Vlasov moment flows and geodesics on the Jacobi group

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.