From fluctuations in particle systems to their scaling limits and applications

We study particle systems and analyse their fluctuations. A motivation for this study work of Onsager, where he describes macroscopic systems by linear relations between forces and fluxes in this system. This Onsager structure is closely related to the modern mathematical theory of gradient flows. We consider dynamical fluctuations in systems described by Markov chains, and discuss a canonical structure that provides a unifying description of Markov chains, Onsager theory, and so-called macroscopic fluctuation theory. As Onsager theory, the theory involves a relation between probability currents (fluxes) and their conjugate forces. However, we will explain that on the level of Markov chains, the relation is non-linear. The tool will be dynamical large deviations, and the formulation is applicable to irreversible Markov chains. We discuss the resulting variational structure, which leads to generalised gradient flows. It is shown that various physically natural splittings can be introduced, which can help to derive applications such as an understanding of acceleration of convergence to equilibrium and dissipation bounds. We will sketch relations between the underlying variational structure on the particle level and its continuum counterpart, which describes the gradient flow of the macroscopic density associated with the particle system. This is joint work with Marcus Kaiser (Bath) and Rob Jack (Cambridge).