Hydrodynamic limit for a spin system on a multidimensional lattice
The hydrodynamic limit for a Markov process of \([0,\infty)\)-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 their 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 nonlinear diffusion equation \(\partial \rho/ \partial t=2^{-1} \Delta P (\rho)\), where \(P\) is an increasing function which in a typical case equals \(\text{const} \cdot \rho^ 2\).
scientific article; zbMATH DE number 569038