How to obtain higher-order multivariate Hermite expansion of Maxwell-Boltzmann distribution by using Taylor expansion?

Abstract

We obtain the higher-order multivariate Hermite expansion of the Maxwell-Boltzmann distribution by using a new, compact tensorial notation and present a method to obtain the nth order multivariate Taylor expansion, which is identical to the nth order multivariate Hermite expansion of the Maxwell-Boltzmann distribution. This study enables us to find higher-order models of discrete kinetic theories such as the lattice Boltzmann theory.