Nonextensive Statistical Mechanics

Chapter 2Learning with Boltzmann–Gibbs StatisticalMechanics ˜ ´o ´ , o [78]Kleoboulos of Lindos (6th century B.C.)2.1 Boltzmann–Gibbs Entropy2.1.1 Entropic FormsThe entropic forms (1.1) and (1.3) that we have introduced in Chapter 1 correspondto the case where the (microscopic) states of the system are discrete. There are,however, cases in which the appropriate variables are continuous. For these, the BGentropy takes the form∫S BG =−kdx p(x) ln[σ p(x)] , (2.1)with∫dx p(x) = 1 , (2.2)where x/σ ∈ R D , D ≥ 1 being the dimension of the full space of microscopicstates (called Gibbs phase-space for classical Hamiltonian systems). Typically xcarries physical units. The constant σ carries the same physical units as x, so thatx/σ is a dimensionless quantity (we adopt from now on the notation [x] = [σ ],hence [x/σ ] = 1). For example, if we are dealing with an isolated classical N-bodyHamiltonian system of point masses interacting among them in d dimensions, wemay use σ = Nd . This standard choice comes of course from the fact that, ata sufficiently small scale, Newtonian mechanics becomes incorrect and we mustrely on quantum mechanics. In this case, D = 2dN, where each of the d pairs ofcomponents of momentum and position of each of the N particles has been takenC. Tsallis, Introduction to Nonextensive Statistical Mechanics,DOI 10.1007/978-0-387-85359-8 2, C○ Springer Science+Business Media, LLC 200919