Abstract
The modified-moment method is applied to solve the Fokker-Planck equation in such a way that the solution is consistent with the requirements of the thermodynamic laws. The formal solution has an attendant mathematical structure for the entropy which is similar to that obtained from the Boltzmann equation. A generalized Gibbs relation is obtained for such a thermodynamic branch of solution. The formalism therefore may be used to study thermodynamic aspects of stochastic processes underlying the Fokker-Planck equation. The method is shown to yield the same analytical results as other methods in the case of linear processes, and in the case of nonlinear processes the moment series converges sufficiently fast to justify a lower-order truncation of the moment series for the distribution function, if the diffusion constant (D) is less than a critical value. In this case the solution of the Fokker-Planck equation tends to the steady-state distribution function. It is shown that the second variation of the nonequilibrium part of the entropy may be used as a Lyapunov function, which provides local criteria of evolution only.
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