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

Abstract

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 $$N\gg 1$$ N ≫ 1 . 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 $$N^{-1}$$ N - 1 (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.

Global density equations for a population of actively switching particles

“…The Dean-Kawasaki (DK) equation for non-switching systems is a stochastic partial differential equation (SPDE) that describes fluctuations in the global density ρ(x, t) = N −1 N j =1 δ(x − X j (t)) of N over-damped Brownian particles (Brownian gas) with positions X j (t) ∈ R d at time t [32,33]. It is an exact equation for the global density in the distributional sense, and is of considerable current interest within the context of stochastic and numerical analysis [34][35][36][37][38][39][40]. The DK equation is also used extensively in non-equilibrium statistical physics, where it is combined with dynamical density functional theory (DDFT) in order to derive hydrodynamical models of interacting particle systems [41][42][43][44].…”

Quantitative homogenization of interacting particle systems

“…For gradient models and small perturbations thereof, out-of-equilibrium fluctuation results have been derived in [12,14,21,32,47,49]. Recent progress on higher-order approximations and large deviations include [13,17,20].…”