Metriplectic relaxation to equilibria

Metriplectic dynamical systems consist of a special combination of a Hamiltonian and a (generalized) entropy-gradient flow, such that the Hamiltonian is conserved and entropy is dissipated/produced (depending on a sign convention). It is natural to expect that, in the long-time limit, the orbit of a metriplectic system should converge to an extremum of entropy restricted to a constant-Hamiltonian surface. In this paper, we discuss sufficient conditions for this to occur. Then, we construct a class of metriplectic systems inspired by the Landau operator for Coulomb collisions in plasmas, which is included as special case. For this class of brackets, checking the conditions for convergence reduces to checking two usually simpler conditions, and we discuss examples in detail. We apply these results to the construction of relaxation methods for the solution of equilibrium problems in fluid dynamics and plasma physics.