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Analysis of the compatibility between the Maxwell-Boltzmann distribution and the Rayleigh-Jeans spectrum for classical systems.

We analyze in detail the validity of the compatibility between the Maxwell-Boltzmann (MB) distribution and the Rayleigh-Jeans (RJ) spectrum obtained in recent works by the authors for classical relativistic systems. We show that the MB distribution and the RJ spectrum are not compatible if we do not remove high enough frequencies. By analyzing the applicability of the approximation methods used in previous works to obtain the MB distribution from the RJ spectrum, we conclude that these methods are valid only if we introduce a high-frequency cutoff in the RJ spectrum. A short discussion is made on the meaning of this cutoff.