# Equilibration properties of small quantum systems: further examples

## TL;DR

This paper examines the equilibration behavior of small quantum systems through analytical analysis of specific models, revealing how parameters influence the transition from good to poor equilibration.

## Contribution

It provides analytical insights into the equilibration properties of small quantum systems, including random Hamiltonians and specific physical models, highlighting parameter-dependent transitions.

## Key findings

- Trace T varies between 1 and N, indicating degrees of equilibration.
- Random Hamiltonians show statistical properties of T under symmetry groups.
- Parameter changes induce continuous transitions in equilibration quality.

## Abstract

It has been proposed to investigate the equilibration properties of a small isolated quantum system by means of the matrix of asymptotic transition probabilities in some preferential basis. The trace $T$ of this matrix measures the degree of equilibration of the system prepared in a typical state of the preferential basis. This quantity may vary between unity (ideal equilibration) and the dimension $N$ of the Hilbert space (no equilibration at all). Here we analyze several examples of simple systems where the behavior of $T$ can be investigated by analytical means. We first study the statistics of $T$ when the Hamiltonian governing the dynamics is random and drawn from a distribution invariant under the group U$(N)$ or O$(N)$. We then investigate a quantum spin $S$ in a tilted magnetic field making an arbitrary angle with the preferred quantization axis, as well as a tight-binding particle on a finite electrified chain. The last two cases provide examples of the interesting situation where varying a system parameter -- such as the tilt angle or the electric field -- through some scaling regime induces a continuous transition from good to bad equilibration properties.