Entropy as a metric generator of dissipation in metriplectic systems

This lecture is a short review on the role entropy plays in dissipative dynamics formulated in terms of Leibniz bracket algebræ (LBA). While conservative dynamics is given an LBA formulation in the Hamiltonian framework, with total energy H generating the motion via classical Poisson brackets or quantum commutation brackets, an LBA formulation can be given to classical dissipative dynamics through the metriplectic bracket algebra (MBA): the conservative component of the dynamics is still generated via a Poisson algebra by the total energy H, while S, the entropy of the degrees of freedom statistically encoded in friction, generates dissipation via a metric bracket. Here a (necessarily partial) overview on the types of systems subject to MBA formulation is presented, and the physical meaning of the quantity S involved in each is discussed. Then, the parallel between the classical MBA and the commutator-anticommutator algebræ of open quantum systems is examined: the role of the quantum environment is analysed in view of a thorough comparison of it with the role of classical microscopic degrees of freedom giving rise to dissipation through their entropy in MBA.