General Non-Applicability of the Liouville Equation in Statistical Mechanics and a New μ-space Stochastic Equation for Dynamical Systems

Abstract

By examining both the divergence of the velocity vector in orthogonal Cartesian coordinate space $\mathbfΓ $ of dimension $\R^{\textrm {2fN}}$ and the structure of the Hamiltonian determining a system trajectory, it is shown that the standard Liouville equation cannot describe anything more than linear motion for typical Hamiltonians with separated momentum and space variables and some significant consequences such as the Poincare recurrence theorem do not obtain for such Hamiltonians. A new stochastic equation which is everywhere in principle discontinuous is developed for dynamical systems described by a general Hamiltonian which is a functional of the space and momentum variables, and where the average trajectory of a system point is proven to be orthogonal to any constant energy surface consonant with the system energy at equilibrium. This equation does not assume the presence of binary collision only, as required in the standard first-order Boltzmann equation, and is therefore suitable to describe dense systems as well, and may be viewed as an alternative to the latter. Some new macroscopic variational principles for non-equilibrium thermodynamical systems are proposed, where one object for future work would be to relate the microscopic description given here with the macroscopic principles.