  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, $ \Delta S_{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 $
\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
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.
