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Physica A Statistical Mechanics and its Applications

Quantum-kinetic equations for time correlation functions in higher-order perturbation theory

M.C.J. Leermakers, L.G. Suttorp

Published Sep 1, 1981• 0 citations
Publisher
DOI
10.1016/0378-4371(81)90136-9

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The memory kernel of the kinetic equation for the time correlation function of a quantum fluid is determined both in third order of the interaction strength and in the low-density approximation. The results are obtained with the help of a diagram representation for the kernel. The connexion with the kinetic theory for classical fluids is established. The properties of a nonequilibrium fluid may be studied with the help of the kinetic theory of time correlation functions. These functions are determined by specifying the memory kernel in the kinetic equation that governs their time evolution. For classical fluids a suitable formalism that permits the approximate evaluation of this kernel has been developed several years ago (for a review see ref. 1). The generalization of these methods to the case of quantum fluids was considered in a few recent papers2f3). In particular, the memory kernel was established for a normal quantum fluid in the weakcoupling approximation. In the long-time and large-distance limit this kernel reduces to that of the linearized Uehling-Uhlenbeck equation4) in which the scattering amplitude is replaced by its first Born approximation. The purpose of the present paper is to extend the available results by employing higher-order perturbation theory for the evaluation of the memory kernel. An essential ingredient of the treatment will consist in the introduction of four-point diagrams that represent the various contributions to the kernel. With the use of this diagram representation we shall derive the memory kernel in third order of the interaction strength. In this order the kernel still has the same general form as the weak-coupling kernel, but with shifted energies and a generalized transition probability. Its relation to the kernel of the linearized Bogoliubov-Gurov’) equation will be established. An alternative approach to the approximate evaluation of the memory kernel makes use of an expansion in the density. The low-density limit of the
There exists the well known approximate expression describing the large time behaviour of matrix elements of the evolution operator in quantum theory: =exp(at)+... This expression plays the crucial role in considerations of problems of quantum decoherence, radiation, decay, scattering theory, stochastic limit, derivation of master and kinetic equations etc. This expression was obtained in the Weisskopf-Wigner approximation and in the van Hove (stochastic) limit. We derive the exact general formula which includes the higher order corrections to the above approximate expression: =exp(At+B+C(t)). The constants A and B and the oscillating function C(t) are computed in perturbation theory. The method of perturbation of spectra and renormalized wave operators is used. The formula is valid for a general class of Hamiltonians used in statistical physics and quantum field theory.
This paper extends an earlier quantum kinetics treatment for dilute, weakly interacting, partially Bose-Einstein condensed gases, presented by the author elsewhere (Proukakis N P and Burnett K 1996 J. Res. Natl Inst. Stand. Technol. 101 457), by consistently treating the dynamics of the uncondensed atoms to the same level of approximation as the condensed atoms. Our method is based on a hierarchy of coupled equations of motion for the condensate mean field and fluctuations around this mean field, truncated to second order in the (effective two-body) interatomic potential, and with suitable decoupling approximations for higher-order correlations. By applying perturbation theory in the Hartree-Fock-Bogoliubov basis, we re-derive the quantum kinetic theory of Walser et al (Walser R, Williams J, Cooper J and Holland M 1999 Phys. Rev. A 59 3878), which further indicates the consistency of our treatment with the Kadanoff-Baym non-equilibrium Green function formalism for trapped gases.