Thermodynamic Properties of Non-Equilibrium States Subject to Fokker-Planck Equations

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We investigate the relation between fluctuation theory and thermodynamic theory of non-equilibrium states assuming the Fokker-Planck equation for the evolution law, and deduce two formulas corresponding to the entropy and entropy-production balance equations in thermodynamics I. ∂/(∂t)(-logψ/(ψ_0))ψ+∂/(∂x_λ)(J_λ(-logψ/(ψ_0))-V_λ)ψ=V_μX_μψ≥0, II. ∂/(∂t)(V_μX_μ)ψ+∂/(∂x_λ)(J_λV_μX_μ-2(∂V_λ)/(∂x_μ)V_μ)ψ=-2(T_<μν>V_μV_ν+(∂V_μ)/(∂x_ν)(∂V_ν)/(∂V_μ)ψ≤0. (φ and φ_0 are time-dependent and steady-state solutions of the F-P equation. They are also contained in J_λ, V_μ, X_μ and T_<μν>). The inequality in I represents the second law of thermodynamics (ensured automatically by the F-P equation) while that in II the evolution criterion for stability (not automatically ensured) reflecting the Glansdorff-Prigogine theory, and are considered to realize the thermodynamic proposition without recourse to the local-equilibrium assumption. We discuss briefly the laser instability as an example.
理論物理学刊行会の論文
1977-05-25