  We study the behavior of R\'enyi entropies for pure states from standard
assumptions about chaos in the high-energy spectrum of the Hamiltonian of a
many-body quantum system. We compute the exact long-time averages of R\'enyi
entropies and show that the quantum noise around these values is exponentially
suppressed in the microcanonical entropy. For delocalized states over the
microcanonical band, the long-time average approximately reproduces the
equilibration proposal of H. Liu and S. Vardhan, with extra structure arising
at the order of non-planar permutations. We analyze the equilibrium
approximation for AdS/CFT systems describing black holes in equilibrium in a
box. We extend our analysis to the situation of an evaporating black hole, and
comment on the possible gravitational description of the new terms in our
approximation.

High Energy Physics - Theory