Current reservoirs in the simple exclusion process

Abstract

We consider the symmetric simple exclusion process in the interval $[-N,N]$ with additional birth and death processes respectively on $(N-K,N]$, $K>0$, and $[-N,-N+K)$. The exclusion is speeded up by a factor $N^2$, births and deaths by a factor $N$. Assuming propagation of chaos (a property proved in a companion paper "Truncated correlations in the stirring process with births and deaths") we prove convergence in the limit $N\to \infty$ to the linear heat equation with Dirichlet condition on the boundaries; the boundary conditions however are not known a priori, they are obtained by solving a non linear equation. The model simulates mass transport with current reservoirs at the boundaries and the Fourier law is proved to hold.

References in corpus (1)

• A diffusive system driven by a battery or by a smoothly varying field

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• Super-hydrodynamic limit in interacting particle systems
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• Large Deviations in the Symmetric Simple Exclusion Process with Slow Boundaries: A Hydrodynamic Perspective
• Thermodynamics of nonequilibrium driven diffusive systems in mild contact with boundary reservoirs
• The boundary driven zero-range process
• Fick law and sticky Brownian motions
• Exponential rate of convergence in current reservoirs
• Extinction time for a random walk in a random environment
• Hydrodynamics of a particle model in contact with stochastic reservoirs
• Hydrodynamics for asymmetric simple exclusion on a finite segment with Glauber-type source