Dynamical approach to anomalous diffusion: Response of Lévy processes to a perturbation

Trefán, Gy.; Floriani, Elena; West, Bruce J.; Grigolini, Paolo

doi:10.1103/physreve.50.2564

Cited by 65 publications

(60 citation statements)

References 30 publications

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“…In a sense this analysis shows that thermodynamics is possible also in the region where the central limit theorem holds in the generalized form established by Lévy [35,36]. It is interesting to remark that earlier research work [37,38] has established that the dynamical approach to diffusion, based on the stationary assumption on the fluctuations responsible for diffusion, is incompatible with the condition µ < 2. In this region a diffusion process must rest on a continuous-time random walk method implying the breakdown of the stationary assumption [37].…”

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confidence: 81%

A non extensive approach to the entropy of symbolic sequences

Buiatti

Grigolini

Palatella

1999

Physica A: Statistical Mechanics and its Applications

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Symbolic sequences with long-range correlations are expected to result in a slow regression to a steady state of entropy increase. However, we prove that also in this case a fast transition to a constant rate of entropy increase can be obtained, provided that the extensive entropy of Tsallis with entropic index q is adopted, thereby resulting in a new form of entropy that we shall refer to as Kolmogorov-Sinai-Tsallis (KST) entropy. We assume that the same symbols, either 1 or −1, are repeated in strings of length l, with the probability distribution p(l) ∝ 1 l µ . The numerical evaluation of the KST entropy suggests that at the value µ = 2 a sort of abrupt transition might occur. For the values of µ in the range 1 < µ < 2 the entropic index q is expected to vanish, as a consequence of the fact that in this case the average length < l > diverges, thereby breaking the balance between determinism and randomness in favor of determinism. In the region µ ≥ 2 the entropic index q seems to depend on µ through the power law expression q = (µ − 2) α with α ≈ 0.13 (q = 1 with µ > 3). It is argued that this phase-transition like property signals the onset of the thermodynamical regime at µ = 2.

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confidence: 81%

A non extensive approach to the entropy of symbolic sequences

Buiatti

Grigolini

Palatella

1999

Physica A: Statistical Mechanics and its Applications

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“…This scaling is the same scaling as that of the second moment [5,16] and it is different from the Lévy scaling, which is, as we shall see in Section IV, α = 1/(β + 1). Thus, in the late time region the exact solution of Eq.…”

Section: Late Time Behaviormentioning

confidence: 92%

Strange kinetics: conflict between density and trajectory description

2002

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We study a process of anomalous diffusion, based on intermittent velocity fluctuations, and we show that its scaling depends on whether we observe the motion of many independent trajectories or that of a Liouville-like equation driven density. The reason for this discrepancy seems to be that the Liouville-like equation is unable to reproduce the multi-scaling properties emerging from trajectory dynamics. We argue that this conflict between density and trajectory might help us to define the uncertain border between dynamics and thermodynamics, and that between quantum and classical physics as well.

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“…By means of the stationary assumption, we recover the results of Refs. [4,19,20] and we realize why these earlier results go beyond the Green-Kubo prediction [21,22].…”

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confidence: 83%

Linear Response to Perturbation of Nonexponential Renewal Processes

2005

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We study the linear response of a two-state stochastic process, obeying the renewal condition, by means of a stochastic rate equation equivalent to a master equation with infinite memory. We show that the condition of perennial aging makes the response to coherent perturbation vanish in the long-time limit. DOI: 10.1103/PhysRevLett.95.220601 PACS numbers: 05.40.Fb, 02.50.ÿr, 82.20.Uv Many complex processes generate erratic jumps back and forth from a state ''on'' to a state ''off.'' We limit ourselves to quoting ionic channel fluctuations [1][2][3], currently triggering the search for a form of stochastic resonance valid also in the nonexponential case [4], and the intermittency of blinking nanocrystals [5,6]: It has been assessed that the intermittent fluorescence of these materials obeys the renewal theory [7]; namely, a jump from one to the other state has the effect of resetting the system's memory to zero. The nonexponential nature of the distribution of sojourn times makes this renewal process nonergodic and generates aging effects that are the object of an increasing theoretical interest [8,9]. Similar properties are found with surface-enhanced Raman spectra of single molecules [10].The authors of [11][12][13] studied the joint effect of aging and perturbation on a process of subdiffusion, without establishing, however, a direct connection with the issue of non-Poisson stochastic resonance [4]. Here we fill this gap by means of a stochastic master equation, with no time convolution. This method might be extended to an arbitrarily large number of states and to the case of arbitrarily intense perturbation.Let us assume that the distributions of on and off times are identical. We assign to the survival probability (SP) of this process, t, the inverse power law formwith > 1. This corresponds to the joint action of the time-dependent rate [14,15] qt q 0 =1 q 1 t, with q 0 ÿ 1=T and q 1 1=T, and of a resetting prescription. To illustrate this condition, let us imagine the random drawing of a number from the interval I 0; 1 at discrete times i 1; 2; . . . . The interval I is divided into two parts, I 1 and I 2 , with I 1 ranging from 0 to p i , and I 2 ranging from p i to 1. Note that p i 1 ÿ q i < 1 and q i 1, and, as a consequence, the number of times we keep drawing numbers from I 1 , without moving to I 2 , is very large.Let us evaluate the distribution of these persistence times, and let us discuss under which conditions we get the SP of Eq. (1). The SP function is the probability of remaining in I 1 after n drawings and is consequently given byUsing the condition q i 1, and evaluating the logarithm of both terms of Eq. (2), we obtainThe condition q i 1 implies that i and n of Eq. (3) are so large as to make q i virtually identical to a function of the continuous time t, q i qt q 0 =r1 q 1 t, with t t. Thus, Eq. (3) yields the SP of Eq. (1), and the corresponding waiting time distribution density, , readsWe denote as collisions the rare drawings of a number from I 2 , followed by resetting. Thus the col...

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Dynamical approach to anomalous diffusion: Response of Lévy processes to a perturbation

Cited by 65 publications

References 30 publications

A non extensive approach to the entropy of symbolic sequences

A non extensive approach to the entropy of symbolic sequences

Strange kinetics: conflict between density and trajectory description

Linear Response to Perturbation of Nonexponential Renewal Processes

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