Conditions for Equilibrium at Negative Absolute Temperatures

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Abstract
In the classical phenomenological thermodynamic theory of Gibbs, equilibrium is defined as the state of maximum entropy at constant energy, and a theorem is proved (the energy theorem) which asserts that the equilibrium state is the state of minimum energy at fixed entropy. This theorem is not true for systems at negative absolute temperatures. By examining a familiar statistical model which can exhibit negative absolute temperatures we find the correct form of the energy theorem. It turns out that at negative absolute temperatures the state of equilibrium of a system with a given entropy is that in which the system has its highest energy.