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# Photo-Carnot engine

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 Title: Photo-Carnot engine Author: World Heritage Encyclopedia Language: English Subject: Collection: Publisher: World Heritage Encyclopedia Publication Date:

### Photo-Carnot engine

A photo-Carnot engine is a Carnot cycle engine in which the working medium is a photon inside a cavity with perfectly reflecting walls. Radiation is the working fluid, and the piston is driven by radiation pressure.

A quantum Carnot engine is one in which the atoms in the heat bath are given a small bit of quantum coherence. The phase of the atomic coherence provides a new control parameter.

The deep physics behind the second law of thermodynamics is not violated; nevertheless, the quantum Carnot engine has certain features that are not possible in a classical engine.

## Derivation

The internal energy of the photo-Carnot engine is proportional to the volume (unlike the ideal-gas equivalent) as well as the 4th power of the temperature (see Stefan–Boltzmann law):

U = \varepsilon\sigma T^{4} \,.

The radiation pressure is only proportional to this 4th power of temperature but no other variables, meaning that for this photo-Carnot engine an isotherm is equivalent to an isobar:

P = \frac{U}{3 V} = \frac{\varepsilon \sigma T^{4}}{3 V} \,.

Using the first law of thermodynamics ( dU = dW + dQ ) we can determine the work done through an adiabatic ( dQ = 0 ) expansion by using the chain rule ( dU = \varepsilon \sigma dV T^{4} + 4 \varepsilon \sigma V T^{3} dT ) and setting it equal to dW = -P dV = -\frac{1}{3} \varepsilon \sigma T^{4} dV \,.

Combining these gives us \frac{2}{3} T^{4} dV = 4 V T^{3} dT which we can solve to find \frac{V^{1/6}}{T} = const \,.

....

The efficiency of this reversible engine must be the Carnot efficiency, regardless of the mechanism and so \eta = \frac{T_H - T_C}{T_H} \,.