MINI COURSES
André Schlichting (University Ulm) - Breakdown of the mean-field description of interacting particle systems: Phase transitions, metastability and coarsening
Abstract: Cluster formation and coarsening are ubiquitous phenomena in systems governed by attractive interactions. We investigate these processes for weakly interacting diffusions on the one-dimensional torus subject to locally attractive, finite-range interactions. In particular, we study the emergence of clusters and the subsequent coarsening of multi-cluster configurations into a single-cluster state. Depending on the parameter regime, this coarsening is driven by two competing mechanisms:
(1) coalescence, whereby whole clusters move and merge in a manner reminiscent of diffusion-driven coalescence; and
(2) mass exchange, in which individual particles detach from one cluster and attach to another, leading to an effective transfer of mass between clusters.
Based on an Eyring–Kramers-type asymptotic analysis, we establish a unified effective model capturing the interplay between these mechanisms and show that the associated deterministic mean-field PDE exhibits dynamical metastability induced by mass exchange. In addition, we introduce a new variant of the strict Riesz rearrangement inequality to characterize the global minimizers of the free energy, showing that they are either spatially uniform or single-cluster states, i.e., symmetric and decreasing about their center of mass.
Marielle Simon (Aix-Marseille Université) - The mathematical derivation of thermodynamics laws from interacting oscillators
Abstract: The heat equation is known to be a macroscopic phenomenon, emerging after a diffusive rescaling in both space and time. However, deriving Fourier's law from a microscopic dynamics consistent with Newton's laws remains a mathematical challenge. In this mini-course I will introduce a class of models for which several mathematical results have been proved in the last decades, namely the chain of interacting oscillators with some stochastic perturbation. Thanks to sufficient mixing properties of the microscopic dynamics, one can show that the macroscopic energy density diffuses, with various boundary conditions which depend on the microscopic boundary mechanisms. If I have time I will also show how modifying some properties of the noise can drastically change the macroscopic behavior of the energy, from diffusive to superdiffusive, evolving according to some fractional Laplacian.