Linear and Nonlinear Fractional PDEs from interacting particle systems

TL;DR

This paper discusses the derivation of hydrodynamic limits for long-range interacting particle systems, resulting in fractional PDEs, including linear and nonlinear cases like the fractional porous medium equation.

Contribution

It provides a novel derivation of fractional PDEs from particle systems, highlighting the transition from linear to nonlinear equations based on interaction rates.

Findings

Derivation of fractional PDEs from particle systems.

Identification of linear and nonlinear fractional equations.

Connection between particle interactions and PDE types.

Abstract

In these notes, we describe the strategy for the derivation of the hydrodynamic limit for a family of long range interacting particle systems of exclusion type with symmetric rates. For $m \in \mathbb{N}:=\{1, 2, \ldots\}$ fixed, the hydrodynamic equation is $\partial_t \rho(t,u)= [-(-\Delta)^{\gamma /2} \rho^m](t,u) $. For $m=1$, this {is} the fractional equation, which is linear. On the other hand, for $m \geq 2$, this is the fractional porous medium equation (which is nonlinear), obtained by choosing a rate which depends on the number of particles next to the initial and final position of a jump.