Conceptual Inadequacy of the Shore and Johnson Axioms for Wide Classes of Complex Systems

Abstract

It is by now well known that the Boltzmann-Gibbs-von Neumann-Shannon logarithmic entropic functional (\(S_{BG}\)) is inadequate for wide classes of strongly correlated systems: see for instance the 2001 Brukner and Zeilinger's {\it Conceptual inadequacy of the Shannon information in quantum measurements}, among many other systems exhibiting various forms of complexity. On the other hand, the Shannon and Khinchin axioms uniquely mandate the BG form \(S_{BG}=-k\sum_i p_i \ln p_i\); the Shore and Johnson axioms follow the same path. Many natural, artificial and social systems have been satisfactorily approached with nonadditive entropies such as the \(S_q=k \frac{1-\sum_i p_i^q}{q-1}\) one (\(q \in {\cal R}; \,S_1=S_{BG}\)), basis of nonextensive statistical mechanics. Consistently, the Shannon 1948 and Khinchine 1953 uniqueness theorems have already been generalized in the literature, by Santos 1997 and Abe 2000 respectively, in order to uniquely mandate \(S_q\). We argue here that the same remains to be done with the Shore and Johnson 1980 axioms. We arrive to this conclusion by analyzing specific classes of strongly correlated complex systems that await such generalization.

Keywords

• nonadditive entropies
• nonextensive statistical mechanics
• strongly correlated random variables
• Shore and Johnson axioms