2005/02/28 by Constantino Tsallis, Murray Gell-Mann, Murray Gell‐Mann +1 · 3 citations
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Combinatorics #Complex Systems and Time Series Analysis #Conjecture #Discrete mathematics #Entropy (arrow of time) #Mathematics #Physics #Quantum mechanics #Scale invariance #Statistical Mechanics and Entropy #Statistics #cond-mat.stat-mech
paper · pdf · doi:10.1073/pnas.0503807102
published as Proc.Nat.Acad.Sci. 102 (2005) 15377-15382 · 15 pages, including 9 figures and 8 Tables. The new version is considerably enlarged with regard to the previous ones. New examples and new references have been included
arxiv created 2005/07/24 · openalex publication_date 2005/10/17 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
Phase space can be constructed for N equal and distinguishable subsystems that could be probabilistically either weakly correlated or strongly correlated. If they are locally correlated, we expect the Boltzmann-Gibbs entropy S(BG) identical with -k Sigma(i) p(i) ln p(i) to be extensive, i.e., S(BG)(N) proportional, variant N for N --> infinity. In particular, if they are independent, S(BG) is strictly additive, i.e., S(BG)(N) = NS(BG)(1), for allN. However, if the subsystems are globally correlated, we expect, for a vast class of systems, the entropy S(q) identical with k[1 - Sigma(i) p(q)(i)]/(q - 1) (with S(1) = S(BG)) for some special value of q not equal 1 to be the one which is extensive [i.e., S(q)(N) proportional, variant N for N --> infinity]. Another concept which is relevant is strict or asymptotic scale-freedom (or scale-invariance), defined as the situation for which all marginal probabilities of the N-system coincide or asymptotically approach (for N --> infinity) the joint probabilities of the (N - 1)-system. If each subsystem is a binary one, scale-freedom is guaranteed by what we hereafter refer to as the Leibnitz rule, i.e., the sum of two successive joint probabilities of the N-system coincides or asymptotically approaches the corresponding joint probability of the (N - 1)-system. The kinds of interplay of these various concepts are illustrated in several examples. One of them justifies the title of this paper. We conjecture that these mechanisms are deeply related to the very frequent emergence, in natural and artificial complex systems, of scale-free structures and to their connections with nonextensive statistical mechanics. Summarizing, we have shown that, for asymptotically scale-invariant systems, it is S(q) with q not equal 1, and not S(BG), the entropy which matches standard, clausius-like, prescriptions of classical thermodynamics.