Abstract
The Dean–Kawasaki equation – one of the most fundamental SPDEs
of fluctuating hydrodynamics – has been proposed as a model for density
fluctuations in weakly interacting particle systems. In its original form it is
highly singular and fails to be renormalizable even by approaches such as
regularity structures and paracontrolled distributions, hindering mathematical
approaches to its rigorous justification. It has been understood recently that
it is natural to introduce a suitable regularization, e. g., by applying a formal
spatial discretization or by truncating high-frequency noise: This yields wellposed equations that should still precisely approximate the law of the particle
density fluctuations.
In the present work, we prove that a regularization in the form of a formal
discretization of the Dean–Kawasaki equation indeed accurately describes
density fluctuations in systems of weakly interacting diffusing particles: We
show that in suitable weak metrics, the law of fluctuations as predicted by
the discretized Dean–Kawasaki SPDE approximates the law of fluctuations
of the original particle system, up to an error that is of arbitrarily high order in
the inverse particle number and a discretization error. In particular, the Dean–
Kawasaki equation provides a means for efficient and accurate simulations of
density fluctuations in weakly interacting particle systems.
of fluctuating hydrodynamics – has been proposed as a model for density
fluctuations in weakly interacting particle systems. In its original form it is
highly singular and fails to be renormalizable even by approaches such as
regularity structures and paracontrolled distributions, hindering mathematical
approaches to its rigorous justification. It has been understood recently that
it is natural to introduce a suitable regularization, e. g., by applying a formal
spatial discretization or by truncating high-frequency noise: This yields wellposed equations that should still precisely approximate the law of the particle
density fluctuations.
In the present work, we prove that a regularization in the form of a formal
discretization of the Dean–Kawasaki equation indeed accurately describes
density fluctuations in systems of weakly interacting diffusing particles: We
show that in suitable weak metrics, the law of fluctuations as predicted by
the discretized Dean–Kawasaki SPDE approximates the law of fluctuations
of the original particle system, up to an error that is of arbitrarily high order in
the inverse particle number and a discretization error. In particular, the Dean–
Kawasaki equation provides a means for efficient and accurate simulations of
density fluctuations in weakly interacting particle systems.
| Original language | English |
|---|---|
| Journal | Annals of Probability |
| DOIs | |
| Publication status | Acceptance date - 29 Jan 2025 |
Acknowledgements
We thank the anonymous referees for their careful reading of themanuscript and valuable suggestions.
Funding
All authors gratefully acknowledge funding from the Austrian Science Fund (FWF) through the project F65. CR gratefully acknowledges support from the Austrian Science Fund (FWF), grants P30000, P33010, W1245. FC gratefully acknowledges funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 75