Entropy Non-Conservation and Boundary Conditions for Hamiltonian Dynamical Systems

Gerard McCaul, Alexander Pechen, Denys I. Bondar

Published 2019-04-06Version 1

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 particle and harmonic systems evolving in a bounded phase-space in such a way that entropy is non-conserved. While these non-conserving states are classically forbidden, they may be interpreted as states of a quantum system tunnelling 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 tunnelling 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.