Canonical phase space approach to the noisy Burgers equation: Probability distributions

TL;DR

This paper introduces a canonical phase space method for analyzing stochastic systems like the noisy Burgers and KPZ equations, deriving probability distributions through a least action principle, with results matching known models.

Contribution

It develops a nonperturbative Hamilton-Jacobi framework for stochastic PDEs, providing new insights into their probability distributions and long-time behaviors.

Findings

Derived long-time skew distribution approaching Gaussian

Obtained short-time distribution with soliton contributions

Discussed higher-dimensional distribution behavior

Abstract

We present a canonical phase space approach to stochastic systems described by Langevin equations driven by white noise. Mapping the associated Fokker-Planck equation to a Hamilton-Jacobi equation in the nonperturbative weak noise limit we invoke a {\em principle of least action} for the determination of the probability distributions. We apply the scheme to the noisy Burgers and KPZ equations and discuss the time-dependent and stationary probability distributions. In one dimension we derive the long-time skew distribution approaching the symmetric stationary Gaussian distribution. In the short-time region we discuss heuristically the nonlinear soliton contributions and derive an expression for the distribution in accordance with the directed polymer-replica and asymmetric exclusion model results. We also comment on the distribution in higher dimensions.