Time evolution of an initial density discontinuity in a Knudsen gas

Abstract

The equilibrium and non-equilibrium behavior of gases are old and incompletely solved problems in classical statistical mechanics. Attempts to solve these problems have inspired a variety of numerical and analytical approaches. The usual methods for studying the time evoution of gases usually start with kinetic and transport equations derived from the Boltzmann equation, e.g. Navier-Stokes equation and linear diffusion equation, or with transport equations derived from perturbative 'corrections' to the Bolzmann equation, e.g. Burnett and super-Burnett equations. Just a year ago, however, Du, Li, and Kadanoff [Phys. Rev. Lett. 74, 1268] exhibited that, in one dimension, the results of macroscopic hydrodynamics do not agree with the results of molecular dynamics in one-dimension. Because of this, any analytical method not starting from traditional bases like the Navier-Stokes and diffusion equations is of current interest.
In this paper, we derive exact expressions for the time evolution of the Knudsen gas — a confined gas whose constituents pass through each other and which react, through collisions, only with their container. The approach to equilibrium of the Knudsen gas has never been shown analytically.