Limitations of Boltzmann's kinetic equation

Abstract

It is often assumed that Boltzmann's kinetic equation (BKE) for the evolution of the velocity distribution function f ( r ,  w ,  t ) in a gas is valid regardless of the magnitude of the Knudsen number defined by ϵ ≡ τ d ln  ϕ /d t , where ϕ is a macroscopic variable like the fluid velocity v or temperature T , and τ is the collision interval. Almost all accounts of transport theory based on BKE are limited to terms in O ( ϵ )≪1, although there are treatments in which terms in O ( ϵ 2 ) are obtained, classic examples being due to Burnett and Grad. The mathematical limitations that arise are discussed, for example, by Kreuzer and Cercignani. However, as we shall show, the physical limitation to BKE is that it is not valid for the terms of order higher than ϵ because the assumption of ‘molecular chaos’, which is the basis of Boltzmann's collision integral, is an approximation that applies only up to first order in ϵ . Another difficulty with Boltzmann's collision integral is that it is defined at a point, so that the varying ambient conditions upon which transport depends must be found by Taylor series expansions along particle trajectories. This fails in a strong-field magnetoplasma where, in a single collision interval, the trajectories are almost infinitely repeating gyrations; we shall illustrate this by deriving a dominant O ( ϵ 2 ) transport equation for a magnetoplasma that cannot be found from Boltzmann's equation. A further problem that sometimes arises in BKE occurs when an external force is present, the equilibrium state being constrained by the stringent Maxwell–Boltzmann conditions. Unless this is removed by a transformation of coordinates, confusion between convection and diffusion is probable. A mathematical theory for transport in tokamaks, termed neoclassical transport , is shown to be invalid, one of several errors being the retention of an electric field component in the drift kinetic equation.