Nonextensive Statistical Mechanics

2.1 Boltzmann–Gibbs Entropy 232.1.2.6 Lesche-Stability or Experimental RobustnessAn entropic form S({p i }) (or any other functional of the probabilities, in fact) is saidLesche-stable or experimentally robust [79] if and only if it satisfies the followingcontinuity property. Two probability distributions {p i } and {pi ′ } are said close if theysatisfy the metric property:D ≡W∑|p i − p i ′ |≤d ɛ , (2.14)i=1where d ɛ is a small real number. Then, experimental robustness is verified if, for anyɛ>0, a d ɛ exists such that D ≤ d ɛ impliesR ≡∣ S({p i}) − S({p ′ i })S max∣ ∣∣ 0) . (2.17)Equation (2.14) corresponds to μ = 1. The Pythagorean metric corresponds toμ = 2. What about other values of μ? It happens that only for μ ≥ 1 the triangularinequality is satisfied, and consequently it does constitute a metric. Still, why not