Nonextensive Statistical Mechanics

24 2 Learning with Boltzmann–Gibbs Statistical Mechanics0R BG-0.2-0.4-0.6d = 0.1d = 0.001d = 0.01-0.8-1QC0 0.1 0.2 0.3 0.4 0.51/W0.1R BG d = 0.10d = 0.001-0.1d = 0.01-0.2-0.3QEP0 0.1 0.2 0.3 0.4 0.51/WFig. 2.1 Illustration of the Lesche-stability of S BG . QC and QEP stand for quasi-certainty andquasi-equal-probabilities, respectively (see details in [110, 113]).using values of μ>1? Because, only for μ = 1, the distance D does not dependon W , which makes it appropriate for a generic property [80].We come back in Section 3.2.2 onto this interesting property introduced byLesche.2.1.2.7 Shannon Uniqueness TheoremLet us assume that an entropic form S({p i }) satisfies the following properties:(i) S({p i }) is a continuous f unction of {p i }; (2.18)(ii) S(p i = 1/W, ∀i) monotonically increases with the totalnumber of possibilities W; (2.19)(iii) S(A + B) = S(A) + S(B) if p A+Bij= pi A p B j ∀(i, j) , (2.20)