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, $\Phi_{\bf{k}}(t)$. 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 $\Phi_{\bf{k}}(t)$, which is derived from a Peierls-Boltzmann type transport equation for its Fourier transform in time $\Phi_{\bf{k}, \omega}$. That transport equation is a natural extension of the steady state transport equation, we previously derived for $\Phi_{\bf{k}}(0)$. We find a new and remarkable result which applies to all the non-linear systems studied here. The long time decay of $\Phi_{\bf{k}}(t)$ is described by $\Phi_{\bf{k}}(t) \sim \exp(-a|{\bf k}|t^{\gamma})$, where $a$ is a constant and $\gamma$ is system dependent.