Algebrizing friction: a brief look at the Metriplectic Formalism Massimo Materassi (1) Emanuele Tassi (2) (1) Istituto dei Sistemi Complessi ISC-CNR, Sesto Fiorentino, Firenze, Italy E-mail: massimo.materassi@fi.isc.cnr.it (2) Centre de Physique Théorique, CNRS -Aix-Marseille Universités, Campus de Luminy, Marseille, France E-mail: tassi@cpt.univ-mrs.fr Abstract: The formulation of Action Principles in Physics, and the introduction of the Hamiltonian framework, reduced dynamics to bracket algebræ of observables. Such a framework has great potentialities, to understand the role of symmetries, or to give rise to the quantization rule of modern microscopic Physics. Conservative systems are easily algebrized via the Hamiltonian dynamics: a conserved observable H generates the variation of any quantity f via the Poisson bracket {f,H}. Recently, dissipative dynamical systems have been algebrized in the scheme presented here, referred to as metriplectic framework: the dynamics of an isolated system with dissipation is regarded as the sum of a Hamiltonian component, generated by H via a Poisson bracket algebra; plus dissipation terms, produced by a certain quantity S via a new symmetric bracket. This S is in involution with any other observable and is interpreted as the entropy of those degrees of freedom statistically encoded in friction. In the present paper, the metriplectic framework is shown for two original “textbook” examples. Then, dissipative Magneto-Hydrodynamics (MHD), a theory of major use in many space physics and nuclear fusion applications, is reformulated in metriplectic terms. Keywords: Dissipative systems, Hamiltonian systems, Magneto-Hydrodynamics. 1. Introduction Hamiltonian systems play a key role in Physics, since the dynamics of elementary particles appear to be Hamiltonian. Hamiltonian systems are endowed with a bracket algebra (that of quantum commutators, or classically of Poisson brackets): such a scheme is of exceptional clarity in terms of symmetries [1], offering the opportunity of retrieving most of the information about the system without even trying to solve the equations of motion. Despite their central role, Hamiltonian systems are far from covering the main part of real systems: indeed, Hamiltonian systems are intrinsically conservative and reversible, while, as soon as one zooms out from the level of elementary particles, the real world appears to be made of dissipative, irreversible processes [2]. In most real systems there are couplings bringing energy from processes at a certain time- or space-scale, treated deterministically, to processes evolving at much “smaller” and “faster” scales, to be treated statistically, as “noise”. This is exactly what friction does, and this transfer appears to be irreversible. IntellectualArchive Vol.1, No.3 45