Nonextensive Statistical Mechanics

1.2 Background and Indications in the Literature 9lower bound. Without such a property there can be no rigorous basis for the statisticalmechanics of such a system (Fisher and Ruelle 1966). Basically it is that simple.One can ignore the fact that one knows that there is no rigorous basis for one’scomputer manipulations; one can try to improve the situation, or one can look foranother job.Needless to say that the very existence of the present book constitutes but anattempt to improve the situation!The same point is addressed by W.C. Saslaw in his 1985 Gravitation Physics ofStellar and Galactic Systems [31]:When interactions are important the thermodynamic parameters may lose their simpleintensive and extensive properties for subregions of a given system. [...] Gravitationalsystems, as often mentioned earlier, do not saturate and so do not have an ultimate equilibriumstate.Radu Balescu, in his 1975 Equilibrium and Nonequilibrium Statistical Mechanics[32], wrote:It therefore appears from the present discussion that the mixing property of a mechanicalsystem is much more important for the understanding of statistical mechanics thanthe mere ergodicity. [...] A detailed rigorous study of the way in which the concepts ofmixing and the concept of large numbers of degrees of freedom influence the macroscopiclaws of motion is still lacking.David Ruelle wrote in page 1 of his 1978 Thermodynamical Formalism [33] (andmaintains in page 1 of his 2004 Edition):The formalism of equilibrium statistical mechanics – which we shall call thermodynamicformalism – has been developed since G.W. Gibbs to describe the properties of certainphysical systems. [...] While the physical justification of the thermodynamic formalismremains quite insufficient, this formalism has proved remarkably successful at explainingfacts.The mathematical investigation of the thermodynamic formalism is in fact not completed:the theory is a young one, with emphasis still more on imagination than on technicaldifficulties. This situation is reminiscent of pre-classic art forms, where inspiration has notbeen castrated by the necessity to conform to standard technical patterns. We hope that someof this juvenile freshness of the subject will remain in the present monograph!He wrote also, in page 3:The problem of why the Gibbs ensemble describes thermal equilibrium (atleastfor“large systems”) when the above physical identifications have been made is deep and incompletelyclarified.The basic identification he is referring to is between β and the inverse temperature.Consistently, the first equation in both editions (page 3) is dedicated to definethe entropy to be associated with a probability measure. The BG form is introducedafter the words “we define its entropy” without any kind of justification or physicalmotivation.The same theme is retaken by Floris Takens in the 1991 Structures in Dynamics[34]. Takens wrote:The values of p i are determined by the following dogma : if the energy of the system inthe ith state is E i and if the temperature of the system is T then: p i = e −Ei /kT /Z(T ),