# Nonequilibrium Thermodynamics and Steady State Density Matrix for Quantum Open Systems

## TL;DR

This paper investigates the nonequilibrium thermodynamics of a finite quantum electron system connected to reservoirs, deriving the steady state density matrix and analyzing entropy production, confirming its positivity and expanding theoretical understanding.

## Contribution

It introduces a detailed analysis of the nonequilibrium density matrix for quantum open systems driven by reservoir differences, extending previous theoretical frameworks.

## Key findings

- Entropy production rate is always positive.
- Different expressions for entropy production are consistent.
- Results expand and corroborate earlier theoretical work.

## Abstract

We consider the generic model of a finite-size quantum electron system connected to two (temperature and particle) reservoirs. The quantum open system is driven out of equilibrium by the presence of both a temperature and a chemical potential differences between the two reservoirs. The nonequilibrium (NE) thermodynamical properties of such a quantum open system are studied for the steady state regime. In such a regime, the corresponding NE density matrix is built on the so-called generalised Gibbs ensembles. From different expressions of the NE density matrix, we can identify the terms related to the entropy production in the system. We show, for a simple model, that the entropy production rate is always a positive quantity. Alternative expressions for the entropy production are also obtained from the Gibbs-von Neumann conventional formula and discussed in detail. Our results corroborate and expand earlier works found in the literature.