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 $S_{univ}$ increases in spontaneous processes, $ Delta S_{univ} u003e 0$. Here we seek an alternative entropy of the $S_{univ}$ 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 $ Delta F_{sys} = - T Delta S_{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 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.