Existence of the Chapman-Enskog solution and its relation with first-order dissipative fluid theories

TL;DR

This paper investigates the conditions for the existence of the Chapman-Enskog first-order solution to the Boltzmann equation, comparing traditional and formal approaches, and explores their implications for relativistic and non-relativistic fluid theories.

Contribution

It demonstrates the equivalence of different methods for deriving first-order solutions and clarifies the relation between Chapman-Enskog solutions and relativistic dissipative fluid theories.

Findings

Both methods lead to the same integral equation in non-relativistic cases.

In relativistic systems, the source term varies but yields equivalent constitutive equations.

Invariant definitions of transport coefficients are crucial for consistency.

Abstract

The conditions for the existence of the Chapman-Enskog first-order solution to the Boltzmann equation for a dilute gas are examined from two points of view. The traditional procedure is contrasted with a somehow more formal approach based on the properties of the linearized collision operator. It is shown that both methods lead to the same integral equation in the non-relativistic scenario. Meanwhile, for relativistic systems, the source term in the integral equation adopts two different forms. However, as we explain, this does not lead to an inconsistency. In fact, the constitutive equations that are obtained from both methods are shown to be equivalent within relativistic first-order theories. The importance of stating invariant definitions for the transport coefficients in this context is emphasized.