Two- parameter deformations of logarithm, exponential, and entropy: a consistent framework for generalized statistical mechanics

A consistent generalization of statistical mechanics is obtained by applying the maximum entropy principleto a trace-form entropy and by requiring that physically motivated mathematical properties are preserved. Theemerging differential-functional equation yields a two-parameter class of generalized logarithms, from whichentropies and power-law distributions follow: these distributions could be relevant in many anomalous systems.Within the specified range of parameters, these entropies possess positivity, continuity, symmetry, expansibility,decisivity, maximality, concavity, and are Lesche stable. The Boltzmann-Shannon entropy andsome one-parameter generalized entropies already known belong to this class. These entropies and theirdistribution functions are compared, and the corresponding deformed algebras are discussed.