 Negative temperature

A negative temperature describes an internally equilibrated distribution that favors higher energy levels. For Boltzmann entropy $S=k_B\ln \Omega$, with $\Omega$ counting states in an energy window, defineA decreasing branch of $S(E)$ therefore has $T<0$. In a finite collection of independent two-level systems, the number of configurations is largest near equal populations and decreases as the upper level becomes more fully occupied. For a level separation $\Delta >0$, the Boltzmann factors give $p_{\mathrm{upper}}/p_{\mathrm{lower}}=e^{-\beta \Delta }>1$ when $\beta <0$.

A bounded upper energy makes these inverted canonical weights normalizable in the usual finite-spin models. An ordinary free-particle kinetic energy is unbounded above: negative $\beta$ would make its momentum integral diverge. The spin subsystem must also equilibrate internally before uncontrolled exchange with such other degrees of freedom destroys the inversion. Heat flow from negative to positive temperature shows why this state is hotter than every positive-temperature reservoir.

Negative-temperature motional states have also been prepared for ultracold bosons in a tailored Bose-Hubbard model. The original experiment reports a quasi-momentum distribution peaked at the upper band edge. ^[ref]

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.