Nonextensive Statistical Mechanics

112 4 Stochastic Dynamical Foundations of Nonextensive Statistical Mechanicsp(x, t)t 2 [p(x, t)] 2−q= D qx 2= (2 − q) D qx{1−qp(x, t)}[p(x, t)]x(q ∈ R; (2− q)D q > 0) .(4.11)Its solution for the initial condition (4.2) is given bywherep q (x) =p q (x, t) = p q (x/[D q t] 13−q )) , (4.12)1√ e −x2 /A q11q = √ π Aq π Aq[] 11 + (q − 1) x2 q−1A q, (4.13)with⎧√ (q − 11)q−1⎪⎨ ( 3−q) if 1 < q < 3,2(q−1)A q = 2 ifq = 1,√ (1 − q5−3q)⎪⎩2(1−q) ( 2−q) if q < 1.1−q(4.14)See Fig. 4.1. See the text for several remarks.i. The upper bound q = 3 arrives from the imposition of normalization. In otherwords, ∫ ∞−∞ dx p q(x) diverges if q ≥ 3, and converges otherwise.ii. If 1 ≤ q < 3, these distributions have an infinite support. If q < 1, they havea compact support; indeed, for q < 1, they vanish for |x| > √ A q /(1 − q) .iii. The variance 〈x 2 〉 ≡ ∫ ∞−∞ dx x2 p q (x) of these distributions is finite forq < 5/3 and divergent for 5/3 ≤ q < 3. This implies that, if we convolutethem N times with N →∞, they approach a Gaussian distribution forq < 5/3 and a Lévy distribution for 5/3 < q < 3. This corresponds toindependence between the N random variables. The situation is completelydifferent if strong correlation exists between them. We focus on this interestingcase later on, when q-generalizing the Central Limit Theorem.iv. The q-variance 〈x 2 〉 q ≡ {∫ ∞−∞ dx x2 [p q (x)] q} / {∫ ∞−∞ dx [p q(x)] q} of thesedistributions is finite for q < 3. Indeed, ∫ ∞constant >0 dx x2 x −2q/(q−1) is finitefor q < 3.v. These distributions extremize (maximize for q > 0, and minimize for q < 0)S q under the appropriate constraints (see Section 3.5.2).