LisMath Seminar 

13/07/2023, 14:30 — 15:00 — Online
Gabriel Nahum, Instituto Superior Técnico, Universidade de Lisboa

Non-linear problems in Interacting Particle Systems

Interacting Particle Systems provide a framework, in a Markovian universe, for the study of phenomena arising from the collective behaviour of a very large number of agents interacting with each other. In this talk I am going to present two results of my PhD work in the context of "non-linear" dynamics.

The SSEP is a nearest neighbour exclusion process defined in a lattice, where particles exchange positions with its empty direct neighbours only, and at most one particle is allowed per site.
The PMM is a kinetically constrained model with exclusion constraints, but depending also on the particular configuration of all the windows, with equal length, that contain the pair of sites that are to exchange occupations. Precisely, fixed some length, the rate corresponds to the number of completely full windows. If this number is zero, the exchange is suppressed. For both the SSEP and the PMM one can prove that "density field $\approx$ differential equation" in probability. This result, in varied topology, is known as the Hydrodynamic Limit. The SSEP yields the heat equation, while the PMM the porous media equation. We define a Markov generator that interpolates continuously the one of the PMM and the SSEP, extending further to a fast diffusion model. The Hydrodynamic Limit is shown and as a result, under a scaling limit, the density field $\rho$ is associated with the diffusion coefficient $ \rho^m $ for any $m > -1$.

This is further generalized in the direction of $\rho^n(1-\rho)^k$ for any $ n , k $ positive integers, through combinatoric arguments. Specifically, it is presented a linear system that characterizes the rates for which the resulting models satisfy the gradient property. This is then extended in a natural way to a long range dynamics.