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We take the view that the standard von Neumann definition, in which theentropy $S^{vN}$ of a pure state is zero, is in evident conflict with thestatement of the second law that the entropy of the universe $S_{univ}$increases in spontaneous processes, $ \Delta S_{univ} > 0$. Here we seek analternative entropy of the universe $S_{univ}$ that is in accord with thesecond law, in a spirit not dissimilar to von Neumann himself in lesser-knownwork. We perform simulations of time dependent dynamics for a previouslydeveloped \cite{polyadbath} model quantum system becoming entangled with aquantum environment. We test the new definition of the entropy of thesystem-environment "universe" against the standard thermodynamic relation $\Delta F_{sys} = - T \Delta S_{univ} $, calculating system properties using thereduced density matrix and standard von Neumann entropy. Good agreement isobtained, showing the compatibility of an entropy for a pure state of auniverse with the statement of the second law and the concept of free energy.Interesting deviation from microcanonical behavior within the zero order energyshell is observed in a context of effectively microcanonical behavior withinthe much larger total basis of the time dependent universe.

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