  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.
