Linear and Nonlinear Fractional PDEs from interacting particle systems

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 $m\in \mathbb{N}:=\{1,2,\dots \}$ fixed, the hydrodynamic equation is ${\partial }_t\ddot{I}(t,u)=[-(-\hat{I}”{)}^{{\hat{I}}^3/2}\ddot{I}{}^m](t,u)$. For $m=1$, this {is} the fractional equation, which is linear. On the other hand, for $m\ge 2$, 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.

27 pages, 1 figure