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

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