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 <i>N</i> independent diffusing particles in the regime of large particle numbers <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>N</mml:mi><mml:mo>≫</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math>. 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 <i>N</i> non-interacting diffusing particles to arbitrary order in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>N</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math> (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.