  Compared to other formulations of the second law of thermodynamics, the
Clausius statement that heat does not spontaneously flow from cold to hot
concerns a system in non-equilibrium states, and in that respect is more
ambitious but also more ambiguous. We discuss two scenarios when the Clausius
statement in its plain form does not hold. First, for ergodic systems, the
energy transfer may be consistent with the statement on a coarse-grained time
scale, but be anomalously directed during time intervals shorter than the
thermalization time. In particular, when an initially colder system is brought
in contact to a hotter bath, the internal energy of the former increases with
time in a long run but not monotonically. Second, the heat transfer may not
respect the Clausius statement on any time-scale in non-ergodic systems due to
the formation of localized vibrational modes. We illustrate the two scenarios
with a familiar model of an isotope atom attached to a semi-infinite harmonic
atomic chain. Technically, the discussion is based on a Langevin equation for
the isotope, using the initial condition when the isotope and chain are
initially prepared in uncorrelated canonical states under the constraint that
the boundary atom between the isotope and chain is initially fixed and later
released. In such setting, the noise in the Langevin equation is
non-stationary, and the fluctuation-dissipation relation has a non-standard
form.
