Nonextensive Statistical Mechanics

228 7 Thermodynamical and Nonthermodynamical Applicationsin airports [389, 429, 430], turbulence at the level of the trees of the Amazon forest,[431,432], turbulent Couette–Taylor flow and related situations [433,439–443],Lagrangian turbulence [444], two-dimensional turbulence in pure electron plasma[445, 446], the so-called one-dimensional “turbulence” [447, 448], among others.Criticism has also been advanced [449]. For several of the experimental situationsthat have been studied, q-statistics appears to be a quite good approximation. However,for some experiments, further improvement becomes possible (see, for instance,[389]) whenever many experimental decades are accessible to measurement.Naturally, we do not intend here to exhaustively review the subject, and thereader is referred to the above literature for details. In what follows we have selectedinstead only a few of those studies, with the aim of characterizing the typesof approaches that have been developed.7.1.3.1 Lattice-Boltzmann Models for FluidsThe incompressible Navier–Stokes equation has been considered [425] on a discretizedD-dimensional Bravais lattice of coordination number b. It is further assumedthat there is a single value for the particle mass, and also for speed. Thebasic requirement for the lattice-Boltzmann model is to be Galilean-invariant (i.e.,invariant under change of inertial reference frame), like the Navier–Stokes equationitself. It has been proved [425] that an H-theorem is satisfied for a trace-form entropy(i.e., of the form S({p i }) = ∑ Wif (p i )) only if it has the form of S q withq = 1 − 2 D . (7.4)Therefore, q < 1 in all cases (q > 0ifD > 2, and q < 0ifD < 2), andapproaches unity from below in the D →∞limit. This study has been generalizedby allowing multiple masses and multiple speeds. Galilean invariance once againmandates [426] an entropy of the form of S q , with a unique value of q determinedby a transcendental equation involving the dimension and symmetry properties ofthe Bravais lattice as well as the multiple values of the masses and of the speeds. Ofcourse, Eq. (7.4) is recovered for the particular case of single mass and single speed.7.1.3.2 Defect TurbulenceExperiments have been done [427] in a convection cell which is heated from belowand cooled from above, and which is tilted a certain angle with respect to gravity.In such circumstances, defects spontaneously appear in the undulations of the fluid:see Fig. 7.7. The distribution of velocities of these defects as well as their diffusionhas been measured: see Figs. 7.8 and 7.9, respectively. The experimental conditionis characterized by the dimensionless driving parameter ɛ ≡ TT c− 1 ≥ 0, whereT is the temperature difference maintained between bottom and top of the cell,and T c is a characteristic temperature difference of the system. Under many differentexperimental conditions (in particular, many values of ɛ), it was found that