- Published 2008

We show that the dynamic approach to Lévy statistics is characterized by aging and multifractality, induced by an ultra-slow transition to anomalous scaling. We argue that these aspects make it a protoptype of complex systems. PACS numbers: 05.40.fb; 05.45.Df; 05.45.Tp; 89.75.Da Typeset using REVTEX 1 The dynamic approach to Lévy statistics is still a poorly understood problem in spite of several attempts made at deriving it from intermittency [1–3] with several techniques ranging from Continuous Time Random Walk (CTRW) [4] to the Nakajima-Zwanzig projection method [5,6]. This letter establishes that the dynamic approach implies aging, multiscaling and multifractality, thereby explaining, among other things, why this issue escaped so far a satisfactory understanding. The aim of this letter is to contribute to the comprehension of this delicate issue. Before addressing the problem from a more technical perspective that will be by necessity less accessible to a general audience, we want to illustrate the problem, and the solution of it afforded by this letter as well, with intuitive and qualitative arguments. To make our illustration as clear as possible, we adopt the same tutorial approach as that used by the authors working in the field of Cryptography [7], and we introduce two new archetypal individuals, Bob and Jerry. We hope that Bob and Jerry might have, with aging induced multifractality, the same fortune as Alice and Bob with Cryptography. These two individuals aim at realizing a process of diffusion of Lévy type, with an intensity that depends on secrete numbers known only to them. Since the width of the probability distribution does not depend only on time, but also on the intensity of the jumps made by the random walkers at any unit time, it is hard, in principle, to establish the time at which the diffusion process began, if the intensities of the jumps are not known. However, as we shall see, this is possible with Bob’s experiment, while it is impossible with Jerry’s experiment. In fact, Bob and Jerry realize Lévy diffusion in two different ways, and, as we shall see, Bob, who adopts the dynamic approach, allows us to predict the exact time at which he started the experiment, while it is not possible to guess the right starting time in the case of Jerry’s experiment. Bob generates a random sequence of pairs {τi, si}, with i = 0, 1, ...∞. The first number of each pair, τi, is randomly drawn from the distribution ψ(τ) = (μ− 1)T μ−1 (T + τ)μ . (1) The index μ has to fit the condition μ > 2, which ensures that the mean waiting time 〈τ〉, 2 〈τ〉 = T μ− 2 , (2) is finite. Thus the number T controls the intensity of 〈τ〉. The second symbol, si, is a sign, + or −, and it is obtained by tossing a fair coin. Let us imagine that this distribution is used to generate a diffusion process according to the following prescription. Bob, who has at his diposal a virtual infinite number of walkers, creates, for any of them, a sequence {τi, si} and makes her travel with velocity siW for the time τi. Then Bob selects a number r belonging to the interval (0, τ0]. The space travelled by the walker at time t > τ0 − r is given by x(t) = W [(τ0 − r)s0 + τ1s1 + · · · +τN−1sN−1 + (τN − r ′)sN ], (3) where r ≡ (τ0 + · · ·+ τN )− r − t and N denotes the number of drawings made by Bob for his walker within the time interval [0, t]. At times t < τ0 − r (N = 0) the space is given by x(t) =W (τ0 − r − r ′)s0, (4) with r = τ0 − r− t, yielding x(0) = 0. Bob keeps secret both the value of W and the value of T . He can try to make his diffusion process look older by increasing either both or only one of these secrete numbers. Furthermore, to erase any possible form of aging, he selects the number r randomly. In fact, this has the effect of ensuring the stationary condition used by the authors of Refs [1,4] to realize Lévy diffusion, under the form of Lévy walk [1], which seems to be more realistic than the flight prescription [2,3]. To appreciate the properties of the Lévy walk, realized by Bob’s experiment, it is convenient to contrast it with Jerry’s experiment. Also Jerry has at his disposal a virtually infinite number of random walkers, whose position at t = 0 is x = 0, and he too, for any of his random walkers, selects an infinite sequence {xi}. The numbers xi are drawn from a symmetric distribution, the positive numbers having the same probability as the negative numbers. For this reason Jerry does need the coin tossing to select si. Furthermore, Jerry makes his walker jump at any time step, by a jump of intensity |xi|, in the positive or negative direction according to the sign of xi. To be more precise, let us say that the distribution used by Jerry is Π(x), defined through its Fourier transform, 3 Π̂(k) = exp(−b|k|), (5) and only Jerry knows the secrete value of b. The distribution of numbers used by Jerry is the well known Lévy distribution [8]: a stable distribution yielding a diffusion process, with the probability distribution pL(x, t), whose Fourier transform is p̂L(k, t) = exp(−b|k| t). (6) The time t is the number of drawings, but it is so large as to be virtually indistiguishable from a continuous number. It is clear that the observation of Jerry’s diffusion process does not allow any observer to establish when he began his experiments. We assume that the observer, which might be a third archetypal individual, does not know the time at which Jerry began his diffusion. By means of the experimental observation he/she can only establish bt, and since b is not known to him/her, he/she cannot determine the value of t. In other words, a broad distribution can be the consequence of Jerry starting his diffusion process at a very early time, but it can also be the consequence of a late beginning with much more intense jumps. It is not so with Bob’s diffusion process. Let us see why. Let us consider, as in the case of Jerry’s experiment, a time t very large. In the case of interest here, μ > 2, the mean waiting time, Eq. (2), is finite. Thus, the number of random drawings and coin tossings is very well approximated by N = t/〈τ〉. Using the Generalized Central Limit Theorem (GCLT) [9] we predict that Jerry’s experiment yields the same statistics as Bob’s experiment, Lévy statistics. However, this important theorem does not afford any clear indication about the time necessary to realize this statistics. The predictions of the GCLT theorems are realized [10] by the following expression for p(x, t): p(x, t) = K(t)pT (x, t)θ(Wt− |x|) + 1 2 δ(|x| −Wt)Ip(t), (7) where θ denotes the Heaviside step function. We also note that lim t→∞ Ip(t) = Φξ(t) = ( T T + t )μ−2 , (8)

@inproceedings{Allegrini2008AgingIM,
title={Aging Induced Multifractality},
author={Paolo Allegrini},
year={2008}
}