Negative absolute temperature can occur only for a system with an energy spectrum bounded above. Since , higher-energy states are more populated than lower-energy states.
Instantaneously reversing a Hamiltonian satisfying leaves populations unchanged but rewrites them as a canonical distribution with . When the partition function is unchanged, the free energy changes sign and the heat capacity is unchanged.
A negative-temperature system is hotter than every positive-temperature system. If heat passes from it to a positive-temperature body, the total entropy change is
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Negative temperature is a concept primarily found in statistical mechanics and thermodynamics, and it can be somewhat counterintuitive. While temperatures are usually thought of as being positive (0 K and above, where 0 K is absolute zero), negative temperatures can occur in systems with a limited number of energy states, such as certain magnetic systems or some types of dissipative systems.