  We study the problem of the approach to equilibrium in a macroscopic quantum
system in an abstract setting. We prove that, for a typical choice of
"nonequilibrium subspace", any initial state (from the energy shell)
thermalizes, and in fact does so very quickly, on the order of the Boltzmann
time $\tau__\mathrm{B}:=h/(k_\mathrm{B}T)$. This apparently unrealistic, but
mathematically rigorous, conclusion has the important physical implication that
the moderately slow decay observed in reality is not typical in the present
setting.
  The fact that macroscopic systems approach thermal equilibrium may seem
puzzling, for example, because it may seem to conflict with the
time-reversibility of the microscopic dynamics. According the present result,
what needs to be explained is, not that macroscopic systems approach
equilibrium, but that they do so slowly.
  Mathematically our result is based on an interesting property of the maximum
eigenvalue of the Hadamard product of a positive semi-definite matrix and a
random projection matrix. The recent exact formula by Collins for the integral
with respect to the Haar measure of the unitary group plays an essential role
in our proof.
