Effective Action Method for Langevin Equation

Abstract

In this paper we present a formulation of the nonlinear stochastic differential equation which allows for systematic approximations. The method is not restricted to the asymptotic, i.e., stationary, regime but can be applied to derive effective equations describing the relaxation of the system from arbitrary initial conditions. The basic idea is to reduce the nonlinear Langevin equation to an equivalent equilibrium problem, which can then be studied with the methods of conventional equilibrium statistical field theory. A particular well suited perturbative scheme is that developed in quantum field theory by Cornwall, Jackiw and Tomboulis. We apply this method to the study of $N$ component Ginzburg-Landau equation in zero spatial dimension. In the limit of $N\to\infty$ we can solve the effective equations and obtain closed forms for the time evolutions of the average field and of the two-time connected correlation functions.