https://cea.hal.science/cea-01484658Luck, J. M.J. M.LuckIPHT - Institut de Physique Théorique - UMR CNRS 3681 - CEA - Commissariat à l'énergie atomique et aux énergies alternatives - Université Paris-Saclay - CNRS - Centre National de la Recherche ScientifiqueEquilibration properties of small quantum systems: further examplesHAL CCSD2017[PHYS.COND.CM-SM] Physics [physics]/Condensed Matter [cond-mat]/Statistical Mechanics [cond-mat.stat-mech]De Laborderie, Emmanuelle2017-03-07 15:32:392023-03-24 14:53:042017-03-08 09:15:03enPreprints, Working Papers, ...https://cea.hal.science/cea-01484658/documenttext/html; charset=utf-81It 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.