  A finite quantum system evolving unitarily equilibrates in a probabilistic
fashion. In the general many-body setting the time-fluctuations of an
observable \mathcal{A} are typically exponentially small in the system size. We
consider here quasi-free Fermi systems where the Hamiltonian and observables
are quadratic in the Fermi operators. We first prove a novel bound on the
temporal fluctuations \Delta\mathcal{A}^{2} and then map the equilibration
dynamics to a generalized classical XY model in the infinite temperature limit.
Using this insight we conjecture that, in most cases, a central limit theorem
can be formulated leading to what we call Gaussian equilibration: observables
display a Gaussian distribution with relative error
\Delta\mathcal{A}/\bar{\mathcal{A}}=O(L^{-1/2}) where L is the dimension of the
single particle space. The conjecture, corroborated by numerical evidence, is
proven analytically under mild assumptions for the magnetization in the quantum
XY model and for a class of observables in a tight-binding model. We also show
that the variance is discontinuous at the transition between a quasi-free model
and a non-integrable one.
