Finite reservoirs lead to Wentzell boundary conditions for independent random walks and exclusion process
We analyze the scaling limits (hydrodynamics/propagation of local equilibrium) of two particle systems, namely independent random walks (IRW) and symmetric exclusion processes (SEP), both in the discrete one-dimensional segment where the left boundary is in contact with a reservoir, which may hold any number of particles. At rate one a particle jumps from site $1$ to the reservoir, and at rate $αη(0) N^{-θ}$ a particle jumps from the reservoir to site $1$ (if site $1$ is empty in SEP), where $η(0)$ is the total number of particles in the reservoir and $θ\geq 0$ is a parameter whose tuning leads to a dynamical phase transition. For every $θ\geq 0$, the hydrodynamic equation is the heat equation with Neumann boundary conditions at the right boundary for both processes, while the boundary conditions at the left boundary depend on the value of $θ$. For $0\leq θ<1$, it is given by Neumann boundary conditions, meaning the deposit is asymptotically empty, acting as a barrier. For $θ>1$, and for IRW (resp. SEP) it is given by a non-homogeneous (resp. homogeneous) Dirichlet boundary condition, meaning the reservoir becomes asymptotically infinite, acting as a heat bath (resp. the reservoir behaves as a sink). For $θ=1$, we obtain a non-local Dirichlet boundary condition, which is additionally non-linear for SEP. As a by-product, we obtain an equivalence between solutions to the heat equation with Wentzell boundary conditions and with non-local Dirichlet boundary conditions related to the total mass of the system.