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

Abstract

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.

A systematic study on immiscible binary systems undergoing thermal/photo reversible chemical reactions

“…s o = h o/T b . 36,39 This theoretical result immediately leads to the conclusion that the negative entropy input into the system can be easily computed as…”

“…s ω = ħω / T b . 36,39 This theoretical result immediately leads to the conclusion that the negative entropy input into the system can be easily computed as with Δ E the total energy of the photons that are emitted from the high-temperature blackbody and have been absorbed by the system. Then the total entropy production (including those of the environment and the system) due to the photon emission and absorption can be evaluated as (detailed derivation is referred to Section VI in ESI†)By noting that the above integration also has the mathematical forms like ( X − Y )( e X − e Y ) with X = ln ϕ M + Δ E MA/B / k B T x and Y = ln ϕ A/B , it is easy to prove the above entropy production is non-negative.…”

Testing the Minimum System Entropy and the Quantum of Entropy

“…The situation changes in beams of light, where all photons have the same frequency and similar directions [57]. Scully [58] estimated the entropy change of a laser with one additional photon at the threshold and found that the change can easily be as low as 10 −6 k. However, as pointed out by Li et al [59], the definition of single-photon entropy is involved and not unique. At equilibrium, the entropy s of a single monochromatic photon with frequency ω can be argued to be s = hω/T.…”