Metriplectic 4-bracket dynamics: A curvature-like framework for describing dissipation*

An inclusive framework  [1] for joined Hamiltonian and dissipative dynamical systems, which builds on early work [2,3,4], will be described.  The framework describes dynamical systems that preserve energy and produce entropy. The dissipative dynamics of the framework is based on the metriplectic 4-bracket, a quantity like the Poisson bracket defined on phase space functions, but unlike the Poisson bracket has four slots with symmetries and properties motivated by Riemannian curvature. Metriplectic 4-bracket dynamics is generating using two generators, the Hamiltonian and the entropy, with the entropy being a Casimir of the Hamiltonian part of the system. The formalism includes all known previous binary bracket theories for dissipation or relaxation as special cases. Rich geometrical significance of the formalism and methods for constructing metriplectic 4-brackets are explored. Examples of both finite and infinite dimensions will  be discussed including  plasma, fluid,  and kinetic descriptions.

[1] P. J. Morrison and  M. H. Updike, An inclusive curvature-like framework for describing dissipation: metriplectic 4-bracket dynamics, arXiv:2306.06787v1 [math-ph] 11 Jun 2023.

[2] P. J. Morrison, Bracket formulation for irreversible classical fields, Phys. Lett. A 100, 423 (1984).

[3] P. J. Morrison, Some Observations Regarding Brackets and Dissipation, Tech. Rep. PAM--228, Univ.  Cal.  Berkeley (1984).

[4] P. J. Morrison, A Paradigm for joined Hamiltonian and dissipative systems, Physica D 18, 410 (1986).