Effective Large-Scale Models for Random Diffusive Systems

In this project we are interested in the mathematical study and approximation of systems exhibiting scale separation. As scale separations (i.e., a measurable distinction between the scale of microscopic structures and a relevant macroscale of interest) are ubiquitous throughout nature, such systems are of fundamental interest in Applied Mathematics. In two parts of the project we consider two different types of scale separation: a physical scale separation or one in terms of the number of particles being considered in a system. In the first part we are first interested in approximating the physical properties of a material sample with random heterogeneities that are much smaller than the size of the sample. From a mathematical point of view, under suitable conditions, the randomness allows for a certain `homogenization` at the material scale -i.e., at the scale of the material sample the heterogenous material can be approximated by a certain homogeneous homogenized medium. Major developments in the last decade have opened the door to the possibility of proving quantitative homogenization results -e.g., convergence rates (in terms of the scale) of the heterogeneous equations to their homogenized counterpart or error estimates for approximations of the homogenized operator- in a wide range of random settings. In this project, we are specifically interested in material samples with corners, edges, and/ or internal interfaces. In the second part of our project we consider interacting particle systems. The dynamics of each particle is given in terms of the pairwise interaction with the other particles on top of a random diffusion. Often systems consist of many particles, making simulations of the particle system computationally prohibitively expensive. This has led to the desire to develop computationally simpler effective models that approximate the particle system as the number of particles becomes infinite -these models often come in the form of PDE (partial differential equation) models. In our project we are interested in the mesoscopic regime of `large but finite` particle numbers. Since, by assuming that there are only finitely many particles, we stop short of the convergence to the infinite-particle effective model, it is necessary to compensate for this by describing the resulting fluctuations of the particle system around the effective model. We specifically study two particular classes of particle systems describing interacting populations, deriving equations of fluctuating hydrodynamics for these situations.