  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.
