Organizers: Ana Djurdjevac, Nicolas Perkowski, Rupert Klein

Abstract:

Interacting particle systems serve as powerful and versatile models across various domains, including molecular dynamics, biology, and social sciences. While these systems oer rich modeling capabilities, their analytical and computational complexity grows rapidly with the number of particles. A common approach to overcome this challenge is to derive eective equations for the empirical particle density. This mini-symposium focuses on the derivation and analysis of nonlinear SPDEs that describe uctuations in particle systems. Key topics will include various SPDEs arising from such uctuation descriptions, along with their well-posedness and the error with respect to the particle system.

Speakers:

14:30h: Federico Cornalba (University of Bath)

A discontinuous Galerkin approximation of the Dean-Kawasaki equation

We introduce and analyse an arbitrary order spatial discontinuous Galerkin (dG) method for the Dean-Kawasaki equation, a highly singular SPDE modelling density fluctuations in large but finite systems of diffusing particles. Our starting point is a general procedure for discretising multiplicative, divergence-form noise on finite element spaces whilst preserving its cross-variation structure at the discrete level; the construction is explicit, elementwise, applies to continuous and discontinuous spaces alike, and extends to general mobilities. Using it, we prove weak error estimates of order $O(h^p)$ between fluctuations of the semi-discrete scheme and those of the underlying particle system, together with a correction that is exponentially small in the scaling regime $Nh^d \gg 1$ and arises because the scheme does not preserve positivity. The resulting method is locally and globally conservative and applies on unstructured simplicial meshes. Quantitative numerical experiments support the analysis, and further experiments illustrate the method beyond the scope of the theory: indicator function observables, external and interaction potentials, convection-dominated regimes and reflecting boundary conditions.

Joint work with Kamran Arora and Tony Shardlow.

14:50h: Rishabh Gvalani

Ergodicity of McKean - Vlasov equations with common noise

We study the ergodic behaviour of the McKean-Vlasov equations driven by common, divergence-free transport noise. In particular, we show that in dimension d≥2, if the noise is white-in-time, mixing, and sufficiently strong it can enforce the uniqueness of invariant probability measures, even if the deterministic gradient flow has multiple steady states. This is joint work with Benjamin Gess (Berlin/Leipzig) and Adrian Martini (Berlin).

15:10h: Helena Kremp (TU Berlin / WIAS)

Stochastic homogenisation for Dean-Kawasaki-type SPDEs

We consider Dean-Kawasaki-type SPDEs with random, oscillating coefficients of the form

∂_t u = ∇⋅ (a(x/ε,ω) ∇Φ(u)) - N^-1/2 ∇⋅ (Φ^1/2 (u) ◊ σ(x/ε,ω)ξ^δ)

on a smooth, bounded domain U with Neumann boundary conditions, where ξ^δ is an ultraviolet cutoff of a vector-valued space time white noise, N a natural number, ε the homogenisation parameter, ◊ denotes Statonovich integration and a=σσ^T is the ergodic environment, which is assumed to be uniformely elliptic. In particular we cover the case of a linear nonlinearity, Φ(u)=u. In that case, the equation is motivated by the system of N iid particles with reflecting boundary conditions in an inhomogeneous environment a. In a specific scaling regime for (N,ε,δ) we prove a quenched convergence result towards a deterministic homogenised limit. A solution to the above equation is understood in the sense of stochastic kinetic solutions (cf. Gess, Fehrman '24 and Fehrman '26), which overcome the issue of singularities of the coeffcients and low integrability of the solution. We further prove a uniform LDP for (u=u^N,ε,δ ) by considering the homogenised limit for a shifted version of the above SPDE, where interestingly the homogenised coefficients ā and ō are in general not related to each other by taking the square-root. The talk is based on a work in progress together with Benjamin Fehrman.

15:30h: Max von Renesse

Well-posedness for Dean–Kawasaki models of Vlasov–Fokker–Planck type 

We consider systems of interacting particles whichare described by a second-order Langevin equation. The class of equations considered includes the situation where the particle evolution is governed by Hamiltonian dynamics with additional damping and noise satisfying a fluctuation–dissipation relation. 

Also covered are systems of two equations describing an evolution of interacting agents, as arising in several descriptions of active matter, including models for flocking and swarming. We first show that such particle systems can be represented exactly by so-called equations of fluctuating hydrodynamics, which in this case are stochastic versions of a Vlasov–Fokker–Planck type equation. While the derivation given here is simple, it is a blueprint for the rigorous derivation of equations of fluctuating hydrodynamics.

We then show a dichotomy previously known for purely diffusive (first-order) systems carries over to the second-order setting considered here: solutions exist for suitable atomic initial data, in which case the solution is, properly scaled, the empirical density describing the particle system. For smooth initial data, however, we prove that no solution exists. 

Joint work with Fenna Müller and Johannes Zimmer