Extensivity and entropy production

dent, then SBG is extensive whereas Sq is, for q ≠ 1, nonextensive. This fact led to its current denomination as “nonextensive entropy”. However, if what we compose are subsystems that generate a nontrivial (strictly or asymptotically) scale-invariant system (in other words, with important global correlations), then it is generically Sq for a particular value of q ≠ 1, and not SBG, which is extensive. Asking whether the entropy of a system is or is not extensive without indicating the composition law of its elements, is like asking whether some body is or is not in movement without indicating the referential with regard to which we are observing the velocity. The overall picture which emerges is that Clausius thermodynamical entropy is a concept which can accomodate with more than one connection with the set of probabilities of the microscopic states. SBG is of course one such possibility, Sq is another one, and it seems plausible that there might be others. The specific one to be used appears to be univocally determined by the microscopic dynamics of the system. This point is quite important in practice. If the microscopic dynamics of the system is known, we can in principle determine the corresponding value of q from first principles. As it happens, this precise dynamics is most frequently unknown for many natural systems. In this case, a way out that is currently used is to check the functional forms of various properties associated with the system and then determine the appropriate values of q by fitting. This has been occasionally a point of – understandable but nevertheless mistaken – criticism against nonextensive theory, but it is in fact common practice in the analysis of many physical systems. Consider for instance the determination of the eccentricities of the orbits of the planets. If we knew all the initial conditions of all the masses of the planetary system and had access to a colossal computer, we could in principle, by using Newtonian mechanics, determine a priori the eccentricities of the orbits. Since we lack that (gigantic) knowledge and tool, astronomers determine those eccentricities through fitting. More explicitly, astronomers adopt the mathematical form of a Keplerian ellipse as a first approximation, and then determine the radius and eccentricity of the orbit through their observations. Analogously, there are many complex systems for which one may reasonably argue that they belong to the class that is addressed by nonextensive statistical concepts, but whose microscopic (sometimes even mesoscopic) dynamics is inaccessible. For such systems, it appears as a sensible attitude to adopt the mathematical forms that emerge in the theory, e.g. q-exponentials, and then obtain through fitting the corresponding value of q and of similar characteristic quantities. Coming back to names that are commonly used in the literature, we have seen above that the expression “nonextensive entropy” can be misleading. Not really so the expression “nonextensive statistical mechanics”. Indeed, the many-body mechanical systems that are primarily addressed within this theory include long-range interactions, i.e., interactions that are not integrable at infinity. Such systems clearly have a total energy which increases quicker than N, where N is the number of its microscopic elements. This is to say a total energy which indeed is nonextensive.