The relativistic quantum Boltzmann equation near equilibrium

TL;DR

This paper establishes the first rigorous proof of the global existence and uniqueness of classical solutions to the relativistic quantum Boltzmann equation near equilibrium for both bosons and fermions.

Contribution

It provides the first mathematical proof of solution existence for the relativistic quantum Boltzmann equation close to equilibrium.

Findings

Proves global existence of solutions near equilibrium.

Establishes uniqueness of solutions for the relativistic quantum Boltzmann equation.

Applies to both bosonic and fermionic particles.

Abstract

The relativistic quantum Boltzmann equation (or the relativistic Uehling-Uhlenbeck equation) describes the dynamics of single-species fast-moving quantum particles. With the recent development of the relativistic quantum mechanics, the relativistic quantum Boltzmann equation has been widely used in physics and engineering such as in the quantum collision experiments and the simulations of electrons in graphene. In spite of such importance, there has been no mathematical theory on the existence of solutions for the relativistic quantum Boltzmann equation to the best of authors' knowledge. In this paper, we prove the global existence of a unique classical solution to the relativistic Boltzmann equation for both bosons and fermions when the initial distribution is nearby a global equilibrium.