From variational to bracket formulations in nonequilibrium thermodynamics of simple systems (Q5918993)

Among the approaches that generalize the fundamental principles of thermodynamics to the case of irreversible processes, one may mention the variational and bracket formulations. The first approach is a systematic one: it allows to tackle many different situations from a common point of view. The second one is a somewhat case-to-case approach: different types of brackets (Poisson bracket is the most known) may be used in different situations. In this paper the authors unities both approaches. The generalization of the Lagrange-d'Alembert principle itself begins from the mechanical system with friction; for this system, the notions of entropy and temperature are defined. Then for systems with internal mass transfer, the thermodynamic displacements are defined besides the well-known thermodynamic forces and fluxes. It allows to derive the evolution equations for such systems. Later three main bracket formalisms, namely single generator, double generator and GENERIC formalism, are derived from the the ,obtained principles and equations. Explicit expressions for all these types of brackets in terms of entropy, Lagrangian and Hamiltonian of the system under discussion are obtained. The authors also discuss case when configuration space is a Lie group and the system has some symmetry connected with this group. For that case reduced expressions for the brackets are derived. The behaviour of coadjoint orbits in this case is discussed, too.