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

Carlos Gabarrete, Ana Laura García-Perciante, Olivier Sarbach

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 Jüttner 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.

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

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 Jüttner 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.

Paper Structure

This paper contains 25 sections, 6 theorems, 166 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Local equilibrium configurations
  4. Nonrelativistic and ultrarelativistic limits
  5. Monotonicity properties with respect to the temperature
  6. Knudsen number, hydrodynamic limit and Chapman-Enskog approximation
  7. Rescaling of the Boltzmann equation
  8. The weak field hydrodynamic approximation and the Hilbert expansion
  9. The linearized collision operator (formal definition)
  10. The Hilbert method
  11. The Chapman-Enskog projection method
  12. Changes of frame and representation
  13. Mathematical discussion of the linearized collision operator
  14. The first-order theory resulting from the Chapman-Enskog projection method
  15. The trace-fixed particle frame
  16. ...and 10 more sections

Key Result

Lemma 1

Let $d\geq 2$. Then, the function $G_{-1}: [0,\infty)\to [0,1)$ is strictly monotonously increasing and invertible. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (6)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Lemma 6