Bose-Einstein Condensation and Intermediate State of the Photon Gas

Possibility of establishment of equilibrium between the photon and the dense photon bunch is studied. In the case, when the density of plasma does not change, the condition of production of the Bose-Einstein condensate is obtained. It is shown that the inhomogeneity of density of photons leads to a new intermediate state of the photon gas. 52.25.Os, 52.27.Ny, 52.40.Db Typeset using REVTEX 1 Importance of scattering processes in interaction of photons with an electron gas was noted for the first time by Kompaneets [1]. Who has shown that the establishment of equilibrium between the photons and the electrons is possible through the Compton effect. The role of the Compton effect in setting equilibrium between the photons and the electrons was discussed by Kompaneets in the nonrelativistic approximation. In his consideration, since the free electron does not absorb and emit, but only scatters the photon, the total number of photons is conserved. Taking into account the fact that for the high frequencies of photons the probability of Compton scattering exceeds the absorption probability, Kompaneets has derived the kinetic equation for the photon distribution function. Using the kinetic equation of Kompaneets, Zel’dovich and Levich [2] have shown that in the absence of absorption the photons undergo Bose-Einstein condensation. Such a possibility for the Bose-Einstein condensation occurs in the case, when the processes of change of energy and momentum in scattering dominates over the processes involving change of photon number in their emission and absorption. Thus, in the above papers [1], [2] the authors demonstrated that the usual Compton scattering leads to the establishment of equilibrium state and Bose-Einstein condensation in the photon-plasma medium. In the present letter, we show that exists an another new mechanism of creation of equilibrium state and Bose-Einstein condensation in a nonideal dense photon gas. In our consideration we assume that the intensity of radiation (strong and super-strong laser pulse, non-equilibrium cosmic field radiation, etc.) is sufficiently large, so that the photon-photon interaction can become more likely than the photon-electron interaction. Under these conditions in the case of relativistic intensities of radiation of electromagnetic (EM) waves, we may consider the system of two weakly interacting subsystems: the photon gas and the plasma, which slowly exchange energy between each other. In other words, relaxation in the photon-plasma system is then a two-stage process: firstly the local equilibrium is established in each subsystem independently, corresponding to some average energies Eγ = KBTγ and Ep = KBTp, where KB is the Boltzmann constant, Tγ is the characteristic ”temperature” of the photon gas, and Tp is the plasma temperature. 2 As it was mentioned above, in the case of the strong radiation of EM waves there is an appreciable probability that the photon-photon interaction takes place, the phases of the waves will, in general, be random functions of time. We need therefore not be interested in the phases and can average over them. Under these conditions the perturbation state of the photon gas can be described in terms of the occupation number N(~k, ω,~r, t) of photons and study how these numbers change due to the processes of interaction of the photons with others or with plasma electrons. Here, N(~k, ω,~r, t) is the slowly varying function in space and time. A new version of kinetic equation for the occupation number N(~k, ω,~r, t) of photons, for modes propagating with wavevector ~k and frequency ω, at a position ~r and time t, was derived in Refs. [3], [4], [5] ∂ ∂t N(~k, ω,~r, t) + c ω (~k · ∇)N(~k, ω,~r, t)− ω psin 1 2 ( ∇~r · ∇~k − ∂ ∂t · ∂ ∂ω )