Differential-Operator Approximations to the Linear Boltzmann Equation

Abstract

A measure of deviation from equilibrium of an ensemble of particles is proposed, which is physically appropriate and of especially simple form when expressed in terms of the expansion coefficients of the ensemble distribution function with respect to the system of orthogonal polynomials obtained by using the equilibrium distribution function as weight function. The linear Boltzmann operator can then be expanded in a series of terms which, under certain circumstances, may be regarded as of successively diminishing magnitude in their effect on the rate of approach to equilibrium. This expansion of the operator is different from the expansion due to Kramers (later discussed by Moyal) in derivate moments, commonly used in approximate stochastic treatments of irreversible processes. With the aid of a theorem on definite operators, it is possible to break off the series at any point and thereby obtain a correspondingly accurate approximation to the linear Boltzmann operator, whose temporal solutions tend to the correct equilibrium distribution function. The first approximation is the Fokker-Planck operator, exactly. The next approximation would be the appropriate operator to use when the stochastic variable begins to deviate appreciably from a linear dissipation law, etc. The method is applied to the ``Rayleigh process'' (ensemble of particles in a rarefied gas medium, the medium itself being in internal equilibrium), and the second approximation to the linear Boltzmann operator for this case is explicitly derived. A possible form for the second approximation in more general processes, suggested by this, is also given.