Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 70
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(3.794). It is easy to realize that this cannot be true. We have shown inSection 2.9 that for high temperatures, the partition function is given by the integral[recall (2.345)]Zcl =Z∞−∞dx −V (x)/kB Te.le (h̄β)(3.795)This integral can in principle be treated by the same background field method as thepath integral, albeit in a much simpler way. We may write x = X + δx and find aloop expansion for an effective potential. This expansion evaluated at the extremumwill yield a good approximation to the integral (3.795) only if the potential is veryclose to a harmonic one.
For any more complicated shape, the integral at small βwill cover the entire range of x and can therefore only be evaluated numerically.Thus we can never expect a good result for the partition function of anharmonicsystems at high temperatures, if it is calculated from Eq.
(3.794).It is easy to find the culprit for this problem. In a one-dimensional system,the correlation functions of the fluctuations around X are given by the correlationfunction [compare (3.301), (3.248), and (3.687)]DEh̄ pG 2(τ − τ 0 )M Ω (X),eh̄1 cosh Ω(X)(|τ − τ 0 | − h̄β/2)=,M 2Ω(X)sinh[Ω(X)h̄β/2](2)δx(τ )δx(τ 0 ) = GΩ2 (X) (τ, τ 0 ) =|τ − τ 0 | ∈ [0, h̄β], (3.796)3263 External Sources, Correlations, and Perturbation Theorywith the X-dependent frequency given bygΩ2 (X) = ω 2 + 3 X 2 .6(3.797)At equal times τ = τ 0 , this specifies the square width of the fluctuations δx(τ ):DE[δx(τ )]2 =h̄Ω(X)h̄β1coth.M 2Ω(X)2(3.798)The point is now that for large temperatures T , this width grows linearly in TD[δx(τ )]2ET →∞−−−→kB T.MΩ2(3.799)The linear behavior follows the historic Dulong-Petit law for the classical fluctuationwidth of a harmonic oscillator [compare with the Dulong-Petit law (2.579) for thethermodynamic quantities].
It is a direct consequence of the equipartition theoremfor purely thermal fluctuations, according to which the potential energy has anaverage kB T /2:MΩ2 D 2 E kB T.x =22(3.800)If we consider the spectral representation (3.245) of the correlation function,GpΩ2 ,e (τ∞1 X1−iωm (τ −τ 0 )−τ ) =,2e2h̄β m=−∞ ωm + Ω0(3.801)we see that the linear growth is entirely due to term with zero Matsubara frequency.The important observation is now that if we remove this zero frequency termfrom the correlation function and form the subtracted correlation function [recall(3.250)]GpΩ02 ,e (τ ) ≡ GpΩ2 ,e (τ )−111 cosh Ω(|τ |−h̄β/2)−,2 =2Ω sinh[Ωh̄β/2]h̄βΩh̄βΩ2(3.802)we see that the subtracted square widtha2Ω ≡ GpΩ02 ,e (0) =Ωh̄β11coth−2Ω2h̄βΩ2(3.803)decrease for large T . This is shown in Fig. 3.14. Due to this decrease, there existsa method to substantially improve perturbation expansions with the help of theso-called effective classical potential.H.
Kleinert, PATH INTEGRALS3273.25 Effective Classical PotentialFigure 3.14 Local fluctuation width compared with the unrestricted fluctuation widthof harmonic oscillator and its linear Dulong-Petit approximation. The vertical axis showsunits of h̄/M Ω, a quantity of dimension length2 .3.25.1Effective Classical Boltzmann FactorThe above considerations lead us to the conclusion that a useful approximation forpartition function can be obtained only by expanding the path integral in powers ofthe subtracted fluctuations δ 0 x(τ ) which possess no zero Matsubara frequency.
Thequantity which is closely related to the effective potential V eff (X) in Eq. (3.663)but allows for a more accurate evaluation of the partition function is the effectiveclassical potential V eff cl (x0 ). Just as V eff (X), it contains the effects of all quantumfluctuations, but it keeps separate track of the thermal fluctuations which makes ita convenient tool for numerical treatment of the partition function. The definitionstarts out similar to the background method in Subsection 3.23.6 in Eq. (3.769).We split the paths as in Eq. (2.435) into a time-independent constant backgroundx0 and a fluctuation η(τ ) with zero temporal average η̄ = 0:x(τ ) = x0 + η(τ ) ≡ x0 +∞ Xxm eiωm τ + cc ,m=1x0 = real,x−m ≡ x∗m ,(3.804)and write the partition function using the measure (2.440) asZ=IDx e−Ae /h̄ =Z∞−∞dx0le (h̄β)ID 0 x e−Ae /h̄ ,(3.805)whereI0−Ae /h̄D xe=∞Ym=1"Z∞−∞Z∞−∞#d Re xm d Im xm −Ae /h̄e.2πkB T /Mωm(3.806)Comparison of (2.439) with the integral expression (2.344) for the classical partitionfunction Zcl suggests writing the path integral over the components with nonzeroMatsubara frequencies as a Boltzmann factorB(x0 ) ≡ e−Veff cl (x0 )/kB T(3.807)3283 External Sources, Correlations, and Perturbation Theoryand defined the quantity V eff cl (x0 ) as the effective classical potential.
The fullpartition function is then given by the integralZZ=∞−∞dx0 −V eff cl (x0 )/kB Te,le (h̄β)(3.808)where the effective classical Boltzmann factor B(x0 ) contains all information onthe quantum fluctuations of the system and allows to calculate the full quantumstatistical partition function from a single classically looking integral. At hightemperature, the partition function (3.808) takes the classical limit (2.454). Thus,by construction, the effective classical potential V eff cl (x0 ) will approach the initialpotential V (x0 ):T →∞V eff cl (x0 ) −−−→ V (x0 ).(3.809)This is a direct consequence of the shrinking fluctuation width (3.803) for growingtemperature.The path integral representation of the effective classical Boltzmann factorB(x0 ) ≡ID 0 x e−Ae /h̄(3.810)can also be written as a path integral in which one has inserted a δ-function toensure the path average1 Z h̄βx̄ ≡dτ x(τ ).(3.811)h̄β 0Let us introduce the slightly modified δ-function [recall (2.345)]δ̃(x̄ − x0 ) ≡ le (h̄β)δ(x̄ − x0 ) =s2πh̄2 βδ(x̄ − x0 ).M(3.812)Then we can writeB(x0 ) ≡ e−Veff cl (x0 )/kB T==IID 0 x e−Ae /h̄ =IDx δ̃(x̄ − x0 ) e−Ae /h̄Dη δ̃(η̄) e−Ae /h̄ .(3.813)As a check we evaluate the effective classical Boltzmann factor for the harmonicaction (2.437).
With the path splitting (3.804), it readsMω 2 2 Mx +Ae [x0 + η] = h̄β2 02Z0h̄βhidτ η̇ 2 (τ ) + ω 2 η 2 (τ ) .(3.814)After representing the δ function by a Fourier integralδ̃(η̄) = le (h̄β)Zi∞−i∞dλ1exp λ2πih̄βZ!dτ η(τ ) ,(3.815)H. Kleinert, PATH INTEGRALS3293.25 Effective Classical Potentialwe find the path integralBω (x0 ) =I−Ae /h̄Dη δ̃(η̄) edλ−i∞ 2πi#)("Z h̄βIλ1M 2η̇ (τ ) − η(τ ) . (3.816)× Dη exp −dτh̄ 02β−βM ω 2 x20 /2=ele (h̄β)Zi∞The path integral over η(τ ) in the second line can now be performed without therestriction h̄β = 0 and yields, recalling (3.552), (3.553), and inserting there j(τ ) =λ/β, we obtain for the path integral over η(τ ) in the second line of (3.816):1λ2exp2 sinh(βh̄ω/2)2Mh̄β 2(Zh̄β0dτZh̄β0dτ0Gpω2 ,e (τ0)−τ ) .(3.817)The integrals over τ, τ 0 are most easily performed on the spectral representation(3.245) of the correlation function:Z0h̄βdτZ0h̄βdτ 0 Gpω2 ,e (τ − τ 0 ) =Zh̄β0dτZ0h̄βdτ 0∞1h̄β1 X−iωm (τ −τ 0 )= 2.22eh̄β m=−∞ ωm + ωω(3.818)The expression (3.817) has to be integrated over λ and yieldsZ i∞dλ1λ2exp2 sinh(βh̄ω/2) −i∞ 2πi2Mω 2 β!=11ωh̄β.2 sinh(βh̄ω/2) le (h̄β)(3.819)Inserting this into (3.816) we obtain the local Boltzmann factor−Vωeff cl (x0 )/kB TBω (x0 ) ≡ e=IDη δ̃(η̄) e−Ae /h̄ =βh̄ω/22 2e−βM ω x0 .sinh(βh̄ω/2)(3.820)The final integral over x0 in (3.805) reproduces the correct partition function (2.401)of the harmonic oscillator.3.25.2Effective Classical HamiltonianIt is easy to generalize the expression (3.813) to phase space, where we define theeffective classical Hamiltonian H eff cl (p0 , x0 ) and the associated Boltzmann factorB(p0 , x0 ) by the path integralDpδ(x0 − x)2πh̄δ(p0 − p) e−Ae [p,x]/h̄,2πh̄(3.821)R h̄βR h̄βwhere x = 0 dτ x(τ )/h̄β and p = 0 dτ p(τ )/h̄β are the temporal averages ofposition and momentum, and Ae [p, x] is the Euclidean action in phase spacehB(p0 , x0 ) ≡ exp −βHeff cli(p0 , x0 ) ≡Ae [p, x] =Z0h̄βIDxIdτ [−ip(τ )ẋ(τ ) + H(p(τ ), x(τ ))].(3.822)3303 External Sources, Correlations, and Perturbation TheoryThe full quantum-mechanical partition function is obtained from the classicallooking expression [recall (2.338)]Z=Z∞−∞dx0Z∞−∞dp0 −βH eff cl (p0 ,x0)e.2πh̄(3.823)The definition is such that in the classical limit, H eff cl (p0 , x0 )) becomes the ordinaryHamiltonian H(p0 , x0 ).For a harmonic oscillator, the effective classical Hamiltonian can be directlydeduced from Eq.
(3.820) by “undoing” the p0 -integration:eff cl (pBω (p0 , x0 ) ≡ e−Hω0 ,x0 )/kB T= le (h̄β)22 2βh̄ω/2e−β(p0 /2M +M ω x0 ) .sinh(βh̄ω/2)(3.824)Indeed, inserting this into (3.823), we recover the harmonic partition function(2.401).Consider a particle in three dimensions moving in a constant magnetic field Balong the z-axis. For the sake of generality, we allow for an additional harmonicoscillator centered at the origin with frequencies ωk in z-direction and ω⊥ in thexy-plane (as in Section 2.19).
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