Analytical Solution and Lie Algebra of the Relativistic Boltzmann Equation

TL;DR

This work tackles analytical solutions of the relativistic Boltzmann equation, focusing on exact BKW-type relaxation and symmetry structures. It introduces a physics-informed ansatz to reconstruct the relativistic BKW solution efficiently, reducing the problem to a finite set of time-dependent coefficients and an analytically tractable collision integral. The Lie algebra of invariant transformations is derived, comprising Poincaré generators plus two scale generators, yielding a closed algebra that clarifies how non-equilibrium, self-similar structures relate to fundamental symmetries. The results connect classical scaling behavior to non-thermal fixed-point dynamics and point to future directions, including extensions to expanding spacetimes and exploration of linearized dynamics around the BKW state.

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.