Lévy-type diffusion on one-dimensional directed Cantor graphs

Single-big-jump principle in physical modeling

The big-jump principle is a well-established mathematical result for sums of independent and identically distributed random variables extracted from a fat-tailed distribution. It states that the tail of the distribution of the sum is the same as the distribution of the largest summand. In practice, it means that when in a stochastic process the relevant quantity is a sum of variables, the mechanism leading to rare events is peculiar: Instead of being caused by a set of many small deviations all in the same direction, one jump, the biggest of the lot, provides the main contribution to the rare large fluctuation. We reformulate and elevate the big-jump principle beyond its current status to allow it to deal with correlations, finite cutoffs, continuous paths, memory, and quenched disorder. Doing so we are able to predict rare events using the extended big-jump principle in Lévy walks, in a model of laser cooling, in a scattering process on a heterogeneous structure, and in a class of Lévy walks with memory. We argue that the generalized big-jump principle can serve as an excellent guideline for reliable estimates of risk and probabilities of rare events in many complex processes featuring heavy-tailed distributions, ranging from contamination spreading to active transport in the cell.

Light diffusion in quenched disorder: role of step correlations

We present a theoretical and experimental study of light transport in disordered media with strongly heterogeneous distribution of scatterers formed via nonscattering regions. Step correlations induced by quenched disorder are found to prevent diffusivity from diverging with increasing heterogeneity scale, contrary to expectations from annealed models. Spectral diffusivity is measured for a porous ceramic where nanopores act as scatterers and macropores render their distribution heterogeneous. Results agree well with Monte Carlo simulations and a proposed analytical model.

Superdiffusion and transport in two-dimensional systems with Lévy-like quenched disorder

We present an extensive analysis of transport properties in superdiffusive twodimensional quenched random media, obtained by packing disks with radii distributed according to a Lévy law. We consider transport and scaling properties in samples packed with two different procedures, at fixed filling fraction and at self-similar packing, and we clarify the role of the two procedures in the superdiffusive effects. Using the behavior of the filling fraction in finite size systems as the main geometrical parameter, we define an effective Lévy exponent that correctly estimates the finite size effects. The effective Lévy exponent rules the dynamical scaling of the main transport properties and identifies the region where superdiffusive effects can be detected.

Random walk in chemical space of Cantor dust as a paradigm of superdiffusion

We point out that the chemical space of a totally disconnected Cantor dust K(n) [Symbol: see text E(n) is a compact metric space C(n) with the spectral dimension d(s) = d(ℓ) = n > D, where D and d(ℓ) = n are the fractal and chemical dimensions of K(n), respectively. Hence, we can define a random walk in the chemical space as a Markovian Gaussian process. The mapping of a random walk in C(n) into K(n) [Symbol: see text] E(n) defines the quenched Lévy flight on the Cantor dust with a single step duration independent of the step length. The equations, describing the superdiffusion and diffusion-reaction front propagation ruled by the local quenched Lévy flight on K(n) [Symbol: see text] E(n), are derived. The use of these equations to model superdiffusive phenomena, observed in some physical systems in which propagators decay faster than algebraically, is discussed.

Scattering lengths and universality in superdiffusive Lévy materials

We study the effects of scattering lengths on Lévy walks in quenched, one-dimensional random and fractal quasilattices, with scatterers spaced according to a long-tailed distribution. By analyzing the scaling properties of the random-walk probability distribution, we show that the effect of the varying scattering length can be reabsorbed in the multiplicative coefficient of the scaling length. This leads to a superscaling behavior, where the dy-namical exponents and also the scaling functions do not depend on the value of the scattering length. Within the scaling framework, we obtain an exact expression for the multiplicative coefficient as a function of the scattering length both in the annealed and in the quenched random and fractal cases. Our analytic results are compared with numerical simulations, with excellent agreement, and are supposed to hold also in higher dimensions.

Transmission probability through a Lévy glass and comparison with a Lévy walk

Recent experiments on the propagation of light over a distance L through a random packing of spheres with a power-law distribution of radii (a so-called Lévy glass) have found that the transmission probability T∝1/L(γ) scales superdiffusively (γ<1). The data has been interpreted in terms of a Lévy walk. We present computer simulations to demonstrate that diffusive scaling (γ≈1) can coexist with a divergent second moment of the step size distribution [p(s)∝1/s(1+α) with α<2]. This finding is in accord with analytical predictions for the effect of step size correlations, but deviates from what one would expect for a Lévy walk of independent steps.

Transport and scaling in quenched two- and three-dimensional Lévy quasicrystals

We consider correlated Lévy walks on a class of two- and three-dimensional deterministic self-similar structures, with correlation between steps induced by the geometrical distribution of regions, featuring different diffusion properties. We introduce a geometric parameter α, playing a role analogous to the exponent characterizing the step-length distribution in random systems. By a single-long-jump approximation, we analytically determine the long-time asymptotic behavior of the moments of the probability distribution as a function of α and of the dynamic exponent z associated with the scaling length of the process. We show that our scaling analysis also applies to experimentally relevant quantities such as escape-time and transmission probabilities. Extensive numerical simulations corroborate our results which, in general, are different from those pertaining to uncorrelated Lévy-walk models.

Role of quenching on superdiffusive transport in two-dimensional random media

Transport in random media is known to be affected by quenched disorder. From the point of view of random walks, quenching induces correlations between steps that may alter the dynamical properties of the medium. This paper is intended to provide more insight into the role of quenched disorder on superdiffusive transport in two-dimensional random media. The systems under consideration are disordered materials called Lévy glasses that exhibit large spatial fluctuations in the density of scattering elements. We show that in an ideal Lévy glass the influence of quenching can be neglected, in the sense that transport follows to very good approximation that of a standard Lévy walk. We also show that, by changing sample parameters, quenching effects can be increased intentionally, thereby making it possible to investigate systematically diverse regimes of transport. In particular, we find that strong quenching induces local trapping effects which slow down superdiffusion and lead to a transient subdiffusivelike transport regime close to the truncation time of the system.

Lévy walks and scaling in quenched disordered media

We study Lévy walks in quenched disordered one-dimensional media, with scatterers spaced according to a long-tailed distribution. By analyzing the scaling relations for the random-walk probability and for the resistivity in the equivalent electric problem, we obtain the asymptotic behavior of the mean-square displacement as a function of the exponent characterizing the scatterers distribution. We demonstrate that in quenched media different average procedures can display different asymptotic behavior. In particular, we estimate the moments of the displacement averaged over processes starting from scattering sites. Our results are compared with numerical simulations, with excellent agreement.