BGK model of the multi-species Uehling-Uhlenbeck equation

TL;DR

This paper introduces a BGK model for the quantum Boltzmann equation in multi-species gases, ensuring conservation laws and the H-theorem, with a focus on the unique determination of equilibrium coefficients through nonlinear relations.

Contribution

It develops a novel BGK model for quantum multi-species gases with conditions guaranteeing equilibrium coefficients are well-defined and consistent with quantum conservation laws.

Findings

Established a sufficient condition for equilibrium coefficients.

Proved the uniqueness of solutions for the nonlinear relations.

Ensured the model shares conservation laws and H-theorem with the quantum Boltzmann equation.

Abstract

We propose a BGK model of the quantum Boltzmann equation for gas mixtures. We also provide a sufficient condition that guarantees the existence of equilibrium coefficients so that the model shares the same conservation laws and $H$-theorem with the quantum Boltzmann equation. Unlike the classical BGK for gas mixtures, the equilibrium coefficinets of the local equilibiriums for quantum multi-species gases are defined through highly nonlinear relations that are not explicitly solvable. We verify in a unified way that such nonlinear relations uniquely determine the equilibrium coefficients under the condition, leading to the well-definedness of our model.