Levy-walk

T. Zavada, R. Kimmich 1998, (The anomalous adsorbate dynamics at surfaces in porous media studied by nuclear magnetic resonance methods. The orientational structure and Levy walks), J. Chem. Phys. 109, 6929. 

Kimmich and coworkers have studied the magnetic relaxation dispersion of liquids adsorbed on or contained in microporous inorganic materials such as glasses and packed silica (34-43) and analyze the relaxation dispersion data using Levy walk statistics for surface diffusion to build... 

How general are our results From a stochastic point of view ergodicity breaking, Levy statistics, anomalous diffusion, aging, and fractional calculus, are all related. In particular ergodicity breaking is found in other models with power-law distributions, related to the underlying stochastic model (the Levy walk). For example, the well known continuous time random walk model also... 

G. Margolin and E. Barkai, Nonergodicity of blinking nanocrystals and other Levy-walk processes. Phys. Rev. Lett. 94 080601 (2005). 

As a step toward the study of thermodynamic equilibrium in the case of anomalous statistical physics, in Section VII we study how the generators of anomalous diffusion respond to external perturbation. The ordinary linear response theory is violated and, in some conditions, is replaced by a different kind of linear response. In Section VIII we review the results of an ambitious attempt at deriving thermodynamics from dynamics for the main purpose of exploring a dynamic approach to the still unsettled issue of the thermodynamics of Levy statistics. The Levy walk perspective seems to be the only possible way to establish a satisfactory connection between dynamics and thermodynamics in... 

The Levy walk is physically more plausible than the Levy flight. How to derive the Levy walk from a Liouville approach of the kind described in Section III Here, we illustrate a path explored some years ago, to establish a connection between GME and this kind of superdiffusion [49,50]. We assume that there exists a waiting time distribution v /(x), prescribed, for instance, by the dynamic model illustrated in Section V. This function corresponds to a distribution of uncorrelated times. We can imagine the ideal experiment of creating the sequence x,, by drawing in succession the numbers of this distribution. Then we create the fluctuating velocity E,(f), according to the procedure illustrated in Section V. 

Let us now show the CTRW in action. The picture of Section IV has to be properly modified so as to fit the physical condition of the Levy walk. Hence following Ref. 70 we have to define the probability of moving by a quantity x in either the positive, x > 0, or negative direction, x < 0, in a time t. This quantity is given by... 

Let us illustrate, in short, the results found by the authors of Ref. 71. Let us study the dynamic approach to Levy diffusion using the Levy walk perspective. Let us use the condition p > 2, which is compatible with the existence of the stationary correlation function (t). As we have shown earlier, the Levy scaling can be derived using the arguments behind Eq. (118). This equation implies that the number of events is proportional to time. This is not quite correct. The exact formula was found by Feller [72], and it reads... 

In the Levy walk case, this is true for q < p — 1. For q > p — 1, the slope becomes equal to 1, ballistic motion, according to... 

In Section VI we have seen that the Levy walk is physically more attractive than the Levy flight. It is therefore reasonable to address the issue of the derivation of noncanonical equilibrium by using the Levy walk perspective. On the other... 

The adoption of Eq. (195) yields an equilibrium distribution that is similar to the WS statistics, with the main difference, however, that the inverse power law is truncated by two peaks, at = W/y and = —W/y. Note that the Levy walk noise i (f) is generated according to the renewal prescriptions of Section VI that is, we use the waiting time distribution /(f) of Eq. (92) and, according to the... 

We note that in Ref. 87 the projection method was used to study the effect of friction on a process that, in the free case, is known to produce a Levy walk. As already discussed in Section VIII, the key point has been that the violation of the Green-Kubo relation must imply a different form of linear response, used later to justify the WS form of noncanonical equilibrium. It is worth mentioning that an interesting result of that article has been the following equation of motion ... 

Physically more interesting is the Levy walk model [21], which is still described by Eq. (11) but now (7(f) is a random variable with finite variance but nontrivial time correlations—for example, such that C (x) t-P with (3 < 1. In other words, condition (ii) is violated. This can be obtained as follows. Let us assume that (7(f) takes the values uq, and maintains its value for a duration T that is random and with probability density i(T) 7 -( +1). Then standard... 

A Levy walk, on the other hand, takes into account the fact that longer steps take longer times to complete than do shorter steps. The recognition of this simple fact ties the distribution of step sizes to the distribution of time intervals, which in the case of turbulence was determined by the fluctuations in the fluid velocity [62]. In the present example the continuum form of the Levy walk process is described by Eq. (42), with the autocorrelation function for the random driver being given by the inverse power law Eq. (66) and W is the constant speed of the walker. The asymptotic form of the second moment for this process is... 

We give two examples of fractal time series. The first is fractal Gaussian intermittent noise characterized by a long-time correlated waiting-time sequence, and the second is a Levy-walk intermittent noise. These examples were developed in an environmental context to explain the observed distribution of earthquakes in California [66]. 

Figure 11. DEA and SDA for (A) a fractal Gaussian intermittent noise with /(t) = exp[—t/y] with y = 25 and H—d — 0.75 the fractal Gaussian relation of equal exponents is satisfied. (B) A Levy-walk intermittent noise with /(t) oc and p = 2.5 note the bifurcation between H — 0.75 and 5 = 0.67 caused by the Levy-walk diffusion relation [66].

The Levy-walk intermittent noise is characterized by an uncorrelated waiting-time sequence and a Levy or an inverse power-law waiting-time distribution function such as given by Eq. (97) with 1 < a < 2. This interval for the scaling index insures that although the second moment diverges, the first moment is finite. The presence of a Levy-walk process in a given time series can be detected by means of the asymptotic relation Eq. (99), which we refer to as... 

It should be stressed that even though the second moment for a Levy walk is finite, the scaling index obtained from the second moment does not give the correct scaling properties of the time series. This is a word of caution regarding the application of FVSMs to determine the scaling properties of a time series. Even when the second moment has the form... 

Figure 3. Comparison of the trajectories of a Gaussian (left) and a Levy (right) process, the latter with index a = 1.5. While both trajectories are statistically self-similar, the Levy walk trajectory possesses a fractal dimension, characterizing the island structure of clusters of smaller steps, connected by a long step. Both walks are drawn for the same number of steps ( 7000).

There are certain ways of overcoming this difficulty (i) by a time cost through coupling between x and t, producing Levy walks [45,98], or (ii) by a cutoff in the Levy noise to prevent divergence [99,100]. While (i) appears a natural choice, it gives rise to a nonMarkov process. Conversely, (ii) corresponds to an ad hoc measure. 

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