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Nonequilibrium Quantum Statistics

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Nonequilibrium Quantum Statistics H. Kleinert, PATH INTEGRALS January 6, 2016 (/home/kleinert/kleinert/books/pathis/pthic18.tex) Path, motive, guide, original, and end Samuel Johnson (1709-1784), The Rambler 18 Nonequilibrium Quantum Statistics Quantum statistics described by the theoretical tools of the previous chapters is quite limited. The physical system under consideration must be in thermodynamic equilibrium, with a constant temperature enforced by a thermal reservoir. In this situation, partition function and the density matrix can be calculated from an ana- lytic continuation of quantum-mechanical time evolution amplitudes to an imaginary time tb ta = ih/k¯ BT . In this chapter we want to go beyond such equilibrium physics− and extend− the path integral formalism to nonequilibrium time-dependent phenomena. The tunneling processes discussed in Chapter 17 belong really to this class of phenomena, and their full understanding requires the theoretical frame- work of this chapter. In the earlier treatment this was circumvented by addressing only certain quasi-equilibrium questions. These were answered by applying the equilibrium formalism to the quantum system at positive coupling constant, which guaranteed perfect equilibrium, and extending the results to the quasi-equilibrium situation analytic continuation to small negative coupling constants. Before we can set up a path integral formulation capable of dealing with true nonequilibrium phenomena, some preparatory work is useful based on the traditional tools of operator quantum mechanics. 18.1 Linear Response and Time-Dependent Green Functions for T =/0 If the deviations of a quantum system from thermal equilibrium are small, the easiest description of nonequilibrium phenomena proceeds via the theory of linear response. In operator quantum mechanics, this theory is introduced as follows. First, the system is assumed to have a time-independent Hamiltonian operator Hˆ . The ground state is determined by the Schr¨odinger equation, evolving as a function of time according to the equation Ψ (t) = e−iHtˆ Ψ (0) (18.1) | S i | S i (in natural units withh ¯ = 1, kB = 1). The subscript S denotes the Schr¨odinger picture. 1274 18.1 Linear Response and Time-Dependent Green Functions for T =0 1275 6 Next, the system is slightly disturbed by adding to Hˆ a time-dependent external interaction, Hˆ Hˆ + Hˆ ext(t), (18.2) → ext ext where Hˆ (t) is assumed to set in at some time t0, i.e., Hˆ (t) vanishes identically for t < t0. The disturbed Schr¨odinger ground state has the time dependence Ψdist(t) = e−iHtˆ Uˆ (t) Ψ (0) , (18.3) | S i H | S i where UˆH (t) is the time translation operator in the Heisenberg picture. It satisfies the equation of motion ˆ˙ ˆ ext ˆ iUH (t)= HH (t)UH (t), (18.4) with1 Hˆ ext(t) eiHtˆ Hˆ ext(t)e−iHtˆ . (18.5) H ≡ To lowest-order perturbation theory, the operator UˆH (t) is given by t ˆ ′ ˆ ext ′ UH (t)=1 i dt HH (t )+ . (18.6) − Zt0 ··· In the sequel, we shall assume the onset of the disturbance to lie at t0 = . Consider an arbitrary time-independent Schr¨odinger observable Oˆ whose Heisenberg−∞ representation has the time dependence iHtˆ −iHtˆ OˆH (t)= e Oeˆ . (18.7) Its time-dependent expectation value in the disturbed state Ψdist(t) is given by | S i dist ˆ dist ˆ † iHtˆ ˆ −iHtˆ ˆ ΨS (t) O ΨS (t) = ΨS(0) UH (t)e Oe UH (t) ΨS(0) h | | i ht | | i ′ ˆ ext ′ ˆ ΨS(0) 1+ i dt HH (t )+ ... OH (t) ≈h | Z−∞ t ′ ˆ ext ′ 1 i dt HH (t )+ ... ΨS(0) (18.8) × − Z−∞ | i t ˆ ′ ˆ ˆ ext ′ = ΨH OH(t) ΨH i ΨH dt OH (t), HH (t ) ΨH + .... h | | i − h | −∞ | i Z h i We have identified the time-independent Heisenberg state with the time-dependent Schr¨odinger state at zero time in the usual manner, i.e., Ψ Ψ (0) . Thus the | H i≡| S i expectation value of Oˆ deviates from equilibrium by ˆ dist ˆ dist ˆ δ ΨS(t) O ΨS(t) ΨS (t) O(t) ΨS (t) ΨS(t) O(t) ΨS(t) h | | i ≡ h t | | i−h | | i ′ ˆ ˆ ext ′ = i dt ΨH OH (t), HH (t ) ΨH . (18.9) − −∞ h | | i Z h i 1 ext int Note that after the replacements H H0, HH HI , Eq. (18.4) coincides with the equation for the time evolution operator in the interaction→ picture→ to appear in Section 18.7. In contrast to that section, however, the present interaction is a nonpermanent artifact to be set equal to zero at the end, and H is the complicated total Hamiltonian, not a simple free one. This is why we do not speak of an interaction picture here. 1276 18 Nonequilibrium Quantum Statistics If the left-hand side is transformed into the Heisenberg picture, it becomes δ Ψ (t) Oˆ Ψ (t) = δ Ψ Oˆ (t) Ψ = Ψ δOˆ (t) Ψ , h S | | S i h H | H | H i h H | H | Hi so that Eq. (18.9) takes the form t ˆ ′ ˆ ˆ ext ′ ΨH δOH(t) ΨH = i dt ΨH OH (t), HH (t ) ΨH . (18.10) h | | i − −∞ h | | i Z h i ′ It is useful to use the retarded Green function of the operators OˆH (t) and HˆH (t ) in the state Ψ [compare (3.40)]: | Hi GR (t, t′) Θ(t t′) Ψ Oˆ (t), Hˆ (t′) Ψ . (18.11) OH ≡ − h H| H H | Hi h i Then the deviation from equilibrium is given by the integral ∞ ˆ ′ R ′ ΨH δOH (t) ΨH = i dt GOH (t, t ). (18.12) h | | i − Z−∞ Suppose now that the observable OˆH (t) is capable of undergoing oscillations. Then an external disturbance coupled to OˆH(t) will in general excite these oscilla- tions. The simplest coupling is a linear one, with an interaction energy Hˆ ext(t)= Oˆ (t)δj(t), (18.13) − H where j(t) is some external source. Inserting (18.13) into (18.12) yields the linear- response formula ∞ ˆ ′ R ′ ′ ΨH δOH (t) ΨH = i dt GOO(t, t )δj(t ), (18.14) h | | i Z−∞ R ˆ where GOO is the retarded Green function of two operators O: GR (t, t′)=Θ(t t′) Ψ Oˆ (t), Oˆ (t′) Ψ . (18.15) OO − h H | H H | H i h i ′ At frequencies where the Fourier transform of GOO(t, t ) is singular, the slightest disturbance causes a large response. This is the well-known resonance phenomenon found in any oscillating system. Whenever the external frequency ω hits an eigen- frequency, the Fourier transform of the Green function diverges. Usually, the eigen- frequencies of a complicated N-body system are determined by calculating (18.15) and by finding the singularities in ω. It is easy to generalize this description to a thermal ensemble at a nonzero temperature. The principal modification consists in the replacement of the ground state expectation by the thermal average Tr (e−H/Tˆ Oˆ) Oˆ T . h i ≡ Tr (e−H/Tˆ ) H. Kleinert, PATH INTEGRALS 18.2 Spectral Representations of Green Functions for T =0 1277 6 Using the free energy F = T log Tr (e−H/Tˆ ), − this can also be written as Oˆ = eF/T Tr (e−H/Tˆ Oˆ). (18.16) h iT In a grand-canonical ensemble, Hˆ must be replaced by Hˆ µNˆ and F by its grand- − canonical version FG (see Section 1.17). At finite temperatures, the linear-response formula (18.14) becomes ∞ ˆ ′ R ′ ′ δ O(t) T = i dt GOO(t, t )δj(t ), (18.17) h i Z−∞ R ′ where GOO(t, t ) is the retarded Green function at nonzero temperature defined by [recall (1.306)] GR (t, t′) GR (t t′) Θ(t t′) eF/T Tr e−H/Tˆ Oˆ (t), Oˆ (t′) . (18.18) OO ≡ OO − ≡ − H H n h io ˆi In a realistic physical system, there are usually many observables, say OH (t) for i =1, 2,...,l, which perform coupled oscillations. Then the relevant retarded Green function is some l l matrix × GR(t, t′) GR(t t′) Θ(t t′) eF/T Tr e−H/Tˆ Oˆi (t), Oˆj (t′) . (18.19) ij ≡ ij − ≡ − H H n h io After a Fourier transformation and diagonalization, the singularities of this matrix render the important physical information on the resonance properties of the system. The retarded Green function at T = 0 occupies an intermediate place between the real-time Green function of field theories6 at T = 0, and the imaginary-time Green function used before to describe thermal equilibria at T = 0 (see Subsection 3.8.2). The Green function (18.19) depends both on the real time6 and on the temperature via an imaginary time. 18.2 Spectral Representations of Green Functions for T =/0 The retarded Green functions are related to the imaginary-time Green functions of ˆ1 equilibrium physics by an analytic continuation. For two arbitrary operators OH , ˆ2 OH , the latter is defined by the thermal average G (τ, 0) G (τ) eF/T Tr e−H/Tˆ Tˆ Oˆ1 (τ)Oˆ2 (0) , (18.20) 12 ≡ 12 ≡ τ H H h i where OˆH(τ) is the imaginary-time Heisenberg operator Oˆ (τ) eHτˆ Oeˆ −Hτˆ . (18.21) H ≡ 1278 18 Nonequilibrium Quantum Statistics R To see the relation between G12(τ) and the retarded Green function G12(t), we take a complete set of states n , insert them between the operators Oˆ1, Oˆ2, and expand | i G (τ) for τ 0 into the spectral representation 12 ≥ F/T −En/T (En−E ′ )τ 1 ′ ′ 2 G12(τ)= e e e n n Oˆ n n Oˆ n .
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