The Dean-Kawasaki equation and the structure of density fluctuations in systems of diffusing particles

Cornalba F, Fischer JL. The Dean-Kawasaki equation and the structure of density fluctuations in systems of diffusing particles. arXiv, .

Preprint | Submitted | English
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Abstract
he 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≫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 (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.
Publishing Year
Date Published
2021-01-01
Journal Title
arXiv
Acknowledgement
FC gratefully acknowledges funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No. 754411.
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Cite this

Cornalba F, Fischer JL. The Dean-Kawasaki equation and the structure of density fluctuations in systems of diffusing particles. arXiv.
Cornalba, F., & Fischer, J. L. (n.d.). The Dean-Kawasaki equation and the structure of density fluctuations in systems of diffusing particles. arXiv.
Cornalba, Federico, and Julian L Fischer. “The Dean-Kawasaki Equation and the Structure of Density Fluctuations in Systems of Diffusing Particles.” ArXiv, n.d.
F. Cornalba and J. L. Fischer, “The Dean-Kawasaki equation and the structure of density fluctuations in systems of diffusing particles,” arXiv. .
Cornalba F, Fischer JL. The Dean-Kawasaki equation and the structure of density fluctuations in systems of diffusing particles. arXiv, .
Cornalba, Federico, and Julian L. Fischer. “The Dean-Kawasaki Equation and the Structure of Density Fluctuations in Systems of Diffusing Particles.” ArXiv.
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