On the geometry and entropy of non-Hamiltonian phase space

Abstract

We analyse the equilibrium statistical mechanics of canonical, non-canonical andnon-Hamiltonian equations of motion, throwing light on the peculiar geometric structure ofphase space. Some fundamental issues regarding time translation and phase space measureare clarified. In particular, we emphasize that a phase space measure should be defined bymeans of the Jacobian of the transformation between different kinds of coordinates sincesuch a determinant is different from zero in the non-canonical case even if thephase space compressibility is null. Instead, the Jacobian determinant associatedwith phase space flows is unity whenever non-canonical coordinates lead to avanishing compressibility, so its use for defining a measure may not always be correct.To better illustrate this point, we derive a mathematical condition for definingnon-Hamiltonian phase space flows with zero compressibility. The Jacobian determinantassociated with the time evolution in phase space is very useful for analysing timetranslation invariance. The proper definition of a phase space measure is particularlyimportant when defining the entropy functional in the canonical, non-canonical, andnon-Hamiltonian cases. We show how the use of relative entropies can circumvent somesubtle problems that are encountered when dealing with continuous probabilitydistributions and phase space measures. Finally, a maximum (relative) entropyprinciple is formulated for non-canonical and non-Hamiltonian phase space flows.