Abstract: The theory of fluctuating hydrodynamics aims to describe dynamics of particle systems for large, but finite numbers of particles. The individual random dynamics of particles are replaced with a stochastic PDE (SPDE) for the overall density. This SPDE formally captures the density fluctuations for the fixed particle number on top of the deterministic limiting dynamics. However, providing and justifying correct interpretations for the resulting SPDEs is mathematically challenging.
This talk aims to illustrate this concept with the Dean—Kawasaki equation as an example. This highly singular SPDE was proposed to describe systems of weakly interacting, diffusing particles. I will discuss how interpreting the Dean--Kawasaki equation as a "recipe" leads to efficient and numerical simulations, that capture the density fluctuations of the particle system to arbitrary order. Based on joint work with Federico Cornalba, Julian Fischer, and Claudia Raithel.