Generalized entropies and logarithms and their duality relations
Abstract
For statistical systems that violate one of the four ShannonKhinchin axioms, entropy takes a more general form than the BoltzmannGibbs entropy. The framework of superstatistics allows one to formulate a maximum entropy principle with these generalized entropies, making them useful for understanding distribution functions of nonMarkovian or nonergodic complex systems. For such systems where the composability axiom is violated there exist only two ways to implement the maximum entropy principle, one using escort probabilities, the other not. The two ways are connected through a duality. Here we show that this duality fixes a unique escort probability, which allows us to derive a complete theory of the generalized logarithms that naturally arise from the violation of this axiom. We then show how the functional forms of these generalized logarithms are related to the asymptotic scaling behavior of the entropy.
 Publication:

Proceedings of the National Academy of Science
 Pub Date:
 November 2012
 DOI:
 10.1073/pnas.1216885109
 arXiv:
 arXiv:1211.2257
 Bibcode:
 2012PNAS..10919151H
 Keywords:

 Physics  Classical Physics;
 Condensed Matter  Statistical Mechanics;
 Mathematical Physics
 EPrint:
 4 pages, 1 page supporting information