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

TL;DR

This paper studies how small quantum systems reach equilibrium by analyzing transition probabilities, using a 1D particle in a random potential to reveal how disorder affects equilibration.

Contribution

It introduces a matrix-based measure of equilibration and applies it to a 1D quantum particle, uncovering universal scaling laws and the impact of disorder on equilibration.

Findings

In ballistic regime, the trace saturates, indicating good equilibration.

In localized regime, the trace grows linearly with system size, indicating poor equilibration.

A universal finite-size scaling law describes the crossover between regimes.

Abstract

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.