An Additive-Noise Approximation to Keller–Segel–Dean–Kawasaki Dynamics

The theory of fluctuating hydrodynamics aims to describe density fluctuations of interacting particle systems as so-called Dean–Kawasaki stochastic partial differential equations. However, Dean–Kawasaki equations are ill-posed and the focus has shifted towards finding well-posed approximations that retain the statistical properties of the particle system. In this talk, we consider the fluctuating hydrodynamics of a system in which particles are attracted to one another through a Coulomb force (Keller–Segel dynamics). We propose an additive-noise approximation and show that it retains the same law of large numbers and central limit theorem as (conjectured for) the particle system. We further deduce a large deviation principle and show that the approximation error lies in the skeleton equation that drives the rate function. Based on joint work with Avi Mayorcas.