Hamiltonian reductions of the one-dimensional Vlasov equation using phase-space moments

Abstract

We consider Hamiltonian closures of the Vlasov equation using the phase-space moments of the distribution function. We provide some conditions on the closures imposed by the Jacobi identity. We completely solve some families of examples. As a result, we show that imposing that the resulting reduced system preserves the Hamiltonian character of the parent model shapes its phase space by creating a set of Casimir invariants as a direct consequence of the Jacobi identity.

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• Statistical moments for classical and quantum dynamics: formalism and generalized uncertainty relations
• Geometry of Vlasov kinetic moments: a bosonic Fock space for the symmetric Schouten bracket
• Hamiltonian fluid closures of the Vlasov-Amp{è}re equations: from water-bags to N moment models