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Compared to other formulations of the second law of thermodynamics, theClausius statement that heat does not spontaneously flow from cold to hotconcerns a system in non-equilibrium states, and in that respect is moreambitious but also more ambiguous. We discuss two scenarios when the Clausiusstatement in its plain form does not hold. First, for ergodic systems, theenergy transfer may be consistent with the statement on a coarse-grained timescale, but be anomalously directed during time intervals shorter than thethermalization time. In particular, when an initially colder system is broughtin contact to a hotter bath, the internal energy of the former increases withtime in a long run but not monotonically. Second, the heat transfer may notrespect the Clausius statement on any time-scale in non-ergodic systems due tothe formation of localized vibrational modes. We illustrate the two scenarioswith a familiar model of an isotope atom attached to a semi-infinite harmonicatomic chain. Technically, the discussion is based on a Langevin equation forthe isotope, using the initial condition when the isotope and chain areinitially prepared in uncorrelated canonical states under the constraint thatthe boundary atom between the isotope and chain is initially fixed and laterreleased. In such setting, the noise in the Langevin equation isnon-stationary, and the fluctuation-dissipation relation has a non-standardform.

Non-Clausius heat transfer: The method of the nonstationary Langevin equation