Auxiliary field loop expansion of the effective action for a class of stochastic partial differential equations

Abstract

We present an alternative to the perturbative (in coupling constant) diagrammatic approach for studying stochastic dynamics of a class of reaction diffusion systems. Our approach is based on an auxiliary field loop expansion for the path integral representation for the generating functional of the noise induced correlation functions of the fields describing these systems. The systems we consider include Langevin systems describable by the set of self interacting classical fields ϕi(x,t) in the presence of external noise ηi(x,t), namely (∂t−ν∇2)ϕ−F[ϕ]=η, as well as chemical reaction annihilation processes obtained by applying the many-body approach of Doi–Peliti to the Master Equation formulation of these problems. We consider two different effective actions, one based on the Onsager–Machlup (OM) approach, and the other due to Janssen–deGenneris based on the Martin–Siggia–Rose (MSR) response function approach. For the simple models we consider, we determine an analytic expression for the Energy landscape (effective potential) in both formalisms and show how to obtain the more physical effective potential of the Onsager–Machlup approach from the MSR effective potential in leading order in the auxiliary field loop expansion. For the KPZ equation we find that our approximation, which is non-perturbative and obeys broken symmetry Ward identities, does not lead to the appearance of a fluctuation induced symmetry breakdown. This contradicts the results of earlier studies.