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Hamiltonian hydrodynamic reductions of one-dimensional Vlasov equations

Rayan Oufar, Cristel Chandre

TL;DR

This work presents a unified Hamiltonian framework for reducing the 1D Vlasov-Poisson system to hydrodynamic-like fluid models with an arbitrary number of variables. By exploiting hydrodynamic Poisson brackets and Casimir invariants, the authors identify normal variables in which all known Hamiltonian closures (multi-delta, waterbag, Burby, and Chandre four-field) are polynomial and share a single generating moment, $μ_2$, which is cubic. The closures are transparently described parametrically via a set of normal variables, revealing a structural connection among diverse models and enabling a compact Hamiltonian description of reduced dynamics with preserved Casimir invariants. This structural unification suggests avenues for data-driven or sparsity-based discovery of closure relations while maintaining the Hamiltonian character of the reduced Vlasov-Poisson dynamics.

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 $μ$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.

Hamiltonian hydrodynamic reductions of one-dimensional Vlasov equations

TL;DR

This work presents a unified Hamiltonian framework for reducing the 1D Vlasov-Poisson system to hydrodynamic-like fluid models with an arbitrary number of variables. By exploiting hydrodynamic Poisson brackets and Casimir invariants, the authors identify normal variables in which all known Hamiltonian closures (multi-delta, waterbag, Burby, and Chandre four-field) are polynomial and share a single generating moment, , which is cubic. The closures are transparently described parametrically via a set of normal variables, revealing a structural connection among diverse models and enabling a compact Hamiltonian description of reduced dynamics with preserved Casimir invariants. This structural unification suggests avenues for data-driven or sparsity-based discovery of closure relations while maintaining the Hamiltonian character of the reduced Vlasov-Poisson dynamics.

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 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.
Paper Structure (17 sections, 15 theorems, 152 equations)

This paper contains 17 sections, 15 theorems, 152 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hydrodynamic Poisson brackets for fluid and plasmas
  4. Definition and properties of hydrodynamic Poisson brackets
  5. Partial decoupling of hydrodynamic Poisson brackets
  6. Centered moments
  7. Classification of known Hamiltonian fluid closures
  8. The multi-delta distribution
  9. The waterbag distribution
  10. Burby's Hamiltonian closure (see Ref. [Burby, 2023])
  11. Expression of the Casimir invariants
  12. Four-field Hamiltonian closure of Ref. [Chandre 2022]
  13. Expression of the bracket in the mu-variables
  14. Proofs of statements in Section 1
  15. Proof of Lemma \ref{['prop:murel']}
  16. ...and 2 more sections

Key Result

Theorem 3.1

(see Ref. dubrovin_hamiltonian_1983) For the bracket eqn:PBh with non-degenerate $\alpha$ to satisfy the Jacobi identity, a necessary and sufficient condition is that there exists a locally invertible change of field variables ${\bf u}\mapsto {\bm \nu}={\bm \nu}({\bf u})$ such that the bracket expre where $\overline{F}[{\bm \nu}]=F[{\bf u}]$ and $\overline{F}_n={\delta \overline{F}}/{\delta \nu_n}

Theorems & Definitions (31)

  • Theorem 3.1
  • Lemma 3.1
  • proof
  • Corollary 3.2
  • proof
  • Definition 3.1: Kinetic hydrodynamic bracket
  • Corollary 3.3
  • proof
  • Definition 3.2: Partially decoupled bracket
  • Proposition 3.1
  • ...and 21 more