The hydrodynamic limit for a system with interactions prescribed by Ginzburg-Landau type random Hamiltonian

As a microscopic model we consider a system of interacting continuum like spin field over R^d. Its evolution law is determined by the Ginzburg-Landau type random Hamiltonian and the total spin of the system is preserved by this evolution. We show that the spin field converges, under the hydrodynamic space-time scalling, to a deterministic limit which is a solution of a certain nonlinear diffusion equation. This equation describes the time evolution of the macroscopic field. The hydrodynamic scaling has an effect of the homogenization on the system at the same time.