Hydrodynamical behavior of symmetric exclusion with slow bonds

TL;DR

This paper investigates how slow bonds in a one-dimensional exclusion process affect the hydrodynamic limit, revealing different PDE behaviors depending on the strength of the slow bonds.

Contribution

It characterizes the hydrodynamic limits of the exclusion process with slow bonds across different regimes of the parameter , providing a comprehensive phase diagram of the limiting PDEs.

Findings

For , the limit is the heat equation.

At =1, the limit involves a differential operator with a measure including slow bonds.

For >1, the limit is the heat equation with Neumann boundary conditions.

Abstract

We consider the exclusion process in the one-dimensional discrete torus with $N$ points, where all the bonds have conductance one, except a finite number of slow bonds, with conductance $N^{-\beta}$, with $\beta\in[0,\infty)$. We prove that the time evolution of the empirical density of particles, in the diffusive scaling, has a distinct behavior according to the range of the parameter $\beta$. If $\beta\in [0,1)$, the hydrodynamic limit is given by the usual heat equation. If $\beta=1$, it is given by a parabolic equation involving an operator $\frac{d}{dx}\frac{d}{dW}$, where $W$ is the Lebesgue measure on the torus plus the sum of the Dirac measure supported on each macroscopic point related to the slow bond. If $\beta\in(1,\infty)$, it is given by the heat equation with Neumann's boundary conditions, meaning no passage through the slow bonds in the continuum.