"The Dean-Kawasaki equation and the structure of density fluctuations in systems of diffusing particles"Fischer, JulianThe theory of fluctuating hydrodynamics attempts to describe density fluctuations in systems of interacting particles in the regime of large particle numbers by means of suitable SPDEs. The Dean-Kawasaki equation - a strongly singular SPDE - is perhaps the most basic equation of fluctuating hydrodynamics; in its simplest formulation, 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 N. The singular nature of the Dean-Kawasaki equation presents a substantial challenge for both its analysis and its rigorous mathematical justification: Besides being non-renormalizable by Hairer's theory of regularity structures, it has recently been shown to not even admit nontrivial martingale solutions. In this talk, we give a rigorous quantitative justification of the Dean-Kawasaki equation: We show that structure-preserving formal discretizations of the Dean-Kawasaki equation may approximate the density fluctuations of N non-interacting diffusing particles to arbitrary order in 1/N (in suitable weak metrics), the accuracy being only limited by the numerical scheme. In other words, the Dean-Kawasaki equation may be interpreted as a "recipe" for accurate and efficient numerical simulations of the density fluctuations of diffusing particles. We expect our approach to generalize to the case of weakly interacting particles and possibly beyond. |
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