Abstract
We consider discrete lattice gas models in a finite interval with stochastic jump dynamics in the interior, which conserve the particle number, and with stochastic dynamics at the boundaries chosen to model infinite particle reservoirs at fixed chemical potentials. The unique stationary measures of these processes support a steady particle current from the reservoir of higher chemical potential into the lower and are non-reversible. We study the structure of the stationary measure in the hydrodynamic limit, as the microscopic lattice size goes to infinity. In particular, we prove as a law of large numbers that the empirical density field converges to a deterministic limit which is the solution of the stationary transport equation and the empirical current converges to the deterministic limit given by Fick's law.
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Communicated by A. Jaffe
Dedicated to Res Jost and Arthur Wightman
Supported in part by NSF Grants DMR 89-18903 and INT 8521407. H.S. also supported by the Deutsche Forschungsgemeinschaft
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Eyink, G., Lebowitz, J.L. & Spohn, H. Hydrodynamics of stationary non-equilibrium states for some stochastic lattice gas models. Commun.Math. Phys. 132, 253–283 (1990). https://doi.org/10.1007/BF02278011
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DOI: https://doi.org/10.1007/BF02278011