Nonextensive Statistical Mechanics

4.8 Generalizing the Langevin Equation 147Fig. 4.27 Top: Probability distributions P (Y N ) vs. Y N ,withY N ≡ ∑ Ni=1 X i , X i being ( )q = 7 3 -independent random variables associated with a G 9 (X) distribution with β = 1(left), and the5respective ( )q = 9 5 -Fourier Transform, ˜P (k), vs. k (right). Middle: Same as above, in ln 9 -squared5scale (left), and ln 7 -squared scale (right). The straight lines indicate that P (Y N ) and ˜P (k) are3q-Gaussians with q = 9 5 and q = 7 3and βq ′ ∗=9/5(N) for right panel curves. Bottom: β−1, respectively. Their slopes are β−1q ∗=9/5(N) for left panel curvesq (N) vs. N 5 ∗=9/5, which is a straight line with∗ ∣=slope 1 (left); βq ′ 3−q∗=9/5(N) vs. N, which is also a straight line, but with slope0.030995 ...(right) (from [253]).8 C 2(q∗−1)q∗∣q ∗=9/5〈ξ(t) ξ(t ′ )〉=2 M δ(t − t ′ ) . (4.99)The noise amplitude M ≥ 0 stands for multiplicative. The noises ξ(t) and η(t)are assumed uncorrelated. 9 The stochastic differential equation is not completelydefined and must be complemented by an additional rule. This is due to the fact thateach pulse of the stochastic noise produces a jump in x, then the question arises:9 It is possible to combine two such noises into a single effective multiplicative one [304], but, forclarity purposes, here we shall keep track of both sources separately.