  We study the fluctuations of subsystem entropies in closed quantum many-body
systems after thermalization. Using a combination of analytics and numerics for
both random quantum circuits and Hamiltonian dynamics, we find that the
statistics of such entropy fluctuations is drastically different than in the
classical setting. For instance, shortly after a system thermalizes, the
probability of entropy fluctuations for a subregion is suppressed in the
dimension of the Hilbert space of the complementary subregion. This suppression
becomes increasingly stringent as a function of time, ultimately depending on
the exponential of the Hilbert space dimension, until extremely late times when
the amount of suppression saturates. We also use our results to estimate the
total number of rare fluctuations at large timescales. We find that the
"Boltzmann brain" paradox is largely ameliorated in quantum many-body systems,
in contrast with the classical setting.
