Matched pair analysis of the Vlasov plasma

Abstract

We present the Hamiltonian (Lie-Poisson) analysis of the Vlasov plasma, and the dynamics of its kinetic moments, from the matched pair decomposition point of view. We express these (Lie-Poisson) systems as couplings of mutually interacting (Lie-Poisson) subdynamics. The mutual interaction is beyond the well-known semi-direct product theory. Accordingly, as the geometric framework of the present discussion, we address the matched pair Lie-Poisson formulation allowing mutual interactions. Moreover, both for the kinetic moments and the Vlasov plasma cases, we observe that one of the constitutive subdynamics is the compressible isentropic fluid flow, and the other is the dynamics of the kinetic moments of order ≥ 2. In this regard, the algebraic/geometric (matched pair) decomposition that we offer, is in perfect harmony with the physical intuition. To complete the discussion, we present a momentum formulation of the Vlasov plasma, along with its matched pair decomposition.

Lifts of Symmetric Tensors: Fluids, Plasma, and Grad Hierarchy

“…which is called the momentum-Vlasov equation in the literature [19,25,33]. We remark that this system is precisely equal to conformal kinetic dynamics (2.33) with c being zero.…”

“…Keeping the same line of thought, further analysis has also been carried out on fluid dynamics [17,18]. Additionally, rich algebraic structure of momentum-Vlasov dynamics is examined in [25], inspired from the moment algebra of Vlasov dynamics, see for instance [28,29,36]. This present work consists of three main sections, in which we propose novel geometries and kinetic theories generalizing the ones in the literature, along with an appendix.…”

On the role of geometry in statistical mechanics and thermodynamics II: Thermodynamic perspective

“…We have shown that the time evolution generated by (11) terminates at equilibrium states x(e * , n * ) ∈ M ⊂ M (see (18)). The manifold M is the pattern that we have recognized in the phase portrait generated by (11).…”

“…The reduced dynamics is then the compressible fluid motion, whereas M/N has also its own dynamics determined by a Poisson bracket, called Kupershmidt-Manin bracket [42,20]. In a recent study [18,16], it is shown that the relationship between N and M/N can be investigated through the matched-pair geometry, permitting also mutual interactions between N and M/N . This strategy identifies the individual motions of N and M/N as subsystems of M while labeling properly the rest of the terms in the dynamics in M in terms of the mutual actions of N and M/N .…”