Kinetic theory and irreversible thermodynamics of nonlinear transport processes in quantum systems

Abstract

A theory of nonlinear irreversible processes in quantum systems (gases) is developed by using a quantum mechanical Boltzmann equation which is a generalization of the equation originally used by Nordholm, Uehling, and Uhlenbeck. The Boltzmann equation is solved by the modified moment method and quantum mechanical extensions are thereby obtained for various macroscopic equations, including the extended Gibbs relation, derived previously by means of the classical Boltzmann equation. The theory can yield a nonmonotonic entropy production as a function of fluxes in the system and thus provides us with possibility of describing instability in transport processes in the nonlinear regime. In order to demonstrate it with a simplified model, we consider the threshold switching phenomenon in semiconductors at low temperatures and show that, when the impact ionization recombination of donors is postulated as an underlying microscopic process for the phenomenon, the theory can indeed give a qualitatively correct result for the current-field characteristic and thus furnishes a nonequilibrium statistical mechanical description of the phenomenon. It also lays a foundation for studying the electron gas model for various nonlinear transport processes in metals and semiconductors at low temperatures in full conformation with the requirement of the thermodynamic laws.