Nonextensive Statistical Mechanics

1.2 Background and Indications in the Literature 11[...] More generally, in areas of physics where Gibbsianness is not put in by hand,one should expect non-Gibbsianness to be ubiquitous. This is probably the case innonequilibrium statistical mechanics.Since one cannot expect all measures of interest to be Gibbsian, the question thenarises whether there are weaker conditions that capture some or most of the “good”physical properties characteristic of Gibbs measures. For example, the stationary measureof the voter model appears to have the critical exponents predicted (under the hypothesisof Gibbsianness) by the Monte Carlo renormalization group, even though this measureis provably non-Gibbsian.One may also inquire whether there is a classification of non-Gibbsian measuresaccording to their “degree of non-Gibbsianness”.The authors make in this paper no reference whatsoever to nonextensive statisticalmechanics (proposed in fact 5 years earlier [39]). It will nevertheless becomeevident that, interestingly enough, several among their remarks neatly apply to thecontent of the present book. Particularly, it will become obvious that (q − 1) representsa possible measure of “non-Gibbsianness,” where q denotes the entropic indexto be soon introduced.From the viewpoint of the dynamical foundations of statistical mechanics, a recentremark (already quoted in the Preface of this book) by Giulio Casati and TomazProsen [9] is worth to be reproduced at this point:While exponential instability is sufficient for a meaningful statistical description, it is notknown whether or not it is also necessary.Let us anticipate that it belongs to the aim of the present book to convince thereader precisely that it is not necessary: power-law instability appears to do the jobsimilarly well, if we consistently adopt the appropriate entropy.Many more statements exist in the literature along similar lines. But we believethat the ones that we have selected are enough (both in quality and quantity!) fordepicting, at least in an “impressionistic” way, the epistemological scenario withinwhich we are evolving. A few basic interrelated points that emerge include:(i) No strict physical or mathematical reason exists (or, at least, is known) for notexploring the possible generalization of the BG entropy and its consequences.(ii) The BG entropy and any of its possible generalizations should conform to themicroscopical dynamical features of the system, very specifically to properties suchas sensitivity to the initial conditions and mixing. The relevant rigorous necessaryand sufficient conditions are still unknown. The ultimate justification of any physicalentropy is expected to come from microscopic dynamics and detailed geometricalconditions.(iii) No physical or mathematical reason exists (or, at least, is known) for notexploring, in natural, artificial and even social systems, distributions differing fromthe BG one, very specifically for stationary or quasi-stationary states differing fromthermal equilibrium, such as metastable states, and other nonequilibrium ones.(iv) Long-range microscopic interactions (and long-range microscopic memory),as well as interactions exhibiting severe (e.g., nonintegrable attractive) singularitiesat the origin, appear as a privileged field for the exploration and understanding ofanomalous thermostatistical behavior.