Nonextensive Statistical Mechanics

318 8 Final Comments and Perspectivesdard probabilistic mean value) defines the entropy is precisely the inverse of thefunction (namely eq x ) which provides the energy distribution at the stationary state.We are unaware of the existence of any other entropic functional form having allthese properties in common with S BG .Let us stress, at this point, that a property that S BG and S q do not share is additivity.This difference is extremely welcome. It is precisely this fact which makespossible for both entropies to be thermodynamically extensive for a special valueof q, more specifically q = 1 for extensive systems (i.e., those whose correlationsare generically short-ranged), and q < 1 for nonextensive ones (i.e., a large classamong those whose correlations are generically long-ranged).(k) Is nonextensive statistical mechanics necessary or just convenient?Let us first address a somewhat simpler question, namely: Is Boltzmann–Gibbsstatistical mechanics necessary or just convenient? The most microscopic level atwhich collective properties of a system can be answered is that of mechanics (classical,quantum, or any other that might be appropriate for the case). Let us illustratethis with classical Hamiltonian systems. Let us consider a system constituted ofN well-defined interacting particles. Its time evolution is fully determined by theinitial conditions. So, for every admissible set of initial conditions we have a pointevolving along a unique trajectory in the full phase-space . We can in principlecalculate all its mechanical properties, its time averages, its ensemble averages (overwell-defined sets of initial conditions). For example, its time-dependent “temperature”can be defined as being proportional to the average total kinetic energy of Nparticles divided by N. If we wish to approach a more thermodynamical definitionof temperature, we might wish to consider the average of this quantity over an ensembleof initial conditions. This ensemble can be uniformly distributed over theentire space, or be as special or particular as we wish. Of course, in practice,this road is almost always analytically untractable; moreover, it quickly becomescomputationally untractable as well when N increases above some number... wellbelow the Avogadro number!Another approach, which is not so powerful but surely is more tractable (both analyticallyand computationally), consists in considering the projection of the intothe single-particle phase-space μ, where the coordinates and momenta of only oneparticle are taken into account. In other words, we might be interested in discussingonly those properties that are well defined in terms of the single-particle marginalprobabilities. Such is the case of the Vlasov equation (see, for instance, [299]),and analogous approaches such as the Boltzmann transport equation itself. Theseprocedures are expected to be very useful whenever mixing and ergodic hypothesisare (strictly or nearly) verified in space. This surely is the case of almostall Hamiltonian systems whose many elements interact through a potential whichis nowhere singular, and which decays quickly enough at long distances. In othercases, the situation might be more complex. For example, such an approach is notexpected to be very reliable if the microscopic dynamics are such that structures(e.g., hierarchical ones) emerge in space, which might or might not reflect into