Finite reservoirs lead to Wentzell boundary conditions for independent random walks and exclusion process

TL;DR

This paper studies the hydrodynamic limits of particle systems with reservoirs, revealing how boundary conditions depend on a parameter and establishing an equivalence between Wentzell and non-local Dirichlet boundary conditions.

Contribution

It characterizes the boundary conditions for hydrodynamic limits of independent random walks and exclusion processes with finite reservoirs, including a phase transition at a critical parameter value.

Findings

Hydrodynamic limit yields heat equation with Neumann boundary at the right.

Boundary condition at the left depends on parameter , showing phase transition.

At critical parameter, boundary condition becomes non-local and nonlinear.

Abstract

We analyze the scaling limits (hydrodynamic limit/propagation of local equilibrium) of two particle systems in the discrete one-dimensional segment where the left boundary is in contact with a reservoir, which may stow any (finite) number of particles. These two particle systems are independent random walks and the symmetric exclusion process. At rate one a particle (if there is one there) jumps from site $1$ to a finite reservoir, and at rate $\alpha \eta(0)N^{-\theta}$ a particle jumps from the finite reservoir to the site $1$ (if the site $1$ is empty in the exclusion case), where $\eta(0)$ is the total number of particles in the reservoir at that moment and $\theta\geq 0$ is a parameter whose tuning leads to a dynamical phase transition. For all values of $\theta$, the hydrodynamic equation is the heat equation with Neumann b.c. at the right boundary for both systems. On the other hand, the left boundary condition depends on the chosen value of $\theta$. For $\theta\in [0,1)$, it is given by the Neumann b.c., which means that the deposit is asymptotically empty, acting as a barrier. For $\theta\in (1,\infty)$, in the random walk scenario, it is given by a non-homogeneous Dirichlet boundary condition, which means that the reservoir becomes asymptotically infinite, acting as a heat bath, while in the exclusion scenario it is given by a homogeneous Dirichlet boundary condition, meaning that the reservoir behaves as a sink. Finally, at the critical value $\theta=1$, we obtain a non-local Dirichlet boundary condition relating the value at zero to the total mass of the system, which is additionally non-linear in the exclusion scenario. As a by-product of these results, we find an equivalence between solutions to the heat equation with Wentzell boundary conditions and solutions to the heat equation with certain non-local Dirichlet boundary conditions related to the total mass of the system.