Moment Closure Hierarchies for Kinetic Theories

TLDR: In this article, a hierarchy of closed systems of moment equations corresponding to any classical kinetic theory is derived, and the first member of the hierarchy is the Euler system based on Maxwellian velocity distributions, while the second member is based on nonisotropic Gaussian velocity distributions.

Abstract: This paper presents a systematicnonperturbative derivation of a hierarchy of closed systems of moment equations corresponding to any classical kinetic theory. The first member of the hierarchy is the Euler system, which is based on Maxwellian velocity distributions, while the second member is based on nonisotropic Gaussian velocity distributions. The closure proceeds in two steps. The first ensures that every member of the hierarchy is hyperbolic, has an entropy, and formally recovers the Euler limit. The second involves modifying the collisional terms so that members of the hierarchy beyound the second also recover the correct Navier-Stokes behavior. This is achieved through the introduction of a generalization of the BGK collision operator. The simplest such system in three spatial dimensions is a “14-moment” closure, which also recovers the behavior of the Grad “13-moment” system when the velocity distributions lie near local Maxwellians. The closure procedure can be applied to a general class of kinetic theories.