7th Asia-Pacific Conference on Plasma Physics, 12-17 Nov, 2023 at Port Messe Nagoya

The metriplectic 4-bracket: An inclusive framework for consistently joining
Hamiltonian and dissipative systems
Philip J. Morrison
Department of Physics and Institute for Fusion Studies
University of Texas at Austin
e-mail: morrison@physics.utexas.edu
Although an early generalization of Lagrangianmechanics to include dissipation was proposed byRayleigh (1894) and subsequently various specificframeworks for dissipation have been given, e.g., forphase separation in Cahn-Hilliard (1958) and Ricci flows
in Hamilton (1982), here we discuss bracket descriptions
for dissipation that were motivated by the noncanonicalPoisson bracket formulation of plasma models presented
in [1,2] and reviewed in [3]. The motivation of earlyattempts [3-7] was to place dissipation in a kind ofbracket formalism that complements the nondissipative
Poisson bracket formalism. The axioms of the formalism
were first presented in [6,7] and later in [8] called
metriplectic dynamics. Subsequent aspects of
metriplectic dynamics were described by the author andcollaborators in [9-11].
Here we present the metriplectic 4-bracket [12], an
inclusive framework for joined Hamiltonian anddissipative dynamical systems, which preserve energy
and produce entropy. The dissipative dynamics of theframework is based on a 4-bracket, a quantity like thePoisson 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 twogenerators, the Hamiltonian and the entropy, with theentropy being a Casimir of the Hamiltonian part of thesystem. The formalism includes all known previous
binary bracket theories for dissipation or relaxation as
special cases. Rich geometrical significance of theformalism and methods for constructing metriplectic 4brackets are explored. Many examples of both finite andinfinite dimensions are given.
We speculate about metriplectic computation. Just
as symplectic integrators preserve canonical Hamiltonian
structure, with some advantages, there have been recent
efforts to preserve noncanonical Hamiltonian structure(see e.g. [13]) and even preserve or use metriplecticstructure [14-16]. Another kind of dissipative dynamics,
double bracket dynamics, proposed in [17,18] andimproved and used in [19-22], provides a method forcomputation of equilibrium states. This formalism
emerges from the metriplectic 4-bracket. To summarize,in this talk I will describe metriplectic 4-bracket
dynamics, its construction, reduction toprevious
brackets, and its possible use for designingcomputational algorithms.
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Munich (2023)[17] R. W. Brockett, Proc. IEEE 27, 799 (1988)[18] G. K. Vallis, G. Carnevale, and W. R. Young, J.
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Ichiguchi, Phys. Plasmas 25, 082506 (2018)[22] M. Furukawa and P. J. Morrison, Phys. Plasmas 29,102504 (2022)