Séminaire MACS (Modélisation, Analyse et Calcul Scientifique). Commun ICJ & UMPA.

Continuum limit of a hard-sphere particle system by large deviations

by Mark Peletier (TU Eindhoven)

Many stochastic particle systems have well-defined continuum limits: as the number of particles tends to infinity, the density of particles converges to a deterministic limit that satisfies a partial differential equation. In this talk I will discuss one example of this. 
The particle system consists of particles that have finite size: in two and three dimensions they are spheres, in one dimension rods. The particles can not overlap each other, leading to a strong interaction with neighbouring particles.
Such systems of particles have been much studied, but for the continuum limit in dimensions two and up there is currently no rigorous result. There are conjectures about the form of the limit equation, often in the form of Wasserstein gradient flows, but to date there are no proofs.
We also can not give a proof of convergence in higher dimensions, but in the one-dimensional situation we can give a complete picture, including both the convergence and the gradient-flow structure that derives from the large-deviation behaviour of the particles. This gradient-flow structure shows clearly the role of the free energy and the Wasserstein-metric dissipation, and how they derive from the underlying stochastic particle system. 
The proof is based on a special mapping of the particle system to a system of independent particles, that is unique to the one-dimensional setup. This mapping is an isometry for the Wasserstein metric, leading to a beautiful connection between limit equations for interacting and non-interacting particle systems. 
This is a joint work with Nir Gavish and Pierre Nyquist.