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Most familiar systems cannot achieve negative temperatures because adding energy always increases their entropy.
However, some systems have a maximum amount of energy that they can hold, and as they approach that maximum energy their entropy actually begins to decrease.
Because temperature is defined by the relationship between energy and entropy, such a system's temperature becomes negative, even though energy is being added.
As a result, the Boltzmann factor for states of systems at negative temperature increases rather than decreases with increasing state energy.
Therefore no complete system, i. e. including the electromagnetic modes, can have negative temperatures, since there is no highest energy state, so that the sum of the probabilities of the states would diverge for negative temperatures.
However, for quasi-equilibrium systems ( e. g. spins out of equilibrium with the electromagnetic field ) this argument does not apply, and negative effective temperatures are attainable.

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