Irreversible evolution of many-electron systems coupled to a statistical environment

Abstract

We study the evolution of a many-electron system that is confined in a finite spatial region and coupled to a statistical environment. The latter may be composed of several independent bath subsystems, which are held at some statistical equilibrium. From the master equations describing the evolution of the coarse-grained $N$-particle density matrix, we obtain the equations, which describe the evolution of the $n$-particle density operators ${\overline{D}}^{(n)}$ for $n<N$. These equations are hierarchically coupled through the electron-electron interaction. We show that the hierarchy can be truncated under the assumption that the residual interaction of the electrons in the considered system with the environment introduces a memory loss, which hinders the electronic system to build up more than two-particle correlations. We first consider a weakly excited electronic system, which looses memory but where energy exchange with the statistical environment can be neglected. This is the quantum analog of the classical Boltzmann gas in a box. We derive the master equations, which describe the irreversible evolution of the coarse-grained one-particle density matrix. Based on this result, we show that, in accord with the second law of thermodynamics, the corresponding von Neumann entropy either increases with time or it remains constant. Finally, allowing also for energy exchange with one or more bath subsystems provided, e.g., by phonons or photons, we obtain the corresponding general master equations that describe the evolution of a spatially confined interacting electron gas of metallic density.