Summary
As a microscopic model we consider a system of interacting continuum like spin field overR 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.
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Funaki, T. The hydrodynamic limit for a system with interactions prescribed by Ginzburg-Landau type random Hamiltonian. Probab. Th. Rel. Fields 90, 519–562 (1991). https://doi.org/10.1007/BF01192142
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DOI: https://doi.org/10.1007/BF01192142