  In this paper, we investigate and compare two well-developed definitions of
entropy relevant for describing the dynamics of isolated quantum systems:
bipartite entanglement entropy and observational entropy. In a model system of
interacting particles in a one-dimensional lattice, we numerically solve for
the full quantum behavior of the system. We characterize the fluctuations, and
find the maximal, minimal, and typical entropy of each type that the system can
eventually attain through its evolution. While both entropies are low for some
"special" configurations and high for more "generic" ones, there are several
fundamental differences in their behavior. Observational entropy behaves in
accord with classical Boltzmann entropy (e.g. equilibrium is a condition of
near-maximal entropy and uniformly distributed particles, and minimal entropy
is a very compact configuration). Entanglement entropy is rather different:
minimal entropy "empties out" one partition while maximal entropy apportions
the particles between the partitions, and neither is typical. Beyond these
qualitative results, we characterize both entropies and their fluctuations in
some detail as they depend on temperature, particle number, and box size.
