  We study the time evolution of the reduced density matrix of a system of
spin-1/2 particles interacting with an environment of spin-1/2 particles. The
initial state of the composite system is taken to be a product state of a pure
state of the system and a pure state of the environment. The latter pure state
is prepared such that it represents the environment at a given finite
temperature in the canonical ensemble. The state of the composite system
evolves according to the time-dependent Schr{\"{o}}dinger equation, the
interaction creating entanglement between the system and the environment. It is
shown that independent of the strength of the interaction and the initial
temperature of the environment, all the eigenvalues of the reduced density
matrix converge to their stationary values, implying that also the entropy of
the system relaxes to a stationary value. We demonstrate that the difference
between the canonical density matrix and the reduced density matrix in the
stationary state increases as the initial temperature of the environment
decreases. As our numerical simulations are necessarily restricted to a modest
number of spin-1/2 particles ($<36$), but do not rely on time-averaging of
observables nor on the assumption that the coupling between system and
environment is weak, they suggest that the stationary state of the system
directly follows from the time evolution of a pure state of the composite
system, even if the size of the latter cannot be regarded as being close to the
thermodynamic limit.
