text stringlengths 0 188k |
|---|
For Γ<2ER the particle oscillates in response to the external force while for Γ≥2ER the oscillations become over-damped. In further to be specific we always assume that Γ<2ER. |
Note that the Green's function G0(ω) satisfies exact sum-rule |
∫∞−∞A2ωdω2π=1,A=−2IG0. (2.32) |
This sum-rule is actually a general property of the retarded Green's function for the stationary system of relativistic bosons, see [48] and our further considerations in Sect. 6. |
The solution (2.26) of the homogeneous equation can be also represented through the Green's function convoluted with the source term w0(t) expressed through the δ-function and its derivative |
z0(t) = −t∫0dt′G0(t−t′)w0(t′),w0(t)=a0ERsin(β−α0)δ(t−0)−a0cosα0δ′(t−0), (2.34) |
β=arctan(Γ2ωR). |
In Fourier representation we have z0(ω)=−G0(ω)w0(ω), where w0(ω)=a0(ERsin(β−α0)+iωcosα0) . |
Now we are at the position to include effects of anharmonicity, Λ≠0. In the leading order with respect to a small parameter Λ the Fourier transform of the solution z(ω) of the equation of motion acquires the form |
z(ω,Λ) = (2.35) |
where ˜w(ω)=w0(ω)+w(ω). Eq. (2.35) has a straightforward diagrammatic interpretation |
z(ω)=\parbox227.622047pt\includegraphics[width=227.622047pt]Class−diag−eq.eps, (2.36) |
where the thin line stands for the free Green's function iG0(ω), the cross depicts the source i˜w(ω), and the dot represents the coupling constant −iΛ. The integration is to be performed over the source frequencies with the δ-function responsible for the proper frequency addition. The diagrammatic representation can, o... |
z(ω)=\parbox42.679134pt\includegraphics[width=42.679134pt]Class−full−sol.eps, (2.37) |
where the thick line stands for the full Green's function iG(ω) satisfying the Dyson equation shown in Fig. 1. |
Figure 1: Dyson equation for the full Green's function of the anharmonic oscillator described by the equation of motion (2.22). |
Let us consider another aspect of the problem. For simplicity consider a linear oscillator (Λ=0). Assume that in vacuum oscillations are determined by equation |
¨z(t)+E20z(t)=0, (2.38) |
The Fourier transform of the retarded Green's function describing these oscillations is as follows |
G00(ω)=1ω2−ω20+i0ω. (2.39) |
Being placed in an absorbing medium the oscillator changes its frequency and acquires the width, which can be absorbed in the quantity RΣ=E2R−E20, IΣ=−Γω heaving a meaning of a retarded self-energy. Then we rewrite (2.30) as |
G0(ω)=1ω2−ω20−Σ=1(G00)−1−Σ, (2.40) |
and we arrive at equation |
G0=G00+G00ΣG0 (2.41) |
known in quantum field theory, as the Dyson equation for the retarded Green's functions. |
Now we illustrate the above general formula at hand of examples. To be specific we assume that the oscillator was at rest initially, and we start with the case Λ=0. |
Example 1. Consider a response of the system to a sudden change of an external constant force |
F(t)≡F1(t)=F0θ(−t). (2.42) |
The solution of Eq. (2.22) for Λ=0 is |
z(t)≡z1(t) = −+∞∫−∞dω2πie−iωtG0(ω)F0/mω+iϵ=F0/mERωRe−12Γtcos(ωRt−β)θ(t)+F0mE2Rθ(−t), (2.43) |
here β is defined as in Eq. (2.34). The solution is purely causal, meaning that there are no oscillations for t<0 and that they start exactly at the moment when the force ceases. This naturally follows from the retarded properties of the Green's function (2.31), which has the θ-function cutting off any response for neg... |
Solution (2.43) is characterized by three time scales. Two time scales, the period of oscillations P=2πωR, cf. (2.5), and the time of the amplitude quenching, i.e. the decay time t(cl)dec=2/Γ, cf. (2.27), appear already in the free solution (2.26). Another time scale appears as the phase time delay in the response of t... |
δt(cl)ph=β/ωR>0. (2.44) |
The solution (2.43) is depicted on the left panel of Fig. 2 for three values of Γ. Arrows demonstrate that for Γ≠0 the response of the oscillator on the action of the external perturbation is purely causal. The larger Γ is the smaller is t(cl)dec and the larger is δt(cl)ph, i.e. the larger is the time shift of the osci... |
Figure 2: Response of the oscillator to the external force. Left panel – Example 1: the external force is given by (2.42). Solution (2.43) is shown for different values of Γ. Right panel – Example 2: the external force (2.45) is shown by the solid line. Dash-dotted lines depict solutions (2.47). Values of Γ and Tf are ... |
Example 2. Interestingly, the same oscillating system, being placed in another external field, can exhibit apparently acausal reaction. To demonstrate this possibility consider the driving force acting within a finite time interval [−Tf,+Tf] and having a well defined peak occurring at t=0: |
F(t)≡F2(t)=F0cos2(πt2Tf)θ(Tf−|t|). (2.45) |
The oscillator response to this pulse-force is given by |
z(t)≡z2(t) = −F0m+∞∫−∞dω2πe−iωtG0(ω)sin(ωTf)ω+iϵπ2/T2f(ω+iϵ)2−π2/T2f. (2.46) |
After some manipulations the solution acquires the form |
z2(t) = F0mE2R[ζ(t+Tf)θ(t+Tf)−ζ(t−Tf)θ(t−Tf)], |
ζ(t) = 12[1−E2Rr+r−cos(πTft−β−+β+)+ERωR(π2/T2f)r+r−e−12Γtcos(ωRt−β−β−−β+)], |
r± = √(ωR±π/Tf)2+14Γ2,β±=arctan(12Γ/[ωR±π/Tf]), (2.47) |
and the phase shift β here is given by Eq. (2.34). The first two terms in ζ(t) are operative only for −Tf≤t≤Tf and cancel out exactly for t>Tf . If the interval of the action of the force is very short, i.e. TfER≪1, then for t>Tf the oscillator moves like after a single momentary kick similarly to that in Example 1, an... |
(mE2R/F0)z2(t) = 1F0F2(t−Γ/E2R)+π22T2fE2R{(1−12Γ2E2R)cos(πTf[t−ΓE2R]) (2.48) |
+ e−12Γ(t+Tf)ERωRcos(ωR(t+Tf)−3β)}. |
In the given example besides P and t(cl)dec the system is characterized by the initial pulse-time |
tpulse=2Tf (2.49) |
and by two phase time scales |
δt(1)ph=Tf(β−−β+)/πandδt(2)ph=(β+β−+β+)/ωR. (2.50) |
The solution (2.47) is shown in Fig. 2, right panel. As we see from the lower panel, for some values of Tf and Γ the maximum of the oscillator response may occur before the maximum of the driving force. Therefore, if for the identification of a signal we would use a detector with the threshold close to the pulse peak, ... |
Example 3. The temporal response of the system depends on characteristic frequencies of the driving force variation. For a monochromatic driving force |
F(t)≡F3(t)=F0cos(Ept) (2.51) |
the solution of the equation of motion for t>0 is |
z(t)=z3(t)=F0m|G0(Ep)|cos(Ept−δ(Ep))=(F0/m)cos(Ept−δ(Ep))√(E2R−E2p)2+Γ2E2p, (2.52) |
where the phase shift of the oscillations compared to the oscillations of the driving force, δ(Ep), is determined by the argument of the Green's function |
δ(Ep)=π+argG0(Ep)=i2(log[(E2R−E2p)/(EpΓ)−i]−log[(E2R−E2p)/(EpΓ)+i]). (2.53) |
The phase shift δ is determined such that δ(Ep=0)=0. In Eq. (2.53) the logarithm is continued to the complex plane as log(±i)=±π so that the function δ(Ep) is continuous at Ep=ER, see Fig. 3a, and in other points |
tanδ(Ep)=−EpΓ/(E2p−E2R). (2.54) |
The amplitude of the solution (2.52) has a resonance shape peaking at Ep=ER with a width determined by the parameter Γ . In contrast to Examples 1 and 2 solution (2.52) does not contain the time-scale t(cl)dec, since the external force does not cease with time and continuously pumps-in the energy in the system. So, two... |
δt(1)ph=δ(Ep)/Ep (2.55) |
fully control the dynamics. Note that in difference with (2.14), here Ep is the frequency rather than the particle energy. |
We have seen in Example 2 that for some choices of the external force restricted in time the oscillating system can provide an apparently advanced response. The anharmonicity can produce a similar effect. For the case of small anharmonicity, Λ≠0, the solution (2.52) acquires a new term (an overtone) |
which oscillates on the double frequency 2Ep and the phase is shifted with respect to the solution (2.52) by δ(2Ep) . The Fourier transform of this solution is given by Eq. (2.35) provided w0 is put zero. Respectively, there appears an additional phase time scale |
δt(2)ph=(δ(Ep)+12δ(2Ep))/Ep (2.57) |
characterizing dynamics of the overtone. |
In Fig. 3b we show the solution (2.56) for several frequencies Ep. If we watch for maxima in the system response z(t) (shown by arrows) and compare how their occurrence is shifted in time with respect to maxima of the driving force, we observe that for most values of Ep the overtone in (2.56) induces a small variation ... |
Figure 3: Panel a): Phase shift δ(Ep) given by Eq. (2.53). Panel b): Response of the damped anharmonic oscillator to a harmonic external force (2.51) for different values of the force frequency Ep shown by line labels in units of ER for Γ=0.2ER and Λ=0.3E4Rm/F0 . Arrows show response maxima. Vertical dotted line shows ... |
Example 4. In realistic cases the driving force can rarely be purely monochromatic, but is usually a superposition of modes grouped around a frequency Ep: |
F(t)≡F4(t)=F0+∞∫−∞dEg(E−Ep;γ)cos(Et), (2.58) |
where an envelope function g(ϵ;γ), ϵ=E−Ep, is a symmetrical function of frequency deviation picked around ϵ=0 with a width γ and normalized as ∫+∞−∞dϵg(ϵ;γ)=1 . The integral (2.58) can be rewritten as |
F4(t) = F0cos(Ept)+∞∫−∞dϵg(ϵ;γ)cos(ϵt)=AF(γt)cos(Ept), (2.59) |
that allows us to identify Ep as the carrier frequency and AF(γt), as the amplitude modulation depending on dimension-less variable γt. |
For Λ=0, the particle motion is described by the function |
z4(t) = −+∞∫−∞dω2πe−iωtG0(ω)1mF(ω)=−F0m+∞∫−∞dω2πe−iωtG0(ω)π[g(ω+Ep;γ)+g(ω−Ep;γ)] (2.60) |
= −F0mR+∞∫−∞dϵe−i(Ep+ϵ)tG0(Ep+ϵ)g(ϵ;γ). |
The last integral can be formally written as |
mz4(t) = |G0(Ep)|Re−i(Ept−δ(Ep))e−12∂2ElogG0(Ep)∂2t+O(∂3t)AF(γ(t+i∂ElogG0(Ep))). (2.61) |
Here O(∂3t) represents time derivatives of the third order and higher. We used the relation logG0(E)=log|G0(E)|+iδ(E)−iπ, where δ(E) is defined as in Eq. (2.53), but now as function of E rather than Ep. The first-order derivatives generate the shift of the argument of the amplitude modulation via the relation exp(a∂t)A... |
To proceed further with Eq. (2.61) one may assume that the function AF(t) varies weakly with time so that the second and higher time derivatives can be neglected. In terms of the envelop function, this means that g(ϵ) is a very sharp function falling rapidly off for ϵ≳γ while γ≪Γ. A typical time, on which the function ... |
tγ,(cl)dec=1/γ, (2.62) |
If, additionally, the oscillator system has a high quality factor, i.e., Γ≪ER and |∂Elog|G0(Ep)||≪δ′(Ep), that is correct for Ep very near ER, we arrive at the expression |
We see that in this approximation there are five time scales determining the response of the system. The oscillations are characterized by the period P=<eot>Find 7 Bedroom Houses for Sale in Leyland - Zoopla |
7 Bedroom Houses to Buy in Leyland |
***immaculate seven double bedroom house*** This is a superb modern detached family home with superior well proportioned and highly versatile accommodation, finished to the highest standard throughout. This breath-taking property has been built to the ... |
Seven bedroom hmo registered property situated within Plungington offering excellent access to amenities and transport links. No chain delay and viewing highly recommended. (contd...) |
An outstanding three bedroomed barn conversion with A four bedroomed annex! Clayton Hey Fold Barn is a striking barn conversion providing versatile accommodation in the form of a converted, four bedroomed annex! Situated in the popular Ribble Valley ... |
Deceptively spacious traditional detached house set on a generous plot in a much sought after location. Convenient for access to the local amenities, schools, Preston City Centre and main motorway connections. On internal inspection the property ... |
7 6 3 3,843 sq. ft* |
A rare opportunity to purchase a truly exceptional detached family house of character, offering accommodation over four floors including four en-suite bedrooms, three reception rooms and a cinema suite. |
**an impressive victorian home suited for A large family!** Situated in a popular Village on the cusp of the Ribble Valley, stands this luxuriously finished, characterful family home! With impeccably maintained original features throughout a number ... |
This impressive seven bedroomed home in a popular and sought after area of Blackburn, is conveniently located for commuter routes and school links, looks out over Corporation park, and has recently undergone partial refurbishment and renovation, ...<eot>GitHub has a super cool feature called “Pages” which lets you host... |
Tom wanted to use his Pages site to host a web version of his resume which was such a great idea that I totally copied it. Tom’s is really awesome though – he built it using LaTeX to create both the html and a really neatly-formatted pdf from the same markdown file. |
I helped a little with the html side of things. Continue reading Playing with Pages → |
Leave a comment |
Pronoms |
January 4, 2015 development, internetapps, coding, developmentCat B |
I’ve been working on a new app lately. It’s a sort of grocery list creator based on recipes that you enter into the application. |
The inspiration is that I’m sick and tired of going to the supermarket every day and trying to think about what to cook each evening. Neither Tom or I have time these days for such foolishness, so the app helps you plan a menu of meals and then gives you a list of ingredients required for that menu. We’re using it so t... |
Continue reading Pronoms → |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.