Entropy nonconservation and boundary conditions for Hamiltonian dynamical systems

Gerard McCaul, Alexander Pechen, and Denys I. Bondar
Phys. Rev. E 99, 062121 – Published 20 June 2019

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.

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  • Received 19 September 2018
  • Revised 21 May 2019

DOI:https://doi.org/10.1103/PhysRevE.99.062121

©2019 American Physical Society

Physics Subject Headings (PhySH)

Statistical Physics & Thermodynamics

Authors & Affiliations

Gerard McCaul1,2,*, Alexander Pechen3,4,†, and Denys I. Bondar1,‡

  • 1Tulane University, New Orleans, Louisiana 70118, USA
  • 2King's College London, London WC2R 2LS, United Kingdom
  • 3Steklov Mathematical Institute of Russian Academy of Sciences, Gubkina str. 8, Moscow 119991, Russia
  • 4National University of Science and Technology “MISIS”, Leninski prosp. 4, Moscow 119049, Russia

  • *gerard.mccaul@kcl.ac.uk
  • apechen@gmail.com
  • dbondar@tulane.edu

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Issue

Vol. 99, Iss. 6 — June 2019

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