Effective Action Method for the Langevin Equation

A general approach to nonlinear stochastic equations with white noise is proposed. It consists of a path integral representation of the nonlinear Langevin equation and allows for systematic approximations. The present method is not restricted to the asymptotic, i.e., stationary, regime and is suited for deriving equations describing the relaxation of a system from arbitrary initial conditions. After reducing the nonlinear Langevin equation to an equivalent equilibrium problem for the generating functional, we are able to apply known techniques of conventional equilibrium statistical field theory. We extend the effective action method developed in quantum field theory by Cornwall, Jackiw, and Tomboulis [Phys. Rev. D 10, 2428 (1974)] to nonequilibrium processes. Arguments are given as to its superiority over perturbative schemes. These are illustrated by studying an N-component Ginzburg-Landau equation in zero spatial dimension in the limit of large N. Within this limit we show the equivalence of the lowest order approximation, i.e., the dressed loop expansion, with the Gaussian variational ansatz for the effective potential, which leads to the dynamical Hartree approximation