# Entropy non-conservation and boundary conditions for Hamiltonian dynamical systems

## TL;DR

This paper explores how boundary conditions influence entropy conservation in Hamiltonian systems, revealing classical-quantum distinctions and identifying conditions for entropy preservation through self-adjoint extensions.

## Contribution

It introduces a framework using self-adjoint extensions to determine entropy conservation in classical mechanics and links boundary conditions to quantum tunneling effects.

## Key findings

- Entropy is conserved only for specific probability distributions.
- Nonconserving states can be interpreted as quantum tunneling.
- Boundary conditions determine the classical-quantum transition.

## 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.