Analytical Solution and Lie Algebra of the Relativistic Boltzmann Equation

TL;DR

This paper introduces a new method for solving the relativistic Boltzmann equation using ansatz functions, and derives its Lie algebra of symmetries, providing a systematic way to construct invariant transformations.

Contribution

It presents an efficient approach to the relativistic BKW solution and derives the Lie algebra of the relativistic Boltzmann equation, revealing its symmetry structure.

Findings

Exact solutions include equilibrium and BKW-type forms.

The Lie algebra of invariant transformations is explicitly derived.

Symmetry group transformations can be systematically constructed.

Abstract

In this work, we present a novel and more efficient approach to constructing the relativistic BKW (Bobylev, Krook, and Wu) solution. By introducing a class of ansatz functions for the distribution function, we demonstrate that within this specific ansatz space, only the equilibrium and BKW-type forms yield exact solutions to the nonlinear Boltzmann equation. Furthermore, guided by physical insight and drawing upon the framework of relativistic kinetic theory, we derive the Lie algebra of invariant transformations admitted by the relativistic Boltzmann equation. From this algebra, the corresponding symmetry group transformations can be systematically constructed.