Scaling limits of particle systems and microstructural disorder

Introduction

The present proposal focuses on the role of microstructural disorder in the dynamics of many-particle systems. Due to the complexity of such systems, any practical description relies on simplified effective theories. In the tradition of Hilbert’s sixth problem, I aim at the rigorous large-scale derivation of effective theories from fundamental microscopic descriptions.

Importance of Microstructural Disorder

In those derivations, the role of microstructural disorder has often been overlooked for simplicity. However, disorder is key to many systems and can lead to new behaviors. Understanding its effects in scaling limits of particle systems is, therefore, of fundamental interest.

Model Problems

I have selected five model problems illustrating important aspects of the topic:

1. Homogenization: The simplest regime is that of homogenization, where the effect of the disordered background averages out on large scales.
2. Particle Suspensions: For systems like particle suspensions in fluids, microstructural disorder is itself induced by particle positions; as these evolve over time, adapting to external forces, it can lead to nonlinear effects.
3. Irreversibility: Another aspect is the emergence of irreversibility: the transport of mechanical particles in a disordered background typically becomes diffusive on large scales, which gives, for instance, a microscopic explanation for electrical resistance in metals.
4. Self-Diffusion: I also consider the more intricate problem of self-diffusion, where irreversibility rather results from interactions with the ensemble of other particles themselves.
5. Glassiness: A last important aspect concerns the emergence of glassiness, which results from the competition between interactions and disordered background.

Mathematical Framework

Mathematically, this proposal is at the crossroads between the analysis of partial differential equations and probability theory, and it builds on tremendous recent progress in two of my fields of expertise: homogenization and mean-field theory. Their combination provides a timely and innovative framework for new breakthroughs on scaling limits of disordered particle systems.