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, Φk(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 Φk(t), which is derived from a Peierls–Boltzmann type transport equation for its Fourier transform in time Φk,ω. That transport equation is a natural extension of the steady state transport equation, we previously derived for Φk(0). We find a new and remarkable result which applies to all the non-linear systems studied here. The long time decay of Φk(t) is described by Φk(t)∼exp(−a|k|tγ), where a is a constant and γ is system dependent.