The DeanKawasaki equation and the structure of density fluctuations in systems of diffusing particles
Abstract
The DeanKawasaki 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 $N\gg 1$. The singular nature of the DeanKawasaki equation presents a substantial challenge for both its analysis and its rigorous mathematical justification. Besides being nonrenormalisable 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 DeanKawasaki equation by considering the natural regularisation provided by standard numerical discretisations: We show that structurepreserving discretisations of the DeanKawasaki equation may approximate the density fluctuations of $N$ noninteracting diffusing particles to arbitrary order in $N^{1}$ (in suitable weak metrics). In other words, the DeanKawasaki equation may be interpreted as a "recipe" for accurate and efficient numerical simulations of the density fluctuations of independent diffusing particles.
 Publication:

arXiv eprints
 Pub Date:
 September 2021
 DOI:
 10.48550/arXiv.2109.06500
 arXiv:
 arXiv:2109.06500
 Bibcode:
 2021arXiv210906500C
 Keywords:

 Mathematics  Analysis of PDEs;
 Mathematics  Numerical Analysis;
 Mathematics  Probability;
 60H15;
 35R60;
 65N99
 EPrint:
 65 pages, 6 figures