Hydrodynamics for symmetric exclusion in contact with reservoirs

TL;DR

This paper studies how the symmetric exclusion process behaves under different reservoir strengths and jump distributions, deriving various hydrodynamic equations including reaction-diffusion and fractional equations.

Contribution

It characterizes the hydrodynamic limits of the symmetric exclusion process with reservoirs, revealing new boundary conditions and fractional equations based on jump variance and reservoir strength.

Findings

Finite variance jumps lead to reaction or heat equations with boundary conditions.

Infinite variance jumps result in fractional reaction-diffusion equations.

Reservoir strength determines boundary conditions and type of hydrodynamic limit.

Abstract

We consider the symmetric exclusion process with jumps given by a symmetric, translation invariant, transition probability $p(\cdot)$. The process is put in contact with stochastic reservoirs whose strength is tuned by a parameter $\theta\in\mathbb R$. Depending on the value of the parameter $\theta$ and the range of the transition probability $p(\cdot)$ we obtain the hydrodynamical behavior of the system. The type of hydrodynamic equation depends on whether the underlying probability $p(\cdot)$ has finite or infinite variance and the type of boundary condition depends on the strength of the stochastic reservoirs, that is, it depends on the value of $\theta$. More precisely, when $p(\cdot)$ has finite variance we obtain either a reaction or reaction-diffusion equation with Dirichlet boundary conditions or the heat equation with different types of boundary conditions (of Dirichlet, Robin or Neumann type). When $p(\cdot)$ has infinite variance we obtain a fractional reaction-diffusion equation given by the regional fractional Laplacian with Dirichlet boundary conditions but for a particular strength of the reservoirs.