On the classical statistical mechanics of non-Hamiltonian systems

A consistent classical statistical mechanical theory of non-Hamiltonian dynamical systems is presented. It is shown that compressible phase space flows generate coordinate transformations with a nonunit Jacobian, leading to a metric on the phase space manifold which is nontrivial. Thus, the phase space of a non-Hamiltonian system should be regarded as a general curved Riemannian manifold. An invariant measure on the phase space manifold is then derived. It is further shown that a proper generalization of the Liouville equation must incorporate the metric determinant, and a geometric derivation of such a continuity equation is presented. The manifestations of the nontrivial nature of the phase space geometry on thermodynamic quantities is explored.