Nonextensive Statistical Mechanics

140 4 Stochastic Dynamical Foundations of Nonextensive Statistical Mechanics[ ] [dFq [ f ](ξ)(iv) If we consider that F q [ f ](ξ) = 1 +dξξ + 1 d 2 F q [ f ](ξ)ξ=02 dξ]ξ=0 ξ 2 +[ 21 d 3 F q [ f ](ξ)3! dξ]ξ=0 ξ 3 +... ,then F 3 q [ f ](ξ) is uniquely determined by the knowledgeof the sets {〈x n 〉 n } and {ν qn } (n = 0, 1, 2, 3,...). Finally, since theinverse q-Fourier transform exists and, under some conditions, possibly isunique [247], the same knowledge determines in principle f (x) itself [258].(v) If f (x) ∼ 1/|x| γ (|x| → ∞; γ > 0), then we define q = 1 + 1 γ .Thisdetermines q n = 1 + (q − 1)n, hence all the moments 〈x n 〉 n . For example,if f (x) isaQ-Gaussian, we have that γ = 2/(Q − 1), hence 1/(q − 1) =2/(Q − 1). Therefore, the upper admissible limit q = 2 precisely correspondsto the well-known upper admissible value Q = 3.Let us now introduce another definition. A random variable is said to have a(q,α)-stable distribution L q,α (x) 8 if its q-Fourier transform has the formwithi.e., if−b |ξ|αaeq α,1(a > 0, b > 0, 0 0, b > 0, 0