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Motivated by recent discussion about the possibility of non-Clausius (fromcold to hot) heat flow, we revisit the familiar model of an impurity atom ofmass $M$ embedded in an otherwise uniform one-dimensional harmonic lattice ofhost atoms of mass $m$. Assuming that the initial distributions for theimpurity and the rest of the lattice have canonical forms with giventemperatures, we show that the average kinetic energy of the impurity mayincrease with time even if its initial temperature is higher than or equal tothat of the lattice. Such an increase is only temporary in uniform lattices andin lattices with a heavy impurity ($M\ge m$), but may be permanent in latticeswith a localized vibrational mode generated by a light impurity ($M<m$). Thusthe model shows a non-Clausius spontaneous heat flow directed from a colderlattice to a hotter impurity.

Non-Clausius heat transfer: the example of harmonic chain with an impurity