Uniqueness and characterization theorems for generalized entropies

Abstract

The requirement that an entropy function be composable is key: it means that the entropy of a compound system can be calculated in terms of the entropy of its independent components. We prove that, under mild regularity assumptions, the only composable generalized entropy in trace form is the Tsallis one-parameter family (which contains Boltzmann-Gibbs as a particular case).This result leads to the use of generalized entropies that are not of trace form, such as Rényi's entropy, in the study of complex systems. In this direction, we also present a characterization theorem for a large class of composable non-trace-form entropy functions with features akin to those of Rényi's entropy.

A new class of entropic information measures, formal group theory and information geometry

“…Under suitable hypotheses, the problem of composability of S h,f was also consider in Theorem 2 of [6], obtaining f (t) = at + bt q . In the previous Theorem 6 these additional hypotheses are not assumed.…”

“…In Theorem 1 of[6] the following result has been proved: suppose to have a trace-form entropy S = i f (p i ) where f is a function of class C 2 ((0, 1)) ∩ C 1 ([0, 1]), with f ′ not vanishing in any open interval, and assume that the condition (5) holds with χ of class C 1 , then S coincides with the (trace-form version of) Tsallis entropy S q…”

Group entropies: from phase space geometry to entropy functionals via group theory

“…The mathematical apparatus will be kept to a minimum. For an extensive discussion, the reader is referred to the original papers [14], [15], [16], [17]. We start out with the composition requirement in Eq.…”

“…Both families of entropies are assumed to satisfy the first three Shannon-Khinchin axioms for suitable choices of G(t). The main difference between the trace form and the non-trace form class is encoded in a theorem proved in [17], where it was shown that the most general trace-form entropy satisfying Eq. (2) is Tsallis entropy, with Botzmann's entropy as an important special case.…”

“…The derivatives of S[p] for the three trace forms Eqns. (16), (17) and (18) are Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 13 September 2018 doi:10.20944/preprints201809.0247.v1…”