Extensive and nonextensive statistical mechanics: Exp and log distribution functions

Abstract

The dominance of Boltzmann–Gibbs distribution (BG) in statistical physics appears to endow an impression that this would be the only statistical approach available. In fact, a large class of statistical approaches already exists. This is generalizing BG statistics referring to various types of nonextensivity, nonadditivity, nonequilibrium, nonlinearity, etc. For instance, [Formula: see text] and [Formula: see text] functions play crucial roles in both extensive and nonextensive domains of statistical mechanics. Emerging in a physical system, BG statistics defines extensive entropy which relates the number of microstates to thermodynamic quantities or macroscopic states. In this regard, the Boltzmann distribution, extensive statistics, refers to a well-defined probability distribution, while the various types of nonextensive statistics categorically violate the fourth Shannon–Khinchen additivity axiom. The [Formula: see text] and [Formula: see text] distribution functions in BG, Tsallis, and generic statistics are systematically compared. We focus on the mathematical properties of both distribution functions and conclude that their compatibility exclusively depends on the nonextensive parameters, i.e. the mathematical properties of both distribution functions depend on the nonextensive parameters either that of Tsallis- or that of the generic-type of nonextensivity. We also conclude that the statistical nature of the underlying ensemble should be taken into consideration when applying the statistical approach.