Moment closure approximations of the Boltzmann Equation based on {\phi}-divergences

TL;DR

This paper introduces a generalized moment closure method for the Boltzmann equation using {\

Contribution

It extends classical entropy-based closures to {\

Findings

Enables construction of extended thermodynamic theories.

Avoids limitations like inadmissibility and singularities.

Produces symmetric hyperbolic systems preserving Boltzmann structure.

Abstract

This paper is concerned with approximations of the Boltzmann equation based on the method of moments. We propose a generalization of the setting of the moment-closure problem from relative entropy to {\phi}-divergences and a corresponding closure procedure based on minimization of {\phi}-divergences. The proposed description encapsulates as special cases Grad's classical closure based on expansion in Hermite polynomials and Levermore's entropy-based closure. We establish that the generalization to divergence-based closures enables the construction of extended thermodynamic theories that avoid essential limitations of the standard moment-closure formulations such as inadmissibility of the approximate phase-space distribution, potential loss of hyperbolicity and singularity of flux functions at local equilibrium. The divergence-based closure leads to a hierarchy of tractable symmetric hyperbolic systems that retain the fundamental structural properties of the Boltzmann equation.