Sala Seminari (Dip. Matematica).
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, some of which one can prove are false, there are extensive numerical simulations, and there is a formal asymptotic result for small volume fraction, 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 dissipation metric, and how they derive from the underlying stochastic particle system. It also gives clear indications of what one should expect in the higher-dimensional case. This is joint work with Nir Gavish and Pierre Nyquist.