Asymptotically scale-invariant occupancy of phase space makes the entropy S_q extensive

Abstract

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 ≡ -k Σ_i p_i ln p_i to be extensive, i.e., S_BG(N) ∝ N for N → ∞. In particular, if they are independent, S_BG is strictly additive, i.e., S_BG(N) = NS_BG(1), ∀N. However, if the subsystems are globally correlated, we expect, for a vast class of systems, the entropy S_q ≡ k[1 - Σ_i p^q_i]/(q - 1) (with S₁ = S_BG) for some special value of q ≠ 1 to be the one which is extensive [i.e., S_q(N) ∝ N for N → ∞]. 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 → ∞) 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 ≠ 1, and not S_BG, the entropy which matches standard, clausius-like, prescriptions of classical thermodynamics.