On the entropy minimization problem in Statistical Mechanics

In the works on Statistical Mechanics and Statistical Physics, when deriving the distribution of particles of ideal gases, one uses the method of Lagrange multipliers in a formal way. In this paper we treat rigorously this problem for Bose--Einstein, Fermi--Dirac and Maxwell--Boltzmann entropies and present a complete study in the case of the Maxwell--Boltzmann entropy. Our approach is based on recent results on series of convex functions.