Action Principle in Nonequilibrium Statistical Dynamics

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 H"{o}lder 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.