Functional and graphical methods for classical statistical dynamics. I. A formulation of the Martin–Siggia–Rose method

Abstract

A formulation of the Martin–Siggia–Rose (MSR) method for describing the statistical dynamics of classical systems is presented. The present formulation is very similar in structure to the original MSR “operator” formalism and very different from the alternative functional integral formalism of Janssen, de Dominicis, Peliti, and others. The need for imposing certain boundary conditions in the MSR formalism, as pointed out by Deker, is clarified. The basic results of this paper include: a construction of the MSR formalism in a way that demonstrates its internal consistency; a definition of a functional whose functional derivatives give all the correlation functions and response functions of an ensemble of mechanical systems; a graphical expression for the correlation functions and response functions; a graphical expression for the Legendre transform of the functional and of the resulting vertex functions; and a graphical derivation of the appropriate Dyson equation. The present formulation is applicable to systems with highly non-Gaussian statistics, including systems of classical particles described in terms of the particle density in single-particle phase space. In this paper, we consider only the case of ensembles of systems whose coordinates are continuous and whose time evolution is described by deterministic first order differential equations that are local in time. The method is easily extended to systems whose dynamics is governed by stochastic differential equations and to spin systems.