New μ‐Space Stochastic Equation for Dynamical Systems in Liouville Form

Abstract

By examining both the divergence of the velocity vector in standard orthogonal Cartesian coordinate Γ space of dimension ℝ2ptN for the reasonably general case of a Hamiltonian with (a) continuous partial derivatives to at least second order in its variables and with (b) additively separable momentum and space variables, it is shown that the corresponding Liouville equation for such a class of dynamical processes cannot describe non‐linear motion in general in the spaces considered, although hitherto it has been assumed to be applicable for all general motion in this broad class, which encompasses a major portion of physical representation. To extend the scope of dynamical system description for these classes, a new stochastic equation which is everywhere in principle discontinuous is developed for a general Hamiltonian which is a functional of the space and momentum variables, where the form of the new equation is strikingly similar to the Liouville equation which utilizes continuous functions and variables. In this development, the average trajectory of a system point is proved to be orthogonal to any constant energy surface consonant with the system energy at equilibrium, which is analogous to a similar result for the Liouville equation where no such averages are implied. For the general class of Hamiltonians considered, the celebrated Poincaré recurrence theorem does not obtain, strongly suggesting that ergodicity or quasiergodicity may hold, in accordance with the fundamental assumptions of at least equilibrium statistical mechanics, and the Birkhoff theorem would have to be appropriately recast. Due to the fortunate similarity of form to the Liouville equation, it is conjectured that the algebraic techniques developed for solving the Liouville equation may greatly simplify or aid in the development of methods that can be used to solve the present stochastic equation. This equation does not assume the presence of binary collision only, as required in the standard first‐order Boltzmann equation as derived from the Liouville equation, and is therefore suitable to describe dense systems, and functions as another equation for dynamical systems. A discussion of some other new and adapted proposals resembling the Liouville equation is presented. 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. All the standard conservation and dynamical laws of classical mechanics are observed in this work.