Nonequilibrium density matrix for quantum transport: Hershfield approach as a McLennan-Zubarev form of the statistical operator

TL;DR

This paper shows that Hershfield's non-equilibrium density matrix can be expressed as a McLennan-Zubarev ensemble, linking entropy production to non-equilibrium quantum transport and enabling the derivation of non-equilibrium distributions.

Contribution

It demonstrates the equivalence of Hershfield's approach with the McLennan-Zubarev ensemble, providing a formal foundation for analyzing non-equilibrium quantum systems.

Findings

Hershfield's density matrix has a McLennan-Zubarev form.

The correction term relates to entropy production.

Derived a non-equilibrium electron distribution function.

Abstract

In this paper, we formally demonstrate that the non-equilibrium density matrix developed by Hershfield for the steady state has the form of a McLennan-Zubarev non-equilibrium ensemble. The correction term in this pseudo equilibrium Gibbs-like ensemble is directly related to the entropy production in the quantum open system. The fact the both methods state that a non-equilibrium steady state can be mapped onto a pseudo-equilibrium, permits us to develop non-equilibrium quantities from formal expressions equivalent to the equilibrium case. We provide an example: the derivation of a non-equilibrium distribution function for the electron population in a scattering region in the context of quantum transport.