Symmetry Properties of the Linear Boltzmann or Master Operator in the Form of a Differential-Operator Expansion

Abstract

A method for obtaining nonlinear characteristics of the linear Boltzmann or Master operator is given. In order that the probability distribution be the standardized Gaussian distribution at equilibrium, the Hermite coefficients of derivate moments satisfy algebraic relations which are altogether equivalent to the principle of detailed balance. The number of independent Hermite coefficients of the tensorial order n is n⁄2 when n is even and (n−1)⁄2 when n is odd. The relation of the tensorial order two reduces to the Einstein relation and corresponds to linear fluctuations. Relations of the higher order represent nonlinear effects. These relations are independent of the detailed mechanism of processes. They hold for N-dimensional processes, where N is finite. The theory is then applied to the three-dimensional Rayleigh model.