Hydrodynamic limit of gradient exclusion processes with conductances

TL;DR

This paper establishes that the macroscopic behavior of gradient exclusion processes with conductances converges to solutions of a specific nonlinear PDE involving a generalized derivative operator, extending understanding of hydrodynamic limits.

Contribution

It introduces a novel analysis of the hydrodynamic limit for exclusion processes with conductances defined by a general increasing function W, proving convergence to a nonlinear PDE and establishing solution uniqueness.

Findings

The empirical density evolution converges to the PDE solution.

Properties of the operator (d/dx)(d/dW) are characterized.

Uniqueness of weak solutions to the PDE is proven.

Abstract

Fix a strictly increasing right continuous with left limits function $W: \bb R \to \bb R$ and a smooth function $\Phi : [l,r] \to \bb R$, defined on some interval $[l,r]$ of $\bb R$, such that $0<b \le \Phi'\le b^{-1}$. We prove that the evolution, on the diffusive scale, of the empirical density of exclusion processes, with conductances given by $W$, is described by the weak solutions of the non-linear differential equation $\partial_t \rho = (d/dx)(d/dW) \Phi(\rho)$. We derive some properties of the operator $(d/dx)(d/dW)$ and prove uniqueness of weak solutions of the previous non-linear differential equation.