Asymptotic Theory of the Boltzmann Equation

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Abstract
The precise mathematical relation that the Hilbert and Chapman-Enskog expansions bear to the manifold of solutions of the Boltzmann equation is described. These expansions yield inherently imprecise descriptions of a gas in terms of macroscopic fluid variables instead of a molecular distribution function. It is shown that these expansions are asymptotic to a very special class of solutions of the Boltzmann equation for sufficiently small mean free path. Next, a generalization of the Hilbert and Chapman-Enskog expansions is described in terms of extended sets of macroscopic state variables, each governed by partial differential equations similar to those found in fluid dynamics, but sufficiently general to approximate an arbitrary distribution function. The generalized expansions are shown to be asymptotic to quite arbitrary solutions of the Boltzmann equation. It is then shown that the ordinary Hilbert and Chapman-Enskog expansions can also be made asymptotic to very general solutions of the Boltzmann equation by reinterpreting the variables that enter these expansions as certain well-defined replacements for the actual fluid state of the gas. In this way the scope of the Euler, Navier-Stokes, Burnett equations, etc., is greatly extended by interpreting them as governing the artificial variables. Not only are general solutions of the Boltzmann equation shown to be approximated by fluid dynamics (in the limit of small mean free path), but the rapid decay of an arbitrary initial distribution function to a special Hilbert distribution function is also governed by sets of partial differential equations similar to those found in fluid dynamics.

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