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Variational Approach in Studies with Fokker-Planck Equations

H. Hasegawa

1977
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Progress of theoretical physics
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vVc 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 :ohow
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... a specific point expressed as subsidiary conditions in the above forms originates from a determini:;tic 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. § I. Introduction The extremum property of a certain function or functional expressed in the form of variational principle, like in other fields of physics, is of fundamental significance in thermodynamic theory of irreversible processes.*) In 1931 On sager discussed in his paper on the reciprocity theorem 1 ) a variational principle which he attributed to the symmetry relations of the kinetic coefficients; he named it the princi}Jle of least dissipation of energy. It is expressed as a maximum principle of the form A(J) =S(J) -@(J, J) =maximum, where S denotes the time derivative of the entropy S (called the entrojJy produe tion), and {]) a positive quantity exhibiting a dissipation of the energy in the irreversible process (called the dissipation function), both represented as functions of the flux vector J as a variable of the above variation problem.**) Onsager pointed out that it could be compared with Boltzmann's entropy maximum principle for thermal equilibrium, predicting 2 l that a relation should exist which yields a probabilistic interpretation of the result of variation (1) just like S =!?log lV *l See Note added in proof 1. **l By the postulate that the irreversible process be represented by a path of deterministic motion in the space of state variables {x"}, the flux vector J1, is meant by dxjdt=:i, along the path, and S= (iJS/iJx1,):i"=X1,J1,(X"=iJS/iJx1,: the force vector), which shows that S(J) is a linear function of J. The variation (l) is to select such a path, called the "most probable path". For linear proces:;eo; the dissipation function mnst be quadratic, which is meant by @(], J).

doi:10.1143/ptp.58.128
fatcat:63yii6bqb5gptalc3leeovm3wa