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

  • Matching Large Deviation Bounds of the Zero-Range Process in the whole space

    We consider the large deviations of the hydrodynamic rescaling of the zero-range process on $\mathbb{Z}^d$ in any dimension $d\ge 1$. Under mild and canonical hypotheses on the local jump rate, we obtain matching upper and lower bounds, thus resolving the problem opened…

  • A Dynamical Systems Perspective on the Analysis of Neural Networks

    In this chapter, we utilize dynamical systems to analyze several aspects of machine learning algorithms. As an expository contribution we demonstrate how to re-formulate a wide variety of challenges from deep neural networks, (stochastic) gradient descent, and related topics into dynamical statements.…

  • Fluctuation behaviour for interacting particle systems with common noise

    Published · Stochastics and Partial Differential Equations: Analysis and Computations (2026). We consider the asymptotic behaviour of the fluctuation process for large stochastic systems of interacting particles driven by both idiosyncratic and common noise with an interaction kernel \(k \in L^2(\R^d) \cap L^\infty(\R^d)\). Our analysis relies on uniform relative entropy estimates and Kolmogorov’s…

  • Conservative stochastic PDEs on the whole space

    Published · Stochastics and partial differential equations : analysis and computations, 14 (2026) 1, pp. 350-388. The purpose of this paper is to establish a well-posedness theory for conservative stochastic partial differential equations on the whole space. This class of stochastic PDEs arises in fluctuating hydrodynamics, and includes…

  • A quantitative central limit theorem for the simple symmetric exclusion process

    A quantitative central limit theorem for the simple symmetric exclusion process (SSEP) on a $d$-dimensional discrete torus is proven. The argument is based on a comparison of the generators of the density fluctuation field of the SSEP and the generalized Ornstein-Uhlenbeck process,…

  • Higher Order Fluctuation Expansions for Nonlinear Stochastic Heat Equations in Singular Limits

    Published · Stochastic processes and their applications, 193 (2026), p. 104847. Higher order fluctuation expansions for stochastic heat equations (SHE) with nonlinear, non-conservative and conservative noise are obtained. These Edgeworth-type expansions describe the asymptotic behavior of solutions in suitable joint scaling regimes of…

  • Stochastic Modified Flows for Riemannian Stochastic Gradient Descent

    Published · SIAM journal on control and optimization, 62 (2024) 6, pp. 3288-3314. We give quantitative estimates for the rate of convergence of Riemannian stochastic gradient descent (RSGD) to Riemannian gradient flow and to a diffusion process, the so-called Riemannian stochastic modified flow (RSMF). Using…

  • Landau-Lifshitz-Navier-Stokes Equations: Large Deviations and Relationship to The Energy Equality

    Accepted / forthcoming · Annals of Applied Probability. The dynamical large deviations principle for the three-dimensional incompressible Landau-Lifschitz-Navier-Stokes equations is shown, in the joint scaling regime of vanishing noise intensity and correlation length. This proves the consistency of the large…