  Closed quantum systems evolve unitarily and therefore cannot converge in a
strong sense to an equilibrium state starting out from a generic pure state.
Nevertheless for large system size one observes temporal typicality. Namely,
for the overwhelming majority of the time instants, the statistics of
observables is practically indistinguishable from an effective equilibrium one.
In this paper we consider the Loschmidt echo (LE) to study this sort of unitary
equilibration after a quench. We draw several conclusions on general grounds
and on the basis of an exactly-solvable example of a quasi-free system. In
particular we focus on the whole probability distribution of observing a given
value of the LE after waiting a long time. Depending on the interplay between
the initial state and the quench Hamiltonian, we find different regimes
reflecting different equilibration dynamics. When the perturbation is small and
the system is away from criticality the probability distribution is Gaussian.
However close to criticality the distribution function approaches a double
peaked, "batman-hood" shaped, universal form.
