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

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.

Large Fluctuations for Spatial Diffusion of Cold Atoms

We use a new approach to study the large fluctuations of a heavy-tailed system, where the standard large-deviations principle does not apply. Large-deviations theory deals with tails of probability distributions and the rare events of random processes, for example, spreading packets of particles. Mathematically, it concerns the exponential falloff of the density of thin-tailed systems. Here we investigate the spatial density P_{t}(x) of laser-cooled atoms, where at intermediate length scales the shape is fat tailed. We focus on the rare events beyond this range, which dominate important statistical properties of the system. Through a novel friction mechanism induced by the laser fields, the density is explored with the recently proposed non-normalized infinite-covariant density approach. The small and large fluctuations give rise to a bifractal nature of the spreading packet. We derive general relations which extend our theory to a class of systems with multifractal moments.

Localization in one-dimensional chains with Lévy-type disorder

We study Anderson localization of the classical lattice waves in a chain with mass impurities distributed randomly through a power-law relation s-(1+α) with s as the distance between two successive impurities and α>0. This model of disorder is long-range correlated and is inspired by the peculiar structure of the complex optical systems known as Lévy glasses. Using theoretical arguments and numerics, we show that in the regime in which the average distance between impurities is finite with infinite variance, the small-frequency behavior of the localization length is ξ{ω)∼ω-α. The physical interpretation of this result is that, for small frequencies and long wavelengths, the waves feel an effective disorder whose fluctuations are scale dependent. Numerical simulations show that an initially localized wave-packet attains, at large times, a characteristic inverse power-law front with an α-dependent exponent which can be estimated analytically.

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.

Holey random walks: optics of heterogeneous turbid composites

We present a probabilistic theory of random walks in turbid media with nonscattering regions. It is shown that important characteristics such as diffusion constants, average step lengths, crossing statistics, and void spacings can be analytically predicted. The theory is validated using Monte Carlo simulations of light transport in heterogeneous systems in the form of random sphere packings and good agreement is found. The role of step correlations is discussed and differences between unbounded and bounded systems are investigated. Our results are relevant to the optics of heterogeneous systems in general and represent an important step forward in the understanding of media with strong (fractal) heterogeneity in particular.

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.

Weak localization of light in superdiffusive random systems

Lévy flights constitute a broad class of random walks that occur in many fields of research, from biology to economy and geophysics. The recent advent of Lévy glasses allows us to study Lévy flights-and the resultant superdiffusion-using light waves. This raises several questions about the influence of interference on superdiffusive transport. Superdiffusive structures have the extraordinary property that all points are connected via direct jumps, which is expected to have a strong impact on interference effects such as weak and strong localization. Here we report on the experimental observation of weak localization in Lévy glasses and compare our results with a recently developed theory for multiple scattering in superdiffusive media. Experimental results are in good agreement with theory and allow us to unveil the light propagation inside a finite-size superdiffusive system.

Heterogeneous and anisotropic dynamics of a 2D gel

We report x-ray photon correlation spectroscopy (XPCS) results on bidimensional (2D) gels formed by a Langmuir monolayer of gold nanoparticles. The system allows an experimental determination of the fourth order time correlation function, which is compared to the usual second order correlation function and to the mechanical response measured on macroscopic scale. The observed dynamics is anisotropic, heterogeneous and superdiffusive on the nanoscale. Different time scales, associated with fast heterogeneous dynamics inside 2D cages and slower motion of larger parts of the film, can be identified from the correlation functions. The XPCS results are discussed in view of other experimental results and models of three-dimensional gel dynamics.

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.