# What does blackbody radiation look like for negative-temperature systems?

Using the thermodynamic definition of temperature,

$$\frac{1}{T}=\frac{\partial S}{\partial U}\bigg|_{V,N}$$

negative temperatures are possible, in systems where the entropy decreases when energy is added. A few examples have been synthesized in the laboratory, such as population inversions in lasing systems and (in terms of motional degrees of freedom) by adiabatically manipulating a collection of cold atoms using a Feshbach resonance (see Braun, S.; Ronzheimer, J. P.; Schreiber, M.; Hodgman, S. S.; Rom, T.; Bloch, I.; Schneider, U. (2013). "Negative Absolute Temperature for Motional Degrees of Freedom". Science. 339 (6115): 52–55.).

The spectral radiance of blackbody radiation for a system at temperature $$T$$ is described by Planck's law:

$$B(\nu,T)=\frac{2h\nu^3}{c^2}\frac{1}{e{\frac{h\nu}{kT}}-1}$$

and the total luminosity, for a body of area $$A$$ and emissivity $$\epsilon$$, is described by the Stefan-Boltzmann equation:

$$L=\sigma A \epsilon T^4$$

It is clear that the spectral radiance is negative when you plug in negative temperatures, while the total luminosity is still positive. Do these equations still hold for negative-temperature systems? If so, how do we make sense of their results? If not, what do we replace them with?

• Best in mind the derivation of your formula for $B$ uses a geometric series obtained from a partition function $Z$ that won't converge with $T<0$ if $Z$ contains arbitrarily large frequencies. – J.G. Sep 23 '18 at 21:15

• Conventionally, the relation $\partial U = T \partial S$ only describes temperature for systems at equilibrium. – Al Nejati Sep 23 '18 at 21:41