Variational Approach in Studies with Fokker-Planck Equations

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We show that, if a system in non-equilibrium states is to be described by a Fokker-Planck equation, its thermodynamic properties can be statistically formulated in such a way that two forms of the extremum condition on a Lagrangian functional to characterize the equation are constructed, which put the variational principles by Onsager and Prigogine into the statistical mechanical framework, in particular, that the second form is reduced to Prigogine's local potential method. We also show that a specific point expressed as subsidiary conditions in the above forms originates from a deterministic variational principle which results in the general Onsager-Machlup formula, and that it extends the Onsager-Machlup theory of founding the least dissipation principle to a scope beyond the linear Brownian motion. We demonstrate how the two forms can be utilized for solving Fokker-Planck equations in the spirit of the Rayleigh-Ritz method.
理論物理学刊行会の論文
1977-07-25