Numerical and analytical aspects for stochastic equations from Fluctuating Hydrodynamics

Large-scale systems of interacting particles are ubiquitous, with applications ranging from thermal advection phenomena to gradient descent algorithms in machine learning. While the mean-field-limit (MFL) perspective allows to efficiently characterize the ‘average’ evolution of many such particles systems (usually via PDEs), it does not incorporate a macroscopic description of the particles’ intrinsic fluctuations. The theory of Fluctuating Hydrodynamics addresses this aspect by instead describing the particle systems via stochastic PDEs which enrich the underlying MFL dynamics.
For a weakly interacting multi-species system, we will discuss the interplay between MFL dynamics and associated stochastic PDE, and show that numerical discretizations of the latter can approximate the particles’ fluctuations in an accurate and efficient way.
Based on joint works with Julian Fischer (IST Austria), Jonas Ingmanns (IST Austria), Claudia Raithel (TU Dresden).