How can we prove that if a negative-temperature system is in contact with a positive-temperature system, then the heat flow from the first to the second (and finally, the temperature of the second increases) ?

I haven't found a proof on the Internet...

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    $\begingroup$ Possible duplicates: physics.stackexchange.com/q/21851/2451 and links therein. $\endgroup$ – Qmechanic Apr 28 '13 at 8:54
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    $\begingroup$ The entropy increases at least by $\sum dQ/T$ (which must be positive) where $Q$ is heat. This implies that the systems with a larger coefficient $1/T$ have to be multiplied by the (more) positive $dQ$. Colder systems with positive $T$ have a lower positive $T$, and therefore greater positive $1/T$, so they're gaining energy by thermal contact with hotter objects. However, objects with negative temperature have an even smaller value of $1/T$, namely negative one, so they're gaining even more energy. The flows are intuitive if you order the objects by their $1/T$ rather than $T$. $\endgroup$ – Luboš Motl Apr 28 '13 at 9:04
  • $\begingroup$ Two questions : 1) it's false that the temperature of the second system always increases ? 2) why $\sum dQ/T$ must be positive ? If the transformation is not reversible, it could be negative while $\Delta S > 0$ $\endgroup$ – Arnaud Apr 28 '13 at 9:08