Descrizione del progetto
Effetti di disordine nei limiti di scala dei sistemi di particelle
Il progetto PASTIS, finanziato dal CER, si concentra sul ruolo del disordine microstrutturale nella dinamica dei sistemi a molte particelle. Seguendo la tradizione del sesto problema di Hilbert, lo studio punta alla derivazione rigorosa su larga scala delle teorie dalle descrizioni microscopiche fondamentali. Per comprendere gli effetti del disordine nei limiti di scala dei sistemi di particelle disordinate, verranno studiati cinque problemi modello che illustrano diversi aspetti dell’argomento, fra cui gli effetti del disordine sulle sospensioni di particelle nei fluidi, l’irreversibilità nel trasporto di particelle meccaniche in un contesto disordinato, l’autodiffusione e l’emergere della vetrosità. Il progetto combina l’analisi delle equazioni differenziali alle derivate parziali e la teoria della probabilità, sfruttando i recenti progressi nell’omogeneizzazione e nella teoria del campo medio.
Obiettivo
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.
Parole chiave
- 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
Programma(i)
- HORIZON.1.1 - European Research Council (ERC) Main Programme
Argomento(i)
Invito a presentare proposte
(si apre in una nuova finestra) ERC-2022-STG
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