Nonextensive Statistical Mechanics

1.2 Background and Indications in the Literature 7Usually W is set equal to the number of ways (complexions) in which a state,which is incompletely defined in the sense of a molecular theory (i.e., coarsegrained), can be realized. To compute W one needs a complete theory (somethingsuch as a complete molecular-mechanical theory) of the system. For that reasonit appears to be doubtful whether Boltzmann’s principle alone, i.e., without acomplete molecular-mechanical theory (Elementary theory) has any real meaning.The equation S = k log W + const. appears [therefore], without an Elementarytheory – or however one wants to say it – devoid of any meaning from a phenomenologicalpoint of view.By Boltzmann’s principle – expression coined apparently by Einstein himself –,the author refers precisely to the logarithmic form for the entropy that he explicitlywrites down a few words later. It is quite striking the crucial role that Einstein attributesto microscopic dynamics for giving a clear sense to that particular form forthe entropy.Coming back to Gibbs’s book [1], in page 35 he wrote:In treating of the canonical distribution, we shall always suppose the multiple integral inequation (92) [the partition function, as we call it nowadays] to have a finite value, asotherwise the coefficient of probability vanishes, and the law of distribution becomesillusory. This will exclude certain cases, but not such apparently, as will affect the value ofour results with respect to their bearing on thermodynamics. It will exclude, for instance,cases in which the system or parts of it can be distributed in unlimited space [...]. It also excludesmany cases in which the energy can decrease without limit, as when the systemcontains material points which attract one another inversely as the squares of theirdistances. [...]. For the purposes of a general discussion, it is sufficient to call attention tothe assumption implicitly involved in the formula (92).Clearly, Gibbs is well aware that the theory he is developing has limitations. Itdoes not apply to anomalous cases such as gravitation.Enrico Fermi, in his 1936 Thermodynamics [23], wrote:The entropy of a system composed of several parts is very often equal to the sum of theentropies of all the parts. This is true if the energy of the system is the sum of the energiesof all the parts and if the work performed by the system during a transformation is equalto the sum of the amounts of work performed by all the parts. Notice that these conditionsare not quite obvious and that in some cases they may not be fulfilled. Thus, for example,in the case of a system composed of two homogeneous substances, it will be possible toexpress the energy as the sum of the energies of the two substances only if we can neglectthe surface energy of the two substances where they are in contact. The surface energy cangenerally be neglected only if the two substances are not very finely subdivided; otherwise,it can play a considerable role.So, Fermi says “very often,” which virtually implies “not always!”Ettore Majorana, mysteriously missing since 25 March 1938, wrote [24]:This is mainly because entropy is an additive quantity as the other ones. In other words, theentropy of a system composed of several independent parts is equal to the sum of entropyof each single part. [...] Therefore one considers all possible internal determinations asequally probable. This is indeed a new hypothesis because the universe, which is far frombeing in the same state forever, is subjected to continuous transformations. We will thereforeadmit as an extremely plausible working hypothesis, whose far consequences couldsometime not be verified, that all the internal states of a system are a priori equally