Quantum Thermodynamics, Entropy of the Universe, Free Energy, and the Second Law

We take the view that the standard von Neumann definition, in which the entropy S^vN of a pure state is zero, is in evident conflict with the statement of the second law that the entropy of the universe S_univ increases in spontaneous processes, ΔS_univ > 0. Here we seek an alternative entropy of the universe S_univ that is in accord with the second law, in a spirit not dissimilar to von Neumann himself in lesser-known work. We perform simulations of time dependent dynamics for a previously developed \cite{polyadbath} model quantum system becoming entangled with a quantum environment. We test the new definition of the entropy of the system-environment "universe" against the standard thermodynamic relation ΔF_sys = - T ΔS_univ, calculating system properties using the reduced density matrix and standard von Neumann entropy. Good agreement is obtained, showing the compatibility of an entropy for a pure state of a universe with the statement of the second law and the concept of free energy. Interesting deviation from microcanonical behavior within the zero order energy shell is observed in a context of effectively microcanonical behavior within the much larger total basis of the time dependent universe.