The Dean-Kawasaki Equation and the Structure of Density Fluctuations in Systems of Diffusing Particles

The Dean-Kawasaki equation-a strongly singular SPDE-is a basic equation of fluctuating hydrodynamics; it has been proposed in the physics literature to describe the fluctuations of the density of N independent diffusing particles in the regime of large particle numbers $$. The singular nature of the Dean-Kawasaki equation presents a substantial challenge for both its analysis and its rigorous mathematical justification. Besides being non-renormalisable by the theory of regularity structures by Hairer et al., it has recently been shown to not even admit nontrivial martingale solutions. In the present work, we give a rigorous and fully quantitative justification of the Dean-Kawasaki equation by considering the natural regularisation provided by standard numerical discretisations: We show that structure-preserving discretisations of the Dean-Kawasaki equation may approximate the density fluctuations of N non-interacting diffusing particles to arbitrary order in $$ (in suitable weak metrics). In other words, the Dean-Kawasaki equation may be interpreted as a "recipe" for accurate and efficient numerical simulations of the density fluctuations of independent diffusing particles.