17 research outputs found
Well-posedness for a regularised inertial Dean-Kawasaki model for slender particles in several space dimensions
A stochastic PDE, describing mesoscopic fluctuations in systems of weakly
interacting inertial particles of finite volume, is proposed and analysed in
any finite dimension . It is a regularised and inertial version
of the Dean-Kawasaki model. A high-probability well-posedness theory for this
model is developed. This theory improves significantly on the spatial scaling
restrictions imposed in an earlier work of the same authors, which applied only
to significantly larger particles in one dimension. The well-posedness theory
now applies in -dimensions when the particle-width is
proportional to for and is the number of
particles. This scaling is optimal in a certain Sobolev norm. Key tools of the
analysis are fractional Sobolev spaces, sharp bounds on Bessel functions,
separability of the regularisation in the -spatial dimensions, and use of
the Fa\`a di Bruno's formula.Comment: 28 pages, no figure
Multilevel Monte Carlo methods for the Dean-Kawasaki equation from Fluctuating Hydrodynamics
Stochastic PDEs of Fluctuating Hydrodynamics are a powerful tool for the
description of fluctuations in many-particle systems. In this paper, we develop
and analyze a Multilevel Monte Carlo (MLMC) scheme for the Dean-Kawasaki
equation, a pivotal representative of this class of SPDEs. We prove
analytically and demonstrate numerically that our MLMC scheme provides a
significant speed-up (with respect to a standard Monte Carlo method) in the
simulation of the Dean-Kawasaki equation. Specifically, we quantify how the
speed-up factor increases as the average particle density increases, and show
that sizeable speed-ups can be obtained even in regimes of low particle
density. Numerical simulations are provided in the two-dimensional case,
confirming our theoretical predictions.
Our results are formulated entirely in terms of the law of distributions
rather than in terms of strong spatial norms: this crucially allows for MLMC
speed-ups altogether despite the Dean-Kawasaki equation being highly singular.Comment: 23 pages, 9 figure
The Regularised Inertial Dean-Kawasaki equation:discontinuous Galerkin approximation and modelling for low-density regime
The Regularised Inertial Dean-Kawasaki model (RIDK) -- introduced by the
authors and J. Zimmer in earlier works -- is a nonlinear stochastic PDE
capturing fluctuations around the mean-field limit for large-scale particle
systems in both particle density and momentum density. We focus on the
following two aspects. Firstly, we set up a Discontinuous Galerkin (DG)
discretisation scheme for the RIDK model: we provide suitable definitions of
numerical fluxes at the interface of the mesh elements which are consistent
with the wave-type nature of the RIDK model and grant stability of the
simulations, and we quantify the rate of convergence in mean square to the
continuous RIDK model. Secondly, we introduce modifications of the RIDK model
in order to preserve positivity of the density (such a feature only holds in a
''high-probability sense'' for the original RIDK model). By means of numerical
simulations, we show that the modifications lead to physically realistic and
positive density profiles. In one case, subject to additional regularity
constraints, we also prove positivity. Finally, we present an application of
our methodology to a system of diffusing and reacting particles. Our Python
code is available in open-source format.Comment: 35 pages, 13 figure
Multilevel Monte Carlo methods for the Dean-Kawasaki equation from Fluctuating Hydrodynamics
Stochastic PDEs of Fluctuating Hydrodynamics are a powerful tool for the description of fluctuations in many-particle systems. In this paper, we develop and analyze a Multilevel Monte Carlo (MLMC) scheme for the Dean-Kawasaki equation, a pivotal representative of this class of SPDEs. We prove analytically and demonstrate numerically that our MLMC scheme provides a significant speed-up (with respect to a standard Monte Carlo method) in the simulation of the Dean-Kawasaki equation. Specifically, we quantify how the speed-up factor increases as the average particle density increases, and show that sizeable speed-ups can be obtained even in regimes of low particle density. Numerical simulations are provided in the two-dimensional case, confirming our theoretical predictions. Our results are formulated entirely in terms of the law of distributions rather than in terms of strong spatial norms: this crucially allows for MLMC speed-ups altogether despite the Dean-Kawasaki equation being highly singular
From weakly interacting particles to a regularised Dean–Kawasaki model
The evolution of finitely many particles obeying Langevin dynamics is described by Dean-Kawasaki equations, a class of stochastic equations featuring a non-Lipschitz multiplicative noise in divergence form. We derive a regularised Dean-Kawasaki model based on second order Langevin dynamics by analysing a system of particles interacting via a pairwise potential. Key tools of our analysis are the propagation of chaos and Simon's compactness criterion. The model we obtain is a small-noise stochastic perturbation of the undamped McKean-Vlasov equation. We also provide a high-probability result for existence and uniqueness for our model.</p
Well-posedness for a regularised inertial Dean–Kawasaki model for slender particles in several space dimensions
A stochastic PDE, describing mesoscopic fluctuations in systems of weakly interacting inertial particles of finite volume, is proposed and analysed in any finite dimension . It is a regularised and inertial version of the Dean–Kawasaki model. A high-probability well-posedness theory for this model is developed. This theory improves significantly on the spatial scaling restrictions imposed in an earlier work of the same authors, which applied only to significantly larger particles in one dimension. The well-posedness theory now applies in d-dimensions when the particle-width ϵ is proportional to for and N is the number of particles. This scaling is optimal in a certain Sobolev norm. Key tools of the analysis are fractional Sobolev spaces, sharp bounds on Bessel functions, separability of the regularisation in the d-spatial dimensions, and use of the Faà di Bruno's formula
From weakly interacting particles to a regularised Dean–Kawasaki model
The evolution of finitely many particles obeying Langevin dynamics is
described by Dean-Kawasaki equations, a class of stochastic equations featuring
a non-Lipschitz multiplicative noise in divergence form. We derive a
regularised Dean-Kawasaki model based on second order Langevin dynamics by
analysing a system of particles interacting via a pairwise potential. Key tools
of our analysis are the propagation of chaos and Simon's compactness criterion.
The model we obtain is a small-noise stochastic perturbation of the undamped
McKean-Vlasov equation. We also provide a high-probability result for existence
and uniqueness for our model.Comment: 27 pages, no figure