Quantum Fokker-Planck-Kramers equation and entropy production

TL;DR

This paper develops a quantum Fokker-Planck-Kramers equation incorporating dissipation and thermodynamic consistency, providing solutions for quantum harmonic oscillators and analyzing entropy production and thermal conductance in nonequilibrium states.

Contribution

It introduces a quantum Fokker-Planck-Kramers equation with a thermodynamically consistent dissipation term and derives entropy production expressions for quantum systems.

Findings

Derived the quantum Fokker-Planck-Kramers equation with Gibbs state equilibrium.

Provided solutions for quantum harmonic oscillators in contact with heat baths.

Analyzed entropy production and thermal conductance in coupled quantum oscillators.

Abstract

We use a canonical quantization procedure to set up a quantum Fokker-Planck-Kramers equation that accounts for quantum dissipation in a thermal environment. The dissipation term is chosen to ensure that the thermodynamic equilibrium is described by the Gibbs state. An expression for the quantum entropy production is also provided which properly describes quantum systems in a nonequilibrium stationary state. The time-dependent solution is given for a quantum harmonic oscillator in contact with a heat bath. We also obtain the stationary solution for a system of two coupled harmonic oscillators in contact with reservoirs at distinct temperatures, from which we obtain the entropy production and the quantum thermal conductance.