Action Principle in Nonequilibrium Statistical Dynamics

Abstract

We introduce a variational method for approximating distribution functions of dynamics with a ``Liouville operator'' $\hL,$ in terms of a {\em nonequilibrium action functional} for two independent (left and right) trial states. The method is valid for deterministic or stochastic Markov dynamics, and for stationary or time-dependent distributions. A practical Rayleigh-Ritz procedure is advanced, whose inputs are finitely-parametrized ansatz for the trial states, leading to a ``parametric action'' for their evolution. The Euler-Lagrange equations of the action principle are Hamiltonian in form (generally noncanonical). This permits a simple identification of fixed points as critical points of the parametric Hamiltonian. We also establish a variational principle for low-order statistics, such as mean values and correlation functions, by means of {\em least effective action.} The latter is a functional of the given variable, which is positive and convex as a consequence of Hölder realizability inequalities. Its value measures the ``cost'' for a fluctuation from the average to occur and in a weak-noise limit it reduces to the Onsager-Machlup action. In general, the effective action is shown to arise from the nonequilibrium action functional by a constrained variation. This result provides a Rayleigh-Ritz scheme for calculating just the desired low-order statistics, with internal consistency checks less demanding than for the full distribution.

46 pages, LaTex Version 2.09, tar + gzip + uuencode The tar file contains xyzvar.tex, the LaTeX file itself, and seceq.sty, a stylefile for sequential numering of equations by section. The file seceq.sty will automatically be input from the same directory when formatting the LaTeX file. The paper is submitted to Phys. Rev. E

References in corpus (2)

• Rayleigh-Ritz Calculation of Effective Potential Far From Equilibrium
• Effective Action Method for Langevin Equation

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