  We analyze the equilibrium statistical mechanics of canonical, non-canonical
and non-Hamiltonian equations of motion by throwing light into 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 types 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 that its use in order to define a measure may not
be always 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 time evolution in
phase space is altogether useful for analyzing 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.
