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H-theorem and Thermodynamics for generalized entropies that depend only on the probability

We consider a previously proposed non-extensive statistical mechanics in which the entropy depends only on the probability, this was obtained from a f(β) distribution and its corresponding Boltzmann factor. We show that the first term correcting the usual entropy also arises from several f(β) distributions, we also construct the corresponding H-function and demonstrate that a generalized H-theorem is fulfilled. Furthermore, expressing this H-function as function of the simplest Maxwellian state we find, up to a first approximation some modified thermodynamic quantities for an ideal gas. In order to gain some insight about the behavior of the proposed generalized statistics, we present some simulation results for the case of a square-well and Lennard-Jones potentials, showing that an effective repulsive interaction is obtained with the new formalism.