Hydrodynamic limit of particle systems with long jumps

TL;DR

This paper studies long-range interacting particle systems and proves their large-scale behavior converges to a fractional heat equation, highlighting superdiffusive scaling and including results on tagged particles and fractional PDE solutions.

Contribution

It establishes the hydrodynamic limit for zero-range and exclusion processes with long jumps, connecting microscopic dynamics to fractional PDEs.

Findings

Hydrodynamic limit corresponds to a fractional heat equation.

Superdiffusive scaling is confirmed for these processes.

A central limit theorem for a tagged particle is proved.

Abstract

We consider some interacting particle processes with long-range dynamics: the zero-range and exclusion processes with long jumps. We prove that the hydrodynamic limit of these processes corresponds to a (possibly non-linear) fractional heat equation. The scaling in this case is superdiffusive. In addition, we discuss a central limit theorem for a tagged particle on the zero-range process and existence and uniqueness of solutions of the Cauchy problem for the fractional heat equation.