Nonextensive Statistical Mechanics

B.3 Remarks 339B.2 Second ExampleIn the previous example, we have used academically constructed “empiric” distributions.However, exactly the same scenario is encountered if we use random modelssuch as the one introduced in [627]. The variance of q-Gaussian distributions isfinite for q < 5/3, and diverges for 5/3 ≤ q < 3; their norm is finite for q < 3.Two typical cases are shown in Fig. B.3, one of them for q < 5/3, and the otherone for q > 5/3. In both cases, the fluctuations of the variance V [X] ≡ σ 2 areconsiderably larger than those of the q-variance V q [X] ≡ σq 2 .Forq < 5/3, thevariance converges very slowly to its exact asymptotic value; for q > 5/3 does notconverge at all. In all situations, the q-variance quickly converges to its asymptoticvalue, which is always finite, thus constituting a very satisfactory characterization.The reasons for precisely considering in this example the q-variance V q [X], and notany other, are the same that have been indicated in the previous example (see [258]and Section 4.7).B.3 RemarksLet us end by some general remarks. Abe has shown [885] that the q-expectation∑ Wi=1value 〈Q〉 q ≡Q i p q i∑ W , where {Q i } corresponds to any physical quantity, is unstable(in a uniform continuity sense, i.e., similar to the criterion introduced byi=1 pq iLesche for any entropic functional [79], not in the thermodynamic sense) for q ≠ 1,whereas it is stable q = 1. If we consider the particular case Q i = δ i, j , where weuse Kroenecker’s delta function, we obtain as a corollary that the escort distributionitself is unstable for q ≠ 1 3 This fact illustrates a simple property, namely thattwo quantities can be Lesche-stable, and nevertheless their ratio can be Lescheunstable.In the present example, both p q iand ∑ Wi=1 pq iare stable, ∀q > 0, butp q i∑ Wj=1 pq jis unstable for q ≠ 1. The possible epistemological implications of such∑ Wi=1 E i p q i∑ Wi=1 pq isubtle properties for the 1998 formulation [60] of nonextensive statistical mechanicsdeserve further analysis. The fact stands, however, that the characterization of the(asymptotic) power-laws which naturally emerge within this theory undoubtedly isvery conveniently done through q-expectation values, whereas it is not so throughstandard expectation values (which necessarily diverge for all moments whose orderexceeds some specific one, which depends on the exponent of the power-law). Thesituation is well illustrated for the constraints to be used for the canonical ensemble(the system being in contact with some thermostat). If, together with the norm constraint∑ Wi=1 p i = 1, we impose the energy constraint as 〈H〉 q ≡ = U q ,where {E i } are the energy eigenvalues and U q afixedfinite real number, we aredealing (unless we provide some additional qualification) with an unstable quantity.3 This special property was also directly established by Curado [886].