Vlasov moment flows and geodesics on the Jacobi group

TL;DR

This paper characterizes the integrable Bloch-Iserles system as a geodesic flow on the Jacobi group using Vlasov moment algebra, revealing new geometric and algebraic structures underlying the dynamics.

Contribution

It introduces a Lie-Poisson structure with a momentum map for the Bloch-Iserles system, linking Vlasov moments to subgroup inclusions in a central extension of the Jacobi group.

Findings

Particle-type solutions derived from Vlasov interpretation

Identification of subgroup structures in the Vlasov moments

Generalization of the Jacobi group for broader Bloch-Iserles dynamics

Abstract

By using the moment algebra of the Vlasov kinetic equation, we characterize the integrable Bloch-Iserles system on symmetric matrices (arXiv:math-ph/0512093) as a geodesic flow on the Jacobi group. We analyze the corresponding Lie-Poisson structure by presenting a momentum map, which both untangles the bracket structure and produces particle-type solutions that are inherited from the Vlasov-like interpretation. Moreover, we show how the Vlasov moments associated to Bloch-Iserles dynamics correspond to particular subgroup inclusions into a group central extension (first discovered in arXiv:math/0410100), which in turn underlies Vlasov kinetic theory. In the most general case of Bloch-Iserles dynamics, a generalization of the Jacobi group also emerges naturally.