text stringlengths 30 4k | source stringlengths 60 201 |
|---|---|
version available at http://www.math.ucsd.edu/~fan/
wp/localpartfull.pdf. 8-2, 8-3
[2] Gershgorin circle theorem. http://en.wikipedia.org/wiki/Gershgorin_circle_theorem 8-3
[3] Nathan Linial and Avi Wigderson. Expander Graphs And Their Applications. http://www.math.ias.edu/~boaz/
ExpanderCourse/ 8-1
[4] L´aszl´o L... | https://ocw.mit.edu/courses/18-409-topics-in-theoretical-computer-science-an-algorithmists-toolkit-fall-2009/5823d396bfe2ae2c5bea9c431601276e_MIT18_409F09_scribe8.pdf |
MIT 2.852
Manufacturing Systems Analysis
Lecture 14-16
Line Optimization
Stanley B. Gershwin
Spring, 2007
Copyright c�2007 Stanley B. Gershwin.
Line Design
• Given a process, find the best set of machines and buffers on
which it can be implemented.
• Best: least capital cost; least operating cost; least average ... | https://ocw.mit.edu/courses/2-852-manufacturing-systems-analysis-spring-2010/5843df20ba98a98f34469fc47801c1e6_MIT2_852S10_line_opt.pdf |
is generally focused on limiting the number of
iterations by proposing designs efficiently.
• The number of iterations is also limited by choosing a reasonable
termination criterion (ie, required accuracy).
• Reducing computation time per iteration is accomplished by
� using analytical models rather than simulations... | https://ocw.mit.edu/courses/2-852-manufacturing-systems-analysis-spring-2010/5843df20ba98a98f34469fc47801c1e6_MIT2_852S10_line_opt.pdf |
when F (t) is differentiable, and f (t) = dF (t)/dt is
continuous.
• If f (t) is differentiable, maximization or minimization
depends on the sign of d2F (t)/dt2.
Copyright c
�2007 Stanley B. Gershwin.
9
Continuous
Variables and
Objective
Assume f (t) is decreasing.
• Binary search: Guess t0 and
t1 such that f (... | https://ocw.mit.edu/courses/2-852-manufacturing-systems-analysis-spring-2010/5843df20ba98a98f34469fc47801c1e6_MIT2_852S10_line_opt.pdf |
140625
2.00006103515625
2.00006103515625
2.00001525878906
2.00001525878906
2.00000381469727
2.00000381469727
Copyright �2007 Stanley B. Gershwin.
c
11
Continuous
Variables and
Objective
Unconstrained
One-dimensional search
f(t)
• Newton search, exact tangent:
f(t0 )
� Guess t0. Calculate
df (t0)/dt.
� Choose... | https://ocw.mit.edu/courses/2-852-manufacturing-systems-analysis-spring-2010/5843df20ba98a98f34469fc47801c1e6_MIT2_852S10_line_opt.pdf |
0) + (t2 − t0)s = 0.
f(t0 )
f(t2 )
f(t1 )
Unconstrained
One-dimensional search
f(t)
t0
t2
t1
t
� Repeat with t�
0 = t1 and
t� = t2 until |f (t� )| is small
1
enough.
0
Copyright �2007 Stanley B. Gershwin.
c
14
Continuous
Variables and
Objective
Unconstrained
One-dimensional search
Example:
f (t) =... | https://ocw.mit.edu/courses/2-852-manufacturing-systems-analysis-spring-2010/5843df20ba98a98f34469fc47801c1e6_MIT2_852S10_line_opt.pdf |
one-dimensional search ) t� , the value of t that
maximizes Jn(t).
�
n
(xn)
�J
�x
3. Set xn+1 = xn + t� �J (xn).
4. Set n � n + 1. Go to Step 1.
n �x
� also called steepest ascent or steepest descent .
Copyright c
�2007 Stanley B. Gershwin.
17
Continuous
Variables and
Objective
J
Constrained
Constrained ... | https://ocw.mit.edu/courses/2-852-manufacturing-systems-analysis-spring-2010/5843df20ba98a98f34469fc47801c1e6_MIT2_852S10_line_opt.pdf |
x + y ≈ 0
−6
−4
−2
0
2
4
6
−2
0
2
4
6
Solving a linearly-constrained problem is relatively easy. If the
solution is not in the interior, search within the boundary plane.
Copyright �2007 Stanley B. Gershwin.
c
20
Continuous
Variables and
Objective
Constrained
8*(x+y)−.25*(x**4+y**4)
20
16
12
8
4
... | https://ocw.mit.edu/courses/2-852-manufacturing-systems-analysis-spring-2010/5843df20ba98a98f34469fc47801c1e6_MIT2_852S10_line_opt.pdf |
(x)
subject to j(x) ≈ J
Primals and Duals
max j(x)
subject to f (x) � F
f (x), F , j(x), and J are scalars. We will call these problems duals of one
another. (However, this is not the conventional use of the term.) Under certain
conditions when the last inequalities are effective, the same x satisfies both
problem... | https://ocw.mit.edu/courses/2-852-manufacturing-systems-analysis-spring-2010/5843df20ba98a98f34469fc47801c1e6_MIT2_852S10_line_opt.pdf |
The problem is:
Primal problem:
Minimize
k−1
�
i=1
Ni
subject to
P (N1, ..., Nk−1) ≈ P �
Ni ≈ N MIN, i = 1, ..., k − 1.
In the following, we treat the Nis like a set of continuous variables.
Copyright �2007 Stanley B. Gershwin.
c
27
Buffer Space
Allocation
Properties of P (N1, ..., Nk−1)
P (∗, ..., ∗) = min ... | https://ocw.mit.edu/courses/2-852-manufacturing-systems-analysis-spring-2010/5843df20ba98a98f34469fc47801c1e6_MIT2_852S10_line_opt.pdf |
P=0.8975
P=0.9000
200
150
2
N
100
100 50
90
80
70
60
N2
50
40
0
0
50
100
N1
150
200
r1 = .35
p1 = .037
e1 = .904
r2 = .15
p2 = .015
e2 = .909
r3 = .4
p3 = .02
e3 = .952
Copyright �2007 Stanley B. Gershwin.
c
29
Buffer Space
Allocation
Minimize
k−1
�
i=1
Ni
Solution
Primal problem
subjec... | https://ocw.mit.edu/courses/2-852-manufacturing-systems-analysis-spring-2010/5843df20ba98a98f34469fc47801c1e6_MIT2_852S10_line_opt.pdf |
is feasible). (1. Why? 2. When would the
problem be infeasible?) This problem is consequently relatively easy to solve.
Copyright �2007 Stanley B. Gershwin.
c
31
Buffer Space
Allocation
N 2
N 1 +N2 +N3 = N
TOTAL
Solution
Dual problem
Constraint set
(if N MIN = 0).
N 3
N 1
Copyright �2007 Stanley B. Gershwi... | https://ocw.mit.edu/courses/2-852-manufacturing-systems-analysis-spring-2010/5843df20ba98a98f34469fc47801c1e6_MIT2_852S10_line_opt.pdf |
B. Gershwin.
c
34
Buffer Space
Allocation
Solution
Dual Algorithm
• The projected gradient gˆ satisfies
k−1
k−1
k−1
gˆi =
(gi − g¯) =
gi − (k − 1)¯g = 0
�
i=1
�
i=1
�
i=1
• Therefore, if A is a scalar, then
k−1
k−1
k−1
k−1
(Ni + Agˆi) = Ni +
Agˆi = Ni
�
�
�
�
i=1
i=1
i=1
i=1
i=1 Ni = N TOTAL ... | https://ocw.mit.edu/courses/2-852-manufacturing-systems-analysis-spring-2010/5843df20ba98a98f34469fc47801c1e6_MIT2_852S10_line_opt.pdf |
+ A�gˆ is the next guess for N , and the process repeats.
Copyright c
�2007 Stanley B. Gershwin.
36
Buffer Space
Allocation
Solution
Dual Algorithm
Specify initial guess
(N , ..., N )
N =
and search parameters.
k-1
1
Calculate gradient
g.
Calculate search
direction
p.
(Here, p = gˆ.)
A
Find
such that
i... | https://ocw.mit.edu/courses/2-852-manufacturing-systems-analysis-spring-2010/5843df20ba98a98f34469fc47801c1e6_MIT2_852S10_line_opt.pdf |
d
o
r
p
e
g
a
r
e
v
a
m
u
m
x
a
M
i
0.92
0.9
0.88
0.86
0.84
0.82
0.8
0.78
0
50
100
Total buffer space
150
200
P MAX(N TOTAL) as a function of N TOTAL.
Copyright �2007 Stanley B. Gershwin.
c
40
Buffer Space
Allocation
Solution
Primal algorithm
Then, we can find, by 1-dimensional search, N TOT... | https://ocw.mit.edu/courses/2-852-manufacturing-systems-analysis-spring-2010/5843df20ba98a98f34469fc47801c1e6_MIT2_852S10_line_opt.pdf |
First, we show the
average WIP
distribution if all
buffers are the
same size:
Ni = 53,
i = 1, ..., 19
50
40
30
20
10
l
e
v
e
l
r
e
f
f
u
b
e
g
a
r
e
v
A
0
0
2
4
6
8
10
Buffer
12
14
16
18
20
Copyright �2007 Stanley B. Gershwin.
c
43
Buffer Space
Allocation
Example
The “Bowl Phenomena” ... | https://ocw.mit.edu/courses/2-852-manufacturing-systems-analysis-spring-2010/5843df20ba98a98f34469fc47801c1e6_MIT2_852S10_line_opt.pdf |
0.6
0.4
0.2
e
z
S
i
r
e
f
f
u
B
/
l
e
v
e
l
r
e
f
f
u
b
e
g
a
r
e
v
A
=
o
i
t
a
R
0
0
5
10
Buffer
15
20
Copyright �2007 Stanley B. Gershwin.
c
46
Buffer Space
Allocation
Example
• Design the buffers for a 20-machine production line.
• The machines have been selected, and the only
decision rema... | https://ocw.mit.edu/courses/2-852-manufacturing-systems-analysis-spring-2010/5843df20ba98a98f34469fc47801c1e6_MIT2_852S10_line_opt.pdf |
50
Buffer Space
Allocation
Example
• Case 1 MTTF= 200 minutes and MTTR = 10.5
minutes for all machines (P = .95 parts per minute).
• Case 2 Like Case 1 except Machine 5. For
Machine 5, MTTF = 100 and MTTR = 10.5 minutes
(P = .905 parts per minute).
• Case 3 Like Case 1 except Machine 5. For
Machine 5, MTTF = 20... | https://ocw.mit.edu/courses/2-852-manufacturing-systems-analysis-spring-2010/5843df20ba98a98f34469fc47801c1e6_MIT2_852S10_line_opt.pdf |
Allocation
Example
• This shows the optimal distribution of buffer space and the resulting
distribution of average inventory for Case 3.
l
e
v
e
l
r
e
f
f
u
b
e
g
a
r
e
v
A
/
e
z
S
i
r
e
f
f
u
B
55
50
45
40
35
30
25
20
15
10
5
0
0
5
10
Buffer
15
20
Copyright �2007 Stanley B. Gershwin.
c
54
... | https://ocw.mit.edu/courses/2-852-manufacturing-systems-analysis-spring-2010/5843df20ba98a98f34469fc47801c1e6_MIT2_852S10_line_opt.pdf |
r
e
f
f
u
B
55
50
45
40
35
30
25
20
15
10
5
0
0
5
10
Buffer
15
20
Copyright �2007 Stanley B. Gershwin.
c
56
Buffer Space
Allocation
Example
• This shows the ratio of average inventory to buffer size with optimal buffers
for Case 4.
0.75
0.7
0.65
0.6
0.55
0.5
0.45
0.4
0.35
e
z
S
i... | https://ocw.mit.edu/courses/2-852-manufacturing-systems-analysis-spring-2010/5843df20ba98a98f34469fc47801c1e6_MIT2_852S10_line_opt.pdf |
10
Buffer
15
20
Copyright �2007 Stanley B. Gershwin.
c
58
Buffer Space
Allocation
Example
• This shows the ratio of average inventory to buffer size with optimal buffers
for Case 5.
e
z
S
i
r
e
f
f
u
B
/
l
e
v
e
l
r
e
f
f
u
b
e
g
a
r
e
v
A
=
o
i
t
a
R
0.7
0.65
0.6
0.55
0.5
0.45
0.4
0.35
0.3
0
... | https://ocw.mit.edu/courses/2-852-manufacturing-systems-analysis-spring-2010/5843df20ba98a98f34469fc47801c1e6_MIT2_852S10_line_opt.pdf |
z
S
i
r
e
f
f
u
B
/
l
e
v
e
l
r
e
f
f
u
b
e
g
a
r
e
v
A
=
o
i
t
a
R
0.7
0.65
0.6
0.55
0.5
0.45
0.4
0.35
0.3
0
10
20
30
40
50
Buffer
Copyright �2007 Stanley B. Gershwin.
c
61
Buffer Space
Allocation
Example
• Observation from studying buffer space allocation
problems:
� Buffer space is ... | https://ocw.mit.edu/courses/2-852-manufacturing-systems-analysis-spring-2010/5843df20ba98a98f34469fc47801c1e6_MIT2_852S10_line_opt.pdf |
Lecture 5
8.821/8.871 Holographic duality
Fall 2014
8.821/8.871 Holographic duality
MIT OpenCourseWare Lecture Notes
Hong Liu, Fall 2014
Lecture 5
Reminder from last lecture
The vacuum of Minkowski space can be viewed as an entangled state of left Rindler patch and right Rindler patch
|0(cid:105)M ∝
(cid:88)
n
e−πEn |n... | https://ocw.mit.edu/courses/8-821-string-theory-and-holographic-duality-fall-2014/584b62f85f319bd167efb65047336e0c_MIT8_821S15_Lec5.pdf |
5)M is invariant under HRind − HRind (here we have a minus sign, because the time flows oppositely in the
(R)
(L)
left patch).
(R)
eiη(HRind− (L)H
)
Rind |0(cid:105)M = |0(cid:105)M
This can also be seen geometrically: η translation is a boost in (X, T ), i.e. HRind generates a boost. |0(cid:105)M is
clearly invariant u... | https://ocw.mit.edu/courses/8-821-string-theory-and-holographic-duality-fall-2014/584b62f85f319bd167efb65047336e0c_MIT8_821S15_Lec5.pdf |
king vacuum (left) can be expressed as an path integral over the lower half plane of the
Euclidean contention of the black hole spacetime (right).
1.2.4: BLACK HOLE THERMODYNAMICS
From the previous discussion, we know that a black hole has a temperature:
TBH =
(cid:126)
8πGN m
Thus a black hole is a thermodynamic objec... | https://ocw.mit.edu/courses/8-821-string-theory-and-holographic-duality-fall-2014/584b62f85f319bd167efb65047336e0c_MIT8_821S15_Lec5.pdf |
stars have lost (classically).
Now we summarize four laws of black hole mechanics:
• 0th law: surface gravity K is constant over the horizon.
• 1st law:
dM =
K
8πGN
dA + ΩdJ + ΦdQ
where Ω is the angular frequency at the horizon, Φ is the electric potential at the horizon (assume that at ∞
the potential is 0).
• 2nd law... | https://ocw.mit.edu/courses/8-821-string-theory-and-holographic-duality-fall-2014/584b62f85f319bd167efb65047336e0c_MIT8_821S15_Lec5.pdf |
is thermal for m (cid:29) mp, so before m ∼ O(mp), very little information about the
original state can come out. Once m ∼ O(mp), it will be too late for all the information to go out. Then we start
from a pure state and eventually get into a thermal state with density matrix description, i.e. information is lost!
3
M... | https://ocw.mit.edu/courses/8-821-string-theory-and-holographic-duality-fall-2014/584b62f85f319bd167efb65047336e0c_MIT8_821S15_Lec5.pdf |
Lecture #2: Background
READINGS AND FIGURES
Readings: Instructor notes on boundary conditions
Frisk, Chapter 3, sections 3.1, 3.2 and 3.3 (up to Eq. 3.38.)
---------------------------------------------------------
DESCRIPTION OF NOTES
Having the basic wave equation for ocean acoustics in hand, we now need to put ... | https://ocw.mit.edu/courses/2-682-acoustical-oceanography-spring-2012/58540cfbbdd42be15697f7a1523f343b_MIT2_682S12_bglec02.pdf |
the
(pressure) reflection and transmission coefficients, which are also useful, and can be used as boundary
conditions for some calculations. The discussion in this section of notes loosely follows from Clay and
Medwin’s “Acoustical Oceanography.” (Reference in separate attachment.)
The next small class notes secti... | https://ocw.mit.edu/courses/2-682-acoustical-oceanography-spring-2012/58540cfbbdd42be15697f7a1523f343b_MIT2_682S12_bglec02.pdf |
a rather crude, limiting case approximation to the water-
sediment or sediment-basement BC’s, but it is also an incredibly useful one for approximate calculations
due to its simplicity.
One can also pose a boundary condition that is “intermediate” to the two above, called a
“mixed” or impedance BC. Frisk shows this ... | https://ocw.mit.edu/courses/2-682-acoustical-oceanography-spring-2012/58540cfbbdd42be15697f7a1523f343b_MIT2_682S12_bglec02.pdf |
Lecture #1: Background
READINGS AND FIGURES
Readings: Frisk, Chapter 1, sections 1.3 and 1.4; Chapter 2, sections 2.1, 2.2, 2.3, 2.4.
Figures used: COA, Figures 1.1, 1.2, 1.3, 1.5
---------------------------------------------------------
DESCRIPTION OF NOTES
The first two pages of the notes are my own answers to ... | https://ocw.mit.edu/courses/2-682-acoustical-oceanography-spring-2012/58aeef5d8bd077782924c431bad2f2db_MIT2_682S12_bglec01.pdf |
large array of analysis techniques that have
been developed in the context of quantum mechanics, such as WKB theory, mode theory, scattering
theory, and so on. The analogies to quantum mechanical systems are very close in many ways, and many
researchers have exploited this.
The next few pages are my own description... | https://ocw.mit.edu/courses/2-682-acoustical-oceanography-spring-2012/58aeef5d8bd077782924c431bad2f2db_MIT2_682S12_bglec01.pdf |
ally symmetric. The 2D wave equation in
cylindrical coordinates is actually one that has the most immediate uses in ocean acoustics. Section 2.4
discusses the radial part of the 2D solution in cylindrical coordinates, which just turns out to be the
familiar Bessel’s equation. The exact and asymptotic forms of the so... | https://ocw.mit.edu/courses/2-682-acoustical-oceanography-spring-2012/58aeef5d8bd077782924c431bad2f2db_MIT2_682S12_bglec01.pdf |
MIT 3.016 Fall 2005
c
� W.C Carter
Lecture 3
16
Sept. 12 2005: Lecture 3:
Introduction to Mathematica II
Functions and Rules
Besides Mathematica r� ’s large set of builtin mathematical and graphics functions, the
most powerful aspects of Mathematica r� are its ability to recognize and replace patterns
and to ... | https://ocw.mit.edu/courses/3-016-mathematics-for-materials-scientists-and-engineers-fall-2005/58b2b2f2478ab6e14599e1865b1e233e_lecture_03.pdf |
in this case 5) is has no useful meaning anymore. If you had defined a
symbol such as x = 2i previously, then now x would have the value of 10, which is probably
not what was intended. It is much safer to localize variables—in other words, to limit the
scope of their visibility to only those parts of the program that... | https://ocw.mit.edu/courses/3-016-mathematics-for-materials-scientists-and-engineers-fall-2005/58b2b2f2478ab6e14599e1865b1e233e_lecture_03.pdf |
as SomeVariableName , and
then you can refer to what ever pattern matched it with the name SomeVariableName.
This is a bit abstract and probably difficult to understand without the aid of a few examples:
MIT 3.016 Fall 2005
c
� W.C Carter
Lecture 3
19
Mathematica r� Example: Lecture03
Patterns and Replacement
... | https://ocw.mit.edu/courses/3-016-mathematics-for-materials-scientists-and-engineers-fall-2005/58b2b2f2478ab6e14599e1865b1e233e_lecture_03.pdf |
to do something that might be unreasonable. No one would buy a calculator that would try
to return a very big number when division by zero occurs—or would give a real result when
the arccosine of 1.1 is demanded. Functions should probably be defined so that they can be
reused, either by you or someone else. The cond... | https://ocw.mit.edu/courses/3-016-mathematics-for-materials-scientists-and-engineers-fall-2005/58b2b2f2478ab6e14599e1865b1e233e_lecture_03.pdf |
6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007
Lecture 5-1
Lecture 5 - Carrier generation and
recombination (cont.)
February 14, 2007
Contents:
1. G&R rates outside thermal equilibrium (cont.)
2. Dynamics of excess carriers in uniform situations
3. Surface generation and recombination
Reading assig... | https://ocw.mit.edu/courses/6-720j-integrated-microelectronic-devices-spring-2007/58cdd6312da243d9062e56ceee80f955_lecture5.pdf |
equilibrium (cont.)
c) Trap-assisted thermal G&R
Ec
Et
Ev
no
ro,ec=ro,ee
ro,hc=ro,he
po
Ec
Et
Ev
n>no
ro,ec>ro,ee
ro,hc>ro,he
p>po
thermal equilibrium
with excess carriers
Out of equilibrium, if rate constants are not affected:
rec = cen(Nt − nt)
ree = eent = cenint
rhc = chpnt
rhe = eh(Nt − nt) = c... | https://ocw.mit.edu/courses/6-720j-integrated-microelectronic-devices-spring-2007/58cdd6312da243d9062e56ceee80f955_lecture5.pdf |
3)
n = no + n
(cid:3)
p = po + p
LLI: Equilibrium minority carrier concentration overwhelmed but
majority carrier concentration negligibly disturbed
thermal
equilibrium
low-level
injection
high-level
injection
po
ni
no
log n' p'
- for n-type:
- for p-type:
po (cid:4) n (cid:3) (cid:5) p (cid:3) (cid:4... | https://ocw.mit.edu/courses/6-720j-integrated-microelectronic-devices-spring-2007/58cdd6312da243d9062e56ceee80f955_lecture5.pdf |
ite as: Jesús del Alamo, course materials for 6.720J Integrated Microelectronic Devices, Spring 2007.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007
Lecture 5-7
If all G&R processes are i... | https://ocw.mit.edu/courses/6-720j-integrated-microelectronic-devices-spring-2007/58cdd6312da243d9062e56ceee80f955_lecture5.pdf |
as: Jesús del Alamo, course materials for 6.720J Integrated Microelectronic Devices, Spring 2007.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007
Lecture 5-9
Trap recombination (n-type mat... | https://ocw.mit.edu/courses/6-720j-integrated-microelectronic-devices-spring-2007/58cdd6312da243d9062e56ceee80f955_lecture5.pdf |
f
i
l
r
e
i
r
r
a
c
p-Si
T=300 K
1E-8
1/τ=7.8x10-13 ND+1.8x10-31 ND
2
1E-8
1/τ=3.5x10-13 NA+9.5x10-32 NA
2
1E-9
1E-10
1E-9
1E-10
1E+13 1E+14 1E+15 1E+16 1E+17 1E+18 1E+19 1E+20 1E+21
1E+13 1E+14 1E+15 1E+16 1E+17 1E+18 1E+19 1E+20 1E+21
donor concentration (cm-3)
acceptor concentration (cm-3)
For low dopi... | https://ocw.mit.edu/courses/6-720j-integrated-microelectronic-devices-spring-2007/58cdd6312da243d9062e56ceee80f955_lecture5.pdf |
6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007
Lecture 5-11
2. Dynamics of excess carriers in uniform situations
Consider:
• extrinsic uniformly doped semiconductor
• no surfaces nearby
In thermal equilibrium:
n = no
p = po
Go − Ro = 0
no
Ec
Ev
Go
Ro
po
thermal equilibrium
Cite as: Jesús ... | https://ocw.mit.edu/courses/6-720j-integrated-microelectronic-devices-spring-2007/58cdd6312da243d9062e56ceee80f955_lecture5.pdf |
dn
dt
=
dp
dt
= Gext − U
• if Gext > U ⇒ n, p ↑
• if Gext < U ⇒ n, p ↓
Under LLI:
Also:
Then:
U (cid:5)
(cid:3)
n
τ
dn
dt
=
dn(cid:3)
dt
dn(cid:3)
dt
= Gext −
n(cid:3)
τ
Homogeneous solution (Gext = 0) is: e−t/τ
Cite as: Jesús del Alamo, course materials for 6.720J Integrated Microelectronic Devi... | https://ocw.mit.edu/courses/6-720j-integrated-microelectronic-devices-spring-2007/58cdd6312da243d9062e56ceee80f955_lecture5.pdf |
Turn-off transient
Gext
gl
0
n'(t)
τ
gl
0
t=0
t=0
τ
t
t
n (cid:3)(t) = glτ e−t/τ
for t ≥ 0
Technique to measure τ :
Reprinted with permission from Dziewior, J., and W. Schmid. "Auger Coefficients for Highly Doped and Highly Excited Silicon."
Applied Physics Letters 31, no. 5 (1977): 346-348. Copyright 1977... | https://ocw.mit.edu/courses/6-720j-integrated-microelectronic-devices-spring-2007/58cdd6312da243d9062e56ceee80f955_lecture5.pdf |
glT
τ1>>T
τ2
>>T
τ3<<T
0
T
t
τ3
gl
0
• If τ1 (cid:7) T , pulse too short for final value of n(cid:3) to be reached:
n (cid:3)(t) (cid:5) glt
for 0 ≤ t ≤ T
• If τ3 (cid:4) T , final value of n(cid:3) achieved quickly:
n (cid:3)(t) (cid:5) glτ3
for 0 ≤ t ≤ T
shape of n(cid:3)(t) similar to shape of light puls... | https://ocw.mit.edu/courses/6-720j-integrated-microelectronic-devices-spring-2007/58cdd6312da243d9062e56ceee80f955_lecture5.pdf |
2007.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007
Lecture 5-19
3. Surface generation and recombination
Surface: severe disruption of periodic crystal ⇒ lots of traps (G&R
centers)
lot... | https://ocw.mit.edu/courses/6-720j-integrated-microelectronic-devices-spring-2007/58cdd6312da243d9062e56ceee80f955_lecture5.pdf |
trap-assisted recombination),
– τ ∼ N −2 for high N (Auger recombination).
• Order of magnitude of key parameters for Si at 300K:
– τ ∼ 1 ns − 1 ms, depending on doping
Cite as: Jesús del Alamo, course materials for 6.720J Integrated Microelectronic Devices, Spring 2007.
MIT OpenCourseWare (http://ocw.mit.edu/), M... | https://ocw.mit.edu/courses/6-720j-integrated-microelectronic-devices-spring-2007/58cdd6312da243d9062e56ceee80f955_lecture5.pdf |
MIT OpenCourseWare
http://ocw.mit.edu
(cid:10) 6.642 Continuum Electromechanics
Fall 2008
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
(cid:13)
6.642, Continuum Electromechanics, Fall 2004
Prof. Markus Zahn
Lecture 8: Electrohydrodynamic and Ferrohydrodynamic... | https://ocw.mit.edu/courses/6-642-continuum-electromechanics-fall-2008/5938069f486691fdd2376f96b52951ee_lec08_f08.pdf |
⎜
⎜
⎝
+
2
∂ ξ
2
z
∂
⎞
⎟
⎟
⎠
i
= ⇒ =
x
n
i
n
x
=
1
P
0
+
P '
=
T
xx
n
x
+
T
x
y
n
y
+
T n
xz
z
+ γ
2
∂ ξ
2
y
∂
⎛
⎜
⎜
⎝
+
2
∂ ξ
2
z
∂
⎞
⎟
⎟
⎠
1
perturbation perturbation
x yH hμ
x zH hμ
second order
Equilibrium (
)0ξ =
P
0
=
T
xx0
⇒
P
od
−
P
oe
=
1
2
⎡
⎣
μ
2
H
a a
− μ
2
H
b b
⎤
⎦
Perturbations
'
P
d
( )
ξ −
'
P
e
... | https://ocw.mit.edu/courses/6-642-continuum-electromechanics-fall-2008/5938069f486691fdd2376f96b52951ee_lec08_f08.pdf |
of 13
(cid:3)
⎡
p
⎢
(cid:3)
⎢
p
⎣
c
d
⎤
⎥ =
⎥
⎦
a
j
ωρ
k
(cid:3)
⎡
p
⎢
(cid:3)
⎢
p
⎣
e
f
⎤
⎥ =
⎥
⎦
j
ωρ
b
k
⎡
−⎢
⎢
⎢
−⎢
⎣
⎡
−⎢
⎢
⎢
−⎢
⎣
coth ka
1
sinh ka
coth kb
1
sinh kb
1
sinh ka
⎤
⎡
⎥
⎢
⎥
⎢
⎥
⎢
coth ka
⎥ ⎣
⎦
(cid:3)
v
(cid:3)
... | https://ocw.mit.edu/courses/6-642-continuum-electromechanics-fall-2008/5938069f486691fdd2376f96b52951ee_lec08_f08.pdf |
b
a
)
⇒ +
jk
y
(
(cid:108)
(cid:108)
Ψ − Ψ =
d
e
)
(
H
a
−
H
b
)
⇒ +
jk
z
(
(cid:108)
(cid:108)
Ψ − Ψ =
d
e
)
(
jk H H
a
b
−
y
(cid:3)
ξ
)
(
jk H
z
a
−
H
b
(cid:3)ξ
)
(cid:108)
(cid:108)
Ψ − Ψ = +
d
e
(
H
a
−
H
b
ξ(cid:3)
)
k coth ka
(cid:108)
Ψ
μ
a
d
= −
μ
b
k c
oth kb
(cid:108)
Ψ
e
6.642, Continuum Electromechanic... | https://ocw.mit.edu/courses/6-642-continuum-electromechanics-fall-2008/5938069f486691fdd2376f96b52951ee_lec08_f08.pdf |
Equation
(cid:3)
g
ρ ξ −
a
−
2
ω ρ
a
k
coth ka
(cid:3)
ξ + ρ ξ −
b
g
(cid:3)
2
ω ρ
b
k
coth k
b
(cid:3)
ξ
μ
H k coth ka H
a a
a
(
−
H
b
)
=
μ
b
μ
coth kb
(cid:3)
ξ − μ
H k coth kb H
b b
a
(
coth ka
+ μ
b
a
coth kb
−
H
b
(cid:3)
ξμ
)
a
coth ka
− γ
k
2
(cid:3)
ξ
2
ω
k
(
ρ
a
coth ka
+ ρ
b
coth kb
)
= ρ − ρ
b
(
a
)
g
+ γ... | https://ocw.mit.edu/courses/6-642-continuum-electromechanics-fall-2008/5938069f486691fdd2376f96b52951ee_lec08_f08.pdf |
μ μ
a b
⎡
⎣
E. Short Wavelength Limit (
ka
(cid:21)
1, kb
(cid:21)
)
1
tanhka tanhkb 1
≈
≈
2
ω
k
(
ρ + ρ
b
a
)
=
g
(
ρ − ρ + γ
a
b
k
)
2
−
2
(
)
2
B
μ − μ
0
b
(
μ μ μ + μ
b
a b
a
a
k
)
=
f
Incipience of Instability
6.642, Continuum Electromechanics Lecture 8
Prof. Markus Zahn Page 4 of 13
... | https://ocw.mit.edu/courses/6-642-continuum-electromechanics-fall-2008/5938069f486691fdd2376f96b52951ee_lec08_f08.pdf |
⎥ =
⎥
⎦
4g
(
ρ − ρ
b
a
)
γ
k
c
=
1
2
γ
4g
(
ρ − ρ
a
b
)
γ =
g
(
ρ − ρ
a
b
)
γ
Courtesy of MIT Press. Used with permission.
6.642, Continuum Electromechanics Lecture 8
Prof. Markus Zahn Page 5 of 13
Courtesy of MIT Press. Used with permiss... | https://ocw.mit.edu/courses/6-642-continuum-electromechanics-fall-2008/5938069f486691fdd2376f96b52951ee_lec08_f08.pdf |
642, Continuum Electromechanics Lecture 8
Prof. Markus Zahn Page 7 of 13
Courtesy of MIT Press. Used with permission.
III.
Tangential Gradient Fields
Courtesy of MIT Press. Used with permission.
6.642, Continuum Electr... | https://ocw.mit.edu/courses/6-642-continuum-electromechanics-fall-2008/5938069f486691fdd2376f96b52951ee_lec08_f08.pdf |
B. Perturbations
⎡
⎢
⎢
⎢
⎣
(cid:3)
e
α
x
(cid:3)
e
β
x
⎤
⎥
⎥
⎥
⎦
=
k
⎡
⎢
⎢
⎢
⎢
⎣
−
coth k
Δ
−
1
sinh k
Δ
lim Δ → ∞
(cid:3)
e
(cid:3)
e
(cid:108)
k= Φ
a
xa
(cid:108)
k= − Φ
b
xb
1
nh k
si
coth k
⎤
⎡
⎥Δ ⎢
⎥
⎢
⎥ ⎢
⎥ ⎣
⎦
Δ
(cid:3)
v
(cid:3)
v
α
x
β
x
⎤
⎥
⎥
⎥
⎦
⎡
(cid:3)
p
⎢
⎢
(cid:3)
p
⎣
α
β
⎤
⎥
⎥
⎦
=
j
ωρ
k
⎡
⎢
⎢
⎢
⎢
... | https://ocw.mit.edu/courses/6-642-continuum-electromechanics-fall-2008/5938069f486691fdd2376f96b52951ee_lec08_f08.pdf |
ξ(cid:3)
−
2
ω
k
(
ρ + ρ ξ −
b
a
(cid:3)
)
g
(
ρ − ρ ξ =
b
a
(cid:3)
)
−
(
ε
a
−
ε
b
)
E
0
dE
0
dx
(cid:3)
ξ − γ
k
2
(cid:3)
ξ −
jk E
y 0
(
ε
a
−
−
ε
b
k
)(
(
ε
a
) (
ε
a
−
ε
b
)
(cid:3)
ξ
jk E
y 0
)
ε
b
+
2
ω
k
(
ρ + ρ
b
a
)
=
g
(
ρ − ρ
a
b
)
2
+ γ
k
+
(
ε
a
−
ε
b
)
E
0
+
2 2
k E
y 0
k
(
dE
0
x
d
(cid:78)
E
0
R
ε
(
ε... | https://ocw.mit.edu/courses/6-642-continuum-electromechanics-fall-2008/5938069f486691fdd2376f96b52951ee_lec08_f08.pdf |
with permission.
6.642, Continuum Electromechanics Lecture 8
Prof. Markus Zahn Page 13 of 13 | https://ocw.mit.edu/courses/6-642-continuum-electromechanics-fall-2008/5938069f486691fdd2376f96b52951ee_lec08_f08.pdf |
Overview
This will be a mostly self-contained research-oriented course designed for undergraduate students
(but also extremely welcoming to graduate students) with an interest in doing research in theoretical
aspects of algorithms that aim to extract information from data. These often lie in overlaps of
two or more ... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/594e3ae91cc8e865f25d07dcbd2dd460_MIT18_S096F15_Ses1.pdf |
from pair-
wise ratios on compact groups.
11. Some extra material may be added, depending on time available.
Open Problems
A couple of open problems will be presented at the end of most lectures. They won’t necessarily
be the most important problems in the field (although some will be rather important), I have tried... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/594e3ae91cc8e865f25d07dcbd2dd460_MIT18_S096F15_Ses1.pdf |
Nik13], the same paper also has a good accounting
of partial progress on the conjecture.
• It is not so difficult to show that K(n) ≤ n, try it!
√
0.4.2 Matrix AM-GM inequality
We move now to an interesting generalization of arithmetic-geometric means inequality, which has
applications on understanding the differenc... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/594e3ae91cc8e865f25d07dcbd2dd460_MIT18_S096F15_Ses1.pdf |
ative arithmetic-geometric mean
inequality” by b. recht and c. re. 2012.
2
[IKW14] A. Israel, F. Krahmer, and R. Ward. An arithmetic-geometric mean inequality for prod
ucts of three matrices. Available online at arX... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/594e3ae91cc8e865f25d07dcbd2dd460_MIT18_S096F15_Ses1.pdf |
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/594e3ae91cc8e865f25d07dcbd2dd460_MIT18_S096F15_Ses1.pdf |
2
y
+
2
k
z
2
= ω με (cid:22)
k
2
o
Wave Vector k:
Perpendicular to uniform plane wave phase front,
Therefore perpendicular to⎯E and⎯H
L9-2
UPW AT PLANAR BOUNDARY
Case I: TE Wave
x
iE
kz
iH
kx
θi
ik
k
=i
k
o
ok = ω με
rE
θr
rk
ε,μ
kz
εt,μt
θi
y
z
θt
tEtk
“Transverse Electric”
⊥(cid:22) ... | https://ocw.mit.edu/courses/6-013-electromagnetics-and-applications-spring-2009/599acc32617af6d8eac2964c8f42cbb5_MIT6_013S09_lec09.pdf |
:11)(cid:9) (cid:11)(cid:10) (cid:8)(cid:11)(cid:9) (cid:11)(cid:10) (cid:8)(cid:11)(cid:9) (cid:11)(cid:10)
k
r
z
k sin
t
θ =
r
k
i
z
=
=
k
t
z
=
zk
θr = θi Angle of incidence equals angle of reflection
Snell’s Law:
sin
sin
θ
t
θ
i
=
k
o
k
t
=
ω με
ω μ ε
t
t
=
v
t
v
i
=
n
i
n
t
... | https://ocw.mit.edu/courses/6-013-electromagnetics-and-applications-spring-2009/599acc32617af6d8eac2964c8f42cbb5_MIT6_013S09_lec09.pdf |
: θt → 90o ⇒ sin θt → 1 as θi → θc
1
(
n n "critical angle"
−
sin
θ =
c
t
)
i
e.g., [
2
ε = ε ⇒
i
]
o
[n
i
=
2]
[
⇒ θ =
c
45 ]
°
L9-5
NON-UNIFORM PLANE WAVES (NUPW)
Normal refraction: θi < θc
θi
Phase fronts
Glass
Air
θt
λglass
z
Lines of
constant phase
>λo
λo
Beyond the critical ang... | https://ocw.mit.edu/courses/6-013-electromagnetics-and-applications-spring-2009/599acc32617af6d8eac2964c8f42cbb5_MIT6_013S09_lec09.pdf |
z
(x < 0)
=
More generally:
E
t
k
=
t
′
′′•
If lossless medium, k k
where:
−
it
jk r
yTE
ˆ
o
′
(cid:22)
ˆ
z ˆ
j x k
k z
− α
−
′′
jk
=
0
−
E,H e
α
(
j k
′
−
′′
jk
)
i
r
L9-7
EVANESCENT WAVES -- SUMMARY
Names:
“non-uniform plane wave”
“evanescent wave” (
0
=
in direction of decay)
“s... | https://ocw.mit.edu/courses/6-013-electromagnetics-and-applications-spring-2009/599acc32617af6d8eac2964c8f42cbb5_MIT6_013S09_lec09.pdf |
6.867 Machine learning, lecture 4 (Jaakkola)
1
The Support Vector Machine and regularization
We proposed a simple relaxed optimization problem for finding the maximum margin sep
arator when some of the examples may be misclassified:
minimize
�θ�2 + C
1
2
n
�
ξt
t=1
subject to yt(θT xt + θ0) ≥ 1 − ξt and ξt ≥ 0... | https://ocw.mit.edu/courses/6-867-machine-learning-fall-2006/59a63d2efbe8aa01041937ff539a449a_lec4.pdf |
of C
t ξt
as a function of θ and θ0.
�
Cite as: Tommi Jaakkola, course materials for 6.867 Machine Learning, Fall 2006. MIT OpenCourseWare
(http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].(cid:13)(cid:10)
−3−2−10123−1−0.500.511.522.53−3−2−10123−1−0.500.511.522.53 6.867 Mac... | https://ocw.mit.edu/courses/6-867-machine-learning-fall-2006/59a63d2efbe8aa01041937ff539a449a_lec4.pdf |
Another way of dealing with noisy labels in linear classification is to model how the noisy
labels are generated. For example, human assigned labels tend to be very good for “typical
examples” but exhibit some variation in more difficult cases. One simple model of noisy
labels in linear classification is a logistic regr... | https://ocw.mit.edu/courses/6-867-machine-learning-fall-2006/59a63d2efbe8aa01041937ff539a449a_lec4.pdf |
So for example, when we predict the same probability (1/2) for both classes, the log-odds
term is zero and we recover the decision boundary θT x + θ0 = 0. The precise functional
form of the logistic function, or, equivalently, the fact that we chose to model log-odds
with the linear prediction, may seem a little arb... | https://ocw.mit.edu/courses/6-867-machine-learning-fall-2006/59a63d2efbe8aa01041937ff539a449a_lec4.pdf |
example. Assuming each
example is labeled independently from others, this probability of assigning correct labels
to examples is given by the product
L(θ, θ0) =
n
�
P (yt|xt, θ, θ0)
(9)
t=1
Cite as: Tommi Jaakkola, course materials for 6.867 Machine Learning, Fall 2006. MIT OpenCourseWare
(http://ocw.mit.edu/),... | https://ocw.mit.edu/courses/6-867-machine-learning-fall-2006/59a63d2efbe8aa01041937ff539a449a_lec4.pdf |
likelihood function is a bit difficult to work with
directly so we will maximize its logarithm instead:
l(θ, θ0) =
n
�
log P (yt|xt, θ, θ0)
Alternatively, we can minimize the negative logarithm
t=1
− l(θ, θ0) =
=
=
n
� �
log-loss
�
��
− log P (yt|xt, θ, θ0)
t=1
n
�
�
− log g yt(θT xt + θ0)
�
t=1
n
� ... | https://ocw.mit.edu/courses/6-867-machine-learning-fall-2006/59a63d2efbe8aa01041937ff539a449a_lec4.pdf |
materials for 6.867 Machine Learning, Fall 2006. MIT OpenCourseWare
(http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].(cid:13)(cid:10)
6.867 Machine learning, lecture 4 (Jaakkola)
5
correspond to observed frequencies. So, for example, if we group together all the examples
f... | https://ocw.mit.edu/courses/6-867-machine-learning-fall-2006/59a63d2efbe8aa01041937ff539a449a_lec4.pdf |
�, θ0)]
θ ← θ + η · ytxt[1 − P (yt|xt, θ, θ0)]
(17)
(18)
where η is a small (positive) learning rate. Note that P (yt|xt, θ, θ0) is the probability that
we predict the training label correctly and [1 − P (yt|xt, θ, θ0)] is the probability of making a
mistake. The stochastic gradient descent updates in the logisti... | https://ocw.mit.edu/courses/6-867-machine-learning-fall-2006/59a63d2efbe8aa01041937ff539a449a_lec4.pdf |
in this (soft) sense. Another way of understanding this is that the vector of
mistake probabilities is orthogonal to the vector of labels. Similarly, the optimal setting
of θ is characterized by mistake probabilities that are orthogonal to all rows of the label-
example matrix X˜ = [y1x1, . . . , ynxn]. In other word... | https://ocw.mit.edu/courses/6-867-machine-learning-fall-2006/59a63d2efbe8aa01041937ff539a449a_lec4.pdf |
1|xt, θ, θ0)] = 0
=
(21)
(22)
(23)
(24)
t=1
meaning that the prediction errors are orthogonal to any linear function of the inputs.
Let’s try to briefly understand the type of predictions we could obtain via maximum like
lihood estimation of the logistic regression model. Suppose the training examples are
linea... | https://ocw.mit.edu/courses/6-867-machine-learning-fall-2006/59a63d2efbe8aa01041937ff539a449a_lec4.pdf |
7
our uncertainty about what the labels might be. So, when the number of training ex
amples is small we would need to add the regularizer �θ�2/2 just as in the SVM model.
The regularizer helps select reasonable parameters when the available training data fails to
sufficiently constrain the linear classifier.
To estim... | https://ocw.mit.edu/courses/6-867-machine-learning-fall-2006/59a63d2efbe8aa01041937ff539a449a_lec4.pdf |
6.896 Quantum Complexity Theory
September 16, 2008
Lecturer: Scott Aaronson
Lecture 4
1 Review of the last lecture
1.1 BQP
BQP is a class of languages L ⊆ (0, 1)∗, decidable with bounded error probability ( say 1/3 ) by a
uniform family of polynomial-size quantum circuit over some universal family of gate. In to... | https://ocw.mit.edu/courses/6-845-quantum-complexity-theory-fall-2010/59b32d603954c2f22a331ffb689706f3_MIT6_845F10_lec04.pdf |
evolution of all the
�
2.4 BQP ⊆ P SP ACE
In terms of computational complexity, the schrodinger picture ( αx|x�) and Heisenberg’s density
matrix (ρ) both lead to an exponential-space simulation since we need to calculate whole evolution
of state vectors. On the other hand, the Feynman’s path integral, summing up a... | https://ocw.mit.edu/courses/6-845-quantum-complexity-theory-fall-2010/59b32d603954c2f22a331ffb689706f3_MIT6_845F10_lec04.pdf |
∈ L, then M (x) accepts w.p < 1/2.
Note that there is no probability gap, so 1/2 appears instead of 1/3 and 2/3. This class is
physical not realistic for we cannot know whether the probability is 1/2 or 1/2 − 1/2|x| without
running algorithm exponential time. However, in terms of complexity theory, we can prove that... | https://ocw.mit.edu/courses/6-845-quantum-complexity-theory-fall-2010/59b32d603954c2f22a331ffb689706f3_MIT6_845F10_lec04.pdf |
QP
Can we prove that quantum computer exceeds classical computer? The answer is no since it would
imply P =� P SP ACE, which is a great challenge as proving P =� N P .
3.2 Where is N P ?
At first, we still don’t know where N P sits in the diagram and how N P relates to BQP . We
conjecture that N P �⊆ BQP , which me... | https://ocw.mit.edu/courses/6-845-quantum-complexity-theory-fall-2010/59b32d603954c2f22a331ffb689706f3_MIT6_845F10_lec04.pdf |
over work space and output space. Naturally, the states in subrou
tine space(or work space) affect the result of further operation on output space.
4.2 Uncomputing
This smart trick was introduced by Charlie Bennett. At first, we run the subroutine (unitary
operation) and get the answer. Then we apply CNOT-gate to the... | https://ocw.mit.edu/courses/6-845-quantum-complexity-theory-fall-2010/59b32d603954c2f22a331ffb689706f3_MIT6_845F10_lec04.pdf |
(cid:0)(cid:1)(cid:2)(cid:3)(cid:4)(cid:5)
Last
modi(cid:0)ed(cid:1)
September
(cid:2)(cid:3)(cid:4)
(cid:2)(cid:5)(cid:5)(cid:6)
Many(cid:0)body
phenomena
in
condensed
matter
and
atomic
physics
(cid:0)
Lectures
(cid:1)(cid:2)
(cid:3)(cid:4)
Bose
condensation(cid:4)
Symmetry(cid:5)
breaking
and
quasiparticles(cid:4)
In... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/59d41060e232d13d3da729e329a15c73_lec45.pdf |
chemical
potential(cid:5)
At
high
temperature
T (cid:4)
T
(cid:1)
with
BEC
2
h
(cid:10)
(cid:9)(cid:1)
2(cid:1)3
T
(cid:7)
(cid:5) n
(cid:2) (cid:5)
(cid:7)
(cid:7) (cid:12)
(cid:8)(cid:12)(cid:11)(cid:13)(cid:9)(cid:8)(cid:8)(cid:8)
(cid:3)(cid:9)(cid:4)
BEC
m
(cid:6)
(cid:3)(cid:12)(cid:7)(cid:9)(cid:4)
2(cid:1)3
at
... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/59d41060e232d13d3da729e329a15c73_lec45.pdf |
atomic
interaction(cid:5)
We
shall
focus
on
the
problem
of
weakly
nonideal
Bose
gas(cid:5)
This
problem(cid:1)
due
to
the
existence
of
a
simple
analytical
method(cid:1)
serves
well
to
illustrate
the
new
features
of
Bose
condensation
of
interacting
particles(cid:14)
spontaneous
symmetry
breaking(cid:1)
the
o(cid:6)(cid:... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/59d41060e232d13d3da729e329a15c73_lec45.pdf |
U
(cid:7)
U
(cid:3)x
x
(cid:4)dx(cid:5)
A
more
accurate
formula(cid:14)
(cid:11)
(cid:7) (cid:13)
(cid:1)(cid:10) a(cid:7)m(cid:1)
h
where
a
is
(cid:1)
k�0
(cid:18)
2
the
s(cid:0)wave
scattering
length(cid:1)
to
be
discussed
b
e
R
(cid:0)
(cid:11)
l
o
w(cid:5)
... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/59d41060e232d13d3da729e329a15c73_lec45.pdf |
i
j
i
a
B
EC
(cid:7)
(cid:3)(cid:19)(cid:4)
k
�0
N
(cid:0)
j
i
j
i
(cid:4)
(cid:15)(cid:2)
k
(cid:7) (cid:15)
p
N
BEC
(cid:2)
(cid:7) (cid:15)
k
N
1
This
formula(cid:1)
at
large
N
(cid:1)
suggests
to
replace
the
numb
e
r
state
by
a
coherent
state(cid:1)
a
BEC
(cid:7)
N
BEC
(cid:1)
which
is
equivalent
to
replacing
the
o... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/59d41060e232d13d3da729e329a15c73_lec45.pdf |
7)
V
(cid:0)
a
e
turns
into
a
classical
(cid:16)eld
(cid:10)
(cid:7)
N
(cid:7)V
(cid:1)
k
k
(cid:2)
(cid:3)
h
i
(cid:4)
1(cid:1)2
i
kr
where
V
is
system
volume(cid:5)
q
P
To
that
end(cid:1)
we
are
led
to
consider
the
coherent
states
(cid:10)
(cid:7)
exp
V
(cid:10)(cid:17)a
(cid:10)(cid:17)
(cid:10)a
(cid:15)
(cid:7)
e
... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/59d41060e232d13d3da729e329a15c73_lec45.pdf |
(cid:17)
k
k
a
a
k
(cid:1)
while
the
hamiltonian
(cid:3)(cid:13)(cid:4)
commutes
with
N
(cid:5)
One
has
to
understand
why
the
BEC
state
apparently
does
not
respect
the
particle
numb
e
r
conservation(cid:5)
P
We
start
by
noting
that
^
^
^
i(cid:4)
N
i(cid:4)
i(cid:4)
N
i(cid:4)
N
e
(cid:10)
(cid:7)
e
(cid:10)
(cid:2... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/59d41060e232d13d3da729e329a15c73_lec45.pdf |
1)
are
macroscopically
distinct(cid:5)
This
observation
j
i
j
i
(cid:2)
(cid:3)
i(cid:4)
demonstrates
that
the
BEC
states
form
a
degenerate
manifold
parameterized
by
a
phase
j
i
j
i
variable
(cid:15)
(cid:9)
(cid:5) (cid:9)
(cid:9)(cid:1) (cid:5)
To
clarify
the
origin
of
this
degeneracy(cid:1)
let
us
(cid:16)nd
the
sta... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/59d41060e232d13d3da729e329a15c73_lec45.pdf |
cid:1)
i(cid:5)e(cid:5)
the
phase
of
(cid:10)
is
arbitrary(cid:1)
while
the
modulus
(cid:10)
is
(cid:16)xed(cid:1)
thereby
giving
a
j
j
j
j
(cid:9)
(cid:1)
... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/59d41060e232d13d3da729e329a15c73_lec45.pdf |
a
constant
phase
factor
to
the
wavefunction
of
the
system(cid:1)
(cid:10)
(cid:17)
e
(cid:10)(cid:5)
(cid:17)
The
ground
states(cid:1)
however(cid:1)
i(cid:4)
do
not
possess
this
symmetry(cid:14)
adding
a
phase
factor
to
the
state
(cid:10)
produces
a
di(cid:6)erent
(cid:2)
ground
state(cid:5)
This
phenomenon(cid:1)
cal... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/59d41060e232d13d3da729e329a15c73_lec45.pdf |
3)x(cid:4)
(cid:17)
(cid:27)
(cid:3)(cid:28)(cid:4)
(cid:1)
(cid:1)
(cid:5)
(cid:5)
+
h
j
j
i
Using
the
translational
invariance(cid:1)
one
expects
that
the
quantity
(cid:3)(cid:28)(cid:4)
will
depend
only
on
the
distance
x
x
b
e
t
ween
the
two
p
o
i
n
ts(cid:5)
By
going
to
Fourier
representation(cid:1)
one
can
(cid:1)... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/59d41060e232d13d3da729e329a15c73_lec45.pdf |
o
s
e
where
f
(cid:3)
(cid:4)
i
s
a
smooth
function(cid:5)
Accordingly(cid:1)
t
h
e
density
matrix
(cid:3)(cid:11)(cid:15)(cid:4)
has
two
terms(cid:1)
k
R(cid:3)x(cid:2)
x
(cid:4) (cid:7)
n
(cid:8)
f
(cid:3)x
x
(cid:4)
(cid:2)
f
(cid:3)x
x
(cid:4)
e
(cid:1)
(cid:1)
(cid:1)
(cid:0)
(cid:0)
f
k
(cid:3)(cid:11)(cid:9)(cid... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/59d41060e232d13d3da729e329a15c73_lec45.pdf |
10)(cid:3)x(cid:4)
(cid:17)
(cid:27)
suggests
that
the
quantities
(cid:10)(cid:3)x(cid:4)(cid:1)
(cid:17)
0
x
x
(cid:5)
(cid:5)
(cid:1)
0
+
+
i(cid:4)
+
i(cid:4)
h
j
j
i
j
(cid:0)
j(cid:3)(cid:2)
(cid:10)
(cid:17)
(cid:3)x (cid:4)
i
n
s
o
m
e
sen
se
h
a
ve
(cid:16)nite
expectation
values(cid:14)
(cid:10)(cid:3)x(cid:4)... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/59d41060e232d13d3da729e329a15c73_lec45.pdf |
number(cid:1)
the
true
ground
state
(cid:2)TGS(cid:3)
of
a
quantum(cid:4)
mechanical
hamiltonian
is
nondegenerate(cid:5)
This
TGS
is
isotropic
in
(cid:1)
due
to
boundary
e(cid:6)ects
that
split
(cid:0)
the
circular
manifold(cid:5)
The
statement
about
the
absence
of
degeneracy
of
TGS
in
a
(cid:0)nite
system
is
formally
... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/59d41060e232d13d3da729e329a15c73_lec45.pdf |
(cid:8)
(cid:12)
(cid:3)a
a
(cid:8)
a
a
(cid:4) (cid:8)
(cid:11)n(cid:3)a
(cid:8)
a
(cid:4)
(cid:3)a
(cid:8)
a
(cid:4)
(cid:3)(cid:11)(cid:13)(cid:4)
k
k
k
k
k
k
k
k
k
(cid:9)
(cid:0)
(cid:0)
(cid:0)
(cid:0)
(
(cid:6)
)
k
k
X
(cid:0)
(cid:0)
(cid:1)
where
the
sum
is
taken
over
pairs
(cid:3)
(cid:2)
(cid:4)
Here
we... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/59d41060e232d13d3da729e329a15c73_lec45.pdf |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.