Nonextensive Statistical Mechanics

8.2 Frequently Asked Questions 309is a common and convenient mechanism for new ideas to be checked and betterunderstood by the scientific community. Clearly, objections and critiques have frequentlyhelped the progress of science. There is absolutely no reason to expectthat statistical mechanics, and more specifically nonextensive statistical mechanics,would be out of such a process. Quite on the contrary – remember the words ofNicolis and Daems [2] that were cited in the Preface! – given the undeniable fact thatentropy is one among the most subtle and rich concepts in physics. Some frequentlyasked points are addressed here. Indeed, we believe that some space dedicated hereto such issues might well be useful at this stage (see also [803]).This section only includes frequently asked questions, or critiques, the (basic)answer of which is believed to be known. Questions, frequent or not, whose answeris still a matter of research have been considered instead as “open questions,” andas such have been included in Section 8.3.(a) Finally, the entropy S q is extensive or nonextensive?This question is incompletely posed. What can be simply answered is whetherS q is additive or not: S 1 is additive, and S q for q ≠ 1isnonadditive. Extensivityis a more complex question. Indeed, the answer depends not only on the entropicfunctional but also on the system (more precisely, on the nature of the correlationsbetween the elements of the system). If the elements have no correlation at all, oronly local correlations, then typically S 1 is extensive and S q for q ≠ 1isnot. Butifthe correlations are nonlocal, then it can happen (e.g., the quantum magnetic chainanalytically discussed in [201]) that S q is nonextensive for all values of q (includingq = 1), excepting a special value of q ≠ 1 for which S q is extensive.(b) If the entropic index q is chosen such that the entropy S q is extensive, why thistheory is named “nonextensive statistical mechanics?”This kind of mismatch has its historical roots on the fact that, during over onecentury of BG statistical mechanics, the entropy S BG , known to be additive, wasalso extensive for all those systems (known today as extensive systems) forwhichthe BG theory is plainly valid. This led imperceptibly to the abusive use of thewords additive and extensive as practically synonyms. Later on, starting with the1988 paper [39], the distinctive nonadditivity property (Eq. (3.21)) was wrongly,but frequently, referred to as the nonextensivity property. The expression nonextensivestatistical mechanics was coined from there. When, many years later (see,for instance, the end of the Introduction of chapter I in [69]), this matter becamegradually clear, the idea of course emerged to rather call this theory nonadditivestatistical mechanics. But, on the other hand, the expression nonextensive statisticalmechanics was already used in over one thousand papers. Furthermore, statisticalmechanics has to do not only with entropy but also with energy. And the typicalsystems for which the present theory was devised are those involving long-rangetwo-body interactions, for which the total energy is definitively nonextensive. Theexpression nonextensive statistical mechanics was therefore maintained. Nowadays,many authors call nonextensive systems those whose nonequilibrium stationary-statedistribution (or similar properties, such as relaxation functions, and sensitivity to the