay
ed version of the transmitter
ou
tput, information is contained
in the peak of every symbol and
has been
transmitted undistorte
d at FTL speed and hence inform
ation symbols can be detected
wi
th high reliability. The BER pe
rformance of this system will h
ave a degradation of
about 10lo
g
10
(3) = 4.7db compared to the id
eal BPSK system, because we hav
e used a
spectral efficiency fac
tor(SEF=bitrate/bandwidth=
(
)
3/29/233/1≈=
πσσ
ft
) which is 3
times worse than t
he ideal system, in order to en
sure high fidelity of output wa
veform.
Alternatively, we could
use a receive filter with a low
er cut-off frequency, thus redu
cing
noise bandwidth and hence i
mproving BER, but the filter de
lay will be higher and Z(t)
will
have a slightly higher distort
ion due to intersymbol interfer
ence(ISI).
The total processing
delay T
p
is the sum of pulse-shaping de
lay(T
ps
), receive filter
delay(T
f
) and miscellaneous delays suc
h as buffering and other filter
s(T
misc
). If t
g
>
Transmitter
Medium
ReceiverT
p
+T
s
, then we have achieved inform
ation velocity greater than spe
ed of light "c",
between points
A and D, ignoring the length of
transmitter and receiver appar
atus.
Note that the definition o
f information velocity only req
uires zero dispersion over
the r
ange of interest. The notion of
group velocity and group index
are not relevant in
the context
of this paper and hence not in
voked. Even in dispersive media
, it is possible
to recover info
rmation by distortion compensat
ion techniques.
2. 1 An example
of a system with information ve
locity v
i
>c:
Let us take the case of an
inverted two-level medium of am
monia gas. Let
π
ω
2/
r
=24GHz, carrier frequency ,1Ghzf
c
=
signal bandwidth
0
f=100 MHz. Ts=10ns;
Let
3.0*
r p
ω
ω
=
and
03.0*
r
ω
γ
=
, L=300m; ;/103
8
sm c×= L/c=1000 ns.
9.0)(
≈
c
fn .
)(fn
is constant in the interval
0 0
fffff
c c
+≤≤−
t
g
=(1-0.9)*300/c= 100 ns.
T
ps
=10ns; T
f
=10ns; T
p
=20 ns.
Time gained between B a
nd C= t
g
=100 ns= 10 symbols
Time gaine
d between A and D= 100-20= 80
ns = 8 symbols!
This means that
if first symbol (S
1
) arrives at the receiver inpu
t after transmission in a
vaccum
medium, ninth symbol (S
9
) would appear at the receiver
input after transmission
in an
inverted medium at the same ins
tant. We have already shown tha
t S
1
cannot have
any information ab
out S
9
which is in its future, due t
o the fact that the system is
im
plemented as a causal system.
In
formation velocity between A an
d D= v
i
= L/(T
m
+T
p
)=300/(0.9*L/c + 20)=1.087*c
BE
R performance through the inver
ted medium equals that of the s
ame system with
vaccum medium an
d the degradation is less than
0.1 db. BER degradation of the
vaccum
system in this example, i
n comparison to the ideal syste
m is about 3 db. We obtain
BER=
2
10
−
, for a Signal to Noise Ratio(
SNR)=9.6 db.
2.2 Points to remem
ber:
1. Practical digital commu
nication channels such as wirel
ess channels are dispersive
inde
ed. This causes ISI in the rece
ived symbols and the received s
ignal is distorted. This
does no
t mean that information cannot
be recovered. There are two way
s to combat ISI.
The first metho
d is to simply increase transmi
tter power, in appropriate case
s, thus
swamping out ISI with hi
gher signal power. The second m
ethod is equalization, by
estima
ting the channel and compensati
ng for the ISI. This introduces
additional decoding
delay in th
e receiver.
2. When the medium
dispersion in the signal freque
ncy range is negligible, the ti
me
gained in the medium increase
s proportional to medium length
. In the example in Sec.
2.1, if
L=3000m, t
g
=1000 ns=100 symbols. If we u
se multilevel modulation scheme
ssuch as M-ary QAM
1
, we can get more number of bi
ts/symbol, thus increasing the
total
number of bits gained in t
he medium.
We can also increase
t
g
by reducing n(f) in the sign
al frequency range, by increasi
ng
p
ω
.
We can increase t
g
by increasing signal bandwidth
. In the example illustrated in
Sec. 2.1,
if we make
0
f=1 GHz, T
s
=1 ns; L=300m, t
g
=100 ns= 100 symbols! Care sho
uld be
taken to make sure that t
he medium dispersion over the s
ignal bandwidth does not cause
s
evere distortion. As a rule of
thumb, if ,/))()((
0 s c c
TcLfnffn
<<
−
+
then the resulting
distortion
is negligible.
3. Other candidat
e media include:
a. Quantum tunn
elling through a multilayer bar
rier structure in a waveguide
4
.
Nearly zero dispersion over a
0.5 GHz bandwidth at a f
c
=8.7 GHz.
Disadvantage: High at
tenuation of the order of 20 db
over L=0.1142m.
b. Normal, zero
or low dispersion region :
0
ω
ω
>>
. Where
)(
ω
n
<1 and
)(
ω
n
=constant atleast over a small
frequency range. If there is a
small dispersion, it
is combate
d using techniques mentioned in
item 1. Examples include norma
l,
uninverted medium of ammonia
gas.
4. Analog modulation system
s:
Analog modulation systems suc
h as Amplitude modulation(AM) a
nd Frequency
Modulation(FM) also
carry information, the informa
tion being carried by smoothly
v
arying analog modulating signal
m(t), instead of bits. Note th
at m(t), which could be a
live m
usic or video, is not a determi
nistic signal, but a random pro
cess. If m(t) were not
a random
signal, then one could transmit
the music for a short duration
, turn the
transmitter off and e
xpect the receiver to correctly
reproduce the rest of the live
music
program, which is clearly
impossible! The decorrelation
time
1
(
d
τ
) tells us over what
duration o
f time is m(t) correlated. Beyo
nd
d
τ
, m(t) is uncorrelated and hen
ce cannot be
predicted. If the t
ime gained during medium transm
ission is greater than
d
τ
, then indeed
information is tr
ansmitted FTL.
Regarding the exp
eriment
5
done by Dr.Nimtz, transmittin
g through a barrier Mozart's 40
th
symphony at a speed 4.7*c,. t
g
is in the order of nanosecond
s, which is much smaller
than
d
τ
for music which should be in t
he order of milliseconds, hence
it is not possible to
prove bey
ond doubt that FTL information
transfer occurred. If the lengt
h of the medium
is increased suc
h that the time gained t
g
>
d
τ
and distortion is negligible,
then it can be
clearly shown th
at FTL information transfer occ
urred.
3. Negative group velocit
y experiments:
In experiments de
monstrating negative group velo
city
2,6
, it has been argued that this
is
due to pulse reshaping mecha
nism and that no new informatio
n is available in the pulsepeak, which is not already pre
sent in its weak forward tail.
Here another theory is
presented
which could also explain the o
bserved negative shift in the p
ulse peak.
Let us consider a gau
ssian pulse p(n) which is input
to a discrete time LTI system
with
causal impulse response h(n
), yielding an output signal y(
n).
)()(2)(
0
nnnnh
−
−
=
δ
δ
(18)
where
)(n
δ
is the discrete time impulse f
unction. 0
0
≥
nis an integer. We can see tha
t
h(n)=0 for n<0.
The frequency r
esponse of the system is given
by
0
.2)(njejeH
ω ω
−−= ;
f
π
ω
2
=
;periodic with period=
π
2
;(19)
).cos(45)(
0
njeH
ωω
−=
(20)
Angle
−= ∠=
−
)0)0 1
.cos(2.sin(tan)()(nn jeH
ωω ωωθ
(21)
When
0
.n
ω
is very small,
0
.)(n
ω
ω
θ
≈
,
1)(≈
ω
jeH
for
0 00
2;f
πωωω
=≤
.
If the input gaussian pulse
p(t) is chosen as in Eq.(5), th
en its discrete time version is
given by
2
0
)()(mnenp−=
−
π
for
0
20mn
≤
≤
. p(n)=0 for other values of n
.
sftsfttm
t
/3/
00
σ
=
=
. Sampling frequency= f
sf
= 1/t
sf
.
Then
0 0 0
/9)2/(32/32/2 msftsffsfff
t f
=
×
=
×
=
×
=
πσ
π
σ
π
π
ω
.
)(
ω
jeP
has negligible frequency comp
onents for
0
ωω
>
.
π
ω
2
0
<<
when m
0
is large.
When
0
.n
ω
is very small( ;/9
00 00
mnn
=
ω
n
0
<<m
0
), the output signal transform
is given
by
0
.)()(nejePjeY
j
ω
ω
ω
× =
for
0
ωω
≤
= 0
for
0
ωω
>
(22)
Hence the output signal is
given by )()(
0
nnpny
+
≈
! The output peak is negativ
e
shifted w.r.to the peak of the
input gaussian pulse, as shown
below.Another way to derive the same
in time domain is as follows:
T
he impulse response of the syst
em is given by
)()(2)(
0
nnnnh −−=
δδ
The output signal is given by
))()(()()()(2)(
0 0
nnpnpnpnnpnpny
−
−
+
=
−
−
=
If n
0
<<m
0
, then p(n) can be approximate
d to a straight line in the ran
ge [n-n
0
, n+n
0
].
)!()(; )()(; )()(
00 0 0
nnpnynslopenpnynslopennpnp
+=
×
+
=
×
=
−
−
As an example, if m
0
=48 and n
0
=1, ;16/3
0
=
ω
For
0
ωω
≤
,
π
ω
2.
0
<<
n and hence
1)(≈
ω
jeH
.
0
.)(n
ω
ω
θ
≈
. Delay at
.
ω
= 1 /)(
0
−
=
=
−
n
ω
ω
θ
and y(n)=p(n+1);
The frequenc
y response is plotted in the fi
gures in Appendix. Note that th
e frequency
range f=[0:47] corre
sponds to positive frequencies
from f=[0: f
sf
/2] and the range
f=[48:95] co
rresponds to negative frequenci
es from f=[-f
sf
/2:0]
Thus we see that for the
impulse response shown in the
above example, the system adds
t
he input signal to its differen
tiated version and gives an out
put which is advanced w.r.to
the
input pulse! The system
)()(2)(
0
nnnnh −−=
δδ
is clearly causal, since h(n)
=0 for
n<0. The pulse advance is
possible because the pulse is
correlated within the duration
of
the pulse and hence can be ap
proximated to a straight line o
ver a time duration which is
muc
h smaller than pulse duration a
nd can be predicted. It is clea
r that when n
0
becomes
comparable or larger t
han m
0
, prediction is no longer poss
ible because
00 00
/9mnn
=
ω
is
no longer small and hence th
e output pulse is distorted hea
vily. When n
0
>m
0
, if we
convolve p(t) with bina
ry symbols as done before and u
se a symbol duration T
s
=m
0
, it is
clear that it is impos
sible to predict the next rando
m bit, given the present bit!
Hence, in
media with negative gr
oup velocity, no FTL informatio
n transfer is possible. Note
tha
t for the class of impulse resp
onses as given in this example,
extra gain is not required
to a
ssist the observed "superlumina
l" propagation. Unity gain and
negative linear phase
delay over
the frequency range of interes
t are the only requirements.
4.
What about violation of Einstei
n causality?
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0
10
20
30
40
50
60
70
80
90
100
amplitude
time in secondsInput gaussian pulse(line 1) and output pulse(line 2)
line 1
line 2It has been shown that informa
tion can indeed be transmitted
FTL in certain media. What
does
this all mean to Einstein causa
lity? Can we not use FTL inform
ation to travel back
in time and
violate causality and create p
aradoxes? That is an open quest
ion at this point.
The author st
rongly feels that research on s
uperluminal propagation should
go ahead
independently irrespect
ive of what it might mean to po
tential violation of Einstein
ca
usality. It may be that such vi
olations are indeed possible in
the natural world and other
mec
hanisms such as consistent hist
ories and alternate histories h
ypothesis may kick in.
Or it may
be that our interpretation of
the mysterious entity namely,
time, is incorrect and
may need
to revised. Time shown by the
clock may not represent the tru
e time elapsed
while timing an e
vent and it may contain an extr
a component acquired during no
n-
simultaneous synchronisation
mechanism using Einstein synchr
ony, which if subtracted
from th
e final clock reading, will eli
minate time travel and paradoxe
s. The author has
strongly argue
d for this case in a recent pap
er
7
, that T' in Lorentz transform
ation(LT)
must be interpreted di
fferently. The author strongly
believes that , irrespective of
the
correctness of that paper
7
, the current paper on superlu
minal propagation must be
evalua
ted independently and that it w
ould be an incorrect approach t
o rule out
superluminal phenomen
a merely on the basis that it w
ill violate Einstein causality
and
upset our common sense notio
ns. In short, superluminal phen
omena must be researched
indepen
dent of concerns of Einstein ca
usality violations.
5. Conclusio
n:
It has been shown that FTL in
formation transfer is indeed po
ssible using bandlimited
signals
when passed through sufficient
ly long media with unity gain,
a positive index of
refraction l
ess than unity, and zero or low
dispersion ,over the signal fr
equency range.
Information is co
ntained in the peak of the rand
om symbol during a given symbol
duration and information veloci
ty is computed by noting the ti
me spent by the peak
during tran
smission subject to the constra
int that the time gained compar
ed to vaccum
transmission must b
e greater than a symbol duratio
n. The only constraint is place
d on the
refractive index profil
e over the range of interest.
Re
ferences:
1. Simon Haykin "Comm
unication Systems" 3rd edition.
1994.
2. M.W.Mitchell and R.Y.C
hiao "Causality and negative gr
oup delays in a bandpass
amplifi
er", Am. J. Phys. 66, 14 (1998)
.
3. R.Y.Chiao "Amazing Light"
1996.(Chapter 6)
4. G.Nimtz a
nd Heitmann "Superluminal photo
nic tunnelling and quantum elec
tronics"
Progress in Quantu
m Electronics 1997, Vol 21, no.
2,pp 81-108
5. G.Nimtz,Aichmann,
Spieker. Verhandlungen der Deut
schen Physikalischen
Gesell
schaft 7,1258. (1995)
6. L.J.Wan
g,A.Kuzmich,A.Dogariu "Gain-ass
isted superluminal light propag
ation".
Nature Vol.406.,No
.6793 pp277-279. July 2000.
7. A
khila Raman "Special Relativity
and Time Travel Revisited" Jan
2001.
http://xxx.lanl.gov
/abs/physics/0101082
8. M.Mouly
and M.Pautet: The GSM System fo
r Mobile Comuunications.1992.APPENDIX
1
1.5
2
2.5
3
0
10
20
30
40
50
60
70
80
90
100
Magnitude
frequency, 1 unit=1/96 HzMagnitude spectrum of frequency response
line 1
0
2
4
6
8
10
12
14
16
0
10
20
30
40
50
60
70
80
90
100
Magnitude
frequency, 1 unit=1/96 HzMagnitude spectrum of gaussian input pulse
line 1
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0
10
20
30
40
50
60
70
80
90
100
phase in radians
frequency, 1 unit=1/96 HzPhase spectrum of frequency response
line 10
0.05
0.1
0.15
0.2
0.25
0
0.5
1
1.5
2
2.5
3
3.5
4
phase in radians
frequency, 1 unit=1/96 HzPhase spectrum of frequency response
line 1
0
0.5
1
1.5
2
0
0.5
1
1.5
2
2.5
3
3.5
4
Magnitude
frequency, 1 unit=1/96 HzMagnitude spectrum of frequency response
line 1
0
2
4
6
8
10
12
14
16
0
0.5
1
1.5
2
2.5
3
3.5
4
Magnitude
frequency, 1 unit=1/96 HzMagnitude spectrum of gaussian input pulse
line 1 |
arXiv:physics/0102062v1 [physics.bio-ph] 20 Feb 2001Does Recycling in Germinal Centers Exist?
Michael Meyer-Hermann
Institut f¨ ur Theoretische Physik, TU Dresden, D-01062 Dre sden, Germany
E-Mail: meyer-hermann@physik.tu-dresden.de
Abstract: A general criterion is formulated in order to decide if recyc ling of B-cells exists
in GC reaction. The criterion is independent of the selectio n and affinity maturation
process and based solely on total centroblast population ar guments. An experimental test
is proposed to check if the criterion is fulfilled.
11 Introduction
The affinity maturation process in germinal center (GC) react ions has been well charac-
terized in the last decade. Despite a lot of progress concern ing the morphology of GCs
and the stages of the selection process, one fundamental que stion remains unsolved: Does
recycling exist? Recycling means a back differentiation of antibody presenting centroc ytes
(which undergo the selection process and interact with anti gen fragments or T-cells) to
centroblast (which do not present antibodies and therefore do not interact with antigen
fragments but proliferate and mutate).
The recycling hypothesis was first formulated in 1993 [1] and there exists a number
of works to verify it (e.g. [2, 3, 4]). The major conceptual pr oblem in the theoretical
contributions is the number of assumptions which are necess ary to describe the affinity
maturation process. The recycling hypothesis was formulat ed because it seemed unlikely
that a complex optimization process (involving up to about 9 somatic hyper-mutations
[5, 6]) may occur in a one-pass GC, i.e. without recycling: St arting from about 12000
cells which mutate in a (shape) space of high-dimensionalit y the stochastic probability
(mutation is likely to occur stochastically [7, 8]) for at le ast one cell to find the right path
of mutations is very small. It would help, if the already affini ty maturated cells could
reenter the proliferation process in order to multiply and t o enhance the chance to find the
optimal mutation path.
Due to this origin of the recycling hypothesis, the efforts to show its existence were based
on the influence of recycling on the maturation process. Cons equently, it was found that
the number of maturated cells becomes substantially larger with recycling than without
[3]. A quantitative description showed, that in order to bri ng the affinity maturation
process into accordance with all experimental observation s, the recycling probability of
positively selected centrocytes should get as high values a s 80% [4]. Naturally, in both
models important assumptions were made to represent the ant ibody-types in a shape space
(affinity classes, dimension of shape space) and to allow a suc cessive affinity maturation
(affinity neighborhood).
Here, a new and simpler perspective is presented. Not the infl uence of recycling on
the maturation process is examined but its influence on a magn itude which is completely
independent of affinity maturation or selection: the total ce ntroblast population. The
analysis starts from the presupposition that no recycling e xists. Under a minimal set
of assumptions (see Sec. 2) the implications which follow ar e described (see Sec. 3). The
robustness of the results is checked (see Sec. 4) and it is pro posed to test these implications
experimentally (see Sec. 5).
2 Requirements
2.1 General assumptions
In this Gedanken experiment the total centroblast populati onB(t) in a one-pass GC
(without recycling) is considered. Total means, that the encoded antibody-type is ignored
and all centroblasts enter the total population with equal w eight (despite any differences
2in affinity to the antigen).
Presuming a one-pass GC no centrocytes are recycled to centr oblasts, so that the total
centroblast population becomes independent of the number o f centrocytes. Basically, only
centrocytes present antibodies and thus may undergo a selec tion process. As recoil effects
to the total centroblast population are excluded, this magn itude is also independent of
any properties or dynamics of the selection process. Especi ally, no influence of the already
achieved affinity maturation during the optimization proces s exists. Also, neither the
number of positively selected centrocytes, nor the speed of the selection process, nor the
interaction with T-cells, nor the fate of the rejected centr ocytes, nor the production of
plasma and memory cells have any influence on the total centro blast population.
The total centroblast population changes exclusively acco rding to centroblast prolif-
eration and the differentiation of centroblasts to centrocy tes. Therefore, B(t) follows a
differential equation of the following type
dB
dt(t) =−f(t)B(t)≡p(t)B(t)−g(t)B(t), (1)
with time t, the centroblast proliferation rate p(t), the rate of centroblast differentiation
to centrocytes g(t), and the total rate function f(t) =g(t)−p(t).
Cell proliferation has unquestionably to be described by an exponential increase of the
population. Therefore, it is more than reasonable to assume a linear term for proliferation
at each moment of the GC reaction. This applies in the same way to the differentiation term
which should be directly proportional to the centroblast po pulation at each moment. So
it is assumed that nonlinear terms do not enter the total cent roblast population dynamics
Eq.(1).
Note that no place is determined where proliferation or diffe rentiation occurs. There-
fore, the following argument is not restricted by the appear ance or not-appearance of dark
and light zones [9, 10] during a GC reaction.
2.2 About the rates
It is assumed that after an initial pure proliferation phase of some seeder cells, the total
centroblast population does not further increase. This is i n accordance with measurements
of the follicle center volume, which show a peak 3 or 4 days aft er immunization [11], i.e. after
the first pure proliferation phase. As a consequence, the fun ctionf(t) in Eq.(1) is positive
during the whole GC reaction.
The proliferation rate is probably constant during the whol e GC reaction. This has
not been shown experimentally in all details but is well esta blished by a lot of independent
data taken at different moments of the GC reaction [11, 12, 13] . All these data lead to the
same large proliferation rate of
p(t)
ln(2)≈1
6h. (2)
Nevertheless, for the following considerations the value o f the proliferation rate will not be
fixed. In order to assure generality of the argument prolifer ation rates according to Eq.(2)
will be discussed as special case only.
3The situation is more complicated in the case of the different iation rate g(t). This value
is not known experimentally, so its absolute number as well a s its time course are unclear.
From the injection of BrdUrd (5-bromo-2’-deoxyuridine) on e is able to identify the cells
which were in cell cycle in the last 6 hours. In this way it was f ound in GCs with established
dark and light zones, that the centrocyte population is rene wed from centroblasts every 7
hours [11]. It follows for the differentiation rate that
g(t)
ln(2)>1
7h. (3)
Because of the lack of more detailed information, different p ossibilities for the time depen-
dence of the centroblasts to centrocytes differentiation ra te have to be considered in the
following.
2.3 A standard germinal center
Before analyzing the implication in this Gedanken experime nt a typical GC has to be
defined as reference system. The GC reaction is initiated by a few seeder cells in the
environment of follicular dendritic cells (FDC) [5, 11, 14, 15]. These cells proliferate for
about 3 days, and give rise to about 12000 centroblasts [11]. At this time t0= 0hthe
intrinsic GC reaction starts, including a continuation of p roliferation and, additionally, the
differentiation of centroblasts to centrocytes. The analys is made here does not consider
the first phase of pure proliferation, but deals solely with t he GC properties in this second
phase. The life time Tof a germinal center is about 18 days ( T= 432 h) [11, 16], where
the first phase of pure proliferation is not included. At the e nd of the GC reaction, only a
few cells remain in the FDC environment [11, 16], which is ass umed to be 5 for a standard
GC. The sensitivity of the results on these assumptions will be analyzed (see Sec. 4).
3 Analysis of the centroblast population
The different time courses considered for the total centrobl ast population of a standard
GC (defined in Sec. 2.3) are shown in Fig.1. An exponential dec rease corresponds to a
constant rate function (see Fig.2 dashed line). A linear pop ulation decrease is achieved
with the rate function Eq.(15) (see Fig.2 dotted line). Fina lly, using a quadratic rate
function Eq.(17) (see Fig.2 full line) the centroblast popu lation remains unchanged for
the major duration of the GC reaction and is steeply reduced a t the very end.
Experimentally, the time course of the GC total centroblast population seems to lie
in between these three scenarios [11]. The scenario with Eq. (17) is certainly a reasonable
upper bound for monotonically decreasing centroblast popu lations. A population decrease
which is faster than exponentially is unlikely.
It follows that during the major life time of GCs ( t <15d) the condition
f(t) =g(t)−p(t)<0.018/h (4)
holds (see Fig.2). On the other hand the requirement of a mono tonically decreasing
centroblast population leads to the bound f(t)>0, i.e. g(t)> p(t). Taking these results
4Figure 1: Possible time courses of the centroblast population during GC reactions.
together one is lead to
p(t)< g(t)< p(t) + 0.018/h , (5)
which is a very powerful condition. This means that in an expe riment the centroblast to
centrocyte differentiation rate g(t) has to fulfill this condition during the first 15 days of
the GC reaction. If it turns out that the above condition is no t guaranteed experimentally,
then – assuming the rather weak requirement made to be valid – a GC reaction without
recycling of centrocytes to centroblasts is ruled out.
Note that assuming the proliferation rate to adopt the value Eq.(2), the above condition
can be reformulated. It follows a lower bound for the differen tiation rate which is in
accordance with Eq.(3):
1
6h<g(t)
ln(2)<1
5.19h. (6)
This means that if GCs without recycling exist, a centroblas t shall (in average) take not
less than 5 .19 hours to differentiate into centrocytes.
Supposing that the linear centroblast population behavior is the realistic one, the upper
bound becomes even more powerful. During the first 10 days the condition
p(t)< g(t)< p(t) + 0.005/h (7)
5Figure 2: The function f(t)corresponding to the different time courses of the centrobla st
population in Fig.1.
must hold in a GC reaction without recycling. With the prolif eration rate in Eq.(2) this
corresponds to an average centroblast differentiation time of at least 5 .75 hours. This result
demonstrates, that the condition for a one-pass GC may becom e stronger than in Eqs.(5)
and (6), depending on the characteristics of the total centr oblast population decrease.
4 Robustness of the results
4.1 The case of constant rates
In order to check the robustness of the results Eqs.(5) and (6 ) the most critical scenario
(with weakest conditions) is considered in some more detail : the scenario with constant
rates. If the proliferation and differentiation rate are con stant, the centroblast population
model reduces to a pure exponential decrease of the populati on
B(t) =B(0) exp (( p−g)t), (8)
6where the constant rate gis unknown. Focussing on the population after T= 18 days of
the GC reaction, the rate function
f=g−p=−1
Tln/parenleftBiggB(T)
B(0)/parenrightBigg
(9)
is found. The resulting values for the differentiation rate a re shown in Fig.3 in dependence
of the GC life time and for different numbers of remaining cell s at the end of the reaction.
As expected the rate becomes larger for shorter GC life times T. Even supposing a rather
Figure 3: The rate function f=g−pin dependence of the life time of the GC for different
numbers of remaining centroblasts at the end of the GC reacti on.
short life time of 12 days and further supposing that only one cell remains at the end of
the GC reaction, one gets an upper bound of
p(t)< g(t)< p(t) + 0.033/h (10)
for possible values of the differentiation rate. Incorporat ing the proliferation rate Eq.(2)
it follows
1
6h<g(t)
ln(2)<1
4.7h. (11)
7As a consequence, the statement that in one-pass GCs (withou t recycling) the centroblast
to centrocyte differentiation rate should respect the above bounds Eqs.(5) and (6) is not
altered dramatically by variation of the GC properties.
The initial number of centroblasts B(0) was not varied until now. Note that only the
ratioB(T)/B(0) enters Eq.(9), so that a variation of the initial number o f centroblast is
equivalent to the variation of the final number of remaining c ellsB(T). Therefore, the
above constraints Eqs.(10) and (11) are also valid with e.g. 60000 initial and 5 remaining
cells. Lower numbers of initial cells allow stronger bounds than the above ones.
4.2 The case of linear centroblast decrease
Figure 4: The time dependence of the rate function f=g−p(corresponding to a linear
centroblast population decrease) for different GC life time sTand for different numbers of
remaining centroblasts B(T)at the end of the GC reaction.
In the case of a linear centroblast population decrease (see Fig.1 pointed line) the
corresponding rate function f(t) (see Eq.(15)) is analyzed for various border conditions.
The result is shown in Fig.4. Condition Eq.(7) is not affected at all by a variation of
the number of remaining cells at the end of the GC reaction in a range of three orders
8of magnitude (the values chosen for B(T) are 0 .1, 5, and 100). The three corresponding
curves are practically not distinguishable (see Fig.4).
The situation is different for the variation of the GC life tim eT. Here, the period in
which condition Eq.(7) remains valid is prolonged for long l iving GCs and shortened for
short living GCs (see Fig.4 dashed and pointed line, respect ively). For a very short life
time of 12 days (plus 3 days of pure proliferation) the condit ion remains valid for 4 days
only (7 days after immunization). However, condition Eq.(5 ) remains valid even in this
case for 11 days, so for approximately the whole GC life time.
5 Conclusions
The above analysis is summarized with the statement that in a one-pass GC (without
recycling) the centroblast to centrocyte differentiation r ate should respect the condition
Eq.(5) p(t)< g(t)< p(t) + 0.018/hduring the first 15 days of the GC reaction. For a
standard proliferation rate Eq.(2) of ln(2) /6hthis translates into an average time for a
centroblast to differentiate into a centrocyte which cannot be substantially shorter than
5 hours. If this condition is not fulfilled, recycling has nec essarily to be present in GC
reactions. Note that, the other way round, if the average diff erentiation of centroblasts to
centrocytes takes more than 5 hours, recycling is not necess arily absent.
This statement is based basically on exclusively two (weak) assumptions: Firstly, a
linear population dynamic (see Eq.(1)) for the total centro blast number in GCs. Secondly,
a monotonic decrease of the total centroblast population. T herefore, the conclusion is
independent of all speculation on shape spaces, selection p rocesses, and affinity maturation.
In order to decide if recycling exists in GC reactions one has to check the centroblast
to centrocyte differentiation rate experimentally at least at one representative moment of
the GC reaction. The moment has to be chosen such that on one ha nd the GC reaction is
already well established. On the other hand, the later the me asurement is done the less is
known on the proliferation rate. In addition, one approache s the regime where the above
condition looses its validity (see Fig.2). Therefore, an op timal moment will be about 8
days after immunization. Technically, this may be achieved by labeling studies of cells in
the GC which are in cell cycle as it was done in [10, 11].
Alternatively, a systematic and time resolved analysis of t he total centroblast population at
different moments of the GC reaction may improve the criterio n (see e.g. Eq.(7)) because
one could distinguish which of the cell population scenario s presented in Fig.1 corresponds
to existing GC reactions. This may be done by the measurement of the GC volume at
several times of the GC reaction as in [11, 17].
Appendix
The differential equation Eq.(1) is formally solved by
B(t) =B(t0= 0h) exp/parenleftbigg
−/integraldisplayt
0dxf(x)/parenrightbigg
. (12)
9In the following, the border conditions (life time of GC T, and centroblast population at
t0= 0handT) are incorporated into the rate function f.
To achieve a linear decrease of the centroblast population, the second derivative of the
population function is required to vanish. From Eq.(1) and u sing Eq.(12) it follows
d2B
dt2=−df
dtB−fdB
dt
=−df
dtB(0) exp/parenleftbigg
−/integraldisplayt
0dxf(x)/parenrightbigg
+fB(0)fexp/parenleftbigg
−/integraldisplayt
0dxf(x)/parenrightbigg
.(13)
This expression vanishes for
df
dt=f2, (14)
which is solved by
f(t) =−1
t−c1. (15)
The integration constant c1is fixed by the condition that the final population at time T
is given by B(T). With Eq.(12) this leads to
c1=T
1−B(T)
B(0), (16)
which is a positive constant for B(T)< B(0).
In the case of a quadratic rate function
f(t) =(1h)
(t−c2)2(17)
an analogous calculation which incorporates the final centr oblast populations leads to
c2=T
2+/radicaltp/radicalvertex/radicalvertex/radicalbtT2
4−(1h)T
ln/parenleftBigB(T)
B(0)/parenrightBig. (18)
Other solutions are formally possible but do not lead to biol ogically reasonable centroblast
populations.
References
[1] Kepler, T.B., and Perelson, A.S., Cyclic re-entry of ger minal center B cells and the
efficiency of affinity maturation. Immunol. Today 14(1993), 412-415.
[2] Han, S.H., Zheng, B., Dal Porto, J., Kelsoe, G., In situ St udies of the Primary Immune
Response to (4–Hydroxy–3–Nitrophenyl) Acetyl IV. Affinity- Dependent, Antigen-
Driven B-Cell Apoptosis in Germinal Centers as a Mechanism f or Maintaining Self-
Tolerance. J. Exp. Med. 182(1995), 1635-1644.
10[3] Oprea, M., Nimwegen, E.van, and Perelson, A.S., Dynamic s of One-pass Germinal
Center Models: Implications for Affinity Maturation. Bull. Math. Biol. 62(2000),
121-153.
[4] Meyer-Hermann, M., Deutsch, A., and Or-Guil, M., Recycl ing probability and dy-
namical properties of germinal center reactions. e-print: physics/0101015, submitted
toJ. Theor. Biol. (2001).
[5] K¨ uppers, R., Zhao, M., Hansmann, M.L., and Rajewsky, K. , Tracing B Cell Develop-
ment in Human Germinal Centers by Molecular Analysis of Sing le Cells Picked from
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12 |
arXiv:physics/0102063v1 [physics.flu-dyn] 20 Feb 2001Equations relating structure functions of all orders
Reginald J. Hill
NOAA/Environmental Technology Laboratory, 325 Broadway, Boulder CO 80305 USA
(October 14, 2013)
The hierarchy of exact equations are given that relate two-s patial-point velocity structure func-
tions of arbitrary order with other statistics. Because no a ssumption is used, the exact statistical
equations can apply to any flow for which the Navier-Stokes eq uations are accurate and no mat-
ter how small the number of samples in the ensemble. The exact statistical equations can be
used to verify DNS computations and to detect their limitati ons because if DNS data are used
to evaluate the exact statistical equations, then the equat ions should balance to within numerical
precision, otherwise a computational problem is indicated . The equations allow quantification of
the approach to local homogeneity and to local isotropy. Tes ting the balance of the equations
allows detection of scaling ranges for quantification of ine rtial-range exponents. The second-order
equations lead to Kolmogorov’s equation. All higher-order equations contain a statistic composed
of one factor of the two-point difference of the pressure grad ient multiplied by factors of velocity
difference. Investigation of this pressure-gradient-diffe rence statistic can reveal much about two
issues: 1) whether or not different components of the velocit y structure function of given order
have differing exponents in the inertial range, and 2) the inc reasing deviation of those exponents
from Kolmogorov scaling as the order increases. Full disclo sure of the mathematical methods is in
xxx.lanl.gov/list/physics.flu-dyn/0102055.
I. INTRODUCTION
Kolmogorov’s (1941) equation and Yaglom’s equation were th e first two equations of the “dynamic theory”
of the local structure of turbulence. The name “dynamic theo ry” was originated by Monin & Yaglom (1975) (their
Sec. 22) to mean the derivation of equations relating struct ure functions by use of the Navier-Stokes equation and/or
the scalar conservation equation, and the investigation of the resulting statistical equations. Monin & Yaglom
(1975) pointed out that the dynamic theory gives important r elationships among structure functions, and that these
relationships provide extensions of predictions based on d imensional analysis. Theoretical studies (Lindborg 1996;
Hill 1997a) clarified the assumptions that are the basis of Ko lmogorov’s equation and give equations that are valid
for anisotropic and locally homogeneous turbulence as well as for the case of local isotropy and local homogeneity.
Antonia et al. (1983) and Chambers & Antonia (1984) used experimental dat a to study of the balance of the classic
equations of Kolmogorov and Yaglom. There is renewed intere st in examining the balance of those equations using
both experimental and DNS data and in generalizing the equat ions to cases of inhomogeneous, nonstationary, and
anisotropic turbulence (Lindborg 1999; Danaila et al., 1999a,b,c; Antonia et al. 2000). Whereas Kolmogorov’s
(1941) equation relates 2nd- and 3rd-order velocity struct ure functions, the next-order dynamic equation relates 3rd -
and 4th-order structure functions and a pressure-gradient , velocity-velocity structure function. The balance of tha t
next-order equation has been examined by means of experimen tal and DNS data; this showed the behavior of the
pressure-gradient, velocity-velocity structure functio n (Hill & Boratav 2001). There is now interest in dynamic-the ory
equations of arbitrarily high order N(Yakhot 2001). Such equations relate velocity structure fu nctions of order N
andN+ 1 and other statistics. Those equations are given in this pa per.
Using the assumptions of local homogeneity, local isotropy and the Navier-Stokes equation, Yakhot (2001)
derived the equation for the characteristic function of the probability distribution of two-point velocity difference s.
He uses that equation to derive higher-order dynamic equati ons. Equations for arbitrarily high-order structure
functions can be obtained by repeated application of his diff erentiation procedure. Yakhot (2001) studies the inertial -
range, deduces a closure, and thereby determines the inerti al-range scaling exponents of velocity structure function s.
Yakhot’s study is the first to make significant use of dynamic- theory equations to determine scaling exponents.
The purposes and theoretical method of the present paper diff er from those of Yakhot (2001), but one purpose
is to verify Yakhot’s equations from our distinctly differen t derivation. That verification is given in Sec. 5. In
Sec. 2, exact statistical equations relating velocity stru cture functions of any order are derived from the Navier-
Stokes equation. “Exact” means that no assumptions are made other than the assumption that the Navier-Stokes
equation and incompressibility are accurate. Since the equ ations are exact, they apply to any flow, including laminar
flow and inhomogeneous and anisotropic turbulent flow. The ex act statistical equations can be used to verify DNS
1computations and detect their limitations. New experiment al methods of Dahm and colleagues (Su & Dahm 1996)
can also be tested. For example, if DNS data are used to evalua te the exact statistical equations, then the equations
should balance to within numerical precision, otherwise a c omputational problem is indicated. In Sec. 3, statistical
equations valid for locally homogeneous and anisotropic tu rbulence are obtained from the exact equations; those
equations can be used with DNS or experimental (Su & Dahm 1996 ) data to study the approach to local homogeneity
of a particular flow. This can be done by quantifying the terms that are neglected when passing from exact equations
to the locally homogeneous case, and by quantifying changes in the retained terms as local homogeneity is approached
when the spatial separation vector is decreased. In Sec. 4, s tatistical equations valid for locally isotropic and local ly
homogeneous turbulence are obtained from those for the loca lly homogeneous case. The approach to local isotropy
can be studied by means analogous to the above described eval uation of local homogeneity. Such studies might
shed light on the observed persistence of anisotropy (Pumir & Shraiman 1995; Shen & Warhaft 2000). All dynamic-
theory equations are now available to extend the above-ment ioned previous studies of the balance of dynamic-theory
equations.
Incompressibility requires that the different components o f the second-order velocity structure function have
the same scaling exponent in the inertial range. The same is t rue for the third-order structure function. However,
at fourth and higher order there is no such requirement. Ther e have been many studies of the possibility that the
inertial-range scaling exponents of the various structure -function components are unequal (e.g., Chen et al. 1997;
Boratav & Pelz 1997; Boratav 1997; Grossmann et al. 1997; van de Water & Herweijer 1999; Camussi & Benzi 1997;
Dhruva et al. 1997; Antonia et al. 1998; Kahaleras et al. 1996; Noullez et al. 1997; Nelkin 1999; Zhou & Antonia
2000; Kerr et al. 2001). The usefulness of applying the higher-order dynamic -theory equations to those investigations
is considered in Sec. 6.
Derivation of the equations produces substantial mathemat ical detail. Matrix-based algorithms are invented
such that the isotropic formulas for the divergence and Lapl acian of isotropic tensors of any order can be generated
by computer. The details of this mathematics are available a nd are herein referred to as the Archive.1
II. EXACT TWO-POINT EQUATIONS
The Navier-Stokes equation for velocity component ui(x, t) is
∂tui(x, t) +un(x, t)∂xnui(x, t) =−∂xip(x, t) +ν∂xn∂xnui(x, t), (1)
and the incompressibility condition is ∂xnun(x, t) = 0. In (1), p(x, t) is the pressure divided by the density (density
is constant), νis kinematic viscosity, and ∂denotes partial differentiation with respect to its subscri pt variable.
Summation is implied by repeated Roman indices. Consider an other point x′such that x′andxare independent
variables. For brevity, let ui=ui(x, t),u′
i=ui(x′, t), etc. Require that xandx′have no relative motion. Then
∂xiu′
j= 0,∂x′
iuj= 0, etc., and ∂tis performed with both xandx′fixed. The change of independent variables from
xandx′to the sum and difference independent variables is:
X≡(x+x′)/2 and r≡x−x′,and define r≡ |r|. (2)
The relationship between the partial derivatives is
∂xi=∂ri+1
2∂Xi,∂x′
i=−∂ri+1
2∂Xi,∂Xi=∂xi+∂x′
i,∂ri=1
2/parenleftBig
∂xi−∂x′
i/parenrightBig
. (3)
The change of variables organizes the equations in a reveali ng way because of the following properties. In the case of
homogeneous turbulence, ∂Xioperating on a statistic produces zero because that derivat ive is the rate of change with
respect to the place where the measurement is performed. Con sider a term in an equation composed of ∂Xioperating
on a statistic. For locally homogeneous turbulence, that te rm becomes negligible as ris decreased relative to the
integral scale. For the homogeneous and locally homogeneou s cases, the statistical equations retain their dependence
onr, which is the displacement vector of two points of measureme nt. Subtracting (1) at x′from (1) at x, performing
the change of variables (2), and using (3) gives
1The document “Mathematics of structure-function equation s of all orders” by R. J. Hill is available from the editorial a rchive
of the Journal of Fluid Mechanics and at xxx.lanl.gov/list/ physics.flu-dyn/0102055.
2∂tvi+Un∂Xnvi+vn∂rnvi=−Pi+ν/parenleftbig
∂xn∂xnvi+∂x′n∂x′nvi/parenrightbig
, (4)
where vi≡ui−u′
i, Un≡(un+u′
n)/2, Pi≡∂xip−∂x′
ip′. (5)
Now multiply (4) by the product vjvk· · ·vlwhich contains N−1 factors of velocity difference, each factor
having a distinct index. Sum the Nequations as required to produce symmetry under interchang e of each pair of
indices, excluding the summation index n. French braces, i.e., {◦}, denote the sum of all terms of a given type that
produce symmetry under interchange of each pair of indices. The differentiation chain rule gives
{vjvk· · ·vl∂tvi}=∂t(vjvk· · ·vlvi), (6)
{vjvk· · ·vlUn∂Xnvi}=Un∂Xn(vjvk· · ·vlvi) =∂Xn(Unvjvk· · ·vlvi), (7)
{vjvk· · ·vlvn∂rnvi}=vn∂rn(vjvk· · ·vlvi) =∂rn(vnvjvk· · ·vlvi). (8)
The right-most expressions in (7) and (8) follow from the inc ompressibility property obtained from (3) and the fact
that∂xiu′
j= 0,∂x′
iuj= 0; namely, ∂XnUn= 0, ∂Xnvn= 0, ∂rnUn= 0, ∂rnvn= 0. The viscous term in (4) produces
ν/braceleftbig
vjvk· · ·vl/parenleftbig
∂xn∂xnvi+∂x′n∂x′nvi/parenrightbig/bracerightbig
; this expression is treated in the Archive. Thereby
∂t(vj· · ·vi) +∂Xn(Unvj· · ·vi) +∂rn(vnvj· · ·vi) =
− {vj· · ·vlPi}+ 2ν/bracketleftbigg/parenleftbigg
∂rn∂rn+1
4∂Xn∂Xn/parenrightbigg
(vj· · ·vi)− {vk· · ·vleij}/bracketrightbigg
, (9)
where eij≡(∂xnui)(∂xnuj) +/parenleftbig
∂x′nu′
i/parenrightbig /parenleftbig
∂x′nu′
j/parenrightbig
= (∂xnvi) (∂xnvj) +/parenleftbig
∂x′nv′
i/parenrightbig /parenleftbig
∂x′nv′
j/parenrightbig
. (10)
The quantity {vj· · ·vlPi}can be expressed differently on the basis that (3) allows Pito be written as Pi=
∂Xi(p−p′). The derivation is in the Archive; the alternative formula is
{vjvk· · ·vlPi}={∂Xi[vjvk· · ·vl(p−p′)]} −(N−1)(p−p′)/braceleftbig/parenleftbig
sij−s′
ij/parenrightbig
vk· · ·vl/bracerightbig
, (11)
where the rate of strain tensor sijis defined by sij≡/parenleftbig
∂xiuj+∂xjui/parenrightbig
/2.
A. Hierarchy of exact statistical equations
Consider the ensemble average because it commutes with temp oral and spatial derivatives. The above notation
of explicit indices is burdensome. Because the tensors are s ymmetric, it suffices to show only the number of indices.
Define the following statistical tensors which are symmetri c under interchange of any pair of indices, excluding the
summation index nin the definition of F[N+1]:
D[N]≡ /an}b∇acketle{tvj· · ·vi/an}b∇acket∇i}ht,F[N+1]≡ /an}b∇acketle{tUnvj· · ·vi/an}b∇acket∇i}ht,T[N]≡ /an}b∇acketle{t{vj· · ·vlPi}/an}b∇acket∇i}ht,E[N]≡ /an}b∇acketle{t{vk· · ·vleij}/an}b∇acket∇i}ht, (12)
where angle brackets /an}b∇acketle{t/an}b∇acket∇i}htdenote the ensemble average, and the subscripts NandN+ 1 within square brackets denote
the number of indices. The left-hand side of each definition i n (12) is in implicit-index notation for which only the
number of indices is given; the right-hand sides in (12) are i n explicit-index notation. The argument list for each
tensor is understood to be ( X,r, t). The ensemble average of (9) is
∂tD[N]+∇X•F[N+1]+∇r•D[N+1]=−T[N]+ 2ν/bracketleftbigg/parenleftbigg
∇2
r+1
4∇2
X/parenrightbigg
D[N]−E[N]/bracketrightbigg
, (13)
where, ∇X•F[N+1]≡∂Xn/an}b∇acketle{tUnvj· · ·vi/an}b∇acket∇i}ht,∇r•D[N+1]≡∂rn/an}b∇acketle{tvnvj· · ·vi/an}b∇acket∇i}ht,∇2
r≡∂rn∂rn,∇2
X≡∂Xn∂Xn. The notations
∇X•,∇2
X,∇r•, and ∇2
rare the divergence and Laplacian operators in X-space and r-space, respectively.
III. HOMOGENEOUS AND LOCALLY HOMOGENEOUS TURBULENCE
Consider homogeneous turbulence and locally homogeneous t urbulence; the latter applies for small rand large
Reynolds number. The variation of the statistics with the lo cation of measurement or of evaluation is zero for the
homogeneous case and is neglected for the locally homogeneo us case. Since that location is X,the result of ∇X
3operating on a statistic vanishes or is neglected as the case may be. Thus the terms ∇X•F[N+1]and1
4∇2
XD[N]are
deleted in (13); then (13) becomes,
∂tD[N]+∇r•D[N+1]=−T[N]+ 2ν/bracketleftbig
∇2
rD[N]−E[N]/bracketrightbig
. (14)
Because the X-dependence is deleted, the argument list for each tensor is understood to be ( r, t). Note
that∂tD[N]is not necessarily negligible for homogeneous turbulence. The ensemble average of (11) contains
/an}b∇acketle{t{∂Xi[vjvk· · ·vl(p−p′)]}/an}b∇acket∇i}ht={∂Xi/an}b∇acketle{tvjvk· · ·vl(p−p′)/an}b∇acket∇i}ht}={0}= 0. Thus, (11) gives the alternative that
T[N]=−(N−1)/angbracketleftbig
(p−p′)/braceleftbig/parenleftbig
sij−s′
ij/parenrightbig
vk· · ·vl/bracerightbig/angbracketrightbig
. (15)
One distinction between (14) and the hierarchy equations gi ven in equations (13) and (17) by Arad et al. (1999) is
that their t- andr-derivatives operate on only one velocity difference within their product of such differences, whereas
the derivatives in (9) and thus in (14) operate on all Nof the velocity differences.
IV. ISOTROPIC AND LOCALLY ISOTROPIC TURBULENCE
Consider isotropic turbulence and locally isotropic turbu lence; the latter applies for small rand large Reynolds
number. Locally isotropic flows are a subset of locally homog eneous flows (Monin & Yaglom Sec. 13.3, 1975) and
similarly for the relationship between isotropic and homog eneous flows. Thus, the dynamical equations for locally
isotropic and isotropic turbulence are obtained from (14) s uch that the variable Xand the terms ∇X•F[N+1]and
1
4∇2
XD[N](see 13) do not appear. The tensors D[N],T[N], andE[N]in (12) obey the isotropic formula. The
Kronecker delta δijis defined by δij= 1 if i=jandδij= 0 if i/ne}ationslash=j. Let δ[2P]denote the product of PKronecker
deltas having 2 Pdistinct indices, and let W[N](r) denote the product of Nfactorsri
reach with a distinct index;
the argument ris omitted when clarity does not suffer. Because each tensor i n (12) is symmetric under interchange
of any two indices, their isotropic formulas are particular ly simple. Each formula is a sum of M+ 1 terms where
M=N/2 ifNis even, and M= (N−1)/2 ifNis odd. Each term is the product of a distinct scalar function with
aW[N]and a δ[2P]. From one term to the next a pair of indices is transferred fro m aW[N]to aδ[2P]; examples are
in the Archive. For the tensor D[N], denote the Pth scalar function by DN,P(r, t). The isotropic formula for D[N]is
D[N](r, t) =M/summationdisplay
P=0DN,P(r, t)/braceleftbig
W[N−2P](r)δ[2P]/bracerightbig
, (16)
and the isotropic formulas for T[N]andE[N]have the analogous notation. Recall from Sec. 2 the meaning o f the
notation {◦}whereby/braceleftbig
W[N−2P](r)δ[2P]/bracerightbig
denotes the sum of all terms of the type W[N−2P](r)δ[2P]that produce
symmetry under interchange of each pair of indices. An examp le is/braceleftbig
W[1](r)δ[2]/bracerightbig
=rk
rδij+rj
rδki+ri
rδjk.
A special Cartesian coordinate system simplifies the isotro pic formulas. This coordinate system has the positive
1-axis parallel to the direction of r, and the 2- and 3-axes are therefore perpendicular to r. Let N1,N2, andN3be the
number of indices of a component of D[N]that are 1, 2, and 3, respectively; such that N=N1+N2+N3. Because of
symmetry, the order of indices is immaterial so that a compon ent ofD[N]can be identified by N1,N2, andN3. Thus,
denote a component of D[N]byD[N1,N2,N3]which is a function of randt. Likewise,/braceleftbig
W[N−2P](r)δ[2P]/bracerightbig
[N1,N2,N3]is a
specific component of the tensor/braceleftbig
W[N−2P](r)δ[2P]/bracerightbig
. If, in (16) N1of the indices are assigned the value 1, and N2and
N3of the indices are assigned the values 2 and 3, respectively, thenD[N1,N2,N3]and/braceleftbig
W[N−2P](r)δ[2P]/bracerightbig
[N1,N2,N3]will
appear on the left-hand and right-hand sides of (16), respec tively. The/braceleftbig
W[N−2P](r)δ[2P]/bracerightbig
[N1,N2,N3]are numerical
coefficients that do not depend on rbecauser1
r=r
r= 1,r2
r=r3
r= 0. From the Archive, the values of the coefficients
are:
if 2P < N 2+N3then/braceleftbig
W[N−2P](r)δ[2P]/bracerightbig
[N1,N2,N3]= 0,otherwise, (17)
/braceleftbig
W[N−2P](r)δ[2P]/bracerightbig
[N1,N2,N3]=N1!N2!N3!//bracketleftbigg
(N−2P)!2P/parenleftbiggN2
2/parenrightbigg
!/parenleftbiggN3
2/parenrightbigg
!/parenleftbigg
P−N2
2−N3
2/parenrightbigg
!/bracketrightbigg
. (18)
By applying (16-18) for all combinations of indices, one can determine which components D[N1,N2,N3]are zero
and which are nonzero, identify M+ 1 linearly independent equations that determine the DN,Pin terms of M+ 1 of
4theD[N1,N2,N3], and find algebraic relationships between the remaining non zeroD[N1,N2,N3]. The derivations are in
the Archive; a summary follows.
A component D[N1,N2,N3]is nonzero only if both N2andN3are even, and therefore N1is odd if Nis odd, and
N1is even if Nis even. Thereby, ( M+ 1)(M+ 2)/2 components are nonzero. There are 3Ncomponents of D[N];
thus the other 3N−(M+ 1)(M+ 2)/2 components are zero.
There exist ( M+ 1)M/2 kinematic relationships among the nonzero components of D[N]. For each of the
M+ 1 cases of N1, these relationships are expressed by the proportionality
D[N1,2L,0]:D[N1,2L−2,2]:D[N1,2L−4,4]:· · ·:D[N1,0,2L]=
[(2L)!0!/L!0!] : [(2 L−2)!2!/(L−1)!1!] : [(2 L−4)!4!/(L−2)!2!] : · · ·: [0! (2 L)!/0!L!]. (19)
ForN= 4 with L= 2, (19) gives D[0,4,0]:D[0,2,2]:D[0,0,4]= 12 : 4 : 12. In explicit-index notation this can be written
asD2222= 3D2233=D3333, which was discovered by Millionshtchikov (1941) and is the only previously known such
relationship. Now, all such relationships are known from (1 9).
There remain M+ 1 linearly independent nonzero components of D[N]. This must be so because there are
M+ 1 terms in (16) and the M+ 1 scalar functions DN,Ptherein must be related to M+ 1 components. Consider
theM+ 1 linearly independent equations that determine the DN,Pin terms of M+ 1 of the D[N1,N2,N3]. For
simplicity, the chosen components can all have N3= 0; i.e., the choice of linearly independent components can be
D[N,0,0],D[N−2,2,0],D[N−4,4,0],· · ·,D[N−2M,2M,0]. As described above, assigning index values in (16) results in the
chosen components on the left-hand side and algebraic expre ssions on the right-hand side that contain the coefficients
(17-18). In the Archive, those equations are expressed in ma trix form and solved by matrix inversion methods. Given
experimental or DNS data or a theoretical formula for the cho sen components, the solution of the algebraic equations
determines the functions DN,Pin (16); then (16) completely specifies the tensor D[N].
The matrix algorithm in the Archive is an efficient means of det ermining isotropic expressions for the terms
∇r•D[N+1]and∇2
rD[N]in (14). From the example for N= 2 in the Archive, (14) gives the two scalar equations
∂tD11+/parenleftbigg
∂r+2
r/parenrightbigg
D111−4
rD122=−T11+ 2ν/bracketleftbigg/parenleftbigg
∂2
r+2
r∂r−4
r2/parenrightbigg
D11+4
r2D22/bracketrightbigg
−2νE11
= 2ν/bracketleftbigg
∂2
rD11+2
r∂rD11+4
r2(D22−D11)/bracketrightbigg
−4ε/3, (20)
∂tD22+/parenleftbigg
∂r+4
r/parenrightbigg
D122=−T22+ 2ν/bracketleftbigg2
r2D11+/parenleftbigg
∂2
r+2
r∂r−2
r2/parenrightbigg
D22/bracketrightbigg
−2νE22
= 2ν/bracketleftbigg
∂2
rD22+2
r∂rD22−2
r2(D22−D11)/bracketrightbigg
−4ε/3, (21)
where use was made of the fact (Hill 1997a) that local isotrop y gives T11=T22= 0 and 2 νE11= 2νE22= 4ε/3, where
εis the average energy dissipation rate. Since (20-21) are th e same as equations (43-44) of Hill (1997a), and since
Hill (1997a) shows how these equations lead to Kolmogorov’s equation and his 4/5 law, further discussion of (20-21)
is unnecessary. From the example for N= 3 in the Archive,
∂tD111+/parenleftbigg
∂r+2
r/parenrightbigg
D1111−6
rD1122=−T111+ 2ν/bracketleftbig/parenleftbig
∇2
rD/parenrightbig
111−E111/bracketrightbig
, (22)
∂tD122+/parenleftbigg
∂r+4
r/parenrightbigg
D1122−4
3rD2222=−T122+ 2ν/bracketleftbig/parenleftbig
∇2
rD/parenrightbig
122−E122/bracketrightbig
, (23)
/parenleftbig
∇2
rD/parenrightbig
111≡/parenleftbigg
∂2
r+2
r∂r−6
r2/parenrightbigg
D111+12
r2D122=/parenleftbigg
−4
r2+4
r∂r+∂2
r/parenrightbigg
D111, (24)
/parenleftbig
∇2
rD/parenrightbig
122≡2
r2D111+/parenleftbigg
∂2
r+2
r∂r−8
r2/parenrightbigg
D122=1
6/parenleftbigg4
r2−4
r∂r+ 5∂2
r+r∂3
r/parenrightbigg
D111. (25)
The incompressibility condition, D122=1
6(D111+r∂rD111), was used in (24-25). Since Hill & Boratav (2001) discuss
these equations and evaluate them using data, further discu ssion of (22-25) is unnecessary.
The terms ∂tD[N],−T[N], and −2νE[N]in (14) have a repetitive structure in the isotropic equatio ns; e.g., for
N= 4 the 3 equations are
5∂tD1111+/parenleftbig
∇r•D[5]/parenrightbig
1111=−T1111+ 2ν/bracketleftbig/parenleftbig
∇2
rD[4]/parenrightbig
1111−E1111/bracketrightbig
, (26)
∂tD1122+/parenleftbig
∇r•D[5]/parenrightbig
1122=−T1122+ 2ν/bracketleftbig/parenleftbig
∇2
rD[4]/parenrightbig
1122−E1122/bracketrightbig
, (27)
∂tD2222+/parenleftbig
∇r•D[5]/parenrightbig
2222=−T2222+ 2ν/bracketleftbig/parenleftbig
∇2
rD[4]/parenrightbig
2222−E2222/bracketrightbig
. (28)
Thus, it suffices to give the isotropic formulas for the diverg ence ∇r•D[N+1]and Laplacian ∇2
rD[N]; forN= 4 to 7,
those isotropic formulas are given in Table 1. For N= 4 and 5 there are M+ 1 = 3 equations; there are M+ 1 = 4
equations for both N= 6 and 7.
/parenleftbig
∂r+2
r/parenrightbig
D[5,0,0]−8
rD[3,2,0]/parenleftbig
∂2
r+2
r∂r−8
r2/parenrightbig
D[4,0,0]+14
r2D[2,2,0]+10
3r2D[0,4,0] /parenleftbig
∂r+4
r/parenrightbig
D[3,2,0]−8
3rD[1,4,0]2
r2D[4,0,0]+/parenleftbig
−52
3r2+∂2
r+2
r∂r/parenrightbig
D[2,2,0]+34
9r2D[0,4,0] /parenleftbig
∂r+6
r/parenrightbig
D[1,4,0]2
r2D[2,2,0]+/parenleftbig
−2
3r2+∂2
r+2
r∂r/parenrightbig
D[0,4,0]
− − − − − −/parenleftbig
∂r+2
r/parenrightbig
D[6,0,0]−10
rD[4,2,0]/parenleftbig
∂2
r+2
r∂r−10
r2/parenrightbig
D[5,0,0]−14
r2D[3,2,0]+54
r2D[1,4,0] /parenleftbig
∂r+4
r/parenrightbig
D[4,2,0]−4
rD[2,4,0]2
r2D[5,0,0]+/parenleftbig
−154
5r2+∂2
r+2
r∂r/parenrightbig
D[3,2,0]+94
5r2D[1,4,0] /parenleftbig
∂r+6
r/parenrightbig
D[2,4,0]−6
5rD[0,6,0]6
5r2D[3,2,0]+/parenleftbig
−16
5r2+∂2
r+2
r∂r/parenrightbig
D[1,4,0]
− − − − − −/parenleftbig
∂r+2
r/parenrightbig
D[7,0,0]−12
rD[5,2,0]/parenleftbig
∂2
r+2
r∂r−12
r2/parenrightbig
D[6,0,0]−108
r2D[4,2,0]+920
3r2D[2,4,0]−416
15r2D[0,6,0] /parenleftbig
∂r+4
r/parenrightbig
D[5,2,0]−16
3rD[3,4,0]2
r2D[6,0,0]+/parenleftbig
−242
5r2+∂2
r+2
r∂r/parenrightbig
D[4,2,0]+824
15r2D[2,4,0]−248
75r2D[0,6,0] /parenleftbig
∂r+6
r/parenrightbig
D[3,4,0]−12
5rD[1,6,0]4
5r2D[4,2,0]+/parenleftbig
−112
15r2+∂2
r+2
r∂r/parenrightbig
D[2,4,0]+4
3r2D[0,6,0] /parenleftbig
∂r+8
r/parenrightbig
D[1,6,0]2
3r2D[2,4,0]+/parenleftbig
−2
15r2+∂2
r+2
r∂r/parenrightbig
D[0,6,0]
− − − − − −/parenleftbig
∂r+2
r/parenrightbig
D[8,0,0]−14
rD[6,2,0]/parenleftbig
∂2
r+2
r∂r−14
r2/parenrightbig
D[7,0,0]−316
r2D[5,2,0]+3376
3r2D[3,4,0]−1376
5r2D[1,6,0] /parenleftbig
∂r+4
r/parenrightbig
D[6,2,0]−20
3rD[4,4,0]2
r2D[7,0,0]+/parenleftbig
−1472
21r2+∂2
r+2
r∂r/parenrightbig
D[5,2,0]+7808
63r2D[3,4,0]−304
15r2D[1,6,0] /parenleftbig
∂r+6
r/parenrightbig
D[4,4,0]−18
5rD[2,6,0]4
7r2D[5,2,0]+/parenleftbig
−206
15r2+∂2
r+2
r∂r/parenrightbig
D[3,4,0]+1132
175r2D[1,6,0] /parenleftbig
∂r+8
r/parenrightbig
D[2,6,0]−8
7rD[0,8,0]2
7r2D[3,4,0]+/parenleftbig
−76
35r2+∂2
r+2
r∂r/parenrightbig
D[1,6,0]
Table 1: The isotropic formulas for ∇r•D[N+1]are on the left and those for ∇2
rD[N]are on the right. Cases
N= 4 and 5 are the top 3 and second 3 rows, respectively. Cases N= 6 and 7 are the third 4 and bottom 4 rows,
respectively.
V. COMPARISON WITH PREVIOUS RESULTS
The expression/parenleftbig
∂r+2
r/parenrightbig
D111−4
rD122in (20) is the same as equation (9) of Yakhot (2001), and (41) o f Hill
(1997a). The expression/parenleftbig
∂r+2
r/parenrightbig
D1111−6
rD1122in (22) is the same as in the equation that follows Yakhot’s eq .11
, and in equation (16) of Hill & Boratav (2001) and in equation (8) of Kurien (2001);/parenleftbig
∂r+4
r/parenrightbig
D1122−4
3rD2222in
(23) is the same as in equation (13) of Hill & Boratav (2001) an d equation (10) of Kurien (2001). The expressions/parenleftbig
∂r+2
r/parenrightbig
D[6,0,0]−10
rD[4,2,0]and/parenleftbig
∂r+6
r/parenrightbig
D[2,4,0]−6
5rD[0,6,0]for the case N= 5 in Table 1 are the same as in
equations (9) and (10) of Kurien (2001). More generally, the isotropic formulas for ∇r•D[N+1]for the case N1=N,
N2=N3= 0 are/parenleftbig
∂r+2
r/parenrightbig
D[N,0,0]−2(N−1)
rD[N−2,2,0]which agrees with the left-hand side of equation (7) of Yakho t
(2001). The other components of ∇r•D[N+1]were not given by Yakhot (2001). The expressions from the Lap lacian
in (20-21) are the same as in (41-42) of Hill (1997a); and (24- 25) are the same as (7-8) of Hill & Boratav (2001). All
of the remaining results do not appear to have been given prev iously. The above comparisons are sufficient to verify
the matrix algorithm for generating the structure-functio n equations to any desired order, as well as to independently
validate the derivation of Yakhot (2001).
VI. SUMMARY AND DISCUSSION
The third paragraph of the introduction summarizes part of t his paper and is not repeated here. In addition:
All of the kinematic relationships (19) between components of isotropic, symmetric structure functions of arbitrary
order have been identified, whereas previously only one was k nown. All of the components that are zero have been
identified (a recent experimental evaluation of some of them is given by Kurien & Sreenivasan 2000). The kinematic
relationships show that the scaling exponents of certain di fferent components must be equal; if the exponents are not
equal when evaluated using one’s data, then the kinematic re lationships (19) provide a measure of either the error
in the exponents or the deviation from local isotropy. The dy namic equations of order Ncan be used to test the
6extent of a scaling range for evaluation of scaling exponent s of velocity structure functions of order N+1 because the
time-derivative and viscous terms should be zero in an inert ial range. The graphical presentations of the balance of
Kolmogorov’s equation by Antonia et al. (1983), Chambers & Antonia (1984), Danaila et al. (1999a,b), and Antonia
et al. (2000) show the extent of, or deviation from, inertial-ran ge exponents. The higher-order equations given here
can be used in an analogous manner.
The energy dissipation rate εplays an essential role at all rin Kolmogorov’s equation. In our formulation ε
arises in (20-21) from the tensor components 2 νE11and 2νE22. On the other hand, for the next-order equations (22-
23) Hill (1997b) showed that the corresponding terms 2 νE111and 2νE122are negligible in the inertial range. Yakhot
(2001) shows that the components E[N,0,0]are negligible in the inertial range for all of the higher-or der equations for
which Nis odd. Kolmogorov’s (1941) inertial-range scaling using εandras the only relevant parameters can be
used to estimate the relative magnitudes of the term ∇r•D[N+1]in (14) to the terms 2 ν∇2
rD[N]and 2νE[N]. Doing
so, the ratio of any nonzero component of 2 ν∇2
rD[N]or 2νE[N]to the corresponding component of ∇r•D[N+1]is
proportional to ν/r4/3ε1/3= (r/η)−4/3, which asymptotically vanishes in the inertial range ( η≡/parenleftbig
ν3/ε/parenrightbig1/4). Thus,
both terms 2 ν∇2
rD[N]and 2νE[N]are to be neglected in an inertial range if N >2.
One concludes that all equations of order higher than Kolmog orov’s equation reduce to the isotropic formula
for∇r•D[N+1]=−T[N]in the inertial range. This formula shows that T[N]is at the heart of two issues that
have received much attention: 1) whether or not different com ponents of the velocity structure function D[N+1]have
differing exponents in the inertial range, and 2) the increas ing deviation of those exponents from Kolmogorov scaling
asNincreases. The physical basis for the importance of T[N]is the importance of vortex tubes to the intermittency
phenomenon (Pullin & Saffman 1998) combined with the fact tha t the pressure-gradient force is essential to the
existence of vortex tubes; the pressure-gradient force pre vents a vortex tube from cavitating despite the centrifugal
force. Pressure gradients are the sinews of vortex tubes. Di rect investigation of T[N]using DNS can reveal much
about the two issues.
Acknowledgement 1 The author thanks the organizers of the Hydrodynamics Turbu lence Program held at the In-
stitute for Theoretical Physics, University of California at Santa Barbara, whereby this research was supported in par t
by the National Science Foundation under grant number PHY94 -07194.
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arXiv:physics/0102064v1 [physics.acc-ph] 20 Feb 2001
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/CT/B8 /DB/CW/CX
/CW
/CP/D2 /CQ /CT /D3/CQ/D8/CP/CX/D2/CT/CS /CQ /DD /CP /D7/CT/D6/CX/CT/D7 /D3/CU
/CP/D2/D3/D2/CX
/CP/D0/BD/D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2 /CU/D6/D3/D1 /D8/CW/CT /D0/CP/CQ /CU/D6/CP/D1/CT/CJ/BD ℄/BM
H=p2
x
2+p2
y
2+p2
z
2−γ2βxpz
R+β2γ2/parenleftbigg
κx−γ2
R2/parenrightbiggx2
2+β2γ2κyy2
2+ Φ RF(z) /B8/B4/BD/B5/DB/CW/CT/D6/CT γ /CX/D7 /D8/CW/CT /D6/CT/D0/CP/D8/CX/DA/CX/D7/D8/CX
/CU/CP
/D8/D3/D6/B8 κx,y
/D8/CW/CT /B4/D4 /D3/D7/D7/CX/CQ/D0/DD /D0/D3
/CP/D0/B5 /CU/D3
/D9/D7/CX/D2/CV /D7/D8/D6/CT/D2/CV/D8/CW/D7/B4/CX/D2 /D8/CW/CT
/CP/D7/CT /D3/CU /D1/CP/CV/D2/CT/D8/CX
/D5/D9/CP/CS/D6/D9/D4 /D3/D0/CT/D7/B8 /D3/D2/CT /CW/CP/D7κx=−κy
/B5 /CP/D2/CSΦ(z) /D8/CW/CT/CT/DC/D8/CT/D6/D2/CP/D0 /CT/D0/CT
/D8/D6/CX
/D4 /D3/D8/CT/D2 /D8/CX/CP/D0/BA /CC/CW/CT /CS/CX/D6/CT
/D8/CX/D3/D2/D7 x, y, z /CP/D6/CT /D6/CP/CS/CX/CP/D0/B8 /D8/D6/CP/D2/D7/DA /CT/D6/D7/CP/D0/B8 /CP/D2/CS/D8/CP/D2/CV/CT/D2 /D8/CX/CP/D0/B8 /D6/CT/D7/D4 /CT
/D8/CX/DA /CT/D0/DD /BA /C6/D3/D8/CT /D8/CW/CP/D8 /DB /CT /D9/D7/CT /D9/D2/CX/D8/D7 /DB/CX/D8/CW /planckover2pi1=c=kB=m0/D8/CW/D6/D3/D9/CV/CW/D3/D9/D8/B8 /D7/D3 /CP/D0/D0 /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7 /CP/D6/CT /CT/DC/D4/D6/CT/D7/D7/CT/CS /CX/D2 /D4 /D3 /DB /CT/D6/D7 /D3/CU /D8/CW/CT /BV/D3/D1/D4/D8/D3/D2 /D0/CT/D2/CV/D8/CW/D3/CU /D8/CW/CT /D4/CP/D6/D8/CX
/D0/CT/D7
/D3/D2/D7/CX/CS/CT/D6/CT/CS/BA/CC/CW/CT /D0/D3/D2/CV/CX/D8/D9/CS/CX/D2/CP/D0 /D4/CP/D6/D8 /D3/CU /D8/CW/CT /C0/CP/D1/CX/D0/D8/D3/D2/CX/CP/D2 /CS/CT/D4 /CT/D2/CS/D7 /D3/D2 /D8/CW/CT /D4/CW /DD/D7/CX
/CP/D0 /D7/CT/D8/D9/D4/BA/CC/CW/CT /D4/CP/D6/D8/CX
/D0/CT/D7 /D1/CX/CV/CW /D8 /CT/CX/D8/CW/CT/D6 /CQ /CT
/D3/D2/AS/D2/CT/CS /CQ /DD /D8/CW/CT /D2/CT/CP/D6/D0/DD /CW/CP/D6/D1/D3/D2/CX
/D4 /D3/D8/CT/D2 /D8/CX/CP/D0 /D3/CU/D8/CW/CT /CA/BY /CQ/D9
/CZ /CT/D8/B8 /D3/D6 /DB /CT /CW/CP /DA /CT
/CP/D7/CT /D3/CU /CP
/D3/CP/D7/D8/CX/D2/CV /CQ /CT/CP/D1/B8 /DB/CW/CT/D6/CT /D8/CW/CT /D3/D2/D0/DD
/D3/D2/D7/D8/D6/CP/CX/D2 /D8/D7/CX/D1/D4 /D3/D7/CT/CS /D3/D2 /D8/CW/CT /D0/D3/D2/CV/CX/D8/D9/CS/CX/D2/CP/D0 /D1/D3/D8/CX/D3/D2 /CP/D6/CT /D8/CW/CT /D3/D2/CT/D7 /CS/D9/CT /D8/D3 /D8/CW/CT /D4 /CT/D6/CX/D3 /CS/CX
/CX/D8 /DD /D3/CU/D8/CW/CT /D4/D6/D3/CQ/D0/CT/D1/BA /C1/D2 /D8/CW/CT /D7/CT/D5/D9/CT/D0/B8 /DB /CT /DB/CX/D0/D0
/D3/D2/D7/CX/CS/CT/D6 /CQ /D3/D8/CW
/CP/D7/CT/D7/BA/BE /BT/D2/CX/D7/D3/D8/D6/D3/D4/CX
/C7/D7
/CX/D0/D0/CP/D8/D3/D6/C4/CT/D8 /D9/D7 /CP/D7/D7/D9/D1/CT /D8/CW/CP/D8 /D8/CW/CT /D0/D3/D2/CV/CX/D8/D9/CS/CX/D2/CP/D0 /D1/D3/D8/CX/D3/D2 /CX/D7 /CS/CT/D8/CT/D6/D1/CX/D2/CT/CS /CQ /DD /D8/CW/CT /D4/D6/CT/D7/CT/D2
/CT /D3/CU/CP/D2 /CA/BY /CQ/D9
/CZ /CT/D8/BA /CF /CT
/CP/D2 /CP/D4/D4/D6/D3 /DC/CX/D1/CP/D8/CT /D8/CW/CT /D4 /D3/D8/CT/D2 /D8/CX/CP/D0 Φ /CQ /DD /CT/DC/D4/CP/D2/CS/CX/D2/CV /CX/D8 /D8/D3 /BE/D2/CS/D3/D6/CS/CT/D6 /CX/D2z /BA /BY /D3/D6 /D6/CT/CP/D7/D3/D2/D7 /D3/CU /D7/CX/D1/D4/D0/CX
/CX/D8 /DD /B8 /DB /CT /D3/D2/D0/DD /D8/CP/CZ /CT /CX/D2 /D8/D3 /CP
/D3/D9/D2 /D8 /D8/CW/CTO(z2)/D8/CT/D6/D1/B8 /CX/BA /CT/BA/B8 /DB /CT /CP/D7/D7/D9/D1/CT /D8/CW/CP/D8 /D8/CW/CT /D4/CP/D6/D8/CX
/D0/CT /CX/D7 /D3/D2 /D8/CW/CT /D3/D6/CQ/CX/D8 /CP/D2/CS /CX/D7 /D2/D3/D8 /D0/D3/D7/CX/D2/CV /CT/D2/CT/D6/CV/DD /BA/CF /CT
/CP/D2 /D3/CQ/D8/CP/CX/D2 /D8/CW/CT /D7/D8/CX/AR/D2/CT/D7/D7 /D3/CU /D8/CW/CT /D0/D3/D2/CV/CX/D8/D9/CS/CX/D2/CP/D0 /D3/D7
/CX/D0/D0/CP/D8/D3/D6 /CX/D2 /D8/CW/CT /D0/D3
/CP/D0
/CP/D2/D3/D2/B9/CX
/CP/D0
/D3 /GU/D6/CS/CX/D2/CP/D8/CT/D7 /CQ /DD /C4/D3/D6/CT/D2 /D8/DE/B9/CQ /D3 /D3/D7/D8/CX/D2/CV /D8/CW/CT(t, z)
/D3/D1/D4 /D3/D2/CT/D2 /D8/D7 /D3/CU /D8/CW/CT /D1/D3/D1/CT/D2 /D8/D9/D1/DA /CT
/D8/D3/D6/BM
m0
γ(1 +δ)
γ(1 +δ)/radicalbig
1−(γ(1 +δ))−2
→m0
0
δ
+ O(δ2) /B4/BE/B5/CC/CW /D9/D7/B8 /CU/D3/D6 /CP /CQ/D9/D2
/CW/CT/CS /CQ /CT/CP/D1 /DB/CX/D8/CW /CS/CX/D1/CT/D2/D7/CX/D3/D2/D7 σz, σδ
/B8 /DB /CT /CW/CP /DA /CTωlγσl=σδ
/BA/CC/CW/CT /D0/D3/D2/CV/CX/D8/D9/CS/CX/D2/CP/D0 /CP/D2/CS /D6/CP/CS/CX/CP/D0 /D4/CP/D6/D8 /D3/CU /D8/CW/CT /CW/CP/D1/CX/D0/D8/D3/D2/CX/CP/D2 /B4/BD/B5 /CW/CP /DA /CT /D8/CW/CT /CU/D3/D6/D1
H=2/summationdisplay
i=1p2
i
2+ω2
iq2
i
2−µq1p2
/BA /B4/BF/B5/BE/CC/CW/CT /CP/D7/D7/D3
/CX/CP/D8/CT/CS /CX/D2/AS/D2/CX/D8/CT/D7/CX/D1/CP/D0 /D7/DD/D1/D4/D0/CT
/D8/CX
/D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2 /D1/CP/D8/D6/CX/DC /B4/CX/D2(q1, p1, q2, p2) /B9/D7/D4/CP
/CT/B5 /D6/CT/CP/CS/D7
0 1 0 0
−ω2
10 0 µ
−µ0 0 1
0 0 −ω2
20
/B4/BG/B5/CP/D2/CS /CX/D8/D7 /CT/CX/CV/CT/D2 /DA /CP/D0/D9/CT/D7 /CP/D6/CT /CS/CT/D8/CT/D6/D1/CX/D2/CT/CS /CQ /DD /D8/CW/CT /CT/D5/D9/CP/D8/CX/D3/D2
λ4+λ2(ω2
1+ω2
2) +ω2
2(µ2−ω2
1) = 0 /B8 /B4/BH/B5/DB/CW/CX
/CW /DB/CX/D0/D0 /CW/CP /DA /CT /D4/D9/D6/CT/D0/DD /CX/D1/CP/CV/CX/D2/CP/D6/DD /D7/D3/D0/D9/D8/CX/D3/D2/D7 /CU/D3/D6
(ω2
1+ω2
2)>4ω2
2(µ2−ω2
1)>0 /BA /B4/BI/B5/CC/CW/CT /AS/D6/D7/D8
/D3/D2/CS/CX/D8/CX/D3/D2 /DB/CX/D0/D0 /CP/D0/DB /CP /DD/D7 /CQ /CT /CU/D9/D0/AS/D0/D0/CT/CS /CU/D3/D6 /D6/CT/CP/D0/CX/D7/D8/CX
/D1/CP
/CW/CX/D2/CT/D7/BA /CC/CW/CT/D7/CT
/D3/D2/CS /D3/D2/CT
/D3/D6/D6/CT/D7/D4 /D3/D2/CS/D7 /D8/D3 /D8/CW/CT /D1/CP
/CW/CX/D2/CT /CQ /CT/CX/D2/CV /CQ /CT/D0/D3 /DB /D3/D6 /CP/CQ /D3 /DA /CT /D8/D6/CP/D2/D7/CX/D8/CX/D3/D2/BM /CX/CU/D8/CW/CT /D7/CT
/D3/D2/CS /CU/CP
/D8/D3/D6
/CW/CP/D2/CV/CT/D7 /D7/CX/CV/D2/B8 /D8/CW/CT /CT/CX/CV/CT/D2/CU/D6/CT/D5/D9/CT/D2
/DD
/CP/D2 /CQ /CT /D1/CP/CS /D6/CT/CP/D0 /CP/CV/CP/CX/D2 /CQ /DD/AT/CX/D4/D4/CX/D2/CV /D8/CW/CT /D7/CX/CV/D2 /D3/CUω2
z∝Φ′′
RF
/BA /C0/D3 /DB /CT/DA /CT/D6/B8 /D8/CW/CT /CP/CQ/D7/D3/D0/D9/D8/CT /D7/CX/CV/D2 /D3/CU /CQ /D3/D8/CW /D8/CW/CT /CZ/CX/D2/CT/D8/CX
/CP/D2/CS /D8/CW/CT /D4 /D3/D8/CT/D2 /D8/CX/CP/D0 /D8/CT/D6/D1 /DB/CX/D0/D0
/CW/CP/D2/CV/CT/B8 /D0/CT/CP/CS/CX/D2/CV /D8/D3 /D8/CW/CT /B4/CU/D3/D6 /D4/D9/D6/D4 /D3/D7/CT/D7 /D3/CU
/D3/D2/D7/D8/D6/D9
/D8/B9/CX/D2/CV /D8/CW/CT /D5/D9/CP/D2 /D8/D9/D1/B9/D1/CT
/CW/CP/D2/CX
/CP/D0 /CV/D6/D3/D9/D2/CS /D7/D8/CP/D8/CT/B5 /D4/CP/D8/CW/D3/D0/D3/CV/CX
/CP/D0
/CP/D7/CT /D3/CU /CP /CW/CP/D1/CX/D0/D8/D3/D2/CX/CP/D2/D2/D3/D8 /D0/CX/D1/CX/D8/CT/CS /CU/D6/D3/D1 /CQ /CT/D0/D3 /DB/BA /C1/D2 /D8/CW/CT /D7/CT/D5/D9/CT/D0/B8 /DB /CT /DB/CX/D0/D0 /CP/D7/D7/D9/D1/CT /D8/CW/CT /D1/CP
/CW/CX/D2/CT /CX/D7 /CQ /CT/D0/D3 /DB/D8/D6/CP/D2/D7/CX/D8/CX/D3/D2/BA/CC/CW /D9/D7/B8 /CT/DC/D4/CP/D2/CS/CX/D2/CV /B4/BD/B5 /D8/D3 /AS/D6/D7/D8 /D2/D3/D2/B9/D8/D6/CX/DA/CX/CP/D0 /D3/D6/CS/CT/D6 /CX/D2 /D8/CW/CT
/CP/D2/D3/D2/CX
/CP/D0
/D3 /GU/D6/CS/CX/B9/D2/CP/D8/CT/D7 /CP/D2/CS /CP/D4/D4/D0/DD/CX/D2/CV /D8/CW/CT
/CP/D2/D3/D2/CX
/CP/D0 /D8/D6/CP/D2/D7/CU/D3/D6/D1/CP/D8/CX/D3/D2 /D6/CT/D1/D3 /DA/CX/D2/CV /D8/CW/CT /D1/CX/DC/CT/CS /D8/CT/D6/D1/CX/D2 /B4/BF/B5/B8 /DB /CT /D3/CQ/D8/CP/CX/D2 /D8/CW/CT /C0/CP/D1/CX/D0/D8/D3/D2/CX/CP/D2 /D3/CU /CP /BF/B9/CS/CX/D1/CT/D2/D7/CX/D3/D2/CP/D0 /CW/CP/D6/D1/D3/D2/CX
/D3/D7
/CX/D0/D0/CP/D8/D3/D6 /DB/CX/D8/CW
/D3/D6/D6/CT
/D8/CT/CS /CU/D6/CT/D5/D9/CT/D2
/CX/CT/D7 /CV/CX/DA /CT/D2 /CQ /DD /B4/BH/B5/BN /D8/CW/CT /CV/D6/D3/D9/D2/CS /D7/D8/CP/D8/CT /CX/D7
/CW/CP/D6/CP
/D8/CT/D6/CX/DE/CT/CS /CQ /DD /D8/CW/CT/D3
/D9/D4/CP/D8/CX/D3/D2 /D2 /D9/D1 /CQ /CT/D6/D7 nd
n∈ {0,1} /DB/CW/CT/D6/CT/summationtextni
d=N /D3/CU /D8/CW/CT /D3/D7
/CX/D0/D0/CP/D8/D3/D6 /D0/CT/DA /CT/D0/D7/BA /BY /D3/D6/D7/CP/CZ /CT /D3/CU /CV/CT/D2/CT/D6/CP/D0/CX/D8 /DD /B8 /DB /CT
/D3/D2/D7/CX/CS/CT/D6 /D8/CW/CT
/CP/D7/CT /D3/CUd /CS/CX/D1/CT/D2/D7/CX/D3/D2/D7/BA /CC/CW/CT /CV/D6/D3/D9/D2/CS /D7/D8/CP/D8/CT /CU/D3/D6/CP /CV/CX/DA /CT/D2 /D4/CP/D6/D8/CX
/D0/CT /D2 /D9/D1 /CQ /CT/D6
/CP/D2 /CQ /CT
/D3/D2/D7/D8/D6/D9
/D8/CT/CS /CQ /DD /D7/D9
/CT/D7/D7/CX/DA /CT/D0/DD /AS/D0/D0/CX/D2/CV /D7/D8/CP/D8/CT/D7 /DB/CX/D8/CW/D8/CW/CT /D0/D3 /DB /CT/D7/D8 /CT/D2/CT/D6/CV/DD /B4/DB /CT /CS/CX/D7/D6/CT/CV/CP/D6/CS /D7/D4/CX/D2 /CW/CT/D6/CT/B8 /DB/CW/CX
/CW
/CP/D2 /CQ /CT /CT/CP/D7/CX/D0/DD /D6/CT/CX/D2 /D8/D6/D3 /CS/D9
/CT/CS/CQ /DD /D6/CT/D4/D0/CP
/CX/D2/CV N→2N /CX/D2 /D8/CW/CT /AS/D2/CP/D0 /CU/D3/D6/D1 /D9/D0/CP/CT/B5/BA/C1/D2E
ǫF
/B9/D7/D4/CP
/CT/B8 /D8/CW/CT /BY /CT/D6/D1/CX /D7/CT/CP /CX/D7 /CY/D9/D7/D8 /CP /D9/D2/CX/D8d /B9/D7/CX/D1/D4/D0/CT/DC/B8 /CX/D2ni
/B9/D7/D4/CP
/CT/B8 /CPd /B9/D7/CX/D1/D4/D0/CT/DC/DB/CX/D8/CW /CP/DC/CT/D7 /D3/CU /D0/CT/D2/CV/D8/CWω1
ǫF, . . .ωd
ǫF
/BA /CC/CW /D9/D7/B8 /D8/CW/CT /D4/CP/D6/D8/CX
/D0/CT /D2 /D9/D1 /CQ /CT/D6 /CU/D3/D6 /CP /CV/D6/D3/D9/D2/CS /D7/D8/CP/D8/CT/AS/D0/D0/CT/CS /D9/D4 /D8/D3 /D8/CW/CT /BY /CT/D6/D1/CX /CT/D2/CT/D6/CV/DD ǫF
/B8 /DB/CW/CT/D6/CT /DB /CT /CW/CP /DA /CT /CS/CX/D7/D6/CT/CV/CP/D6/CS/CT/CS /D8/CW/CT /DE/CT/D6/D3/B9/D1/D3 /CS/CT/CT/D2/CT/D6/CV/DD1
2/summationtext
iωi
/D3/CU /D8/CW/CT /D3/D7
/CX/D0/D0/CP/D8/D3/D6/B8
N=ǫFd
ω1· · ·ωdd!=1
Ωdd!
/B8 /B4/BJ/B5/BF/D8/CW/CT /DA /D3/D0/D9/D1/CT /D3/CU /CP/D2 /D9/D2/CX/D8d /B9/D7/CX/D1/D4/D0/CT/DC /CQ /CT/CX/D2/CV1
d!
/CP/D2/CSΩ =d√ω1· · ·ωd
/BA/CC/CW/CT /CT/D2/CT/D6/CV/DD /CX/D2 /D8/CW/CTi /D8/CW /CS/CT/CV/D6/CT/CT /D3/CU /CU/D6/CT/CT/CS/D3/D1 /CX/D2 /D8/CW/CP/D8
/CP/D7/CT /CX/D7 /CV/CX/DA /CT/D2 /CQ /DD /CP /D7/D9/D1/D3 /DA /CT/D6 /D8/CW/CTd /B9/D7/CX/D1/D4/D0/CT/DC/BM
Ei=ǫF
ω1/summationdisplay
i1=1ǫF−n1ω1
ω2/summationdisplay
i2=1· · ·ǫF
ωd−1
ωd/summationtextd−1
k=1nkωk/summationdisplay
id=1ωiii
/BA /B4/BK/B5/CA/CT/D4/D0/CP
/CX/D2/CV /CP/D0/D0 /D8/CW/CT /D7/D9/D1/D7 /CQ /DD /CX/D2 /D8/CT/CV/D6/CP/D0/D7/B8 /DB /CT /CW/CP /DA /CT
Ei=ǫFd
Ωd/integraldisplay1
0/integraldisplay1−q1
0· · ·/integraldisplay1−/summationtextd−1
k=1qk
0ǫFqidq1· · ·dqd=ǫFd+1
Ωd(d+ 1)!=NǫF
d+ 1
/B8/B4/BL/B5/DB/CW/CX
/CW/B8 /D3/CU
/D3/D9/D6/D7/CT/D7/B8 /CY/D9/D7/D8 /CT/DC/D4/D6/CT/D7/D7/CT/D7 /CT/D5/D9/CX/CS/CX/D7/D8/D6/CX/CQ/D9/D8/CX/D3/D2 /D3/CU /CT/D2/CT/D6/CV/DD /BA/BY/CX/D2/CP/D0/D0/DD /B8 /D8/CW/CT /CT/D1/CX/D8/D8/CP/D2
/CT /CX/D7 /CV/CX/DA /CT/D2 /CX/D2 /D8/CT/D6/D1/D7 /D3/CU /D8/CW/CT /CP /DA /CT/D6/CP/CV/CT/CS /D7/CX/D2/CV/D0/CT/B9/D4/CP/D6/D8/CX
/D0/CT/CT/DC/D4 /CT
/D8/CP/D8/CX/D3/D2 /DA /CP/D0/D9/CT /D3/CU /D8/CW/CT /CP
/D8/CX/D3/D2/BA /BY /D3/D6 /CP /CW/CP/D6/D1/D3/D2/CX
/D3/D7
/CX/D0/D0/CP/D8/D3/D6/B8 /D8/CW/CT /CP
/D8/CX/D3/D2 /CX/D7 /CY/D9/D7/D8
I=n /B8 /DB/CW/CT/D6/CT n /CX/D7 /D8/CW/CT /CT/DC
/CX/D8/CP/D8/CX/D3/D2 /D2 /D9/D1 /CQ /CT/D6 /D3/CU /D8/CW/CT /CT/D2/CT/D6/CV/DD /D0/CT/DA /CT/D0/B8 /D7/D3/BM
εi=∝an}bracketle{tni∝an}bracketri}ht=Ei
Nωi=ǫF
ωi(d+ 1)=Ω
ωi(d+ 1)d√
Nd! /BA /B4/BD/BC/B5/CP/D2/CS
ε(d)=d/productdisplay
i=1εi=Nd!/parenleftbigg2π
d+ 1/parenrightbiggd
ε(1)=πN /B8ε(2)=8
9π2N /B8ε(3)=3
4π3N
/B4/BD/BD/B5/CC/CW /D9/D7/B8 /D8/CW/CT /D4/D6/D3 /CY/CT
/D8/CT/CS /CT/D1/CX/D8/D8/CP/D2
/CT/D7 /D7
/CP/D0/CT /CP/D7N1
d/B8 /CP/D7 /D3/D2/CT /DB /D3/D9/D0/CS /D2/CP/GN/DA /CT/D0/DD /CP/D7/D7/D9/D1/CT/BA/BY /D9/D6/D8/CW/CT/D6/D1/D3/D6/CT/B8 /CS/D9/CT /D8/D3 /D8/CW/CT /D3
/D9/D6/D6/CT/D2
/CT /D3/CU /D8/CW/CT /CV/CT/D3/D1/CT/D8/D6/CX
/D1/CT/CP/D2 /D3/CU /D8/CW/CT /CU/D6/CT/D5/D9/CT/D2/B9
/CX/CT/D7 /CX/D2 /B4/BD/BC/B5/B8 /D8/CW/CT /D4/D6/D3 /CY/CT
/D8/CT/CS /CT/D1/CX/D8/D8/CP/D2
/CT /CX/D2 /D3/D2/CT /CS/CX/D1/CT/D2/D7/CX/D3/D2
/CP/D2 /CQ /CT /D0/D3 /DB /CT/D6/CT/CS /CQ /DD/D7/CW/CP/D0/D0/D3 /DB/CX/D2/CV /D8/CW/CT /D4 /D3/D8/CT/D2 /D8/CX/CP/D0 /CX/D2 /D8/CW/CT /D3/D8/CW/CT/D6 /CS/CX/D1/CT/D2/D7/CX/D3/D2/D7/BA/C6/D3/D8/CT /D8/CW/CP/D8 /CP /D7/CX/D1/CX/D0/CP/D6 /CP/D4/D4/D6/D3/CP
/CW /CW/CP/D7 /CQ /CT/CT/D2
/CW/D3/D7/CT/D2 /CT/D0/D7/CT/DB/CW/CT/D6/CT/BN /CJ/CJ/BE℄℄ /CV/CX/DA /CT/D7 /CP/D2/CT/D7/D8/CX/D1/CP/D8/CT /CU/D3/D6ǫmin
/CU/D6/D3/D1 /CP /D7/CX/D1/CX/D0/CP/D6 /D6/CT/CP/D7/D3/D2/CX/D2/CV/B8 /CQ/D9/D8 /CT/D2/CS/D7 /D9/D4 /B4/CS/D9/CT /D8/D3 /CP /D1/CX/D7
/D3/D9/D2 /D8/CX/D2/CV/D3/CU /D8/CW/CT /D7/D8/CP/D8/CT/D7/B5 /DB/CX/D8/CW /CP /D7
/CP/D0/CX/D2/CV /CS/CX/AR/CT/D6/CT/D2 /D8 /CU/D6/D3/D1 /D3/D9/D6 /D6/CT/D7/D9/D0/D8/BA/BF /C5/CX/DC/CT/CS /BV/CP/D7/CT/BM /C4/D3/D2/CV/CX/D8/D9/CS/CX/D2/CP/D0/D0/DD /BY /D6/CT/CT /C8 /CP/D6/D8/CX
/D0/CT/D7/CB/D3 /CU/CP/D6/B8 /DB /CT /CW/CP /DA /CT /CP/D7/D7/D9/D1/CT/CS /CP/D2 /CP/D2/CX/D7/D3/D8/D6/D3/D4/CX
/D3/D7
/CX/D0/D0/CP/D8/D3/D6/BA /BU/D9/D8 /CV/CX/DA /CT/D2 /D8/CW/CT
/CP/D7/CT /D3/CU /CP/D4/CP/D6/D8/CX
/D0/CT /D1/D3 /DA/CX/D2/CV /CU/D6/CT/CT/D0/DD /D0/D3/D2/CV/CX/D8/D9/CS/CX/D2/CP/D0/D0/DD /B8 /D8/CW/CT /CT/D2/CT/D6/CV/DD
/D3/D2 /D8/CT/D2 /D8 /D3/CU /D8/CW/CP/D8 /CS/CT/CV/D6/CT/CT /D3/CU/BG/CU/D6/CT/CT/CS/D3/D1 /DB/CX/D0/D0 /CQ /CT /CV/CX/DA /CT/D2 /CQ /DD /D8/CW/CT /D7/D5/D9/CP/D6/CT /D3/CU /D8/CW/CT /B4/CP/D2/CV/D9/D0/CP/D6/B5 /D1/D3/D1/CT/D2 /D8/D9/D1/BA /B4/CF /CT /D1/CX/CV/CW /D8
/D3/D2/D7/CX/CS/CT/D6 /D8/CW/CT /CQ /D3/D9/D2/CS/CP/D6/DD
/D3/D2/CS/CX/D8/CX/D3/D2/D7 /CX/D1/D4 /D3/D7/CT/CS /CQ /DD /CP /D4 /CT/D6/CX/D3 /CS/CX
/CQ /D3 /DC /CX/D2/D7/D8/CT/CP/CS /D3/CU /CP
/CX/D6
/D9/D0/CP/D6 /CP/D6/D6/CP/D2/CV/CT/D1/CT/D2 /D8/BA/B5 /CF /CT /D8/D6/CT/CP/D8 /D8/CW/CT /CV/CT/D2/CT/D6/CP/D0
/CP/D7/CT/B8 /CX/BA /CT/BA /CP /C0/CP/D1/CX/D0/D8/D3/D2/CX/CP/D2
H=˜d/summationdisplay
˜i=1˜ω2
˜i˜n2
˜i+d/summationdisplay
i=1ωini
/BA /B4/BD/BE/B5/CC/CW/CT /CX/D2 /D8/CT/CV/D6/CP/D8/CX/D3/D2 /D3 /DA /CT/D6 /D8/CW/CT /CU/D6/CT/CT /CS/CT/CV/D6/CT/CT/D7 /D3/CU /CU/D6/CT/CT/CS/D3/D1 /D6/D9/D2/D7 /D3 /DA /CT/D6 /CP/D2 /CT/D0/D0/CX/D4/D7/D3/CX/CS/BN/CQ /DD /D6/CT/D7
/CP/D0/CX/D2/CV /D8/D3 /CP /D9/D2/CX/D8 /D7/D4/CW/CT/D6/CT/B8 /DB /CT /CV/CT/D8
N=ǫF˜d
2+d
˜Ω˜dΩd/integraldisplay
Sphere/integraldisplay1−˜q2
Simplexdqd˜q=ǫF˜d
2+d
˜Ω˜dΩd/integraldisplay1
0S˜dq˜d−1(1−˜q2)d
d!d˜q
=SdǫF˜d
2+dB/parenleftBig
d+ 1,˜d
2/parenrightBig
2˜Ω˜dΩdd!=π˜d
2ǫF˜d
2+d
˜Ω˜dΩdΓ/parenleftBig
2d+˜d+2
2/parenrightBig/B8 /B4/BD/BF/B5/DB/CW/CT/D6/CT Sd=2πd
2
Γ(d
2)
/CX/D7 /D8/CW/CT /D7/D9/D6/CU/CP
/CT /D3/CU /D8/CW/CTd /B9/CS/CX/D1/CT/D2/D7/CX/D3/D2/CP/D0 /D9/D2/CX/D8 /D7/D4/CW/CT/D6/CT /CP/D2/CSB /CX/D7 /D8/CW/CT/D9/D7/D9/CP/D0 /BU/CT/D8/CP /CU/D9/D2
/D8/CX/D3/D2 B(x, y) = Γ( x)Γ(y)/Γ(x+y) /BA/CF /CT
/CP/D2 /D6/CT/CP/CS/CX/D0/DD /DB/D6/CX/D8/CT /CS/D3 /DB/D2 /D8/CW/CT /CT/DC/D4 /CT
/D8/CP/D8/CX/D3/D2 /DA /CP/D0/D9/CT/D7 /D3/CU /CT/D2/CT/D6/CV/DD /CX/D2 /D8/CW/CT /CS/CX/AR/CT/D6/CT/D2 /D8/CS/CT/CV/D6/CT/CT/D7 /D3/CU /CU/D6/CT/CT/CS/D3/D1/BM
E˜i
εF=/angbracketleftbig
˜q2/angbracketrightbig
=/integraltext1
0˜q˜d+1/radicalbig
1−˜q2dd˜q
/integraltext1
0˜q˜d−1/radicalbig
1−˜q2dd˜q=B/parenleftBig
d+ 1,˜d
2+ 1/parenrightBig
B/parenleftBig
d+ 1,˜d
2/parenrightBig=˜d
2d+ 2 + ˜d
/B4/BD/BG/B5/CP/D2/CS
Ei
εF=∝an}bracketle{tq∝an}bracketri}ht=/integraltext1
0qd(1−q)˜d
2dq
/integraltext1
0qd−1(1−q)˜d
2dq=B/parenleftBig
d+ 1,˜d
2+ 1/parenrightBig
B/parenleftBig
d,˜d
2+ 1/parenrightBig=2d
2d+ 2 + ˜d
/B4/BD/BH/B5/BY /D3/D6 /D8/CW/CT /D0/D3/D2/CV/CX/D8/D9/CS/CX/D2/CP/D0 /CT/D1/CX/D8/D8/CP/D2
/CT/B8 /DB /CT /D2/CT/CT/CS /D8/CW/CT /CT/DC/D4 /CT
/D8/CP/D8/CX/D3/D2 /DA /CP/D0/D9/CT /D3/CU /D8/CW/CT/CP
/D8/CX/D3/D2/BN /CU/D3/D6 /CP /D4/CP/D6/D8/CX
/D0/CT /CX/D2 /CP /CQ /D3 /DC /DB/CX/D8/CW /D4 /CT/D6/CX/D3 /CS/CX
/CQ /D3/D9/D2/CS/CP/D6/DD
/D3/D2/CS/CX/D8/CX/D3/D2/D7/B8 /D8/CW/CX/D7 /CX/D7/CP/CV/CP/CX/D2 /CY/D9/D7/D8∝an}bracketle{txp∝an}bracketri}ht= 2πn /CX/D2 /D8/CW/CTn /D8/CW /CQ /D3 /DC /CT/CX/CV/CT/D2/D1/D3 /CS/CT/B8 /D7/D3 /DB /CT /D2/CT/CT/CS
∝an}bracketle{t˜q∝an}bracketri}ht=/integraltext1
0˜q˜d/radicalbig
1−˜q2dd˜q
/integraltext1
0˜q˜d−1/radicalbig
1−˜q2dd˜q=B/parenleftBig
d+ 1,˜d+1
2/parenrightBig
B/parenleftBig
d+ 1,˜d
2/parenrightBig/B8 /B4/BD/BI/B5/BH/CP/D2/CS /D8/CW/CT /CU/D3/D6/D1 /CU/CP
/D8/D3/D6 /CU/D3/D6 /D8/CW/CT /D4/D6/D3 /CS/D9
/D8 /CT/D1/CX/D8/D8/CP/D2
/CT /CX/D7
ε(d,˜d)= (2π)d+˜dNB˜d/parenleftBig
d+ 1,˜d+1
2/parenrightBig
Bd/parenleftBig
d+ 1,˜d
2+ 1/parenrightBig
B˜d/parenleftBig
d+ 1,˜d
2/parenrightBig
Bd/parenleftBig
d,˜d
2+ 1/parenrightBig/B8 /B4/BD/BJ/B5/CP/D2/CS/B8 /CU/D3/D6 /D6/CT/CP/D0/B9/DB /D3/D6/D0/CS /D7/CX/D8/D9/CP/D8/CX/D3/D2/D7/B8
ε(2,1)=5
49N /BA /B4/BD/BK/B5/C8/D9/D8/D8/CX/D2/CV /CX/D2 /CP /D6/CT/CP/D0 /D6/CX/D2/CV/B8 /DB /CT
/CP/D2 /CT/DC/D4/D6/CT/D7/D7 /D8/CW/CT /D6/CT/D7/D8/B9/CU/D6/CP/D1/CT /CU/D6/CT/D5/D9/CT/D2
/CX/CT/D7 /CQ /DD /D8/CW/CT/D8/D9/D2/CT/CJ/BF℄/BM ωx≈ωy=βνy
γL
/B8 /DB/CW/CT/D6/CT L /CX/D7 /D8/CW/CT /D0/CT/D2/CV/D8/CW /D3/CU /D8/CW/CT /D6/CX/D2/CV/BA /CC/CW/CT /D0/D3/D2/CV/CX/D8/D9/CS/CX/D2/CP/D0/D1/D3/D1/CT/D2 /D8/D9/D1 /CX/D7 /D5/D9/CP/D2 /D8/CX/DE/CT/CS /CX/D2 /D9/D2/CX/D8/D7 /D3/CU2π
γL
/B8 /D7/D3ωl=2π√
2γL
/BM
εx=∝an}bracketle{tq∝an}bracketri}htεF
ωx=5/radicalBigg
7200π2N2
16807γLν
/BA /B4/BD/BL/B5/BY /D3/D6 /CP /D6/CX/D2/CV /DB/CX/D8/CWL= 2π /D1 /B8γ= 10 /B8N= 1010/B8ν= 100 π /DB /CT /CV/CT/D8 /CP/D2/CT/D1/CX/D8/D8/CP/D2
/CT /D3/CU≈5.3 /BV/D3/D1/D4/D8/D3/D2 /DB /CP /DA /CT/D0/CT/D2/CV/D8/CW/D7/BA/BG /CC/CW/CT /CV/D6/D3/D9/D2/CS /D7/D8/CP/D8/CT /CP/D7 /CP /BY /CT/D6/D1/CX /D0/CX/D5/D9/CX/CS/C1/D2 /D3/D9/D6
/D3/D2/D7/D8/D6/D9
/D8/CX/D3/D2/B8 /DB /CT /D8/CP
/CX/D8/D0/DD /CP/D7/D7/D9/D1/CT /D8/CW/CP/D8 /D8/CW/CT /D4/CP/D6/D8/CX
/D0/CT/B9/D4/CP/D6/D8/CX
/D0/CT /CX/D2 /D8/CT/D6/CP
/D8/CX/D3/D2/CS/D3 /CT/D7 /D2/D3/D8 /D1/D3 /CS/CX/CU/DD /D8/CW/CT /D4/CP/D6/D8/CX
/D0/CT
/D3/D2 /D8/CT/D2 /D8 /D3/CU /D8/CW/CT /CV/D6/D3/D9/D2/CS /D7/D8/CP/D8/CT/BA /CC/CW/CX/D7
/D3/D6/D6/CT/D7/D4 /D3/D2/CS/D7/D4/D6/CT
/CX/D7/CT/D0/DD /D8/D3 /D8/CW/CT /D2/D3/D8/CX/D3/D2 /D3/CU /CP /BY /CT/D6/D1/CX /D0/CX/D5/D9/CX/CS /B4/CX/D2 /D3/D9/D6
/CP/D7/CT/B8 /CP /CW/CX/CV/CW/D0/DD /CP/D2/CX/D7/D3/D8/D6/D3/D4/CX
/D3/D2/CT/B5/B8 /CX/D2 /DB/CW/CX
/CW /D8/CW/CT /CU/D6/CT/CT /D4/CP/D6/D8/CX
/D0/CT /D7/D4 /CT
/D8/D6/D9/D1 /D7/D1/D3 /D3/D8/CW/D0/DD /CS/CT/CU/D3/D6/D1/D7 /CX/D2 /D8/D3 /D8/CW/CT /D5/D9/CP/D7/CX/B9/D4/CP/D6/D8/CX
/D0/CT /D7/D4 /CT
/D8/D6/D9/D1 /D3/CU /CT/D5/D9/CP/D0 /D4/CP/D6/D8/CX
/D0/CT
/D3/D2 /D8/CT/D2 /D8 /DB/CW/CT/D2 /D8/CW/CT /CX/D2 /D8/CT/D6/CP
/D8/CX/D3/D2 /CX/D7 /D7/DB/CX/D8
/CW/CT/CS/D3/D2 /CP/CS/CX/CP/CQ/CP/D8/CX
/CP/D0/D0/DD /BA /CC/CW/CX/D7 /D2/CP/GN/DA /CT /CP/D7/D7/D9/D1/D4/D8/CX/D3/D2 /D3/CU /D8/CW/CT /CT/DC/CX/D7/D8/CT/D2
/CT /D3/CU /CP /BY /CT/D6/D1/CX /D7/D9/D6/CU/CP
/CT/D1/CP /DD /CQ/D6/CT/CP/CZ /CS/D3 /DB/D2 /CX/CU /DB /CT /D8/CP/CZ /CT /CX/D2 /D8/D3 /CP
/D3/D9/D2 /D8 /D4/CP/D6/D8/CX
/D0/CT/B9/D4/CP/D6/D8/CX
/D0/CT /CX/D2 /D8/CT/D6/CP
/D8/CX/D3/D2/D7/BA /CC/CW/CT/CQ /CT/CW/CP /DA/CX/D3/D6 /D3/CU /CP/D2 /D9/D0/D8/D6/CP/B9
/D3/D0/CS /CQ/D9/D2
/CW /CP/CQ /D3 /DA /CT /D8/D6/CP/D2/D7/CX/D8/CX/D3/D2 /DB /D3/D9/D0/CS /CQ /CT /D3/CU /D7/D4 /CT
/CX/CP/D0 /CX/D2 /D8/CT/D6/CT/D7/D8/CW/CT/D6/CT/B8 /D7/CX/D2
/CT /CX/D8 /CT/DC/CW/CX/CQ/CX/D8/D7 /CP /D2/CT/CV/CP/D8/CX/DA /CT/B9/D1/CP/D7/D7 /CQ /CT/CW/CP /DA/CX/D3/D6 /CX/D2 /D8/CW/CT /D0/D3/D2/CV/CX/D8/D9/CS/CX/D2/CP/D0 /CS/CT/CV/D6/CT/CT /D3/CU/CU/D6/CT/CT/CS/D3/D1/BA/BY /D3/D6 /D8/CW/CT
/CP/D7/CT /D3/CU /D8/CW/CT /D7/DD/D7/D8/CT/D1 /CQ /CT/CX/D2/CV /CQ /CT/D0/D3 /DB /D8/D6/CP/D2/D7/CX/D8/CX/D3/D2/B8 /DB /CT
/CP/D2 /D1/CP/CZ /CT /D8/CW/CT /CU/D3/D0/B9/D0/D3 /DB/CX/D2/CV /D7/CT/D1/CX/B9/D5/D9/CP/D2 /D8/CX/D8/CP/D8/CX/DA /CT /CP/D6/CV/D9/D1/CT/D2 /D8 /CU/D3/D6 /D8/CW/CT /CT/DC/CX/D7/D8/CT/D2
/CT /D3/CU /CP /BY /CT/D6/D1/CX /D0/CX/D5/D9/CX/CS/BM /CC/CW/CT/D4/CP/D6/D8/CX
/D0/CT /CQ /CT/CP/D1 /DB/CX/D0/D0 /CW/CP /DA /CT /CP/D2 /CP /DA /CT/D6/CP/CV/CT /D6/CP/CS/CX/CP/D0 /CS/CX/D1/CT/D2/D7/CX/D3/D2 /CV/CX/DA /CT/D2 /CQ /DD /D8/CW/CT /CT/DC
/D9/D6/D7/CX/D3/D2/D3/CU /CP /D4/CP/D6/D8/CX
/D0/CT /D3/D2 /D8/CW/CT /BY /CT/D6/D1/CX /CT/CS/CV/CT /CX/D2 /D8/CW/CT /D6/CP/CS/CX/CP/D0 /D3/D6 /D8/D6/CP/D2/D7/DA /CT/D6/D7/CT /D3/D7
/CX/D0/D0/CP/D8/D3/D6 /D4 /D3/D8/CT/D2 /D8/CX/CP/D0/BM
me−¯ω2∝an}bracketle{tx∝an}bracketri}ht2≈ǫF
/BA /B4/BE/BC/B5/BI/C1/D2 /CP /AG/C5/CT/CP/D2 /BY/CX/CT/D0/CS/AH
/CP/D0
/D9/D0/CP/D8/CX/D3/D2/B8 /DB /CT /CT/D7/D8/CX/D1/CP/D8/CT /D8/CW/CT /CT/AR/CT
/D8/CX/DA /CT /D8/D6/CP/D2/D7/DA /CT/D6/D7/CT /CU/D3
/D9/D7/CX/D2/CV/D7/D8/D6/CT/D2/CV/D8/CW ¯ω2/D8/D3 /CQ /CT /D8/CW/CT /D7/D9/D1 /D3/CU /D8/CW/CT /CT/DC/D8/CT/D6/D2/CP/D0 /CU/D3
/D9/D7/CX/D2/CV /CP/D2/CS /CP /D7/D4/CP
/CT/B9
/CW/CP/D6/CV/CT /D8/D9/D2/CT/CS/CT/D4/D6/CT/D7/D7/CX/D3/D2 /CS/D9/CT /D8/D3 /CP
/CX/D6
/D9/D0/CP/D6 /CQ /CT/CP/D1 /D3/CU /D6/CP/CS/CX/D9/D7√
x2/BM
ω2=ω2
ext−ω2
sc=ω2
ext−Ne2
γL∝an}bracketle{tx∝an}bracketri}ht2. /B4/BE/BD/B5/CC/CW/CT /CU/CP
/D8 /D8/CW/CP/D8 /D8/CW/CT /D7/DD/D7/D8/CT/D1 /CX/D7 /CX/D2 /D8/CW/CT /CV/D6/D3/D9/D2/CS /D7/D8/CP/D8/CT /CP/D0/D0/D3 /DB/D7 /D3/D2/CT /D8/D3 /CT/D0/CX/D1/CX/D2/CP/D8/CT
ǫF
/BN /CU/D3/D6 /D8/CW/CT
/CP/D7/CT /D3/CU /CP
/D3/CP/D7/D8/CX/D2/CV /CQ /CT/CP/D1/B8 /DB /CT /D3/CQ/D8/CP/CX/D2 /CP
/D3/D2/D7/CX/D7/D8/CT/D2
/DD
/D3/D2/CS/CX/D8/CX/D3/D2/BM
/parenleftbigg¯ω
ωext/parenrightbigg2/bracketleftBigg
1 +/parenleftbigge3N
γLω ext/parenrightbigg2/3/parenleftbigg¯ω
ωext/parenrightbigg−2/3/bracketrightBigg
= 1 /B8 /B4/BE/BE/B5/DB/CW/CX
/CW /CW/CP/D7 /CP /D7/D3/D0/D9/D8/CX/D3/D2¯ω
ωext<1 /CU/D3/D6 /CU/D3/D6 /CP/D0/D0N /BA /CC/CW/CT /D6/CT/D7/D9/D0/D8/CX/D2/CV /CT/AR/CT
/D8/CX/DA /CT /CU/D6/CT/D5/D9/CT/D2
/DD/CX/D7 /D7/CW/D3 /DB/D2 /CX/D2 /BY/CX/CV/BA /BD/BA/BH /BY/CX/D2/CX/D8/CT /CC /CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT/CC/CW/CT /CP/CQ /D3 /DA /CT
/D3/D2/D7/CX/CS/CT/D6/CP/D8/CX/D3/D2/D7 /DB /CT/D6/CT /CU/D3/D6 /D8/CW/CT
/CP/D7/CT /D3/CU /DE/CT/D6/D3 /D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT/BA /CC /D3 /CV/CT/D2/CT/D6/CP/D0/CX/DE/CT/D8/D3 /AS/D2/CX/D8/CT /D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT/D7/B8 /DB /CT /CU/D3/D0/D0/D3 /DB /D8/CW/CT /D9/D7/D9/CP/D0 /D4/D6/CT/D7
/D6/CX/D4/D8/CX/D3/D2 /CP/D2/CS /CX/D2 /D8/D6/D3 /CS/D9
/CT /CP
/CW/CT/D1/B9/CX
/CP/D0 /D4 /D3/D8/CT/D2 /D8/CX/CP/D0/BA /C4/CT/D8/B3/D7 /D8/D6/CT/CP/D8 /D8/CW/CT /CP/D0/D0/B9/D3/D7
/CX/D0/D0/CP/D8/D3/D6
/CP/D7/CT /AS/D6/D7/D8/BN /D8/CW/CT /D5/D9/CP/D2 /D8/CX/D8 /DD /DB /CT /DB /CP/D2 /D8/D8/D3
/CP/D0
/D9/D0/CP/D8/CT /CX/D7 /D8/CW/CT /D0/D3/CV/CP/D6/CX/D8/CW/D1 /D3/CU /D8/CW/CT /D4/CP/D6/D8/CX/D8/CX/D3/D2 /CU/D9/D2
/D8/CX/D3/D2 /D3/CU /D8/CW/CT /CV/D6/CP/D2/CS/B9
/CP/D2/D3/D2/CX
/CP/D0/CT/D2/D7/CT/D1 /CQ/D0/CT/BM
logZ= log/summationdisplay
ni∈{0,1}e−β/summationtext
ini(/summationtextd
k=1ωk(ik+1
2)−µ)
=∞/summationdisplay
ik=0log/parenleftBig
1 +e−β(/summationtextωk(ik+1
2)−µ)/parenrightBig/BA /B4/BE/BF/B5/BT/CV/CP/CX/D2/B8 /DB /CT /D8/D6/CP/D2/D7/CU/D3/D6/D1 /D8/CW/CT /D7/D9/D1 /CX/D2 /D8/D3 /CP/D2 /CX/D2 /D8/CT/CV/D6/CP/D0/BA /CC/CW/CT /D3/D2/D0/DD /D2/D3/D2/B9/D8/D6/CX/DA/CX/CP/D0 /CX/D2/B9/D8/CT/CV/D6/CP/D8/CX/D3/D2 /DB /CT /CW/CP /DA /CT /D8/D3 /CS/D3 /CX/D7 /D8/CW/CT /D3/D2/CT /D4 /CT/D6/D4 /CT/D2/CS/CX
/D9/D0/CP/D6 /D8/D3 /D8/CW/CT /D7/D9/D6/CU/CP
/CT/D7 /D3/CU
/D3/D2/D7/D8/CP/D2 /D8/CT/D2/CT/D6/CV/DD E /BN /D8/CW/CT /CX/D2 /D8/CT/CV/D6/CP/D8/CX/D3/D2 /CX/D2 /CP/D0/D0 /D3/D8/CW/CT/D6 /CS/CX/D6/CT
/D8/CX/D3/D2/D7 /CV/CX/DA /CT/D7 /D8/CW/CT /CP/D6/CT/CPEd−1
(d−1)!
/D3/CU /D8/CW/CP/D8/D7/D9/D6/CU/CP
/CT/BM
logZ=1
Ω(d−1)!/integraldisplay∞
0log/parenleftbig
1 +e−β(E−µ)/parenrightbig
Ed−1dE /B8 /B4/BE/BG/B5/BJ00.20.40.60.81
0 2 4 6 8 10ω2
eff/ω2
ext
ωint/ωext/BY/CX/CV/D9/D6/CT /BD/BM /BX/AR/CT
/D8/CX/DA /CT /CU/D3
/D9/D7/CX/D2/CV /D7/D8/D6/CT/D2/CV/D8/CW /D3/CU /CP
/D3/CP/D7/D8/CX/D2/CV /CQ /CT/CP/D1 /CX/D2 /CX/D8/D7 /CV/D6/D3/D9/D2/CS /D7/D8/CP/D8/CT/BN
ωint=e3N/(γL)/BK/DB/CW/CT/D6/CT /DB /CT /CW/CP /DA /CT /D7/D9/CQ/D8/D6/CP
/D8/CT/CS /D8/CW/CT /DE/CT/D6/D3/B9/D4 /D3/CX/D2 /D8 /CT/D2/CT/D6/CV/DD /BA /CF /CT /CX/D2 /D8/CT/CV/D6/CP/D8/CT /CQ /DD /D4/CP/D6/D8/D7 /BM
logZ=β
Ωd!/integraldisplayµ
0Ed
1 +e−β(E−µ)dE /BA /B4/BE/BH/B5/BY /D3/D6 /D7/D1/CP/D0/D0 /D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT/D7/B8 /CX/D2 /D8/CT/CV/D6/CP/D0/D7 /D3/CU /D8/CW/CX/D7 /D7/D8 /DD/D0/CT
/CP/D2 /CQ /CT /CP/D4/D4/D6/D3 /DC/CX/D1/CP/D8/CT/CS /D9/D7/CX/D2/CV /D8/CW/CT/CB/D3/D1/D1/CT/D6/CU/CT/D0/CS /D8/D6/CX
/CZ/BA /CF /CT /AS/D2/CS
logZ=β
Ωd!/parenleftbiggµd+1
d+ 1+dµd−1π2
6β2+. . ./parenrightbigg/BA /B4/BE/BI/B5/BT/D7
∝an}bracketle{tεi∝an}bracketri}ht=−1
Nβ∂
∂ωilogZ
∝an}bracketle{tE∝an}bracketri}ht=−1
N∂
∂βlogZ
/B8 /B4/BE/BJ/B5/DB /CT
/CP/D2 /DB/D6/CX/D8/CT /D8/CW/CT /D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT/B9/CS/CT/D4 /CT/D2/CS/CT/D2 /D8
/D3/D2 /D8/D6/CX/CQ/D9/D8/CX/D3/D2/D7 /D8/D3 /CT/D1/CX/D8/D8/CP/D2
/CT /CP/D2/CS /CT/D2/B9/CT/D6/CV/DD/BM
∆∝an}bracketle{tE∝an}bracketri}ht=1
Nµd−1π2
6β2Ω(d−1)!=dπ2
6β2d√
ΩNd!
∆∝an}bracketle{tεi∝an}bracketri}ht=1
ω∆∝an}bracketle{tE∝an}bracketri}ht
/B8 /B4/BE/BK/B5/DB/CW/CT/D6/CT /DB /CT /CW/CP /DA /CT /D9/D7/CT/CS /D8/CW/CT /DE/CT/D6/D3/B9/D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT EF
/CP/D7
/CW/CT/D1/CX
/CP/D0 /D4 /D3/D8/CT/D2 /D8/CX/CP/D0/BA/BI /C4/CX/D1/CX/D8/CP/D8/CX/D3/D2/D7 /CP/D2/CS /C8/D6/D3/D7/D4 /CT
/D8/D7/CB/D3 /CU/CP/D6/B8 /DB /CT /CW/CP /DA /CT
/D3/D2/D7/CX/CS/CT/D6/CT/CS /D8 /DB /D3 /D0/CX/D1/CX/D8/CX/D2/CV
/CP/D7/CT/D7 /D3/CU /CP
/CX/D6
/D9/D0/CP/D6 /D7/CT/D8/D9/D4/BM /BT/D2 /CX/D2/AS/D2/CX/D8/CT/D0/DD/CT/DC/D8/CT/D2/CS/CX/D2/CV /CW/CP/D6/D1/D3/D2/CX
/D4 /D3/D8/CT/D2 /D8/CX/CP/D0 /CP/D2/CS /CP /CU/D6/CT/CT /D4/CP/D6/D8/CX
/D0/CT /D7/D9/CQ /CY/CT
/D8 /D8/D3 /D4 /CT/D6/CX/D3 /CS/CX
/CQ /D3/D9/D2/CS/CP/D6/DD
/D3/D2/CS/CX/D8/CX/D3/D2/D7/BA /CF/CW/CX/D0/CT /D8/CW/CX/D7 /CX/D7 /CP /D6/CT/CP/D0/CX/D7/D8/CX
/CP/D4/D4/D6/D3/CP
/CW /CU/D3/D6 /D8/CW/CT /D8/D6/CP/D2/D7/DA /CT/D6/D7/CT /CS/CT/CV/D6/CT/CT/D7 /D3/CU/CU/D6/CT/CT/CS/D3/D1/B8 /D8/CW/CT /D0/D3/D2/CV/CX/D8/D9/CS/CX/D2/CP/D0 /CS/DD/D2/CP/D1/CX
/D7 /CX/D7 /D1/D3/D6/CT
/D3/D1/D4/D0/CT/DC/BA /C1/D2 /D8/CW/CT
/CP/D7/CT /D3/CU /CW/CX/CV/CW/D0/D3/D2/CV/CX/D8/D9/CS/CX/D2/CP/D0 /CS/CT/D2/D7/CX/D8/CX/CT/D7/B8 /D8 /DB /D3 /CU/CP
/D8/D3/D6/D7 /D0/CX/D1/CX/D8 /D3/D9/D6 /D1/D3 /CS/CT/D0/BM/BD/BA /CC/CW/CT /D0/D3/D2/CV/CX/D8/D9/CS/CX/D2/CP/D0 /CQ/D9
/CZ /CT/D8 /CX/D7 /CP/D2/CW/CP/D6/D1/D3/D2/CX
/CP/D2/CS /D0/CX/D1/CX/D8/CT/CS /CU/D6/D3/D1 /CP/CQ /D3 /DA /CT/BA /CF/CW/CX/D0/CT/D8/CW/CT /CU/D3/D6/D1/CT/D6 /CU/CP
/D8 /CX/D7 /D4/D6/D3/CQ/CP/CQ/D0/DD /CQ /CT/D2/CX/CV/D2/B8 /D8/CW/CT /D0/CP/D8/D8/CT/D6 /D4 /D3/D7/CT/D7 /D8/CW/CT /D5/D9/CT/D7/D8/CX/D3/D2 /D3/CU /CW/D3 /DB/D8/CW/CT /CX/D2
/D6/CT/CP/D7/CX/D2/CV /D7/D4 /CT
/D8/D6/CP/D0 /CS/CT/D2/D7/CX/D8 /DD /CP/D8 /D8/CW/CT /D9/D4/D4 /CT/D6 /CQ /D3/D9/D2/CS/CP/D6/DD /D3/CU /D8/CW/CT /CQ/D9
/CZ /CT/D8/BL/CP/AR/CT
/D8/D7 /D8/D6/CP/D2/D7/DA /CT/D6/D7/CP/D0 /CT/D1/CX/D8/D8/CP/D2
/CT /CX/D2 /D8/CW/CT
/CP/D7/CT /D3/CU /CP/D2 /CP/D0/D1/D3/D7/D8 /CU/D9/D0/D0 /CA/BY /CQ/D9
/CZ /CT/D8/BN/D8/CW/CX/D7 /CW/CP/D7 /D8/D3 /CQ /CT /CU/D9/D6/D8/CW/CT/D6 /CX/D2 /DA /CT/D7/D8/CX/CV/CP/D8/CT/CS/BA /BT/D2 /CP/D4/D4/D6/D3/D4/D6/CX/CP/D8/CT /CP/D4/D4/D6/D3/CP
/CW /DB /D3/D9/D0/CS /CQ /CT/D8/D3 /D8/D6/CT/CP/D8 /D8/CW/CT /D4 /CT/D6/CX/D3 /CS/CX
/CW/CP/CX/D2 /D3/CU /CA/BY /CQ/D9
/CZ /CT/D8 /CP/D7 /CP /D4 /CT/D6/CX/D3 /CS/CX
/D4 /D3/D8/CT/D2 /D8/CX/CP/D0/B8 /D7/D3 /D8/CW/CT/D0/D3/D2/CV/CX/D8/D9/CS/CX/D2/CP/D0 /CT/CX/CV/CT/D2/CU/D9/D2
/D8/CX/D3/D2 /DB /D3/D9/D0/CS /CQ /CT /BU/D0/D3
/CW /CU/D9/D2
/D8/CX/D3/D2/D7/BA/BE/BA /CC/CW/CT /D4/CP/D6/D8/CX
/D0/CT/B9/D4/CP/D6/D8/CX
/D0/CT /CX/D2 /D8/CT/D6/CP
/D8/CX/D3/D2 /CP/AR/CT
/D8/D7 /D8/CW/CT /D7/D8/CX/AR/D2/CT/D7/D7 /D3/CU /D8/CW/CT /D8/D6/CP/D2/D7/DA /CT/D6/D7/CT/D3/D7
/CX/D0/D0/CP/D8/D3/D6/D7/BA /C1/D2 /D8/CW/CT /D0/CX/D1/CX/D8/CX/D2/CV
/CP/D7/CT /D3/CU /DE/CT/D6/D3 /D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT /CP/D2/CS /D7/D9Ꜷ
/CX/CT/D2 /D8/D0/DD/D7/D8/D6/D3/D2/CV /CU/D3
/D9/D7/CX/D2/CV /CU/D3/D6
/CT/D7 /CX/D2 /CP /D7/D1/D3 /D3/D8/CW /D0/CP/D8/D8/CX
/CT/B8 /D8/CW/CT /CV/D6/D3/D9/D2/CS /D7/D8/CP/D8/CT /CX/D7 /CQ /CT/D0/CX/CT/DA /CT/CS/D8/D3 /CQ /CT /CP
/D6/DD/D7/D8/CP/D0/D0/CX/D2/CT /D7/D8/CP/D8/CT/BA /CC/CW/CT /D1/D3/D7/D8 /D7/CX/D1/D4/D0/CT /D6/CT/CP/D0/CX/DE/CP/D8/CX/D3/D2 /D3/CU /CP
/D6/DD/D7/D8/CP/D0/D0/CX/D2/CT/D7/D8/CP/D8/CT /DB /D3/D9/D0/CS /CQ /CT /CP /D3/D2/CT/B9/CS/CX/D1/CT/D2/D7/CX/D3/D2/CP/D0 /CT/D0/CT
/D8/D6/D3/D2
/CW/CP/CX/D2/B8 /DB/CW/CX
/CW /CX/D7 /CZ/D2/D3 /DB/D2 /CP/D7 /CP/BD/B9/BW /CF/CX/CV/D2/CT/D6
/D6/DD/D7/D8/CP/D0/CJ/BG℄ /CX/D2 /D8/CW/CT
/D3/D2 /D8/CT/DC/D8 /D3/CU /D7/D3/D0/CX/CS /D7/D8/CP/D8/CT /D4/CW /DD/D7/CX
/D7 /B4/CX/D2 /CP /CF/CX/CV/D2/CT/D6
/D6/DD/D7/D8/CP/D0/B8 /D8/CW/CT /D2/CT/D9/D8/D6/CP/D0/CX/DE/CX/D2/CV /AS/CT/D0/CS /CX/D7 /D4/D6/D3 /DA/CX/CS/CT/CS /CQ /DD /D8/CW/CT /CX/D3/D2/D7 /D3/CU /D8/CW/CT
/D6/DD/D7/D8/CP/D0/D0/CP/D8/D8/CX
/CT/B8 /DB/CW/CX
/CW /CP/D6/CT /AG/D7/D1/CT/CP/D6/CT/CS /D3/D9/D8/AH /CW/D3/D1/D3/CV/CT/D2/CT/D3/D9/D7/D0/DD /D8/D3 /CU/D3/D6/D1 /D8/CW/CT /AG/CY/CT/D0/D0/CX/D9/D1/AH/B8/DB/CW/CT/D6/CT/CP/D7 /CX/D2 /D3/D9/D6
/CP/D7/CT /D7/D8/CP/CQ/CX/D0/CX/D8 /DD /CX/D7 /CP
/CW/CX/CT/DA /CT/CS /CQ /DD /CT/DC/D8/CT/D6/D2/CP/D0 /CU/D3
/D9/D7/CX/D2/CV /CT/D0/CT/D1/CT/D2 /D8/D7/B5/BA/CC/CW/CX/D7
/CP/D7/CT /CX/D7 /CW/CX/CV/CW/D0/DD /CS/CT/CV/CT/D2/CT/D6/CP/D8/CT/BM /DB /CT /CT/DC/D4 /CT
/D8 /D8/CW/CT /CT/D0/CT
/D8/D6/D3/D2/D7 /D8/D3 /CQ /CT /CX/D2 /DB /CT/D0/D0/B9/D0/D3
/CP/D0/CX/DE/CT/CS /D7/D8/CP/D8/CT/D7 /DB/CX/D8/CW /CT/D5/D9/CP/D0 /D4 /D3/D8/CT/D2 /D8/CX/CP/D0 /CT/D2/CT/D6/CV/DD /B8 /D8/CW/CT /D3/D2/D0/DD /D5/D9/CP/D2 /D8/D9/D1 /CT/AR/CT
/D8/CQ /CT/CX/D2/CV /D8/CW/CT /D4/CW/D3/D2/D3/D2/CX
/D3/D7
/CX/D0/D0/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /CT/D0/CT
/D8/D6/D3/D2 /D0/CP/D8/D8/CX
/CT/CJ/BH /B8 /BI/B8 /BJ ℄/BN
/D3/D2/D7/CT/B9/D5/D9/CT/D2 /D8/D0/DD /B8 /D8/CW/CT /D8/D6/CP/D2/D7/DA /CT/D6/D7/CT /CT/D1/CX/D8/D8/CP/D2
/CT
/CP/D2 /CQ /CT /CP/D8 /CX/D8/D7 /D5/D9/CP/D2 /D8/D9/D1/B9/D1/CT
/CW/CP/D2/CX
/CP/D0/D1/CX/D2/CX/D1 /D9/D1 /D3/CU /BD /BV/D3/D1/D4/D8/D3/D2 /DB /CP /DA /CT/D0/CT/D2/CV/D8/CW/BA /C5/D3 /CS/CT/D0/CX/D2/CV /D8/CW/CT
/D6/D3/D7/D7/D3 /DA /CT/D6 /CQ /CT/CW/CP /DA/CX/D3/D6/CU/D6/D3/D1 /CU/D6/CT/CT/B9/D4/CP/D6/D8/CX
/D0/CT /CT/CX/CV/CT/D2/D7/D8/CP/D8/CT/D7 /CP/D7 /CX/D2 /D8/CW/CX/D7 /D4/CP/D4 /CT/D6 /D8/D3 /D0/D3
/CP/D0/CX/DE/CT/CS /D7/D8/CP/D8/CT/D7 /CP/D2/CS /D8/CW/CT/D8/D6/CP/D2/D7/CX/D8/CX/D3/D2 /CU/D6/D3/D1 /AS/D2/CX/D8/CT /D8/D3 /DE/CT/D6/D3 /CT/D1/CX/D8/D8/CP/D2
/CT /D6/CT/D5/D9/CX/D6/CT/D7 /CU/D9/D6/D8/CW/CT/D6 /CX/D2 /DA /CT/D7/D8/CX/CV/CP/D8/CX/D3/D2/BA/BJ /BT
/CZ/D2/D3 /DB/D0/CT/CS/CV/D1/CT/D2 /D8/D7/C1 /DB/CX/D7/CW /D8/D3 /D8/CW/CP/D2/CZ /CA/BA /CA/D9/D8/CW /CP/D2/CS /C5/BA /CE /CT/D2 /D8/D9/D6/CX/D2/CX /CU/D3/D6 /D9/D7/CT/CU/D9/D0 /CS/CX/D7
/D9/D7/D7/CX/D3/D2/D7/BA/CA/CT/CU/CT/D6/CT/D2
/CT/D7/CJ/BD℄ /C2/BA /CF /CT/CX/B8 /CG/CX/CP/D3/B9/C8/CX/D2/CV /C4/CX/B8 /CP/D2/CS /BT/D2/CS/D6/CT/DB /C5/BA /CB/CT/D7/D7/D0/CT/D6/BA /BV/D6/DD/D7/D8/CP/D0/D0/CX/D2/CT /CQ /CT/CP/D1 /CV/D6/D3/D9/D2/CS/D7/D8/CP/D8/CT/BA /BU/C6/C4 /C8/D6 /CT/D4/D6/CX/D2/D8/B8 /BU/C6/C4/B9/BH/BE/BF/BK/BD/B8 /CD/BV/B9/BG/BD/BG /B8 /BD/BL/BL/BF/BA/CJ/BE℄ /BT/BA /BT/BA /C5/CX/CZ/CW/CP/CX/D0/CX
/CW/CT/D2/CZ /D3/BA /CC /D3 /D8/CW/CT /D5/D9/CP/D2 /D8/D9/D1 /D0/CX/D1/CX/D8/CP/D8/CX/D3/D2/D7 /CX/D2 /CQ /CT/CP/D1 /D4/CW /DD/D7/CX
/D7/BA /C1/D2/C8/D6 /D3
/CT /CT /CS/CX/D2/CV/D7 /D3/CU /D8/CW/CT /BD/BL/BL/BL /C8/CP/D6/D8/CX
/D0/CT /BT
/CT/D0/CT/D6 /CP/D8/D3/D6 /BV/D3/D2/CU/CT/D6 /CT/D2
/CT/B8 /C6/CT/DB /CH /D3/D6/CZ /B8 /BD/BL/BL/BL/BA/CJ/BF℄ /C5/BA /CE /CT/D2 /D8/D9/D6/CX/D2/CX /CP/D2/CS /CA/BA /CA/D9/D8/CW/BA /CB/CX/D2/CV/D0/CT /D4/CP/D6/D8/CX
/D0/CT /D5/D9/CP/D2 /D8/D9/D1 /CS/DD/D2/CP/D1/CX
/D7 /CX/D2 /CP /D1/CP/CV/B9/D2/CT/D8/CX
/D0/CP/D8/D8/CX
/CT/BA /C1/D2 /C8/D6 /D3
/CT /CT /CS/CX/D2/CV/D7 /D3/CU /D8/CW/CT /BV/D3/D2/CU/CT/D6 /CT/D2
/CT /D3/D2 /C9/D9/CP/D2/D8/D9/D1 /BT/D7/D4 /CT
/D8/D7 /D3/CU/BU/CT /CP/D1 /C8/CW/DD/D7/CX
/D7/B8 /BV/CP/D4/D6/CX/B8 /C1/D8/CP/D0/DD/B8 /BD/BH/B9/BE/BC /C7
/D8/D3/CQ /CT/D6 /BE/BC/BC/BC /B8 /BE/BC/BC/BC/BA/BD/BC/CJ/BG℄ /BX/BA /CF/CX/CV/D2/CT/D6/BA /C8/CW/DD/D7/BA /CA /CT/DA /B8 /BG/BI/BM/BD/BC/BC/BE/B8 /BD/BL/BF/BG/BA/CJ/BH℄ /C0/BA /C2/BA /CB
/CW /D9/D0/DE/BA /CF/CX/CV/D2/CT/D6
/D6/DD/D7/D8/CP/D0 /CX/D2 /D3/D2/CT /CS/CX/D1/CT/D2/D7/CX/D3/D2/BA /C8/CW/DD/D7/BA /CA /CT/DA/BA /C4 /CT/D8/D8/CT/D6/D7 /B8/BJ/BD/BM/BD/BK/BI/BG/B8 /BD/BL/BL/BE/BA/CJ/BI℄ /C0/BA /C2/BA /CB
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/D6/DD/D7/D8/CP/D0 /CX/D2 /D3/D2/CT /CS/CX/D1/CT/D2/D7/CX/D3/D2/BA /C4 /D3/D7 /BT /D0/CP/D1/D3/D7 /BX/D0/CT
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/D3/D9/D0/D3/D1 /CQ
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/D9/D7/CX/D2/CV /AS/CT/D0/CS/BA/C1/D2 /C8/D6 /D3
/CT /CT /CS/CX/D2/CV/D7 /D3/CU /D8/CW/CT /BV/D3/D2/CU/CT/D6 /CT/D2
/CT /D3/D2 /C9/D9/CP/D2/D8/D9/D1 /BT/D7/D4 /CT
/D8/D7 /D3/CU /BU/CT /CP/D1 /C8/CW/DD/D7/CX
/D7/B8/BV/CP/D4/D6/CX/B8 /C1/D8/CP/D0/DD/B8 /BD/BH/B9/BE/BC /C7
/D8/D3/CQ /CT/D6 /BE/BC/BC/BC /B8 /BE/BC/BC/BC/BA /B4/D7/D9/CQ/D1/CX/D8/D8/CT/CS/B5/BA/BD/BD |
arXiv:physics/0102065v1 [physics.plasm-ph] 21 Feb 2001Strong “quantum” chaos in the global ballooning mode spectr um of three-dimensional
plasmas
R. L. Dewar∗
Princeton University Plasma Physics Laboratory, P.O. Box 4 51, Princeton N.J. 08543
P. Cuthbert and R. Ball
Department of Theoretical Physics and Plasma Research Labo ratory, Research School of Physical Sciences & Engineering , The
Australian National University, Canberra 0200 Australia
(July 24, 2013)
The spectrum of ideal magnetohydrodynamic (MHD)
pressure-driven (ballooning) modes in strongly nonaxisym -
metric toroidal systems is difficult to analyze numerically o w-
ing to the singular nature of ideal MHD caused by lack of
an inherent scale length. In this paper, ideal MHD is regu-
larized by using a k-space cutoff, making the ray tracing for
the WKB ballooning formalism a chaotic Hamiltonian bil-
liard problem. The minimum width of the toroidal Fourier
spectrum needed for resolving toroidally localized balloo ning
modes with a global eigenvalue code is estimated from the
Weyl formula. This phase-space-volume estimation method
is applied to two stellarator cases.
PACS numbers: 52.35.Py, 52.55.Hc, 05.45.Mt
In design studies for new magnetic confinement devices
for fusion plasma experiments (e.g. investigations [1,2]
leading to the proposed National Compact Stellarator
Experiment, NCSX [3]), the maximum pressure that can
stably be confined in any proposed magnetic field con-
figuration is routinely estimated by treating the plasma
as an ideal magnetohydrodynamic (MHD) fluid. One
linearizes about a sequence of equilibrium states with in-
creasing pressure, and studies the spectrum of normal
modes (frequency ω) to determine when there is a com-
ponent with Im ω>0, signifying instability.
Even with the simplification obtained by using the
ideal MHD model, the computational task of determining
the theoretical stability of a three-dimensional (i.e. non -
axisymmetric) device, such as NCSX or the four currently
operating helical axis stellators [4], remains a challengi ng
one.
The problem can be posed as a Lagrangian field the-
ory, with the potential term being the energy functional
δW[5]. For a static equilibrium, the kinetic energy is
quadratic in ω, so thatω2is real. Thus instability oc-
curs when ω2<0. There are two main approaches to
analyzing the spectrum—local and global.
∗Permanent address: Research School of Physical Sciences
& Engineering, The Australian National University. E-mail :
robert.dewar@anu.edu.au.In the localapproach, which is used for analytical sim-
plification, one orders the scale length of variation of the
eigenfunction across the magnetic field lines to be short
compared with equilibrium scale lengths [6]. Both inter-
change and ballooning stability can be treated by solving
the general ballooning equations [7], a system of ordinary
differential equations defined on a given magnetic field
line.
Theglobal (Galerkin) approach is to expand the
plasma displacement field in a finite basis set, inserting
this ansatz in the Lagrangian to find a matrix eigen-
value representation of the spectral problem. This ap-
proach has been implemented for ideal MHD in three-
dimensional plasmas in two codes, TERPSICHORE [8]
and CAS3D [9].
Although the Galerkin approach is potentially exact, if
one could use a complete, infinite basis set, it is in prac-
tice computationally challenging due to the large number
of basis functions required to resolve localized instabili -
ties. This leads to very large matrices which must be
diagonalized by iterative methods. There is a need for
analytical insight to determine a suitable truncated ba-
sis set and to predict the nature of the spectrum, e.g.
whether it is continuous or discrete.
Such insight may be obtained by a hybrid local-
global approach, in which one uses a Wentzel–Kramers–
Brillouin (WKB) representation of the eigenfunction. In
the short-wavelength limit, the same analytical simplifi-
cations as are obtained in the local approach are found to
give a local dispersion relation that can be used to give
information on the global spectrum by using ray tracing
and semiclassical quantization.
In axisymmetric systems [10] or in cases where helical
ripple can be averaged out, giving an adiabatic invariant,
[11,12], the ray equations are integrable and hence the
spectrum is characterized by “good quantum numbers”.
However, it has been known for many years [7] that the
ray-tracing problem in strongly three-dimensional sys-
tems is singular because, in the absence of an adiabatic
invariant, the phase-space motion is not bounded—the
rays escape to infinity in the wavevector sector. Dewar
and Glasser [7] argued that this gives rise to a contin-
uous unstable spectrum, with correspondingly singular
generalized eigenfunctions. (A more rigorous treatment
1involves the concept of the essential spectrum and Weyl
sequences [13,14].)
Our proposed regularization of this singularity can be
understood using a simple quantum analogy. Consider
the one-dimensional time-independent Schr¨ odinger equa-
tionHψ=Eψin the limit as the mass of the par-
ticle goes to infinity. Then the kinetic energy disap-
pears and the Hamiltonian becomes H=V(x), where
Vis the potential energy, assumed here to be the har-
monic oscillator potential,1
2x2in suitable units. In the
usual Hilbert space the energy spectrum is continuous:
E≥0 and the (generalized) eigenfunctions singular:
ψ(x) =δ(x−xE)±δ(x+xE), whereV(xE)≡E.
We now seek a regularization of this problem by re-
strictingψto the space of functions with a finite band-
width in wavenumber k:
ψ(x) =/integraldisplaykmax
−kmaxdk
2πψkexpikx. (1)
This truncated Fourier-integral representation models
what occurs when one seeks to find the spectrum numer-
ically using a truncated Fourier-series representation.
We take as starting point a Lagrangian for the wave-
function,
L=/integraldisplay∞
−∞ψ∗[E−V(x)]ψdx. (2)
Inserting Eq. (1) in Eq. (2) gives
L=/integraldisplaykmax+0
−kmax−0/bracketleftbigg
E|ψk|2−/vextendsingle/vextendsingle/vextendsingle/vextendsingledψk
dk+ψkδ(k+kmax)
−ψkδ(k−kmax)|2/bracketrightBigdk
2π.(3)
This is infinite unless we require the coefficients of the
δ-functions to vanish. That is, ψk= 0 atk=±kmax.
The Euler–Lagrange equation is ( d2/2dk2+E)ψk= 0,
which has the solutions exp ±i(2E)1/2k. These waves
would propagate to infinity if it were not for the reflecting
boundary conditions at ±kmaxwe have just derived.
That is, we have removed the continuum by box quan-
tization ink-space. In the following we shall do the same
for the ballooning mode problem.
As in [7] we write the magnetic field of an arbi-
trary three-dimensional toroidal equilibrium plasma with
nested magnetic flux surfaces labeled by an arbitrary pa-
rametersasB=∇ζ×∇ψ−q∇θ×∇ψ≡∇α×∇ψ,
whereα≡ζ−qθ. Here,θandζare the poloidal and
toroidal angles, respectively, ψ(s) is the poloidal flux
function, and q(s) is the inverse of the rotational trans-
form. Since B·∇s=B·∇α= 0,sandαserve to label
an individual field line.
We take the stream function [6] to be given by ϕ=
/hatwideϕexp(iS−iωt),where /hatwideϕ(θ|s,α) is assumed to vary on
the equilibrium scale. The phase variation is taken to berapid, so k≡∇Sis ordered to be large. The frequency
ωis orderedO(1), which requires that the wave vector be
perpendicular to B:k·B≡0. (In this study we consider
unstable ideal MHD modes, ω2<0.)
It immediately follows that the eikonal is constant on
each field line: S=S(α,s). From the definition of the
wave vector, k=kα∇α+ks∇s≡kα[∇α+θkq′(s)∇s]
wherekα≡∂S/∂α andks≡∂S/∂s . Here the anglelike
ballooning parameter θkappears naturally as the ratio
ks/q′(s)kα[10].
The ballooning equation emerges in the large |k|ex-
pansion [7,6] as an ordinary differential equation to be
solved on each field line ( α,s) with given ( kα,ks) under
the boundary condition /hatwideϕ(θ)→0 at infinity to give the
eigenvalueλ(α,s,k α,ks). This constitutes a local disper-
sion relation λ≡ρω2(the mass density ρbeing assumed
constant everywhere).
The ray equations are the characteristics of the eikonal
equationλ(α,s,∂ αS,∂sS) =ρω2. These are Hamiltonian
equations of motion with α,sthe generalized coordinates,
kα,ksthe canonically conjugate momenta, and λas the
Hamiltonian.
In axi- or helically symmetric systems all field lines on
a given magnetic surface are equivalent— αis ignorable
andkαis a constant of the motion. In this case the equa-
tions are integrable and semiclassical quantization can be
used to predict the approximate spectrum of global bal-
looning instabilities [10]. This technique can sometimes
be applied successfully, even in nonsymmetric systems, if
there are regions of phase space with a large measure of
invariant tori [15,11]. In [11] this was verified using the
global eigenvalue code TERPSICHORE [8].
At the other extreme, if the ray orbits are chaotic (but
still bounded) then the global spectrum is not regularly
structured, but must rather be described statistically by
the density of states and the probability distribution of
level spacings using the techniques of quantum chaos the-
ory (see e.g. [16,17]).
However, because of the scale invariance of the ideal
MHD equations, λdepends only on the direction of k,
noton its magnitude: λ=λ(α,s,θ k). This has the
consequence that the ray orbits are unbounded in phase
space, so, strictly speaking, ideal MHD gives rise to a
quantum chaotic scattering [16,17] problem rather than
a straight quantum chaos problem. This leads to the con-
tinuous spectrum [7] with singular generalized eigenfunc-
tions that cannot really be represented using the simple
eikonal ansatz.
On the other hand, the absence of a natural length
scale in ideal MHD is a mathematical artifact. Phys-
ically, the ion Larmor radius provides a lower cutoff in
space, or an upper cutoff in |k|, beyond which ideal MHD
ceases to apply. The ballooning equation is also physi-
cally regularized by inclusion of diamagnetic drift [18,15 ].
However, since in general it leads to a complex ray
tracing problem [19], we shall not attempt to model dia-
2magnetic drift stabilization in this paper. Rather, we
regularize the ray equations simply by adding a barrier
term to the effective ray “Hamiltonian” H(α,s,k α,ks),
H=λ(α,s,k α,ks) +U(kα), (4)
where the barrier potential we use is U(kα)≡K(|kα| −
kmax)2for|kα|>kmaxand 0 for |kα|<kmax. In the limit
of the constant K→ ∞, this infinite box potential gives
the ideal MHD ray equations for |kα|<kmaxand reflect-
ing boundary conditions at |kα|=kmax. Thus we have a
two-degree of freedom Hamiltonian billiard problem.
Although overly crude for modeling FLR stabilization,
the cutoff at |kα|=kmaxprovides a reasonable model for
representing the finite spectral bandwidth in the toroidal
Fourier mode number ( n) representation used in the
global eigenvalue codes TERPSICHORE [8] and CAS3D
[9].
−0.075 0 0.075
α0.8860.8880.890.8920.894q
02468101214
θk−0.15−0.1−0.0500.050.10.15
α
FIG. 1. The sections θk= 0 and q= 0.893 of the topo-
logically spherical isosurfaces of the central, (0,0), bal looning
mode branch, bounded by the isosurface λ=−6 (arbitrary
units). The darker shades denote higher growth rates, the
peak corresponding to λ≈ −8.
Using ballooning-unstable plasma equilibria calculated
for the H-1NF heliac [20,4] using the VMEC code [21],
detailed parameter scans have been undertaken for two
cases. The first case studied [22] was obtained by increas-
ing the pressure gradient of a marginally stable equilib-
rium [23] uniformly across the plasma and thus was bal-
looning unstable at the edge of the plasma. The ray
tracing problem for this case would involve consideration
of the effect of the plasma boundary.
Thus a second equilibrium, ballooning stable near the
edge of the plasma, was calculated for the purposes of
the present paper. This case has a more peaked pressure
profile than the first, but both have average β≈1%,
whereβis the ratio of plasma pressure to magnetic field
pressure.
Theq-profiles are not monotonic—in the peaked pres-
sure profile case studied in this paper, qwas 0.8895 on
the magnetic axis, rising to a maximum value of 0.8964
quite close to the magnetic axis, then falling monoton-
ically to 0.8675. Clearly the (global) magnetic shear is
very weak. Despite this fact and the non-monotonicity,
there is some formal simplification in choosing s≡q, andwe have taken s=qsince the region of plasma studied
is in a monotonic-decreasing part of the q-profile (the
decreasing region outside the maximum- qsurface).
In these scans the most unstable ballooning eigen-
value was tabulated on a three-dimensional grid in s,α,θ k
space. The dependence on αwas found to be rapid. The
dependence on θkwas much slower, but the variation was
sufficient that the higher-growth-rate isosurfaces formed
a set of distinct, topologically spherical branches. It was
argued in [22] that this branch structure is produced by
Anderson localization in bad curvature regions due to
the strong breaking of both helical and axisymmetry in
H-1NF.
According to the perturbation expansion in q′de-
scribed in [22], a quadratic form in α,θkshould form a
good approximation to λ−λmin(q) in the neighborhood
of the central branch. Accordingly a least-squares fit on
each surface was performed to provide a simple analytical
description of the (0 ,0) [22] branch.
The radial dependence of the fitting coefficients was
approximated by fitting to third-degree polynomials in
q. Sections of the resulting approximation to the cen-
tral branch are shown in Fig. 1. The isosurface spans a
substantial range of magnetic surfaces within the plasma
— the narrow range of variation in qis due to the low
magnetic shear in H-1NF.
In order to establish the nature of the ray dynamics de-
scribed by the regularized Hamiltonian, Eq. (4), a numer-
ical integration with cutoff at kmax= 50 was performed
with initial conditions q=q2,α= 0, andkα= 5, where
[q1,q2] = [0.8852,0.8951] is the q-range spanned by the
λ=−6 isosurface as seen in Fig. 1. (A run with kα= 10
was also performed, with similar results.) Choosing the
valueK= 1 gave a good compromise between the sharp
boundary potential to be modeled, and the smooth po-
tential required for the numerical integration. The orbit
remained on the “energy shell” λ=−6 to within an ac-
curacy of one part in 106over the “time” interval of the
integration, 7500.
0.8860.888 0.8920.894q
-400-200200400kq
0.8860.888 0.8920.894q1020304050kα
FIG. 2. Two views of intersections with the Poincar´ e sur-
face of section α= 0.
The two Poincar´ e plots in Fig. 2 show the orbit to be
strongly chaotic, filling the “energy shell” ergodically, e x-
cept that the regions kα>0 andkα<0 are dynamically
disjoint. The solid curve shown surrounding the outer
limits of the “energetically accessible” region is calcu-
lated by solving λ(0,q,k q/kmax) =−6.
3According to the Weyl formula [16, pp. 257–261],
the number, N(λmax), of global eigenmodes with eigen-
values below the eigenvalue λmaxis given, asymp-
totically in the limit N→ ∞ , asN(λmax)∼
v4D(λmax)/(2π)2. Herev4D(λmax) is the volume of the
dynamically acessible 4-dimensional phase-space region
λ(α,q,k q/kα)< λmax, 0< kα< kmax. Thekαintegra-
tion can be performed analytically, giving v4D(λmax) =
1
2k2
maxv3D(λmax),wherev3D(λmax) is the volume within
the isosurface λ(α,q,θ k) =λmax. Thus
N(λmax)∼1
8π2k2
maxv3D(λmax). (5)
We can make a rather rough estimate of the minimum
value ofnmaxrequired for CAS3D or TERPSICHORE
to find even one eigenvalue with λ < λ maxby setting
N(λmax) = 1 and calculating kmax≈nmaxfrom Eq. (5).
This givesnmax(N= 1)∼(8π2/v3D)1/2.
The isosurface λ=−6 studied above is about the
largest of the disjoint topologically spherical isosurfac es
corresponding to the highly toroidally localized strongly
ballooning unstable regions of α,q,θ kspace. (For λ>−6
the isosurfaces are no longer topologically spherical.) Us -
ing the polynomial fits described above, we calculate
v3D(−6) = 0.02158. This gives nmax(N= 1)≈60. As-
suming that the dominant contributions to the MHD en-
ergyδWcome from the rational surfaces intersecting the
λ=−6 isosurface, we thus predict that it would be neces-
sary to include, as a minimum set, basis functions corre-
sponding to one of the two “mode families” [9] contained
in the set ( n,m) = (9,8), (18,16), (19,17), (27,24),
(28,25), (35,31), (36,32), (37,33), (38,34), (44,39),
(45,40), (46,41), (47,42), (53,47), (54,48), (55,49,),
(56,50), and (57 ,51) to resolve a toroidally localized bal-
looning mode. (Here n,mare the toroidal and poloidal
Fourier mode numbers, respectively.)
The large value of nmax(N= 1) required, and the un-
usual spread in nrequired in the basis set, will make
these modes difficult to resolve using global eigenvalue
codes (e.g. the simplifying phase factor method some-
times used in CAS3D studies [1] would not be appropri-
ate). It is hoped that the Weyl formula estimate above
will act as a guide in a future more extensive study us-
ing such a code. Physically, the large value of nmaxsug-
gests that toroidally localized ballooning modes in H-1NF
should be subject to strong FLR stabilization.
We can also apply the same approach to the toroidally
localized ballooning branches found in the Large Heli-
cal Device (LHD) study [12]. From the plots in [12] we
estimatev3D∼0.05, which gives nmax(N= 1)≈40.
The ballooning calculations were carried out on the
Australian National University Supercomputer Facility’s
Fujitsu VPP300 vector processor. We thank Dr. H. J.
Gardner for providing the H-1 heliac VMEC input files
and Dr. S. P. Hirshman for use of the VMEC equilibrium
code. Some of this work was done while one of us (RLD)was a visiting scientist at Princeton University Plasma
Physics Laboratory, supported under US DOE contract
No. DE-AC02-76CH0-3703. Useful conversations with
Drs. M. Redi and A.H. Boozer are gratefully acknowl-
edged.
[1] A. H. Reiman et al., Plasma Physics Reports 23, 472
(1997).
[2] A. Reiman et al., Plasma Phys. Control. Fusion 41, B273
(1999).
[3] G. H. Nielson et al., Phys. Plasmas 7, 1911 (2000).
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[invited paper, to be published in Phys. Plasmas (2001)].
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[9] C. Schwab, Phys. Fluids B 5, 3195 (1993).
[10] R. L. Dewar, J. Manickam, R. C. Grimm, and M. S.
Chance, Nucl. Fusion 21, 493 (1981), corrigendum: Nucl.
Fusion, 22 (1982) 307.
[11] W. A. Cooper, D. B. Singleton, and R. L. Dewar, Phys.
Plasmas 3, 275 (1996), erratum: Phys. Plasmas 3, 3520
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[12] P. Cuthbert et al., Phys. Plasmas 5, 2921 (1998).
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ory(Kluwer, Dordrecht, The Netherlands, 1989), pp.
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4 |
arXiv:physics/0102066v1 [physics.gen-ph] 21 Feb 2001An iterative algorithm for generating selected eigenspace s of large matrices
F. Andreozzi, A. Porrino, and N. Lo Iudice
Dipartimento di Scienze Fisiche, Universit` a di Napoli Fed erico II,
and Istituto Nazionale di Fisica Nucleare.
Complesso Universitario di Monte S. Angelo, Via Cintia, 801 26 Napoli, Italy
We propose a new iterative algorithm for generating a subset of eigenvalues and eigenvectors of
large matrices and give convergence criteria for the iterat ive process. We show that the method
can be turned naturally into an importance sampling algorit hm which greatly reduces the number
of basis states needed for an accurate determination of the e igenvectors. Finally, we argue that,
because of its extreme simplicity and efficiency, the method m ay represent a valid alternative to the
much more sophisticated importance sampling approaches cu rrently adopted.
02.70.-c 21.60.-n 71.10.-w
It has become customary in many branches of physics to resort to the diagonalization of an Hamiltonian matrix
in large-dimensional spaces as a tool for an accurate determ ination of the properties of complex quantum systems.
Lattice models in more than one dimension, in all their varia nts, quantum dots with many electrons and nuclear
shell-model problems involving many valence nucleons in a m ajor shell are widely known examples. For most of the
systems to be studied, however, the space dimensions are so l arge as to render a complete diagonalization out of reach
of the computational resources existing now as well as in a fo reseeable future. It is therefore compulsory to make
the hypothesis that sampling a small fraction of the basis states suffices to generate the eigensol utions to the desired
accuracy.
This is the underlying assumption of the quantum Monte Carlo methods [1] where one uses a properly defined
function of the Hamiltonian as a stochastic matrix which gui des a Markov process to sample the basis. These
techniques are quite effective for computing ground states p roperties [2] only if the stochastic matrix is positive. Thi s
condition, however, is not fulfilled in many cases. The same M onte Carlo methods, when used to generate a truncated
basis for diagonalizing the many-body Hamiltonian [3], bec ome quite involved. One has in fact to deal with the
redundancy of the basis states, inherent to the stochastic p rocess, which may slow considerably the convergence of
the procedure, and with the problem of the restoration of the symmetries generally broken in stochastic approaches.
The importance sampling inspires also approaches dealing w ith the direct diagonalization of the Hamiltonian matrix.
Stochastic diagonalization [4,5] is an example. This metho d samples the basis states relevant to the ground state
through a combination of plane (Jacobi) rotations and matri x inflation. It is therefore free of minus-sign problems.
In the same spirit, we developed an iterative method, extrem ely easy to implement, for generating a subset of
eigenvectors of a large matrix. Under given conditions, the iterative process converges to any selected set of eigen-
vectors, whatever is the selection criterion adopted. The c onvergence conditions become also sufficient if we generate
the lowest or the highest eigenvalues. The method can be natu rally turned into an importance sampling algorithm
which greatly enhances the efficiency of the iterative proces s.
We assume first that the matrix Ais symmetric and is obtained from a self-adjoint operator ˆAin an orthonormal
basis {|1/an}bracketri}ht,|2/an}bracketri}ht,...,|N/an}bracketri}ht}. Its matrix elements are therefore aij=/an}bracketle{ti|ˆA|j/an}bracketri}ht. For the sake of simplicity, we illustrate
the procedure for a one-dimensional eigenspace, selected a ccording to a given prescription rule. The algorithm consis ts
of a first approximation loop and a subsequent iteration of re finement loops. The first loop goes through the following
steps:
1a)Start with the first two vectors of the basis and diagonalize t he matrix/parenleftbigg
λ(1)
1a12
a12a22/parenrightbigg
, where we have put
λ(1)
1=a11.
1b)Select the eigenvalue λ(1)
2and the corresponding eigenvector |φ(1)
2/an}bracketri}ht=K(1)
2,1|φ(1)
1/an}bracketri}ht+K(1)
2,2|2/an}bracketri}htsatisfying the
assigned rule, where |φ(1)
1/an}bracketri}ht ≡|1/an}bracketri}ht.
forj= 3,...,N
1c)computeb(1)
j=/an}bracketle{tφ(1)
j−1|ˆA|j/an}bracketri}ht.
1d)Diagonalize the matrix/parenleftBigg
λ(1)
j−1b(1)
j
b(1)
jajj/parenrightBigg
.
1e)Select the eigenvalue λ(1)
jand the corresponding eigenvector |φ(1)
j/an}bracketri}htsatisfying the given prescription.
endj
1The first loop yields an approximate eigenvalue λ(1)
N≡E(1)≡λ(2)
0and an approximate eigenvector |ψ(1)/an}bracketri}ht ≡|φ(1)
N/an}bracketri}ht ≡|
φ(2)
0/an}bracketri}ht=/summationtextN
i=1K(1)
N,i|i/an}bracketri}ht. With these new entries we start an iterative procedure whic h goes through the following
refinement loops:
forn= 2,3,... till convergence (if any)
forj= 1,2,...,N
2a)Computeb(n)
j=/an}bracketle{tφ(n)
j−1|ˆA|j/an}bracketri}ht.
2b)Solve the generalized eigenvalue problem
det/bracketleftBigg/parenleftBigg
λ(n)
j−1b(n)
j
b(n)
jajj/parenrightBigg
−λ/parenleftBigg
1K(n)
j−1,j
K(n)
j−1,j1/parenrightBigg/bracketrightBigg
= 0.
2c)Select the eigenvalue λ(n)
jand the corresponding eigenvector |φ(n)
j/an}bracketri}htsatisfying the assigned criterion.
endj
endn.
It is worth to point out that, since the current eigenvector i s not orthogonal to any of the basis vectors, the generalized
eigenvalue problem 2b) replaces the standard one 1d). The n-th loop yields an approximate eigenvalue λ(n)
N≡E(n)≡
λ(n+1)
0. As for the eigenvector, we observe that, at any step of the j-loop, we have
|φ(n)
j/an}bracketri}ht=p(n)
j|φ(n)
j−1/an}bracketri}ht+q(n)
j|j/an}bracketri}ht, (1)
with the appropriate normalization condition [ p(n)
j]2+ [q(n)
j]2+ 2p(n)
jq(n)
jK(n)
j−1,j= 1. The iteration of Eq. (1)
yields then-th eigenvector
|ψ(n)/an}bracketri}ht ≡|φ(n)
N/an}bracketri}ht=P(n)
0|ψ(n−1)/an}bracketri}ht+N/summationdisplay
i=1P(n)
iq(n)
i|i/an}bracketri}ht, (2)
where the numbers P(n)
iare defined as
P(n)
i=N/productdisplay
k=i+1p(n)
k(i= 0,1,...,N −1) ;P(n)
N= 1. (3)
The algorithm defines therefore the sequence of vectors (2), whose convergence properties we can now examine. The
numbersq(n)
jandp(n)
jcan be expressed as
q(n)
j=|B(n)
j|
/bracketleftBig
(ajjK(n)
j−1,j−b(n)
j)2+ 2K(n)
j−1,j(ajjK(n)
j−1,j−b(n)
j)B(n)
j+ (B(n)
j)2/bracketrightBig1
2, (4)
p(n)
j= (ajjK(n)
j−1,j−b(n)
j)q(n)
j
B(n)
j, (5)
where
B(n)
j=/bracketleftBig
λ(n)
j−1−λ(n)
j/bracketrightBig
−K(n)
j−1,j/bracketleftBig
(ajj−λ(n)
j)(λ(n)
j−1−λ(n)
j)/bracketrightBig1
2. (6)
It is apparent from these relations that, if
|λ(n)
j−1−λ(n)
j| →0,∀j, (7)
the sequence |ψ(n)/an}bracketri}hthas a limit |ψ/an}bracketri}ht, which is an eigenvector of the matrix A. In fact, defining the residual vectors
|r(n)/an}bracketri}ht= (ˆA−E(n))|ψ(n)/an}bracketri}ht, (8)
a direct computation gives for their components
2r(n)
l=p(n)
N/bracketleftBig
(all−λ(n)
l)(λ(n)
l−1−λ(n)
l)/bracketrightBig1
2+q(n)
N/braceleftBig
alN−λ(n)
NδlN/bracerightBig
−p(n)
N/braceleftBig/parenleftbig
λ(n)
l−1−λ(n)
l/parenrightbig
K(n)
l,l−1+/parenleftbig
λ(n)
N−1−λ(n)
N/parenrightbig
K(n)
l,N−1/bracerightBig
. (9)
In virtue of Eq. (7), the norm of the n-th residual vector converges to zero, namely ||r(n)||→0. Eq. (7) gives therefore
a necessary condition for the convergence of the |ψ(n)/an}bracketri}htto an eigenvector |ψ/an}bracketri}htofA, with a corresponding eigenvalue
E=limE(n). This condition holds independently of the prescription ad opted for selecting the eigensolution. Indeed,
we never had to specify the selection rule in steps 1b), 1e) an d 2c). Eq. (7) is not only a necessary but also a sufficient
condition for the convergence to the lowest or the highest ei genvalue of A. In fact, the sequence λ(n)
jis monotonic
(decreasing or increasing, respectively), bounded from be low or from above by the trace and therefore convergent.
Having proved the convergence of the iterative procedure to an exact eigenvector, let us now show that the algorithm
lends itself to straightforward extensions and improvemen ts which may widen its range of applicability and may
render the diagonalization process extremely effective. Ob vious extensions are obtained by removing some of the
initial assumptions. The same iterative procedure suggest s that there is no need to assume the orthogonality of
the basis states we started with. A non-orthogonal basis can be treated by simply substituting steps 1a) and 1d)
of the first loop with the appropriate generalized eigenvalu e problem. Also, we can relax the assumption that Ais
symmetric (Hermitian). We have only to update both right and left eigenvectors and perform steps 1c) and 2a) for
both non-diagonal matrix elements.
The method can be turned into an importance sampling algorit hm. We need just to impose in the first loop that
a state |j/an}bracketri}htwith eigenvalue λjis to be retained only if
∆j=|λj−λj−1|≃(b(1)
j)2
|ajj−λj−1|>ǫ,
whereǫis an arbitrarily fixed parameter. As we shall see, the import ance sampling leads to an effective drastic
reduction of the matrix dimensions.
Last, but not least, the algorithm can be easily reformulate d in a context which allows to compute at once any
numbermof eigenvectors. We have simply to replace the two-dimensio nal matrices with multidimensional ones
having the following block structure: A m×msubmatrix diagonal in the selected meigenvalues, which replaces
λ(n)
j, am′×m′submatrix corresponding to ajjand twom×m′off-diagonal blocks replacing b(n)
jorK(n)
j−1,j. We
developed an importance sampling algorithm also for the mul tidimensional case. Such an extension will be outlined
in an extended version of this work.
In order to test the efficiency and the convergence rate of the i terative procedure, we applied the algorithm to a
system of 20 particles distributed over 20 doubly-degenera te equispaced single-particle levels with a level spacing o f
1 MeV and interacting through a two-body pairing interactio n of constant strength G= 0.32 MeV. The resulting
Hamiltonian matrix is of the order 184756. This many-body pr oblem, which is relevant to many mesoscopic systems
like nuclei and superconducting grains [6], represents a se vere test. Indeed, because of the off-diagonal long range
order of the system [7] and the uniform spacing of the single p article levels, we are far from a perturbative regime
and have no a priori prescription for cutting the dimensions of the basis space.
We adopted first the single vector iterative procedure, whic h converges to a single eigenvector, and have computed
the lowest, the first excited and the highest eigenstates (Ta ble I). The first excited state was obtained by selecting, at
each step, the approximate eigenvector having the second ba sis state as dominant component. The rapid convergence
of the algorithm is quite apparent. It is however worth notin g that the convergence rate depends crucially on the
initial ordering of the basis. We found that the convergence is much faster if we order the diagonal matrix elements
in a monotonic sequence.
The results obtained by adopting the multidimensional vers ion of the algorithm are shown in Table II, where the
first five eigenvalues are reported. Its faster convergence w ith respect to the one-dimensional case is to be noticed.
Such a greater efficiency is most likely due to the following fe ature. The orthogonality constraint implicit in the
diagonalization procedure, when enforced within a multidi mensional space, allows to identify and characterize in mor e
detail and with much less ambiguity the selected eigensolut ions to be determined. Such a more precise characterization
allows to discriminate even between eigenvectors having th e same basis states as dominant components.
The effectiveness of the importance sampling algorithm is il lustrated in Table III, where the results obtained for the
lowest and the highest eigenvalue by using decreasing value s of the sampling parameter ǫare shown. The corresponding
number of selected basis states is also given. Clearly the us e of such an algorithm has made possible a substantial
reduction of the dimensions of the problem. We found a simila r reduction also in the multidimensional case.
3The just outlined test suggests that this algorithm may be us efully adopted for generating the ground state or a
selected number of eigenstates. In the first case, it may repr esent an alternative to Monte Carlo techniques. Indeed,
the method is immune to the fermion sign problem, is much more straightforward and more direct, yielding explicitly
the ground state wave function. It may be also a simpler and mo re efficient alternative to stochastic diagonalization
since, in all the steps of our procedure, we deal only with 2 ×2 matrices. In the second case, it may represent an
efficient tool for an accurate study of mesoscopic systems, li ke quantum dots with many electrons and heavy nuclei.
Studies of quantum dots based on exact diagonalization of th e Hamiltonian were feasible only for a small number
of electrons or when the calculation was confined within rela tively small subspaces [8]. These restrictions should be
largely removed by the present algorithm implemented with t he importance sampling. As for the nuclear systems,
we expect that the method will enable us to carry out a practic ally exact and exhaustive study of the spectroscopic
properties of heavy nuclei, for which full shell model studi es are still unfeasible.
ACKNOWLEDGMENT. The work was partly supported by the Prin 99 of the Italian MURST
[1] A comprehensive account of the existing techniques can b e found in Quantum Monte Carlo Methods in Physics and Chem-
istry, M. P. Nightingale and C. J. Umrigar eds.,(Kluwer Academic P ublishers, the Netherlands, 1999).
[2] See for instance J.A. White, S.E. Koonin, and D.J. Dean, P hys. Rev. C 61, 034303 (2000).
[3] T. Otsuka, M. Honma, and T. Mizusaki, Phys. Rev. Lett. 81, 1588 (1998).
[4] H. de Raedt and W. von der Linden, M. Frick, Phys. Rev. B 45, 8787 (1992).
[5] H. de Raedt and M. Frick, Phys. Rep. 231, 107 (1993).
[6] J. Dukelsky, and G. Sierra, Phys. Rev. Lett. 83, 172 (1999).
[7] C. N. Yang, Rev. Mod. Phys. 34, 694 (1962).
[8] J. J. Palacios, L. Martin-Moreno, G. Chiappe, E. Louis, a nd C. Tejedor, Phys. Rev. B 50, 5760 (1994)-II.
TABLE I. Results obtained by the single vector iteration alg orithm for the two lowest and the highest eigenvalues of the
pairing Hamiltonian matrix. The first row gives the order of t he iteration.
2 4 6 8
105.4879 105.4730 105.4727 105.4727
107.8235 107.8385 107.8395 107.8396
307.2632 307.2827 307.2827 307.2827
TABLE II. Results obtained by the multidimensional algorit hm for the lowest five eigenvalues of the pairing Hamiltonian
matrix. The first row gives the order of the iteration.
2 4 6
105.4729 105.4727 105.4727
107.8398 107.8396 107.8396
109.8008 109.8006 109.8006
109.8011 109.8006 109.8006
111.7699 111.7697 111.7697
TABLE III. Results obtained by the single vector algorithm f or the lowest and the highest eigenvalue using importance
sampling. The rows marked ngive the number of sampled basis states.
n 121 480 2554 5772 19115
105.7336 105.5937 105.4943 105.4802 105.4735
n 102 177 1952 2186 3241
307.2611 307.2675 307.2819 307.2822 307.2825
4 |
1Quasi-quantization of writhe in ideal knots
Piotr /G33/G4C/G48/G55/G44 /G14/G56/G4E/G4C/G03/G44/G51/G47/G03Sylwester /G33/G55/G5D/G5C/G45/G5C/GE1/G0F
Faculty of Technical Physics
/G33/G52/G5D/G51/G44 /G14 /G38/G51 /G4C/G59/G48/G55 /G56/G4C /G57/G5C /G52 /G49 /G37/G48 /G46/G4B/G51/G52 /G4F/G52/G4A/G5C
Piotrowo 3, 60 965 /G33/G52 /G5D/G51/G44 /G14 /G0F
e-mail: Piotr.Pieranski@put.poznan.pl
ABSTRACT
The values of writhe of the most tight conformations, found by the
SONO algorithm, of all alternating prime knots with up to 10 crossings are
analysed. The distribution of the writhe values is shown to be concentrated
around the equally spaced levels. The “writhe quantum” is shown to be
close to the rational 4/7 value. The deviation of the writhe values from the
n*(4/7) writhe levels scheme is analysed quantitatively.
PACS: 87.16.AC21. Introduction
Knot tying is not only a conscious activity reserved for humans. In nature knots are
often tied by chance. Thermal fluctuations may entangle a polymeric chain in such a
manner that an open knot becomes tied on it. This possibility and its physical
consequences were considered by de Gennes [1].
Whenever the ends of a polymer molecule become connected – a closed knot is tied.
Understanding the topological aspects of the physics of polymers is an interesting and
challenging problem [2, 3]. Formation of knotted polymeric molecules can be simulated
numerically [4, 5]. A simple algorithm creating random walks in the 3D space provides a
crude model of the polymeric chain. Obviously, knots tied on random walks are of
various topological types. The more complex a knot is, the less frequently it occurs. The
probability of formation of various knot types was studied [6] and the related problem of
the size of knots tied on long polymeric chains was also analysed [7].
DNA molecules are not protected from becoming knotted. Knots of various types and
catenanes are easily formed as intermediate products of DNA replication and
recombination processes [8]. The probability of the knot formation within the DNA
molecules was analysed [9]. In appropriate experimental conditions knots tied on DNA
molecules of identical molecular weight can be created [10]. Although being of the same
molecular weight, the topoisomers display different physical properties. For instance,
their electrophoretic migration rate is different and proportional to the average crossing
numbers of the so-called ideal conformations of the knotted molecules [11].
Tying a knot on a rope needs closing its ends. In the case of DNA, the closing is easier
if meeting ends are appropriately oriented. The geometrical parameter of a knot
responsible for the relative orientation of its ends is the writhe of the knot [12, 13, 14]. As
shown recently, writhe of the ideal conformations of prime knots displays a curious
quasi-quantization properties. Below, we present results of our study of the phenomenon.
2. Ideal knots
From the topological point of view a conformation of a knot is of no importance [15].
From the physical point of view it matters a lot. For knots tied on the ideal, i.e. the utterly
flexible but infinitely hard in its circular cross-section rope there exists a minimum value
of the rope length at which knot of the given type can still be tied. The particular3conformation of a knot for which the minimum is reached is called ideal [16]. It is
assumed that there exists but a single conformation which minimizes the rope length.
Fig. 1. Evolution of a loose conformation of the 10 120 knot towards its most tight
conformation. The evolution was enforced by SONO algorithm. Calculations started with a
number of segments N=91. At the end of the tightening process the number of segments was
doubled 3 times up to N=712. Numerical errors within the final value of writhe are smaller
than 1% and the deviation of the writhe value from 16 predicted by the Cerf-Stasiak is anintrinsic property of the conformation found by SONO.
Finding ideal conformations is not a trivial task. There exist a few algorithms which
are aimed to perform it. One of them is SONO (Shrink-On-No-Overlaps) described in our
earlier papers [17]. Fig.1 presents how simulating the process in which the rope shrinks
slowly SONO arrives at the ideal conformation of the 10 120 knot. As seen in the figure,4the knot changes considerably its conformation. The changes of conformation are
accompanied by changes of its writhe value.
Initially, to speed up the evolution, the number of segments, of which the equilateral
knot is constructed, is kept as low as possible. At the end of the calculation it is raised to
a value at which the inaccuracy of the writhe calculation is better than 1%.
Although for the sake of brevity we talk in what follows about the ideal
conformations, the “ideal” must be understood as “the most tight one, found by the
SONO algorithm ”. There is no escape from this uncertainty - the ideal conformation is
known at present only for a single prime knot: the trivial knot.
3. Writhe
One of the essential parameters which distinguish between the shapes of various
conformations of the same knot is the 3D writhe. (In what follows we shall refer to it in
brief as writhe.) If r1 and r2 are two points within conformation K of a knot and r1,2 = r2 –
r1 is the vector which joins the points, then writhe of the conformation is given by the
value of the double integral:
∫∫⋅×=
KK rd dWr3
2,12 2,11)(
41 rrr
π (1)
As shown by Georges /G26/G03/G4F/G58/G4A/G03/G55/G48/G44/G51/G58, in spite that at the r1 = r2 diagonal of the K×K
integration domain the denominator of the integrated ratio goes to zero, the integral does
not diverge.
Writhe calculation formula defined by eq.1 is valid for continuous knots. Knots
processed by the SONO algorithm are discrete, they are represented by tables of ( x,y,z)
coordinates of n vertices. Thus, the calculation of their writhe must be performed using
discrete sum formulae [18].
4. Writhe quantization hypothesis
In the first paper on the geometry of ideal knots [19], it has been indicated that the
writhe within some families torus and twist prime knots grows with the crossing number
in a linear manner. Ideal conformations of knots discussed in ref. 19 were found via
simulated annealing. Using the more efficient SONO algorithm we performed an
extensive search for the most tight conformations of all prime knots with up to 105crossings. Preliminary analysis of the results we obtained for knots with up to 9 crossings
was described in ref. 17 where we pointed out (see fig. 8 there) that the writhe values of
prime knots with up to 9 crossings show a visible tendency to gather around a few, well
defined levels. No hypothesis concerning their spacing was formulated.
The observation stimulated a series of theoretical considerations [20], which lead to
the hypothesis that for alternating prime knots the quantum of writhe is 4/7, and showed
that the actual writhe values can be predicted by a topological invariant computable from
any minimum crossing number diagram. It is the aim of the present paper to verify
quantitatively the hypothesis of the 4/7 writhe quantum on the set of all alternating prime
knots with up to 10 crossings.
0 50 100 150 20005101520Wr/(4/7)
knot number
Fig. 2. Wr/(4/7) versus the knot number for ideal conformations of all alternating prime
knots with up to 10 crossings. Horizontal lines indicate the writhe levels suggested by Cerf
and Stasiak. The gap visible in the set of plotted points in the vicinity of the knot number=80
corresponds to the non-alternating knots with 9 crossings. A smaller gap localized aroundknot number=40 corresponds to 3 non-alternating knots with 8 crossings.
5. Quantitative verification of the 4/7 hypothesis
If the Cerf-Stasiak 4/7 writhe quantum hypothesis is right, the values of Wr/(4/7)
should for the alternating knots be located close to integer levels. Fig. 2 presents the plot6of the Wr/(4/7) values of the ideal conformations of all alternating prime knots with up to
10 crossings. The values are plotted versus the knot number which localizes a particular
knot in the Rolfsen table of prime knots. See, e.g. ref. 21. Confirming our earlier
observations reported in ref. 3 the plot reveals that the writhe values are distributed in a
highly inhomogeneous manner. The values of the Wr/(4/7) variable are clearly gathering
around the integer levels; the Cerf-Stasiak 4/7 writhe quantum hypothesis seems to be
qualitatively confirmed. The n*(4/7) writhe levels scheme was plotted in Fig. 2 according
to the suggestions of Cerf and Stasiak. It seems to fit well the data provided by SONO.
But, is it really the best writhe levels scheme? A simple test convinced us, that this is
indeed the case.
Fig. 3. The average relative deviation 〈|dW|〉 of the writhe values of all alternating knots
with up to 10 crossings from n*qW writhe levels versus the writhe quantum qW.
To check quantitatively which value of the writhe quantum qW fits best the set of our
writhe data, we calculated the dependence of the average relative deviation 〈|dWi|〉 of the
writhe values Wri from n*qW levels, where qW was swept throughout the [0.1, 1.0]
interval. The 〈|dW|〉 value was calculated as follows:
∑∑
∈ ∈−
= =
Aibest
ii
AiiqWqWnWr
NdWNdW1 1
(2)7
Fig. 4. The deviations dWi of the writhe values of all alternating knots with up to 10
crossings from levels of the optimal scheme defined by qW=4/7.
Fig. 5. The probability P of finding a the dW deviations within 20 counting bins. The
Gaussian function which fits the data is also plotted; its half-width σ equals 0.2.
Wri is the writhe value of the i-th knot. nibest is the number of the level, nibestqW ,
closest to Wri. The summation runs over all alternating prime knots with up to 108crossings. As seen in Fig.3, the relative average deviation 〈|dW|〉 of the writhe values from
the levels separated by qW displays a clear minimum at qW=0.5702, a value very close to
the rational 4/7 ≅0.5714. At the minimum 〈|dW|〉=0.075, significantly less than 0.25
expected in absence of the quantization tendency.
Having checked that suggested by Cerf and Stasiak 4/7 writhe quantum produces a
writhe levels scheme which fits best the writhe values found by SONO we performed a
quantitative analysis of the distribution of observed deviations. Thus, assuming qW=4/7
we calculated for each of the analyzed knots the deviation dWi of its writhe value Wri
from the closest n*qW writhe level. The plot of the deviations versus the knot number is
shown in Fig.4. As seen in the figure, the deviations are spread in an almost uniform
manner; their absolute value is never larger than 0.25. The width of the spread is only
slightly smaller for smaller knots. To analyze the distribution in a quantitative manner,
we divided the [-0.5, 0.5] interval into bins of 0.05 width. Counting knots whose dWi
value were located within consecutive bins and dividing the counts by the total number of
analyzed knots we obtained the probability of finding the writhe value within each of the
counting bins. The shape of the probability distribution is shown in Fig.5. The half-width
σ of the Gaussian which fits best the distribution equals 0.203. Moreover, as seen in fig.4,
the writhe value of none of the studied knots deviates from the Cerf-Stasiak quantization
scheme more than 0.25 qW.
6. Discussion
Let us summarize results described above.
1. The existence of the writhe quasi-quantization tendency noticed previously within a
very limited set of knots [17] has been confirmed within a much broader set of knots:all alternating prime knots with up to 10 crossings.
2. The qW=4/7 separation of the writhe levels suggested by Cerf and Stasiak [20] was
shown to fit best the writhe data obtained with the use of the SONO algorithm.
The half width σ of the distribution of the writhe deviations from the closest Cerf-
Stasiak levels was shown to be equal 0.2. In view of the analysis of numerical errors we
performed, deviations of such a magnitude cannot be attributed to the inaccuracy of the
writhe calculations; they must be seen as the intrinsic property of the most tight
conformations found by the SONO algorithm. Will a different algorithm of the
determination of the ideal conformations substantially reduce the value of σ? Will it9reduce it to zero? Is the writhe quasi-quantization an approximate or an exact rule? The
questions posed above remain open. An independent analysis, performed with the use of
a different knot tightening algorithm, could shed more light on them. It seems also that
further theoretical considerations along the line presented in ref. 22 should help to
understand the origin of the writhe quantization phenomenon. But, whatever the answers
to the questions fromulated above, results of the present study confirm beyond any doubt
that the writhe of ideal conformations of prime knots shows a strong tendency to group
close to well defined equidistant levels.
As mentioned in the introduction, there exist practical implications of the writhe quasi-
quantization phenomenon, which were not noticed before. Let us assume that a knot is
tied on a rope having a certain internal structure; let for the sake of simplicity it be a
bundle of parallel threads. When forming the rope into a knot one wants to perturb as
little as possible its internal bundle structure, one should follow the procedure known as
the parallel transport. As indicated by Maggs [23], one can show that the writhe of a knot
is closely related to the Berry ’s phase [24].
Fig. 6. The ideal 3 1 (left) and 4 6 (right) knots tied on a rope with an internal
structure. The parallel transport of the internal structure of the rope leads in the case
of the 3 1 knot to a distinct misfit of the orientation of the meeting ends. The ends fit
perfectly well in the case of the 4 1 knot – the arrow indicates the hardly visible
meeting point.
If connecting the ends of the rope one wants to keep identity of the threads, the angle
of the relative orientation of the meeting ends should be equal to a multiple of 2 π. This
happens when the writhe value of the knot is integer. From such a point of view, the tight10conformations of the knots whose writhe values are grouped around a non-integer writhe
level are more difficult to tie than the tight conformations of knots whose writhe value
stays close to an integer level. Figure 6 presents the ideal 3 1 and the 4 1 knots tied on a
rope with an internal structure. As seen in the figure, ends of the rope formed using to the
parallel transport procedure into the 3 1 knot do not meet at a proper relative orientation.
The knot can be closed only when an additional twist is introduced into the rope. It seems
to us that the effect should be taken into consideration in the analysis of knots tied on e.g.
the DNA molecules.
This work was carried out within project BW 63-013/2000.
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Singapore, 1998.
17 P. /G33/G4C/G48/G55/G44 /G14/G56/G4E/G4C/G0F/G03/G4C/G51/G03/G55/G48/G49/G11/G03/G14/G18/G1111
18 D. Cimasoni, Computing the writhe of a knot , accepted for publication in J. Knot
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arXiv:physics/0102068v1 [physics.atom-ph] 21 Feb 2001Thermalization of magnetically trapped metastable helium
A. Browaeys, A. Robert, O. Sirjean, J. Poupard, S. Nowak, D. B oiron, C. I. Westbrook and A. Aspect
Laboratoire Charles Fabry de l’Institut d’Optique,
UMR 8501 du CNRS, B.P. 147, F-91403 ORSAY CEDEX, FRANCE
We have observed thermalization by elastic collisions of ma gnetically trapped metastable helium
atoms. Our method directly samples the reconstruction of a t hermal energy distribution after the
application of an RF knife. The relaxation time of our sample towards equilibrium gives an elastic
collision rate constant of α∼5×10−9cm3/s at a temperature of 1 mK. This value is close to the
unitarity limit.
Bose-Einstein condensation (BEC) of dilute atomic va-
pors has been observed in Rb [1], Na [2], Li [3] and H [4].
Atoms in these gases are in their electronic ground state.
Metastable helium in the 23S1state (He∗), which has
long been of interest to the laser cooling community, is
by contrast in a state 20 eV above the ground state. This
situation presents new possibilities for the study of cold
dilute atomic gases. First, the large internal energy per-
mits efficient detection by ionization of other atoms and
surfaces: it is possible to study very small samples. Sec-
ond, Penning ionization by both the background gas and
between trapped atoms offers a high time resolution mon-
itor of the number and density of trapped atoms. Third,
the possibility of using the large internal energy of He*
for atomic lithography has already been demonstrated
[5], and this application as well as atom holography [6]
may benefit from highly coherent sources. Finally, much
theoretical work has already been devoted to estimation
of the elastic collision cross sections on the one hand and
Penning ionizing rates on the other [7, 8]. Experiments
such as the one reported here can test this work.
BEC is achieved in dilute gases by evaporative cool-
ing of a magnetically trapped sample [9]. In He∗, it is
hampered by the fact that in a magneto-optical trap, the
typical starting point of magnetic trapping, the achiev-
able atomic density is limited by a large light-assisted
Penning ionization rate [10, 11, 12, 13, 14, 15]. On the
other hand, the scattering length for low energy elastic
collisions is predicted to be quite large, and the Penning
ionization rate highly suppressed in a spin polarized sam-
ple [7, 8]. He* in a magnetic trap necessarily constitutes
a spin polarized sample and experiments have already
demonstrated a suppression of more than one order of
magnitude [16, 17, 18]. If the theoretical estimates are
right, efficient evaporative cooling may still be possible in
spite of the low initial trap density. We report here the
observation of the thermalization of He∗due to elastic
collisions which appears to roughly bear out the predic-
tions.
To perform a thermalization experiment, a trapped
cloud is deliberately placed out of equilibrium and its
relaxation due to the elastic collisions between trapped
particles is observed. Usually the observations are made
by imaging the spatial distribution as a function of time[19, 20, 21]. In our experiment the relaxation is observed
in the energy distribution of the atoms in the magnetic
trap. First this distribution is truncated above ERF=hν
by a radio-frequency pulse (or RF knife) of frequency ν.
The cloud rethermalizes by elastic collisions and the pop-
ulation of the states of energy higher than ERFincreases
from zero; for large times compared to the thermalization
timeτththe distribution reaches a thermal distribution
[22]. With the help of an analytical model and numerical
simulations, we deduce τthfrom the time dependence of
the number of atoms with energy above ERF. We mea-
sure this time dependence by applying a second RF knife
after a delay time t, and with a frequency slightly above
that of the first one. Our model also allows us to relate
τthto the elastic collision rate per atom in the trap.
Much of our setup has been described previously [14,
17]. Briefly, we use a LN 2cooled DC discharge source
to produce a beam of metastable He atoms. The beam
is slowed down to ∼100 m/s using Zeeman slowing and
loads a magneto-optical trap. Typically, 3 ×108atoms are
trapped at a peak density of 3 ×109at/cm3, limited by
light-induced Penning ionization. The temperature of the
cloud is about 1 mK and the cloud is roughly spherical
with an RMS size of 2.5 mm. We then apply a 5 ms
Doppler molasses to cool the atoms down to 300 µK. This
is achieved by switching off the magnetic field, decreasing
the detuning close to resonance and lowering the intensity
to 10 % of its value in the MOT. An optical pumping
step allows us to trap up to 1 .5×108atoms in a Ioffe-
Pritchard trap. We use a ”cloverleaf” configuration [23]
withB′= 85 G/cm, B′′= 25 G/cm2and a bias field
B0= 200 G. The two sets of coils are outside the vacuum,
separated by 4 cm. After lowering the bias field to 4 G,
the temperature of the compressed atomic sample reaches
1 mK. The lifetime of the trap is 60 s.
We use a 2 stage microchannel plate (MCP) to detect
the atoms. The MCP is placed 5 cm below the trap-
ping region and has an active area of 1.4 cm diameter.
Two grids above the MCP allow us to repel all charged
particles and detect only the He*. After turning off the
magnetic trap, the MCP signal corresponds to a time of
flight spectrum (TOF) which gives the temperature of
the atoms. The area of this spectrum is proportional to
the number of atoms in the trap at the time it was turned2
/K30/K2E/K30/K30/K2E/K32/K30/K2E/K34/K30/K2E/K36/K30/K2E/K38/K31/K2E/K30
/K30 /K31/K30/K30 /K32/K30/K30 /K33/K30/K30 /K34/K30/K30 /K35/K30/K30
/K52/K46/K20/K6B/K6E/K69/K66/K65/K20/K66/K72/K65/K71/K75/K65/K6E/K63/K79/K20/K28/K4D/K48/K7A/K29/K52/K46/K20/K6B/K6E/K69/K66/K65/K20/K66/K72/K65/K71/K75/K65/K6E/K63/K79/K20/K28/K4D/K48/K7A/K29/K30 /K31/K30/K30 /K32/K30/K30 /K33/K30/K30 /K34/K30/K30/K65/K6E/K65/K72 /K67/K79/K20/K64/K69/K73/K74/K72/K69/K62/K75/K74/K69/K6F/K6E/K20/K28/K61/K2E/K75/K2E/K29/K66/K72/K61/K63/K74/K69/K6F/K6E/K20/K6F/K66/K20/K72/K65/K6D/K61/K69/K6E/K69/K6E/K67/K20/K74/K72/K61/K70/K70/K65/K64/K20/K61/K74/K6F/K6D/K73
FIG. 1: RF spectrum of atoms in the magnetic trap. a) Fraction of remaining trapped atoms after the RF pulse as a function
of the RF frequency ν. b) Derivative of these data, i.e. the energy distribution i n the magnetic trap. The solid line is the
prediction for a cloud at a temperature of 1.1 mK, the tempera ture measured by time of flight (TOF). The dashed line indicat es
the frequency corresponding to the bias field.
off. The collection and detection efficiency of the MCP
varies by roughly a factor of two depending on the mag-
netic field configuration we use, and so one must take
care to only use data corresponding to the same mag-
netic field when making comparisons. We also use the
MCP to monitor the atoms falling out of the trap while
applying an RF knife. The area of the MCP signal in this
case measures the number of atoms with an energy above
that of the RF knife. Finally, when we bias the grids so
as to attract positive ions, the MCP signal can be used
to observe the products of Penning ionization with the
background gas while the trap is on. This signal is pro-
portional to the number of trapped atoms. We observe
an exponential decay, indicating that two body loss (He*
+ He*) is negligible.
Two parallel coils in the vacuum system produce an RF
magnetic field perpendicular to the bias field and consti-
tutes the RF knife. To understand the effect of the RF
knife on the trapped cloud and to assure that our sam-
ple is at thermal equilibrium, we first performed an RF
spectroscopy measurement of the energy of the atoms in
the trap [24]. We apply an RF pulse at a frequency hν
which changes the Zeeman sublevel of the atoms from the
trapped M= +1 state to M= 0. The duration of the
knife is 3 s, which is necessary to expel all the atoms with
energy above hνover the entire range which we explore.
We then turn off the magnetic trap to measure the num-
ber of remaining atoms. Observation of the atoms falling
onto the MCP during the RF knife shows that the flux of
atoms expelled is negligible at the end of the pulse. An
example of the RF spectrum is shown in Fig. 1a). The
derivative of the data gives the energy distribution. In
Fig. 1b) we compare this distribution with a thermal one
at 1.1 mK, the temperature measured by an independent
TOF measurement. We conclude that our atomic sample
is close to thermal equilibrium.
We begin the thermalization experiment with a 2 sRF knife of frequency ν1= 135 MHz (corresponding to
η= (hν−2µBB0)/kBT∼6). Next we measure the
number of atoms falling onto the MCP during a second
RF knife at a slightly higher frequency ( ν2= 138 MHz)
and delayed by a time t. Assuming that the angular dis-
tribution of the atoms expelled by the second RF knife
is constant during the thermalization process, the MCP
signal is proportional to the number of expelled atoms.
Plots of the number of expelled atoms as a function of t
are shown in Fig. 2 for samples having different numbers
of atoms but the same temperature to within 10%. Fig. 2
shows that the number of atoms above the RF knife in-
creases rapidly and then falls again with a time constant
close to the trap lifetime as atoms are lost. If the ini-
tial increase is indeed due to thermalizing collisions, the
initial slope of each curve should be proportional to the
square of the number of atoms. Our data roughly confirm
this dependence.
To be more quantitative, and to determine the ther-
malization time τth, we use a model based on the Boltz-
mann equation under the sufficient ergodicity hypothesis
and inspired by [25]. We divide the sample into two en-
ergy regions, E−andE+, with energies below and above
ηkBTrespectively and denote by N−andN+the num-
ber of atoms belonging to the two regions. We assume
thatη≫1. Immediately after truncation, N+= 0, and
we seek a differential equation governing the time depen-
dence of N+. Since η≫1, we only take into account
collisions of the type ( E−) + (E−)↔(E−) + (E+), and
neglect all collisions involving two atoms in E+in either
the final or initial state. The corresponding flux ˙N+is
thus of the form
˙N+= ∆1N−2−∆2N−N+. (1)
The coefficients ∆ 1and ∆ 2are calculated using Boltz-
mann equation [27]. In particular ∆ 1N−is exactly the
evaporation rate in an evaporative cooling process [25].3
/K30 /K31/K30 /K32/K30 /K33/K30 /K34/K30 /K35/K30/K30/K2E/K30/K30/K2E/K35/K31/K2E/K30/K31/K2E/K35
/K74/K69/K6D/K65/K20/K62/K65/K74/K77/K65/K65/K6E/K20 /K74/K68/K65/K20/K32/K20/K52/K46/K20/K6B/K6E/K69/K76/K65/K73/K20/K28/K73/K29/K4D/K43/K50 /K20/K73/K69/K67/K6E/K61/K6C/K20/K28/K6D/K56 /K2E/K73/K29
FIG. 2: Integrated MCP signal during the RF probe pulse
as a function of the delay between the truncation and probe
pulses. The three curves correspond to 5 ×107, 7×107and
10×107atoms in the trap, varied by changing the power in
the Zeeman slowing laser. The lifetime of the trap is 38 ±4 s,
and the temperature is 0 .9±0.1 mK.
If we make the further approximations that the atoms
inE−andE+have thermal distributions [26], neglect
variations of the temperature during thermalization and
assume that the collision cross section σis independent
of velocity, ∆ 1and ∆ 2are analytic functions of the trap
parameters, atomic mass m,σ,ηandµBB0/kBT. This
latter parameter appears because our trap cannot be
approximated by an harmonic trap; we use the semi-
linear form [25]. It is straightforward to take into ac-
count the finite lifetime τof the atomic sample since
N−(t) +N+(t) =N−(0)exp ( −t/τ). The solution of the
resulting differential equation is :
N+(t) =Nthe−t/τ[1 +q
1−q−exp[τ
τth(1−e−t/τ)]] (2)
where τ−1
th=γel√
2q
1−qe−ηVev
VeandNth= (1−q)N−(0).
The elastic collision rate is γel=nσvwithndefined at
the center of the trap and v= 4/radicalbig
kBT/πm . The quan-
tities Vev,Veandqare defined as in [25, 27]; they are
analytic functions of ηandµBB0/kBT. The quantity q
is the ratio of the number of atoms below the RF knife to
the total for a thermal distribution (about 0.9 under our
conditions), and Nthis the asymptotic value of N+for
infinite trap lifetime. Numerical simulations of the en-
ergy form of Boltzmann equation are in good agreement
with our model for η >10; for η= 6 the quantity γelτth
is 1.8 times larger meaning that our assumption about
the distribution function fails for small η[27]. We take
this factor into account in calculating γel.
To fit the data of Fig. 2 with eq. (2), we fix the lifetime
τat its measured value and use τthandNthas adjustable
parameters. The uncertainty in τthis estimated by vary-
ing the lifetime of the trap within its uncertainty range
and looking at the resulting dispersion in τth. The uncer-/K30/K31/K30 /K38/K30/K39/K30/K35/K30 /K36/K30/K34/K30/K37/K30 /K33/K30 /K32/K30/K30/K2E/K30/K30/K2E/K31/K30/K2E/K35
/K30/K2E/K32/K30/K2E/K33/K30/K2E/K34
/K54 /K4F/K46/K20/K61/K72/K65/K61/K20/K28/K6D/K56 /K2E/K73/K29/K74
/K74/K68/K2D/K31/K20
/K20/K28/K73/K2D/K31
/K29
FIG. 3: Thermalization rate τ−1
thversus the area of the cor-
responding TOF spectrum (proportional to the number of
trapped atoms). The solid line shows a linear fit constrained
to pass through the origin.
tainty in the number of trapped atoms is estimated from
the dispersion of the TOF area measurements before and
after taking a curve as in Fig. 2. The exact value of q
has little influence on the fit.
We have made several tests to check the consistency of
our results. First, we have checked that the fitted value of
Nthcorresponds to the expected fraction of atoms above
the knife for our temperature. Second, Fig. 3 shows that
τ−1
this proportional to the number of trapped atoms, as
it must be if the process of refilling of the upper energy
class is due to two body collisions. We can exclude any
effect independent of the number of atoms. The line pass-
ing through the origin uses the slope as a fit parameter
and has χ2= 5 for 8 degrees of freedom. Third we have
done an additional experiment that confirms the presence
of elastic collisions: in a trap decay rate experiment, in
the presence of the RF knife, the ion signal exhibits a
clear non-exponential behaviour at short times. This ef-
fect can be satisfactorily interpreted as elastic collisio ns
bringing atoms above the RF knife and hence allows a
measurement of the evaporation rate. This rate is con-
sistent with the results obtained in our thermalization
experiment. Fourth we have checked that heating can-
not explain the repopulation of the upper energy classes.
With the trap undisturbed, we can place an upper limit
on the heating rate of 25 µK in 60 s. This limit is two or-
ders of magnitude too low to explain our data. Lastly, we
have performed the thermalization experiment for differ-
ent lifetimes of the magnetic trap (20, 40 and 60 s) and
found consistent results.
From our data in Fig. 3, we can deduce an accurate
measurement of the thermalization time; the fit gives
τth= 3.0±0.3 s for the densest sample. Using the mea-
sured temperature and bias field, this value of τthleads
toγel= 6±1 s−1; this result depends on the accuracy
of our thermalization model. To find the rate constant
α=γel/n, we must estimate the density. Since the data4
show that our sample is close to thermal equilibrium, we
can calculate the volume of the trap knowing the trap
parameters. The absolute measurement of the number
of atoms is performed by measuring the total power ab-
sorbed from a saturating laser beam, similar to [15]. A
TOF area of 75 mV.s corresponds to 108atoms in the
magnetic trap with an uncertainty of a factor of 2. This
leads to α= 5×10−9cm3/s to within a factor 3 at
T= 1±0.1 mK. The ENS-Paris group has obtained a
similar result with a different measurement [28]. The uni-
tarity limit at that temperature is α∼10−8cm3/s. This
means that it is probably not valid to use a constant elas-
tic cross section in our model and some deviation might
appear in the quantity γelτth. We are currently investi-
gating refinements to our thermalization model.
The results shown here are very encouraging for
evaporative cooling of He* in search of BEC.
We thank P. Leo and P. Julienne and the ENS helium
group for helpful discussions. This work was partially
supported by the EC under contracts IST-1999-11055
and HPRN-CT-2000-00125, and DGA grant 99.34.050
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arXiv:physics/0102069v1 [physics.chem-ph] 21 Feb 2001Trial function optimization for excited states of van der Wa als
clusters
M. P. Nightingale and Vilen Melik-Alaverdian
Department of Physics, University of Rhode Island, Kingsto n RI 02881, USA
A method is introduced to optimize excited state trial wave f unctions.
The method is applied to ground and vibrationally excited st ates of bosonic
van der Waals clusters of upto seven particles. Employing op timized trial
wavefunctions with three-body correlations, we use correl ation function Monte
Carlo to estimate the corresponding excited state energies .
I. INTRODUCTION
In this paper we address the problem of computing energies of vibrationally excited
states by means of quantum Monte Carlo methods. We propose a m ethod and apply it to
bosonic van der Waals clusters. As do other quantum Monte Car lo methods, our approach
has the ability to deal with systems with strong anharmonici ties due to quantum mechanical
fluctuations. For such systems conventional variational, n ormal mode, and basis set methods
fail in the treatment of vibrational states. In contrast to o ther Monte Carlo methods, the
one discussed here does not require a priori knowledge of nod al surfaces.
Our method1relies on the use of optimized trial functions for the excite d states. In this
paper the optimization method is explained in detail. In app lications, once the optimized
trial functions have been constructed, we use correlation f unction Monte Carlo to reduce
systematically the variational bias of the energy estimate s. In principle, the imaginary time
spectral evolution (POITSE) method,2(also see the paper by Whaley in this volume) could
be used with the optimized excited state wavefunctions we di scuss here, and it would be
interesting to compare the relative merits of these two proj ection methods.
II. ONE STATE
We consider clusters of atoms of mass µ, interacting pairwise via a Lennard-Jones poten-
tial. In dimensionless form, the pair potential can be writt en asv(r) =r−12−2r−6and the
Hamiltonian as H=P2/2m+V, whereP2/2mandVare the total kinetic and potential
energy operators. The only parameter is the dimensionless massm−1= ¯h2/µσ2ǫ, which
is proportional to the square of the de Boer parameter,3a dimensionless measure of the
importance of quantum fluctuations.
We use the position representation and denote by R, the 3NcCartesian coordinates of
theNcatoms in the cluster. Suppose we have a real-valued trial fun ction ˜ψ(R). Typically,
this trial function may have 50-100 parameters and it may dep end non-linearly on these
parameters. First we recall how this state can be optimized b y minimization of the variance
of the local energy E(R) defined by
H˜ψ(R)≡ E(R)˜ψ(R). (1)
Following Umrigar et al.,4one can minimize the varianceχ2=/angbracketleft(H − /angbracketleftH/angbracketright )2/angbracketright, (2)
which in the position representation can be written as the va riance of the local energy. Note
thatχ2is nothing but the square of the uncertainty in the energy, so thatχ2= 0 for any
eigenstate of the Hamiltonian H.
The minimization of χ2can be done by means of a Monte Carlo procedure with the
following steps:
1. Select a sample of configurations R1,...,R sfrom the probability density ψ2
g(to be
defined).
2. Evaluate:
B=
ˆψ(R1)
...
ˆψ(Rs)
andB′=
ˆψ′(R1)
...
ˆψ′(Rs)
, (3)
where
ˆψ(R) =˜ψ(R)
ψg(R)andˆψ′(R) =H˜ψ(R)
ψg(R)(4)
3. Find Efrom least-squares solution of
Hˆψ(Rσ) =Eˆψ(Rσ), (5)
forσ= 1,...,s :
E=s/summationdisplay
σ=1ˆψ(Rσ)Hˆψ(Rσ)
s/summationdisplay
σ=1ˆψ(Rσ)2. (6)
4. Vary the parameters in the trial function to minimize χ2, the normalized sum of
squared residues defined by the previous step:
χ2=s/summationdisplay
σ=1[Hˆψ(Rσ)−Eˆψ(Rσ)]2
s/summationdisplay
σ=1ˆψ(Rσ)2. (7)
For the purpose of optimizing only the groundstate, the best choice for the guiding func-
tionψg, which is used to generate the sample of configurations, is th e optimized groundstate
wavefunction itself. Since this function is only known at th e end of the optimization, one
uses a reasonable initial guess, if available. Otherwise, a few bootstrap iterations may be
required.
For optimization of excited states, one can use a power of the optimized groundstate trial
wavefunction. We use a power which is roughly in the range fro m one half to one third. This
has the effect of increasing the range of configurations sampl ed with appreciable probability.
The goal is to produce a sample that has considerable overlap with the all excited states of
interest.III. SEVERAL STATES
Next we consider the problem of finding the ‘best’ linear comb ination of a number of
given elementary basis functions β1,...,β n. Before we continue, we should explain our
terminology, since it reflects the procedure that will be use d. We shall form linear combina-
tions of the elementary basis functions . These linear combinations depend on any non-linear
parameters that appear in the elementary basis functions; t he linear combinations will be
optimized with respect to the non-linear parameters by mean s of the general non-linear
trial function optimization procedure described in Sectio n II. Finally, these optimized ba-
sis functions which will serve as the basis functions in a correlation function Monte Carlo
calculation,5,6and we shall return to this in more detail later.
If the ‘best’ linear combinations of elementary basis funct ions are defined in the sense that
for such linear combinations the expectation value of the en ergy is stationary with respect
to variation of the linear coefficients, the solution to this p roblem is well known. Being a
linear problem, the solution requires for its implementati on traditional linear algebra.7The
featured matrices consist of the matrix of overlap integral s of the elementary basis functions,
and the matrix of the Hamiltonian sandwiched between them. T he trouble, of course, is
that the required matrix elements can only be estimated by me ans of Monte Carlo methods
for the cluster problem and the elementary basis functions w e employ.
Stationarity of the energy is equivalent to the least-squar es principle that is used in the
following algorithm. The latter can be used with a very small sample of configurations, but
in the limit of an infinite sample it produces precisely the so lution for which the energy is
stationary.
To find the optimal linear coefficients perform the following s teps:
1. Select a sample of configurations R1,...,R sfrom the probability density ψ2
g(as dis-
cussed previously).
2. Evaluate:
B=
ˆβ1(R1)· · ·ˆβn(R1)
.........
ˆβ1(Rs)· · ·ˆβn(Rs)
, (8)
and
B′=
ˆβ′
1(R1)· · ·ˆβ′
n(R1)
.........
ˆβ′
1(Rs)· · ·ˆβ′
n(Rs)
, (9)
where
ˆβi(R) =βi(R)
ψg(R)andˆβ′
i(R) =Hβi(R)
ψg(R)(10)3. Find
E= (Eij)n
i,j=1 (11)
from least-squares fit to
Hˆβi(Rσ) =n/summationdisplay
j=1ˆβj(Rσ)Eji, (12)
forσ= 1,...,s andi= 1,...,n .
4. Find the eigensystem of Eand write
Eij=n/summationdisplay
i=1d(k)
i˜Ekˆd(k)
j, (13)
where the ˆd(k)
iandd(k)
jare components of left and right eigenvectors of Ewith eigen-
value ˜Ek.
This algorithm yields an approximate expression for the eig enstate for energy Ek:
ψ(k)(R)≈˜ψ(k)(R) =/summationdisplay
iβi(R)d(k)
i. (14)
In addition, one has the following approximate inequality7
Ek<∼˜Ek. (15)
The inequality holds rigorously in the absence of statistic al noise, i.e., if an infinite Monte
Carlo sample is used or if by other means the quantum mechanic al overlap integrals and
matrix elements, corresponding to the matrices NandHdefined in Eq. (17), are evaluated
exactly.
In the ideal case that the basis functions are linear combina tions of no more than ntrue
eigenfunctions of the Hamiltonian, the previous algorithm yields the true eigenvalues, even
for a finite Monte Carlo sample, unless it fails altogether fo r lack of sufficient independent
data.
To find the least-squares solution for Efrom Eq. (12) we write the latter in the form
B′=BE. (16)
Multiply through from the left by BT, the transpose of B, and invert to obtain
E= (BTB)−1(BTB′)≡N−1H. (17)
It is simple to verify that indeed this yields the least-squa res solution of Eqs. (12).
It is is well-known that the solution for Eas written in Eq. (17) is numerically unstable.8
This is a consequence of the fact that the matrix Nis ill conditioned if the βkare nearly
linearly dependent, which indeed is the case for our element ary basis functions. The solutionto this problem is to use a singular value decomposition to ob tain a numerically defined and
regularized inverse B−1. In terms of the latter, one finds from Eq. (17)
E=B−1B′. (18)
More explicitly, one uses a singular value decomposition to write9
B=USVT, (19)
whereUandVare square orthogonal matrices respectively of order s, the sample size, and
n, the number of elementary basis functions, while Sris a rectangular s×nmatrix with
zeroes everywhere except for its leading diagonal elements σ1≥σ2≥σr>0;ris chosen
such that the remaining singular values are sufficiently clos e to zero to be ignored. In our
applications, we ignored all singular values σkwithσk<103σ1ǫdbl, whereǫdblis the double
precision machine accuracy. This seems a reasonable choice , but we have no compelling
argument to justify it.
From Eq. (19) one obtains
E=VrS−1
rUT
rB′, (20)
where Uris thes×rmatrix consisting of the first rcolumns of U;Vris then×rmatrix
likewise obtained from V; andSris ther×rupper left corner of S.
IV. ELEMENTARY BASIS FUNCTIONS
We used elementary basis functions of the general form intro duced in Ref. 10. Rotation
and translation symmetry are built into these functions by w riting them as functions of
all interparticle distances. First of all, we introduce a sc aling function with values that
change appreciably only in the range of interparticle dista nces that occur in the cluster
configurations with appreciable probability. For this purp ose we first introduce a piecewise
linear function f. This function has three parameters: x1<x2<x3, which define the four
linear segments of the continuous function f:
f(x) =
−1 forx≤x1,
0 forx=x2,
1 forx≥x3.(21)
The parameters are determined by the relevant length scales of the system. The parameter
x1sets the scale for how close two atoms can get with reasonable probability; x2roughly
equals the most likely interparticle distance; and x3is the distance at which one expects the
onset of the long distance asymptotic regime. Possibly, one could drop x2and use a simpler
function consisting of three linear segments only.
The function fhas no continuous derivatives and cannot be used directly as a scaling
function. Instead, we use the generalized Gaussian transfo rm
ˆf(x) =/integraldisplay∞
−∞f(x′) exp−(x−x′)2
2cxdx′, (22)withc= 0.1.
In their most general form the wavefunctions in Ref. 10 conta in five-body correlations,
but in the work reported here we have only used three-body cor relations and for completeness
we shall describe the construction of these functions expli citly.
Choose three of the Ncatoms. Suppose they have labels α,β,andγand Cartesian
coordinates rα,rβandrγ. This defines three scaled interatomic distances
ˆrα=ˆf(|rβ−rγ|)
ˆrβ=ˆf(|rγ−rα|)
ˆrγ=ˆf(|rα−rβ|)
(23)
Define three invariants as sums of powers of these variables
Ip= ˆrp
α+ ˆrp
β+ ˆrp
γ (24)
withp= 1,2,3. Clearly, any polynomial in the invariants I1,I2andI3is symmetric with
respect to permutation of the labels α,β,andγ. A convenient property of these variables is
that the reverse is also true: any symmetric polynomial in th e three scaled distances can be
written as a polynomial in the invariants I1,I2andI3. This makes it simple to parameterize
these symmetric polynomials.
In terms of the invariants we define ‘minimal polynomials’ sias follow: pick a monomial
inI1,I2,andI3and sum over all possible ways of choosing three atoms α,β,andγ. These
polynomials are minimal in the sense that one cannot omit any single term without violating
the bosonic symmetry.
In addition to bosonic symmetry, we impose short and long-di stance boundary conditions.
This yields the following form for the elementary basis func tions
βi(R) =si(R) exp
/summationdisplay
jajsj(R)−/summationdisplay
σ<τ/parenleftigg
κkrστ+1
5√mr5
στ/parenrightigg
(25)
with
κk=2
Nc−1/radicaltp/radicalvertex/radicalvertex/radicalbt−m˜Ek
Nc. (26)
As discussed in detail in Ref. 10, the r−5
στterm in the exponent and its coefficient are chosen
so that, when two atoms approach each other, the strongest di vergence in the local energy,
i.e. the Lennard-Jones r−12divergence, is canceled by the divergence in the local kinet ic
energy. The energy ˜Ekis determined self-consistently by iteration; one or two it erations
typically suffice. The specific form of the decay constant is ch osen on the basis of two
assumptions. The energy is assumed to be proportional to the number of atom pairs in the
cluster.11This is reasonable for small clusters, but for larger ones th is should probably be
modified to reflect the expectation that the energy is proport ional to the average number
of nearest neighbor pairs. The second assumption is that if o ne atom is far away from all
others, the wave function can be written as the product of an Nc−1 cluster wavefunction
and an exponentially decaying part that carries a fraction o f the total energy equal to the
number of bonds connecting that atom to the others.Theajin Eq. (25) are non-linear variational parameters. Their op timal values are re-
optimized for each excited state. In principle, one could op timize all non-linear parameters,
including those that appear in the scaling function and the f actors that impose the boundary
conditions. However, it has been our experience that this pr oduces strongly correlated
variational parameters and results in unstable fits.
V. REDUCTION OF VARIATIONAL ERRORS
The linear and non-linear optimization procedures describ ed above are used to gener-
ate basis functions for a correlation function Monte Carlo c alculation,12which increases the
statistical accuracy of the energy estimates and reduces th e systematic errors due to imper-
fections of the variational functions. The number of these b asis functions is much smaller
than the number of elementary basis functions that appear in the linear combinations. The
advantage of not using allelementary basis can be understood as follows.
Suppose that the optimization phase yields states |˜ψ(k)/angbracketrightwithk= 1,...,n′<n. Corre-
lation function Monte Carlo in a statistical sense yields th e basis functions
|˜ψ(i)(t)/angbracketright ≡e−Ht|˜ψ(i)/angbracketright. (27)
Astincreases, the spectral weight of undesirable excited stat es, i.e., states kwithEk>E n′is
decreased. That is desirable, but at the same time all basis s tates approach the groundstate
and therefore become more nearly linearly dependent. More e xplicitly, one has Monte Carlo
estimates of the following the generalization of Eq. (17)
E(t) =N(t)−1H(t), (28)
with
Nij(t) =/angbracketleft˜ψ(i)(t)|˜ψ(j)(t)/angbracketright (29)
and
Hij(t) =/angbracketleft˜ψ(i)(t)|H|˜ψ(j)(t)/angbracketright. (30)
Again, trouble is caused by an ill-conditioned matrix, whic h in this case is N(t), and in-
creasingly so for increasing values of the projection time t. Obviously, the better are the
trial states |˜ψ(i)/angbracketrightand the fewer is their number, the less severe is this problem . We should
also point out in this context that the singular value decomp osition cannot be used in this
case. The reason is that the analogs of the matrices BandB′become too big to store for
Monte Carlo samples of the size required in the correlation f unction Monte Carlo runs.
VI. RESULTS
In Table I we compare results obtained with our Monte Carlo me thod with results of
Leitner et al.,11which were obtained by the discrete variable representatio n (DVR) method.
With the exception of the fifth state of Ne, the Monte Carlo res ults agree with or improvethe DVR results. In some cases, the disagreement can be attri buted to lack of convergence
of the DVR results.13The discrepancy for the fifth state of Ne may be an illustratio n of a
weakness of the correlation function Monte Carlo method, as it is commonly implemented,
namely the difficulty of estimating the statistical and syste matic errors.
There can be problems both with obtaining reliable estimate s of the statistical errors and
with making sure that one has convergence as a function of pro jection time t[cf.Eq. (28)].
This is a consequence of the fact that the data for different va lues of the projection time
are strongly correlated since they are obtained from the sam e Monte Carlo data. Correlated
noise may introduce false trends or obscure true ones, a prob lem that in principle can be
solved by performing independent runs for different project ion times, but that would greatly
increase the computation time.
Unreliable statistical error estimates may come about beca use the correlation function
Monte Carlo calculation takes the form of a pure-diffusion Mo nte Carlo14,15calculation.
The algorithm used for the latter features weights consisti ng of a number of fluctuating
factors proportional to the projection time t. Consequently, as the projection time tin-
creases, the variance of the estimators increases and they a cquire a significantly non-Gaussian
distribution,16which renders error bars computed in the standard way increa singly mislead-
ing. Conceivably, one could reduce the severity of this effec t by using branching random
walks,17as is done in standard diffusion Monte Carlo, or by means of rep tation Monte
Carlo.18
In Table II we present results for the energies of the first five levels of Ar clusters of sizes
four through seven. Our method allows one to go beyond seven a tom clusters, but, as one can
see from Table II, the statistical errors increase with syst em size. To obtain more accurate
results for larger clusters it would probably be helpful to i nclude higher order correlations
in the wavefunction, since the degrees of the polynomials we re chosen sufficiently high that
increasing then further no longer improves the quality of th e trial functions.
Figure 1 contains three energy levels as a function of mass fo r four particle clusters. The
harmonic approximation implies that for large masses the en ergy will be a linear function of
m−1
2. We expect the energy to vanish quadratically in the vicinit y of the dissociation limit.
The results are therefore plotted using variables that yiel d linear dependence both for large
masses and for energies close to zero.19As the of the energy levels approaches zero, both the
optimization and the projection methods begin to fail, and c orrespondingly data points are
missing. Again, the use of trial wavefunctions with four-bo dy correlations is likely to yield
more accurate results for smaller masses.
In the elementary basis functions, we typically used polyno mials of degree ten in the
prefactors and of degree three in the exponent. The diffusion Monte Carlo runs used on the
order of a million steps with a time step of a couple of tenths. The longer runs typically
took a few hours on a four processor SGI Origin 200.TABLE I. Vibrational energy levels Ekof noble gas trimers; the estimated errors are a few units
in the least significant decimal.
k Ne3 Ar3
MC DVR MC DVR
1 -1.719 560 -1.718 -2.553 289 43 -2.553
2 -1.222 83 -1.220 -2.250 185 5 -2.250
3 -1.142 0 -1.138 -2.126 361 -2.126
4 -1.038 -1.035 -1.996 43 -1.996
5 -0.890 -0.898 -1.946 7 -1.947
TABLE II. Vibrational energy levels Ekof Ar clusters; the estimated errors are a few units in
the least significant decimal.
k Ar4 Ar5 Ar6 Ar7
1 -5.118 11 -7.785 1 -10.887 9 -14.191
2 -4.785 -7.567 -10.561 -13.969
3 -4.674 -7.501 -10.51 -13.80
4 -4.530 -7.39 -10.46 -13.74
5 -4.39 -7.36 -10.35 -13.71
-2.5-2-1.5-1-0.50
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35−|Ek|1
2
m−1
2 KrAr Ne He
✛
✻❄ ❄ ❄k= 1 :
✸✸✸✸✸✸✸✸✸✸✸✸✸
k= 2 :
+++++++++++ +
k= 3 :
✷✷✷✷✷✷✷✷✷✷
FIG. 1. −√−Ekfor lowest three vibrational levels ( k= 1,2,3) of four particle clusters vs m−1
2.
The estimated errors for most energies are smaller, than the plot symbols. Results for level k= 3
become unreliable near He and have not been included. The ver tical arrows indicate Kr, Ar, Ne,
and He; the horizontal arrow indicates the classical value -√
6.ACKNOWLEDGMENTS
This research was supported by the (US) National Science Fou ndation (NSF) through
Grant DMR-9725080. It is our pleasure to thank David Freeman and Cyrus Umrigar for
valuable discussions.
1M. P. Nightingale and V. Melik-Alaverdian, submitted to Phy s. Rev. Lett. .
2D. Blume, M. Lewerenz, P. Niyaz, and K.B. Whaley, Phys. Rev. E 55, 3664 (1997). D. Blume
and K. B. Whaley, J. Chem. Phys. 112, 2218 (2000) and references therein.
3J. de Boer, Physica ,14, 139 (1948).
4C.J. Umrigar, K.G. Wilson, and J.W. Wilkins, Phys. Rev. Lett .60, 1719 (1988); C.J. Umrigar,
K.G. Wilson, and J.W. Wilkins, in Computer Simulation Studies in Condensed Matter Physics,
Recent Developments, edited by D.P. Landau K.K. Mon and H.B. Sch¨ uttler, Springer Proceedings
in Physics (Springer, Berlin, 1988).
5D.M. Ceperley and B. Bernu, J. Chem. Phys. 89, 6316 (1988).
6B. Bernu, D.M. Ceperley, and W.A. Lester, Jr., J. Chem. Phys. 93, 552 (1990); W.R. Brown,
W.A. Glauser, and W.A. Lester, Jr., J. Chem. Phys. 103, 9721 (1995).
7J.K.L. MacDonald, Phys. Rev. 43, 830 (1933).
8W.H. Press, B.P. Flannery, S.A. Teukolsky, and W.T. Vetterl ing,Numerical Recipes , (Cambridge
University Press, Cambridge 1992), Section 2.6.
9G.H. Golub and C.F. van Loan, Matrix computations (Second Edition), (Johns Hopkins Univer-
sity Press, 1989) Chapter 5.5.
10Andrei Mushinski and M. P. Nightingale, J. Chem. Phys. 101, 8831, (1994).
11D.M. Leitner, J.D. Doll, and R.M. Whitnell, J. Chem. Phys. 94, 6644 (1991).
12D.M. Ceperley and B. Bernu, J. Chem. Phys. 89, 6316 (1988).
13D.M. Leitner, private communication.
14M. Caffarel and P. Claverie, J. Chem. Phys. 88, 1088 (1988); ibid.p. 1100.
15C.J. Umrigar, M.P. Nightingale, and K.J. Runge, J. Chem. Phy s.99, 2865 (1993) and references
therein; we used a single, weighted walker, and an accept rej ect step with τeff=τfor accepted
moves and τeff= 0 for rejected ones.
16J.H. Hetherington, Phys. Rev. A 30, 2713, 1984.
17M. P. Nightingale in Quantum Monte Carlo Methods in Physics and Chemistry, edited by M. P.
Nightingale and C. J. Umrigar, (NATO Science Series, Kluwer Academic Publisher, Dordrecht,
1999), p. 1.
18S. Baroni and S. Moroni, in Quantum Monte Carlo Methods in Physics and Chemistry, edited
by M. P. Nightingale and C. J. Umrigar, (NATO Science Series, Kluwer Academic Publisher,
Dordrecht, 1999), p. 313.
19M. Meierovich, A. Mushinski, and M.P. Nightingale, J. Chem. Phys.105, 6498 (1996). |
arXiv:physics/0102070v1 [physics.atom-ph] 22 Feb 2001,
High-precision calculations of electric-dipole amplitud es for
transitions between low-lying levels of Mg, Ca, and Sr
S. G. Porsev,∗M. G. Kozlov, and Yu. G. Rakhlina
Petersburg Nuclear Physics Institute,
Gatchina, Leningrad district, 188300, Russia
A. Derevianko
Physics Department, University of Nevada, Reno, Nevada 895 57
(Dated: January 7, 2014)
Abstract
To support efforts on cooling and trapping of alkaline-earth atoms and designs of atomic clocks,
we performed ab initio relativistic many-body calculations of electric-dipole t ransition amplitudes
between low-lying states of Mg, Ca, and Sr. In particular, we report amplitudes for1Po
1→
1S0,3S1,1D2, for3Po
1→1S0,1D2, and for3Po
2→1D2transitions. For Ca, the reduced matrix
element /angbracketleft4s4p1Po
1||D||4s2 1S0/angbracketrightis in a good agreement with a high-precision experimental va lue
deduced from photoassociation spectroscopy [Zinner et al., Phys. Rev. Lett. 85, 2292 (2000) ].
An estimated uncertainty of the calculated lifetime of the 3 s3p1Po
1state of Mg is a factor of
three smaller than that of the most accurate experiment. Cal culated binding energies reproduce
experimental values within 0.1-0.2%.
PACS numbers: 31.10.+z, 31.15.Ar, 31.15.Md, 32.70.Cs
∗Electronic address: porsev@thd.pnpi.spb.ru
1I. INTRODUCTION
Many-body methods have proven to be a highly accurate tool fo r determination of atomic
properties, especially for systems with one valence electr on outside a closed-shell core [1]. For
alkali-metal atoms a comparison of highly-accurate experi mental data with calculations [2]
allows one to draw a conclusion that modern ab initio methods are capable of predicting
basic properties of low-lying states with a precision bette r than 1%.
Fordivalent atoms such a comprehensive comparison was previously hinde red by a lack
of high-precision measurements of radiative lifetimes. De spite the lifetimes of the lowest
nsnp1Po
1andnsnp3Po
1states were repeatedly obtained both experimentally and th eoretically
[3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], persistent
discrepancies remain. Only very recently, Zinner et al. [3] have achieved 0.4% accuracy for
the rate of 4 s4p1Po
1→4s2 1S0transition in calcium. This high-precision value was deduc ed
from photoassociation spectroscopy of ultracold calcium a toms. One of the purposes of the
present work is to test the quality of many-body techniques f or two-valence electron systems
by comparing our result with the experimental value from Ref . [3].
We extend the earlier work [25] and report results of relativ istic many-body calcula-
tion of energy levels and electric-dipole transition ampli tudes for Mg, Ca and Sr. The
calculations are performed in the framework of configuratio n-interaction approach coupled
with many-body perturbation theory [26, 27]. We tabulate el ectric-dipole amplitudes for
1Po
1→1S0,3S1,1D2, for3Po
1→1S0,1D2, and for3Po
2→1D2transitions and estimate
theoretical uncertainties.
Cooling and trapping experiments with alkaline-earth atom s were recently reported for
Mg [4], Ca [3, 29], and Sr [28, 30]. The prospects of achieving Bose-Einstein condensation
were also discussed [3, 31]. Our accurate transition amplit udes will be helpful in designs
of cooling schemes and atomic clocks. In addition, these amp litudes will aid in determi-
nation of long-range atomic interactions, required in calc ulation of scattering lengths and
interpretation of cold-collision data. For example, dispe rsion (van der Waals) coefficient
C6characterizes the leading dipole-dipole interaction of tw o ground-state atoms at large
internuclear separations [32]. The coefficient C6is expressed in terms of energy separations
and electric-dipole matrix elements between the ground and excited atomic states. Approx-
imately 80% of the total value of C6arises from the principal transition nsnp1Po
1−ns2 1S0,
requiring accurate predictions for the relevant matrix ele ment. Therefore our results will be
also useful in determination of dispersion coefficients.
II. METHOD OF CALCULATIONS
In atomic-structure calculations, correlations are conve ntionally separated into three
classes: valence-valence, core-valence, and core-core co rrelations. A strong repulsion of
valence electrons has to be treated non-perturbatively, wh ile it is impractical to handle the
two other classes of correlations with non-perturbative te chniques, such as configuration-
interaction (CI) method. Therefore, it is natural to combin e many-body perturbation theory
(MBPT) with one of the non-perturbative methods. It was sugg ested [26] to use MBPT to
construct an effective Hamiltonian Heffdefined in the model space of valence electrons. En-
ergies and wavefunctions of low-lying states are subsequen tly determined using CI approach,
i.e. diagonalizing Heffin the valence subspace. Atomic observables are calculated with ef-
fective operators [27]. Following the earlier work, we refe r to this method as CI+MBPT
2formalism.
In the CI+MBPT approach the energies and wavefunctions are d etermined from the
Schr¨ odinger equation
Heff(En)|Φn/angbracketright=En|Φn/angbracketright, (1)
where the effective Hamiltonian is defined as
Heff(E) =HFC+ Σ(E). (2)
HereHFCis the two-electron Hamiltonian in the frozen core approxim ation and Σ is the
energy-dependent correction, involving core excitations . The operator Σ completely accounts
for the second order of perturbation theory. Determination of the second order corrections
requires calculation of one– and two–electron diagrams. Th e one–electron diagrams describe
an attraction of a valence electron by a (self-)induced core polarization. The two-electron
diagrams are specific for atoms with several valence electro ns and represent an interaction
of a valence electron with core polarization induced by anot her valence electron.
Already at the second order the number of the two–electron di agrams is large and their
computation is very time-consuming. In the higher orders th e calculation of two-electron
diagrams becomes impractical. Therefore we account for the higher orders of MBPT indi-
rectly. It was demonstrated [33] that a proper approximatio n for the effective Hamiltonian
can substantially improve an agreement between calculated and experimental spectra of
multielectron atom. One can introduce an energy shift δand replace Σ( E)→Σ(E−δ)
in the effective Hamiltonian, Eq. (2). The choice δ=0 corresponds to the Brillouin-Wigner
variant of MBPT and the Rayleigh-Schr¨ odinger variant is re covered setting δ=En−E(0)
n,
where E(0)
nis the zero-order energy of level n. The latter is more adequate for multielectron
systems [34]; for few-electron systems an intermediate val ue ofδis optimal. We have deter-
mined δfrom a fit of theoretical energy levels to experimental spect rum. Such an optimized
effective Hamiltonian was used in calculations of transitio n amplitudes.
To obtain an effective electric-dipole operator we solved ra ndom-phase approximation
(RPA) equations, thus summing a certain sequence of many-bo dy diagrams to all orders
of MBPT. The RPA describes a shielding of externally applied field by core electrons. We
further incorporated one- and two-electron corrections to the RPA to account for a difference
between the VNand VN−2potentials and for the Pauli exclusion principle. In additi on, the
effective operator included corrections for normalization and structural radiation [27]. The
RPA equations depend on transition frequency and should be s olved independently for each
transition. However, the frequency dependence was found to be rather weak and we solved
these equations only at some characteristic frequencies. T o monitor a consistency of the
calculations we employed both length (L) and velocity (V) ga uges for the electric-dipole
operator.
The computational procedure is similar to calculations of h yperfine structure constants
and electric-dipole amplitudes for atomic ytterbium [35, 3 6]. We consider Mg, Ca and Sr as
atoms with two valence electrons above closed cores [1 s,...,2p6], [1s,...,3p6], and [1 s,...,4p6],
respectively [37]. One-electron basis set for Mg included 1 s–13s, 2p–13p, 3d–12d, and 4 f–
11forbitals, where the core- and 3,4 s, 3,4p, 3,4d, and 4 forbitals were Dirac-Hartree-Fock
(DHF) ones, while all the rest were virtual orbitals. The orb itals 1 s–3swere constructed by
solving the DHF equations in VNapproximation, 3 porbitals were obtained in the VN−1ap-
proximation, and 4 s, 4p, 3,4d, and 4 forbitals were constructed in the VN−2approximation.
We determined virtual orbitals using a recurrent procedure , similar to Ref. [38] and employed
in previous work [26, 27, 35, 36]. The one-electron basis set for Ca included 1 s–13s, 2p–13p,
33d–12d, and 4 f–11forbitals, where the core- and 4 s, 4p, and 3 dorbitals are DHF ones,
while the remaining orbitals are the virtual orbitals. The o rbitals 1 s–4swere constructed by
solving the DHF equations in the VNapproximation, and 4 pand 3dorbitals were obtained
in the VN−1approximation. Finally, the one-electron basis set for Sr i ncluded 1 s–14s, 2p–
14p, 3d–13d, and 4 f–13forbitals, where the core- and 5 s, 5p, and 4 dorbitals are DHF ones,
and all the rest are the virtual orbitals. The orbitals 1 s–5swere constructed by solving
the DHF equations in the VNapproximation, and 5 pand 4dorbitals were obtained in the
VN−1approximation. Configuration-interaction states were for med using these one-particle
basis sets. It is worth emphasizing that the employed basis s ets were sufficiently large to
obtain numerically converged CI results. A numerical solut ion of random-phase approxima-
tion equations required an increase in the number of virtual orbitals. Such extended basis
sets included 1 s–ks, 2p–kp, 3d–(k-1)d, 4f–(k-4)f, and 5 g–(k-8)gorbitals, where k=19,20,21
for Mg, Ca, and Sr, respectively. Excitations from all core s hells were included in the RPA
setup.
III. RESULTS AND DISCUSSION
A. Energy levels
In Tables I – III we present calculated energies of low-lying states for Mg, Ca, and
Sr and compare them with experimental values. The two-elect ron binding energies were
obtained both in the framework of conventional configuratio n-interaction method and using
the formalism of CI coupled with many-body perturbation the ory. Already at the CI stage
the agreement of the calculated and experimental energies i s at the level of 5%. The addition
of many-body corrections to the Hamiltonian improves the ac curacy by approximately an
order of magnitude. Finally, with an optimal choice of param eterδthe agreement with
experimental values improves to 0.1–0.2%.
Compared to the binding energies, fine-structure splitting of triplet states and singlet-
triplet energy differences represent a more stringent test o f our method. For the3Po
1,2,3-states
the fine-structure splitting is reproduced with an accuracy of several per cent in the pure CI
for all the three atoms, while the3Po
1–1Po
1energy differences are less accurate (especially
for Ca and Sr). As demonstrated in Ref. [33], the two-electro n exchange Coulomb integral
Rnp,ns,ns,np (n=3,4,5 for Mg, Ca, and Sr, respectively) determining the spl itting between3Po
1
and1Po
1states is very sensitive to many-body corrections. Indeed, with these corrections
included, the agreement with the experimental data improve s to 1-2% for all the three atoms.
The case of the even-parity3,1DJ-states is even more challenging. For Ca, these four states
are practically degenerate at the CI stage. A repulsion of th e level1D2from the upper-lying
levels of np2configuration pushes it down to the level3D2and causes their strong mixing.
As seen from Table II these states are separated only by 10 cm−1, while the experimental
energy difference is 1550 cm−1. As a result, an accurate description of superposition of
3D2and1D2states is important. The3D2–1D2splitting is restored when the many-body
corrections are included in the effective Hamiltonian. Thes e corrections properly account
for core polarization screening an interaction between sdandp2configurations.
For Sr, the fine-structure splittings of3DJstates and energy difference between the3DJ
and the1D2levels are also strongly underestimated in the pure CI metho d. Again the
inclusion of the many-body corrections substantially impr oves the splittings between the D-
states. It is worth emphasizing, that for such an accurate an alysis a number of effects was
4taken into account, i.e., spin-orbit interaction, configur ation interaction, and core-valence
correlations. A proper account for all these effects is of par ticular importance for determi-
nation of electric-dipole amplitudes forbidden in LS-coupling, such as for3Po
J→1S0,1D2
transitions.
B. Transition amplitudes
In this section we present calculations of electric-dipole (E1) amplitudes for3,1Po
1→1S0,
3,1Po
1→1D2,3Po
2→1D2, and1Po
1→3S1transitions. The calculated reduced matrix
elements for Mg, Ca, and Sr are presented in Tables IV and V. Fo r a transition I→Fthe
Einstein rate coefficients for spontaneous emission (in 1 /s) are expressed in terms of these
reduced matrix elements /angbracketleftF||D||I/angbracketright(a.u.) and wavelengths λ(˚A) as
AFI=2.02613 ×1018
λ3|/angbracketleftF||D||I/angbracketright|2
2JI+ 1. (3)
A number of long-range atom-atom interaction coefficients co uld be directly obtained from
the calculated matrix elements. At large internuclear sepa rations Ran atom in a state |A/angbracketright
predominantly interacts with a like atom in a state |B/angbracketrightthrough a potential V(R)≈ ±C3/R3,
provided an electric-dipole transition between the two ato mic states |A/angbracketrightand|B/angbracketrightis allowed.
The coefficient C3is given by
|C3|=|/angbracketleftA||D||B/angbracketright|21/summationdisplay
µ=−1(1 +δµ,0)/parenleftBigg
JA1JB
−Ω+µ
2µΩ−µ
2/parenrightBigg2
, (4)
where Ω is the conventionally defined sum of projections of to tal angular momenta on in-
ternuclear axis.
From a solution of the eigen-value problem, Eq. (1), we obtai ned wave functions, con-
structed effective dipole operators, and determined the tra nsition amplitudes. The calcula-
tions were performed within both traditional configuration -interaction method and CI cou-
pled with the many-body perturbation theory. The compariso n of the CI and the CI+MBPT
values allows us to estimate an accuracy of our calculations . As it was mentioned above, to
monitor the consistency of the calculations, we determined the amplitudes using both length
and velocity gauges for the dipole operator. In general, dip ole amplitudes calculated in the
velocity gauge are more sensitive to many-body corrections ; we employ the length form of
the dipole operator in our final tabulation.
We start the discussion with the amplitudes for the principa lnsnp1Po
1→ns2 1S0tran-
sitions ( n= 3 for Mg, n= 4 for Ca, and n= 5 for Sr). Examination of Table IV reveals
that the many-body effects reduce the L-gauge amplitudes by 1 .6% for Mg, 5.5% for Ca,
and 6.4% for Sr. Further, the MBPT corrections bring the leng th and velocity-form results
into a closer agreement. For example, for Sr at the CI level th e velocity and length forms
differ by 2.7% and this discrepancy is reduced to 0.8% in the CI +MBPT calculations.
A dominant theoretical uncertainty of the employed CI+MBPT method is due to im-
possibility to account for all the orders of many-body pertu rbation theory. It is worth
emphasizing that in our CI calculations the basis sets were s aturated and the associated
numerical errors were negligible. We expect that the theore tical uncertainty is proportional
to the determined many-body correction. In addition, we tak e into account the proximity
of the amplitudes obtained in the L- and V-gauges. We estimat e the uncertainties for the
5nsnp1Po
1→ns2 1S0transition amplitudes as 25–30% of the many-body correctio ns in the
length gauge. The final values for /angbracketleftnsnp1Po
1||D||ns2 1S0/angbracketright, recommended from the present
work, are 4.03(2) for Mg, 4.91(7) for Ca, and 5.28(9) a.u. for Sr.
We present a comparison of our results for /angbracketleftnsnp1Po
1||D||ns2 1S0/angbracketrightwith experimental data
in Table IV and in Fig. 1. Our estimated accuracy for Mg is a fac tor of three better than that
of the most accurate experiment and for Sr is comparable to ex perimental precision. For
Ca, the dipole matrix element of the1Po
1→1S0was recently determined with a precision of
0.2% by Zinner et al. [3] using photoassociation spectroscopy of ultracold Ca at oms. While
our result is in harmony with their value, the experimental a ccuracy is substantially better.
An updated analysis [40] of photoassociation spectra of Ref . [3] leads to a somewhat better
agreement with our calculated value.
A very extensive compilation of earlier theoretical result s for the1Po
1→1S0transition
amplitudes can be found in Ref. [6] for Mg and in Ref. [9] for Ca . In a very recent mul-
ticonfiguration Hartree-Fock (MCHF) calculations for Mg [7 ] the authors have determined
/angbracketleft3s3p1Po
1||D||3s2 1S0/angbracketright= 4.008 a.u. This value agrees with our final result of 4.03(2) a.u .
For heavier Sr the correlation effects are especially pronou nced and only a few calculations
were performed. For example, MCHF calculations for Sr [8] fo und in the length gauge
/angbracketleft5s5p1Po
1||D||5s2 1S0/angbracketright= 5.67 a.u. By contrast to the present work, the core-polarizati on
effects were not included in this analysis. As a result, this c alculated value is in a good
agreement with our result 5.63 a.u. obtained at the CI stage, but differs from the final value
5.28(9) a.u.
Another nonrelativistically allowed transition is1Po
1→1D2and one could expect that
this amplitude can be determined with a good accuracy. For Mg this is really so. However,
for Ca and Sr an admixture of the configuration p2brings about large corrections to this
amplitude, especially in the velocity gauge. Another compl ication is the following. The
matrix element of electric-dipole operator can be represen ted in the V-gauge as (atomic
units ¯h=|e|=me= 1 are used):
/angbracketleftF|D|I/angbracketright=i c/angbracketleftF|α|I/angbracketright/(EI−EF). (5)
Herecis the speed of light, EIandEFare the energies of initial and final states, and α
are the Dirac matrices. For the1Po
1→1D2transition in Ca and Sr the energy denominator
is approximately 0.01 a.u. Because the E1-amplitudes of the se transitions ∼1 a.u. (see
Table IV), the respective numerators are of the order of 0.01 a.u. Correspondingly the matrix
elements /angbracketleftF|α|I/angbracketrightare small and are very sensitive to corrections, i.e., the V- gauge results are
unstable. As a result we present only the L-gauge values for1Po
1→1D2E1 amplitudes for
Ca and Sr. An absence of reliable results in V-gauge hampers a n estimate of the accuracy,
so we rather conservatively take it to be 25%. Note that even w ith such a large uncertainty
our value for Sr significantly differs from the experimental v alue [24]. The measurement in
[24] has been carried out on the1D2→1S0transition and an interference between electric-
quadrupole (E2) and Stark-induced dipole amplitudes was ob served. In order to determine
the transition rate a theoretical value of the E2-amplitude for the1D2→1S0transition was
taken from [41]. It may be beneficial either to measure direct ly the rate of the E1-transition
1Po
1→1D2or to measure the rate of the E2-transition1D2→1S0.
For the3Po
J→1S0,1D2transitions the respective E1-amplitudes are small; these are non-
relativistically forbidden intercombination transition s and consequently their amplitudes are
proportional to spin-orbit interaction. The calculated re duced matrix elements are presented
in Table V.
6One can see from Tables I – III that the MBPT corrections to the fine structure splittings
are large, amplifying significance of higher order many-bod y corrections. In addition, higher
order corrections in the fine-structure constant αto the Dirac-Coulomb Hamiltonian are
also important here. As demonstrated in Ref. [6], the Breit i nteraction reduces the dipole
amplitude of3Po
1→1S0transition in Mg by 5%. At the same time for all the intercom-
bination transitions the agreement between L- and V-gauges is at the level of 6-8%. We
conservatively estimate the uncertainties of the calculat ed intercombination E1 amplitudes
to be 10–12%.
To reiterate, we carried out calculations of energies of low -lying levels and electric-dipole
amplitudes between them for divalent atoms Mg, Ca, and Sr. We employed ab initio rela-
tivistic configuration interaction method coupled with man y-body perturbation theory. The
calculated removal energies reproduce experimental value s within 0.1-0.2%. A special em-
phasis has been put on accurate determination of electric-d ipole amplitudes for principal
transitions nsnp1Po
1→ns2 1S0. For these transitions, we estimated theoretical uncertai nty
to be 0.5% for Mg, 1.4% for Ca, and 1.7% for Sr. For Ca, the reduc ed matrix element
/angbracketleft4s4p1Po
1||D||4s2 1S0/angbracketrightis in a good agreement with a high-precision experimental va lue [3].
An estimated uncertainty of the calculated lifetime of the l owest1Po
1state for Mg is a
factor of three smaller than that of the most accurate experi ment. In addition, we evalu-
ated electric-dipole amplitudes and estimated theoretica l uncertainties for1Po
1→3S1,1D2,
3Po
1→1S0,1D2, and for3Po
2→1D2transitions. Our results could be useful in designs of
cooling schemes and atomic clocks, and for accurate descrip tion of long-range atom-atom
interactions needed in interpretation of cold-collision d ata.
Acknowledgments
We would like to thank H. Katori, C. Oates, and F. Riehle for st imulating discussions.
This work was supported in part by the Russian Foundation for Basic Researches (grant
No 98-02-17663). The work of A.D. was partially supported by the Chemical Sciences,
Geosciences and Biosciences Division of the Office of Basic En ergy Sciences, Office of Science,
U.S. Department of Energy.
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7TABLE I: Two-electron binding energies E valin a.u. and energy differences ∆ (cm−1) for low-lying
levels of Mg.
CI CI+MBPT Experiment
Config. Level E val ∆ E val ∆ E val ∆
3s2 1S0 0.819907 — 0 .833556 — 0 .833518a—
3s4s3S1 0.635351 40505 0 .645853 41196 0 .645809 41197.4
3s4s1S0 0.624990 42779 0 .635283 43516 0 .635303 43503.1
3s3d1D2 0.613603 45278 0 .621830 46469 0 .622090 46403.1
3s3p3Po
0 0.724170 21012 0 .733896 21879 0 .733961 21850.4
3s3p3Po
1 0.724077 21032 0 .733796 21901 0 .733869 21870.4
3s3p3Po
2 0.723889 21073 0 .733596 21945 0 .733684 21911.1
3s3p1Po
1 0.662255 34601 0 .674226 34975 0 .673813 35051.4
aTwo electron binding energy of the ground state is determine d as a sum of the first two
ionization potentials IP (Mg) and IP (Mg+), where IP (Mg) = 61669.1 cm−1and
IP (Mg+)= 121267 .4 cm−1[39].
TABLE II: Two-electron binding energies in a.u. and energy d ifferences ∆ in cm−1for the low-lying
levels of Ca.
CIaCI+MBPT Experiment
Config. Level E val ∆ E val ∆ E val ∆
4s2 1S0 0.636590 — 0 .661274 — 0 .660927b—
4s3d3D1 0.528838 23649 0 .567744 20527 0 .568273 20335.3
4s3d3D2 0.528868 23642 0 .567656 20547 0 .568209 20349.2
4s3d3D3 0.528820 23653 0 .567517 20577 0 .568110 20371.0
4s3d1D2 0.528824 23652 0 .559734 22285 0 .561373 21849.6
4s5s3S1 0.498205 30372 0 .517490 31557 0 .517223 31539.5
4s4p3Po
0 0.574168 13700 0 .591521 15309 0 .591863 15157.9
4s4p3Po
1 0.573942 13750 0 .591274 15363 0 .591625 15210.1
4s4p3Po
2 0.573486 13850 0 .590774 15473 0 .591143 15315.9
4s4p1Po
1 0.530834 23211 0 .553498 23654 0 .553159 23652.3
aNote that the conventional CI fails to recover the correct or dering of D-states.bFor the ground
state E val= IP (Ca)+IP (Ca+), where IP (Ca) = 49304.8 cm−1and IP (Ca+) = 95752.2 cm−1
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8TABLE III: Two-electron binding energies in a.u. and energy differences ∆ in cm−1for the low-
lying levels of Sr.
CI CI+MBPT Experiment
Config. Level E val ∆ E val ∆ E val ∆
5s2 1S0 0.586538 — 0 .614409 — 0 .614601a—
5s4d3D1 0.497148 19619 0 .532110 18063 0 .531862 18159.1
5s4d3D2 0.497077 19635 0 .531809 18129 0 .531590 18218.8
5s4d3D3 0.496941 19664 0 .531298 18242 0 .531132 18319.3
5s4d1D2 0.494339 20235 0 .522311 20213 0 .522792 20149.7
5s6s3S1 0.460940 27566 0 .481533 29162 0 .482291 29038.8
5s5p3Po
0 0.529636 12489 0 .548754 14410 0 .549366 14317.5
5s5p3Po
1 0.528850 12662 0 .547896 14598 0 .548514 14504.4
5s5p3Po
2 0.527213 13021 0 .546079 14997 0 .546718 14898.6
5s5p1Po
1 0.491616 20833 0 .515901 21621 0 .515736 21698.5
aFor the ground state E val= IP (Sr)+IP (Sr+), where IP (Sr) = 45925.6 cm−1and IP (Sr+) =
88964.0 cm−1[39].
TABLE IV: Reduced electric-dipole matrix elements for tran sitions allowed in LS-coupling. nis
the principal quantum number of the first valence sandpshells and mcorresponds to the first
valence dshell; n= 3 for Mg, 4 for Ca, and 5 for Sr; m= 3 for Mg and Ca, and 4 for Sr. All values
are in a.u.
Mg Ca Sr
CI CI+MBPT CI CI+MBPT CI CI+MBPT
/angbracketleftnsnp1Po
1||D||ns2 1S0/angbracketright
L-gauge 4 .09 4 .03 5 .20 4 .91 5 .63 5 .28
V-gauge 4 .06 4 .04 5 .11 4 .93 5 .48 5 .32
Final value 4 .03(2) 4 .91(7) 5 .28(9)
Experiment 4 .15(10)a4.967(9)b5.57(6)c
4.06(10)d4.99(4)c5.40(8)e
4.12(6)f4.93(11)g
/angbracketleftnsnp1Po
1||D||nsmd1D2/angbracketright
L-gauge 4 .43 4 .62 1 .16 1 .75 1 .92
V-gauge 4 .47 4 .59
Final value 4 .62(5) 1 .2(3) 1 .9(4)
Experiment 1.24(18)h
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principal quantum number of the first valence sandpshells and mcorresponds to the first valence
dshell; n= 3 for Mg, 4 for Ca, and 5 for Sr; m= 3 for Mg and Ca, and 4 for Sr. All values are in
a.u.
Mg Ca Sr
CI CI+MBPT CI CI+MBPT CI CI+MBPT
/angbracketleftnsnp3Po
1||D||ns2 1S0/angbracketright
L-gauge 0 .0055 0 .0064 0 .027 0 .034 0 .12 0 .16
V-gauge 0 .0062 0 .0062 0 .030 0 .032 0 .13 0 .17
Final value 0 .0064(7) 0 .034(4) 0 .160(15)
Experiment 0 .0053(3)a0.0357(4)b0.1555(16)c
0.0056(4)d0.0352(10)e0.1510(18)e
0.0061(10)f0.0357(16)g0.1486(17)h
/angbracketleftnsnp1Po
1||D||ns(n+ 1)s3S1/angbracketright
L-gauge 0 .0088 0 .0097 0 .035 0 .043 0 .15 0 .19
V-gauge 0 .0089 0 .0101 0 .035 0 .045 0 .15 0 .20
Final value 0 .0097(10) 0 .043(5) 0 .19(2)
/angbracketleftnsnp3Po
1||D||nsmd1D2/angbracketright
L-gauge 0 .0052 0 .0049 0 .059 0 .33 0 .19
V-gauge 0 .0050 0 .0047 0 .061 0 .36 0 .18
Final value 0 .0049(5) 0 .059(6) 0 .19(2)
/angbracketleftnsnp3Po
2||D||nsmd1D2/angbracketright
L-gauge 0 .0039 0 .0031 0 .028 0 .15 0 .10
V-gauge 0 .0041 0 .0032 0 .024 0 .16 0 .06
Final value 0 .0031(4) 0 .028(3) 0 .10(2)
aRef. [16];bRef. [17];cRef. [18];dRef. [19];eRef. [20];fRef. [21];gRef. [22];hRef. [23].
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[35] S. G. Porsev, Yu. G. Rakhlina, and M. G. Kozlov, J. Phys. B 32, 1113 (1999).
[36] S. G. Porsev, Yu. G. Rakhlina, and M. G. Kozlov, Phys. Rev . A60, 2781 (1999).
103.9 4 4.1 4.2 4.3/K54/K68/K69/K73/K20/K77/K6F/K72/K6B/K4C/K69/K6C/K6A/K65/K62/K79/K20 /K65/K74/K20/K61/K6C/K2E /K20/K28/K31/K39/K38/K30/K29/K4C/K75/K6E/K64/K69/K6E/K20 /K65/K74/K20/K61/K6C/K2E /K20/K28/K31/K39/K37/K33/K29/K53/K6D/K69/K74/K68/K20/K61/K6E/K64/K20/K47/K61/K6C/K6C/K61/K67/K68/K65/K72/K20/K28/K31/K39/K36/K36/K29
4.84.854.94.95 5 5.05 5.1/K54/K68/K69/K73/K20/K77/K6F/K72/K6B/K5A/K69/K6E/K6E/K65/K72/K20 /K65/K74/K20/K61/K6C/K2E /K20/K28/K32/K30/K30/K30/K29/K4B/K65/K6C/K6C/K79/K20/K61/K6E/K64/K20/K4D/K61/K74/K68/K75/K72/K20/K28/K31/K39/K38/K30/K29/K48/K61/K6E/K73/K65/K6E/K20/K28/K31/K39/K38/K33/K29
5.1 5.2 5.3 5.4 5.5 5.6 5.7/K54/K68/K69/K73/K20/K77/K6F/K72/K6B/K4B/K65/K6C/K6C/K79/K20/K61/K6E/K64/K20/K4D/K61/K74/K68/K75/K72/K20/K28/K31/K39/K38/K30/K29/K50/K61/K72/K6B/K69/K6E/K73/K6F/K6E/K20 /K65/K74/K20/K61/K6C/K2E /K20/K28/K31/K39/K37/K36/K29/K4D/K67
/K43/K61
/K53/K72
FIG. 1: Comparison of calculated reduced matrix elements /angbracketleftnsnp1Po
1||D||ns2 1S0/angbracketrightwith experi-
mental data in a.u.
[37] Although the calculations were ab initio relativistic, for brevity we suppress total angular
momentum jin the designations of orbitals.
[38] P. Bogdanovich and G. ˇZukauskas, Sov. Phys. Collection, 23, 13 (1983).
[39] C. E. Moore, Atomic Energy Levels , Natl. Bur. Stand. (U.S.) Circ. No. 467 (U.S., Washington,
1958).
[40] E. Tiemann, private communication.
[41] C. W. Bauschlicher Jr, S. R. Langhoff, and H. Partridge, J . Phys. B 18, 1523 (1985).
11 |
arXiv:physics/0102071v1 [physics.plasm-ph] 22 Feb 2001Electron Mobility Maximum in Dense Argon
Gas at Low Temperature
A.F.Borghesani
Istituto Nazionale per la Fisica della Materia
Department of Physics, University of Padua
Via F. Marzolo8, I-35131 Padua, Italy1
Abstract
We report measurements of excess electron mobility in dense Argon gas at the
two temperatures T= 152 .15 and 162 .30 K, fairly close to the critical one ( Tc=
150.7 K), as a function of the gas density Nup to 14 atoms ·nm−3(Nc= 8.08
atoms·nm−3). For the first time a maximum of the zero-field density-norma lized
mobility µ0Nhas been observed at the same density where it was detected in
liquid Argon under saturated vapor pressure conditions. Th e existence of the µ0N
maximum in the liquid is commonly attributed to electrons sc attering off long–
wavelength collective modes of the fluid, while for the low–d ensity gas a density–
modified kinetic model is valid. The presence of the µ0Nmaximum also in the gas
phase raises therefore the question whether the single scat tering picture valid in the
gas is valid even at liquid densities.
Key words: electron mobility, kinetic theory, dense gases, multiple s cattering
effects, disordered systems.
PACS: 51.50+v, 52.25.Fi
1 Introduction
The study of the transport properties of excess electrons in dense non
polar gases gives important pieces of information on the nat ure, dynamics,
and energetics of the states of excess electrons in disorder ed systems.
In the neighborhood of the critical point of the liquid–vapo r transition
the density can be varied in a large interval with a reasonabl y small change of
1E-mail:borghesani@padova.infm.it
Preprint submitted to Elsevier Preprint 2 February 2008pressure and therefore it is relatively easy to investigate how the nature of the
electronic states evolve starting from the dilute gas regim e, where the kinetic
theory is appropriate, towards the liquid regime.
The central quantity of interest is the electron mobility µ.It is defined
as the ratio of the mean velocity vD,acquired by an electron drifting in the
medium under the action of an externally applied and uniform electric field,
and the strength of the field E:µ=vD/E.Its zero–field limit, µ0= lim
E→0µ,is
related to the fundamental properties of the e−atom interaction. In the low
density regime, the kinetic theory relates µ0to the electron–atom momentum
transfer scattering cross section σmtaccording to the following equation [1]
µ0N=4e
3 (2πm)1/2(kBT)5/2∞/integraldisplay
0ǫ
σmt(ǫ)e−ǫ/kBTdǫ (1)
where Nis the number density of the gas. eandmare the electron charge
and mass, respectively. Tis the gas temperature and ǫthe electron energy.
µ0Nis the zero–field density–normalized mobility.
For a given electron–atom cross section, Eq. (1) predicts th atµ0Ndepends
only on the gas temperature and is independent of the gas dens ity. In contrast
with the prediction of the classical kinetic theory [1], the electron mobility
shows a strong dependence on the density of the medium [2]. It is therefore
important to study the effect of the environment on the electr on–atom scat-
tering mechanisms in a relatively dense phase.
Large deviations from the classical prediction, called anomalous density
effects , have been observed even in the simplest systems represente d by the
noble gases [2]. In gases, such as He [3] and Ne [4,5], where th e short–range
repulsive exchange forces dominate the electron–atom inte raction, there is a
negative density effect , namely, µ0Ndecreases with increasing Nand even-
tually drops rapidly to very low values because of the format ion of localized
electron states at high enough densities and in the liquid [6 ,7].
The situation in Ar is different. Here, the e−atom scattering at low en-
ergies is essentially determined by the long–range attract ive polarization in-
teraction because the atomic polarizability of Ar is quite l arge. Owing to this
feature, µ0Nshows a positive density effect , i.e., it increases with increasing N
[8]. A further relevant feature of Ar and of the heavier noble gases is that in the
liquid the electron mobility has a value comparable to that i n the crystalline
state [9]. It is commonly believed that this characteristic s of the mobility is
due to the existence, also in the liquid, of a conduction band . Therefore, it
is interesting to investigate, as a function of the density o f the medium, the
transition from the classical single scattering situation in the dilute gas to the
2multiple scattering scenario at higher densities and the ev entual formation of
extended (or localized) electron states in the liquid.
The explanation of the different density effects in the mobility is commonly
based on the realization that the average interatomic dista nce at high densities
becomes comparable to the electron de Broglie wavelength λ.In this situation,
the conditions for single scattering break down and quantum effects become
important. Moreover, also the mean free path ℓbecomes comparable to λand
multiple scattering effects come into play, too [2].
Recent and accurate measurements of mobility in Ne [4,5] and Ar [10,11]
have put into evidence that the different behavior of the mobi lity in different
gases can be rationalized into an unified picture, where all t he multiple scat-
tering effects are taken into account in a heuristic way. A mod el (henceforth
known as the BSL model) has been developed that incorporates all features
of the several models proposed to interpret the different den sity effects and
merges the several multiple scattering effects into the sing le scattering picture
of kinetic theory [10].
Three main multiple scattering effects have been identified a nd all of them
stem from the fact that the electron mean free path becomes co mparable to
its wavelength and that the latter may also become larger tha n the average
interatomic distance if the density is large enough. The firs t effect is a density–
dependent quantum shift V0(N) of the ground state energy of an excess elec-
tron immersed in the medium. According to the SJC model [12] V0(N),can
be written as
V0(N) =UP(N) +Ek(N) (2)
UPis a potential energy contribution arising from the screene d polarization
interaction of the electron with the surrounding atoms. Ek(N) is a kinetic
energy term, essentially due to excluded volume effects beca use the volume
accessible to electrons shrinks as the density is increased . Owing to its nature,
Ekis positive and increases with increasing N.An expression for it is obtained
by imposing on the electron ground–state wave function the c onditions of
average traslational simmetry about the equivalent Wigner –Seitz (WS) cell
centered about each atom of the gas. V0may be either >0 (this is the case
of He [13] and Ne [14]) or <0 (as for Ar [15,16]), depending on the relative
sizes of UPandEk.However, the experimental mobility results indicate that
only the kinetic energy term Ekhas to be added to the true electron kinetic
energy when the scattering properties (namely, the cross se ctions) have to be
calculated. Differently stated, the bottom of the electron e nergy distribution
function is shifted by the amount Ek[5].
The second multiple scattering effect is an enhancement of el ectron backscat-
3tering due to quantum self–interference of the electron wav e function scattered
off atoms located along paths which are connected by time–rev ersal simmetry
[17]. This phenomenon is closely related to the weak localization regime of the
electronic conduction in disordered solids and to the Ander son localization
transition [18]. It depends on the ratio of the electron wave length to its mean
free path λ/ℓ=Nσmtλ.For Ar, at the density of the experiments, Nσmtλ <1.
Therefore, a linearized treatment of this effect due to Atraz hevet al. can be
adopted [19]. The net result is that the scattering cross sec tion is enhanced
by the factor (1 + Nσmtλ/π).
Finally, the third multiple scattering effect is due to corre lations among
scatterers. The electron wave packet encompasses a region c ontaining several
atoms, especially at low temperature and high density, and i s scattered off all of
them simultaneously. The total scattered wave packet is obt ained by summing
up coherently all partial scattering amplitudes contribut ed by each atoms. The
net result is that the cross section is weighted by the static structure factor of
the fluid which is related to the gas isothermal compressibil ity [20].
In the density–modified kinetic model (BSL model) [10], the d ensity–nor-
malized mobility is calculated according to the classical k inetic theory equa-
tions [1] with the modifications necessary to take into accou nt the mentioned
multiple scattering effects
µN=−/parenleftbigge
3/parenrightbigg/parenleftbigg2
m/parenrightbigg1/2∞/integraldisplay
0ǫ
σ⋆
mt(ǫ+Ek)dg
dǫdǫ (3)
g(ǫ) is the Davydov–Pidduck electron energy distribution func tion [21,22]
g(ǫ) =Aexp
−ǫ/integraldisplay
0
kBT+M
6mz/parenleftBiggeE
Nσ⋆
mt/parenrightBigg2
−1
dz
(4)
where Mis the Ar atomic mass. gis normalized as/integraltext∞
0z1/2g(z) dz= 1.
The multiple scattering effects act by dressing the cross sec tion so that
the effective momentum–transfer scattering cross section i s given by [10]
σ⋆
mt(w) =F(w)σmt(w)/bracketleftBigg
1 +2/planckover2pi1NF(w)σmt(w)
(2mw)1/2/bracketrightBigg
(5)
w=ǫ+Ek(N) is the electron energy shifted by the kinetic zero–point en ergy
contribution Ek.The group velocity of the wave packet is v= [2(w−Ek)/m]1/2
and it only contributes to the energy equipartition value ar ising from the gas
4temperature [22]. The Lekner factor F[20] takes into account correlations
among scatterers
F(k) =1
4k42k/integraldisplay
0q3S(q) dq (6)
withk2= 2mǫ//planckover2pi12.This is equivalent to the formulation given elsewhere [30].
An expression for the static structure factor in the Ornstei n–Zernicke form in
the near–critical region of Ar has been found in literature [ 23]
S(q) =S(0) + ( qL)2
1 + (qL)2(7)
where S(0) is related to the gas isothermal compressibility χTby the re-
lation S(0) = NkBTχTand the correlation length Lis defined as L2=
0.1l2[S(0)−1]. l≈10˚A is the so–called short–range correlation length [23].
Experiments in Ne at T≈45 K [4,5] and in Ar at T= 162 .7 K up to
N≈6.5 atoms ·nm−3[10] proved that the kinetic energy shift can be calculated
according to the Wigner–Seitz model [24] as Ek=EWS≡/planckover2pi12k2
0/2m,where the
wavevector k0is obtained by self–consistently solving the eigenvalue eq uation
tan[k0(rs−˜a(k0))]−k0rs= 0 (8)
rs= (3/4πN)1/3is the radius of the WS cell and ˜ ais the hard–core radius of
the Hartree–Fock potential for rare gas atoms. In the BSL mod el, according to
a suggestion found in literature [12], ˜ ais estimated from the total scattering
cross section as ˜ a=/radicalBig
σT/4π.
The BSL model has been successfully used to analyze the exper imental
data in Ar at T= 152 .5 K up to N≈10 atoms ·nm−3[11] by assuming that for
the highest densities the WS model for the calculation of Ekis inappropriate
andEkmust be deduced from the experiment.
It is natural to ask if the reason of the deviation of the exper imentally de-
termined Ekvalues from the WS model is that the BSL model has been used
beyond its limits of applicability, or if different mechanis ms become active for
momentum transfer processes at high N.It is known, in fact, that in liquid Ar
the mobility of thermal electrons shows a maximum not very di stant from the
highest density investigated in previous experiments [11] . The mobility maxi-
mum occurs practically at the same density where V0(N) has a minimum [15].
The existence of this maximum, that indicates a situation of minimum scat-
tering, has been interpreted within the deformation–poten tial theory [25,26].
5Intrinsic density fluctuations of the fluid modulate the ener gyV0at the bottom
of the conduction band. The spatial disuniformity of the gro und–state energy
of electrons is the source of scattering. This is an intrinsi c multiple–scattering
theory because electrons are scattered off long–wavelength collective modes
of the fluid. These sorts of phononic models [25–27] do correctly predict the
existence of the mobility maximum at the right value N≈12.5 atoms ·nm−3,
but they fail to predict the density- and electric field depen dence of the µN
data as the BSL model does.
Owing to these reasons, we have investigated an extended den sity range
in Ar at T= 152 .15 K up to a maximum density N= 14 atoms ·nm−3,well
above the density of the mobility maximum in liquid, in order to see if the
mobility maximum is a feature typical of the liquid only or if it appears also
in the gas. In the latter case, there would be sound motivatio ns to extend the
density-modified kinetic approach even to the liquid [28–30 ].
2 Experimental Details
The electron mobility measurements have been carried out by using the
well–known pulsed photoemission technique [5,31]. A swarm of electrons drifts
in the gas under the action of an externally applied uniform e lectric field. The
timeτespent by the particles crossing the drift distance dis measured and the
drift velocity is calculated as vD=d/τe.The mobility is simply obtained as µ=
vD/E,where Eis the strength of the electric field. A simplified schematics of
the experimental apparatus is shown in Fig. 1. A high–pressu re cell is mounted
on the cold head of a cryocooler inside a triple–shield therm ostat. The cell
can be operated in the range 25 < T < 330 K and its temperature can be
stabilized within ±0.01 K. The cell can withstand pressures up to 10 MPa and
the pressure Pis measured with an accuracy of ±1 kPa. The gas density N
is calculated from TandPby means of an accurate equation of state [32].
A parallel–plate capacitor is contained in the cell and is po wered by a D.C.
high–voltage generator [33]. A gold–coated fused silica wi ndow is placed in the
center of one of the plates and can be irradiated with a short p ulse (≈4µs)
of VUV light produced by a Xe flashlamp. Thus, a bunch of electr ons, whose
number ranges between 4 and 400 fC depending on the gas pressu re and on
the applied electric field strength, is photoinjected into t he drift space. During
the drift motion of the charges towards the anode a current is induced in the
external circuit. In order to improve the signal–to–noise r atio the current is
integrated by a passive RC network. The resulting voltage si gnal is amplified
and recorded by a digital oscilloscope. The drift time is obt ained by analyzing
the signal waveform with a personal computer. Typical signa l waveforms are
shown in Fig. 2. The estimated error on the mobility is less th an 5 % .
6Fig. 1. Simplified schematics of the experimental setup. FL: Xe flashlamp, CN: cell,
PD: photodiode, BC: bellow circulator, OX: Oxisorb cartrid ge, DV: digital volt-
meter, -V: high–voltage generator, CT: LN2–cooled active c harcoal trap, DG: pres-
sure gauge, PC: personal computer, E: emitter, C: collector , A: amplifier, DS:digital
scope.
Fig. 2. Signal waveforms induced by electrons uniformly dri fting in a gas. At left:
pure gas, at right: gas containing a few p.p.m. of O 2electron–attaching impurities.
The gas used is ultra–high purity Argon with a nominal O 2content of 1 part
per million. Further purification is accomplished by recirc ulating the gas in a
closed loop through a LN2–cooled activated–charcoal trap a nd a commercial
Oxisorb cartridge. The final O 2amount is estimated to be a fraction of a part
per billion.
73 Experimental Results and Discussion
We have carried out measurements at two different temperatur es in the
neighborhood of the critical temperature, namely, at T= 162 .30 K and T=
152.15 K ( Tc= 150 .7 K).We have investigated the dependence of the density–
normalized mobility µNas a function of the density–reduced electric field
strength E/Nand of the density N.The density range explored encompasses
the critical density Nc= 8.08 atoms ·nm−3.In Fig. 3 we show sample µN
data at T= 162 .30 K. These data agree well with previous measurements at
1.010
0.01 0.1 1 10µN (1026 V-1 m-1 s-1)
E/N (10-24 V m2)
Fig. 3. µNas a function of the reduced electric field strength E/N for
T= 162 .30 K. The densities are (from top) N= 11.74,11.15,9.96,9.07,8.08,7.58,
7.06,5.03 atoms ·nm−3.
T= 162 .7 K [10]. The data for T= 152 .15 K are qualitatively similar to those
shown in Fig. 3.
The behavior of the reduced mobility µNof excess electrons in Ar as a
function of the reduced electric field is quite complicated, although it is now
well understood for not too high densities ( N < 7.0 atoms ·nm−3) [10]. At
low field strength µNis a constant independent of E/N. This constant value
is the zero–field density–normalized mobility µ0Nand pertains to thermal
electrons. In fact, at such low fields, the energy gained by el ectrons from
the field is negligible in comparison with their thermal ener gy. According to
the prediction of the classical kinetic theory [1], µ0Nshould be constant and
independent of N,while, experimentally, a completely different behavior of
µ0Nis observed, as it is easily realized by observing Fig. 3.
8At small and medium N, µN displays a maximum as a function of E/N
in the range (2 ≤E/N≤4)×10−24V m2,whose position depends on the
density. Since the maximum is observed from the dilute gas up to the present
densities, it is obvious that it has to be attributed to the Ra msauer–Townsend
minimum of the e−Ar atomic momentum–transfer scattering cross section,
which is located at an electron energy ǫRT≈230 meV [35,36]. When E/N
has the value ( E/N)maxcorresponding to the mobility maximum, the average
electron energy is equal to the energy of the Ramsauer minimu m/angbracketleftǫ/angbracketright=ǫRT.
For larger N,the mobility maximum at ( E/N)maxgradually disappears.
Finally, for even larger E/N≥4×10−24V m2,theµNcurves for all
densities merge into a single curve that is well described by the classical kinetic
equations with the given cross section. For large E/N, the behavior of µN
becomes therefore independent of density and is easily expl ained in terms
of the BSL model. At low E/N, i.e., at small electron energies ǫ,the de
Broglie wavelength of the electron, λ=h/√
2mǫ,is pretty large and the
electron wavepacket is so much extended as to encompass many atoms at
once. In this situation, multiple scattering effects are ver y important. As the
mean electron energy /angbracketleftǫ/angbracketrightis increased by increasing E/N, the extension of the
electron wave function, as measured by λ,decreases and the importance of
multiple scattering is reduced. Therefore, the experiment al points converge to
the prediction of the classical kinetic theory. This behavi or is present also in
the liquid [11] and has been observed also in Neon gas [5], whe re the energy
dependence of the cross section is completely different from that of Ar and
where the temperature of the experiment ( T≈45 K) was much lower than in
the present case.
As already observed for several temperatures [10,11], the m obility max-
imum related to the Ramsauer minimum of the cross section shi fts to lower
E/N values as the density is increased, as shown in Fig. 4, where t he re-
duced field of the mobility maximum, ( E/N)max,is plotted as a function of
NforT= 152 .15 K. This behavior is also consistent with previous measure -
ments in gaseous Ar at room temperature for densities up to 2 a toms·nm−3
[34]. On approaching the critical density, for N >6.0 atoms ·nm−3,the de-
crease of ( E/N)maxproceeds at a much faster rate than before, until, for
N≈Nc= 8.08 atoms ·nm−3,(E/N)max→0 and the mobility maximum
disappears, as can be seen in Fig. 3.
The observed behavior can be explained by noting that, for ( E/N)max,
the average electron energy equals that of the Ramsauer mini mum of the cross
section, /angbracketleftǫ/angbracketright=ǫRT.Generally speaking, it turns out that /angbracketleftǫ/angbracketrightcan be written in
the form
/angbracketleftǫ/angbracketright=3
2kBT+Ek(N) +f(E/N) (9)
90.01.02.03.04.05.0
0 2 4 6 8 10(E/N)max (10-24 V m2)
N (atoms nm-3)
Fig. 4. Decrease of ( E/N)maxwith increasing NforT= 152 .15 K. The line has no
theoretical meaning.
where f(E/N) is a monotonically increasing function of the reduced elec tric
field [22]. Since the density–dependent electron kinetic en ergy shift Ek(N)
increases with increasing N,(E/N)maxmust decrease in order to fulfill the
condition /angbracketleftǫ/angbracketright=ǫRTwith increasing Nat constant temperature. [We will
furthermore show that the change of slope of ( E/N)maxas a function of Nfor
N >6.0 atoms ·nm−3is related to the change of slope of Ek(N) at the same
density.] Eventually, for N > N c,the electron energy distribution function is
so largely shifted by Ekas to sample the scattering cross section well beyond
the Ramsauer–Townsend minimum. This is the reason of the dis appearing of
the mobility maximum.
A cornerstone for the understanding of the electron scatter ing in dense
gases is represented by the analysis of behavior of the zero– field density–
normalized mobility µ0Nas a function of the density, because, as already
pointed out, the classical kinetic theory predicts that µ0Nis independent of
N,while the experiment gives a strongly density–dependent µ0Nfor every
explored temperature. In Fig. 5 we therefore show the zero–fi eld density–
normalized mobility µ0Nas a function of Nfor the two investigated temper-
atures T= 162 .30 K and T= 152 .15 K. Previous data taken at T= 162 .7 K
with a different apparatus [10] are reported in order to show t he consistency
of the present data. Moreover, also data obtained in liquid A r [11] are shown
for comparison, although it must be remembered that the meas urements in
the liquid are taken along the liquid-vapor coexistence lin e and are therefore
not isothermal.
Two relevant features emerge from Fig. 5. The first one is the i mpres-
1005101520253035
0 2 4 6 8 10 12 14 16µ0N (1026 V-1 m-1 s-1)
N (atoms nm-3)Nm ≈12.5
Fig. 5. µ0Nas a function of NforT= 162 .30 (open squares), T= 162 .7 (open
circles) [10], and 152 .15 K (closed circles). The closed squares are the results in l iquid
Ar [11]. The arrow indicates the value of the density Nmof the µ0Nmaximum in
gas. The lines are only guides for the eye.
sive increase of µ0Nwith increasing Nfor both temperatures up to N≈
11.0 atoms ·nm−3.This behavior is present also at room temperatures [8] and
has been one the primary motivations for the development of m ultiple scat-
tering theories. The BSL model explains quantitatively thi s feature of µ0Nfor
N≤10atoms ·nm−3.ForE/N→0,electrons are in thermal equilibrium with
the host gas and do not gain energy from the field. Therefore, t heir average
energy is /angbracketleftǫ/angbracketright ≪ǫRT.In this energy range, the momentum–transfer cross sec-
tion decreases rapidly with increasing electron energy [35 ,36], as shown in Fig.
6. Roughly speaking, µ0Nis a sort of weighted average of the inverse cross
0.1110
0 0.1 0.2 0.3 0.4 0.5σmt (10-20 m2)
ε (eV)
Fig. 6. Momentum–transfer scattering cross section of Ar [3 5].
11section, as it can be realized by inspecting Eq. 1. To a first ap proximation,
/angbracketleft1/σmt/angbracketright ≈1/σmt(/angbracketleftǫ/angbracketright).At constant T,this average should be constant and
should not depend on the density N,unless /angbracketleftǫ/angbracketrightdepends on it. In particular,
owing to the shape of σmt(ǫ), µ0Ncan increase with increasing Nonly if /angbracketleftǫ/angbracketright
does the same. Therefore, the positive density effect of µ0Nsupports the con-
clusion that the average electron kinetic energy includes a density–dependent
contribution that is positive and increases with increasin gN,as expressed
by Eq. 9. This conclusion immediately rationalizes the obse rvations described
by Christophorou et al. [34]. In fact, they note that the minimum of the in-
verse density–normalized mobility at zero–field, i.e. a qua ntity proportional
to the cross section, shifts to lower energies as the density is increased. They
calculate the average electron energy at zero field accordin g to the Nernst–
Einstein–Townsend relation
/angbracketleftǫ/angbracketright=3
2DL
µ0=3
2kBT
where DLis the longitudinal diffusion coefficient. By so doing, they ne glect
the density–dependent zero–point electron energy Ek(N),and therefore the
Ramsauer–Townsend minimum seemingly appears at lower ener gies, as can be
seen by inspecting Eq. 9 with E/N= 0,i.e.(f(E/N) = 0) ,and by neglecting
Ek.
The second most relevant feature is the presence of a sharp ma ximum of
µ0NforN≡Nm≈12.5 atoms ·nm−3atT= 152 .15 K. [Probably, such max-
imum would exist also at the higher temperature, but the limi ted pressure
range of the experimental cell ( P≤10.0 MPa) did not allow the investiga-
tion of larger densities at higher temperatures.] The µ0Nmaximum occurs
at nearly the same density where it was observed in liquid Ar [ 11], as also
shown in Fig. 5. A similar behavior has been previously obser ved in liquid
and gaseous CH 4[37].
It is well–known [25,27] that the maximum of µ0Nin liquid has been
attributed to scattering of electrons off long-wavelength c ollective modes of
the fluid. The intrinsic density fluctuations of the liquid mo dulate the elec-
tron energy at the bottom of the conduction band of the liquid ,V0(N),and
the spatial fluctuations of V0(N) act as the potential for the scattering, just
as lattice deformation–potentials from acoustic phonons s catter carriers in
semiconductors. Within the deformation potential theory, the potential for
scattering is linear in the density deviations about the ave rage value. Since it
happens that V0(N) has a minimum at nearly the same density, Nm,where
µ0Nis maximum [15], the scattering potential vanishes to first o rder at Nm.
Therefore, at this density the scattering of electrons is ve ry much reduced and
the electron mobility turns out to be maximal. For any other N/negationslash=Nm,the
slope of V0(N) as a function of Nis nonzero and deformation potential is large
12enough to efficiently scatter electrons, thus reducing their mobility.
However, a gas does not support phonons as a liquid does. Ther efore, the
presence of the µ0Nmaximum in dense Ar gas at the same density Nmas
in the liquid raises the question if, beyond a given density t hreshold, there
is a change in the physical mechanisms which determine the mo bility in the
gas, or, rather, if the single scattering picture of the dens ity–modified kinetic
theory can be extended to the liquid.
On one hand, one could argue that, at low and medium densities , the single
scatterer approximation is valid and electrons can be descr ibed as scattering
off individual atoms, though the scattering properties have to be density–
modified in order to account for multiple scattering effects. On increasing the
density, a conduction band might develop and electrons migh t be scattered off
long–wavelength collective modes of the dense gas, though t hey might not be
true phonons.
On the other hand, there are several reasons to extend the sin gle scattering
picture to the liquid. First of all, the phononic theories ar e developed for
thermal electrons only, i.e., they make predictions only on µ0Nand completely
disregard the electric field dependence of the experimental µNdata. This
dependence is very important because it is intimately relat ed to the shape
of the atomic cross section. There have indeed been more or le ss successful
attempts to use the classical kinetic theory even in the liqu id [28–30], though
the cross section has been taken as an adjustable parameter. Moreover, the
phononic models do not even predict accurately µ0N,unless higher–order
scattering processes are taken into account [26,27].
It is therefore challenging to investigate the possibility that the density–
modified kinetic model can account for the µ0Nmaximum and that a single–
scattering picture can be retained even in the liquid, owing to its conceptual
simplicity. The BSL model has been thus used for the analysis of the experi-
mental data at high density.
First of all, we do not make any assumptions on the value of the kinetic
energy shift Ek(N).We treat it as an adjustable parameter to be determined
by fitting the equations of the model Eqns. 3–5 in the limit E/N→0 to
the experimental data. Literature data for the cross sectio n have been used
[35]. Once Ek(N) has been determined by this fitting procedure, the average
electron energy (Eq. 9) could be evaluated, if necessary, as
/angbracketleftǫ/angbracketright=Ek(N) +∞/integraldisplay
0z3/2g(z)dz
where gis given by Eq. 4, and Eq. 9 is recovered. In Fig. 7 the resultin gEk
130.000.050.100.150.20
0 2 4 6 8 10 12EK (eV)
N (atoms nm-3)
Fig. 7. Values of Ekplotted as a function of NforT= 162 .30 K (open circles) and
T= 152 .15 K (closed circles). Previous values determined from data atT= 162 .7
K [10] are shown (open squares). The solid line is the predict ion of the WS model.
The arrow indicates the density where ( E/N)maxchanges slope.
values are shown as a function of Nfor the two investigated temperatures.
Previous determinations of EkforT= 162 .7 K [10] are also shown for compar-
ison to assess the consistency of these new sets of measureme nts with previous
ones.
The data at T= 162 .30 K agree very well with the data taken at T= 162 .7
K. There are small differences between the results for T= 152 .15 K and
T= 162 .30 K, which might be attributed to the larger gas compressibi lity for
the temperature close to Tcand to the uncertainty with which the short–range
correlation length lis known [23]. Nonetheless, the experimentally determined
Ekvalues agree pretty well with the prediction of the WS model ( shown as
a solid line in Fig. 7) for densities up to ≈7.0 atoms ·nm−3.For larger N,
starting at practically the same density where ( E/N)maxchanges slope (see
Fig. 4), up to N≈10 atoms ·nm−3,the values of Ekthat reproduce the
experimental values of µ0Nincrease faster with Nthan the prediction of the
WS model. This is not a failure of the BSL model. It is just rela ted to the
fact that the WS model is only valid when rs≫˜a.Unfortunately, up to now
there are no theoretical calculations of Ekwith which the present results can
be compared.
In the density range of the present experiment, the BSL model does not
reproduce only µ0Nwith the proper choice of Ek(N).It also shows a high de-
14gree of internal consistency because the value of Ekdetermined from the data
at low field allows to reproduce quite accurately the full E/N−dependence
ofµN.This result is shown in Fig. 8, where the curves for several de nsities
110
0.1 1 10µΝ (1026 m-1 V-1 s-1)
E/N (10-24 V m2)
Fig. 8. Comparison of the results of the BSL model with the exp erimental
µN(E/N) data for some densities (dotted lines). The densities are ( from top):
N= 13.33,9.935,5.144,0.502 atoms ·nm−3.The solid lines are obtained by us-
ing the effective cross section σeff,as described in the text.
are compared with the prediction of the BSL model (dotted lin es). The dot-
ted curves are obtained by using the Ek(N) values determined by fitting the
model to the zero–field data. It can be realized that, at small and medium
N,the features of the mobility are all reproduced well. Namely , the position
and strength of the mobility maximum as well as its behavior a s a function of
the density are described accurately. The behavior at small – and high–fields
ofµNis correctly reproduced, with the curves for different densi ties merging
into a single one at large E/N. All these observations are consistent with the
hypothesis that the kinetic–energy shift Ek(N) increases with increasing N,
and that it can grow so large as to shift the average electron e nergy beyond
ǫRT.
However, it is also evident that, for even larger N(≥10 atoms ·nm−3),the
BSL model as such does neither reproduce the µ0Nmaximum for N=Nm,
nor the decrease of µ0Nwith increasing NforN > N m.On one hand, it is
clear, of course, that the overall behavior of µ0Nas a function of Nmust be
related to the shape of the atomic cross section and to the den sity–dependent
quantum shift of the electron energy distribution function . Unfortunately, the
scattering cross sections are known with limited accuracy a s far as strength
15and position of the Ramsauer minimum are concerned. Differen t choices of σmt
give different strength and position of the µ0Nmaximum [11] or they may not
even give a maximum at all.
On the other hand, the use of an effective density–modified sca ttering
cross section σ⋆
mthas proven so powerful giving a nice agreement between
model and data up to fairly large densities that it is interes ting to extend this
paradigm to higher densities. According to a suggestion pro posed in literature
[11,28], at high Na good agreement between data and model can be obtained
by scaling σ⋆
mtby a factor c0(N).The adjustable parameter c0depends only
onN,but it is independent of E/N and of the electron energy. Therefore, it
has no influence on the dependence of µNonE/Nat constant density. While
c0is introduced as an adjustable parameter, the energy shift Ekis no longer
determined from the experimental data. Instead of Ekit is rather used the
value EWSgiven by the WS model and calculated according to Eq. 8 (solid
line in Fig. 7). σ⋆
mtis everywhere replaced by σeff=c0(N)σ⋆
mtandc0is so
0.060.080.10.3
9 10 11 12 13 14σeff (10-20 m2)
N (atoms nm-3)0.17 0.18 0.19EWS + 3/2 kBT (eV)
Fig. 9. Effective scattering cross section σeff=c0(N)σ⋆
mtas a function of N(lower
scale) and energy (upper scale). The upper scale has been obt ained by converting
density to energy by means of the WS model.
adjusted as to reproduce the behavior of µ0Nas a function of N. c 0turns out
to be of order unity c0≈ O(1).With this choice the electric field dependence of
µNis reproduced even at the highest densities. The shape of the effective cross
section at thermal energies σeff=c0(N)σ⋆
mt[(3/2)kBT+EWS(N)] is shown in
Fig. 9. The energy scale on the upper horizontal axis has been obtained by
converting density to energy by means of the WS model Eq. 8. By comparing
Fig. 9 to Fig. 6 there is undoubtely a strong similarity betwe enσeffand
the atomic cross section σmt.In particular, it is interesting the presence of
a Ramsauer–type minimum also in the effective cross section. Moreover, the
strength of σeffis very close to its atomic companion, though the position
16of the minimum occurs at a somewhat lower energy and the minim um itself
appears to be narrower. This feature might be due to the use of the WS model
for the N→ǫconversion. If the experimentally determined values of Ek(N)
had been used instead of the WS model, the σeffminimum would be broader
and shifted to larger energies because Ek(N)> E WSforN≥7 atoms ·nm−3.
This effective cross section can be compared with the effectiv e one/angbracketleftσL/angbracketright
obtained in liquid Ar by Christophourou et al. [34]. Even there, /angbracketleftσL/angbracketrightappears
to be much narrower and its minimum occurs at an energy much sm aller than
in the atomic one. An agreement with the two effective cross se ctions could
be obtained by adding Ek(N) to and by using F(k) rather than S(0) in the
data of Christophorou.
4 Conclusions
In this paper measurements of the excess electron mobility i n dense Ar gas
in the neighborhood of the liquid–vapor critical point are p resented. The most
important result of the present experiment is the observati on of a sharp max-
imum of the zero–field density–normalized mobility µ0Nat the same density
where it occurs in the liquid.
The interpretation of these measurements is challenging be cause two op-
posite points of views must be reconciled. In fact, in the low –density gas it is
customary to adopt a single scattering picture, while in the dense liquid the
electron transport properties are described in terms of sca ttering of electrons
off collective excitations of the medium. The interesting po int is to under-
stand if the physical mechanisms underlying the scattering processes gradu-
ally change at some density between the dilute gas and the den se liquid or
if the kinetic picture valid at low density can be still retai ned, with obvious
modifications so as to include multiple scattering, also in t he liquid phase.
To this goal, the present data have been analyzed by extendin g the heuris-
tic model proposed by Borghesani et al.to explain the mobility measurements
in moderately dense Ar gas [10]. The model is a density–modifi ed kinetic model
based on the classical kinetic theory [1], where density–de pendent multiple–
scattering effects are included in a heuristic way. The model is based on the
quantum density-dependent shift of the ground state energy of the electrons
in a dense and disordered medium, on the accounting for corre lations among
scatterers described by the static structure factor of the m edium, and on the
quantum self–interference of the electron wavepacket scat tered off randomly
located scattering centers along paths connected by time–r eversal. The kinetic
term of the ground state energy shift must be added to the usua l kinetic en-
ergy of the electrons when the cross section and other dynami c properties of
17scattering have to be evaluated. All these effects concur to d ress the atomic
cross section resulting in an effective density–dependent m omentum transfer
cross section, that nonetheless is intimately related to th e atomic one.
Although the data for quite high densities fit well in this mod el, the
description of the newly discovered feature, namely the mob ility maximum,
requires the introduction of a density–dependent adjustab le parameter that
scales the cross section. In any case, however, the kinetic p icture is preserved
and the resulting effective cross section turns out to be very similar to the
atomic one. In particular, it shows a Ramsauer–type minimum .
The overall success of this kinetic description is striking , even more if one
takes into account the limited accuracy with which the atomi c cross section
is known, especially in the region of the Ramsauer minimum, a nd the uncer-
tainty with which the energy–dependent structure function F(ǫ) is known,
particularly in the neighborhood of the critical point. It a lso appears that
more refined theoretical calculations of the kinetic energy shift at high density
are needed, as well as a treatment of the effect of the density fl uctuations on
the electron energy distribution function.
Moreover, these data still raise the question of how to treat theoretically
the scattering processes for momentum transfer at very larg e densities, possi-
bly including contributions from mechanisms different from density–modified
kinetic processes.
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21 |
arXiv:physics/0102072v1 [physics.flu-dyn] 22 Feb 2001Interaction of two deformable viscous drops under
external temperature gradient
V. Berejnov, O. M. Lavrenteva, and A. Nir
Department of Chemical Engineering, Technion - Israel Inst itute of Technology,
Haifa 32000, Israel
December 21, 2013
Abstract
The axisymmetric deformation and motion of interacting dro plets in an im-
posed temperature gradient is considered using boundary-i ntegral techniques for
slow viscous motion. Results showing temporal drop motion, deformations and
separation are presented for equal-viscosity fluids. The fo cus is on cases when the
drops are of equal radii or when the smaller drop trails behin d the larger drop. For
equal-size drops, our analysis shows that the motion of a lea ding drop is retarded
while the motion of the trailing one is enchanced compared to the undeformable
case. The distance between the centers of equal-sized defor mable drops decreases
with time. When a small drop follows a large one, two patterns of behavior may
exist. For moderate or large initial separation the drops se parate. However, if
the initial separation is small there is a transient period i n which the separation
distance initially decreases and only afterwards the drops separate. This behavior
stems from the multiple time scales that exist in the system.
Key Words: droplets interaction; thermocapillary migration; drop de forma-
tion; boundary integral equations
11 Introduction
It has long been known that a viscous droplet submerged into a nother immiscible non-
isothermal liquid migrates in the direction of the temperat ure gradient, ∇T. This
droplet’s migration is due to the temperature-induced surf ace tension gradient ∇s(γ)
at the droplet interface Sand it is addressed as a thermocapillary migration. Young et
al(1) were first to investigate the mechanism of such migration both theoretically and
experimentally. These authors found conditions for the sta bilization of a buoyant bubble
under gravity force by the thermocapillary force which was d irected oppositely. They
obtained a theoretical prediction for the migration veloci ty of a single spherical bubble
of radius a, which is placed in a viscous fluid of viscosity η1, with an imposed constant
temperature gradient A. In the limit of zero Marangoni number/vextendsingle/vextendsingle/vextendsingle∂γ
∂T/vextendsingle/vextendsingle/vextendsinglea2Aρ1
η12(i.e., with neg-
ligible convective transport effects) the droplet velocity Uin the laboratory coordinate
frame is related to the uniform temperature gradient by
U=aA
η1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂γ
∂T/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle¯U,¯U=2
(2 + 3 η) (2 + κ). [1]
Hereη=η2/η1andκ=κ2/κ1are the ratios of viscosities and thermal conductivities
between the internal and the surrounding fluids, respective ly,/vextendsingle/vextendsingle/vextendsingle∂γ
∂T/vextendsingle/vextendsingle/vextendsingleis a constant derivative
of the surface tension with respect to the local temperature Ton the interface, and ¯Uis
a dimensionless migration velocity. In the case when the dro plet and the external fluid
have equal viscosity and thermal properties ¯U= 2/15≃0.133. A review of the further
development of the subject can be found in (2).
Bratukhin (3) considered the migration of a deformable visc ous droplet at the small
Marangoni number. Using perturbation techniques in a spher ical coordinate system ( r, θ)
with origin at the center of the droplet, he found an expressi on for the deformation of
the droplet’s shape r=R(θ):
R(θ) = 1 + Ma2s2P2(θ) +O(Ma3),
s2=−3
4γ0/parenleftBigg5
2¯U2(1−ρ)(1 +η)
4 +η+Pr¯U
63χ49χ−7χκ+ 18κ
(3 + 2 κ)(2 +κ)/parenrightBigg
. [2]
2HerePr=ν1/χ1is the Prandtl number, ρ=ρ2/ρ1andχ=χ2/χ1are the ratios of density
and thermal diffusivity between the internal and the surroun ding fluids, respectively,
γ0=γρ1a/η1is the dimensionless surface tension corresponding to the c enter of the
drop, and P2(θ) is the second order Legendre polynomial.
The first term of s2is related to the inertia term in the Navier Stokes equation, while
the second one is related to the convective term in the energy transport equation. It is
interesting to note that the expansion of the droplet deform ation begins with the term
ofO(Ma2), and that this term vanishes in the case that combines equal densities of the
phases ( ρ= 1) and negligible convective heat transport ( Pr= 0). It follows from [2]
that the deformations are small also for large values of the r eference surface tension γ0
(small capillary number).
Balasubramaniam and Chai (4) showed that the Young solution satisfies the full
Navier-Stokes equations if the temperature field on the sphe rical interface is proportional
to cos θ, i.e. in the case of negligible convective transport. The pe rturbation of the
spherical form constructed in (4) under the assumption of sm all capillary number has
the same form as [2].
In real applications, however, fluid drops are not isolated a nd their motion is influ-
enced by the presence of neighboring particles. This intera ction of drops substantially
alters the migration velocities and it may cause drops’ defo rmations even in the case of
negligible convective transport . On the other hand, the second mechanism of the defor-
mations’ suppression (large surface tension) is expected t o be valid in the multi body
case as well.
Most of the numerous theoretical studies of the interaction of bubbles and droplets
in the course of their thermocapillary migration were perfo rmed under the assumption
of nondeformable drops (zero capillary number). Meyyappan et al(5) and Feuillbois (6)
studied the Marangoni induced axisymmetric migration of tw o bubbles and Meyyappan
and Subramanian (7) studied the non-axisymmetric case. Acr ivoset al(8), Satrape (9)
and Wang et al(10) examined ensembles of more than two bubbles. The remark able
3finding of these studies is that a bubble migrates in the prese nce of another bubbles of
the same size with exactly the same velocity as when isolated . Bubbles of different sizes
affect the motion of the neighboring bubbles. The influence of the presence of a larger
bubble on the motion of a smaller one is more pronounced than i ts influence on the
motion of the larger bubble.
These analyses were extended to the case of liquid drops by An derson (11), Keh and
Chen (12, 13, 14) and Wei and Subramanian (15). In this case, t he migration velocity of
a drop in the presence of another drops differs from the veloci ty of an isolated drop even
when the drops are of equal size. Similar to the case of intera cting bubbles, a larger drop
exerts more pronounced influence on the motion of a smaller on e. The thermocapillary
motion of two non-conducting drops was studied by Loewenber g and Davis (16).
The literature on the thermocapillary motion of deformable drops is limited in contrast
to a somewhat related problem concerning the interaction of drops undergoing a buoyancy
driven motion. For the latter case numerous recent studies r evealed a rich variety of
interaction patterns of deformable drops depending on the B ond number and the initial
configuration of the system ( see e.g. Manga and Stone (17), Zi nchenko et al(18)) and the
literature cited). Numerical simulations of an axisymmetr ic buoyancy-driven interaction
of a leading drop and a smaller trailing drop were recently re ported by Davis (19). It
was demonstrated that the trailing drop elongates consider ably due to the hydrodynamic
influence of the leading one. Afterwards, depending on the go verning parameters, the
drops may either separate and return to spherical shape or th e trailing drop may be
captured by the leading one, or one of the drops may break up.
For the thermocapillary induced motion, the effect of deform ability was studied mostly
by a perturbation technique assuming small deformations. R ecently, Rother and Davis
(25) applied lubrication approximation to study the effect o f a slight deformability of the
interfaces on the thermocapillary-driven migration of two drops at close proximity.
Zhou and Davis (20) presented a recent study in which the drop s are free to deform
in the course of their thermocapillary interaction. They us ed a boundary-integral tech-
4nique to study the thermocapillary interaction of a deforma ble viscous drop with a larger
trailing drop making no a priori assumptions regarding the magnitude of deformations.
It was demonstrated that in this case the deformations are sm all and have a small effect
on the drops motion at moderate separation. The influence of d eformability becomes
significant only when the drops are close together. Zhou and D avis (20) did not adjust
the surface tension to follow the migration of the drops in th e external temperature field
that would be stationary in the absence of a drop. Hence, the b oundary conditions that
they used were not compatible with the continuous change of t he positions of the drops.
The goal of the present work is to extend the analysis of the th ermocapillary interac-
tion of deformable drops to the cases of equal-sized drops an d the leading larger drop and
to study the influence of the deformations on the relative mot ion of the droplets. We have
applied the boundary integral method for the simulations of the thermocapillary induced
motion of two deformable viscous drops in the case of moderat e capillary numbers and
equal viscosity and thermal properties of the dispersed and continuous phases.
2 Problem formulation
Consider two drops embedded in an immiscible viscous fluid wi th a temperature gradient
applied along their line of center. In the absence of gravity the drops migrate in the
direction of the applied temperature gradient one beyond an other due to thermocapillary
forces. The trailing and the leading drop and the continuous fluid are marked by indices
1, 2 and 3, respectively. All the fluids are assumed to be Newto nian and incompressible.
The interface tension γis assumed to depend linearly on temperature γ(T) =γ(T0)−
∂γ
∂T(T−T0). The temperature field approaches a given linear function T=Az+T0far
from the dispersed species. The problem is considered in a la boratory coordinate system,
see Fig. 1.
Non-dimensional variables are introduced using the follow ing scaling: the radius a1
of the first droplet for the length, V∗=/vextendsingle/vextendsingle/vextendsingle∂γ
∂T/vextendsingle/vextendsingle/vextendsinglea1A/η3for the velocity, a1/V∗for the time,
5/vextendsingle/vextendsingle/vextendsingle∂γ
∂T/vextendsingle/vextendsingle/vextendsingleAfor the pressure and Aa1for the temperature. The problem is governed by the
following dimensionless parameters: the Marangoni number ,Ma, the Prandtl number,
Pr, and the capillary number, Ca, where
Ma=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂γ
∂T/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglea12A ρ
η32;Pr=ν3
χ3;Ca=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂γ
∂T/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglea1A
γ0. [3]
The other dimensionless parameters are the ratios of the mat erial properties of the fluids,
η=ηi/η3, ρi/ρ3, κ=κi/κ3and the geometric parameters R=a2/a1andd=L/a1,
where Lis the distance between the drops along their line of centers .
We shall be interested in the case of small Marangoni numbers and moderate values
of the other parameters. Following (3) one can see that, at th e leading order in Ma, the
inertial and convective transport effects are negligible, a nd the motion inside and outside
the droplets is governed by the Stokes equations:
∇p3=△v3,x∈Ω3,
∇pi=η△vi,x∈Ωi, i= 1,2,
∇vi= 0,x∈Ωi, i= 1,2,3, [4]
where piandviare the pressure and velocity field, respectively. The tempe rature field
inside and outside of droplets satisfies the Laplace equatio n
△Ti= 0,x∈Ωi, i= 1,2,3. [5]
The continuous fluid is quiescent far from the drops
v3→0,asx→ ∞. [6]
At the interfaces the velocity field is continuous,
v0(x) =vi(x),forx∈Σi, i= 1,2, [7]
and the kinematic boundary conditions require that
ni·vi=Vni,forx∈Σi, i= 1,2, [8]
6where Vniis a velocity of the interface Σ iin the normal direction. The tangential stress
balance may be written as
τi·(Π3−Πi)·ni=−∂Ti(x)
∂τ,x∈Σi, i= 1,2, [9]
and the balance of the normal stresses reads
ni·(Π3−Πi)·ni=/parenleftbigg1
Ca−Ti(x)/parenrightbigg
Hi,x∈Σi, i= 1,2, [10]
HereΠ3=−p3I+T3andΠi=−piI+ηTiwithTibeing the rate of deformation tensor,
τiis a unit vector tangential to the ithinterface, and Hi=∇ ·niis the mean curvature
on the ithinterface.
The temperature and the heat flux are continuous
Ti=T3, κ∂Ti
∂n=∂T3
∂nforx∈Σi, i= 1,2, [11]
and at infinity the temperature tends to a given linear functi on
T→zat|x| → ∞ . [12]
3 Boundary-integral formulation. Numerical
method
The boundary-value problem outlined in the previous sectio n may also be formulated in
an equivalent integral form (21). Following the classical p otential theory (22) we can
obtain a system of boundary-integral equations for the temp erature and velocities on the
interfaces. When the thermal properties and viscosities of the phases are equal for all
Ωiphases the boundary integral equations simplify considera bly, and the solution to the
problem acquires an explicit form. Thus, the temperature an d the velocities everywhere
are given by
T(x) =z, [13]
7v(x) = −1
8π/contintegraldisplay
ΣJ·/bracketleftBigg/parenleftbigg1
Ca−T(y)/parenrightbigg
H(y)n+∂T(y)
∂ττ/bracketrightBigg
dS(y). [14]
Herey∈Σ = Σ 1+ Σ2and the kernels for the single layer potentials for Stokes flo w are
J(ξ) =I
ξ+ξξ
ξ3,
withξ=x−yandξ=|ξ|. Note that the dimensionless surface tension equals 1 /Ca−z,
and the natural restriction γ≥0 implies that unequality z≤1/Cahold for all points
on the interfaces during the entire time of the process. Appl ying this unequality to the
initial moment t= 0 provides an upper bound for the capillary number
Ca <1
2R1+ 2R2+d0, [15]
where d0is an initial separation between the surfaces of the drops.
In the axisymmetric case, the integrations over the azimuth al angle in [14] can be per-
formed analytically, and the problem is reduced to the compu tation of a one-dimensional
singular boundary-integral for the interfacial velocity
v(x) =/integraldisplay
σB(x,y)·f(y)dl(y). [16]
Herex= (rx, zx) andy= (r, z) are vectors on an azimuthal ( r, z) plane in a cylindrical
coordinate system, σ=σ1+σ2is the union of curves describing the drops’ surfaces in
this plane, and dl(y) denotes the differential arc length of the corresponding co ntour.
f(y) =/parenleftbigg1
Ca−T(y)/parenrightbigg
Hn+∂T(y)
∂ττ=fnn+fττ
is a traction jump across the interface. The complete expres sions for components of the
kernel B(x,y) are given in the Appendix. Note that the kernel [16] have a lo garithmic
singularity at y→x. In the absence of Marangoni effect, when the tangential stre sses
are continuous ( ∂T/∂τ = 0), these singularities are often eliminated making use of a
subtraction technique (21), that is based on a well known equ ality
/integraldisplay
σiB(x,y)·n(y)dl(y) = 0. [17]
8When the tangential Marangoni stresses associated with the temperature gradients on
the drops surfaces are present, this procedure cannot be sim ply incorporated into the
singularity-subtraction formulation. For the purpose of a study of the droplets migration
it is sufficient to compute the normal component of surface vel ocity Vn=n(x)·v(x).
We employed a subtraction technique (21, 24) in which the nor mal component of the
velocity on the interfaces can be evaluated as
Vn=n(x)·v(x) =n(x)·/integraldisplay
σB(x,y)·f(y)dl(y)
=/integraldisplay
σfτ(y)n(x)·B(x,y)·τ(y)dl(y)
+n(x)·/integraldisplay
σ[fn(y)−fn(x∗]B(x,y)·n(y)dl(y), [18]
where x∗is an arbitrary chosen point in the bulk or on the interface. T he choice x∗=x
removes a singularity in the integral. The integrand of the fi rst summand in the right-
hand side of [18] tends to a finite limit at y→x(see the Appendix).
When the separation between the drops surfaces is small, ’ne ar-singularities’ appear in
the integrand if the reference point xis located in the gap region. Following Loewenberg
and Hinch (24) we facilitate the calculation of integrals at such ’near singular’ using
[18], where x∗is the image point across the gap region to x. This approach provides a
regularization of the ’near-singularity’ for the regions o n the surfaces that are in close
proximity.
In our numerical computations, the drop boundaries in the az imuthal plane are dis-
cretized using marker points and are approximated by cubic s plines. The number of
points on the two interfaces is equal. The marker points are d istributed uniformly with
respect to the arc length (non-uniform distribution was not needed in our simulation
because the curvature of the interface remains moderate at a ll time). The normal and
tangent unit vectors at the interfaces and the mean curvatur es are computed by taking
the derivatives of the position vector with respect to the ar c length. When the normal
velocities on the interfaces are obtained, the positions of the marker points are advances
9using Euler or Runge Kutta method. After that the cubic splin e approximation of the
interfaces is reconstructed and the marker points are redis tributed on the new surfaces
uniformly with respect to the arc length. The Krasny (23) filt ering procedure was used
to smooth the spatial frequency spectrum and to remove noise that was introduced by
the computation of high derivatives in the code. All Fourier modes below some ’tolerant
number’, which depends on the number of marker points were se t to 0. At every step
the positions of the drops’ centers of mass were obtained. Th e velocities of the centers of
mass were consequently calculated via numerical differenti ation with respect to time. The
calculations are stopped when the interfaces of the drops in tersect or when the surface
tension at any point on the leading drop becomes negative.
4 Results and discussion
We consider the axisymmetric drift of two deformable drople ts with different size. For
simplicity we assume that the droplets and the continuous me dia have identical viscosity,
densities and thermal properties. This ensures that the tem perature field is continuous
and has continuous derivatives across the interfaces. The d roplets are initially spherical
and start to move from an initial separation distance. The pa rameters considered include
the capillary number Ca, the initial droplets’ radii R1andR2and the initial distance
d0between surfaces of the droplets. During the motion we obser ve the evolution of the
positions and of the form of the droplets, and the flow pattern s inside the droplets and in
the external fluid. The relative position was measured by the distance between centers
of mass L(t) and by the distance between the nearest points on the surfac es of the two
drops d(t). The deformation of a droplet was characterized by the Tayl or deformation
parameter
D=2rmax−∆z
2rmax+ ∆z, [19]
where ∆ zis the length of the drop along the axis of symmetry, and rmaxis the maximum
radial dimension. Positive and negative values of Dcorrespond to oblate and prolate
10shapes, respectively.
4.1 Migration velocities. The case of non-deformable drops
We have tested our numerical code by comparing calculated dr ift velocities with available
analytic results. As it was mentioned above, there are analy tical results (11, 13) for the
thermocapillary induced motion of non-deformable spheric al drops in a uniform external
temperature gradient. These results can be applied for the a xisymmetric cases of drift of
drops with different radii R1andR2, and for different separation distances between the
centers of mass.
For the test of our code we considered a case of small capillar y number, Ca= 0.001,
in which the expected deformation of initial spherical shap e of the drops during the
evolution of the positions is negligibly small. The case of w ell-separated non-deformable
drops was consider by Anderson (11). He showed that for dropl ets of equal radii, the
velocities are equal and have the form
U=/parenleftbigg
1 +1
L2−7
3·1
L6/parenrightbigg
|U|. [20]
Keh and Chen (13) considered also drops at small distance and derived a general formula
for the velocities
U1=M11|U1|+M12|U2|
U2=M21|U1|+M22|U2| [21]
where M11, M12, M21, M22are mobility coefficients that were calculated numerically, and
|U1|,|U2|are droplets’ velocities that each drop would have in the abs ence of another
drop.
The velocities of equal droplets calculated using the formu lae [20], [21] and our
Boundary-Integral code are depicted in Fig. 2 versus the sep aration distance. The
good agreement in all distance-limits is visible. In the cas e of large separations our cal-
culations, denoted by solid circles, coincide with Anderso n’s asymptotic formula (curve
11d), and in the limit of infinite distance between the drops we r ecover the value of the drift
velocity 0 .133 as in the Young-Bratukhin problem (1, 3). In the limit of a small separa-
tion distance our results are in good agreement with the calc ulations of Keh and Chen
as well (dashed curve). A comparison for the case of drops of d ifferent radii is presented
in Fig. 3. We observe that the case of the drift velocities of t he two non-deformable
different droplets is in a good agreement with Keh and Chen pre dictions.
4.2 Deformation of equal-sized drops. Effects of Ca and veloc -
ity patterns
Consider now the case of larger capillary numbers, where con siderable deformations are
expected. The time evolution of the Taylor deformation para meter is shown in Fig. 4 for
different values of the capillary number and of the initial se paration distance. We observe
two qualitatively different scenarios of the deformation de velopment. For sufficiently
large initial separations and small Ca(e.g. curves d and e), the deformation is small
and becomes almost stationary after a short transient perio d. On the other hand, for
the relatively small initial separations and large Ca(e.g. curves a and b), the evolution
of the deformation after the transient period becomes slowe r, but remains essentially
non-steady. During the initial period, with time scale of O(Ca), the deformations occur
because the non-uniform surface tension is not compatible w ith the initial spherical form
of the drop. The motion during this period is driven by the sur face tension, rather than by
its gradient, and the deformation is controlled by the norma l component of the interfacial
stress. At longer time scales the development of the deforma tions is induced mostly by the
changes in the mutual positions of the drops (characterized by the separation distance).
In Fig. 5 the deformation parameter is depicted versus separ ation distance for equal-
size drops and different capillary numbers. The initial sepa ration was kept constant at
L0= 2.05. The leading drop becomes more deformed than the trailing one. This can be
explained by the fact that the leading drop is located in a hot ter region and, thus, it has
a lower surface tension. The dependence of well developed de formations on the capillary
12number is demonstrated in Fig. 6. Here we plotted the deforma tion parameter of the
more deformed leading drop in Fig. 5 when the separation dist ance reached L= 2.04.
The dependence on small Caappears to be linear.
The deviation of the surfaces from the spherical shape/radicalBig
r2
i(θ) + (zi(θ)−Zi)2−Riis
demonstrated in Fig. 7 for different values of time. One can se e considerable deformations
in the gap region while the outer regions of the drops remain a lmost spherical. It is evident
from Figs. 4, 5 and 7 that the leading drop deforms to an oblate shape while the trailing
one becomes prolate. The mechanism of this deformation is sh own in Figure 8 where
we plot the velocity field for two equal drops at close proximi ty for time t= 10. The
reference frame moves with the rear point on the axis of the tr ailing drop as marked in
the figure. In this reference frame, the drops are almost at re st except for the gap region.
The motion is composed of a flow past an aggregate of two drops a nd the deformation of
the gap region in the opposite direction. The elongation of t he trailing drop to a prolate
shape and the flattening of the leading drop to the oblate shap e are clearly evident.
Similar to the deformation, the velocities of the drops’ cen ters of mass achieve quasi-
steady slowly changing values after a short transient perio d. For two equal drops at
Ca= 0.2 these are plotted in Fig. 2 versus the distance between the s urfaces (points on
the curves aandc). One can see that, at large separations, the velocities of t he two equal-
sized drops are almost equal and approach the velocities of t he non-deformable drops.
In contrast to this, for sufficiently small separation, the ve locities of the two deformed
drops are different. The leading oblate drop moves slower tha n the trailing prolate one.
The time evolution of the surface-to-surface distance d(t) and center-to center distance
L(t) forCa= 0.2 and different initial separations is shown in Fig. 9. We obse rve two
different behaviors depending on the initial separation. At small initial separations,
surface to surface distance decreases with time (Figure 9 (e )). For larger separations,
surface-to surface distance shows a transient behavior in w hich it grows initially and
decreases after some period (Fig. 9(b)). This initial perio d of repulsion increases with
the initial separation, and for d0= 0.5 (see Fig 9(c)) the surface-to-surface distance grows
13during the entire time of the process. To understand this beh avior recall that the leading
drop deforms into an oblate shape, thus, moving apart from th e trailing one. The latter,
in turn, deforms into a prolate shape, however, its deformat ions are lower due to a higher
surface tension. As a result, the separation distance incre ases. The difference in the
effective surface tension increases with the separation dis tance, hence, when the distance
is sufficiently small, this effect is overcome by the simultane ous approach of the droplets’
centers of mass (Fig. 9 (d) and (f)) and, after a period of repu lsion, the drops surfaces
approach each other. The period of initial ’repulsion’ beco mes wider for larger initial
separations, while when the initial distances are small eno ugh, the difference in the rate
of deformation becomes negligible, the initial repulsion p eriod vanishes (see Fig. 9 e) and
the drops approach during the entire time of the process. Not e that the motion of the
droplets towards each other is quite slow, and in the cases th at we considered drops that
were initially very close together do not progress to a compl ete collision.
4.3 Interaction of Drops of Different Size.
The interaction of equal size drops is governed by the interp lay of two effects: rela-
tive motion caused by the oblate(prolate) shape of the leadi ng(trailing) drop, and more
pronounced deformations of the leading drop in the gap regio n due to its lower surface
tension. These effects take place also for unequal drops, but in this case they are combined
with the relative motion caused by the different sizes. Simil ar to the case of equal drops,
after a short transient period the migration velocity of the trailing drop is enchanced
and the leading drop motion is retarded compared to the case w ith no deformations. If
the leading drop is smaller than the trailing one, the separa tion distance decreases with
time and the drops come to near contact. For this case, the lar ger drop remains almost
spherical, and the smaller one deforms considerably. The fo rm of the drops in close prox-
imity and the velocity field is shown in Fig. 10 (a) for Ca= 0.2, R 1= 1, R 2= 0.5.
The reference frame moves with the rear point of the trailing drop. The motion is a
superposition of a flow past an aggregate of the drops and a mot ion and deformation of
14the surface of the leading drop. The larger trailing drop def orms mostly in the near gap
region. The deformation pattern in this case agrees qualita tively with the one reported
by Zhou and Davis (20).
The situation is different when the trailing drop is smaller t han the leading one. In
this case the gap between nondeformable drops increases and the drops separate. The
deformability may result in a more complex interaction betw een such drops if the initial
separation is small enough. The velocity field is shown in Fig . 10 (b) for Ca= 0.2, R 1=
1, R2= 1.5. The reference frame moves with the point at the center of th e gap region.
One can observe a very slow motion in the gap and considerable normal velocities at the
interfaces in the outer regions, indicating the developing deformations. The evolution
of the separation distances and the deformation parameter a re shown in Figs 11, 12
and 13 for various drop size ratio and Ca. Different curves on each plot correspond to
different initial separations. It is seen that for the large i nitial separations, the surface-to-
surface distance increases during the entire duration of th e process, as in the case when
the deformations are absent. When the initial separation is small the distance is first
reduced and only increase at the later stages. The transitio n is controled by the capillary
number (see Fig.14) and it seems to occur at separation dista nce of O(10−2).
To understand this unusual behavior, consider the evolutio n of the deformation factor
that is shown in these figures. The smaller drop deforms much s tronger than the larger
one assuming a prolate form. This is an anticipated result si nce the deformation of the
trailing drop is induced by the pulling presence of the leadi ng one and vice versa. The
larger drop induces larger deformations. Since in our case t he leading drop is larger,
higher deformations of the trailing drop are expected. At th e depicted size ratios this
effect overcomes the effect of a lower surface tension over the leading drop that has an
opposite effect in the case of equal drops, where the leading d rop is more deformed.
The deformation of the smaller drop develops very fast durin g an initial transient
period and becomes slower afterwards. The interface of a lar ger drop is initially pushed
by the trailing one and assumes an oblate shape. These deform ations are combined with
15the relative motion of the drops apart from each other due to t he different radii. Unless
the drops are very close in size, after the initial period the elongation of the trailing
drop becomes slower, and the distance between the interface s begins to grow. The drops
separate and the influence of the trailing drop on the leading one causes its deformation
to reverse itself and it also becomes prolate (see Figs. 12(c ) and 13(c)). When the drops
are very close in size (see Fig. 11(c)), the evolution their s hape is more symmetric and
resembles the one depicted in Fig. 4 for the equal-size case.
During the initial period the drops may come very close toget her and attractive
London van der Waals forces may results in the rupture of the i nter-droplet film and
cause an eventual coalescence. This process, however, is be yond the scope of our work.
The distance between the centers of mass of the drops increas es monotonically while the
surface exhibits a transient behavior. During the early sta ges of the dynamics the gap
shrinks and the surfaces come to close proximity, while at la ter stages the drops separate
as in the case with no deformations. The time in which the init ial perturbation of the
shape is relaxed and the gap acquires a minimum appears to dep end linearly on the
capillary number. In this case the system may result in a high er coalescence rate due to
diminishing gap in the transient period.
5 Concluding remarks
The interaction between two drops in a temperature gradient shows a variety of behav-
ior patterns. Equal drops always approach each other, howev er, the surface-to-surface
distance has a transient behavior during which the separati on first grows and later di-
minishes. When two unequal drops are moving, with the bigger one leading, an opposite
transient behavior is observed if the initial separation di stance is small. Here, the surfaces
first approach each other and later depart while the drops are moving away from each
other. This non-trivial behavior takes place during an init ial period of the process. Since
our calculations are based on the creeping flow approximatio n that neglect inertia of the
16fluid, these results are applicable only if the flow field in the fluid is established much
faster than an equilibrium form of the deformed surface. Thr ee different time scales are
present in the process under consideration: a viscous relax ation time tν=a2
1ρ/η3, capil-
lary surface relaxation time ts=η3a1/γ0, and the thermocapillary time t∗=η3//parenleftBig
∂γ
∂TA/parenrightBig
,
that was chosen as the time scale in our considerations. Sinc etν/t∗=Maandts/t∗=Ca,
the condition Ma << Ca is sufficient to ensure that the transient period can be descri bed
in a creeping flow approximation.
It is interesting to compare the magnitude of deformation in our case with different
cases such as buoyancy-driven motion (19) and that resultin g from non-linear dependence
of surface tension on the external field (26). We found that th e deformation in the case
of a thermocapillary migration causes behavior of the drops at close proximity that is
qualitatively different from that observed in the other case s. The calculated magnitude
in the thermocapillary was relatively small for both leadin g and trailing drops and for
different aspect ratios. The rich variety of interaction pat tern typical for a motion driven
buoyancy is not observed. There are several reasons why it is so. The first is a well-known
fact that the disturbances of the velocity field, resulting f rom the thermocapillary-induced
motion of a single drop, decay with the distance from the part icle as 1 /r3. This is much
faster than the 1 /rdecay in the case of the gravity-driven motion. Thus, the infl uence
of the presence of a drop on the motion and deformation of the o ther one is expected to
be smaller and to decay more rapidly with the separation dist ance than the one in the
gravity-driven motion case. The other important difference of these two processes is that
the Bond number
Bo=∆ρga12
γ0,
that governs the deformability of the drops undergoing grav ity-driven migration, may
take any values from 0 to infinity, and the interesting behavi or is observed when this value
is relatively large, Bo > 3 (see (19)). In contrast to this, the value of the capillary n umber
that governs the deformability of the drops in the course of t heir thermocapillary-driven
migration is restricted from above if only physically relev ant cases of positive interfacial
17tension are considered. In our calculations the non-vanish ing values of Cawere typically
about 0 .2.
Note that using the linear profile of a surface tension – tempe rature dependence
results in restriction on the extent of collective drift of d rops. Practically there is some
moment of time at which the calculation must stop, because af ter this time the surface
tension becomes zero at the front edge of the leading drop. In the case of temperature
dependence this situation implies a phase transition. Howe ver, when the surface tension
depends on concentration of a solute, zero surface tension c annot be achieved. In this
case the dependence of surface tension on concentration ( fo r some pairs fluid-solute)
can be divided to two different regions. In the first region the dependence of tension on
concentration is strong and can be described by the linear mo del considered above. In
the second region, when the surface tension is diminished, i ts change with concentration
is very slow and it can be approximated by a constant (26).
We performed a calculation for the case of equal drops using t his simple two-region
dependence of the interfacial tension letting the linear de pendence down the value
γ//parenleftBig
Aa1dγ
dT/parenrightBig
= 10−2and keeping it constant beyond this point. When the drops ent er
the region of low surface tension the deformation pattern ch anges and more pronounced
distortion of the spherical shapes are obtained. This defor mation pattern is accompanied
by a clear reduction of the drops’ speed and the entire migrat ion is halted. A demonstra-
tion of such behavior is given in Figure 15. More detailed dyn amics of such interactions
are left for the future study.
Acknowledgements
This research was supported by The Israel Science Foundatio n founded by the Academy
of Science and Humanities. O. M. L. acknowledges the support of the Israel Ministry
for Immigrant Absorption. The authors wish to thank A. M. Les hansky for helpful
discussions.
18Appendix: Kernels in Eqn. [16]
Ifrx/negationslash= 0, i.e. the point xis not on the axis of symmetry
Bzz= 2k/parenleftbiggr
rx/parenrightbigg1/2/bracketleftBigg
F(k) +(z−zx)2
(rx+r)2+ (zx−z)2E(k)/bracketrightBigg
,
Bzr=kz−zx
(rxr)1/2/bracketleftBigg
F(k)−r2
x−r2+ (z−zx)2
(rx+r)2+ (zx−z)2E(k)/bracketrightBigg
,
Brz=−kz−zx
rx/parenleftbiggr
rx/parenrightbigg1/2/bracketleftBigg
F(k) +r2
x−r2−(z−zx)2
(rx+r)2+ (zx−z)2E(k)/bracketrightBigg
,
Brr=k
rxr/parenleftbiggr
rx/parenrightbigg1/2/braceleftBig
[r2
x+r2+ 2(zx−z)2]F(k)−
2(z−zx)4+ 3(z−zx)2(r2
x+r2) + (r2
x−r2)2
(rx+r)2+ (zx−z)2E(k)/bracerightBigg
,
HereFandEare complete elliptic functions of the first and second kind r espectively,
k2=4rxr
(rx+r)2+ (zx−z)2,
andnzandnrdenote the projection of the normal vector in the azimuthal p lane on the
axiszandr.
If the point xis on the axis of symmetry, rx= 0,k= 0 and the kernels take the form
Bzz= 2πrr2+ 2(z−zx)2
[r2+ (z−zx)2]3/2, B rz= 0,
Bzr=−2πr2(z−zx)
[r2+ (z−zx)2]3/2, B rr= 0,
Asymptotic expansions as ytends to x(rx/negationslash= 0):
Bzz=−ln[r2+ (z−zx)2] +czz+. . . B rz=crz+. . .,
Bzr=czr+. . ., B rr=−ln[r2+ (z−zx)2] +crr+. . .,
n(x)·B(x,y)·τ(y) =crznr(x)τz(x) +czrnz(x)τr(x) +. . .,
where crr,crz,czr, and czzare constants. If the curve σiis parametrized by the arc
length, the kernels are regular for rx= 0.
19References
1. Young, N. O., Goldstein, J. S., and Block, M. J., J. Fluid Mech. 6, 350 (1959).
2. Subramanian, R. S., in “Transport Processes in Bubbles, Drops and Particles” (P.
Chhabra, D. DeKee), p.1. Hemisphere, New York, 1992
3. Bratukhin, Yu. K., Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti, Gaza. 5,
156 (1975)
4. Balasubramaniam, R. and Chai A., J. Colloid Interface Sci. 119531 (1987).
5. Meyyappan, M., Wilcox, W. R., and Subramanian, R. S., J. Colloid Interface Sci.
94, 243 (1983).
6. Feuillbois, F., J. Colloid Interface Sci. 131, 267 (1989).
7. Meyyappan, M., and Subramanian, R. S., J. Colloid Interface Sci. 97, 291 (1984).
8. Acrivos, A., Jeffrey, D. J., and Saville, D. A., J. Fluid Mech. 212, 95 (1990).
9. Satrape, J. V., Phys. Fluids A 41883 (1992).
10. Wang, Y., Mauri, R., and Acrivos, A., J. Fluid Mech. 261, 47 (1994).
11. Anderson, J. L., Int. J. Multiphase Flow 11, 813 (1985).
12. Keh, H. J., and Chen, S. H., Int. J. Multiphase Flow 16, 515 (1990).
13. Keh, H. J., and Chen, L. S., J. Colloid Interface Sci. 151, 1 (1992).
14. Keh, H. J., and Chen, L. S., Chem. Eng. Sci. 48, 3565 (1993).
15. Wei, H., and Subramanian, R. S., Phys. Fluids A 51583 (1993).
16. Loewenberg, M., and Davis, R., J. Fluid Mech. 256, 107 (1993).
17. Manga, M., and Stone H. A., J. Fluid Mech .256, 647 (1993).
2018. Zinchenko, A. Z.. Rother M. A., and Davis, R. H, J. Fluid Mech .391, 249 (1999).
19. Davis, R. H., Phys. Fluids A 111016 (1999).
20. Zhou, H., Davis, R. H, J. Colloid Interface Sci. 181, 60 (1996)
21. Pozrikidis, C., ”Boundary Integral and Singularity Met hods for Linearized Viscous
Flow,” Cambridge Univ. Press, Cambridge, UK, 1992.
22. Ladyzhenskaya, O. A., “ The Mathematical Theory of Visco us Incompressible
Flow.” Gordon and Breach, 1969.
23. Krasny, R., J. Fluid Mech. 167, 65 (1986).
24. Loewenberg, M. and Hinch E. J., J. Fluid Mech. 321395 (1996).
25. Rother, M. A. and Davis, R. H., J. Colloid Interface Sci. 214297 (1999).
26. Adamson A.W., “Physical Chemistry of Surfaces”, 3-Ed., Wiley-Interscience Pub-
lication, New York-London-Sydney-Toronto, (1976)
21Figure legends
Figure 1. Geometric sketch of two viscous drops immersed in an externa l temperature gradi-
ent∇Tparallel of their axis of symmetry.
Figure 2. Center of mass velocities for equal drops R1=R2= 1. The solid line ( d), dashed
line (b) and the points on these lines are the results of Anderson, Ke h and Chen and
our computations for Ca= 0.001, respectively. The points ( a) and ( c) correspond
to leading and trailing deformable drops at Ca= 0.2.
Figure 3. Droplet velocities for Ca= 0.001 and different aspect ratio. In the case ( a, b)
R1= 1.0 and R2= 2.0. In the case ( c, d)R1= 1.0 and R2= 0.5. The lines band
dcorrespond to the larger drop.
Figure 4. The evolution of Taylor deformation parameter defined in Eqn . [19]: ( a)Ca=
0.2, d0= 0.1; (b)Ca= 0.2, d0= 0.5; (c)Ca= 0.1, d0= 0.1; (d)Ca= 0.1, d0=
0.5; (e)Ca= 0.1, d0= 2. Positive and negative values correspond to oblate and
prolate shapes, respectively.
Figure 5. Deformation parameter for equal-size droplets versus the d istance between cen-
ters of mass: ( a)Ca= 0.2, d0= 0.05; (b)Ca= 0.1, d0= 0.05; (c)
Ca= 0.05, d0= 0.05. Positive and negative values correspond to oblate and
prolate shapes, respectively.
Figure 6. Deformation of the leading droplet versus the capillary num ber at L= 2.04 for
equal-size droplets.
Figure 7. The local deformations for trailing ( a) and leading ( b) equal-size droplets for the
caseCa= 0.2 and d0= 0.01. 1, 2 and 3 correspond to 0 .5, 3.3, and 10 units of
time, respectively.
Figure 8. Velocity patterns for Ca= 0.2,R1= 1.0 and R2= 1.0. The reference frame moves
with the speed of the marked point.
22Figure 9. The evolution of the separation and center-to-senter dista nces for equal-size drops
atCa= 0.2 as function of the initial separation ( a). (b) Transient dynamics for
intermediate initial separation. ( c, d) Evolution of surface-to-surface separation
and center-to-center distance for relatively large initia l separation. ( e, f) Evolution
of surface-to-surface separation and center-to-center di stance for relatively small
initial separation.
Figure 10. Velocity patterns for ( a)Ca= 0.2,R1= 1.0 and R2= 0.5; (b)Ca= 0.15,R1= 1.0
andR2= 1.5. The reference frame moves with the speed of the marked poin t.
Figure 11. Interaction between a leading larger drop and a trailing sma ller one at close prox-
imity for Ca= 0.2, R1= 1.0, R2= 1.1: (a) evolution of center-to-center distance;
(b) evolution of surface-to surface separation; ( c) evolution of Taylor deformation
parameter. The numbers 1 to 9 denote the different initial sep aration.
Figure 12. Interaction between a leading larger drop and a trailing sma ller one at close prox-
imity for Ca= 0.15, R1= 1.0, R2= 1.5: (a) evolution of center-to-center distance;
evolution of surface-to surface separation; ( c) evolution of Taylor deformation pa-
rameter. The numbers 1 to 6 denote the different initial separ ation.
Figure 13. Interaction between a leading larger drop and a trailing sma ller one at close prox-
imity for Ca= 0.12, R1= 1.0, R2= 2: ( a) evolution of center-to-center distance;
(b) evolution of surface-to surface separation; ( c) evolution of Taylor deformation
parameter. The numbers 1 to 7 denote the different initial sep aration.
Figure 14. The relaxation time trwith minimum separation distance as function of the capil-
lary number.
Figure 15. Deformation of equal-size drops migrating into a region of v anishing surface tension.
The upper plot shows the dependence of surface tension on dis tance. The lower
plot demonstrates migration and deformation pattern for va rious time. ( a)t= 0;
(b)t= 5; ( c)t= 19; ( d)t= 25.
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arXiv:physics/0102073v1 [physics.gen-ph] 22 Feb 2001Direction Adaptation Nature of Coulomb’s Force and Gravita tional Force in
4-Dimensional Space-time
H. Y. Cui∗
Department of Applied Physics
Beijing University of Aeronautics and Astronautics
Beijing, 100083, China
(February 24, 2013)
It was found that the Coulomb’s force and gravitational forc e on a particle always act in the
direction perpendicular to the 4-velocity of the particle i n 4-dimensional space-time, rather than
along the line joining a couple of particles. This inference is obviously supported from the fact that
the magnitude of the 4-velocity is kept constant. It indicat es there is a new nature here named
as direction adaptation nature. This formulation has signi ficant influence on who to teach physics
taday, for example, we always say to students that the sun’s g ravitational force directs at the sun,
but this sentence is uncorrect according to the direction ad aptation nature of gravitational force.
03.50.De, 04.20.Cv, 03.30.+p
In the world, almost every young people was educated to know t hat the Coulomb’s force and gravitational force
act along the line linking a couple of particles. But this sta tements contradicts to the relativity theory.
In an inertial Cartesian coordinate system S: (x1, x2, x3, x4=ict), the 4-velocity uof a particle has components
(uµ, µ= 1,2,3,4 ) defined as in the relativity theory [1], the magnitude of th e 4-velocity uis given by
|u|=√uµuµ=/radicalbig
−c2=ic (1)
where since the reference frame Sis a Cartesian coordinate system whose axes are orthogonal t o one another, there
is no distinction between covariant and contravariant comp onents, only subscripts need be used. Here and below,
summation over twice repeated indices is implied in all case , Greek indices will take on the values 1,2,3,4. The above
equation stands, any force can never change uin the magnitude but can change uin the direction. We therefore
conclude that the Coulomb’s force and gravitational force o n a particle always act in the direction perpendicular to
the 4-velocity of the particle in the 4-dimensional space-t ime, rather than along the line joining a couple of particles .
Alternatively, any 4-force fsatisfy the following perpendicular relation
uµfµ=uµmduµ
dτ=m
2d(uµuµ)
dτ= 0 (2)
In present paper, Eq.(2) has been elevated to an essential re quirement for definition of any force, which bring out
many new aspects for the Coulomb’s force and gravitational f orce.
Suppose that two charged particle qandq′locate at the positions ( x1, x2, x3, x4) and ( x′
1, x′
2, x′
3, x′
4) in the Cartesian
coordinate system Sand move at 4-velocities uandu′respectively, as shown in Fig.1. Two characters for the
Coulomb’s force must be considered. The first, because elect romagnetic interaction occurs through photon exchange
process, there is a retardation condition, so that the Coulo mb’s force on qcan only happen at time t=t′+ (r/c) ,
where ris the spatial distance of the two particles. The second, the Coulomb’s force is perpendicular to the velocity
direction of q, as illustrated in Fig.1. Like a centripetal force, the Coul omb’s force fonqshould make an attempt
to rotate itself about its path center, the force fshould lie in the plane of u′andX, in agreement with the classical
Coulomb’s formula for two rest particles, so we can write it a s
f=au′+bX (3)
Where aandbare unknown coefficients. Using relation f⊥u, i.e.
u·f=a(u·u′) +b(u·X) = 0 (4)
we rewrite Eq.(3) as
f=a
u·X[(u·X)u′−(u·u′)X] (5)
It follows from the direction of Eq.(5) that the unit vector o f the Coulomb’s force direction is given by
1ˆ
f=1
c2r[(u·X)u′−(u·u′)X] (6)
because
ˆ
f=1
c2r[(u·X)u′−(u·u′)X]
=1
c2r[(u·R)u′−(u·u′)R]
=−[(ˆu·ˆ
R)ˆu′
−(ˆu·ˆu′
)ˆ
R]
=−cosαˆu′
+ sinαˆ
R (7)
|ˆ
f|= 1 (8)
Where αrefers to the angle between uandR,ˆu=u/ic,ˆ
u′=u′/ic,ˆ
R=R/r. Suppose that the magnitude of the force
fhas the classical form
|f|=kqq′
r2(9)
Combination of Eq.(9) with (6), we obtain a modified Coulomb’ s force
f=kqq′
c2r3[(u·X)u′−(u·u′)X]
=kqq′
c2r3[(u·R)u′−(u·u′)R] (10)
It is follows from Eq.(10) that the force can be rewritten in t erns of 4-vector components as
fµ=qFµνuν (11)
Fµν=∂µAν−∂νAµ (12)
Aµ=kq′
c2u′
µ
r=kq′
cu′
µ
u′ν(x′ν−xν)(13)
Where we have used the relations
∂µ/parenleftbigg1
r/parenrightbigg
=−R
r3(14)
r=c∆t=cq′O
ic=cˆu′
·X
ic=u′
ν(x′
ν−xν)
c(15)
The latter represents the retardation relation as illustra ted in Fiq.1. Obviously, Eq.(13) is known as the Lienard-
Wiechert potential for a moving particle.
From Eq.(13), we have
∂µAµ=kq′u′
µ
c2∂µ/parenleftbigg1
r/parenrightbigg
=−kq′u′
µ
c2/parenleftbiggRµ
r3/parenrightbigg
= 0 (16)
because u′⊥R. It is known as the Lorentz gauge condition.
To note that Rhas three degrees of freedom on the condition R⊥u′, so we have
∂µRµ= 3 (17)
∂µ∂µ/parenleftbigg1
r/parenrightbigg
=−4πδ(R) (18)
2From Eq.(12), we have
∂νFµν=∂ν∂µAν−∂ν∂νAµ=−∂ν∂νAµ
=−kq′u′
µ
c2∂ν∂ν/parenleftbigg1
r/parenrightbigg
=kq′u′
µ
c24πδ(R)
=µ0J′
ν (19)
where J′
ν=q′u′
νδ(R). From Eq.(12), by exchanging the Indices and taking the sum mation over, we have
∂λFµν+∂µFνλ+∂νFλµ= 0 (20)
The Eq.(19) and (20) are known as the Maxwell’s equations. Fo r continuous media, they stand as well as.
The above formalism clearly show that the Maxwell’s equatio ns can been derived from the classical Coulomb’s force
and the perpendicular relation of force and velocity in an ax iomatic manner. Specially, Eq.(5) directly accounts for
the geometrical meanings of the curl of vector potential.
The above formalism has a significance on guiding who to devel ope the gravitational theory which exists many
versions, specially in the scope of the general relativity i n a sense.
In analogy with the modified Coulomb’s force of Eq.(10), we di rectly suggest a modified universal gravitational
force as
f=−Gmm′
c2r3[(u·X)u′−(u·u′)X]
=−Gmm′
c2r3[(u·R)u′−(u·u′)R] (21)
for a couple of particles with masses mandm′respectively. Comparing with the statements about the Coul omb’s
force and gravitational force in the most textbooks, the per pendicular relation of force and velocity is named here as
direction adaptation nature of force.
It follows from Eq.(21) that we can predict there are gravita tional radiation and magnetic-like components for the
gravitational force. Obviously, if the Newton’s law of univ ersal gravitation and the direction adaptation nature of
force both are basic axioms, the predicted effects will exist . Particularly, the magnetic-like components will act as a
key role in the geophysics. Because of lacking experimental data to the predicted effects in a sense, in the following
our interest turns to certain nonlinear effects of photon flyi ng [4] concerning with the direction adaptation nature.
We begin by discussing the motion of a single photon. The path in Fig.2 is a graph of the photon’s coordinate x1
plotted as a function of time ict. The photon is emitted at point P1 and received at point P2, tw o reference frames
SandS′are fixed at the two points respectively, and corrected their x1-axes in the same direction according to the
photon traveling direction. To note that c=dx1/dt,umakes an angle of 45 degree with the x1-axis at P1. The
gravitational force acts like a centripetal force deflectin guto the center of the path, so the 4-velocity of the photon
has rotated an angle θafter its travel from P1 to P2 through the action of the gravit ational force. Then there are
coordinate transformations for the two reference frames SandS′, given by
dx1=dx′
1cosθ−dx′
4sinθ (22)
dx4=dx′
1sinθ+dx′
4cosθ (23)
For calculating the work done by the gravitational force fro m P1 to P2, we establish a xg-axis lying along the
gravitational direction, Newton’s second law of motion is m odified in terms of the angle θto
f=mdug
dτ=md(icsinθ)
dτ=−mc2sinθdsinθ
dxg(24)
Where ug=dxg/dτ=icsinθ,dτis the proper time interval. Here, for a photon, m→0 and dτ→0, but they can
be eliminated without introducing any singularity. Definin g the gravitational potential difference Φ between the two
points P1 and P2, and from Eq.(24) we obtain
Φ =/integraldisplay
p1→p2/parenleftbiggf
m/parenrightbigg
dxg=−1
2c2sin2θ (25)
Combination of Eqs.(22,23) with (25), the effect formula
3/parenleftbiggdt
dt′/parenrightbigg
dx′
1=0=/parenleftbiggdx4
dx′
4/parenrightbigg
dx′
1=0=/radicalbig
1−sin2θ≃1 +Φ
c2(26)
gives the spectral shift for the photon. If the photon moves f rom the place of lower gravitational potential to the
higher, then Φ <0, there is the spectral red shift for the photon.
For a particle moving under the Newtonian gravitational att raction of a spherical object of mass Msituated at the
origin of the reference polar coordinate frame S: (r, ϕ, x 4=ict), its orbit is given by
1
r=1
a(1−e)[1 +ecos(ϕ−ϕ0)] (27)
where notations consist with the convention [1]. In the case of planet with the smaller speed comparing to the speed
of light, one never needs to solve its relativistic dynamica l equation, the precision of Eq.(27) is enough for describin g
the motion of the particle (planet).
For an observer at the infinite distance point from the partic le, the 4-velocity of photons before arriving at the
observer have rotated an angle θdue to the effect of the gravitational field in our view of point s. The bent rays form
an image of the particle orbit at the lower location in Fig.3. The reference coordinate system S′: (x′
1, x′
2, x′
3, x′
4=ict′)
fixed at the observer is depicted in Fig.3 with the dot lines as well as. If the position of the particle in S′is measured
with three coordinates: ρ, ϕ, t′′, where ρis the distance from the center O′to the image particle, t′′is the time of the
center O′inS′. Thus ( ρ, ϕ, t′′) consists of an abstract coordinate system S′′, in which OO′is perpendicular to O′P′
because of the center O′with coordinates (0 ,0, t′′). Then there are coordinate transformations between the SandS′:
cosθ=dρ/dr (28)
dt=dt′′cosθ (29)
Eq.(29) is equivalence to Eq.(23) on the condition dρ= 0 for the center O′. Following the Eq..(25), we know
cosθ= 1 +Φ
c2= 1−rs
ρ(30)
Where rsis the Schwarzchild radius of the massive object M. In the abstract frame S′′: (x1≡ρ, x2≡ϕ, x3, x4≡ct′′)
where we introduce metric tensor gµν, the motion of the particle is given by
..xµ+Γµ
νλ.xν.xλ= 0 (31)
The metric tensor gµνcan be directly obtained from
ds2=dr2+r2dθ2+ (dx3)2−c2dt2
= cos−2θdρ2+ρ2dθ2+ (dx3)2−c2cos2θdt2(32)
It is known as the Schwarzchild metric tensor [1] [2]. We then may expect that all conclusions made for the particle
motion here are identical to that in the general relativity. Thus, the motion equation of the particle is given by
/parenleftbiggdρ
dt′′/parenrightbigg2
+/parenleftbigg
1−rs
ρ/parenrightbiggh2
ρ2−2GM
ρ=K=const. (33)
ρ2dϕ
dt′′=h=const. (34)
Specially, the perihelion advance of the particle is given b y
∆ϕ
T=3πrs
Ta(1−e)(35)
It is as the same as that in the general relativity [1] [2].
We emphasize that any measurable quantity can to be defined in various space-time, a suitable space-time will be
determined from the agreement in the experiment and the theo ry.
The path of a photon traveling in an equatorial plane is given by Eq.(33,34). For the photon, Eq.(34) becomes
4h=ρ2dϕ
dt′′=const. =/parenleftbigg
ρ2dϕ
dt′′/parenrightbigg
ρ=∞=∞ (36)
Then Eq.(33) becomes [1]
d2u
dϕ2+u=3
2rsu2(37)
where set u= 1/ρ. So in its flight past the massive object Mof radius Rthe photon is deflected through an angle
δ=2rs
R(38)
in agreement with observations [1] [2].
If there is not the nonlinear effect of photon flying, any strai ght line in the space-time is of the path of photon.
Since the light is bent in the 4-dimensional space-time for a n infinite distance observer, we consequently conclude that
the space will bend due to the gravitational effect mentioned above near a massive object.
The particle at rest has the 4-velocity u= (0,0,0, ic) in 4-dimensional Cartesian coordinate system. The ”rest”
refers to in the space. Considering two rest inertial frames , labeled SandS′, fixed respectively at two rest particles
with their x1-axes lying along the same line. At one moment, the particle o f the frame S′starts into a gravitational
field or electromagnetic field Φ, and goes out with a classical speed valong the x1-axis. In our view of points, the
x′
4-axis of the frame S′has rotated an angle θ, because the force deflects the 4-velocity of the S′particle. From
Eq.(25), the angle is given by.
cosθ=/radicalbigg
1 +2Φ
c2=/radicalbigg
1 +v2
c2≃1/radicalbig
1−v2/c2(39)
If the two origins coincide at the time t=0 after that acceler ation, then the coordinate transformations between the
two frame are
dx1=dx′
1cosθ−dx′
4sinθ
=1/radicalbig
1−v2/c2dx′
1+iv/c/radicalbig
1−v2/c2d′x4 (40)
dx4=dx′
1sinθ+dx′
4cosθ
=−iv/c/radicalbig
1−v2/c2dx′
1+1/radicalbig
1−v2/c2d′x4 (41)
It is known as the Lorentz transformation.
It is noted that the direction adaptation nature stands for a ny force: strong, electromagnetic, weak and gravitational
interactions, so there are new aspects remained for physics to explore.
In conclusion, it was found that the gravitational force and Coulomb’s force on a particle always act in the direction
perpendicular to the 4-velocity of the particle in 4-dimens ional space-time, rather than along the line joining a coupl e
of particles. This inference is obviously supported from th e fact that the magnitude of the 4-velocity is kept constant.
It indicates there is a new nature here named as direction ada ptation nature. The Maxwell’s equations can been
derived from the classical Coulomb’s force and the directio n adaptation nature of force in an axiomatic manner.
Certain gravitational effects can be explained in terms of th e direction adaptation nature of gravitational force for
photon flying.
∗E-mail: hycui@public.fhnet.cn.net
[1] E. G. Harris, Introduction to Modern Theoretical Physic s, Vol.1, (John Wiley & Sons, USA, 1975).
[2] J. Foster, J. D. Nightingale, A Short Course in General Re lativity, (Springer-Verlag, New York, 1995).
[3] J. H. Taylor, J. M. Weisberg, The Astrophysical Journal, 253, 908(1982).
5[4] H. Y. Cui, College Physics (A monthly edited by Chinese Ph ysical Society in Chinese), ”Influences of the Gravitationa l
Field on the Clock and Rod”, 10, 31(1992).
FIG. 1. The Coulomb’s force acting on qis perpendicular to the 4-velocity uofq, and lies in the plane of u′andXwith the
retardation with respect to q′.
FIG. 2. The graph of a photon travelling from P 1to P2with a rotation angle θof its 4-velocity in the gravitational field in
4-dimensional space-time.
FIG. 3. The top view of the orbit of the particle in 4-dimensio nal space-time, and its image formed by the bent rays in the
gravitational field of M.
61Manuscript: “Direction adaptation nature of …” by H. Y. Cui.
Fig.1 The Coulomb’s force acting on q is perpendicular to the 4-velocity u of q, and lies in the plane of u′
and X with the retardation with respect to q′.
S x4 u
O R
q
f
u′ X
u′ ⊥R
q′ X=x-x′
x1,2,32Manuscript: “Direction adaptation nature of …” by H. Y. Cui.
Fig.2 The graph of a photon travelling from P1 to P2 with a rotation angle θ for its 4-velocity in the
gravitational field in 4-dimensional space-time.
S
x4=ict S′
x′4=ict u′
x′1
P2
θ
xg
f u
x1
P13Manuscript: “Direction adaptation nature of …” by H. Y. Cui.
Fig.3 The top view of the orbit of the particle in 4-dimensional space-time, and its image formed by the
bent rays in the gravitational field of M.
S x4=ict x2 S′ x′4 x′2
ray ray x′1
spiral orbit
O r ϕ x1
θ
spiral cone→ P ′
ρ ϕ
the image
O′
|
arXiv:physics/0102074v1 [physics.gen-ph] 23 Feb 2001Effective theory of systems coupled strongly to rapidly-var ying external sources.
R. Huertaaand J. Wudkab
aDepartamento de F´ ısica Aplicada, Cinvestav-IPN Unidad M´ erida.
M´ erida, Yucat´ an 97310, M´ exico
bDepartment of Physics, University of California, Riversid e CA 92521-0413.
(October 29, 2013)
We consider quantum systems which interact strongly with a r apidly varying environment and
derive a Schr¨ odinger-like equation which describes the ti me evolution of the average wave function.
We show that the corresponding Hamiltonian can be taken to be Hermitian provided all states
are rotated using an appropriate unitary transformation. T he formalism is applied to a variety of
systems and is compared and contrasted with related results describing stochastic resonances.
PACS: 05.40, 14.60.P, 42.15, 32.80, 05.10.G
I. INTRODUCTION
The study of quantum systems which interact strongly with th eir environment often presents serious challenges
due to the possibility that these interactions cannot be neg lected and, in addition, also vary rapidly and randomly
with time [1]. The effects of such external fields is often unav oidable and interesting and can lead to unexpected
phenomena such as, for example, those studied under the blan ket term of stochastic resonances [2]. In this paper we
will study one subset of such systems.
We will assume that the interactions with the environment ar e described by a time-dependent contribution to the
Hamiltonian denoted by H′(t) which cannot be treated perturbatively. In analogy with a s imilar problem in mechan-
ics [3] we will assume self-consistently that the states of t he system can be decomposed into a sum of slowly varying
modes and high-frequency components of small amplitude. Us ing this decomposition we will show that the time
evolution of the slow modes is determined by an effective Hami ltonian which, to leading order, depends quadratically
on the external interaction H′. The formalism assumes that the time scales associated with the interactions with the
environment are much shorter than all other frequencies in t he problem. Denoting by Ω a typical frequency of the
interaction H′, we will obtain a solution to Schr¨ odinger’s equation as a se ries in 1/Ω.
The effective Hamiltonian describes the average time evolut ion of the system and can exhibit resonances under
some special circumstances which will be illustrated using simple examples. It is also worth noting that the same
formalism can be applied to anysystem evolving according to a Schr¨ odinger-like equation assuming that the operators
corresponding to the Hamiltonian contain terms which vary r apidly in the evolution parameter. We also provide
examples of this type of generalization: using geometrical optics we study light-ray propagation in a random media,
and, we determine the effects of a time-independent potentia l which varies rapidly with position on the wave functions
of a non-relativistic particle.
The paper is organized as follows: in section II we give a desc ription of the formalism and find the effective Hamil-
tonian that will be used in the applications. The behavior of the effective Hamiltonian under unitary transformations
is studied in section III; the formalism is then applied to va rious illustrative examples in section IV. In section V we
give an alternative view of the problem in terms of the Fokker -Planck equation and the results are then compared and
contrasted with the formalism used in deriving the standard stochastic resonances (section VI). Paring comments
and conclusions are presented in section VII. Finally, a mat hematical detail is relegated to the appendix.
II. QUANTUM SYSTEMS WITH RAPIDLY-VARYING EXTERNAL FIELDS
We consider a generic quantum system with a Hamiltonian of th e form
H=H0+H′, (1)
whereH′is time dependent with characteristic frequencies assumed larger than all the other energy scales in the
system (we take units where ¯ h= 1). In general we will also allow H0to vary with time, but with the restriction
that the time scale(s) associated with H0are much smaller than those associated with the time variati on ofH′. In
addition we assume that H′is larger than H0so that, symbolically
1H0,˙H0/H0<H′<˙H′/H′. (2)
More specifically we assume that H′admits a Fourier expansion of the type
H′=/summationdisplay
|ω|>ΩHωe−iωt, (3)
where the sum is over a set of frequencies {ω}such that the differences also obey |ω−ω′| ≥Ω. In general we will
allow the Fourier coefficients Hωto be time-dependent, but, as for H0, we assume that the corresponding frequencies
are small compared to Ω. Henceforth “slow” will mean “of freq uency ≪Ω”.
In solving the Schr¨ odinger equation for such a system we wil l assume that the wave function can be separated into
a slowly varying piece ψand a rapidly varying (frequency∼>Ω) “ripple” χof small amplitude,
Ψ =ψ/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
slow+χ/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
fast. (4)
We will then match slow and fast terms, noting that even thoug h|χ|is small, ˙χcan be large.
We will assume that all quantities can be written as the sum of a slowly varying piece (containing frequencies ≪Ω)
and a fast piece (in special cases one or the other may vanish) . It then proves convenient to introduce the following
notation: for any quantity Ξ
∝an}b∇acketle{tΞ∝an}b∇acket∇i}ht: slow part of Ξ . (5)
For example, ∝an}b∇acketle{tΨ∝an}b∇acket∇i}ht=ψ,∝an}b∇acketle{tH∝an}b∇acket∇i}ht=H0,∝an}b∇acketle{tH′∝an}b∇acket∇i}ht= 0.
To solve Schr¨ odinger’s equation for this class of systems w e consider a typical term in (3) and define the expansion
parameter
η∼|Hω|
Ω(6)
(alternatively η∼ |/integraltext
dtH′|); the previous restrictions imply that η <1. We will now assume that the wave function
has an expansion in powers of η:
Ψ =ψ+χ1+χ2+· · ·, χ n∼ηn. (7)
This expansion is useful for n≪1/η, beyond this order we typically obtain n−fold products of slowly varying
quantities which can generate terms with frequency ∼Ω, and the separation between slowly and rapidly varying
terms cannot be maintained. These effects are, however, subd ominant since their amplitude are suppressed by a small
factor ∝ηn≪1.
Substituting Hfrom (1) and Ψ from (4) in Schr¨ odinger’s equation
HΨ =i˙Ψ, (8)
we find, to lowest order in 1 /Ω
i˙ψ+i˙χ1+O(Ωη2) =H0ψ+H′ψ+O(H0η,Ωη2), (9)
whence
i˙χ1=H′ψ, i ˙ψ=H0ψ. (10)
Sinceψis slowly varying, the first equation can be solved to this ord er inηby takingψconstant, namely,
χ1=Uψ;U=/parenleftbigg1
i/integraldisplay
dtH′/parenrightbigg
=/summationdisplay
|ω|>Ω1
ωHωe−iωt. (11)
To the next order we write
Ψ =ψ+Uψ+χ2+O(η3), (12)
and obtain
2i˙ψ+iU˙ψ+i˙χ2=H0ψ+H0Uψ+H′Uψ+O(Ωη3,H0η2). (13)
Note thatH′Ucontains both slow and fast terms. Using the notation (5) we fi nd
i˙ψ= (H0+∝an}b∇acketle{tH′U∝an}b∇acket∇i}ht)ψ,
i˙χ2= ([H0,U] +H′U − ∝an}b∇acketle{tH′U∝an}b∇acket∇i}ht)ψ, (14)
where the second equation can be solved (to the present order inη) by neglecting the time variation in H0andψ.
To this order the average wave function then obeys a Schr¨ odinger-like equation with an effective Hamiltonian [4]
Heff=H0+∝an}b∇acketle{tH′U∝an}b∇acket∇i}ht. (15)
It is easy to see that to this order in η Heffis Hermitian, however, to order η2we find
Heff=H0+∝an}b∇acketle{tH′U∝an}b∇acket∇i}ht − ∝an}b∇acketle{tU ([H0,U] +H′U)∝an}b∇acket∇i}ht, (16)
which is not Hermitian:
Heff−Heff†=/angbracketleftbig
H′U+UH′+/bracketleftbig
U2,H0/bracketrightbig/angbracketrightbig
+· · ·.
=i∂t/angbracketleftbig
U2/angbracketrightbig
+/bracketleftbig/angbracketleftbig
U2/angbracketrightbig
,H0/bracketrightbig
+· · · (17)
and equals, to this order the totaltime derivative of the operator/angbracketleftbig
U2/angbracketrightbig
. This property corresponds to the fact that
there is some probability “leakage” of order η2from the slowly varying part of the wave function to the rapid ly varying
ripple. This is to be expected since/angbracketleftbig
|χ|2/angbracketrightbig
=O(η2) and is non-zero in general.
The non-Hermiticity of the effective Hamiltonian for the slo wly varying modes frequently appears in expansions
similar to the one considered here [5]1. This result can be better understood by considering the beh avior of the above
expansion under unitary transformations to which we now tur n.
III. UNITARY TRANSFORMATIONS
In this section we determine the behavior of the effective Ham iltonian (16) under unitary transformations. We show
below that the non-Hermitian piece in (16) is modified under s uch transformations and, in fact, can be completely
eliminated.
For the case of a constant transformation, Ψ → CΨ with ˙C= 0 it is clear that H→ C†HCandHeff→ C†HeffC.
If the unitary transformation is time-dependent, however, the result is more complicated. We will concentrate on
transformations of the form
Ψ =eFˆΨ, (18)
whereFis anti-Hermitian, rapidly varying, and of order η. The Hamiltonian for the transformed states ˆΨ is
ˆH=e−FHeF−ie−F∂teF
=H+ [H,F] +1
2[[H,F],F]−i˙F−i
2/bracketleftBig
˙F,F/bracketrightBig
−i
6/bracketleftBig/bracketleftBig
˙F,F/bracketrightBig
,F/bracketrightBig
+O(η3). (19)
A tedious repetition of the procedure outlined in section II gives the following expression for the corresponding
effective Hamiltonian
ˆHeff=H0+∝an}b∇acketle{tH′U∝an}b∇acket∇i}ht+i
2∂t∝an}b∇acketle{tF(F−2U)∝an}b∇acket∇i}ht+1
2∝an}b∇acketle{t[H0,[F,U]]∝an}b∇acket∇i}ht
+1
2∝an}b∇acketle{t[[H0,U],U]∝an}b∇acket∇i}ht − ∝an}b∇acketle{tUH′U∝an}b∇acket∇i}ht −1
2/angbracketleftbig/bracketleftbig
H0,(F− U)2/bracketrightbig/angbracketrightbig
+O(H0η3,Ωη4). (20)
1In [5] a non-Hermitian term was found already in the first orde r, the discrepancy between this result and the one obtained
here is due to different assumptions concerning the time-dep endence and magnitude of the various terms in the Hamiltonia n,
leading to different expansion parameters.
3For a general choice of Fthis expression is still non-Hermitian. However for the spe cial case
F=U+O(η2), (21)
we obtain
ˆHeff=H0+1
2∝an}b∇acketle{t[H′,U]∝an}b∇acket∇i}ht+1
2∝an}b∇acketle{t[[H0,U],U]∝an}b∇acket∇i}ht − ∝an}b∇acketle{tUH′U∝an}b∇acket∇i}ht+O(H0η3,Ωη4), (22)
which is explicitly Hermitian and, in fact, it is identical t o the Hermitian part of (16). It is, of course, always possibl e
to return to the original frame using ˆΨ = exp( −U+· · ·)Ψ.
The wave function in the new frame, ˆΨ, has an expansion similar to (7)
ˆΨ =ˆψ+ ˆχ1+ ˆχ2+· · ·,ˆχn∼ηn, (23)
where the slowly varying piece ˆψevolves unitarily in time since (22) is Hermitian (at least t o orderη2). Using (11)
we find
ˆψ=/parenleftbigg
1−1
2/angbracketleftbig
U2/angbracketrightbig/parenrightbigg
ψ+O(η3),
ˆχ1= 0. (24)
TheO(η2) difference between ψandˆψquantifies the probability leak into the rapidly varying sec tor for the original
frame. The second order term in the wave function ˆ χ2cannot be determined without a specific choice for the O(η2)
terms inF. The specific form of the relation between ˆψandψcan also be understood form the expression for the
non-Hermitian part of the effective Hamiltonian obtained in (17).
We conjecture that this procedure can be carried order-by-o rder inη, but since we will not need these higher order
corrections we will not pursue this further. The expression (22) is the form of the effective Hamiltonian which will be
used in the following examples.
IV. EXAMPLES
The previous results can be applied to a variety of systems. I n this section we will consider 5 such examples. Our
main concern will be to illustrate a wide range of systems tha t can be studied using the above formalism
A.N-level quantum systems
In this section we consider a quantum system with a finite numb er of states. This serves, for example, as a model
for spin or flavor changes in elementary particles; it also de scribes the basic physics of nuclear magnetic resonance
and related phenomena. The most general Hamiltonian for the se systems can be expanded in terms of the generators
ofSU(N) which we denote by {λa}and which satisfy
/bracketleftbig
λa,λb/bracketrightbig
=iCabcλc. (25)
We can then write
H0=/summationdisplay
afaλa, H ω=/summationdisplay
aga
ω,λa(26)
whereHωare the Fourier coefficients of the rapidly-varying Hamilton ian (see (3)). Note that ga
−ω∗=ga
ωsinceH′is
Hermitian.
Substituting in (22) we find
Heff=/summationdisplay
aϕaλa,
ϕa=fa−i/summationdisplay
|ω|>Ωgc
ω∗gb
ω
2ωCabc+/summationdisplay
|ω|>Ωfdgc
ω∗gb
ω
2ω2CdbeCeca+/summationdisplay
|ω|,|ω′|>Ωgb
ωgc
ω′gd
ω+ω′∗
3ω′(ω+ω′)CdbeCeca+· · ·, (27)
4where the ellipsis denote higher order terms in η.
For the particularly simple case of a two-level system with Cabc= 2ǫabcandλa=σa(the usual Pauli matrices) this
reduces to
Heff=
f−/summationdisplay
|ω|>Ωgω×g∗
ω
ω+· · ·
·σ σσ. (28)
It is clear that there is the possibility for the H′-induced terms to generate resonances if we allow the fato
vary slowly in time. To see this explicitly we consider (28) t aking for simplicity f2=g3
ω= 0, andf1=constant.
Substituting we find
ϕ1=f1, ϕ 2= 0, ϕ 3=f3−E′+O(1/Ω2), (29)
where
E′= 2/summationdisplay
|ω|>Ω1
ωImg2
ω∗g1
ω. (30)
Iff3is allowed to vary slowly with time the effective Hamiltonian will exhibit a resonance when f3=E′if|E′| ≫ |f2|.
This resonance will lead to large transition amplitude prov idedf3varies sufficiently slowly: |f2| ≫/radicalBig
|˙f3|(which is
the usual adiabatic resonance condition [6]). A specific exa mple is presented in fig. 1.
In fig. 1 we also compare the results obtained using the effecti ve Hamiltonian (28) to those obtained solving the
Schr¨ odinger equation exactly with initial conditions Ψ( t= 0) =ψ(t= 0) = 1. It can be seen that the solutions
obtained using Heffdo indeed describe the average behavior of the wave-functio n provided ηis sufficiently small. We
conclude that, at least for the case of a 2-level system, the e ffective Hamiltonian (22) accurately describes the average
evolution of the system.
The presence of noise ( i.e.,H′) can then generate unexpected resonances. The condition fo r these to occur is,
qualitatively,
ΩH0∼H′2; (31)
we will see later that these resonances are related, but not i dentical, to the well-studied stochastic resonances [2].
B.δ-function comb
A simple model where the above formalism can be applied and wh ich is also exactly solvable is provided by a 2-level
system with Hamiltonian
H=/summationdisplay
nN/summationdisplay
i=1Λiδ(t−ti−nT); 0<ti<T, (32)
where the matrices Λ iare Hermitian. This represents a set of N δfunctions which repeats with period T. This type
of potential is of interest in signal processing [7] and is al so similar to the one used in the study the effect of a laser
beam on a set of charged particles [8].
For simplicity we will assume
Λi=/parenleftbigg
0λi
λ∗
i0/parenrightbigg
;N/summationdisplay
i=1λi= 0, (33)
in this case H0= 0,so thatH=H′, we will also take Ω = 2 π/T.
In order to obtain the effective Hamiltonian we first construc t
−i/integraldisplayt
0dtH′=−iN/summationdisplay
i=1ΛiΘ(t−ti),(0<t<T ), (34)
where Θ denotes the step function. Uis then the fast part of this quantity,
51234t
−150−100−5050100150g1(c)
1234t
−150−100−5050100150g2(d)0.511.522.533.54t0.20.40.60.81|Ψ|2,|ψ|2(a)
1234t
−224ϕ3 (b)
FIG. 1. Example of a resonant phenomenon induced by the prese nce of rapidly varying interactions of large amplitude.
(a):comparison of |Ψ|2obtained by an exact (numerical) integration of Schr¨ oding er’s equation (black jagged curve) with |ψ|2
obtained integrating (14) using the effective Hamiltonian i n (28, 29) (light superimposed curve). The top light curve is the result
of integrating Schr¨ odinger’s equation when H′= 0; the dotted vertical line denotes the time at which the dia gonal elements
inHeffvanish. (b):Diagonal element of Heff; for this example we chose E′= 4.78684 andf3=E′[2/(1 +t2) + 1/5].(c,d):
Non vanishing elements of H′; the specific expression used was g1+ig2= 39.9567e926.291t−0.956304 i+ 35.6145e984.461t+0.660091 i+
39.1024e1057.84t−0.732253 i+ 30.4239e1208.99t+2.43462 i+ 29.1863e1953.06t+0.719083 i. For this example, η≃0.03
6U=−i/integraldisplayt
0H′+i/angbracketleftbigg/integraldisplayt
0H′/angbracketrightbigg
=−iN/summationdisplay
i=1Λi/bracketleftbigg
Θ(t−ti)−1 +ti
T/bracketrightbigg
,(0<t<T ), (35)
where the slow part of a quantity is obtained by averaging ove r the period T.
Using then (33) and substituting in (22) we easily find
Heff=1
2∝an}b∇acketle{t[H′,U]∝an}b∇acket∇i}ht+· · ·
=1
T
/summationdisplay
i>jImλiλ∗
j
σ3+· · ·. (36)
For this system we have Ω ∼1/Tso thatη= max {|λi|}; it then follows that (36) will be accurate provided |λi| ≪1.
This model can be solved exactly by elementary means. Replac ingδ(t−ti−nT) by a rectangle of height 1 /τand
widthτcentered at t=ti+nTit is easy to see that in the limit τ→0
Ψ(t+
i) =e−iΛiΨ(t−
i), (37)
wheret±
i=ti±δ, δ→0. It follows that
Ψ(T) =e−iΛNe−iΛN−1· · ·e−iΛ1Ψ(0). (38)
In the limit where the λiare small we obtain
e−iΛNe−iΛN−1· · ·e−iΛ1= exp
−1
2/summationdisplay
i>j[Λi,Λj] +· · ·
= exp [ −iTHeff+· · ·] (39)
which shows that, at least for small λi,Heffdetermines the leading contributions to the average time ev olution of the
wave function.
C. Geometrical Optics example
The calculations in the previous sections referred to quant um systems, but it clear that any system whose dynamical
equations can be cast into a Schr¨ odinger-like form can be tr eated in the same way. In particular for this general case
there is no need to require Hto be a Hermitian operator.
An example of this situation is provided by the description o f light-ray evolution within geometrical optics [9]. For
small angles the position and direction of light ray within g eometrical optics can be described using a two-component
vector
/parenleftbigg
h
α/parenrightbigg
, (40)
wherehdenotes the height with respect to a reference line and αthe tilt (assumed to be small). Any transformation
of a light ray can be described using a 2 ×2 matrix [9]. In particular
translation :/parenleftbigg
1x
0 1/parenrightbigg
,
refraction :/parenleftbigg
1 0
(n1/n2−1)/R n 1/n2/parenrightbigg
, (41)
where the translation is by a distance xand the refraction is from a medium of refraction index n1to another with
indexn2andRdenotes the radius curvature of the interface.
We now assume that Rand the index of refraction change smoothly though rapidly w ith distance. We define
ν=−1
ndn
dx, (42)
7so that the general matrix which transports a ray by a distanc eδxis
M=/parenleftbigg
0 1
ζ′ν/parenrightbigg
. (43)
where primes denote xderivatives and
ζ=/integraldisplay
dxν
R(44)
The system then corresponds to a two-level quantum system wi th “time”xand “Hamiltonian” H=iM. A simple
application of (15) yields
Heff=i/parenleftbigg
0 1
K20/parenrightbigg
, K2=∝an}b∇acketle{tζ′(lnn)∝an}b∇acket∇i}ht= +1
2/angbracketleftbig
(lnn)2(1/R)′/angbracketrightbig
, (45)
where we assumed ∝an}b∇acketle{tζ′∝an}b∇acket∇i}ht=∝an}b∇acketle{tν∝an}b∇acket∇i}ht= 0 and we kept only the first corrections induced by the rapidl y varying terms2. Note
thatK2can be negative and that it vanishes (at least to lowest order ) whenRornare constant.
The effective operator which determines the translation ove r a finite distance Xis then (assuming for simplicity
thatKis position independent)
A=e−iXHeff=/parenleftbigg
cosh(KX) (1/K)sinh(KX)
Ksinh(KX) cosh( KX)/parenrightbigg
(46)
(theK2<0 case is obtained by analytic continuation). This matrix is equivalent to a thick symmetric lens with
radius of curvature ¯Rand thickness ¯dsuch that
1
¯R=K
1−¯ntanh(KX/2) ¯d=¯n
Ksinh(KX), (47)
where ¯ndenotes the index of the lens material.
Thus, within the approximations inherent to geometrical op tics, the effects of a region of rapidly varying index
of refraction and curvature on light rays are equivalent, on average to those of a thick lens of appropriately chosen
characteristics. For example, using a thick lens with a high index of refraction, ¯ n≫1 we find
K|K|=−2¯n2
¯R¯d. (48)
Conversely, a thick lens can be found which completely cance ls the effects of Aand this can be used to measure the
fluctuations in the original system (more specifically, thos e fluctuations which contribute to ζ).
The above expressions suffer corrections form higher-order terms in the expansion in powers of η. Using (16)3we
find that the next-order term in (45) is
i/parenleftbigg
0 0
k20/parenrightbigg
, k2=/angbracketleftbig
ζ2/angbracketrightbig
− ∝an}b∇acketle{tζ∝an}b∇acket∇i}ht2−1
2/angbracketleftbigg
lnn/bracketleftbigg
lnn−1
2∝an}b∇acketle{tlnn∝an}b∇acket∇i}ht/bracketrightbigg
ζ′/angbracketrightbigg
, (49)
where we assumed for simplicity that ∝an}b∇acketle{t(lnn− ∝an}b∇acketle{tlnn∝an}b∇acket∇i}ht)ζ∝an}b∇acket∇i}htand/angbracketleftbig
[lnn− ∝an}b∇acketle{tlnn∝an}b∇acket∇i}ht]2/angbracketrightbig
are independent of x(in general the
averaged quantities may still vary slowly with x). The second order correction is negligible provided |K2| ≫ |k2|.
In addition there are deviations form these predications du e to the inherent limitations of geometrical optics (for
example, it is assumed the light rays lie on a plane, diffracti on is neglected, etc.). In neglecting them we have tacitly
assumed that the scale of all fluctuations is large compared t o the wavelength and that all reflection and refraction
angles are small.
2The second expression for K2follows from 2 ν(lnn)/R= (1/R)′(lnn)2−[(lnn)2/R]′and/angbracketleftbig
[(lnn)2/R]′/angbracketrightbig
= 0 to first order in
the rapidly varying quantities.
3As mentioned previously there is no reason to demand Hermiti city in the effective Hamiltonian for this case
8D. The noisy Jaynes-Cummings model
In this example we consider a simplified version of an atom int eracting with a photon field, as described by the
Jaynes-Cummings model [10], with the addition of two types o f interaction with an external rapidly varying fields.
We will show that this problem is also well suited for study us ing the techniques introduced above. We first assume
that the external field are coupled to the photons, and then di rectly to the atoms. We then show that both situations
are unitarily equivalent.
The unperturbed Hamiltonian for this model is
H0=ω0a†a+1
2Ω0(a†σ−+aσ+) +ǫσ3. (50)
Denoting by |n;↑∝an}b∇acket∇i}htand|n;↓∝an}b∇acket∇i}htthe states with nphotons and atomic spin up and down respectively, the eigens tates of
H0are [10]
|n±∝an}b∇acket∇i}ht=cn∓|n;↓∝an}b∇acket∇i}ht ±sign(κ)cn±|n−1;↑∝an}b∇acket∇i}ht,
cn±=1√
2/parenleftbigg
1±1√
1 +nκ2/parenrightbigg
;κ=Ω0
2ǫ−ω0, (51)
with energies
En±=/parenleftbigg
n−1
2/parenrightbigg
ω0±/vextendsingle/vextendsingle/vextendsingle/vextendsingleǫ−1
2ω0/vextendsingle/vextendsingle/vextendsingle/vextendsingle/radicalbig
1 +nκ2. (52)
a. Noisy photon field We will now couple the photons to external sources which vary rapidly with time. The
interaction Hamiltonian is assumed to be
H′=ξa†+ξ∗a. (53)
Substituting (50) and (53) in (22) yields
ˆHeff=H0+ω0/angbracketleftbig
|θ|2/angbracketrightbig
−Im/angbracketleftBig
θ∗˙θ/angbracketrightBig
+O(η3);ξ=i˙θ. (54)
The difference between ˆHeffandH0is, in this case, trivial and can be eliminated by a simple cha nge in the overall
phase of the states.
Non-trivial terms may arise at higher orders but this would r equire calculating the (Hermitian version of the) effective
Hamiltonian to order η3. Instead of following this uninspiring approach we conside r a different way of introducing the
interaction with the external fields and then show that the co rresponding effective operator corresponds to the order
η3contribution generated by (53). To this end we first consider a unitary transformation of the form
S= exp/bracketleftbig
a†ζ(t)−aζ∗(t)/bracketrightbig
;i˙ζ−ω0ζ+ξ= 0. (55)
Then the transformed Hamiltonian is
Hnew=S(H0+H′)S†+i˙SS†
=H0−1
2Ω0(σ−ζ∗+σ+ζ)−1
2Im(ζ∗˙ζ)−ω0|ζ|2
=H0+H′
new, (56)
which defines H′
new. This shows that the original system is equivalent to one whe re the external sources are coupled
directly to the spin of the atoms through ζ. From its definition it can be seen that ζis of orderηso that, substituting
H′
newin (15), gives the effective Hamiltonian up to and including t erms of order ηζ2∼η3. The effects of this type of
interaction are considered in the next paragraph.
b. Noisy spin interaction We now consider
H′=ζσ++ζ∗σ−, (57)
which to lowest order (using (15)) gives
Heff=H0−Im/angbracketleftBig
ϑ∗˙ϑ/angbracketrightBig
σ3;ζ=i˙ϑ, (58)
9corresponding to the non-trivial replacement
ǫ→ǫeff=ǫ−Im/angbracketleftBig
ϑ∗˙ϑ/angbracketrightBig
. (59)
This replacement also describes the leading (average) effec t of (53) provided we identify ξ=¨ϑ[1 +O(η)]. Ifξ∼η0
then the modification is indeed of order η3. In order to go back to the original problem we act with Son the states.
The effects of the external sources is, to lowest order, summa rized by the simple shift (59) which corresponds to a
change in the energy gap between the two spin states of the “at om” of this model. In particular, for noise of sufficiently
large amplitude, we can have ǫeff=ω0/2 in which case the Rabi frequency vanishes, En+=En−and photon number
is conserved.
The shift in ǫcan also lead to resonant behavior between states of differen tn. For example the energies for the
states |0−∝an}b∇acket∇i}htand|1−∝an}b∇acket∇i}htare equal provided
ǫeff−1
2ω0=Ω2
0
8ω0±1
2ω0 (60)
which has a solution only for 0 <ω0/Ω0≤1/2.
These results are reliable provided η≪1 which corresponds to |ϑ| ≪1 and |ω0|,|Ω0|,|ǫ| ≪Ω. As in the two level
system resonances occur when |H0|/Ω∼η2.
E. Quantum system in an inhomogeneous potential
The ideas presented in the previous sections can be translat ed to the case of a particle whose Hamiltonian is of the
form
H=−1
2m∇2+V0+V1, (61)
whereV1varies rapidly with position. For this case we consider the t ime-independent Schr¨ odinger equation HΨ =EΨ
and look for solutions Ψ = ψ+χwhereχis of small amplitude but exhibits rapid variation in positi on whileψis
slowly varying and of large amplitude. Substituting this An satz we find
/parenleftbigg
−1
2m∇2+V0/parenrightbigg
ψ≃Eψ− ∝an}b∇acketle{tV1χ∝an}b∇acket∇i}ht,
−1
2m∇2χ≃ −V1ψ (62)
which can be solved to lowest order giving
Heffψ=Eψ,
Heff=−1
2m∇2+V0+/angbracketleftbigg
V12m
∇2V1/angbracketrightbigg
. (63)
In this case, for any quantity A,∝an}b∇acketle{tA∝an}b∇acket∇i}htdenotes the part of A(if any) which varies slowly with position.
In one dimension the same result can be obtained by convertin g the time-independent Schr¨ odinger’s equation to
a first order equation for the vector (Ψ ,−idΨ/dx) and substituting in (22) or (16), and using xas the evolution
parameter.
The additional term in Heffisnegative definite and will tend to bind the particle. In particular, ta kingV0= 0
and assuming V1vanishes at infinity the effective Hamiltonian Heffwill always exhibit a bound state (of zero angular
momentum) in ≤2 dimensions, that is, in ≤2 dimensions a rapidly varying potential of zero average wil l always
exhibit localized states. The same is true in higher dimensi ons provided the amplitude of V1is large enough
V. PROBABILISTIC CONSIDERATIONS
In this section we provide an alternative view of the problem using the Fokker-Plank equation. For simplicity we
consider the case of a two-level system with Hamiltonian
10H=H0+H′,
H0=/summationdisplay
afaσa,
H′=/summationdisplay
aGaσa, (64)
where {Ga}are stochastic variables whose probability function will b e described below.
We will study the Fokker-Plank for the polarization vector ψ†σ σσψwhose probability density is given by
P(r,t) =/angbracketleftBig
δ(3)/parenleftbig
ψ†(t)σ σσψ(t)−r/parenrightbig/angbracketrightBig
G, (65)
where the symbol ∝an}b∇acketle{t· · ·∝an}b∇acket∇i}htGdenotes the average over the stochastic variables Ga. In terms of a functional integral we
will use
∝an}b∇acketle{tA∝an}b∇acket∇i}htG=/integraldisplay/productdisplay
a[dGa]Aexp/braceleftBigg
−1
2/integraldisplay /integraldisplay
dtdt′/summationdisplay
abGa(t)Kab(t,t′)Gb(t′)/bracerightBigg
, (66)
withKsymmetric ( Kab(t,t′) =Kba(t′,t)) and positive definite. We denote by σthe inverse kernel K−1,
/integraldisplay
ds/summationdisplay
cKac(t,s)σcb(s,t′) =δb
aδ(t−t′). (67)
It is easy to see that σab(t,t′) =/angbracketleftbig
Ga(t)Gb(t′)/angbracketrightbig
.
We will now restrict further considerations to cases where Ga(t) is correlated with Gb(t′) only fortclose tot′, that
is for cases where σab(t,t′) vanishes except when t∼t′. In this case we define
¯σab(t) =/integraldisplayt
−∞dt′σab(t,t′), (68)
and, following the standard derivation of the Fokker-Plank equation [11,12], we obtain,
i˙P=/parenleftBigg
2/summationdisplay
aLafa−4i/summationdisplay
abLaLb¯σab/parenrightBigg
P, (69)
where thefadetermineH0in (64) and La, a= 1,2,3 denote the usual angular momentum operators in 3 dimension s.
There are corrections to this equation but these can be ignor ed provided σ(t,t′) is sufficiently localized around t=t′.
We will now restrict ourselves to situations where ¯ σtakes the form
¯σab=1
2Dδab−1
2/summationdisplay
cǫabcac. (70)
The first term is commonly used in treating this type of proble ms, the second term implies a correlation between
GaandGbwitha∝ne}ationslash=band is usually assumed to vanish; we will find, however that it is precisely this term that is
responsible for the resonances described previously.
Substituting (70) in (69) yields
i˙P= 2/bracketleftbig
(f−a)·L−iDL2/bracketrightbig
P. (71)
It is important to note that this choice still corresponds to a positive definite kernel Kso that (66) is well defined.
In order to relate these expressions to the ones obtained pre viously we first write ¯ σin terms of the two-point
correlator,
¯σab(t,t′) =/integraldisplayt
−∞dt′/angbracketleftbig
Ga(t)Gb(t′)/angbracketrightbig
G. (72)
We then expand Gain a Fourier series,
Ga(t) =/summationdisplay
ωGa
ωe−ıωt(73)
11(not necessarily restricted to |ω|>Ω) and assume that/angbracketleftbig
Ga
ωGb
ω′/angbracketrightbig
G≃0 forω+ω′∝ne}ationslash= 0 (which is equivalent to assuming
that the correlator σab(t,t′) is non-zero for t∼t′only). In this case
D= lim
δ→02
3/summationdisplay
ωδ
δ2+ω2/angbracketleftbig
|Gω|2/angbracketrightbig
G,
a= lim
δ→0i/summationdisplay
ωω
δ2+ω2∝an}b∇acketle{tGω×G−ω∝an}b∇acket∇i}ht)G. (74)
In obtaining these expressions we included a convergence fa ctoreδt, δ→0 in (73). Note that Dwill vanish unless
theGa
ωare continuously distributed in an interval containing ω= 0.
Comparing this result to (28) we find that the term f−ain (71) corresponds to the leading term in Heff. In
addition, however, the Fokker-Plank equation contains the non-Hermitian diffusion term −2iDL2Pwhich forces P
to decrease exponentially in time for all but the zero-angul ar momentum modes [12]. Qualitatively this implies that
for large times the polarization vector will end up uniforml y distributed and the effects of the f−aterm will be
completely washed-out. For intermediate times, however, t he presence of the f−aterm can lead to interesting effects.
The general solutions to (71) can be obtained in terms of sphe rical harmonics4Yl
m,
P=/summationdisplay
lmξl
m(t)e−2Dl(l+1)t−iv0mtYl
m(ˆr), (75)
where the exponential is introduced for later convenience. Since Pis real the coefficients ξl
mobey
ξl
m∗= (−1)mξl
−m. (76)
The above expansion can be substituted into (71) leading, fo r eachl, to a set of 2 l+ 1 coupled ordinary differential
equations in tthat can in principle be solved for any choice of f,aandD.
To illustrate this procedure we will consider a special case which is similar to the one often studied when considering
stochastic resonances. We take
f−a=1
2v0ˆz+1
2ucos(ω0t)ˆx, (77)
corresponding, for example, to f3=G3= 0 (so that a1=a2= 0),f2= 0,f1=ucos(ω0t) anda3=−v0. The
equations for the coefficients ξl
min (75) then become
˙ξl
m+i
2ucos(ω0t)/bracketleftBig/radicalbig
l(l+ 1)−m(m−1)eiv0tξl
m−1+/radicalbig
l(l+ 1)−m(m+ 1)e−iv0tξl
m+1/bracketrightBig
= 0, (78)
Forl= 0 the solution is simply ξ0
0=constant which is determines the normalization of P; the equations for l∝ne}ationslash= 0
can be solved numerically using standard techniques. The ca sel= 1 is of special interest since the coefficients ξl=1
m
determine the average polarization of the system as a functi on of time:
/angbracketleftbig
ψ†(t)σ σσψ(t)/angbracketrightbig
=/integraldisplay
d2ˆr(Pˆr)/slashbigg/integraldisplay
d2ˆrP
=/radicalbig
2/3
ξ0
0e−4Dt/parenleftbigg
Re/bracketleftbig
ξ1
−1(t)eiv0t/bracketrightbig
,Im/bracketleftbig
ξ1
−1(t)eiv0t/bracketrightbig
,1√
2ξ1
0(t)/parenrightbigg
. (79)
We will be interested in the possibility of the system resona ting at the driving frequency ω0. This can be investigated
by first solving the above equations and then Fourier transfo rming the result. We define,
˜ξl
m(ω) =/integraldisplay
dteiωtξl
m(t), (80)
and we study the behavior of/vextendsingle/vextendsingleξ1
1(ω=v0)/vextendsingle/vextendsingleas a function of v0for various values of ω0(note that ˜ξalso depends
explicitly on v0since this parameter appears in the differential equation (7 8)). The result is presented in Fig. 2
which clearly shows an enhancement in the Fourier coefficient s of frequency v0whenv0=ω0, the shape of the curves
are characteristic of resonant behavior. These resonances are also illustrated by the behavior of/angbracketleftbig
ψ†σ3ψ/angbracketrightbig
for various
values ofv0, ω0. An example is plotted in fig. 3.
4It is easy to see that Pis, in fact, independent of |r|.
125 10 15 20 25
v00.050.10.150.20.250.30.350.4|ξ□
−11(ω=v0)|ω0=5
ω0=10
ω0=15
ω0=20
FIG. 2. Resonances in a two-level system; ω0denotes the external driving frequency.
25 50 75 100 125 150t
−0.5−0.250.250.50.751<z>^
v0≠ω0
ω0=v0
FIG. 3. Resonant behavior of the zcomponent polarization vector (79),/angbracketleftbig
ψ†σ3ψ/angbracketrightbig
=/angbracketleftˆz/angbracketright, for a two level system. The
parameters chosen were u= 0.1,ξ0
0= 1/√
3,D= 0.0025,ω0= 5 and, for the case v0/negationslash=ω0,v0=ω0±0.2
13The same system can be studied using the time averaging proce dure of sections II-III provided we assume that
the corresponding restrictions on the parameters are satis fied. Assuming this is the case, the effective Hamiltonian
corresponding to the choice (77) is readily seen to be, to low est order in η,
Heff=/parenleftbigg
v0/2 cos(ω0t)
cos(ω0t)−v0/2/parenrightbigg
. (81)
The corresponding Schr¨ odinger equation for the slow modes Heffψ=i˙ψcan be solved numerically using standard
techniques and the solutions are seen to exhibit resonances wheneverv0is an integral multiple of ω0.
There are, of course, differences between the solutions to th e Schr¨ odinger equation associated with (81) and the
solutions derived from (78). The quantities Gaused in obtaining (78) are assumed to be stochastic variable s whose
distribution is determined by (66, 67, 68, 70). In contrast, when deriving (81) we assumed the ga
ωare non-zero only
for|ω|>Ω, and we also required Ω ≫ |ω0|,|v0|,|˙gω/gω|.
The values for aobtained in cases become identical if we assume that the aver age over the stochastic variables in
the first case give the same result as the average over time int ervals much larger than 1 /Ω in the second case.
The diffusion term Dwill vanish, as mentioned above, unless the Gωare distributed continuously around ω= 0.
This term corresponds to the non-Hermitian contribution (1 7) which, for this case is simply proportional to the unit
matrix. Taking f2,3=g3= 0, in the example of section IVA (for the case N= 2) we find
Heff−Heff†=−id, d =d
dt/summationdisplay
|ω|>Ω/vextendsingle/vextendsingle/vextendsingle/vextendsingleg1
ω+ig2
ω
ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
, (82)
which is of order η2. Note that d= 0 ifg1,2
ωare time independent; this corresponds to the vanishing of DshouldGω
vanish when ωlies in an interval around ω= 0.
In concluding this section we note that it is possible to gene rate the required correlation between g1andg2by
mixing and filtering two uncorrelated functions n1,2. The details are presented in the Appendix.
VI. COMPARISON WITH THE STANDARD STOCHASTIC RESONANCES
The resonances described above are reminiscent of the well- studied stochastic resonances [2] that are characterized
by an increased sensitivity to small perturbations when noi se of an optimal amplitude is introduced. This feature
is also observed when the condition (31) is satisfied, and is i llustrated by the behavior of the system studied in the
previous section.
More specifically, stochastic resonances occur when there i s a match between a noise-induced transition rate, rNand
the one produced by an external perturbation. If the latter i s assumed to be harmonic of frequency ω0, then typically
resonances occur when ω0∼πrN. This can be understood by considering a system that initial ly has 2 degenerate
minima, such that the harmonic perturbation will first favor one and then the other (alternating with period π/ω0).
If the resonance condition on rNis realized, then the times at which one minima is disfavored will coincide with the
times at which noise-induced transitions to the other minim a are most probable, and this enhances the response of
the system to the external perturbation. This behavior is al so observed in the systems studied above, for example, the
resonances in Fig. 2 occur when v0=ω0where, according to (81), v0is proportional to the noise-induced transition
rate.
There are, however, some technical differences. To illustra te these we consider the following one-dimensional system
that exhibits stochastic resonances
˙x=V′(x,t) +e(t), V(x,t) =V0(x) +uxcos(ω0t), (83)
whereedenotes a stochastic variable, uis a small coupling constant and V0is a potential with two (degenerate)
minima. The noise is assumed to obey
∝an}b∇acketle{te(t)e(t′)∝an}b∇acket∇i}ht=Fδ(t−t′). (84)
Following the same steps [13] described above it is possible to obtain the Fokker-Plank equation for the probability
density P(y,t) =∝an}b∇acketle{tδ(x(t)−y)∝an}b∇acket∇i}hteand the corresponding average ∝an}b∇acketle{tx(t)∝an}b∇acket∇i}hte. The Fourier coefficient of ∝an}b∇acketle{tx(t)∝an}b∇acket∇i}htecorresponding
to frequency ω0has an amplitude proportional to ( λ/F)//radicalbig
ω2
0+λ2whereλdenotes the noise-induced hopping rate
(the Kramers’ rate [14]), ln λ∝ −1/F. For fixed ω0this amplitude also displays an enhancement at a certain val ue
ofF[2]. Comparing these results with those obtained in the prev ious section we note that
14•The usual stochastic resonances occur for uncorrelated noi se obeying (84) while resonance behavior in (64)
requires the correlations implied by having a∝ne}ationslash= 0 in (70).
•The resonances described above have the usual shape (see Fig . 2) for the resonant curve. This is not necessarily
the case for the stochastic resonances usually discussed in the literature.
•Usual stochastic resonances occur whenever the driving fre quency is about half the Kramers’ rate, which depends
exponentially on the noise level F. For the case presented in this paper resonances occur when t he driving
frequency is ∼ |a|as defined in (74), and is proportional to the square of the amp litude of the stochastic
variablesGa.
VII. CONCLUSIONS
In this work we propose a formalism which makes possible to st udy systems under the influence of rapidly-varying
external fields (which are not necessarily perturbative) wh ose typical frequency we denoted by Ω. The formalism
provides a solution as a power series in 1 /Ω and assumes a clear separation of fast (frequencies∼>Ω) and slow
(frequencies ≪Ω) modes.
We showed that the evolution of the slow modes is determined b y an effective Hamiltonian which is not neces-
sarily Hermitian; a point noted in other related calculatio ns [5]. The non-Hermitian contributions to the effective
Hamiltonian, however, can be eliminated by performing an ap propriate unitary transformation.
The formalism was applied to various classical and quantum s ystems. In some examples we found that the external
field can produce a resonant behavior in the system. These res onances are related, but not identical, to the stochastic
resonances studied in the literature [2] In particular the r esonant phenomena studied in this paper occur only when
the interaction with the environment involves several corr elated terms.
In one particular application of the formalism we argued tha t the presence of a random time-independent potential
will necessarily generate bound states in systems of dimens ion 1 and 2, and in other dimensions as well provided
the amplitude of the potential is sufficiently large. The conn ection of this result with the phenomenon of Anderson
localization [15] are tantalizing and will be considered in a future publication.
ACKNOWLEDGMENTS
We would like to thank W. Beyermann, R. deCoss and T.J. Weiler for illuminating comments and insights. This
research was supported in part by US DOE contract number DE-F G03-94ER40837(UCR) and by Conacyt (M´ exico).
APPENDIX
In this appendix we describe a simple construction which gen erates stochastic variables gasatisfying (70,72), in
terms of a set of uncorrelated variables ni, specifically, we assume
∝an}b∇acketle{tni(t)nj(t′)∝an}b∇acket∇i}ht=1
2Diδijδ(t−t′), (85)
and search for new variables gasatisfying
/angbracketleftbig
ga(t)gb(t′)/angbracketrightbig
=1
2FE(t−t′)δab−1
2ǫabcacO(t−t′), (86)
where Eis an even function of its argument while Ois odd and satisfy
/integraldisplay0
−∞E(s)ds=/integraldisplay0
−∞O(s)ds= 1. (87)
In terms of the Fourier transformed quantities,
˜ga(ω) =/integraldisplay∞
−∞dte−iωtga(t),˜ni(ω) =/integraldisplay∞
−∞dte−iωtni(t), (88)
15we require
/angbracketleftbig
˜ga(ω)˜gb(ω′)/angbracketrightbig
=πδ(ω+ω′)/bracketleftBig
Fδab˜E(ω)−ǫabcac˜O(ω)/bracketrightBig
,
∝an}b∇acketle{t˜ni(ω)˜nj(ω′)∝an}b∇acket∇i}ht=πδ(ω+ω′)Diδij, (89)
where
˜E(ω)∗=˜E(−ω) = + ˜E(ω),
˜O(ω)∗=˜O(−ω) =−˜O(ω). (90)
We look for a linear relation between ˜ gaand ˜ni, namely
˜ga(ω) =/summationdisplay
iKai(ω)ni(ω). (91)
Writing
Kia=1/radicalbig
DaF˜E(Q++Q−)ia, (92)
withQ±(±ω) =±Q±(ω) and assuming QT
±=±Q±(whereTindicates the transpose) we find
Q2
++Q2
−= 1,{Q+,Q−}ij=ǫikjνk, (93)
whereνk=ak˜O/(F˜E). These equations are solved, for example, by choosing
Q+= coshu(1 +ˆa⊗ˆa),(Q−)ij= sinhuǫikjˆak(94)
with sinh(2 u) =|ν|.
It follows that given a set of uncorrelated variables niit is possible to generate the desired correlated quantitie sga
through a linear filter defined by the (frequency-dependent) matrixK.
[1] There are many works that treat the problem of systems sub ject to the action a rapidly-varying fields. For example, I.
M. Lifshits, et al.,Electron Theory of Metals (New York; Consultants Bureau, 1973). S. Stenholm, Rev. Mod . Phys. 58,
699 (1986). G. Papanicolaou, editor. Random media (New York; Springer-Verlag, 1987). F. Moss and P.V.E. McCli ntock,
editors. Noise in nonlinear dynamical systems (Cambridge, New York; Cambridge University Press; 1988-19 89). P. Jung,
Phys. Rep. 234, 175 (1993).
[2] R.A. Benzi et al., J. Phys. A 14, L453 (1981). R.A. Benzi et al., Tellus 34, 10 (1982). R.A. Benzi et al., SIAM (Soc. Ind.
Appl. Math.) J. Appl. Math. 43565 (1983). NATO Advanced Research Workshop: Stochastic Re sonance in Physics and
Biology, San Diego, CA, USA, 30 March-3 April 1992.), J. of St atistical Physics, 70(1993). For recent reviews see P. Jung,
ref. [1]. L. Gammaitoni et al., Rev. Mod. Phys., 70, 223(1998).
[3] P.L. Kapitsa, J. Eksp. Theor. Fiz. 21, 588 (1951) see also L.Landau and S. Lifshitz, Mechanics , 3rd ed. (Pergamon Press,
New York, 1991).
[4] J. Vidal and J. Wudka, Phys. Rev. A 44, 5383 (1991).
[5] C. P. Burgess and D. Michaud, Annals Phys. 256, 1 (1997) [hep-ph/9606295].
[6] A. Messiah, Quantum mechanics (Amsterdam; North-Holland. New York; Interscience Publis hers. 1961-62).
[7] See, for example, S.G. Mallat, A wavelet tour of signal processing , 2nd ed. (San Diego; Academic Press; 1999)
[8] S. Stenholm, ref. [1] S.A. Gardiner et al., Phys.Rev.Lett, 79, 4790 (1997).
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[10] E.T.Jaynes and Cummings, Proc. I.R.E. 51, 89 (1963). For a recent review see B.W. Shore and P.L. Knight , J. Mod.
Optics 40, 1195 (1993).
[11] J. Zinn-Justin, Quantum Field Theory and Critical Phenomena , 3rd ed. (Oxford; Clarendon Press; 1996). N.G. Van
Kampen, Stochastic Processes in Physics and Chemistry , rev. and enl. ed. (Amsterdam, New York; North-Holland, 199 2).
C.W. Gardiner, ref. [12]
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New York; Springer-Verlag; 1990, c1985)
[13] Hu Gang et al., Phys. Rev. A 42, 2030 (1990).
[14] H. Kramers, Physica (Utrecht) 7, 284 (1940).
[15] P.W. Anderson, Phys. Rev. 109, 1492 (1958). E. Abrahams et al., Phys. Rev. Lett. 42, 673 (1979). For pedagogical
discussions see C.M. Soukoulis and E.N. Economou, Waves in R andom Media, 9, 255 (1999). J.Callaway, Quantum Theory
of the Solid State , 2nd ed. (Boston; Academic Press; 1991). B.Tanner, Introduction to the Physics of electrons in solids ,
(Cambridge, New York; Cambridge University Press; 1995).
17 |
arXiv:physics/0102075v1 [physics.comp-ph] 23 Feb 2001Molecular dynamics simulation for pressure-induced struc tural transition from C 60fullerene
into amorphous diamond
Akihito Kikuchi and Shinji Tsuneyuki
Institute for Solid State Physics, University of Tokyo,
Kashiwa-no-ha 5-1-5, Kashiwa-shi, Chiba 277-8581, Japan
Abstract
The pressure-induced structural transition in fcc C 60fullerene by shock compres-
sion and rapid quenching is investigated by a semi-empirica l tight-binding molecular
dynamics simulation, adopting a constant-pressure scheme and a method of the order
N electronic structure calculation. At first, the process of the amorphization of C 60is
demonstrated. The simulated results indicated that, in the material fabricated after
the quenching, the remaining dangling bonds have a large infl uence on physical proper-
ties, such as, the density and the presence of the band gap at t he Fermi level. We have
furthermore studied the formation of the short-range order , observed as amorphous
diamond. In order to form the amorphous diamond phase, the bo nding state of sp2
must be turned into that of sp3. The transition process is seriously influenced from the
the external pressure, the temperature, or the presence of h ydrogen. The comparison
to the pressure-induced structural transition in the graph ite is also executed and a brief
discussion on the difference in those carbon crystals is give n.1 Introduction
In the field of high-pressure material science, diverse carb on systems under pressure have
been intensively studied with interest in synthesizing new phases. For example, the pressure-
induced structural transition from C 60fullerene to amorphous diamond is realized by shock
compression and rapid quenching[1]. The shock compression and rapid quenching generate
the high pressure (50-55GPa) and the temperature (2000-300 0K) in a fraction of a microsec-
ond. Consequently the C 60fcc crystal transforms into stable transparent glassy chip s ofµm
in size, which are confirmed to be an amorphous phase of diamon d by electron energy-loss
spectroscopy and electron diffractometry[1]. Though this p hase has a short-range order sim-
ilar to normal diamond, it is amorphous. This amorphous phas e is characterized by being
consisted mostly from sp3bonding, in contrast with previously reported amorphous ca rbon
which is considered to be a disordered phase of graphite with sp2bonds[2, 3]. By an investi-
gation of the sample after the shock compression, the fabric ation process for the amorphous
diamond is postulated in the following way [1]. Once the fcc c ell of C 60is compressed, the
inter-cluster bonding between C60molecules is formed. In other words, the polymerization of
the C 60clusters occurs there, as was observed in several high-pres sure experiments[4, 5, 6].
By the further compression, molecules collapse and change i nto the amorphous. In the
amorphous, the bonding state of sp3is gradually produced and it forms a short-range order
extending up to the volume of the normal diamond unit cell. Th is high temperature phase
is quenched and obtained as the amorphous diamond, in which a greater part of the bonding
state turns into sp3type without the crystal growth in long range.
In fact, there is uncertainty in the understanding of the fab rication process from C 60to
amorphous diamond, since the postulation for the process is obtained by checking the various
1phases of C 60remaining after the compression and the quenching and it is n ot based on
direct observation of the reaction. In order to clarify such a transition process, the aid of the
computer simulation will be needed. As for the structural tr ansformation in carbon systems,
there are a lot of theoretical approaches, from the first prin ciples or semi-empirical way.
However, those studies focus on the transformation between graphite and diamond. In the
present stage, the theoretical simulation on the pressure- induced structural transformation
in C 60has not been executed sufficiently. Therefore, the object of t he present work is
set to be a theoretical understanding of the pressure-induc ed structural transition in C 60
crystal. For this purpose, a molecular dynamics simulation is executed using a tight-binding
Hamiltonian with model parameters for carbon and hydrogen s ystems proposed by Winn ,
Rassinger and Hafner[7], combined with the constant pressu re scheme by Wentzcovotch[8, 9].
The model parameters can reproduce the energy difference and geometries in various carbon
systems including hydrogen, such as molecules and reconstr ucted diamond surfaces, as well
as the bulk property of diamonds and graphites. This model pa rameters also give results
consistent with other ab-initio and semi-empirical calcul ations for liquid carbon phase. In
addition, to save computational costs, a part of calculatio ns, involved in the crystal structure
optimization, was carried out using the density matrix meth od [10], what is called the order
N method.
The contents of this paper are as follows. At first the process of the pressure-induced
amorphization of C 60will be presented. According to the simulation presented he re, the
C60fcc crystal is easily and speedily transformed into the amor phous by the compression,
going through the path postulated by the experiment as above , since the C 60structure is
constructed by bondings of distorted sp2type and it has potentiality in the transition to sp3
2type bondings. However, in the simulated result, the amorph ous phase itself is far from the
state which should be classified as ”amorphous diamond”, esp ecially just after the transition
from C 60, since the collapsed C 60fcc crystal includes a number of dangling bonds which have
not yet turned into sp3bondings. Therefore, the present work will furthermore pur sue the
transition from sp2to sp3in the amorphous phase through the compression, which will g rad-
ually reduce the number of the dangling bonds and drive the sy stem towards the ”amorphous
diamond” phase of randomly packing sp3bondings. The details of those dynamical effects,
such as the dependence on the temperature and the pressure, a re not necessarily clarified
by the experiment alone. Finally, the present paper will giv e a comparison of the structural
transition in C 60to the pressure-induced graphite-diamond transition by th e simulation. It
will explain the reason why C 60fcc crystal shows the transition to amorphous diamond phase
and why graphite directly turns into perfect cubic diamond.
2 The simulation scheme
In the present work, the simulations are executed in the foll owing way.
1)The initial external pressure is set to be 0GPa and the pres sure is gradually raised to
the desired value.
2) Then the simulation is executed at a stationary pressure i n sufficiently long time.
3) To simulate the process of the rapid quenching, the optimi zed structure at 0GPa and
0K is finally obtained, by letting down the applied pressure.
In the present stage, we do not have the detailed information on the physical quantities
in the sample through the shock-compression, such as, the va riation in the pressure and the
temperature related to the propagation of shock waves. For e xample, the temperature is not
3preserved and rapidly escapes from the compressed system wh ile the pressure is kept to be
high. Thus, in some calculations, the temperature is also gi ven as a parameter independent
of the pressure. In the electronic structure calculation, a periodic boundary condition is
applied to the fcc unit cell containing four C 60molecules and the summation over k-space is
estimated at the Γ point alone.
3 Amorphization from the C 60fcc crystal
3.1 Compression at 0K
At first, the C 60fcc crystal was optimized at the pressure of 50-100GPa, keep ing the tem-
perature at 0K. The C 60fcc structure is stable in this pressure range. The initial c ubic cell
parameter including four C 60molecules is set to be 14.15 ˚A. For example, a compression at
50GPa changed the cell into smaller one, in which the cell par ameter shrinks to 13.6 ˚A. In
the compressed cell, the relative positions of atoms are sti ll preserved, while the covalent
bonds are formed between C 60molecules. This is a kind of the polymerization, but not the
amorphization.
3.2 Compression at finite temperature
Obviously, the reason why the C 60crystal is stable in such a high pressure is that the
simulation is executed in an unphysical condition, i.e., ke eping the temperature at 0 K. In
the experiments for fabricating the amorphous diamond, the temperature rapidly rises from
the room temperature to the range no less than 2000-3000 K. Th en the collapse of the crystal
will become drastic owing to the active movements of atoms, a s can be expected from the
phase diagram of carbon[11]. Therefore, in this section, th e simulation is executed with the
condition that the temperature of the system can vary. To do t his, the equation of motion is
4solved in the constant pressure scheme of molecular dynamic s[8] without any scaling of the
kinetic energy.
Figure 1 shows the snapshots of an example of the amorphizati on. In this case, the
pressure is linearly increased from 0GPa to 65GPa in the first 0.45ps and kept constant after
that. The temperature is set to be 300K in the beginning and sp ontaneously increases to
2000-3000K by compression partly because this is adiabatic compression and partly because
of recombination of C-C bonds. In the early stage, the stacki ng structure of the fcc crystal is
still preserved. The C 60molecules come to show deformation and polymerization, if t he inter-
cluster distance becomes much closer to the intra-cluster a tomic distance, as was observed in
the compression study by Duclos et al.[12]. By further compression, amorphization occurs
and it results in the destruction of the molecular structure in the whole cell. At the same
time, the sp2bonding gradually transforms into the sp3.
The structural change in Fig.1 is also quantitatively check ed. Figures 2 and 3 show the
change in the pair-distribution function and the bond-angl e distribution, which correspond
to the snapshot of the transition process in Fig.1. The initi al fcc C 60crystal has already
turned into the amorphous phase in the first 1000 MD steps (1.0 ps), as can be seen from
these figures. The initial bond-angle distribution has two p eaks at 108 degrees and 120
degrees, which mean bond-angles in five- and six-membered ri ngs. As the time goes by,
these two clear peaks become more and more broad and finally me rge into one peak around
110 degrees. This indicates the vanishing of the molecular s tructure of C 60.
Compressions in the pressure higher than 65GPa accompanies more temperature increase
and it turns C 60into amorphous. On the other hand, if the maximum pressures a re lower than
50GPa, and if the simulation starts at the room temperature, the fcc C 60crystal does not
5collapse completely, probably because the final temperatur e(1000-2000K) is not sufficiently
high. However, if the initial temperature is set to be sufficie ntly high, as high as in the order
of 1000K, the amorphization proceeds even at about 50GPa.
4 Formation of amorphous diamond
To gain the frozen phase after the rapid-quenching, we execu te the crystal structure opti-
mization firstly at the high pressure and finally obtain the op timized structure at 0GPa,
gradually letting down the pressure. In the actual process o f the shock compression and
rapid quenching, the high-temperature phase is frozen into the system owing to the rapid
decrease in the temperature, while the high pressure is stil l kept.
Figures 4(a) and 4(b) show the results of the crystal structu re optimization, where the
maximum compression pressures are set to be 65GPa and 125GPa , respectively. (In both
cases, the optimized structures at 0GPa are obtained after t he MD simulations continued as
long as about 10ps, so that the property of the finally fabrica ted material shall not seriously
be affected by the finiteness of the simulation time.) In these structures, the short-range
order, formed in the high temperature and the high pressure, remains almost unchanged.
The pair distribution function (not shown here) shows the pe aks around 1.5 ˚A and 2.5 ˚A,
which correspond to the contributions from the first- and nex t-nearest neighboring atoms,
as is seen in that of the high temperature phase. However, onl y from the location of the
peaks, it is difficult to determine whether the short-range or der comes from a diamond-like
structure or not. It may be likely that the peaks are attribut ed to the fragments of the C 60,
since the distances from one atom to the first- or next-neares t neighboring atoms are of the
same extent both in diamond and in a single C 60molecule. Thus the distribution of the
6azimuthal angle, as is defined in Fig.5(a), was checked. Here the azimuthal angle is defined
to be the relative angle of the two planes, respectively span ned by two bonds, when these two
plane share one common bond. This distribution gives the inf ormation on the networking
structure of the tetrahedron of sp3, and it will have peaks around 60 degrees or 180 degrees
for cubic diamond and around 140 degrees for the C 60structure. (The pairs of six- and five-
membered rings in C 60form the angle of about 140 degrees.) In Fig.5(b), for the cas e of the
compression at 65GPa, there are no prominent peaks around 60 or 140 degrees. It means that
the cage structure of C 60vanishes, while the diamond-like short range order is not cl early
formed. On the other hand, in the structure compressed at 125 GPa, there is a prominent
peak near 60 degrees in Fig.5(c). The cubic-diamond-like sh ort-range order, extending up to
the third-nearest neighbors, is much more developed there, since this structure compressed
by the larger pressure has much more number of sp3bonding.
In fact, the networking structure of bonds is not necessaril y confined to cubic diamond
phase alone, since the energy difference between cubic and he xagonal diamond phase is little
[9]. Several local bonding structures similar to that of hex agonal diamond are found in the
simulated results. In perfect hexagonal diamond structure , the azimuthal angle with respect
to a bond on an atom parallel to the c-axis takes 0 and 120 degre es. On the other hand
azimuthal angles with respect to other three bonds on an atom take 60 and 180 degrees,
which is the same as cubic diamond. Therefore, even in perfec t hexagonal diamond phase,
the contributions to the distribution of azimuthal angles a t 0 and 120 degrees is weak and it
is one third of those at 60 and 180 degrees. Thus contribution from hexagonal-diamond like
short range order around 0 or 120 degrees is hidden due to the r andomness in amorphous.
The density in the structure compressed by the maximum press ure 65GPa [Fig.4(a)] is
7estimated to be about 2.7 g/ cm3. On the other hand, the density is 3.5 g/ cm3after
the compression at the maximum pressure 125GPa [Fig.4(b)], which is comparable to the
experiment. In the actual amorphous diamond, the density is estimated to be larger than
3.3 g/cm3[13].
Figures 6(a) and 6(b) show the density of states of the finally fabricated material (frozen
phase) after the compression at the maximum pressure 65GPa a nd 125GPa. These electronic
structures are quantitatively different from that of the ini tial C 60fcc crystal, which is given in
Ref.[14]. The difference between Fig.6(a) and 6(b) results f rom the number of dangling bonds.
In these figures, the contribution to the DOS from threefold c arbon atoms is compared to the
total DOS. Owing to the contribution from dangling bonds, th e gap between the conduction
and valence bands vanishes. In case of the compression at 65G Pa[Fig.6(a)], the ratio of atoms
with dangling bond amounts to 50 %, while in case of the compre ssion at 125GPa[Fig.6(b)],
the ratio of such atoms deceases to about 10%. Therefore, in t he latter case, the conduction
and valence bands are distinguished by a reduction of the DOS near the Fermi level, which
is featured as a shaded zone in the figure. This reduction in th e DOS is interpreted to be
an analogous of the wide gap in the perfect diamond structure . In Fig.6(a), a narrow gap
at the Fermi level is formed in the dangling bond states and it separates the occupied and
unoccupied states. This is because the presence of a gap, eve n if it is narrow, makes the
system more stable.
Comparing the results corresponding to the compression at 6 5GPa and 125GPa, the
latter case, compressed at the larger pressure, is consider ed to be a better simulation, since
the actual amorphous diamond has a transparent optical prop erty similar to the normal
diamond and it will have an electronic structure with a large gap near the Fermi level. In
8our simulations, at the pressure above 100GPa and the temper ature above 5000K, the ratio
of threefold carbon atoms decreases to 10 −20%, and the valence and conduction bands tend
to be distinguished by a wide range reduction in the DOS near t he Fermi level, owing to the
decrease of the dangling bonds. In the experimentally obtai ned amorphous diamond, the
number of the the dangling bonds will also be reduced, possib ly more than in this simulation.
(It is certain that the dangling bonds still remain in the act ual amorphous diamond, as was
indicated by the EELS spectrum in Ref.[1].)
5 An analysis for the dynamical process in the transi-
tion
After the rapid amorphous transition, it takes a long time un til the short range order is
formed again. In this section, the dynamical process in the r eaction, especially related to the
pressure and the temperature, is investigated. Figure 7 sho ws the results by a simulation
where the external pressure is increased from 0GPa to 65GPa, and after that, the pressure is
furthermore increased to 125 GPa, in order to check the press ure dependence. Figures 7(a)-
(d) show the time dependence of the external pressure, the te mperature, the mean square
of the displacement, and the ratio of carbon atoms with fourf old coordination. The increase
in the ratio of carbon atoms with fourfold coordination stan ds for the transition from sp2
to sp3, and it reflects on the reconstruction of the short-range ord er. In the amorphous
phase, sp2-type bonding is gradually transformed into sp3-type, while the transition speed,
in other word, the speed of the short-range order formation i s going down. However, at least
in the order of the picoseconds, the system has not yet arrive at the stationary state. The
figure furthermore shows the dependence of the reaction on th e external pressure. When the
9external pressure is raised again, the number of the sp3bonding grows much more, since the
transition is accelerated by the higher temperature and the higher density. The density in
the high temperature phase amounts to about 3.0g/cm3at 65GPa and increases to about
3.5g/cm3at 125GPa.
The simulated result supports the postulations for the amor phous diamond formation
process from the experimental data[1]. We should stress her e the importance of the following
phenomena. Since the high temperature is generated through the reconstruction of the
bonding, the reaction is enhanced so that the system can cros s the potential barrier and
transform its structure. (The system obtains work by the com pression and the temperature
increases to some extent, but it is not enough to speed up the r eaction furthermore.) As
an example, figure 8 shows the results of the two simulation wh ere the external pressure is
increased to 125GPa. In the first case, denoted as (A), there i s no restriction to the variation
of the temperature. On the other hand , in the second case, den oted as (B), the temperature
is set to be 2500K after 0.5ps. The transition from sp2to sp3is apparently hindered in the
case of (B), since the movement of atoms is inactive because o f the lower temperature, in
contrast with the case (A). For the same reason, the compress ion at 0K, shown in the previous
section, cannot turn C 60to the amorphous. These simulations also suggest that there are
possibly innumerable quasi-stable configurations by which the system is easily trapped. The
amorphous diamond phase can also be regarded as one of such qu asi-stable transient phases
located in the reaction path from C 60to the bonding state of sp3in whole crystal, i.e. perfect
diamond phase. (In fact, the shock compression applied to lo wer-grade C 60[15] exhibited the
entire transition to the diamond crystallite, probably bec ause of the easier crystal growth in
the presence of defects and impurities.) Since shock compre ssion process continues as long as
10nanoseconds, it is possible that the actual amorphous diamo nd phase has a structure much
closer to perfect diamond, compared to the present simulati on whose time-scale is at most
picoseconds order. In the actual shock compression, a large fluctuation in the pressure and
the temperature through the shock-wave propagation will fa bricate variously altered phases,
as are classified in several states[1], ranging from the slig htly compressed fcc structure to the
amorphous diamond.
It appears that the speed of increasing pressure may have som e influence on the reaction.
For example, if the response of the crystal inner stress cann ot catch up with the rapid
increase of the pressure, the decaying process of the crysta l will become more drastic. The
response of the inner stress is related to the cell deformati on and it is dependent on the
fictitious mass assigned to the cell deformation, as well as t he external pressure and the
inner stress in the constant-pressure scheme. We would like to avoid such dependence of
the simulation on the artificial degree of freedom as possibl e. For this purpose, we adjusted
the fictitious mass heavy enough so that the fluctuation of the ”kinetic energy” assigned
to the cell deformation is kept to be very small and the almost ”isenthalpic” simulation
becomes possible at a stationary pressure. The speed of rais ing the applied pressure is set
to be sufficiently slow in such a way that the inner stress shall rise parallel with the external
pressure. By doing so, the system property is almost determi ned by the the current pressure,
and scarcely dependent on the kinetic contribution from the cell deformation. For example,
in figure 7 and 8, where the pressure is raised to 125GPa in two d ifferent ways, the final
system properties, such as the temperature and the bonding o rder, are similar.
116 Discussions
In the electronic structure of the finally fabricated materi al after the compression, the con-
tribution from dangling bonds is not negligible. If the cont ribution of this kind appearing
between the valence and conduction band is large, the fabric ated material will lose the trans-
parent optical property and it will not be qualified to be call ed amorphous diamond. As we
have seen, the contribution from dangling bonds will be redu ced after the compression at
sufficiently higher temperature and pressure. In fact, there may be other mechanism that
will reduce the contribution from dangling bonds. For examp le, it is well-known that the
presence of hydrogen atoms reduces the number of dangling bo nds in case of amorphous
silicon. The DOS after the shock-compression in presence of hydrogen atoms are given in
Fig.9[16]. A comparison between Fig.9(a) and Fig.6(a), whi ch are the DOS after the com-
pression at 55GPa and 65GPa, shows that the DOS is apparently reduced around the Fermi
level in the presence of the hydrogen, even if the pressure is somewhat lower. However, there
is no clear gap between the valence and conduction band in Fig .9(a). Such a situation is not
improved so much at higher pressure, as in Fig.9(b). In addit ion, according to the simulation
like this, even in the case when much more numbers of the hydro gen atoms are included,
dangling bonds not terminated by hydrogens are still left. T his will be because the reactivity
and the mobility of bonds are weakened by the presence of many hydrogen atoms and the
formation of the short range order of sp3will rather be hindered.
In case of the compression of C 60, there are number of quasi-stable configurations in
amorphous phase, and such a transient phase will easily be fr ozen by the quenching. This
will be the major difference to the pressure-induced structu ral transition from graphite to
diamond. Figure 10 shows the transition of the bonding state from sp2to sp3in the com-
12pression of the graphite. In the figure, the rapid increase in the ratio of fourfold carbon
atoms after 1000fs stands for the transition from graphite t o diamond. The transition speed
is far faster than the cooling speed of the temperature in the shock compression experiment
which is estimated to be from 106to 1010K/s[17]. Graphite and diamond phases are located
in very close configurations in the potential surface and the re is no stable transient state in
the transition path from graphite to diamond. Since graphit e rapidly transforms into the
cubic diamond without being trapped by any quasi-stable str ucture , perfect diamond phase
remains alone after the quenching and amorphous diamond pha se will not be obtained[18].
It should be noted here that, in comparison with ab-initio th eory, there are less accuracy
and less transferability in the tight-binding models, in sp ite of the fact that the model
parameters succeed in reproducing the bulk property of cert ain carbon crystals. This is
because the tight-binding model parameters are obtained by the fitting to the ab-initio
results for several crystal structures. In order to check th e model parameter dependence of
our simulation, we have executed additional calculations u sing another model parameter by
Xu et al. [19] and compared the results to those given in the pr evious sections. According to
the results obtained by the Xu’s parameters, if the initial c ondition and the applied pressure
are the same, the electronic and structural properties, suc h as the transition rate from sp2
to sp3and the increase in the temperature, are of the same extent, c ompared to those given
in the previous sections. For example, when the compression is executed above 100GPa, the
ratio of atoms with dangling bond decreases to 10 −20 % and the DOS shows the reduction
near the Fermi level, as well as the result in the previous sec tion.
It should also be mentioned that the model Hamiltonian used h ere does not include
van der Waals interactions and is not quantitatively sufficie nt for the initial C 60fcc structure
13formed by van der Waals force. However, the purpose of the pre sent work is the investigation
in a compressed carbon system. Since the van der Waals force i s much weaker than covalent
bonds between atoms, it will be negligible in the simulation , especially after the transition
into the high-density amorphous. In other words, the model p arameters adopted here will
be a good, even if not the best, description for the physical p rocess of the pressure-induced
structural transition from the C 60fcc crystal into the amorphous diamond.
7 Conclusion
We have investigated the amorphization of C 60fcc and the formation of amorphous diamond
phase. The electronic property of amorphized phase of C 60just after the transition is far
from so-called amorphous diamond, since the amorphized C 60contains a number of dangling
bonds which have not yet turned into sp3type bonding. Amorphous diamond phase is
being gradually formed through the change in the bonding str ucture from sp2to sp3under
high pressure and high temperature. If dangling bonds are su fficiently reduced, the valence
and conduction bands are distinguished and a transparent op tical property is observed in
amorphous diamond phase as well as perfect cubic diamond. Am orphous phase is interpreted
as one of quasi-stable phases in the transition path from C 60to diamond phase under pressure.
If the temperature and the pressure are not sufficiently high, the reaction will be interrupted
before forming amorphous diamond phase, probably because t here are many quasi-stable
phases between C 60and diamond and the system is liable to be trapped by such phas es.
This is in contrast with pressure-induced graphite-diamon d transition. Since graphite and
diamond phase are located in very near configurations in the p otential surface, the transition
from graphite to diamond proceeds rapidly without passing t hrough any stable transient
14phase. From this reason, the formation of amorphous diamond phase is characteristic in the
compression of C 60and such a phase is not obtained in the compression of graphit e.
References
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15[16] In the compression with the presence of hydrogen, the te mperature grows more than
that does in the case without hydrogen, since the combinatio n energy between C and H
is released together with that between C and C. However the tr ansition speed itself is
not raised so large.
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[18] Even in the pressure-temperature range where the C 60turns into the amorphous, the
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distorts the graphite sheets and causes the fluctuation wher e the weak inter-layer bridg-
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the diamond will start. This accounts for the time-lag betwe en the raise of the pressure
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16Figure captions
•Figure 1: Snapshots of the transition from C 60fcc crystal to amorphous are given. The
pressure is increased from 0GPa to 65GPa in 0.45ps and kept to be 65GPa after that.
The pictures show the system cut out into a 15 ×15×15˚A3cube, which does not stand
for the unit cell.
•Figure 2 : Pair distribution function in each MD step, which c orresponds to the com-
pression given in figure 1.
•Figure 3: Bond-angle distribution in each MD step, which cor respond to the compres-
sion given in figure 1. Although the structures, characteris tic to C 60, remain at first,
they have almost vanished in 0.5ps and the system turns into t he amorphous.
•Figure 4(a): The structure fabricated after the compressio n at the maximum pressure
at 65GPa.
•Figure 4(b): The structure fabricated after the compressio n at the maximum pressure
at 125GPa.
•Figure 5(a):Definition of azimuthal angle.
•Figure 5(b):Distribution of the azimuthal angle after the c ompression at 65GPa, in the
structure of figure 4(a).
•Figure 5(c):Distribution of the azimuthal angle after the c ompression at 125GPa, in the
structure of figure 4(b).
•Figure 6 :Density of states corresponding to figure 5. The sol id line shows the total
DOS, and the dotted line shows the contribution from threefo ld carbon atoms.
•Figure 6(a): DOS after the compression at 65GPa.
•Figure 6(b): DOS after the compression at 125GPa.
•Figure 7: An example of the dynamical effect caused by changes in the pressure through
the compression is given here. Pressure is raised from 0GPa t o 65GPa, and after that,
again raised to 125GPa. The early stage of this simulation, w here the pressure is
confined in the range from 0GPa to 65GPa, corresponds to the sn apshot given in figure
1. These figures show the very slow reconstruction process of the short-range order in
the amorphous at the high pressure and temperature.
•Figure 7(a):The change in the applied pressure.
•Figure 7(b):The variation in the temperature. By raising th e pressure again, the tem-
perature increases again.
•Figure 7(c):The mean square of the displacement. By raising the pressure again, the
atomic movement becomes more active.
•Figure 7(d):The ratio of fourfold carbon atoms. The increas e in this ratio means the
formation of the sp3bonding. By raising the pressure again, much more numbers of
the sp3bonding are generated.
•Figure 8: Another example of the dynamical effect caused by ch ange in the temperature
through the compression is given here. The pressure is raise d from 0GPa to 125GPa.
17In the path, denoted as (A), there is no concentration on the t emperature. In the path
(B), the temperature is scaled at 2500K after 0.5ps.
•Figure 8(a):The change in the applied pressure.
•Figure 8(b):The variation in the temperature.
•Figure 8(c):The mean square of the displacement. In the path (B), owing to the lower
temperature, the atomic movement becomes more inactive.
•Figure 8(d):The ratio of fourfold carbon atoms. In the path ( B), owing to the lower
temperature, the transition speed from sp2to sp3decreases.
•Figure 9: The density of states with the presence of hydrogen . The solid line shows the
total DOS, and the dotted line shows the contribution from th reefold carbon atoms.
The cell includes hydrogen atoms at 12.5% in number. Figure 9 (a) shows the DOS after
the compression at the maximum pressure 55GPa. The contribu tion from threefold C
atoms is about 40% in the total DOS. Figure 9(b) shows the DOS a fter the compression
at the maximum pressure 125GPa. The contribution from three fold C atoms is about
20% in the total DOS.
•Figure 10: This figure shows the ratio of fourfold carbon atom s in the pressure-induced
structural transition from graphite to cubic diamond. The t ransition from sp2to sp3in
the whole crystal can bee seen in the rapid increase in that ra tio, changing from 0 to 1.
In this simulation, the unit cell includes 240 carbon atoms. The pressure is raised from
0GPa to 150GPa in initial 0.15ps and kept constant. The tempe rature is scaled to be
5000K throughout the simulation. The graphite directly tur ns into the stable perfect
cubic diamond, taking a reaction path similar to that demons trated in a first principles
simulation[9].
18/BY/CX/CV/D9/D6/CT /BD/BM /CB/D2/CP/D4/D7/CW/D3/D8/D7 /D3/CU /D8/CW/CT /D8/D6/CP/D2/D7/CX/D8/CX/D3/D2 /CU/D6/D3/D1 /BV/BI/BC
/CU
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/D6/CT/CP/D7/CT/CS /CU/D6/D3/D1 /BC/BZ/C8 /CP /D8/D3 /BI/BH/BZ/C8 /CP /CX/D2 /BC/BA/BG/BH/D4/D7 /CP/D2/CS /CZ /CT/D4/D8 /D8/D3 /CQ /CT /BI/BH/BZ/C8 /CP /CP/CU/D8/CT/D6/D8/CW/CP/D8/BA /CC/CW/CT /D4/CX
/D8/D9/D6/CT/D7 /D7/CW/D3 /DB /D8/CW/CT /D7/DD/D7/D8/CT/D1
/D9/D8 /D3/D9/D8 /CX/D2 /D8/D3 /CP /BD/BH /A2 /BD/BH /A2 /BD/BH
/AN/BT
/BF
/D9/CQ /CT/B8 /DB/CW/CX
/CW /CS/D3 /CT/D7/D2/D3/D8 /D7/D8/CP/D2/CS /CU/D3/D6 /D8/CW/CT /D9/D2/CX/D8
/CT/D0/D0/BA/B4/CP/B5/BC/BA/BC/D4/D7/BB/BC/C8 /CP/BM/C1/D2/CX/D8/CX /CP/D0 /D7/D8/D6/D9
/D8/D9/D6/CT
Active/B4/CQ/B5/BC/BA/BH/D4/D7/BB/BI/BH/BA/BC/C8 /CP/BM/C8 /D3/D0 /DD/D1 /CT/D6 /CX/DE/CT/CS /D4/CW/CP/D7/CT
Active/B4
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Active/BC
/BD
/BE
/BF
/BG
/BH
/BI
/BJ
/BK/BC /BD /BE /BF /BG /BH /BI /BJ /BK /BL /BD/BC/CA/B4
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/B4/CP/B5/D8/CX/D1/CT /BP /BC/BA/BC/BH/D4/D7/BC
/BD
/BE
/BF
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/BK/BC /BD /BE /BF /BG /BH /BI /BJ /BK /BL /BD/BC/CA/B4
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/D8/CX/D3/D2 /CX/D2 /CT/CP
/CW /C5/BW /D7/D8/CT/D4/B8 /DB/CW/CX
/CW
/D3/D6/D6/CT/D7/D4 /D3/D2/CS/D7 /D8/D3 /D8/CW/CT
/D3/D1/D4/D6/CT/D7/D7/CX/D3/D2 /CV/CX/DA /CT/D2 /CX/D2 /AC/CV/D9/D6/CT /BD/BA/BC
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/BF/BC
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/BH/BC
/BI/BC
/BJ/BC/BC /BE/BC /BG/BC /BI/BC /BK/BC /BD/BC/BC /BD/BE/BC /BD/BG/BC /BD/BI/BC /BD/BK/BC/BW/CT/CV/D6/CT/CT/D7
/B4/CP/B5/D8/CX/D1/CT /BP /BC/BA/BC/BH/D4/D7/BC
/BD/BC
/BE/BC
/BF/BC
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/BI/BC
/BJ/BC/BC /BE/BC /BG/BC /BI/BC /BK/BC /BD/BC/BC /BD/BE/BC /BD/BG/BC /BD/BI/BC /BD/BK/BC/BW/CT/CV/D6/CT/CT/D7
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/CW
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/D3/D1/B9/D4/D6/CT/D7/D7/CX/D3/D2 /CV/CX/DA /CT/D2 /CX/D2 /AC/CV/D9/D6/CT /BD/BAActive/BY/CX/CV/D9/D6/CT /BG/B4/CP/B5/BM /CC/CW/CT /D7/D8/D6/D9
/D8/D9/D6/CT /CU/CP/CQ/D6/CX
/CP/D8/CT/CS /CP/CU/D8/CT/D6 /D8/CW/CT
/D3/D1/D4/D6/CT/D7/D7/CX/D3/D2 /CP/D8 /D8/CW/CT /D1/CP/DC/CX/D1 /D9/D1 /D4/D6/CT/D7/B9/D7/D9/D6/CT /CP/D8 /BI/BH/BZ/C8 /CP/BA
Active/BY/CX/CV/D9/D6/CT /BG/B4/CQ/B5/BM /CC/CW/CT /D7/D8/D6/D9
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/CP/D8/CT/CS /CP/CU/D8/CT/D6 /D8/CW/CT
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/LiteDiagLines/LiteDiagLines/LiteDiagLines/LiteDiagLines/LiteDiagLines/LiteDiagLines/LiteDiagLines/LiteDiagLines/LiteDiagLines/LiteDiagLines/LiteDiagLines/LiteDiagLines/LiteDiagLines/LiteDiagLines/LiteDiagLines/LiteDiagLines/LiteDiagLines/LiteDiagLines/LiteDiagLines/LiteDiagLines/LiteDiagLines/LiteDiagLines/LiteDiagLines/LiteDiagLines/LiteDiagLines/LiteDiagLines/LiteDiagLines/LiteDiagLines/LiteDiagLines/LiteDiagLines/LiteDiagLines/LiteDiagLines/LiteDiagLines/LiteDiagLines/LiteDiagLines/LiteDiagLines/LiteDiagLines/LiteDiagLines/LiteDiagLines/LiteDiagLines/LiteDiagLines/LiteDiagLines/LiteDiagLines/LiteDiagLines
/LiteDiagLines/LiteDiagLines/LiteDiagLines/LiteDiagLines
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/LiteDiagLines/LiteDiagLines/LiteDiagLines/LiteDiagLines/LiteDiagLines/LiteDiagLines
/LiteDiagLines/LiteDiagLines/LiteDiagLines/LiteDiagLines/BY/CX/CV/D9/D6/CT /BH/B4/CP/B5/BM /BW/CT/AC/D2/CX/D8/CX/D3/D2 /D3/CU /D8/CW/CT /CP/DE/CX/D1 /D9/D8/CW/CP/D0 /CP/D2/CV/D0/CT/B4/CQ/B5 /BV/D3/D1/D4/D6/CT/D7/D7/CT/CS /CP/D8 /BI/BH/BZ/C8 /CP/BC
/BE/BC
/BG/BC
/BI/BC
/BK/BC
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/D3/D1/D4/D6/CT/D7/D7/CX/D3/D2 /CP/D8 /BI/BH/BZ/C8 /CP/B8 /CX/D2 /D8/CW/CT/D7/D8/D6/D9
/D8/D9/D6/CT /D3/CU /AC/CV/D9/D6/CT /BG/B4/CP/B5/BA/B4
/B5 /BV/D3/D1/D4/D6/CT/D7/D7/CT/CS /CP/D8 /BD/BE/BH/BZ/C8 /CP/BC
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/BG/BC
/BI/BC
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/BD/BC/BC/BC /BE/BC /BG/BC /BI/BC /BK/BC /BD/BC/BC /BD/BE/BC /BD/BG/BC /BD/BI/BC /BD/BK/BC/BW/CT/CV/D6/CT/CT/BY/CX/CV/D9/D6/CT /BH/B4
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/D4 /D4 /D4 /D4 /D4/D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4/D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4/D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4/D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4/D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4/D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4/D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4/D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4/D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4/D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4/D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4/D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4/D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4/D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4/D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4/D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4/D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4/D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4/D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4 /D4
/BC
/BC/BA/BD
/BC/BA/BE
/BC/BA/BF/B9/BE/BH /B9/BE/BC /B9/BD/BH /B9/BD/BC /B9/BH /BC /BH /BD/BC /BD/BH /BE/BC/BX/C6/BX/CA /BZ/CH/B4/CT/CE/B5/BY/CX/CV/D9/D6/CT /BI /BM/BW/CT/D2/D7/CX/D8 /DD /D3/CU /D7/D8/CP/D8/CT/D7
/D3/D6/D6/CT/D7/D4 /D3/D2/CS/CX/D2/CV /D8/D3 /AC/CV/D9/D6/CT /BH/BA /CC/CW/CT /D7/D3/D0/CX/CS /D0/CX/D2/CT /D7/CW/D3 /DB/D7 /D8/CW/CT /D8/D3/D8/CP/D0 /BW/C7/CB/B8/CP/D2/CS /D8/CW/CT /CS/D3/D8/D8/CT/CS /D0/CX/D2/CT /D7/CW/D3 /DB/D7 /D8/CW/CT
/D3/D2 /D8/D6/CX/CQ/D9/D8/CX/D3/D2 /CU/D6/D3/D1 /D8/CW/D6/CT/CT/CU/D3/D0/CS
/CP/D6/CQ /D3/D2 /CP/D8/D3/D1/D7/BA/BY/CX/CV/D9/D6/CT /BI/B4/CP/B5/BM /BW/C7/CB /CP/CU/D8/CT/D6 /D8/CW/CT
/D3/D1/D4/D6/CT/D7/D7/CX/D3/D2 /CP/D8 /BI/BH/BZ/C8 /CP/BA/BY/CX/CV/D9/D6/CT /BI/B4/CQ/B5/BM /BW/C7/CB /CP/CU/D8/CT/D6 /D8/CW/CT
/D3/D1/D4/D6/CT/D7/D7/CX/D3/D2 /CP/D8 /BD/BE/BH/BZ/C8 /CP/BA/BY/CX/CV/D9/D6/CT /BJ/BM /BT/D2 /CT/DC/CP/D1/D4/D0/CT /D3/CU /D8/CW/CT /CS/DD/D2/CP/D1/CX
/CP/D0 /CT/AB/CT
/D8
/CP/D9/D7/CT/CS /CQ /DD
/CW/CP/D2/CV/CT/D7 /CX/D2 /D8/CW/CT /D4/D6/CT/D7/D7/D9/D6/CT/D8/CW/D6/D3/D9/CV/CW /D8/CW/CT
/D3/D1/D4/D6/CT/D7/D7/CX/D3/D2 /CX/D7 /CV/CX/DA /CT/D2 /CW/CT/D6/CT/BA /C8/D6/CT/D7/D7/D9/D6/CT /CX/D7 /D6/CP/CX/D7/CT/CS /CU/D6/D3/D1 /BC/BZ/C8 /CP /D8/D3 /BI/BH/BZ/C8 /CP/B8 /CP/D2/CS/CP/CU/D8/CT/D6 /D8/CW/CP/D8/B8 /CP/CV/CP/CX/D2 /D6/CP/CX/D7/CT/CS /D8/D3 /BD/BE/BH/BZ/C8 /CP/BA /CC/CW/CT /CT/CP/D6/D0/DD /D7/D8/CP/CV/CT /D3/CU /D8/CW/CX/D7 /D7/CX/D1 /D9/D0/CP/D8/CX/D3/D2/B8 /DB/CW/CT/D6/CT /D8/CW/CT/D4/D6/CT/D7/D7/D9/D6/CT /CX/D7
/D3/D2/AC/D2/CT/CS /CX/D2 /D8/CW/CT /D6/CP/D2/CV/CT /CU/D6/D3/D1 /BC/BZ/C8 /CP /D8/D3 /BI/BH/BZ/C8 /CP/B8
/D3/D6/D6/CT/D7/D4 /D3/D2/CS/D7 /D8/D3 /D8/CW/CT /D7/D2/CP/D4/D7/CW/D3/D8/CV/CX/DA /CT/D2 /CX/D2 /AC/CV/D9/D6/CT /BD/BA /CC/CW/CT/D7/CT /AC/CV/D9/D6/CT/D7 /D7/CW/D3 /DB /D8/CW/CT /DA /CT/D6/DD /D7/D0/D3 /DB /D6/CT
/D3/D2/D7/D8/D6/D9
/D8/CX/D3/D2 /D4/D6/D3
/CT/D7/D7 /D3/CU /D8/CW/CT/D7/CW/D3/D6/D8/B9/D6/CP/D2/CV/CT /D3/D6/CS/CT/D6 /CX/D2 /D8/CW/CT /CP/D1/D3/D6/D4/CW/D3/D9/D7 /CP/D8 /D8/CW/CT /CW/CX/CV/CW /D4/D6/CT/D7/D7/D9/D6/CT /CP/D2/CS /D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT/BA/BY/CX/CV/D9/D6/CT /BJ/B4/CP/B5/BM/CC/CW/CT
/CW/CP/D2/CV/CT /CX/D2 /D8/CW/CT /CP/D4/D4/D0/CX/CT/CS /D4/D6/CT/D7/D7/D9/D6/CT/BA/B4/CP/B5/C8/D6/CT/D7/D7/D9/D6/CT
/BZ/C8 /CP/BC
/BE/BC
/BG/BC
/BI/BC
/BK/BC
/BD/BC/BC
/BD/BE/BC
/BD/BG/BC/BC /BD/BC/BC/BC /BE/BC/BC/BC /BF/BC/BC/BC /BG/BC/BC/BC /BH/BC/BC/BC /BI/BC/BC/BC /BJ/BC/BC/BC /BK/BC/BC/BC /BL/BC/BC/BC /BD/BC/BC/BC/BC/D8/CX/D1/CT/B4/CU/D7/B5/B4/CQ/B5/CC /CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT
/B4/C3/B5/BC
/BD/BC/BC/BC
/BE/BC/BC/BC
/BF/BC/BC/BC
/BG/BC/BC/BC
/BH/BC/BC/BC
/BI/BC/BC/BC/BC /BD/BC/BC/BC /BE/BC/BC/BC /BF/BC/BC/BC /BG/BC/BC/BC /BH/BC/BC/BC /BI/BC/BC/BC /BJ/BC/BC/BC /BK/BC/BC/BC /BL/BC/BC/BC /BD/BC/BC/BC/BC/D8/CX/D1/CT/B4/CU/D7/B5/BY/CX/CV/D9/D6/CT /BJ/B4/CQ/B5/BM/CC/CW/CT /DA /CP/D6/CX/CP/D8/CX/D3/D2 /CX/D2 /D8/CW/CT /D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT/BA /BU/DD /D6/CP/CX/D7/CX/D2/CV /D8/CW/CT /D4/D6/CT/D7/D7/D9/D6/CT /CP/CV/CP/CX/D2/B8 /D8/CW/CT/D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT /CX/D2
/D6/CT/CP/D7/CT/D7 /CP/CV/CP/CX/D2/BA/B4
/B5 /BO /B4 /D6 /B4 /D8 /B5 /A0 /D6 /B4/BC/B5/B5
/BE/BQ
/B4
/AN/BT
/BE/B5/BC
/BH
/BD/BC
/BD/BH
/BE/BC
/BE/BH
/BF/BC/BC /BD/BC/BC/BC /BE/BC/BC/BC /BF/BC/BC/BC /BG/BC/BC/BC /BH/BC/BC/BC /BI/BC/BC/BC /BJ/BC/BC/BC /BK/BC/BC/BC /BL/BC/BC/BC /BD/BC/BC/BC/BC/D8/CX/D1/CT/B4/CU/D7/B5/BY/CX/CV/D9/D6/CT /BJ/B4
/B5/BM/CC/CW/CT /D1/CT/CP/D2 /D7/D5/D9/CP/D6/CT /D3/CU /D8/CW/CT /CS/CX/D7/D4/D0/CP
/CT/D1/CT/D2 /D8/BA /BU/DD /D6/CP/CX/D7/CX/D2/CV /D8/CW/CT /D4/D6/CT/D7/D7/D9/D6/CT /CP/CV/CP/CX/D2/B8/D8/CW/CT /CP/D8/D3/D1/CX
/D1/D3 /DA /CT/D1/CT/D2 /D8 /CQ /CT
/D3/D1/CT/D7 /D1/D3/D6/CT /CP
/D8/CX/DA /CT/BA/B4/CS/B5/CA/CP/D8/CX/D3 /D3/CU /CU/D3/D9/D6/CU/D3/D0/CS /CP/D8/D3/D1/D7/BC
/BC/BA/BE
/BC/BA/BG
/BC/BA/BI
/BC/BA/BK
/BD/BC /BD/BC/BC/BC /BE/BC/BC/BC /BF/BC/BC/BC /BG/BC/BC/BC /BH/BC/BC/BC /BI/BC/BC/BC /BJ/BC/BC/BC /BK/BC/BC/BC /BL/BC/BC/BC /BD/BC/BC/BC/BC/D8/CX/D1/CT/B4/CU/D7/B5/BY/CX/CV/D9/D6/CT /BJ/B4/CS/B5/BM/CC/CW/CT /D6/CP/D8/CX/D3 /D3/CU /CU/D3/D9/D6/CU/D3/D0/CS
/CP/D6/CQ /D3/D2 /CP/D8/D3/D1/D7/BA /CC/CW/CT /CX/D2
/D6/CT/CP/D7/CT /CX/D2 /D8/CW/CX/D7 /D6/CP/D8/CX/D3 /D1/CT/CP/D2/D7 /D8/CW/CT/CU/D3/D6/D1/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D7/D4
/BF/CQ /D3/D2/CS/CX/D2/CV/BA /BU/DD /D6/CP/CX/D7/CX/D2/CV /D8/CW/CT /D4/D6/CT/D7/D7/D9/D6/CT /CP/CV/CP/CX/D2/B8 /D1 /D9
/CW /D1/D3/D6/CT /D2 /D9/D1 /CQ /CT/D6/D7 /D3/CU/D8/CW/CT /D7/D4
/BF/CQ /D3/D2/CS/CX/D2/CV /CP/D6/CT /CV/CT/D2/CT/D6/CP/D8/CT/CS/BA/BY/CX/CV/D9/D6/CT /BK/BM /BT/D2/D3/D8/CW/CT/D6 /CT/DC/CP/D1/D4/D0/CT /D3/CU /D8/CW/CT /CS/DD/D2/CP/D1/CX
/CP/D0 /CT/AB/CT
/D8
/CP/D9/D7/CT/CS /CQ /DD
/CW/CP/D2/CV/CT /CX/D2 /D8/CW/CT /D8/CT/D1/D4 /CT/D6/B9/CP/D8/D9/D6/CT /D8/CW/D6/D3/D9/CV/CW /D8/CW/CT
/D3/D1/D4/D6/CT/D7/D7/CX/D3/D2 /CX/D7 /CV/CX/DA /CT/D2 /CW/CT/D6/CT/BA /CC/CW/CT /D4/D6/CT/D7/D7/D9/D6/CT /CX/D7 /D6/CP/CX/D7/CT/CS /CU/D6/D3/D1 /BC/BZ/C8 /CP /D8/D3/BD/BE/BH/BZ/C8 /CP/BA /C1/D2 /D8/CW/CT /D4/CP/D8/CW/B8 /CS/CT/D2/D3/D8/CT/CS /CP/D7 /B4/BT/B5/B8 /D8/CW/CT/D6/CT /CX/D7 /D2/D3
/D3/D2
/CT/D2 /D8/D6/CP/D8/CX/D3/D2 /D3/D2 /D8/CW/CT /D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT/BA/C1/D2 /D8/CW/CT /D4/CP/D8/CW /B4/BU/B5/B8 /D8/CW/CT /D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT /CX/D7 /D7
/CP/D0/CT/CS /CP/D8 /BE/BH/BC/BC/C3 /CP/CU/D8/CT/D6 /BC/BA/BH/D4/D7/BA/BY/CX/CV/D9/D6/CT /BK/B4/CP/B5/BM/CC/CW/CT
/CW/CP/D2/CV/CT /CX/D2 /D8/CW/CT /CP/D4/D4/D0/CX/CT/CS /D4/D6/CT/D7/D7/D9/D6/CT/BA/B4/CP/B5/C8/D6/CT/D7/D7/D9/D6/CT
/BZ/C8 /CP/BC
/BE/BC
/BG/BC
/BI/BC
/BK/BC
/BD/BC/BC
/BD/BE/BC
/BD/BG/BC/BC /BH/BC/BC /BD/BC/BC/BC /BD/BH/BC/BC /BE/BC/BC/BC /BE/BH/BC/BC/D8/CX/D1/CT/B4/CU/D7/B5/B4/CQ/B5/CC /CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT
/B4/C3/B5/B4/BU/B5
/B9
/B4/BT/B5
Ꜵ
Ꜵ
Ꜵ/BM/BC
/BD/BC/BC/BC
/BE/BC/BC/BC
/BF/BC/BC/BC
/BG/BC/BC/BC
/BH/BC/BC/BC
/BI/BC/BC/BC/BC /BH/BC/BC /BD/BC/BC/BC /BD/BH/BC/BC /BE/BC/BC/BC /BE/BH/BC/BC/D8/CX/D1/CT/B4/CU/D7/B5/BY/CX/CV/D9/D6/CT /BK/B4/CQ/B5/BM/CC/CW/CT /DA /CP/D6/CX/CP/D8/CX/D3/D2 /CX/D2 /D8/CW/CT /D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT/BA/B4
/B5 /BO /B4 /D6 /B4 /D8 /B5 /A0 /D6 /B4/BC/B5/B5
/BE/BQ
/B4
/AN/BT
/BE/B5/B4/BU/B5
/B9
/B4/BT/B5
Ꜵ
Ꜵ
Ꜵ/BM/D8/CX/D1/CT/B4/CU/D7/B5
/BC
/BH
/BD/BC
/BD/BH
/BE/BC
/BE/BH/BC /BH/BC/BC /BD/BC/BC/BC /BD/BH/BC/BC /BE/BC/BC/BC /BE/BH/BC/BC/BY/CX/CV/D9/D6/CT /BK/B4
/B5/BM/CC/CW/CT /D1/CT/CP/D2 /D7/D5/D9/CP/D6/CT /D3/CU /D8/CW/CT /CS/CX/D7/D4/D0/CP
/CT/D1/CT/D2 /D8/BA /C1/D2 /D8/CW/CT /D4/CP/D8/CW /B4/BU/B5/B8 /D3 /DB/CX/D2/CV /D8/D3 /D8/CW/CT/D0/D3 /DB /CT/D6 /D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT/B8 /D8/CW/CT /CP/D8/D3/D1/CX
/D1/D3 /DA /CT/D1/CT/D2 /D8 /CQ /CT
/D3/D1/CT/D7 /D1/D3/D6/CT /CX/D2/CP
/D8/CX/DA /CT/BA/B4/CS/B5 /CA/CP/D8/CX/D3 /D3/CU /CU/D3/D9/D6/CU/D3/D0/CS /CP/D8/D3/D1/D7/B4/BU/B5
/B9
/B4/BT/B5
Ꜵ
Ꜵ
Ꜵ/BM/BC
/BC/BA/BE
/BC/BA/BG
/BC/BA/BI
/BC/BA/BK
/BD/BC /BH/BC/BC /BD/BC/BC/BC /BD/BH/BC/BC /BE/BC/BC/BC /BE/BH/BC/BC/D8/CX/D1/CT/B4/CU/D7/B5/BY/CX/CV/D9/D6/CT /BK/B4/CS/B5/BM/CC/CW/CT /D6/CP/D8/CX/D3 /D3/CU /CU/D3/D9/D6/CU/D3/D0/CS
/CP/D6/CQ /D3/D2 /CP/D8/D3/D1/D7/BA /C1/D2 /D8/CW/CT /D4/CP/D8/CW /B4/BU/B5/B8 /D3 /DB/CX/D2/CV /D8/D3 /D8/CW/CT /D0/D3 /DB /CT/D6/D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT/B8 /D8/CW/CT /D8/D6/CP/D2/D7/CX/D8/CX/D3/D2 /D7/D4 /CT/CT/CS /CU/D6/D3/D1 /D7/D4
/BE/D8/D3 /D7/D4
/BF/CS/CT
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/D8/D0/DD/D8/D9/D6/D2/D7 /CX/D2 /D8/D3 /D8/CW/CT /D7/D8/CP/CQ/D0/CT /D4 /CT/D6/CU/CT
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arXiv:physics/0102076v1 [physics.bio-ph] 23 Feb 2001Errors drive the evolution of biological
signalling to costly codes
Gonzalo G. de Polavieja
Dept. Zoology, Downing St.
University of Cambridge, CB2 3EJ, UK.
gg234@cus.cam.ac.uk
1Abstract
Reduction of costs in biological signalling seems an evo-
lutionary advantage, but recent experiments have shown sig -
nalling codes shifted to signals of high cost with a under-
utilisation of low cost signals. Here I show that errors in
the efficient translation of biological states into signals s hift
codes to higher costs, effectively performing a quality con-
trol. The statistical structure of signal usage is predicte d
to be of a generalised Boltzmann form that penalises sig-
nals that are costly and sensitive to errors. This predicted
distribution of signal usage against signal cost has two mai n
features: an exponential tail required for cost efficiency an d
an underutilisation of the low cost signals required to pro-
tect the signalling quality from the errors. These predicti ons
are shown to correspond quantitatively to the experiments i n
which gathering signal statistics is feasible as in visual c ortex
neurons.
KEYWORDS: signalling, cost, noise, neuron , information th eory
SHORT TITLE: Errors drive signalling to costly codes
21 Introduction
Cells, groups of cells and multicellular organisms communi cate their states
using signals. The types of signals and encoding mechanisms used can be
very different but, irrespectively of the mechanism, signal transmission should
have a high efficiency within biological constraints. A unive rsal constraint
is the signalling cost. Have biological signalling codes ev olved to minimise
cost? Cost reduction seems advantageous ( ?????? ) but signalling systems
might be simultaneously optimal not only respect to cost but also to other
constraints resulting in signalling codes very different to the cost efficient
ones. A second universal constraint is communication error s. Here I consider
the extension of information theory ( ??) to include errors and cost together
as constraints of signalling systems and find the optimal sig nal usage under
these constraints. For clarity of exposition and because th e best data sets for
statistical analysis are in neural signals, I will particul arise the discussion to
cell signalling and discuss the relevance of results to othe r signalling systems
afterwards.
Neurons provide an experimentally tractable case of cell si gnalling. The
experimental evidence in neurons is counterintuitive. Neu rons codes can un-
derutilise low cost signals. For neurons using different spi ke rates as signals,
3it has been found that low rates that take lesser metabolic co st to produce are
typically underutilised ( ?). Similarly, neurons using spike bursts as signals
underutilise the bursts of one spike that would take lesser p roduction cost
(??). Theories of cost efficiency cannot explain these experimen tal results.
According to these theories of cost efficiency, signalling sy stems should max-
imise their capacity to represent different states given a co st constraint or
maximise the ratio of this representational capacity and th e cost ( ??). The
optimal distribution for these theories is an exponential d ecaying with signal
cost. In this way the most probable signals are those of lowes t cost in clear
contrast to the underutilisation of the low cost signals obs erved experimen-
tally. For this reason I consider here the evolution of biolo gical signalling
codes towards efficiency of transmission within the biologic al constraints of
both cost and errors.
This paper is organised as follows. Section 2 gives the theor etical frame-
work and the general result of optimal signal usage when both costs and errors
constrain the signalling system. To find this optimal signal usage an iterative
algorithm that can be easily implemented is given. Section 3 shows that the
optimal solutions found predict quantitatively the experi mental results for
signal usage in visual cortex neurons. Section 4 gives the co nclusions and
4discusses the application to a variety of biological signal ling systems includ-
ing animal communication for which it is shown that cheaters shift efficient
codes to high cost.
2 Theoretical treatment
For signal transmission between a signaller and a receiver t o work, the
signaller must use encoding rules that correlate its signal ling states C=
{c1, c2, ..., c N}with the signals S={s1, s2, ..., s N}. For intercellular sig-
nalling, the signals Scan be different values of concentration of the same
chemical, different mixtures of several chemicals, differen t time patterns (say,
different frequencies of spike generation or bursts of differ ent sizes), different
spatial patterns or even different patterns of activation of a group of cells.
The cellular states Care the internal variables representing the ideal sig-
nals without errors. Experimentally, identical stimulati ons of the cell will
produce a distribution of signals were the peak is the ideal n oiseless signal
corresponding to the cellular state and the variance comes f rom the errors.
The correlation of states and signals is subject to the const raints imposed
by cost and errors. We characterise these errors with the err or matrix of
5conditional probabilities Qkj≡p(ck|sj), a matrix given by the probability
that the signal sjcomes from the state ck. When there are no errors present
each signal comes from a single state, and the error matrix Qis diagonal.
When there are errors present, there are nonzero nondiagona l elements. The
costs can be in molecular machinery (a convenient parameter can be the
number of ATP molecules), in transmission times (for exampl e, bursts of
many spikes take longer times to transmit than of fewer spike s) and in risks
(for example by the use of chemicals that can be toxic). We can formally
write the costs of producing the signals as ǫkjwith for example ǫ12the cost for
the conversion of the first state into the second signal. As we are interested
in the signal usage, we refer the costs to the signals as ǫj=/summationtext
kQkjǫkj. We
always label the signals in order of increasing cost, ǫ1≤ǫ2≤...≤ǫN.
We also need to formalise the notion of correlation between t he signallers
states and the signals in order to consider the consequences of cost and errors
for this correlation. We require a general measure of correl ation that is valid
for any nonlinear dependencies, unlike correlation functi ons (?), and that
does not use a metric that measures correlation in an arbitra ry manner. The
averaged distance between the actual joint distribution p(ci, sj) and the dis-
tribution corresponding to complete decorrelation p(ci, sj)decorr ≡p(ci)p(sj)
6gives such a general measure of correlation of the form
I(C;S) =/summationdisplay
i,jp(ci, sj) log/parenleftbiggp(ci, sj)
p(ci)p(sj)/parenrightbigg
, (1)
that is zero for the completely decorrelated case and increa ses with increasing
correlation. This is the standard measure of statistical co rrelation used in
communication theory where it is known as mutual information (?). The
mutual information Itakes care of the errors as a constraint. To see this, we
can write its expression in (1) in terms of the error matrix Qby separating
it into the signal variability and the signal uncertainty te rms as I(C;S) =
H(S)−H(S|C) , with H(S) =−/summationtext
jp(sj) logp(sj) and H(S|C) =/summationtext
jpjξj
with
ξj=−/summationdisplay
kQkjlogPjk (2)
a measure of the signal uncertainty for signal sjandPjk≡p(sj|ck) the
probability that the state ckproduces the signal sj. We can express Pkjin
terms of Qjkusing Bayes’ theorem as Pjk= (p(sj)Qkj)/(/summationtext
jp(sj)Qkj). With
these relations we see that the mutual information can be wri tten as the
difference of a term H(S) that measures the variability of the signal and a
7termH(S|C) that measures the signal uncertainty as the variability of the
signal that comes from the errors in Q. This second term H(S|C) is the
constraint given by the errors.
Using the mutual information Ias the measure of correlation between
states and signals, that includes the constraint given by th e errors, together
with the cost constraint, we can now formulate precisely our problem. With
which frequencies p(si) should the signals Sbe used to have a high mutual
information Ibetween states Cand signals Sgiven the errors Qand the
average cost E=/summationtext
ip(si)ǫias the biological constraints? To answer this
question we use the method of Lagrange multipliers. The solu tion of the
equations obtained by this method can be found using differen t numerical
methods and we have chosen the one given in Algorithm 1 based o n the
Blahut-Arimoto algorithm ( ??), commonly used in rate distortion theory
(?), because it is particularly transparent as to the form of th e solution.
From Algorithm 1, we obtain that the optimal signal usage tak ing errors and
cost as constraints is of the form in (4)
/hatwidep(sj) =/hatwideZ−1exp/parenleftBig
−/hatwideβǫj−/hatwideξj/parenrightBig
, (6)
8Algorithm 1 Optimal signal usage
Initialise the signal usage to a random vector p1.
fort= 1,2,...until convergence do
Pt
jk=pt(sj)Qkj/summationtext
jpt(sj)Qkj(3)
pt+1(sj) =exp−/parenleftbig
βtǫj−/summationtext
kQkjlogPt
jk/parenrightbig
/summationtext
jexp−/parenleftbig
βtǫj−/summationtext
kQkjlogPt
jk/parenrightbig, (4)
where βtin (4) has to be evaluated for each tfrom the cost constraint
/summationtext
jǫjexp−/parenleftbig
βtǫj−/summationtext
kQjklogPt
jk/parenrightbig
/summationtext
jexp−/parenleftbig
βtǫj−/summationtext
kQjklogPt
jk/parenrightbig=E. (5)
end for
where the hat on p,Z,βandξis a reminder that their values are obtained
using the iterative Algorithm 1. The expression for ξis given in (2) and Zis
the normalisation constant. This solution has a number of in teresting charac-
teristics. Both signal cost, through the term βǫj, and the signal uncertainty
from the errors ξj, penalise the usage of the signal sjin an exponential form.
With no errors present the signal usage is a decaying exponen tial with the
signal cost ǫ. And with no cost constraint the signal usage is an exponen-
tial against the signal uncertainty from the errors ξ. The distribution for the
error-free case coincides with the one obtained in Statisti cal Mechanics where
9it is known as the Boltzmann distribution. We name the genera l distribution
including the effect of the errors in (6) as a generalised Bolt zmann distribu-
tion. To obtain the general relationship between statistic al correlation Iand
average cost E, substitute the distribution in (6) in the expression for Iin
(1) to obtain /hatwideI=/hatwideβE+ log /hatwideZ, where the parameter /hatwideβgiven in (5) and the
normalisation constant /hatwideZare nonlinear functions of the average cost E. This
expression is the most general relationship between mutual information and
cost for efficient signalling.
Given the error matrix Q, an average energy Eand signals costs ǫ, that
can be obtained either experimentally or from theoretical m odels, Algorithm
1 gives the optimal signal usage that maximizes signal quali ty while max-
imizing cost-efficiency. We can advance some characteristic s of the signal
usage for optimal communication. In biological systems we e xpect that the
errors produced with highest probability are those with the lowest amplitude.
Two examples illustrate this point. Consider first a cell tha t translates some
states into signals but that when it is in a nonsignalling sta te, spontaneously
produces signals by error. The most probable signals to be pr oduced by error
are those of lowest amplitude and therefore lowest cost. Thi s is the case in
neurons when different values of spike rates are used as differ ent signals and
10spontaneous signalling, say following a Poisson distribut ion, produces the
highest rates with very low probability. The signals of lowe r rate have then a
higher signal uncertainty and according to expression in (6 ) are then under-
utilized. As a second example consider animal communicatio n. According
to the present framework, cheaters that can produce low-cos t signals enter
as errors in the communication between healthy animals. The se errors make
the low-cost signals to have higher uncertainty and, as in th e case of neuronal
signalling, according to (6) the low-cost signals should be underutilized.
3 Comparison with experiments
The signal usage of a small percentage of neurons, 16% in the c ase of neurons
in the visual cortex area MT of macaques ( ?), can be explained with a theory
of cost-efficient signalling ( ??). To explain the signal usage for the totality of
visual cortex neurons we use the formalism presented in the p revious section
that not only requires signal efficiency but signal quality. A s in (?), I assume
maximum signal variability with an energy constraint, the n ovelty here is
to require signal quality by minimizing signal uncertainty . We also assume
that the spike rates are the symbols that the visual cortex ne urons use to
11communicate ( ???) and that the costs of each symbol in ATP molecules can
be taken to be linearly proportional to the rate value. As a si mple model
to the main contribution from noise we assume spontaneous si gnalling when
the cell should be in a nonsignalling state. This random spik e production
is modelled by a Poisson distribution, with the average numb er of spikes
produced by error in an interval as the single parameter that distinguishes
different cells. When this number is low, the optimal signal u sage obtained
from the Algorithm 1 can be approximated as
p(Rate) = Z−1exp (−exp (−Rate/α)−βRate) , (7)
where Zis the normalization constant. Cost efficiency is assured by t he term
−βRate that penalizes signals by their cost. Signal quality is assured by the
term exp ( −Rate/α) that penalizes signals by their signal uncertainty, that
increases with α. The predictions made by the optimal signalling in (7) are:
(a) For high rate values the term required for signal quality in (7) is negligible,
so optimal signal usage reduces to an exponential decaying w ith rate. (b)
Low rate values are expected to be underutilized respect to t he straight line
in (a). Specifically, the difference between the straight lin e and the logarithm
12of the probability, −βRate −log(p) must be a decreasing exponential. We
compare these predictions to the rate distributions of infe rior temporal cortex
neurons of two rhesus macaques responding to video scenes th at have been
recently reported ( ?). The experimental distribution of rates for two of the
cells (labelled as ba001−01 and ay102−02 in ( ?)) are given in Figure 1
using a 400 ms window. As seen in Figure 1 the two predictions c orrespond
to the experimental data. Cost-efficiency is responsible for signal usage at
high rates and both cost-efficiency and signal quality for the lower values of
rate. Different neurons may have different values of the avera ge cost and
different noise properties but the signal usage seems to be ad apted to the
optimal values for each cell.
4 Discussion
We have seen that the effect of errors in the evolution of signa lling sys-
tems towards efficiency is to shift signalling codes to higher cost to minimize
signal uncertainty. The optimal signal usage for a communic ation system
constrained by errors and cost has been shown to have a genera lised Boltz-
mann form in equation (6) that penalises signals that are cos tly and that are
13sensitive to errors. The two main features of this optimal si gnal usage are an
exponential tail at high cost signals needed for cost efficien cy and an under-
utilisation of the low cost signals required to protect the s ignal quality against
errors while maintaining the cost efficiency. The prediction s made by this
optimal signal usage have been shown to correspond to the the experimental
measurements in visual cortex neurons.
We have so far discussed cell signalling, but as we noticed al ready in the
Introduction we have chosen this particular type of signall ing for concrete-
ness. The theoretical framework here proposed does not requ ire knowledge of
the underlying mechanisms of signalling. The theory only us es the notion of
statistical correlation of states and signals without the n eed to make concrete
how this correlation is physically established and without any description of
the types of signals except for the costs and errors. This is e nough to un-
derstand the optimal signal usage with cost and error constr aints. For this
reason, the results apply generally to biological communic ation and also to
non-biological communication. Intracellular communicat ion and machine-
machine communication are two possible domains of applicat ion. Another
important case is animal communication for which game-theo retical models
have predicted that the evolutionary incentive to deceit is overcame increas-
14ing the cost of signals ( ???). These costly signals are called handicaps and
make the communication reliable in the sense of being honest . A different
perspective is gained from the formalism presented here. Ch eaters enter as
errors in the communication between healthy animals and as t hey are only
able to produce low cost signals, the signal uncertainty of t he low cost signals
is higher. According to the general result in (6) these low co st signals should
be underutilized. This means that signal quality requires a shift to high cost
signals, as we saw in the case of neurons. In this case, cost ca n be metabolic,
times or risks. In this way we obtain a statement of the handic ap principle
based on optimal communication without using the theory of g ames. Pro-
vided we have knowledge of the communication symbols, their cost and error
characteristics, the present formalism would give the opti mal use of symbols
according to signal quality and cost-efficiency.
It is interesting to discuss the limits of the theoretical fr amework. First,
we have assumed that errors and cost are the only constraints of the com-
munication system. Although these constraints are univers al, particular sys-
tems might have extra constraints. However, even in the pres ence of new
constraints, the effect of errors would be to shift the signal ling code to higher
cost. Second, we have argued that in biological communicati on systems the
15errors that are produced with highest probability are those of the lowest am-
plitude and therefore of the lowest cost. But this need not al ways be the
case. For example, processing of the signals at the receiver cell might fail
more frequently for the most complex incoming signals, typi cally those with
highest cost. In this case, there would be an extra penalisat ion of the high
cost signals and the decay of the distribution would be faste r than exponen-
tial. There is partial experimental evidence for this type o f code in ( ?) (see
their Figure 4(f,g)). Other types of codes are possible with different error
properties that can still be predicted from the general rela tion (6).
Acknowledgements
Dennis Bray, William Bialek, Fabrizio Gabbiani, John Hopfie ld, Rufus Joh-
stone, and Amotz Zahavi are acknowledged for fruitful discu ssions. I am
especially indebted to Simon Laughlin for many discussions and critical com-
ments on the manuscript. I am also very grateful to Vijay Bala subramanian
and Michael J. Berry for discussing their independent resul ts on metabolic
efficiency prior to publication. I am thankful to Stephano Pan zeri for sending
me the data for Figure 1 and for discussing the results in ( ?). This research
16has been supported by a Wellcome Trust Fellowship in Mathema tical Biology.
17FIGURE CAPTIONS
Figure 1 . The probability distribution of rate usage for visual cort ex
neurons follows the optimal distribution in equation (7) (s olid line) with the
predicted exponential tail (dashed line) for high rates and the underutilisation
at low costs. The exponential tail makes visual cortex neuro ns cost efficient
and the underutilisation of the low cost signals protects th eir signal quality
against errors while remaining cost efficient. The errors are responsible for a
shift to higher cost signals, with a maximum at a rate of value of 10 spikes
in the 400 ms window instead of at a rate of 1 spike if there were no errors
present. The experimental data have been taken from the two v isual cortex
neurons labelled as (a) ba001−01 and (b) ay102−02 in ( ?).
1851020
1030
15204050
25--
23
-
-45
-
-7
6
-8
RateLog(probability)
RateLog(probability)
Figure 1.
Errors drive the evolution of biological
signalling to costly codes
G.G. de Polaviejaa
b |
null |
arXiv:physics/0102078v1 [physics.gen-ph] 23 Feb 2001Fuzzy, Non Commutative SpaceTime: A New
Paradigm for A New Century
B.G. Sidharth
B.M. Birla Science Centre, Adarshnagar, Hyderabad - 500 063 , India
Abstract
Much of twentieth century physics, whether it be Classical o r Quantum,
has been based on the concept of spacetime as a differentiable manifold. While
this work has culminated in the standard model, it is now gene rally accepted
that in the light of recent experimental results, we have to g o beyond the
standard model. On the other hand Quantum SuperString Theor y and a re-
cent model of Quantized Spacetime in which, for example, an e lectron can
be meaningfully described by the Kerr-Newman metric, have s hown promise.
They lead to mathematically identical spacetime-energy-m omenta commuta-
tion relations, and infact an identical non commutative geo metry which is a
departure from the usual concept of spacetime. This could we ll be a new
paradigm for the new century.
1 Introduction
At the beginning of the twentieth century several Physicist s including Poincare and
Abraham amongst others were tinkering unsuccessfully with the problem of the
extended electron[1, 2]. The problem was that an extended el ectron appeared to
contradict Special Relativity, while on the other hand, the limit of a point particle
lead to inexplicable infinities. These infinities dogged phy sics for many decades.
Infact the Heisenberg Uncertainity Principle straightawa y leads to infinities in the
limit of spacetime points. It was only through the artifice of renormalization that
’t Hooft could finally circumvent this vexing problem, in the 1970s (Cf. paper by ’t
Hooft in this volume).
Nevertheless it has been realized that the concept of spacet ime points is only
approximate[3, 4, 5, 6, 7]. We are beginning to realize that i t may be more meaning-
ful to speak in terms of spacetime foam, strings, branes, non commutative geometry,
fuzzy spacetime and so on[8]. This is what we will now discuss .
2 Two Approaches
We now consider the well known theory of Quantum SuperString s and also an ap-
proach in which an electron is considered to be a Kerr-Newman Black Hole, with
the additional input of fuzzy spacetime.
As is well known, String Theory originated from phonenomeno logical considera-
tions in the late sixties through the pioneering work of Vene ziano, Nambu and
1others to explain features like the s-t channel dual resonan ce scattering and Regge
trajectories[9]. Originally strings were conceived as one dimensional objects with
an extension of the order of the Compton wavelength, which wo uld fudge the point
vertices of the s-t channel scattering graphs, so that both w ould effectively corre-
spond to one another (Cf.ref.[9]).
The string itself is governed by the equation[10]
ρ¨y−Ty′′= 0 (1)
where the frequency ωis given by
ω=π
2/radicalBigg
T
ρ(2)
T=mc2
l;ρ=m
l(3)
/radicalbig
T/ρ=c (4)
Tbeing the tension of the string, which has to be introduced in the theory, l
its length and ρthe line density. The identification (3) gives (4) where cis the
velocity of light, and (1) then goes over to the usual d’Alemb ertian or massless
Klein-Gordon equation. It is worth noting that as l→0 the potential energy which
is∼/integraltextl
0T/parenleftBig
∂y
∂x/parenrightBig2
dxrapidly oscillates.
Quantization of the states leads to
/an}b∇acketle{t∆x2/an}b∇acket∇i}ht ∼l2(5)
The string effectively shows up as an infinite collection of Ha rmonic Oscillators [10].
It follows from the above that the length lof the string turns out to be the Compton
wavelength, a circumstance which has been described as one o f the miracles of String
Theory by Veneziano[11].
The above strings are really Bosonic strings. Raimond[12], Scherk[13] and others
laid the foundation for the theory of Fermionic strings. Ess entially the relativistic
Quantized String is given a rotation, when we get back the equ ation for Regge
trajectories,
J≤(2πT)−1M2+a0¯hwitha0= +1(+2)for the open (closed) string (6)
Attention must be drawn to the additional term a0which now appears in (6). It
arises from a zero point energy effect. When a0= 1 we have gauge Bosons while
a0= 2 describes the gravitons. In the full theory of Quantum Sup er Strings, we are
essentially dealing with extended objects rotating with th e velocity of light, rather
like spinning black holes. The spatial extention is at the Pl anck scale while features
like extra space time dimensions which are curled up in the Ka luza Klein sense and,
as we will see, non commutative geometry appear[14, 15].
The above considerations raise the question, can a charged e lementary particle be
pictured as a Kerr Newman Black Hole, though in a Quantum Mech anical context
rather than the General Relativistic case? Indeed it is well known that the Kerr
Newman Black Hole itself mimics the electron remarkably wel l including the purely
2Quantum Mechanical anomalous g= 2 factor[16]. The problem is that there would
be a naked singularity, that is the radius would become compl ex,
r+=GM
c2+ıb,b≡/parenleftbiggG2Q2
c8+a2−G2M2
c4/parenrightbigg1/2
(7)
whereais the angular momentum per unit mass.
This problem has been studied in detail by the author in recen t years[1, 17, 18].
Indeed it is quite remarkable that the position coordinate o f an electron in the Dirac
theory is non Hermitian and mimics equation (7), being given by
x= (c2p1H−1t+a1) +ı
2c¯h(α1−cp1H−1)H−1, (8)
where the imaginary parts of (7) and (8) are both of the order o f the Compton
wavelength.
The key to understanding the unacceptable imaginary part wa s given by Dirac
himself[19], in terms of Zitterbewegung. The point is that a ccording to the Heisen-
berg Uncertainity Principle, space time points themselves are not meaningful- only
space time intervals have meaning, and we are really speakin g of averages over such
intervals, which are atleast of the order of the Compton scal e. Once this is kept in
mind, the imaginary term disappears on averaging over the Co mpton scale.
Indeed, from a classical point of view also, in the extreme re lativistic case, as is
well known there is an extension of the order of the Compton wa velength, within
which we encounter meaningless negative energies[20]. Wit h this proviso, it has
been shown that we could think of an electron as a spinning Ker r Newman Black
Hole. This has received independent support from the work of Nottale[21].
3 Non Commutative Geometry
We are thus lead to the picture where there is a cut off in space t ime intervals as
indicated in the introduction.
In the above two scenarios, the cut off is at the Compton scale ( l,τ). Such discrete
space time models compatible with Special Relativity have b een studied for a long
time by Snyder and several other scholars[22, 23, 24]. In thi s case it is well known
that we have the following non commutative geometry
[x,y] = (ıa2/¯h)Lz,[t,x] = (ıa2/¯hc)Mx,
[y,z] = (ıa2/¯h)Lx,[t,y] = (ıa2/¯hc)My, (9)
[z,x] = (ıa2/¯h)Ly,[t,z] = (ıa2/¯hc)Mz,
whereais the minimum natural unit and Lx,Mxetc. have their usual significance.
Moreover in this case there is also a correction to the usual Q uantum Mechanical
commutation relations, which are now given by
[x,px] =ı¯h[1 + (a/¯h)2p2
x];
[t,pt] =ı¯h[1−(a/¯hc)2p2
t];
3[x,py] = [y,px] =ı¯h(a/¯h)2pxpy; (10)
[x,pt] =c2[px,t] =ı¯h(a/¯h)2pxpt; etc.
wherepµdenotes the four momentum.
In the Kerr Newman model for the electron alluded to above (or generally for a
spinning sphere of spin ∼¯hand of radius l),Lxetc. reduce to the spin¯h
2of a
Fermion and the commutation relations (9) and (10) reduce to
[x,y]≈0(l2),[x,px] =ı¯h[1 +βl2],[t,E] =ı¯h[1 +τ2] (11)
whereβ= (px/¯h)2and similar equations.
Interestingly the non commutative geometry given in (11) ca n be shown to lead to
the representation of Dirac matrices and the Dirac equation [25]. From here we can
get the Klein Gordon equation, as is well known[26, 27], or al ternatively we deduce
the massless string equation (1), using (4).
This is also the case with superstrings where Dirac spinors a re introduced, as indi-
cated in Section 2. Infact in QSS also we have equations mathe matically identical
to the relations (11) containing momenta. This, which impli es (9), can now be seen
to be the origin of non-commutativity.
The non commutative geometry and fuzzyness is contained in ( 11). Infact fuzzy
spaces have been investigated in detail by Madore and others [28, 29], and we are
lead back to the equation (11). The fuzzyness which is closel y tied up with the
non commutative feature is symptomatic of the breakdown of t he concept of the
spacetime points and point particles at small scales or high energies. As has been
noted by Snyder, Witten, and several other scholars, the div ergences encountered
in Quantum Field Theory are symptomatic of precisely such an extrapolation to
spacetime points and which necessitates devices like renor malization. As Witten
points out[30], ”in developing relativity, Einstein assum ed that the space time co-
ordinates were Bosonic; Fermions had not yet been discovere d!... The structure of
space time is enriched by Fermionic as well as Bosonic coordi nates.”
Interestingly, starting from equation (11), we can deduce t hatlis the Compton
wavelength without however assuming it to be so. Let us write the first equation
of (11) as
[x,y] =ıH (12)
The relation (12) shows that yplays a role similar to the xcomponent of the
momentum, and infact mathematically we have
y=ıH
¯hpx≡˜hpx (13)
At the extreme energies and speeds, we would have
y=˜hp=˜hmc,m ˙y=py,x=˜hpy (14)
From (13) it follows that
y=Hd
dx
whence
Ty′′=T
Hyy′=T
H2y·y2(15)
4Further from (14) it follows that
ρ¨y=ρ
md
dt/parenleftbiggx
˜h/parenrightbigg
=ρ
m2˜h·y
˜h=ρ
m2˜h2y (16)
Fusing (15) and (16) in to one we get
H2
˜h21
m2c2≡l2=/parenleftbiggh
mc/parenrightbigg2
=y2
wherelis now the Compton wavelength. This is the explanation for th e so called
miraculous emergence of the Compton wavelength in string th eory, as noted by
Veneziano (Cf.ref.[11]).
Finally it may be pointed out that the tension Tof String Theory, appears as the
energy of the Quantum Mechanical Kerr Newman Black Hole allu ded to in Section
2 via the relation (3).
We next have to see how, from the Compton scale above, we arriv e at the Planck
scale of QSS. For this, we note that from (11), using
∆p·∆x≈h,
we get,
∆p·∆x= ¯h[1 +l2
(∆x)2]
Whence
∆p(∆x)3= ¯h[(∆x)2+l2] (17)
Witten describes it as an extra correction to the Heisenberg Uncertainity Principle.
As long as we are at usual energies, we have the usual Uncertai nity Principle, and
the usual bosonic or commutative spacetime. At high energie s however we encounter
the extra term in (17) viz., ¯ h′= ¯hl2. With this, the Compton scale goes over from
ltol3, the Planck scale (Cf. also [31]). Equally interesting is th e fact that as
can be seen from (17), the single xdimension gets trebled. At these Planck scales,
therefore, a total of six extra dimensions appear, which are curled up in the Kaluza
Klein sense at the Planck scale. This provides an explanatio n for the puzzling six
extra dimensions of QSS.
4 Further Issues
(i) Vortices
As described in detail in [17] the Quantum Mechanical Kerr-N ewman Black Hole
could also be considered to be a vortex. If we take two paralle l spinning vortices
separated by a distance dthen the angular velocity is given by
ω=ν
πd2,
whereν=h/m.
Whence the spin of the system turns out to be h, that is in usual units the spin is
5one, and the above gives the states ±1.
There is also the case where the two above vortices are anti pa rallel. In this case
there is no spin, but rather there is the linear velocity give n by
v=ν/2πd
This corresponds to the state 0 in the spin 1 case.
Together, the two above cases give the three −1,0,+1 states of spin 1 as in the
Quantum Mechanical Theory.
In the case of the Quantum Mechanical Kerr-Newman Black Hole hydrodynamical
vortex pictured above, it is interesting that for the bound s tate, there is really no
interaction in the particle physics sense. The interaction comes in because in the
above description we really identify a background Zero Poin t Field with the hy-
drodynamical flow (Cf.ref.[18] and also[32]). Interesting ly in a simulation involving
vortices, such an ”attraction” was noticed[33].
(ii) Monopoles
It is interesting that the above considerations lead to a cha racterization of the
elusive monopole. Infact a non commutative geometry can be a ssociated with a
powerful magnetic field[34], and specialising to the equati ons (11) we can show that
this fieldBsatisfies,
Bl2∼nhc
2e
which is the celebrated equation of the monopole.
(iii) Duality
A related concept, which one encounters also in String Theor y is Duality. Infact
the relation (11) leads to (Cf. also equation (17),
∆x∼¯h
∆p+α′∆p
¯h(18)
whereα′=l2, which in Quantum SuperStrings Theory ∼10−66. Witten has won-
dered about the basis of (18), but as we have seen, it is a conse quence of (11).
In String Theory this is an expression of the duality relatio n,
R→α′/R
This is symptomatic of the fact that we cannot go down to arbit rarily small space-
time intervals, below the Planck scale in this case (Cf.ref. [14]).
In the Quantum Mechanical Kerr-Newman Black Hole model of th e electron, on
the contrary, we are at Compton scale, and the effect of (18) is precisely that seen
in point 1 above: We go from the electric charge eto the monopole, as in the Olive-
Montonen duality[35], (Cf.also ref.[14]).
(iv) Spin
6One could argue that the non commutative relations (11) are a n expression of Quan-
tum Mechanical spin. To put it briefly, for a spinning particl e the non commutativity
arises when we go from canonical to covariant position varia bles. Zakrzewsk[36] has
shown that we have the Poisson bracket relation
{xj,xk}=1
m2Rjk,(c= 1),
whereRjkis the spin. The passage to Quantum Theory then leads us back t o the
relation (11).
Conversely it was shown that the relations (11) imply Quantu m Mechanical spin[5].
Another way of seeing this is to observe as noted in (13) that ( 11) implies that
y=αˆpy,whereαis a dimensional constant viz [ T/M] and ˆpyis the analogue of the
momentum, but with the Planck constant replaced by l2. So the spin is given by
|/vector r×/vector p| ≈2xpy∼2α−1l2=1
2/parenleftbigg¯h
m2c2/parenrightbigg−1
×h2
m2c2=¯h
2
as required.
(v) Extremal Black Holes
Going back to the relation (7), we can see that if
a=¯h
Mc∼GM
c2
then we are at the Planck scale and have a Planck mass Schwarzc hild Black Hole.
The purely Quantum Mechanical Compton length equals the cla ssical Schwarzchild
radius.
Also if,
Q∼Mc2, (19)
while at the same time the particlee has no spin, so that a= 0, we recover a
Schwarzchild Black Hole. We observe that if the mass M∼electron mass then the
chargeQfrom (19) turns out to be ∼1000e, as in the case of the monopole.
Interestingly these parameters also fit a neutrino, whose ma ss, as recent experiments
indicate is given by
m≤10−8me (20)
It was further argued[37] that a neutrino could in principle have an electric charge,
a millionth that of the electron, while, as the neutrino has n o Compton wavelength
we can apply in principle equation (19). (20) coupled with th is and with the above
electric charge shows that indeed equation (19) is satisfied .
Such particles however have a very high Bekenstein temperat ure
∼10−7/parenleftbiggM0
M/parenrightbigg
K,
M0being the solar mass and would disintegrate into gamma rays w ithin about
10−23M3secs. So these extremal Black Holes would not be detectable, but t his
7could nevertheless provide a rationale for the puzzling cos mic gamma ray emissions.
(vi) Spacetime
We have seen that the spacetime given by (11) is radically diff erent from its usual
description. Infact the usual spacetime is a sort of a statio nary spacetime, a low
energy approximation, as will be clear by the following argu ment. We start with
the Nelsonian theory in which there is a complex velocity pot entialV−ıU, due
to a double Weiner process. This has been shown to lead to the u sual Quantum
Mechanical description[38]. Indeed the diffusion equation ,
∆x·∆x=h
m∆t≡γ∆t
can also be written as
m∆x
∆t·∆x=h= ∆p·∆x
which is the usual Heisenberg description.
Using theWKB approximation, the Nelsonian wave function
ψ=√ρe(ı/¯h)s
becomes
(px)−1
2eı
¯h/integraltext
p(x)dx
whence
ρ=1
px(21)
In this case the condition U≈0 gives
v· ∇ln(√ρ)≈0
that is the probability densityh ρand hence from (21) the momentum varies very
slowly with x.
The continuity equation now gives
∂ρ
∂t+/vector∇(ρ/vector v) =∂ρ
∂t= 0
which shows that ρis independent of talso[18]. This is a scenario of, strictly speak-
ing, a single particle universe, without environmental effe cts, a scenario which is
an approximation valid for small incremental changes. (The more physical scenario
takes all the particles in the universe into account, leadin g to what may be called
stochastic holism[32]). In this case, we can take limits to v anishing spacetime inter-
vals, as in the usual theory (Cf. ’t Hooft loc. cit). Spacetim e in this description is a
differentiable manifold and instead of the relations (11), s pacetime is commutative.
Effectively we are neglecting l2. This has been the backbone of twentieth century
physics.
On the other hand according to Witten[39], ”String Theory is a part of twenty-first
century physics that fell by chance into the twentieth centu ry.” It does appear that
non commutative fuzzy spacetime is a paradigm for the twenty -first century.
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1960, p.364.
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10 |
arXiv:physics/0102079v1 [physics.atom-ph] 23 Feb 2001Operator Ordering in Quantum Radiative Processes
J. L. Tomazelli
Departamento de F´ ısica e Qu´ ımica, Faculdade de Engenhari a, Universidade Estadual Paulista,
Campus da Guaratinguet´ a, Av. Dr. Ariberto Pereira da Cunha 333, 12500-000
Guaratinguet´ a, SP, Brazil.
L. C. Costa
Instituto de F´ ısica Te´ orica, Universidade Estadual Paul ista,
01405-900, S˜ ao Paulo, Brazil.
Abstract
In this work we reexamine quantum electrodynamics of atomic eletrons in the
Coulomb gauge in the dipole approximation and calculate the shift of atomic
energy levels in the context of Dalibard, Dupont-Roc and Coh en-Tannoudji
(DDC) formalism by considering the variation rates of physi cal observables.
We then analyze the physical interpretation of the ordering of operators in
the dipole approximation interaction Hamiltonian in terms of field fluctua-
tions and self-reaction of atomic eletrons, discussing the arbitrariness in the
statistical functions in second order bound-state perturb ation theory.
PACS numbers: 32.80.-t, 31.15.Md, 42.50.-p
1I. INTRODUCTION
In radiative processes, the ordering problem of atomic and fi eld operators in the interac-
tion Hamiltonian of bound state QED has been raised since the works by Senitzki, Milloni
and others [1]. Behind this discussion is the physical inter pretation of atomic radiative ef-
fects such as the radiative line shifts in spontaneous emiss ion. Alternative approaches were
proposed in order to elucidate important issues concerning such problem. Among them are
those based on the complementarity between radiation react ion and vacuum fluctuation ef-
fects, which provide a conceptual basis for the physical int erpretation of different radiative
processes.
In the Dalibard, Dupont-Roc and Cohen-Tannoudji (DDC) form ulation, the ordering
between the operators of the electromagnetic field, conside red as a reservoir ( R), and a
microscopic atomic system ( S) play a fundamental role in the identification of the respect ive
contributions due to the reservoir fluctuation (fr) and the s elf-reaction (sr) [2] [3]. They
showed that the symmetric ordering gives a true physical mea ning to the (fr) and (sr) rates.
In this letter we explore the DDC construct in order to stabli sh a connection between
two distinct treatments, investigating the dependence of t he energy shifts on a more general
ordering, both in the effective Hamiltonian and in the densit y matrix approaches.
II. THE EFFECTIVE HAMILTONIAN FORMULATION
In the dipole approximation, the Hamiltonian of the global s ystem S+Ris given by
H=HS+HR+V, (1)
where HSis the Hamiltonian of the microscopic system S,HRthe Hamiltonian of the
reservoir RandVthe interaction between SandR, which we assume to be of the form
V=−gRS(gis the coupling constant and RandSare, respectively, Hermitian observables
ofRandS).
2The rate of variation for an arbitrary Hermitian observable GofSis given by the
Heisenberg equation of motion, and the contribution of the c oupling Vto this rate can be
writen as
/parenleftBiggdG
dt/parenrightBigg
coupling=−ig
¯h[R(t)S(t), G(t)] =gλN(t)R(t) +g(1−λ)R(t)N(t), (2)
where N(t) =−(i/¯h)[S(t), G(t)] is an Hermitian observable of the microscopic system and
λan arbitrary real number [3]. In the above equation we have us ed the freedom in the
ordering of R(t) and N(t), since they commute.
In order to obtain the contributions of reservoir fluctuatio n (rf) and self-reaction (sr) we
perform the following replacement
X(t) =Xf(t) +Xs(t), (3)
(X=R, S, G ) where Rf(resp. SfandGf) is the solution, to order 0 in g, of the Heisenberg
equation of motion for R(resp. SandG), corresponding to a free evolution between t0and
t, andRs(t) (resp. SsandGs) the solution to first order and higher in g. Then, substituting
(3) in (2) and retaining terms up to second order in g, we obtain
/parenleftBiggdG
dt/parenrightBiggrf
(t) =−ig
¯h{(1−λ)Rf(t)[Sf(t), Gf(t)] +λ[Sf(t), Gf(t)]Rf(t)} −
−g2
¯h2/integraldisplayt
t0dt′[Sf(t′),[Sf(t), Gf(t)]]×
×((1−λ)Rf(t′)Rf(t)) +λRf(t)Rf(t′)), (4)
/parenleftBiggdG
dt/parenrightBiggsr
(t) =−g2
¯h2/integraldisplayt
t0dt′[Rf(t′), Rf(t)]×
×((1−λ)Sf(t′)[Sf(t), Gf(t)] +λ[Sf(t), Gf(t)]Sf(t′)). (5)
Since the rates (4) and (5) contain only free operators, thei r average value in the reservoir
stateσRgives1
1Note that the term in the first line of (4) do not contribute to t he respective rate since it is linear
in the absorption and emission operators of the field.
3/angbracketleftBigg/parenleftBiggdG
dt/parenrightBiggrf
(t)/angbracketrightBigg(R)
=−g′2
¯h2/integraldisplayt
t0dt′C(R)(t, t′, λ) [Sf(t′),[Sf(t), Gf(t)]], (6)
/angbracketleftBigg/parenleftBiggdG
dt/parenrightBiggsr
(t)/angbracketrightBigg(R)
=−g′2
2¯h2/integraldisplayt
t0dt′χ(R)(t, t′)×
× {(1−λ)Sf(t′)[Sf(t), Gf(t)] +λ[Sf(t), Gf(t)]Sf(t′)}, (7)
where we have define g′=√
2gand
C(R)(t, t′, λ) =1
2TrR[σR{λRf(t)Rf(t′) + (1 −λ)Rf(t′)Rf(t)}], (8)
χ(R)(t, t′) =i
¯hTrR{σR[Rf(t′), Rf(t)]}θ(t−t′). (9)
The functions C(R)andχ(R)are statistical functions of the reservoir [5]. C(R)is a kind of
correlation function, describing the “dynamics of fluctuat ions” of Rin the stationary state
σR(t0);χ(R)is the linear susceptibility of the reservoir, determining the linear response of
the averaged observable /an}bracketle{tR(t)/an}bracketri}htwhen the reservoir is acted upon by a perturbation2.
In order to find the energy shifts corresponding to the (rf) an d (sr) rates we rewrite (6)
and (7) in a convenient form, namely
/angbracketleftBigg/parenleftBiggdG
dt/parenrightBiggrf
(t)/angbracketrightBigg(R)
=i
¯h/an}bracketle{t[(Heff(t))rf, G(t)]/an}bracketri}htR+ (10)
+/parenleftBigg−g′2
2¯h2/parenrightBigg/summationdisplay
i/an}bracketle{t[Yi(t, λ),[Si(t), G(t)]] + [Si(t),[Yi(t, λ), G(t)]]/an}bracketri}htR,
/angbracketleftBigg/parenleftBiggdG
dt/parenrightBiggsr
(t)/angbracketrightBigg(R)
=i
¯h/an}bracketle{t[(Heff(t))sr, G(t)]/an}bracketri}htR+
+/parenleftBigg−ig′2
4¯h2/parenrightBigg/summationdisplay
i/an}bracketle{t[Z′
i(t, λ)[Si(t), G(t)] + [Si(t), G(t)]Z′′
i(t, λ)−
−Si(t)[Z′′
i(t, λ), G(t)]−[Z′
i(t, λ), G(t)]Si(t)/an}bracketri}htR (11)
where
(Heff(t))rf=ig′
2¯h[Y(t, λ), S(t)], (12)
(Heff(t))sr=−g′
4[Z′(t, λ)S(t) +S(t)Z′′(t, λ)] (13)
2In (9) θis the Heaviside function, θ(x) = 1 if x >0,θ(x) = 0 if x <0.
4are second order corrections to the Hamiltonian part of Scaused by its interaction with the
reservoir and
Y(t, λ) =/summationdisplay
abqab(t)/an}bracketle{ta|S|b/an}bracketri}ht/integraldisplay∞
0dτ C(R)(τ, λ)e−iωabτ, (14)
Z′(t, λ) = (1 −λ)/summationdisplay
abqab(t)/an}bracketle{ta|S|b/an}bracketri}ht/integraldisplay∞
−∞dτ χ(R)(τ)e−iωabτ, (15)
Z′′(t, λ) =λ/summationdisplay
abqab(t)/an}bracketle{ta|S|b/an}bracketri}ht/integraldisplay∞
−∞dτ χ(R)(τ)e−iωabτ(16)
withqab≡ |a/an}bracketri}ht/an}bracketle{tb|,ωab= (Ea−Eb)/¯handτ=t−t′. Following the same point of view of
[3], expression (12) (resp. (13)) describes the part of the e volution due to reservoir fluctu-
ations (resp. due to self-reaction) and which can be describ ed by an effective Hamiltonian.
The second line of expression (10) (resp. (11)) describes th e non-Hamiltonian part of the
evolution of Gcaused by the reservoir fluctuation (resp. self reaction).
A. The Energy Shifts: Hamiltonian Part
Corrections (12) and (13) to the Hamiltonian HSaffect Sthrough a shifting in its
energy eingenstates. Hence, considering a state |a/an}bracketri}ht(which is an eigenstate of HS) we have
the following energy shifts
(δEa)rf=/an}bracketle{ta|(Heff(t0))rf|a/an}bracketri}ht, (17)
(δEa)sr=/an}bracketle{ta|(Heff(t0))sr|a/an}bracketri}ht. (18)
Using expression (12), and noting that
Y(t0) =/integraldisplay∞
0C(R)(τ, λ)Sf(t0−τ)dτ, (19)
expression (17) for ( δEa)rfbecomes
(δEa)rf=−g′2
2/integraldisplay+∞
−∞C(R)(τ, λ)χ(S,a)(τ)dτ, (20)
where we have introduced a new statistical function, the sus ceptibility of the system obser-
vables
5χ(S,a)(τ) =i
¯h/an}bracketle{ta|[Sf(t0), Sf(t0−τ)]|a/an}bracketri}htθ(τ). (21)
From expression (13) for ( Heff)sr, we can follow the same steps as those from (17) to (20).
As a result we obtain
(δEa)sr=−g′2
2/integraldisplay+∞
−∞χ(R)(τ)C(S,a)(τ, λ)dτ, (22)
where, again, we have introduced a new statistical function , the “correlation” for the system
observables
C(S,a)(τ, λ) =1
2/an}bracketle{ta|λSf(t0)Sf(t0−τ) + (1 −λ)Sf(t0−τ)Sf(t0)|a/an}bracketri}ht. (23)
For future convenience we write (20) and (22) in the frequenc e space. Using the Parseval’s
theorem we have
(δEa)rf=−g′2
2/integraldisplay+∞
−∞C(R)(ω, λ)χ(S,a)(ω)dω, (24)
(δEa)sr=−g′2
2/integraldisplay+∞
−∞χ(R)(ω)C(S,a)(ω, λ)dω, (25)
where we have used the parity properties of Candχ[3] [4].
Formulas (24) and (25) give us the energy shifts which, a priori , depends on λthrough
the “correlation functions”, expressions (8) and (23). In t he next section we use this result
to make a connection between the previous approach and that a dopted in [4], where the
density matrix formulation is employed.
III. THE DENSITY MATRIX FORMULATION
The same energy shifts given by (24) and (25) can also be obtai ned using a matrix
approach based on the evolution equation for the density ope rator of the global system
S+Rin the interaction picture with respect to HS+HR. Following [4], the energy shift
for a state |a/an}bracketri}htofScaused by its interaction with Rthrough Vis
6∆a=1
¯hP/summationdisplay
µ,νpµ/summationdisplay
b|/an}bracketle{tν, b|V|µ, a/an}bracketri}ht|2
Eµ+Ea−Eν−Eb(26)
where pµis a distribution of probability corresponding to the reser voir average in the sta-
tionary state σRand|µ/an}bracketri}ht,|ν/an}bracketri}hteigenstates of HRwith eigenvalue Eµ,Eν. In (26) Pdenotes
the principal value.
From (26) we can factorize the matrix element /an}bracketle{tµ, a|V|ν, b/an}bracketri}htin two parts, one relative to
Sand another relative to R,
∆a=g′2
2¯h2/summationdisplay
µ,νpµ|/an}bracketle{tµ|R|ν/an}bracketri}ht|2/bracketleftBigg/summationdisplay
b|/an}bracketle{ta|S|b/an}bracketri}ht|2P1
ωµν+ωab/bracketrightBigg
. (27)
In this way, since we know the functional structure of C(ω) and χ(ω), namely
C(R)(ω) =/summationdisplay
µ,νpµπ|/an}bracketle{tµ|R|ν/an}bracketri}ht|2[δ(ω+ωµν) +δ(ω−ωµν)], (28)
χ(R)(ω) =χ′(R)(ω) +iχ′′(R)(ω), (29)
χ′(R)(ω) =−1
¯h/summationdisplay
µ,νpµ|/an}bracketle{tµ|R|ν/an}bracketri}ht|2/bracketleftBigg
P1
ωµν+ω+P1
ωµν−ω/bracketrightBigg
, (30)
χ′′(R)(ω) =π
¯h/summationdisplay
µ,νpµ|/an}bracketle{tµ|R|ν/an}bracketri}ht|2[δ(ωµν+ω)−δ(ωµν−ω)], (31)
and analogous expressions for S(where only pa= 1 is nonzero), we can make a mathematical
trick and rewrite the fraction 1 /(ωµν+ωab) as
P1
ωµν+ωab=1
2/integraldisplay
dω×
×/braceleftBigg/parenleftBigg
P1
ωµν+ω+P1
ωµν−ω/parenrightBigg
[λδ(ω+ωab) + (1 −λ)δ(ω−ωab)]+
+/parenleftbigg
P1
ωab+ω+P1
ωab−ω/parenrightbigg
[λδ(ω+ωµν) + (1 −λ)δ(ω−ωµν)]/bracerightbigg
. (32)
In the above identity we have already introduced the paramet erλof last section in order
to stablish a formal connection with the previous approach. Substituting (32) into (27) we
obtain: ∆ a= ∆rf
a+ ∆sr
a, where
¯h∆rf
a=−g′2
2/integraldisplay∞
−∞C(R)(ω, λ)χ(S,a)(ω)dω, (33)
¯h∆sr
a=−g′2
2/integraldisplay∞
−∞χ(R)(ω)C(S,a)(ω, λ)dω. (34)
7In the original formulation, given in [4], the physical mean ing of the above expressions
was simple and clear in terms of (fr) and (sr) effects. However , since the Hermicity of
expressions (33) and (34) is lost due to the λ’s appearence in the correlations, we don’t have
such simple interpretation. But, it must be noted that if we c hoose λ= 1/2 the original
results are recovered. In addition, it can be show that despi te the λ’s presence in C, its
effect on ∆ ais null [7].
IV. CONCLUDING REMARKS
In this work we have applied to the original formulation of DD C construct a more general
ordering between the atomic and electromagnetic field opera tors and calculate the energy
shift due to the effective Hamiltonian part. The result showe d that the freedom in ordering
expression (2) reflects in the energy shifts (24) and (25) thr ough the λ’s appearance in the
correlation functions. Such dependence enables us to stabl ish a formal connection with the
density matrix formulation, where, instead of an arbitrary ordering of operators, we have
made use of a simple mathematical identity.
It must be also noted that our procedure still permit us to fix a posteriori a suitable
ordering which keeps its (rf) and (sr) interpretation, as ca n be seen by looking directly to
expressions (33) and (34). Further, it can be shown that for a practical case (the Lamb shift
and the AC Stark effect) the energy shifts (24) and (25), or (33 ) and (34), give the same
contribuction, independent of the ordering we choose [7].
Once we get a better understanding on the arbitrariness in th e operator ordering in
DDC construct, we expect to find a direct connection with the w orks by Senitzki, Milloni
and others. The main ideia is constructing a similar structu re in the Fock space.
Another interesting application of the present formalism i s a possible generalization of
the operator ordering in the spirit of q-deformed operator a lgebras, subject of a forthcoming
work.
8V. ACKNOWLEDGEMENTS
JLT thanks CNPq for partial financial support and the IFT/UNE SP for the hospitality.
LCC is grateful to FAPESP for the financial support.
9REFERENCES
[1] I. R. Senitzky, Phys. Rev. Lett. 31(1973) 955; J. R. Ackerhalt, P. L. Knight and J. H.
Eberly, Phys. Rev. Lett. 30(1973) 456; P. W. Milonni and W. A. Smith, Phys. Rev. A
11(1975) 814;
[2] J. Dalibard, J. Dupont-Roc and C. Cohen-Tannoudji, J. de Physique 43(1982) 1617;
[3] J. Dalibard, J. Dupont-Roc and C. Cohen-Tannoudji, J. de Physique 45(1984) 637;
[4] C. Cohen-Tannoudji, J. Dupont-Roc and G. Grynberg, “Ato m-Photon Interactions -
Basic Processes and Applications”, J. Wiley, NY (1998);
[5] P. Martin, “Many Body Physics”, les Houches 1967, edited by C. de Witt and R. Balian,
Gordon and Breach, NY (1968), p39;
[6] B. Duplantier, Th´ ese 3ecycle, (1978) Paris (unpublished). Expressions of this typ e can
also be found in the context of QED in K. Huang, Phys. Rev. 101(1956) 1173;
[7] L. C. Costa, master thesis, IFT-D.007/00, IFT/UNESP, S˜ ao Paulo, (2000) (unpublished).
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