On exp and log distributions in extensive and nonextensive statistical mechanics

Abstract In statistical physics, the dominance of Boltzmann--Gibbs distribution seems to endow an impression that this would be the only statistical approach. Indeed, there is a large class of statistical approaches referring to various types of nonextensivity. For instance, $\log$ and $\exp$ distributions play a crucial role in extensive and nonextensive statistical mechanics. Emerging in a physical system, Maxwell--Boltzmann statistics (MB) defines extensive entropy which relates the number of microstates to thermodynamic quantities. In this regard, the Boltzmann distribution is nothing but a probability distribution. The various types of nonextensive statistics categorically violate the fourth Shannon--Khinchen additivity. We compare between $\log$ and $\exp$ distributions in MB, Tsallis, and generic statistics and studied their mathematical properties. We conclude that their compatibility exclusively depends on the nonextensivity parameters. PACS numbers: 05.70.Ln, 05.65.+b