Geometry of Vlasov kinetic moments: a bosonic Fock space for the symmetric Schouten bracket

TL;DR

This paper reveals the geometric structure of Vlasov kinetic moments, linking them to symmetric Schouten brackets and bosonic Fock spaces, and introduces a new momentum map framework for plasma dynamics.

Contribution

It establishes a Lie-Poisson structure for Vlasov moments, identifies the Schouten bracket as the underlying Lie bracket, and connects moments with bosonic Fock space representations.

Findings

Kinetic moments form a Lie-Poisson algebra related to the Schouten bracket.

Moment Lie algebra is connected to a bundle of bosonic Fock spaces.

The framework extends to anisotropic plasma interactions.

Abstract

The dynamics of Vlasov kinetic moments is shown to be Lie-Poisson on the dual Lie algebra of symmetric contravariant tensor fields. The corresponding Lie bracket is identified with the symmetric Schouten bracket and the moment Lie algebra is related with a bundle of bosonic Fock spaces, where creation and annihilation operators are used to construct the cold plasma closure. Kinetic moments are also shown to define a momentum map, which is infinitesimally equivariant. This momentum map is the dual of a Lie algebra homomorphism, defined through the Schouten bracket. Finally the moment Lie-Poisson bracket is extended to anisotropic interactions.