Nonextensive Statistical Mechanics

3.8 About Universal Constants in Physics 103If we assume k = k B in Eq. (3.21), and cancel it on both sides, we obtainS q (A + B) = S q (A) + S q (B) + 1 − qk BS q (A)S q (B) . (3.241)As we see, we go back to the BG situation if (1 − q)/k B = 0. This can occur intwo different manners, namely either q = 1(∀k −1B )ork−1 B= 0(∀q). In this sense,any departure from the BG entropic composition law is equivalent to a departurefrom k −1B ≠ 0.In thermal equilibrium (as well as in other stationary states) k B always appearscoupled together with the temperature T in the form k B T . In other words, smallvalues for k −1Bis equivalent to the high temperature limit. It seems reasonable tothink that this connection is not unrelated to the fact that, for small (k B T ) −1 ,theBG canonical and grand-canonical ensembles asymptotically recover the microcanonicalensemble. The same happens for Bose–Einstein and Fermi–Dirac quantumstatistics, in fact for all statistics [101] which unifies the standard quantum ones.Even more, the same happens for all q-statistics if we take into account the propertyeq x ∼ 1+ x,forx → 0 and all values of q. In other words, for (q −1)/k BT → 0, allFig. 3.22 Physical structure at the 1/k B = 0 plane. The full diagram involves 4 universal constants,and would be a tetrahedron. At the center of the tetrahedron we have the case c −1 = h = G =k −1B= 0, and the overall tetrahedron corresponds to 1/c > 0, h > 0, G > 0, 1/k B > 0 (statisticalmechanics of quantum gravity).