Kompaneyets equation refers to a non-relativistic, Fokker–Planck type, kinetic equation for photon number density with which photons interact with an electron gas via Compton scattering, first derived by Alexander Kompaneyets in 1949 and published in 1957 after declassification.[1][2] The Kompaneyets equation describes how an initial photon distribution relaxes to the equilibrium Bose–Einstein distribution. Kompaneyets pointed out the radiation field on its own cannot reach the equilibrium distribution since the Maxwells equation are linear but it needs to exchange energy with the electron gas. The Kompaneyets equation has been used as a basis for analysis of the Sunyaev–Zeldovich effect.[3]

Mathematical description

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Consider a non-relativistic electron bath that is at an equilibirum temperature  , i.e.,  , where   is the electron mass. Let there be a low-frequency radiation field that satisfies the soft-photon approximation, i.e.,   where   is the photon frequency. Then, the enery exchange in any collision between photon and electron will be small. Assuming homogeneity and isotropy and expanding the collision integral of the Boltzmann equation in terms of small energy exchange, one obtains the Kompaneyets equation.[4]

The Kompaneyets equation for the photon number density   reads[5][6]

 

where   is the total Thomson cross-section and   is the electron number density;   is the Compton range or the scattering mean free path. As evident, the equation can be written in the form of the continuity equation

 

If we introudce the rescalings

 

the equation can be brought to the form

 

The Kompaneyets equation conserves the photon number

 

where   is a sufficiently large volume, since the energy exchange between photon and electron is small. Furthermore, the equilibrium distribution of the Kompaneyets equation is the Bose–Einstein distribution for the photon gas,

 

References

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  1. ^ Kompaneets, A. (1957). The establishment of thermal equilibrium between quanta and electrons. Soviet Journal of Experimental and Theoretical Physics, 4(5), 730-737.
  2. ^ Oliveira, G. E. F., Maes, C., & Meerts, K. (2021). On the derivation of the Kompaneets equation. Astroparticle Physics, 133, 102644.
  3. ^ Sunyaev, R. A., & Zeldovich, Y. B. (1970). Small-scale fluctuations of relic radiation. Astrophysics and Space Science, 7, 3-19.
  4. ^ Bernstein, J., & Dodelson, S. (1990). Aspects of the Zel'dovich-Sunyaev mechanism. Physical Review D, 41(2), 354.
  5. ^ Zel'dovich, Y. B., & Syunyaev, R. A. (1972). Shock wave structure in the radiation spectrum during Bose condensation of photons. Soviet Journal of Experimental and Theoretical Physics, Vol. 35, p. 81, 35, 81.
  6. ^ Milonni, P. W. (2021). Simplified derivation of the Kompaneets equation. Physics of Plasmas, 28(9).