Lagrangian statistical mechanics applied to non-linear stochastic field equations

We consider non-linear stochastic field equations such as the KPZ equation for deposition and the noise driven Navier-Stokes equation for hydrodynamics. We focus on the Fourier transform of the time dependent two-point field correlation, Phi(k)(t). We employ a Lagrangian method aimed at obtaining the distribution function of the possible histories of the system in a way that fits naturally with our previous work on the static distribution. Our main result is a non-linear integro-differential equation for Phi(k)(t), which is derived from a Peierls-Boltzmann type transport equation for its Fourier transform in time Phik. That transport equation is a natural extension of the steady state transport equation, we previously derived for Phi(k)(0). We find a new and remarkable result which applies to all the non-linear systems studied here. The long time decay of Phi(k)(t) is described by Phi(k)(t) similar to exp(-a/k/t(gamma)), where a is a constant and gamma is system dependent. (C) 2002 Elsevier Science B.V. All rights reserved.