Covariant statistical mechanics of non-Hamiltonian systems

In this paper, using the elegant language of differential forms, we provide a covariant formulation of the equilibrium statistical mechanics of non-Hamiltonian systems. The key idea of the paper is to focus on the structure of phase space and its kinematical and dynamical roles. While in the case of Hamiltonian systems, the structure of the phase space is a symplectic structure (a nondegenerate closed two-form), we consider an almost symplectic structure for the more general case of non-Hamiltonian systems. An almost symplectic structure is a nondegenerate but not necessarily closed two-form structure. Consequently, the dynamics becomes non-Hamiltonian and based on the fact that the structure is nondegenerate, we can also define a volume element. With a well-defined volume in hand, we derive the Liouville equation and find an invariant statistical state. Recasting non-Hamiltonian systems in terms of the almost symplectic geometry has at least two advantages: the formalism is covariant and therefore does not depend on coordinates and there is no confusion in the determination of the natural volume element of the system. For clarification, we investigate the application of the formalism in two examples in which the underlying geometry of the phase space is locally conformal symplectic geometry.