First-Principle Derivation of Single-Photon Entropy and Maxwell-Jüttner Velocity Distribution

This work is devoted to deriving the entropy of a single photon in a beam of light from first principles. Based on the quantum processes of light-matter interaction, we find that, if the light is not in equilibrium, there are two different ways, depending on whether the photon is being added or being removed from the light, of defining the single-photon entropy of this light. However, when the light is in equilibrium at temperature T, the two definitions are equivalent and the photon entropy of this light is hν/T. From first principles, we also re-derive the Jüttner velocity distribution showing that, even without interatomic collisions, two-level atoms will relax to the state satisfying the Maxwell-Jüttner velocity distribution when they are moving in blackbody radiation fields.