Condensation of an ideal gas with intermediate statistics on the horizon

We consider a boson gas on the stretched horizon of the Schwartzschild and Kerr black holes. It is shown that the gas is in a Bose-Einstein condensed state with the Hawking temperature $T_c=T_H$ if the particle number of the system be equal to the number of quantum bits of space-time $N \simeq {A}/{ł_{p}^{2}}$. Entropy of the gas is proportional to the area of the horizon $(A)$ by construction. For a more realistic model of quantum degrees of freedom on the horizon, we should presumably consider interacting bosons (gravitons). An ideal gas with intermediate statistics could be considered as an effective theory for interacting bosons. This analysis shows that we may obtain a correct entropy just by a suitable choice of parameter in the intermediate statistics.

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