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We usually see thermodynamic cycles (e g. Carnot) applied to ideal gas particles. Can we also apply these cycles to "gases" of quantum particles such as photons and electrons?

For example, a photon gas has pressure $P(T) = \frac{U}{3V}$. Can I put this gas in a piston cylinder assembly and subject it to a Carnot cycle?

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The key feature of the Carnot cycle isn't so much it's particular shape on a PV diagram, but the fact that it alternately follows adiabats and isotherms through state space. In other words, it is a rectangle on a T S diagram. Cribbing directly from the Wikipedia page on for photon gas, we get the following relationship between entropy and temperature:

$$ S = (\text{something}) \cdot VT^3 $$

So, to change the entropy while holding the temperature constant you just change the volume. To change the temperature while holding entropy constant is a bit harder. You have to slowly vary the temperature of the container holding your photon gas, while at the same time adjusting the volume to ensure that $VT^3$ remains constant. By doing these two things, it is possible to go around in a rectangle in $T, S$ space forever.

This construction should produce the analogous version of the Carnot cycle for a photon gas. You can then apply additional photon gas equations to find $P$, $U$, etc.

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