Fluctuations and the Boltzmann equation. I
- 1 January 1976
- journal article
- research article
- Published by American Physical Society (APS) in Physical Review A
- Vol. 13 (1), 458-470
- https://doi.org/10.1103/physreva.13.458
Abstract
The evolution of a homogeneous dilute gas is treated as a Markov process in the complete set of coarsegrained velocity states of all particles. From the Siegert master equation for the process a Fokker-Planck equation is derived which describes, in the limit , the fluctuations in the occupation numbers , whose average behavior is governed by the (appropriately discretized) Boltzmann equation: The continuum limit corresponds to fluctuations in the usual molecular distribution function . 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.
Keywords
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