Hydrodynamics for the ABC model with slow/fast boundary

TL;DR

This paper derives the hydrodynamic limit equations for the ABC particle model with slow/fast boundary reservoirs, revealing how boundary interactions influence the macroscopic behavior in a weakly asymmetric setting.

Contribution

It establishes the hydrodynamic limit for the ABC model with boundary reservoirs, detailing the boundary conditions based on reservoir strength and particle exchange dynamics.

Findings

Hydrodynamic limit described by coupled nonlinear PDEs.

Boundary conditions depend on reservoir interaction strength.

Results applicable to weakly asymmetric particle systems.

Abstract

In this article, we consider the ABC model in contact with slow/fast reservoirs. In this model, there is at most one particle per site, which can be of type $\alpha\in\{A,B,C\}$ and particles exchange positions in the discrete set of points $\{1,\cdots, N-1\}$ with a weakly asymmetric rate that depends on the type of particles involved in the exchange mechanism. At the boundary points $x=1, N-1$ particles can be injected or removed with a rate that depends on the type of particles involved. We prove that the hydrodynamic limit, in the diffusive time scale, is given by a system of non-linear coupled equation with several boundary conditions, that depend on the strength of the reservoir's action.