Abstract
The large deviation properties of equilibrium (reversible) lattice gases are mathematically reasonably well understood. Much less is known in nonequilibrium, namely for nonreversible systems. In this paper we consider a simple example of a nonequilibrium situation, the symmetric simple exclusion process in which we let the system exchange particles with the boundaries at two different rates. We prove a dynamical large deviation principle for the empirical density which describes the probability of fluctuations from the solutions of the hydrodynamic equation. The so-called quasi potential, which measures the cost of a fluctuation from the stationary state, is then defined by a variational problem for the dynamical large deviation rate function. By characterizing the optimal path, we prove that the quasi potential can also be obtained from a static variational problem introduced by Derrida, Lebowitz, and Speer.
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Bertini, L., De Sole, A., Gabrielli, D. et al. Large Deviations for the Boundary Driven Symmetric Simple Exclusion Process. Mathematical Physics, Analysis and Geometry 6, 231–267 (2003). https://doi.org/10.1023/A:1024967818899
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DOI: https://doi.org/10.1023/A:1024967818899