Abstract
We review recent developments in the rigorous derivation of hydrodynamic-type macroscopic equations from simple microscopic models: continuous time stochastic cellular automata. The deterministic evolution of hydrodynamic variables emerges as the “law of large numbers,” which holds with probability one in the limit in which the ratio of the microscopic to the macroscopic spatial and temporal scales go to zero. We also study fluctuations in the microscopic system about the solution of the macroscopic equations. These can lead, in cases where the latter exhibit instabilities, to complete divergence in behavior between the two at long macroscopic times. Examples include Burgers' equation with shocks and diffusion-reaction equations with traveling fronts.
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Lebowitz, J.L., Presutti, E. & Spohn, H. Microscopic models of hydrodynamic behavior. J Stat Phys 51, 841–862 (1988). https://doi.org/10.1007/BF01014887
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DOI: https://doi.org/10.1007/BF01014887