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

Hans C. Fogedby
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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.

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Hans C. Fogedby. 1999-01-04. Canonical phase space approach to the noisy Burgers equation: Probability distributions. https://doi.org/10.1103/physreve.59.5065

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