On finite-dimensional smoothed-particle Hamiltonian reductions of the Vlasov equation

TL;DR

This paper explores how spatial smoothing in finite-dimensional particle Hamiltonian reductions of the Vlasov equation affects the Hamiltonian structure, using reproducing kernel Hilbert spaces and modifying the Poisson bracket for Maxwell systems.

Contribution

It introduces a novel framework for incorporating spatial smoothing into Hamiltonian reductions of the Vlasov equation, linking it to reproducing kernel Hilbert spaces and modifying the Poisson structure for Maxwell systems.

Findings

Smoothing corresponds to a convolutional regularization of the Hamiltonian.

The shape function acts as a kernel in a reproducing kernel Hilbert space.

Smoothing in Vlasov-Maxwell involves modifying the Poisson bracket and the coupling terms.

Abstract

The inclusion of spatial smoothing in finite-dimensional particle-based Hamiltonian reductions of the Vlasov equation are considered. In the context of the Vlasov-Poisson equation (and other mean-field Lie-Poisson systems), smoothing amounts to a convolutive regularization of the Hamiltonian. This regularization may be interpreted as a change of the inner product structure used to identify the dual space in the Lie-Poisson Hamiltonian formulation. In particular, the shape function used for spatial smoothing may be identified as the kernel function of a reproducing kernel Hilbert space whose inner product is used to define the Lie-Poisson Hamiltonian structure. It is likewise possible to introduce smoothing in the Vlasov-Maxwell system, but in this case the Poisson bracket must be modified rather than the Hamiltonian. The smoothing applied to the Vlasov-Maxwell system is incorporated by inserting smoothing in the map from canonical to kinematic coordinates. In the filtered system, the Lorentz force law and the current, the two terms coupling the Vlasov equation with Maxwell's equations, are spatially smoothed.