A consistent classical statistical mechanical theory of non-Hamiltonian dyn
amical systems is presented. It is shown that compressible phase space flow
s generate coordinate transformations with a nonunit Jacobian, leading to a
metric on the phase space manifold which is nontrivial. Thus, the phase sp
ace of a non-Hamiltonian system should be regarded as a general curved Riem
annian manifold. An invariant measure on the phase space manifold is then d
erived. It is further shown that a proper generalization of the Liouville e
quation must incorporate the metric determinant, and a geometric derivation
of such a continuity equation is presented. The manifestations of the nont
rivial nature of the phase space geometry on thermodynamic quantities is ex
plored.