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Reaction-diffusion equations for interacting particle systems. (English) Zbl 0629.60107
We study interacting spin (particle) systems on a lattice under the combined influence of spin flip (Glauber) and simple exchange (Kawasaki) dynamics. We prove that when the particle-conserving exchanges (stirrings) occur on a fast time scale of order \(\epsilon^{-2}\) the macroscopic density, defined on spatial scale \(\epsilon^{-1}\), evolves according to an autonomous nonlinear diffusion-reaction equation. Microscopic fluctuations about the deterministic macroscopic evolution are found explicitly. They grow, with time, to become infinite when the deterministic solution is unstable.

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
82B05 Classical equilibrium statistical mechanics (general)
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