Nonextensive Statistical Mechanics

8.2 Frequently Asked Questions 319nontrivial structures in the μ space itself. 3 This could be the case if the interactionsdecay very slowly with distance, at least for various classes of initial conditions.A third possible approach is that of stochastic equations. The paradigm of such alevel of description is the Langevin equation. One particle is selected (and followed)from the entire system, and part of the action of all the others is seen as a noise,typically a white Gaussian-like one. Such a description has the advantage of beingrelatively simple. It has however the considerable disadvantage of being partiallyphenomenological, in the sense that one has to introduce quite ad hoc types ofnoises. If we are not interested in following the possible trajectories of a singleparticle, but rather in the time evolution of probability distributions associated withsuch particles, we enter into the level of description of the Fokker–Planck equation,and the alike. At this mesoscopic level, exact analytical calculations, or relativelyeasy numerical ones, are relatively frequent.A fourth possible approach is that of statistical mechanics. It directly connects –and this is where its beauty and power come from – the relevant microscopic informationcontained, for instance, in the Hamiltonian (with appropriate boundaryconditions), to useful macroscopic quantities such as equations of states, specificheats, susceptibilities, and even various important correlation functions. In someepistemological sense, it superseeds all the previous types of approaches, exceptingthe fully microscopic one with which it should always be consistent. This lastpoint is kind of trivial since statistical mechanics is nothing but a “shortcuted path”from the microscopic world to the macroscopic one. Let us precisely qualify thesense in which statistical mechanics “superseeds” other approaches such as thoseof Vlasov, Langevin, and Fokker–Planck. We mean that, whenever the collectivestates (usually at thermal equilibrium) and the quantities that are being calculatedare exactly the same, no admissible mesoscopic description could be inconsistentwith the statistical mechanical one.A fifth possible approach is that of thermodynamics. It directly connects manytypes of macroscopic quantities with sensible simplicity. However, it is incapableof calculating from first principles quantities such as specific heats, susceptibilities,among many others. One expects, of course, that the results and connections obtainedat the thermodynamical level will be consistent with those obtained at any ofthe previous levels, whenever comparison is justified and possible.After this brief overview, it becomes kind of trivial to answer part of our initialquestion. Indeed, statistical mechanics is not necessary, but it can be extremelyconvenient; also, it provides an unifying description of a great variety of usefulquestions. A point which remains to be answered is the following one. Given thefact that we do have – since more than one century – BG statistical mechanics,do we need, or is it convenient, a more general one? We can say that it is notnecessary in the very same sense that, as we saw above, BG statistical mechanicsis not necessary either. Is it convenient? We may say that, whenever possible,3 Nontrivial structures in μ space imply nontrivial ones in space. The other way around is nottrue: structures could exist in space which would not be seen in μ space (the “shadow” of afractal sponge on a wall can be a quite smooth surface).