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Defining temperature $T\equiv\left(∂S/∂U\right)^{-1}$ is universal. It applies even with negative temperature.

Except for negative temperature, when does defining temperature as average kinetic energy not apply?

Ralph Baierlin, Thermal Physics (1999) offers this example: “a gaseous system that is in the quantum domain and yet is nearly classical. For this context . . . fermions and boson—when manifestly at the same temperatures—have different average translational kinetic energies.”

What is the general statement for when temperature is not equal to average kinetic energy, and what are good examples?

(When is temperature not a measure of the average kinetic energy of the particles in a substance? is, I think, just about average kinetic energy in a monatomic gas. I mean here to ask about all cases of average kinetic energy—monatomic/polyatomic, gas/liquid/solid, etc.)

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    $\begingroup$ An Ising model does not have translational motion, yet has temperature. $\endgroup$
    – Jon Custer
    Commented Apr 20, 2020 at 15:29
  • $\begingroup$ Does it have any other motion that allows it to have “kinetic” energy? $\endgroup$
    – JPM
    Commented Apr 20, 2020 at 18:07
  • $\begingroup$ Nope, only spin directions. Yet there is still a temperature that describes the assemblage of spins. $\endgroup$
    – Jon Custer
    Commented Apr 20, 2020 at 18:08
  • $\begingroup$ And T≡1/(∂S/∂U) applies? Thanks. $\endgroup$
    – JPM
    Commented Apr 20, 2020 at 18:28

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