Entropy non-conservation and boundary conditions for Hamiltonian dynamical systems

Abstract

Applying the theory of self-adjoint extensions of Hermitian operators to Koopman von Neumann classical mechanics, the most general set of probability distributions is found for which entropy is conserved by Hamiltonian evolution. A new dynamical phase associated with such a construction is identified. By choosing distributions not belonging to this class, we produce explicit examples of both free particles and harmonic systems evolving in a bounded phase-space in such a way that entropy is nonconserved. While these nonconserving states are classically forbidden, they may be interpreted as states of a quantum system tunneling through a potential barrier boundary. In this case, the allowed boundary conditions are the only distinction between classical and quantum systems. We show that the boundary conditions for a tunneling quantum system become the criteria for entropy preservation in the classical limit. These findings highlight how boundary effects drastically change the nature of a system.

arXiv comment9 pages, 3 figures (corrected types and updated conclusion and fig. 3)

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Cited by in corpus (4)

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• The wave operator representation of quantum and classical dynamics
• Free to Harmonic Unitary Transformations in Quantum and Koopman Dynamics
• On reconstruction of states from evolution induced by quantum dynamical semigroups perturbed by covariant measures