Hydrodynamic limit for a spin system on a multidimensional lattice

The hydrodynamic limit for a Markov process of [0, ∞)-valued spin fields on a periodic multidimensional lattice is studied. In the process a positive real number, called energy, is attached to each site of the lattice and each couple of adjacent sites exchange thier energy by random amounts at random times. The law of the exchange is such that the sum of the total energy is conserved, and that the process is reversible and of gradient type for the energy distribution. We show that under diffusion type scaling of space and time, the macroscopic energy distribution converges to a deterministic limit which is characterized by a non-linear diffusion equation ∂ρ/∂t=2^-1ΔP(ρ), where P is an increasing function which in a typical case equals const·ρ².