On the Metriplectic Dynamics for Systems with Internal Degrees of Freedom

Following the revival of interest in the behavior of classical, nonlinear systems, a variety of new theoretical tools was developed to allow for a description of the essential properties of those systems. One of the most powerful mathematical methods developed recently is the Lie-Poisson bracket technique [1], the use of which has led to considerable progress in our understanding of conservative relativistic fluid and plasma dynamics [2-6]. On the other hand it has been appreciated since quite a time that the nonlinear dissipative systems are of predominant applied interest [7,8]. The phase transformation physics should serve here as an important example. Quite recently it has been argued that one can describe the irreversible or dissipative processes by means of the dissipative bracket technique [ 9,10]. It has also been demonstrated that one can unify both Lie-Poisson and dissipative brackets into a technique called either mixed canonical-dissipative or metriplectic one [ 11 ]. That technique has been employed implicitly in refs. [ 12-14] to study the nonrelativistic Navier-Stokes fluid and magnetic systems. Recently we have discussed metriplectic dynamics of relativistic plasma kinetics [ 15 ]. In this Letter we present a generalization of the metriplectic dynamics for the case of many-particle systems when the particles are equipped with internal degrees of freedom. Previously whe have outlined the Lie-Poisson formulation for such a system [ 16]. The dynamics we shall study here shares similarity with the one called DCB1 in ref. [ 10]. Consider a system of classical, nonrelativistic particles possessing additional degrees of freedom lumped together into a vector S which we shall call spin. We shall assume that spin components obey the usual Poisson bracket relations for the angular momentum generators, that is