Variational Principle for Non-Equilibrium States and the Onsager-Machlup Formula

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Abstract
We consider diffusion processes with a general second-order differential operator of the elliptic type to extend the Onsager-Machlup theory of Gaussian Markoff processes, and establish the two-gate joint probability formula log W(x(1)t1, x(2)t2) = (1/2)(log ρ0(x(1)t1)+log ρ0(x(2)t2)) - ∫t2t1 (1/2)(φ{ρ,V(S)}+ψ{ρ,X(ρ)})optimdt+const, where φ{ρ,V(S)} and ψ{ρ,X(ρ)} are the dissipation functional expressed as a quadratic form of the velocity field V(S) and the force field X(ρ), respectively, averaged over the distribution ρ. Here the ρ and S are the two conjugate scalar fields (functions of state variables x as well as of time t) in the variational formulation for the pair of diffusion equations analogous to those for the Schrödinger equations, whose explicit forms are given in terms of a solution of the diffusion equations under a specified boundary condition.

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