Generalized entropy optimized by a given arbitrary distribution

An ultimate generalization of the maximum entropy principle is presented. An entropic measure, which is optimized by a given arbitrary distribution with the finite linear expectation value of a physical random quantity of interest, is constructed. It is concave irrespective of the properties of the distribution and satisfies the H-theorern for the master equation combined with the principle of microscopic reversibility. This offers a unified basis for a great variety of distributions observed in nature. As examples, the entropies associated with the stretched exponential distribution postulated by Anteneodo and Plastino (1999 J. Phys. A: Math. Gen. 32 1089) and the kappa-deformed exponential distribution by Kaniadaki (2002 Phys. Rev. E 66 056125) and Naudts (2002 Physica A 316 323) are derived. To include distributions with divergent moments (e.g., the Levy stable distributions), it is necessary to modify the definition of the expectation value.