We investigate the equilibration of a small isolated quantum system by means
of its matrix of asymptotic transition probabilities in a preferential basis.
The trace of this matrix is shown to measure the degree of equilibration of the
system launched from a typical state, from the standpoint of the chosen basis.
This approach is substantiated by an in-depth study of the example of a
tight-binding particle in one dimension. In the regime of free ballistic
propagation, the above trace saturates to a finite limit, testifying good
equilibration. In the presence of a random potential, the trace grows linearly
with the system size, testifying poor equilibration in the insulating regime
induced by Anderson localization. In the weak-disorder situation of most
interest, a universal finite-size scaling law describes the crossover between
the ballistic and localized regimes. The associated crossover exponent 2/3 is
dictated by the anomalous band-edge scaling characterizing the most localized
energy eigenstates.Comment: 19 pages, 7 figures, 1 tabl
