An investigation of equilibration in small quantum systems: the example of a particle in a 1D random potential

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.