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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]

Simon Braun and collaborators prepared a state with negative temperature for motional degrees of freedom in an ultracold atomic system. They tailored a lattice model so the occupied high-energy states were compatible with the relevant upper-energy bound.^[ref]

In this regime $\beta =\partial (S/k_B)/\partial E<0$: increasing energy reduces the entropy of the constrained system. The state is not colder than absolute zero. Its population favours high-energy states, and its thermodynamic ordering differs from the positive-temperature branch. The experimental energy bounds and stability conditions are essential.