Nonextensive Statistical Mechanics

1.1 Introduction 5whereand∑W A∑W BS BG (A + B) ≡−ki=1j=1p A+Bij∑W AS BG (A) ≡−ki=1ln p A+Bij(with W = W A W B ), (1.5)piA ln pi A , (1.6)∑W BS BG (B) ≡−k p B j ln p B j . (1.7)Expression (1.1) was first proposed (for simple continuous systems) by Boltzmann[5,6] in the 1870s, and was then refined by Gibbs [1] for more general systems.It is the basis of the usual BG statistical mechanics. In particular, its optimizationunder appropriate constraints (that we shall describe later on) yields, for a system inthermal equilibrium with a thermostat at temperature T , the celebrated BG factoror weight, namelywithj=1p i = e−β E iZ BG(1.8)β ≡ 1/kT , (1.9)W∑Z BG ≡ e −β E j, (1.10)j=1and where {E i } denotes the energy spectrum of the system, i.e., the eigenvaluesof the Hamiltonian of the system with the adopted boundary conditions; Z BG isreferred to as the partition function.Expressions (1.1) and (1.8) are the landmarks of BG statistical mechanics, andare vastly and successfully used in physics, chemistry, mathematics, computationalsciences, engineering, and elsewhere. Since their establishment, about 130 yearsago, they constitute fundamental pieces of contemporary physics. Though notoriouslyapplicable in very many systems and situations, we believe that they need tobe modified (generalized) in others, in particular in most of the so-called complexsystems (see, for instance, [12,15–18]). We believe, in other words, that they are notuniversal, as somehow implicitly (or explicitly) thought until not long ago by manyphysicists. They must have in fact a restricted domain of validity, as any other humanintellectual construct. As Newtonian mechanics, nonrelativistic quantum mechanics,special relativity, Maxwell electromagnetism, and all others. The basic purpose