Moment equations in the kinetic theory of gases and wave velocities

Abstract

We consider the evolution system for N-moments of the Boltzmann equation and we require the compatibility with an entropy law. This implies that the distribution function $f({\bf x},t,{\bf c})$ depends only on a single scalar variable $\chi$ which is a polynomial in ${\bf c}$ . It is then possible to construct the generators such that the system assumes a symmetric hyperbolic form in the main field. For an arbitrary $f(\chi)$ we prove that the systems obtained maximise the entropy density. If we require that the entropy coincides with the usual one of non-degenerate gases, we obtain an exponential function for $f(\chi)$ , which was already found by Dreyer. From these results the behaviour of the characteristic wave velocities for an increasing number of moments is studied and we show that in the classical theory the maximum velocity increases and tends to infinity, while in the relativistic case the wave and shock velocities are bounded by the speed of light.