Kinetic theory for a relativistic charged gas: mathematical foundations of the hydrodynamic limit and first-order results within the projection method

TL;DR

This paper develops a relativistic kinetic theory framework for charged gases, deriving first-order hydrodynamic equations using the projection method and establishing their mathematical and physical properties.

Contribution

It introduces a novel relativistic generalization of the projection method within the Chapman-Enskog approximation for charged gases in arbitrary spacetimes.

Findings

Derived frame-independent transport coefficients.

Established a hyperbolic, causal, and stable relativistic fluid theory.

Provided a systematic microscopic derivation of first-order constitutive equations.

Abstract

In this work, the first-order constitutive equations for a relativistic charged gas are obtained based on the Chapman-Enskog expansion of near-equilibrium solutions to the Boltzmann equation by implementing the projection method. To this purpose we consider an arbitrary fixed background spacetime and electromagnetic field, and present a novel procedure within the Chapman-Enskog approximation which is the relativistic generalization of the projection method developed in the Newtonian case. Motivated by a rigorous study of the linearized collision operator, we argue that the most natural frame to derive a relativistic dissipative fluid theory from kinetic theory is the trace-fixed particle frame. This frame determines the state variables by requiring compatibility of the first few moments of the one-particle distribution function with those of the J\"uttner distribution. The resulting constitutive relations are derived, and it is shown that the corresponding transport coefficients are, in fact, frame-independent if suitably defined. We point out that, in frame, there exists an additional freedom which we refer to as a choice of representation. Moreover, we discuss how both of these choices can be implemented at the microscopic level within the projection method. This allows for a systematic derivation of general first-order constitutive equations starting from the ones obtained in the trace-fixed particle frame. We obtain a fluid theory which, in the trace-fixed particle frame and for a suitable choice of the parameters associated with the representation freedom, is strongly hyperbolic, causal, and for which global equilibrium states are stable.