Nonextensive Statistical Mechanics

102 3 Generalizing What We Learntwhere p(x) can be a distribution of probabilities. We immediately verify thatandF q [p](0) = 1 , (3.239)[ ∫ dFq [p](ξ)∞dξ]ξ=0 = i dx x [p(x)] q . (3.240)As we see, it is the numerator of the q-mean value, and not that of the standardmean value, which emerges naturally. As we shall see in due time, Eqs. (3.239) and(3.240) are the two first elements of an infinite set of finite values which, withinsome restrictions, appear to uniquely determine the distribution p(x) itself.(viii) Last but not least, let us rephrase property (iii) in very elementary terms.We assume that we have the simple case of a stationary-state q-exponential distributionp(x) ∝ eq−βx (x ≥ 0). The characterization of a distribution such as thisone involves two important numbers, namely the decay exponent 1/(q − 1) of thetail, and the overall width 1/β of the distribution. It must be so for any value ofq < 2 (upper bound for the existence of a norm). We easily verify that the standardmean value of x diverges in the region 3/2 ≤ q < 2, and it is therefore useless forcharacterizing the width of the distribution. The q-mean value instead is finite anduniquely determined by the width 1/β up to q < 2. In other words, the robust informationabout the width of the distribution is provided precisely through the escortdistribution.−∞3.8 About Universal Constants in PhysicsI would mention at this place a point which epistemologically remains kind ofmysterious. We shall exhibit and further comment that, for any value of the entropicindex q ≠ 1 and all systems, the stationary-state energy distribution withinnonextensive statistical mechanics becomes that of BG statistical mechanics in thelimit of vanishing inverse Boltzmann constant 1/k B . The physical interpretation ofthis property is, in my opinion, quite intriguing and unavoidably reminds the factssuch as quantum mechanics becoming Newtonian mechanics in the limit of vanishing, special relativity becoming once again Newtonian mechanics in the limitof vanishing 1/c, and general relativity recovering the Newtonian flat space-timein the limit of vanishing G. While we may say that, for these three mechanicalexamples, the corresponding physical interpretations are kind of reasonably wellunderstood (see Fig. 3.22), it escapes to equally clear perception what kind of subtleinformational meaning could be attributed to 1/k B going to zero while q is keptfixed at an arbitrary value. The meaning of the four universal constants , c, G, k Bhas been addressed by G. Cohen–Tannoudji in terms of physical horizon [801] (seealso [802]).