  A small system in contact with a macroscopic environment usually approaches
an asymptotic state, determined only by some macroscopic properties of the
environment such as the temperature or the chemical potential. In the long-time
limit, the state of the small system is thus expected to be independent of its
initial state. In some situations, however, the asymptotic state of the system
is influenced by its initial state and some information about the initial state
is kept for all times. Motivated by this finding, we propose a measure to
quantify the influence of the initial state of an open system on its dynamics.
Using this measure we derive conditions under which the asymptotic state exists
and is unique. We demonstrate our concepts for the dynamics of the spin-boson
model, identify three qualitatively different long-time behaviors, and discuss
how they can be distinguished based on the proposed measure.
