  A system put in contact with a large heat bath normally thermalizes. This
means that the state of the system approaches an equilibrium state, the latter
depending only on macroscopic characteristics of the bath (e.g. temperature),
but not on the initial state of the system. The above statement is the
cornerstone of the equilibrium statistical mechanics; its validity and its
domain of applicability are central questions in the studies of the foundations
of statistical mechanics. In the present paper we concentrate on one aspect of
thermalization, namely, on the system initial state independence (ISI) of the
equilibrium state. A necessary condition for the system ISI is derived in the
quantum framework. We use the derived condition to prove the absence of the
system ISI in a specific model. Namely, we consider a single spin coupled to a
large bath, the interaction being of a specific form. Although the model under
consideration is nontrivial enough to exhibit the decoherence and the approach
to equilibrium, the derived necessary condition is not fulfilled and thus the
equilibrium state depends on the initial state of the spin.
