Quantum Nonequilibrium Ensemble Method

Abstract

The systems considered in the previous chapters were classical gases obeying the Boltzmann statistics and the Boltzmann kinetic equation, but transport phenomena in semiconductors and small quantum devices at low temperatures, radiation interacting with particles, quantum gases at low temperatures, and so on usually require quantum mechanical treatments. The questions then arise of whether quantum systems can be described in a macroscopic formalism similar to the one developed for the classical gases, whether the nonequilibrium ensemble method is still applicable, and whether there is a mathematical structure for a theory of irreversible processes in such quantum systems. The answers to these questions are in the affirmative. In this chapter we apply the nonequilibrium ensemble method to a quantum kinetic equation and develop a theory of irreversible processes in quantum gases. When faced with the task of statistically treating a quantum system, the obvious strategy to take is to use a kinetic equation governing the density matrix or a kinetic equation for the Wigner distribution function [1] of the system. Another method, a little more ad hoc, would be to assume a semiclassical kinetic equation such as the Boltzmann-Nordheim-Uehling-Uhlenbeck (BNUU) kinetic equation [2,3] as a generalization of the classical Boltzmann equation. However, since the latter can be derived from the former under some approximations [4–6] of the collision term, it is not as semiclassical and ad hoc as the words might give the impression. In Appendix B we will establish that the density matrix approach gives rise to generalized hydrodynamics equations which have the same mathematical structures as their classical counterparts, except that the macroscopic variables therein are quantum mechanical averages of dynamical observables (operators). However, the same aim can be more simply achieved if the density matrix and its governing equation—the kinetic equation—are represented by the Wigner distribution function and its governing equation derived from the quantum kinetic equation for the density matrix. Since the generalized hydrodynamics equations thus derived are subjected to the restriction of the second law of thermodynamics in the nonequilibrium ensemble method, the transport properties computed therefrom will be thermodynamically consistent. The thermodynamic compatibility of transport properties is a useful criterion to judge the quality of the results because it is not obvious that nonlinear transport properties calculated are acceptable thermodynamically if the system is in the nonlinear regime. For example, many of the phenomena [7–9] in semiconductors and small quantum devices at high field gradients are essentially nonlinear, but a thermodynamically consistent theory of nonlinear transport processes will not be assured unless special care is exercised to satisfy the requirements of the laws of thermodynamics The nonequilibrium ensemble method will ensure it even for quantum systems, as will be shown.