Fluctuations and the Boltzmann equation. I

Abstract

The evolution of a homogeneous dilute gas is treated as a Markov process in the complete set of $K$ coarsegrained velocity states of all $N$ particles. From the Siegert master equation for the process a Fokker-Planck equation is derived which describes, in the limit $N\ensuremath{\rightarrow}\ensuremath{\infty}$, the fluctuations in the occupation numbers ${n}_{i}(t)$, whose average behavior is governed by the (appropriately discretized) Boltzmann equation: The continuum limit $K\ensuremath{\rightarrow}\ensuremath{\infty}$ corresponds to fluctuations in the usual molecular distribution function $f(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}\stackrel{\ensuremath{\rightarrow}}{\mathrm{v}};t)$. On similar reasoning, a Fokker-Planck equation is obtained for the fluctuation process near equilibrium, where the average is governed by the linearized Boltzmann equation. The theory of linear irreversible processes, which offers a statistical description of fluctuations on a thermodynamical basis, is applied to the linearized Boltzmann equation---treated as a linear phenomenological equation---following the development given recently by Fox and Uhlenbeck: The resulting stochastic equation is seen to be equivalent to the Fokker-Planck equation obtained from the master equation, yielding a multidimensional Ornstein-Uhlenbeck process which describes the fluctuations in molecular phase space.