Entropy nonconservation and boundary conditions for Hamiltonian dynamical systems

Area of Science:

• Theoretical physics
• Mathematical physics
• Quantum mechanics

Background:

• Koopman-von Neumann classical mechanics extends classical dynamics using Hilbert spaces.
• Hamiltonian evolution typically conserves entropy for isolated systems.
• Self-adjoint extensions of Hermitian operators are fundamental in quantum mechanics.

Purpose of the Study:

• To find the most general probability distributions conserving entropy under Hamiltonian evolution in classical mechanics.
• To explore the connection between classical entropy conservation and quantum boundary conditions.
• To identify conditions under which classical systems exhibit non-conserved entropy.

Main Methods:

• Application of self-adjoint extension theory to Koopman-von Neumann mechanics.
• Identification of a novel dynamical phase.
• Construction of explicit examples of entropy non-conservation for free and harmonic systems in bounded phase-space.

Main Results:

• The most general set of entropy-conserving probability distributions was determined.
• Non-entropy-conserving states, classically forbidden, were explicitly constructed.
• These states were interpreted as quantum tunneling phenomena.

Conclusions:

• Boundary conditions in quantum systems dictate entropy preservation in the classical limit.
• Boundary effects fundamentally alter system dynamics, blurring the line between classical and quantum descriptions.
• The study reveals a deep connection between classical entropy conservation and quantum mechanical boundary phenomena.