On the geometry and entropy of non-Hamiltonian phase space

We analyse the equilibrium statistical mechanics of canonical, non- canonical and non-Hamiltonian equations of motion, throwing light on the peculiar geometric structure of phase space. Some fundamental issues regarding time translation and phase space measure are clarified. In particular, we emphasize that a phase space measure should be defined by means of the Jacobian of the transformation between different kinds of coordinates since such a determinant is different from zero in the non-canonical case even if the phase space compressibility is null. Instead, the Jacobian determinant associated with phase space flows is unity whenever non-canonical coordinates lead to a vanishing compressibility, so its use for defining a measure may not always be correct. To better illustrate this point, we derive a mathematical condition for defining non-Hamiltonian phase space flows with zero compressibility. The Jacobian determinant associated with the time evolution in phase space is very useful for analysing time translation invariance. The proper definition of a phase space measure is particularly important when defining the entropy functional in the canonical, non-canonical, and non-Hamiltonian cases. We show how the use of relative entropies can circumvent some subtle problems that are encountered when dealing with continuous probability distributions and phase space measures. Finally, a maximum (relative) entropy principle is formulated for non-canonical and non-Hamiltonian phase space flows.