Lagrangian statistical mechanics applied to non-linear stochastic field equations

Non-linear field equations such as the KPZ equation for deposition and the Navier-Stokes equation for hydrodynamics are discussed by the derivation of transport equations for the correlation function of the field h(r,t), i.e., (h(r,t) h(r',t')), where h satisfies a diffusion equation driven by a noise, f, defined as noise with a given spectrum, and containing a non linear term, Mhh, which couples to the field itself. In previous work, an equation for the steady state correlation function 𝜙(r-r') = (h(r,t) h(r',t')) was derived and solved to give a power law solution in an intermediate range of k, i.e. 𝜙(k) ˜ |k|^α.

In this paper (joint work with Moshe Shwartz), the probability distribution for h(r,t) or h_k,w is derived, the procedure having the same relation to the static distribution as Lagrangian mechanics has to the Hamiltonian. A conservative system has a static solution exp(-H/kT), but there is no equivalent for the distribution of histories, so there is little study of this approach. However, since approximations are essential, the Lagrangian method is used, and is more powerful than the usual Hamiltonian → Liouville's equation → Boltzmann equation route.

The approximate equation is derived and solved in the usual conditions, and takes the form 𝜙_k(t) = 𝜙_kexp(-kt^γ).