Nonextensive Statistical Mechanics

8.2 Frequently Asked Questions 317properties for all q > 0. Such features point S q as being very special, althoughprobably not unique, for thermostatistical purposes.(i) By adjusting the constraints under which the entropy optimization is done, onecan obtain virtually any desired distribution. Is that not a serious problem?Soon after their second critique, Zanette and Montemurro advanced a third one[827]. This time the objection addresses nonthermodynamical systems, in contrast tothe previous critiques which mainly addressed thermodynamical ones. It is arguedby these authors that nonthermodynamical applications of nonextensive statisticsare ill-defined, essentially because of the fact that any probability distribution canbe obtained from the nonadditive entropy S q by conveniently adjusting the constraintused in the optimization. We argue here that, since it is well known to beso for any entropic form and, in particular, for the (additive) Boltzmann–Gibbsentropy S BG (see [828]), the critique brings absolutely no novelty to the area. Inother words, it has nothing special to do with the entropy S q . In defense of the usualsimple constraints, typically averages of the random variable x i or of xi2 (where x iis to be identified according to the nature of the system), we argue, and this forall entropic forms, that they can hardly be considered as arbitrary, as Zanette andMontemurro seem to consider. Indeed, once the natural variables of the system havebeen identified (e.g., constants of motion of the system), the variable itself and, insome occasions, its square obviously are the most basic quantities to be constrained.Such constraints are used in hundreds (perhaps thousands) of useful applicationsoutside (and also inside) thermodynamical systems, along the information theorylines of Jaynes and Shannon, and more recently of A. Plastino and others. Andthis is so for S BG , S q , and any other entropic form. If, however, other quantitiesare constrained (e.g., an average of x σ or of |x| σ ) for specific applications, it isclear that, at the present state-of-the-art of information theory, and for all entropicforms, this must be discussed case by case. Rebuttals of this critique can be foundin [803, 829].As a final comment let us mention that statistical mechanics is much more thatjust a stationary-state (e.g., thermal equilibrium) distribution. Indeed, under exactlythe same constraints, the optimization of S BG and (S BG ) 3 yield precisely thesame distribution. This is obviously not a sufficient reason for using (S BG ) 3 ,insteadof S BG , in a thermostatistical theory which must also satisfy thermodynamicalrequirements.(j) What properties are common to S BG and S q ?The additive entropy S BG and the nonadditive entropy S q share a huge amountof mathematical properties. These include nonnegativity, expansibility (∀q > 0),optimality for equal probabilities, concavity (∀q > 0), extensivity, Lesche-stability(or experimental robustness) (∀q > 0), finiteness of the entropy production per unittime, existence of partition function depending only on temperature, composability,the Topsoe factorizability property [830] (∀q > 0), the mathematical relationship ofthe Helmholtz free energy with the partition function is the same as the microscopicenergies with their probabilities, the function (namely ln q x) which (through a stan-