Abstract

The dynamics of interacting agents subject to binary forces can be often described at the microscopic level by a system of ODEs (with possible stochasticity) with discrete convolutions, with a macroscopic counterpart given by a so called nonlocal interaction pde. Both the microscopic and the macroscopic models can be endowed with a gradient flow structure in a measure space which allows to give a mathematical statement of “concentration” phenomena in finite time. This is done in particular for models with nonlocal attractive force without diffusion (no stochasticity). Models with moderate repulsions and long range attraction give rise to continuum counterpart with a quadratic “porous medium” type term in the continuum pde, which generates a threshold phenomena in the large time behaviour. The main results described are the outcome of collaborations with J. A. Carrillo (Imperial College), M. Burger (Muenster) and other collaborators. I will also briefly discuss the interplay with entropy solution theory (collaboration with D. Matthes, TU Munich, plus some work in preparation).