What do generalized entropies look like? An axiomatic approach for complex, non-ergodic systems

摘要

Shannon and Khinchin showed that assuming four information theoretic axiomsthe entropy must be of Boltzmann-Gibbs type, $S=-\sum_i p_i \log p_i$. Here wenote that in physical systems one of these axioms may be violated. Fornon-ergodic systems the so called separation axiom (Shannon-Khinchin axiom 4)will in general not be valid. We show that when this axiom is violated theentropy takes a more general form, $S_{c,d}\propto \sum_i ^W \Gamma(d+1, 1- c\log p_i)$, where $c$ and $d$ are scaling exponents and $\Gamma(a,b)$ is theincomplete gamma function. The exponents $(c,d)$ define equivalence classes forall interacting and non interacting systems and unambiguously characterize anystatistical system in its thermodynamic limit. The proof is possible because oftwo newly discovered scaling laws which any entropic form has to fulfill, ifthe first three Shannon-Khinchin axioms hold. $(c,d)$ can be used to defineequivalence classes of statistical systems. A series of known entropies can beclassified in terms of these equivalence classes. We show that thecorresponding distribution functions are special forms of Lambert-${\cal W}$exponentials containing -- as special cases -- Boltzmann, stretched exponentialand Tsallis distributions (power-laws). In the derivation we assume trace formentropies, $S=\sum_i g(p_i)$, with $g$ some function, however more generalentropic forms can be classified along the same scaling analysis.