Descripción del proyecto
Efectos de desorden en los límites de escala de los sistemas de partículas
El equipo del proyecto PASTIS, financiado por el Consejo Europeo de Investigación, se centra en el papel del desorden microestructural en la dinámica de los sistemas de muchas partículas. Siguiendo la tradición del sexto problema de Hilbert, el estudio tiene como objetivo la derivación rigurosa a gran escala de teorías a partir de descripciones microscópicas fundamentales. Para comprender los efectos del desorden en los límites de escala de los sistemas de partículas desordenadas, se estudiarán cinco problemas modelo que ilustran diferentes aspectos del tema, incluidos los efectos del desorden en las suspensiones de partículas en fluidos, la irreversibilidad en el transporte de partículas mecánicas en un fondo desordenado, la autodifusión y la aparición de la vidriosidad. En el proyecto se combina el análisis de ecuaciones diferenciales parciales y la teoría de la probabilidad, capitalizando así los recientes avances en homogeneización y teoría del campo medio.
Objetivo
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.
Palabras clave
- 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
Programa(s)
- HORIZON.1.1 - European Research Council (ERC) Main Programme
Régimen de financiación
HORIZON-ERC - HORIZON ERC GrantsInstitución de acogida
1050 Bruxelles / Brussel
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