Nonextensive Statistical Mechanics

1.4 A Few Words on the Foundations of Statistical Mechanics 17weight or mass of the system. In statistical mechanical terms, we expect it to beproportional to N for large N. 3The foundations of any statistical mechanics are, as already said, expected tocover basically all of the above points. There is a wide-spread vague belief amongphysicists that these steps have already been satisfactorily accomplished since longfor the standard, BG statistical mechanics. This is not so! Not so surprising after all,given the enormity of the corresponding task! For example, as already mentioned, atthis date, there is no available deduction, from and only from microscopic dynamics,of the celebrated BG exponential weight (1.8). Neither exists the deduction frommicroscopic dynamics of the BG entropy (1.1).For standard systems, there is not a single reasonable doubt about the correctnessof the expressions (1.1) and (1.8) and of their relationships. But, from the logicaldeductiveviewpoint, there is still pretty much work to be done! This is, in fact,kind of easy to notice. Indeed, all the textbooks, without exception, introduce theBG factor and/or the entropy S BG in some kind of phenomenological manner, oras self-evident, or within some axiomatic formulation. None of them introducesthem as (and only as) a rational consequence of Newtonian, or quantum mechanics,using theory of probabilities. This is in fact sometimes referred to as the Boltzmannprogram. Boltzmann himself died without succeeding its implementation. Althoughimportant progress has been accomplished in these last 130 years, Boltzmann programstill remains in our days as a basic intellectual challenge. Were it not thegenius of scientists like Boltzmann and Gibbs, were we to exclusively depend onmathematically well-constructed arguments, one of the monuments of contemporaryphysics – BG statistical mechanics – would not exist!Many anomalous natural, artificial, and social systems exist for which BG statisticalconcepts appear to be inapplicable. Typically because they live in peculiarstationary or quasi stationary states that are quite different from thermal equilibrium,where BG statistics reigns. Nevertheless, as we shall see, some of them can still behandled within statistical mechanical methods, but with a more general entropy,namely S q , to be introduced later on [39, 59, 60].It should be clear that, whatever is not yet mathematically justified in BG statisticalmechanics, it is even less justified in the generalization to which the presentbook is dedicated. In addition to this, some of the points that are relatively wellunderstood in the standard theory can be still unclear in its generalization. In otherwords, the theory we are presenting here is still in intense evolution (sets of reviewscan be found in [62, 64–76]).3 Let us anticipate that it has been recently shown [55–58] that, if we impose a Poissonian distributionfor visitation times in phase-space, in addition to the first and second principles of thermodynamics,we obtain the BG functional form for the entropy. If a conveniently deformed Poissoniandistribution is imposed instead, we obtain the S q functional form. These results in themselvescannot be considered as a justification from first principles of the BG, or of the nonextensive, statisticalmechanics. Indeed, the visitation distributions are phenomenologically introduced, and thefirst and second principles are just imposed. This connection is nevertheless extremely clarifying,and can help producing a full justification.