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

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.

18 pages Revtex file, including 8 eps-figures, submitted to Phys. Rev. E

References in corpus (7)

• Exact Large Deviation Function in the Asymmetric Exclusion Process
• Levy flights in quenched random force fields
• Quantized Scaling of Growing Surfaces
• Soliton approach to the noisy Burgers equation: Steepest descent method
• Morphology and scaling in the noisy Burgers equation: Soliton approach to the strong coupling fixed point
• Solitons and diffusive modes in the noiseless Burgers equation: Stability analysis
• Dynamical Scaling in Dissipative Burgers Turbulence

Cited by in corpus (39)

• Mapping out of equilibrium into equilibrium in one-dimensional transport models
• Large Deviations of Surface Height in the Kardar-Parisi-Zhang Equation
• Dynamical phase transition in large deviation statistics of the Kardar-Parisi-Zhang equation
• Large deviations of surface height in the $1+1$-dimensional Kardar-Parisi-Zhang equation: exact long-time results for $\lambda H<0$
• Explicit solution of the optimal fluctuation problem for an elastic string in random potential
• Short-time height distribution in 1d KPZ equation: starting from a parabola
• The Kardar-Parisi-Zhang equation in the weak noise limit: Pattern formation and upper critical dimension
• Localized growth modes, dynamic textures, and upper critical dimension for the Kardar-Parisi-Zhang equation in the weak noise limit
• Statistics of large currents in the Kipnis-Marchioro-Presutti model in a ring geometry
• Landau theory of the short-time dynamical phase transitions of the Kardar-Parisi-Zhang interface
• Height distribution tails in the Kardar-Parisi-Zhang equation with Brownian initial conditions
• Full Statistics of Nonstationary Heat Transfer in the Kipnis-Marchioro-Presutti Model
• Exact short-time height distribution for the flat Kardar-Parisi-Zhang interface
• Dynamics of DNA-breathing: Weak noise analysis, finite time singularity, and mapping onto the quantum Coulomb problem
• Finite-size effects in the short-time height distribution of the Kardar-Parisi-Zhang equation
• Large fluctuations of a Kardar-Parisi-Zhang interface on a half-line
• Power laws and stretched exponentials in a noisy finite-time-singularity model
• Large fluctuations of a Kardar-Parisi-Zhang interface on a half-line: the height statistics at a shifted point
• Time-averaged height distribution of the Kardar-Parisi-Zhang interface
• Intermittency of Height Fluctuations and Velocity Increment of The Kardar-Parisi-Zhang and Burgers Equations with infinitesimal surface tension and Viscosity in 1+1 Dimensions
• Solitons in the noisy Burgers equation
• Statistical Theory for the Kardar-Parisi-Zhang Equation in 1+1 Dimension
• Correlations, soliton modes, and non-Hermitian linear mode transmutation in the 1D noisy Burgers equation
• Symmetries and zero modes in sample path large deviations
• Nonequilibrium Steady State of a Weakly-Driven Kardar-Parisi-Zhang Equation
• Domain wall propagation and nucleation in a metastable two-level system
• Nonequilibrium dynamics of a growing interface
• Wind-Driven Gas Networks and Star Formation in Galaxies: Reaction-Advection Hydrodynamic Simulations
• Damped finite-time-singularity driven by noise
• Scaling function for the noisy Burgers equation in the soliton approximation
• Canonical phase space approach to the noisy Burgers equation
• Local average height distribution of fluctuating interfaces
• Weak noise approach to the logistic map
• Soliton-dynamical approach to a noisy Ginzburg-Landau model
• Patterns in the Kardar-Parisi-Zhang equation
• Pattern Selection in a Phase Field Model for Directional Solidification
• Towards a strong coupling theory for the KPZ equation
• Exact short-time height distribution and dynamical phase transition in the relaxation of a Kardar-Parisi-Zhang interface with random initial condition
• Systematic analysis of critical exponents in continuous dynamical phase transitions of weak noise theories