Canonical phase space approach to the noisy Burgers equation

TL;DR

This paper introduces a phase space method to analyze the noisy Burgers equation, deriving probability distributions and connecting to models like directed polymers and exclusion processes.

Contribution

It develops a canonical phase space framework for the stochastic Burgers equation, providing new insights into its probability distributions and nonlinear soliton effects.

Findings

Derived the long-time skew distribution approaching Gaussian

Connected short-time distributions to directed polymer and exclusion models

Analyzed stationary and time-dependent probability distributions

Abstract

Presenting a general phase approach to stochastic processes we analyze in particular the Fokker-Planck equation for the noisy Burgers equation 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 regime we discuss heuristically the nonlinear soliton contributions and derive an expression for the distribution in accordance with the directed polymer-replica model and asymmetric exclusion model results.