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Determination of equilibrium parameters of the Marle model for polyatomic gases

Byung-Hoon Hwang

Abstract

The BGK model is a relaxation-time approximation of the celebrated Boltzmann equation, and the Marle model is a direct extension of the BGK model in a relativistic framework. In this paper, we introduce the Marle model for polyatomic gases based on the Jüttner distribution devised in [Ann. Phys., 377, (2017), 414--445], and show the existence of a unique set of equilibrium parameters of the Jüttner distribution.

Determination of equilibrium parameters of the Marle model for polyatomic gases

Abstract

The BGK model is a relaxation-time approximation of the celebrated Boltzmann equation, and the Marle model is a direct extension of the BGK model in a relativistic framework. In this paper, we introduce the Marle model for polyatomic gases based on the Jüttner distribution devised in [Ann. Phys., 377, (2017), 414--445], and show the existence of a unique set of equilibrium parameters of the Jüttner distribution.
Paper Structure (5 sections, 1 theorem, 45 equations)

This paper contains 5 sections, 1 theorem, 45 equations.

Table of Contents

  1. Introduction
  2. Relativistic extended thermodynamics of polyatomic molecules
  3. Marle model for polyatomic gases
  4. Main result
  5. Proof of theorem \ref{['main']}

Key Result

Theorem 1.1

Let $f$ be non-negative and not trivially zero so that $V_f^\mu$ exists. Assume that the state density is chosen as $\phi(\mathcal{I})=\mathcal{I}^\sigma$ with $\sigma >-1$. Then there exists a unique set of equilibrium parameters $n,U^\mu$ and $\gamma$ of $f_E$ satisfying the following identities: Indeed, $n=n_f$, $U^\mu=U_f^\mu$, and $\gamma$ is determined by the nonlinear relation: where

Theorems & Definitions (3)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2