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 , i.e.

,

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

was derived and solved to give a power law solution in an intermediate

range of k, i.e. .

In this paper (joint work with Moshe Shwartz), the probability distribution for

or 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 .