Nonextensive Statistical Mechanics

20 2 Learning with Boltzmann–Gibbs Statistical Mechanicsinto account (we recall that [momentum][position] = []). For the case of equalprobabilities (i.e., p(x) = 1/, where is the hypervolume of the admissibleD-dimensional space), we haveS BG = k ln(/σ ) . (2.3)A particular case of p(x) is the following one:p(x) =W∑p i (x − x i ) (W ≡ /σ ) , (2.4)i=1where (x −x i ) denotes a normalized uniform distribution centered on x i and whose“width” is σ (hence its height is 1/σ ). In this case, Eqs. (2.1), (2.2) and (2.3) preciselyrecover Eqs. (1.1), (1.2) and (1.3).In both discrete and continuous cases that we have addressed until now, wewere considering classical systems in the sense that all physical observables arereal quantities and not operators. However, for intrinsically quantum systems, wemust generalize the concept. In that case, the BG entropic form is to be written (asfirst introduced by von Neumann) in the following manner:S BG =−kTrρ ln ρ, (2.5)withTrρ = 1 , (2.6)where ρ is the density matrix acting on a W -dimensional Hilbert vectorial space(typically associated with the solutions of the Schroedinger equation with the chosenboundary conditions; in fact, quite frequently we have W →∞).A particular case of ρ is when it is diagonal, i.e., the following one:ρ ij = p i δ ij , (2.7)where δ ij denotes Kroenecker’s delta function. In this case, Eqs. (2.5) and (2.6)exactly recover Eqs. (1.1) and (1.2).All three entropic forms (1.1), (2.1), and (2.5) will be generically referred in thepresent book as BG-entropy because they are all based on a logarithmic measurefor disorder. Although we shall use one or the other for specific purposes, we shallmainly address the simple form expressed in Eq. (1.1).