On the Statistical Mechanics of Non-Hamiltonian Systems: The Generalized Liouville Equation, Entropy, and Time-Dependent Metrics

Several questions in the statistical mechanics of non-Hamiltonian systems are discussed. The theory of differential forms on the phase space manifold is applied to provide a fully covariant formulation of the generalized Liouville equation. The properties of invariant volume elements are considered, and the nonexistence in general of smooth invariant measures noted. The time evolution of the generalized Gibbs entropy associated with a given choice of volume form is studied, and conditions under which the entropy is constant are discussed. For non-Hamiltonian systems on manifolds with a metric tensor compatible with the flow, it is shown that the associated metric factor is in general a time-dependent solution of the generalized Liouville equation.