Stochastic cancellations and the large-scale rheology of random suspensions

The large-scale rheology of random suspensions aims at describing how suspensions of small but many objects influence (sometimes drastically) a fluid flow. In physics this is the realm of complex fluids, with well-established phenomenological models. The derivation of such models from the physics of its constituents (fluid and rigid particles, say) is a widely open question. In this talk I will discuss questions of effective viscosity, sedimentation speed, and reduction of effective viscosity by active particles. The common difficulty in these problems is to understand stochastic cancellations, which one cannot do easily since interactions are not explicit (they are defined via a PDE, which makes the problem nonlinear with respect to the randomness). We propose a strategy to extract the infra-red non-integrable parts explicitly, and treat the (integrable, yet implicit) remainder using quantitative homogenization (and especially large-scale regularity) as a tool to establish multi-particle estimates.