Nonextensive Statistical Mechanics

104 3 Generalizing What We Learntthe stationary-state statistics that we are focusing on asymptotically exhibit confluenceonto a single behavior, namely that corresponding to the BG microcanonicalensemble, which corresponds to even occupancy of the admissible phase-space.But even occupancy is associated to a Lebesgue measure which essentially factorizesinto the Lebesgue measures corresponding to the various degrees of freedom.In other words, it corresponds to independence. The connection ends by recallingthat the appropriate entropy for probabilistically independent subsystemsprecisely is S BG , i.e., q = 1. So, from the entropic viewpoint, the k −1Bplane representedin Fig. 3.22 equivalently corresponds to q = 1. Out of this plane, in somesubtle sense, we start having information corresponding to nontrivially correlatedsubsystems.In this context, it is interesting to focus again on Eq. (3.43). It is precisely theexistence of the extra term that enables [262], for special correlations, to recoverthe Clausius entropy thermodynamical extensivity S q (A + B) ∼ S q (A) + S q (B)forlarge systems A and B.Let us close this digression about the physical universal constants by focusingon the fact that all known constants used in contemporary physics can be expressedin terms of units of length, time, mass, and temperature. Equivalently, each of themcan be expressed as a pure number multiplied by some combination of powers ofc −1 , h, G, and k −1B. No further reduction below four universal constants is possiblein contemporary physics. This point is however quite subtle, as can be seen in[308–311]. It is related to the fact that any fundamental discovery tends to reduce thenumber of units that are necessary to express the physical quantities. For example,in ancient times, there were independent units for area and length. The situationchanged when it became clear that, in Euclidean geometry, an area can be expressedas the square of a length.Consistently with the above, Planck introduced [312, 831] four natural units forlength, mass, time, and temperature, namely√hGunit of length == 4.13 × 10 −33 cm (3.242)c√ 3 hcunit of mass == 5.56 × 10 −5 g (3.243)G√hGunit of time == 1.38 × 10 −43 s (3.244)c 5unit of temperature = 1√hc5k B G = 3.50 × 1032 o K. (3.245)There is no need to add to this list the elementary electric charge e. Indeed,it is related to the already-mentioned constants through the fine-structure constantα ≡ 2πe 2 /hc = 1/137.035999 ...