Lattice Boltzmann and nonextensive diffusion

Statistical physics today is arguably in much the same situation that Euclidean geometry found itself in the early nineteenth century. Over the last decade, an increasing body of evidence has indicated that denying a certain postulate of statistical physics – the extensivity of the entropy – results not in a contradiction, but rather in an entirely new family of mathematically consistent variants of the statistical physics developed by Boltzmann and Gibbs (see Editorial). The mathematical formulation of these variants begins with a generalization of the definition of the entropy in terms of the microscopic state probabilities of the system under study (see Box 1 in the Editorial). A family of such entropies has been posited, parametrized by a single positive number q, such that the usual BoltzmannGibbs formulation is recovered when q = 1. More precisely, whereas the Boltzmann-Gibbs entropy is expressed in terms of the logarithm function, nonextensive variants are expressed in terms of a q-deformed logarithm (defined in Box 1) to which application of l’Hôpital’s rule confirms reduction to the ordinary logarithm as q → 1. Remarkably, many fundamental results of statistical physics, such as the Maxwell relations and Onsager reciprocity, are “q-invariant”; that is, they hold for any statistical physics in the family. Other results, such as the Fluctuation-Dissipation Theorem and the compressible Navier-Stokes equations for viscous fluid dynamics, must be modified by the addition of terms that vanish when q = 1.