Hamiltonian hydrodynamic reductions of one-dimensional Vlasov equations

TL;DR

This paper explores Hamiltonian fluid reductions of the 1D Vlasov-Poisson equation, revealing a unified polynomial parametric closure structure derived from Casimir invariants, which simplifies the understanding of various models.

Contribution

It introduces a systematic framework using hydrodynamic Poisson brackets to unify different Hamiltonian closures of the Vlasov equation through polynomial parametric forms.

Findings

All closures lead to a common polynomial parametric form.

Normal variables derived from Casimir invariants reveal structural similarities.

A unified description of 1D Vlasov-Poisson dynamics is achieved.

Abstract

We investigate Hamiltonian fluid reductions of the one-dimensional Vlasov-Poisson equation. Our approach utilizes the hydrodynamic Poisson bracket framework, which allows us to systematically identify fundamental normal variables derived from the analysis of the Casimir invariants of the resulting Poisson bracket. This framework is then applied to analyze several well-established Hamiltonian closures of the onedimensional Vlasov equation, including the multi-delta distribution and the waterbag models. Our key finding is that all of these seemingly distinct closures consistently lead to the formulation of a unified form of parametric closures: When expressed in terms of the identified normal variables, the parameterization across all these closures is revealed to be polynomial and of the same degree. All these parametric closures are uniquely generated from one of the moments, called $\mu$2, a cubic polynomial in the normal variables. This result establishes a structural connection between these different physical models, offering a path toward a more unified and simplified description of the one-dimensional Vlasov-Poisson dynamics through its reduced hydrodynamic forms with an arbitrary number of fluid variables.