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We study the time evolution of the reduced density matrix of a system ofspin-1/2 particles interacting with an environment of spin-1/2 particles. Theinitial state of the composite system is taken to be a product state of a purestate of the system and a pure state of the environment. The latter pure stateis prepared such that it represents the environment at a given finitetemperature in the canonical ensemble. The state of the composite systemevolves according to the time-dependent Schr{\"{o}}dinger equation, theinteraction creating entanglement between the system and the environment. It isshown that independent of the strength of the interaction and the initialtemperature of the environment, all the eigenvalues of the reduced densitymatrix converge to their stationary values, implying that also the entropy ofthe system relaxes to a stationary value. We demonstrate that the differencebetween the canonical density matrix and the reduced density matrix in thestationary state increases as the initial temperature of the environmentdecreases. As our numerical simulations are necessarily restricted to a modestnumber of spin-1/2 particles ($<36$), but do not rely on time-averaging ofobservables nor on the assumption that the coupling between system andenvironment is weak, they suggest that the stationary state of the systemdirectly follows from the time evolution of a pure state of the compositesystem, even if the size of the latter cannot be regarded as being close to thethermodynamic limit.

Approach to equilbrium in nano-scale systems at finite temperatur