Project description
Disorder effects in scaling limits of particle systems
The ERC-funded PASTIS project focuses on the role of microstructural disorder in the dynamics of many-particle systems. Following the tradition of Hilbert’s sixth problem, the study aims at the rigorous large-scale derivation of theories from fundamental microscopic descriptions. To understand the disorder effects in scaling limits of disordered particle systems, five model problems that illustrate different aspects of the topic, including the effects of disorder on particle suspensions in fluids, irreversibility in the transport of mechanical particles in a disordered background, self-diffusion, and the emergence of glassiness, will be studied.The project combines the analysis of partial differential equations and probability theory, capitalising on recent progress in homogenisation and mean-field theory.
Objective
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. 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.
I have selected five model problems illustrating important aspects of the topic. The simplest regime is that of homogenization, where the effect of the disordered background averages out on large scales. 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. 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. I also consider the more intricate problem of self-diffusion, where irreversibility rather results from interactions with the ensemble of other particles themselves. A last important aspect concerns the emergence of glassiness, which results from the competition between interactions and disordered background.
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.
Fields of science
Keywords
- PDE with random coefficients
- interacting particle systems
- disordered media
- microstructure
- scaling limits
- homogenization
- effective behavior
- mean-field limits
- collective behavior
- kinetic limits
- irreversibility
- random drift
- particle suspensions
- flow-induced microstructure
- active fluids
- quantum diffusion
- self-diffusion
- thermalization
- Lenard-Balescu equation
Programme(s)
- HORIZON.1.1 - European Research Council (ERC) Main Programme
Topic(s)
Funding Scheme
ERC - Support for frontier research (ERC)Host institution
1050 Bruxelles / Brussel
Belgium