The relativistic kinetic dispersion relation: Comparison of the relativistic Bhatnagar-Gross-Krook model and Grad's 14-moment expansion

TL;DR

This paper compares relativistic kinetic models and the 14-moment theory, proposing modifications to improve accuracy and analyzing their dispersion relations to assess their physical validity.

Contribution

It introduces a modified Marle model for better transport coefficients and evaluates the effectiveness of second-order terms in the 14-moment theory.

Findings

Modified Marle model aligns better with Boltzmann equation results.

14-moment theory's dispersion relation does not match kinetic equation outcomes.

Second-order terms in 14-moment theory do not necessarily improve physical description.

Abstract

In this paper, we study the Cauchy problem of the linearized kinetic equations for the models of Marle and Anderson-Witting, and compare these dispersion relations with the 14-moment theory. First, we propose a modification of the Marle model to improve the resultant transport coefficients in accord with those obtained by the full Boltzmann equation. Using the modified Marle model and Anderson-Witting model, we calculate dispersion relations that are kinetically correct within the validity of the BGK approximation. The 14-moment theory that includes the time derivative of dissipation currents has causal structure, in contrast to the acausal first-order Chapman-Enskog approximation. However, the dispersion relation of the 14-moment theory does not accurately describe the result of the kinetic equation. Thus, our calculation indicates that keeping these second-order terms does not simply correspond to improving the physical description of the relativistic hydrodynamics.