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

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 $αη(0)N^{-θ}$ a particle jumps from the finite reservoir to the site $1$ (if the site $1$ is empty in the exclusion case), where $η(0)$ is the total number of particles in the reservoir at that moment and $θ\geq 0$ is a parameter whose tuning leads to a dynamical phase transition. For all values of $θ$, 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 $θ$. For $θ\in [0,1)$, it is given by the Neumann b.c., which means that the deposit is asymptotically empty, acting as a barrier. For $θ\in (1,\infty)$, 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 $θ=1$, 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.

Figures (6)

• Figure 1: Illustration of jump rates (without the diffusive scaling parameter $N^2$) for independent random walks in contact with a finite reservoir.
• Figure 2: Model of diffusion for the parabolic PDE \ref{['weak_solution_non_local']}. The segment of length $1/\alpha$ represents the finite reservoir in contact with the left boundary of the interval $[0,1]$.
• Figure 3: Illustration of jump rates (without the diffusive scaling parameter $N^2$) for the symmetric exclusion process in contact with a finite reservoir.
• Figure 4: Model of diffusion corresponding to the parabolic PDE \ref{['non_linear_Dirichlet']}. The segment of length $1/\alpha$ represents the finite reservoir in contact with the left boundary of the box $[0,1]$.
• Figure 5: A second model of diffusion corresponding to the parabolic PDE \ref{['non_linear_Dirichlet']}. The segment of length $1/(\alpha(1-\rho(t,0)))$ represents the finite reservoir whose capacity is a function of its internal height.

Theorems & Definitions (49)

• Proposition 2.1
• Theorem 2.2: Propagation of Local Equilibrium
• Definition 2.3
• Proposition 2.4
• Definition 2.5
• Proposition 2.6
• Theorem 2.7: Hydrodynamic limit of independent random walks
• Remark 2.8
• Remark 2.9
• Remark 2.10