Is the Entropy S _q Extensive or Nonextensive?

Abstract

The cornerstones of Boltzmann-Gibbs and nonextensive statistical mechanics respectively are the entropies S _BG ≡ −k ∑ _{i = 1} ^W p _i ln p _i and S _q ≡ k (1−∑ _{i = 1} ^W p _i ^q)/(q−1) (q∊ℜ S ₁ = S _BG ). Through them we revisit the concept of additivity, and illustrate the (not always clearly perceived) fact that (thermodynamical) extensivity has a well defined sense only if we specify the composition law that is being assumed for the subsystems (say A and B). If the composition law is not explicitly indicated, it is tacitly assumed that A and B are statistically independent. In this case, it immediately follows that S _BG (A+B) = S _BG (A)+S _BG (B), hence extensive, whereas S _q (A+B)/k = [S _q (A)/k]+[S _q (B)/k]+(1−q)[S _q (A)/k][S _q (B)/k], hence nonextensive for q ≠ 1. In the present paper we illustrate the remarkable changes that occur when A and B are specially correlated. Indeed, we show that, in such case, S _q (A+B) = S _q (A)+S _q (B) for the appropriate value of q (hence extensive), whereas S _BG (A+B) ≠ S _BG (A)+S _BG (B) (hence nonextensive). We believe that these facts substantially improve the understanding of the mathematical need and physical origin of nonextensive statistical mechanics, and its interpretation in terms of effective occupation of the W a priori available microstates of the full phase space. In particular, we can appreciate the origin of the following important fact. In order to have entropic extensivity (i.e., lim _N→∞ S(N)/N < ∞, where N≡ numberof elements of the system), we must use (i) S _BG , if the number W ^eff of effectively occupied microstates increases with N like W ^{{eff}}∼ W ∼ μ^N (μ ≥ 1); (ii) S _q with q = 1−1/ρ, if W ^{{eff}}∼ N^ρ < W (ρ ≥ 0). We had previously conjectured the existence of these two markedly different classes. The contribution of the present paper is to illustrate, for the first time as far as we can tell, the derivation of these facts directly from the set of probabilities of the W microstates.