Relativistic transformation of the canonical distribution function in relativistic statistical mechanics

We deduce a relativistic transformation of the canonical distribution function from basic principles; that is, from relativistic canonical transformations of dynamical variables. The thermodynamics which is obtained from such a distribution coincides with the recent proposal put forward by Ares de Parga et al. [G. Ares de Parga, B. López-Carrera, F. Angulo-Brown, J. Phys. A 38 (2005) 2821] of a relativistic renormalized thermodynamics. The invariances of the canonical distribution and the partition function are obtained. By noticing a mistake committed by Balescu while dealing with the partition function of an ideal gas, the consistency of the theory is showed. The Maxwell distribution function of an ideal gas moving with a constant velocity with respect to a reference frame is calculated. A natural asymmetry appears as a difference with the distribution function at the same temperature but at rest with respect to the same reference frame. The quantum case and the covariance of the theory are analyzed.