Periodic Reporting for period 1 - PASTIS (Scaling limits of particle systems and microstructural disorder)
Período documentado: 2023-05-01 hasta 2025-10-31
This project explores the role of microstructural disorder on the dynamics of many-particle systems. Due to the inherent complexity of these systems, their practical description relies on simplified effective theories. In the tradition of Hilbert's sixth problem, the project seeks to rigorously derive the latter from first-principles microscopic models. In those derivations, disorder is often overlooked for simplicity, yet in many systems disorder is not a mere perturbation and can fundamentally alter system behavior. Uncovering how disorder affects the scaling limits of particle systems is therefore of essential interest.
I have selected five model problems illustrating important aspects of the topic.
- Part I is concerned with homogenization regimes where the effect of the disordered background averages out on large scales.
- Part II addresses more intricate systems like particle suspensions in fluids, where the microstructural disorder is itself induced by particle positions; as these evolve over time, adapting to external forces, it leads to nonlinear (non-Newtonian) effects.
- Part III is concerned with the emergence of irreversibility: the transport of particles or waves in disordered background typically becomes diffusive on large scales, which is key for instance to a microscopic explanation for electrical resistance in metals.
- Part IV addresses the more intricate problem of self-diffusion, where irreversibility rather results from interactions with the ensemble of other particles themselves.
- Part V is concerned with the emergence of glassiness, which results from the competition between interactions and disordered background.
Mathematically, the project lies at the crossroads between the analysis of partial differential equations, probability theory, and mathematical physics, 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.
I have selected five model problems illustrating important aspects of the topic.
- Part I is concerned with homogenization regimes where the effect of the disordered background averages out on large scales.
- Part II addresses more intricate systems like particle suspensions in fluids, where the microstructural disorder is itself induced by particle positions; as these evolve over time, adapting to external forces, it leads to nonlinear (non-Newtonian) effects.
- Part III is concerned with the emergence of irreversibility: the transport of particles or waves in disordered background typically becomes diffusive on large scales, which is key for instance to a microscopic explanation for electrical resistance in metals.
- Part IV addresses the more intricate problem of self-diffusion, where irreversibility rather results from interactions with the ensemble of other particles themselves.
- Part V is concerned with the emergence of glassiness, which results from the competition between interactions and disordered background.
Mathematically, the project lies at the crossroads between the analysis of partial differential equations, probability theory, and mathematical physics, 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.
In the first funding period, my research focused on Parts II, III, and IV of the project, with significant progress on several of the initially planned challenges as well as some more unexpected results.
Part II is concerned with particle suspensions in fluids. These are prototypical systems where flow-induced microstructure creates a nonlinear effective response to external flows, leading to the well-known non-Newtonian rheology of suspensions. The main difficulties in the analysis arise from the long-range, multi-body, and singular nature of hydrodynamic interactions. I devoted a series of works to very dilute suspensions, for which I obtained a comprehensive derivation of non-Newtonian effects: starting from particle dynamics, I derived the so-called Doi model in the dilute mean-field regime and then recovered explicit non-Newtonian fluid models in the hydrodynamic limit. Next, I addressed the sedimentation problem for homogeneous suspensions and derived in particular Batchelor's dilute formula for the mean settling speed, which solved a long-standing problem: this required to develop a rigorous theory for hydrodynamic renormalization, which provides a new tool in the study of suspensions and will be fundamental for the sequel of the project. Among other things, I also solved a related open problem: the homogenization of 2D Euler equations in porous media, based on an unexpected combination of tools from homogenization, elliptic regularity, and dynamical systems.
Part III explores some aspects of the long-term diffusive transport of non-interacting particles or waves in disordered media. I developed a non-diagrammatic spectral strategy allowing for a complete proof of Cherenkov radiation for an electron coupled to a quantized field of scalar massive bosons, which I view as a simplified analogue of the quantum diffusion conjecture. Although unrelated, this also inspired me a non-perturbative approach to the Bourgain-Spencer conjecture in homogenization. Next, I started to explore the long-time behavior of low-energy acoustic waves in disordered media—an intriguing question given the lack of a definite conjecture regarding the nature of the bottom of the spectrum for random acoustic operators. As a first step, I revisited earlier work on long-time homogenization for acoustic waves, I proved some new "large-scale" dispersive estimates, and derived first bounds on the localization length.
Part IV investigates the slow fluctuating diffusive behavior of interacting particle systems, as an effect of particle correlations, beyond the mean-field regime. I obtained a partial proof of this so-called self-diffusion for various systems near thermal equilibrium. Interestingly, for point-vortex systems, I also found out that this mechanism can sometimes break down, in which case the system exhibits some slow conservative evolution: this matches the "resonant relaxation" predicted in the physics literature, for which my work provides the first quantitative description. Unexpectedly, some of the techniques developed for this self-diffusion problem led us to devise a novel dual hierarchical method to solve new cases of the long-standing mean-field limit problem for classical particle systems with singular interactions—which is still one of the major open problems in mathematical physics.
Part II is concerned with particle suspensions in fluids. These are prototypical systems where flow-induced microstructure creates a nonlinear effective response to external flows, leading to the well-known non-Newtonian rheology of suspensions. The main difficulties in the analysis arise from the long-range, multi-body, and singular nature of hydrodynamic interactions. I devoted a series of works to very dilute suspensions, for which I obtained a comprehensive derivation of non-Newtonian effects: starting from particle dynamics, I derived the so-called Doi model in the dilute mean-field regime and then recovered explicit non-Newtonian fluid models in the hydrodynamic limit. Next, I addressed the sedimentation problem for homogeneous suspensions and derived in particular Batchelor's dilute formula for the mean settling speed, which solved a long-standing problem: this required to develop a rigorous theory for hydrodynamic renormalization, which provides a new tool in the study of suspensions and will be fundamental for the sequel of the project. Among other things, I also solved a related open problem: the homogenization of 2D Euler equations in porous media, based on an unexpected combination of tools from homogenization, elliptic regularity, and dynamical systems.
Part III explores some aspects of the long-term diffusive transport of non-interacting particles or waves in disordered media. I developed a non-diagrammatic spectral strategy allowing for a complete proof of Cherenkov radiation for an electron coupled to a quantized field of scalar massive bosons, which I view as a simplified analogue of the quantum diffusion conjecture. Although unrelated, this also inspired me a non-perturbative approach to the Bourgain-Spencer conjecture in homogenization. Next, I started to explore the long-time behavior of low-energy acoustic waves in disordered media—an intriguing question given the lack of a definite conjecture regarding the nature of the bottom of the spectrum for random acoustic operators. As a first step, I revisited earlier work on long-time homogenization for acoustic waves, I proved some new "large-scale" dispersive estimates, and derived first bounds on the localization length.
Part IV investigates the slow fluctuating diffusive behavior of interacting particle systems, as an effect of particle correlations, beyond the mean-field regime. I obtained a partial proof of this so-called self-diffusion for various systems near thermal equilibrium. Interestingly, for point-vortex systems, I also found out that this mechanism can sometimes break down, in which case the system exhibits some slow conservative evolution: this matches the "resonant relaxation" predicted in the physics literature, for which my work provides the first quantitative description. Unexpectedly, some of the techniques developed for this self-diffusion problem led us to devise a novel dual hierarchical method to solve new cases of the long-standing mean-field limit problem for classical particle systems with singular interactions—which is still one of the major open problems in mathematical physics.
A major advance of the project is the development of a general scheme for the rigorous renormalization of long-range hydrodynamic interactions in particle suspensions. The approach relies on a diagrammatic decomposition of multibody interactions, enabling the systematic identification of cancellation structures. Using this framework, I was able to provide the first rigorous proof of Batchelor’s dilute formula for the mean settling speed of homogeneous suspensions—a long-standing open problem. I also extended this result to higher-order semi-dilute corrections, resolving an open question in the physics literature where formal renormalization methods had remained limited to leading order in the dilution.
Another breakthrough is the development of a new dual hierarchical approach to mean-field limits for classical particle systems with singular interactions. Hierarchical methods have usually been considered useless for classical particles due to the loss of velocity derivatives, but we show that they can still be exploited successfully on the dual side: in a nutshell, dual a priori estimates allow to relegate the loss of derivatives to perturbative terms. This opens a promising new direction in the theory of singular mean-field limits.
I also emphasize the non-perturbative approach that I developed for the Bourgain-Spencer conjecture in homogenization, based on a new notion of weak correctors that may prove useful in other homogenization problems.
Finally, a last important advance concerns the study of point-vortex systems near equilibrium. I showed that the expected thermalization of a tagged particle may fail in some circumstances: the system then follows a slow conservative evolution—identified with the resonant relaxation predicted in the physics literature. Despite its significance, a theoretical description of this process remained elusive due to statistical closure problems. My work provides its first quantitative characterization, using a framework reminiscent of quantum field theory.
Another breakthrough is the development of a new dual hierarchical approach to mean-field limits for classical particle systems with singular interactions. Hierarchical methods have usually been considered useless for classical particles due to the loss of velocity derivatives, but we show that they can still be exploited successfully on the dual side: in a nutshell, dual a priori estimates allow to relegate the loss of derivatives to perturbative terms. This opens a promising new direction in the theory of singular mean-field limits.
I also emphasize the non-perturbative approach that I developed for the Bourgain-Spencer conjecture in homogenization, based on a new notion of weak correctors that may prove useful in other homogenization problems.
Finally, a last important advance concerns the study of point-vortex systems near equilibrium. I showed that the expected thermalization of a tagged particle may fail in some circumstances: the system then follows a slow conservative evolution—identified with the resonant relaxation predicted in the physics literature. Despite its significance, a theoretical description of this process remained elusive due to statistical closure problems. My work provides its first quantitative characterization, using a framework reminiscent of quantum field theory.