Linear and Nonlinear Fractional PDEs from interacting particle systems (2407.02246v1)

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 $. 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.