Non-equilibrium Statistical Mechanics, Interacting Particle Systems and Fluctuating Hydrodynamics
Main fields of application
Many macroscopic laws arise from large systems of randomly interacting particles far from equilibrium. We study how microscopic interactions produce deterministic hydrodynamic equations and stochastic fluctuation fields.
The programme combines mean-field and hydrodynamic limits with central-limit corrections, large deviations and entropy methods. Conservative SPDEs provide continuum descriptions of fluctuations, while degenerate diffusion and singular interactions create analytical challenges at every scale. We seek quantitative links between particles and fields, including universality of fluctuations and the role of gradient-flow structures. Applications include exclusion and zero-range processes, vortex and aggregation models, and large stochastic systems related to learning.
Current directions
- Mean-field and hydrodynamic limits
- Gaussian fluctuations and large deviations
- Conservative SPDEs
- Entropy and gradient-flow methods
- Singular and degenerate interactions
Related people
Related projects
Selected publications
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The Porous Medium Equation: Large Deviations and Gradient Flow with Degenerate and Unbounded Diffusion
Published · Communications on pure and applied mathematics, 78 (2025) 9, pp. 1609-1655. The problem of deriving a gradient flow structure for the porous medium equation which is {\em thermodynamic}, in that it arises from the large deviations of some microscopic particle system, is studied…
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Stochastic Modified Flows, Mean-Field Limits and Dynamics of Stochastic Gradient Descent
Published · Journal of machine learning research, 25 (2024) 30, pp. 1-27. We propose new limiting dynamics for stochastic gradient descent in the small learning rate regime called stochastic modified flows. These SDEs are driven by a cylindrical Brownian motion and improve the so-called…
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Conservative SPDEs as fluctuating mean field limits of stochastic gradient descent
Published · Probability theory and related fields, 192 (2025) 3/4, pp. 1447-1515. The convergence of stochastic interacting particle systems in the mean-field limit to solutions of conservative stochastic partial differential equations is established, with optimal rate of convergence. As a second main result, a…