On the derivation of the Kompaneets equation

Dipoles in blackbody radiation: momentum fluctuations, decoherence, and drag force

“…We note that the above drag force is related to the momentum diffusion constant of the sphere (equation ( 33)) via the fluctuation-dissipation relation (equation ( 88)); the momentum diffusion constant in turn is related to the rate of decoherence of the particle via equation (34). Such a connection between decoherence rate, momentum diffusion and thermal drag is pertinent to deducing decoherence rate of dielectric nanospheres in the quantum regime, as was done in [26].…”

“…has exactly the form of the more general collisional decoherence rate [13][14][15][16]. In the long-wavelength limit the decay rate of the reduced density matrix for a dielectric sphere reduces to −Λ(x − x ) 2 , where Λ is defined in equation (34). The 'decoherence factor' F x − x , as it appears in the theory of collisional decoherence [13][14][15][16], represents the overlap between the states of environmental scatterers that scatter off the positions x and x of the particle's center of mass, denoting their distinguishability.…”

“…Suppose equation ( 103) describes the evolution of the photon distribution when the field with some initial distribution n(ω, 0) is brought into interaction with a rarefied electron gas which is assumed to be in thermal equilibrium at the temperature T and to remain so. Since the electrons do not absorb radiation, our assumption of no absorption leading to equation ( 103) is valid and, in the approximation equation (86), n(ω, t) satisfies the Kompaneets equation [29][30][31][32][33][34]:…”

Resetting photons

“…The mc 2 /k B T multiplier to the timescale expresses that electrons are slow-moving and so the average collision barely shifts a photon's frequency. That is what allowed Kompaneets to make a diffusion approximation to the Boltzmann-equation; see [9] for more details. The ω 2 -dependence (hidden in the ω 4 ) is due to the fact that the frequency shift is proportional to the frequency itself, implying that the variance on the shift goes like ω 2 .…”

“…Its origin is that the photon carries momentum proportional to its frequency. A more statistical-mechanical perspective and microscopic derivation of ( 1) is presented in [9]. An important point is made by noting that the Kompaneets equation ( 1) implements only the Compton interaction to write the dynamics for n(τ , ω).…”