Lévy flights in inhomogeneous media

We investigate the impact of external periodic potentials on superdiffusive random walks known as Lévy flights and show that even strongly superdiffusive transport is substantially affected by the external field. Unlike ordinary random walks, Lévy flights are surprisingly sensitive to the shape of the potential while their asymptotic behavior ceases to depend on the Lévy index mu. Our analysis is based on a novel generalization of the Fokker-Planck equation suitable for systems in thermal equilibrium. Thus, the results presented are applicable to the large class of situations in which superdiffusion is caused by topological complexity, such as diffusion on folded polymers and scale-free networks.     

Generalization of the Khinchin theorem to Lévy flights

One of the most fundamental theorems in statistical mechanics is the Khinchin ergodic theorem, which links the ergodicity of a physical system with the irreversibility of the corresponding autocorrelation function. However, the Khinchin theorem cannot be successfully applied to processes with infinite second moment, in particular, to the relevant class of Lévy flights. Here, we solve this challenging problem. Namely, using the recently developed measure of dependence called Lévy correlation cascade, we derive a version of the Khinchin theorem for Lévy flights. This result allows us to verify the Boltzmann hypothesis for systems displaying Lévy-flight-type dynamics.     

Superdiffusion and viscoelastic vortex flows in a two-dimensional complex plasma

Viscoelastic vortical fluid motion in a strongly coupled particle system has been observed experimentally. Optical tracking of particle motion in a complex plasma monolayer reveals high grain mobility and large scale vortex flows coexistent with partial preservation of the global hexagonal lattice structure. The transport of particles is superdiffusive and ascribed to Lévy statistics on short time scales and to memory effects on the longer scales influenced by cooperative motion. At these longer time scales, the transport is governed by vortex flows covering a wide spectrum of temporal and spatial scales.     

Vicious Lévy flights

We study the statistics of encounters of Lévy flights by introducing the concept of vicious Lévy flights--distinct groups of walkers performing independent Lévy flights with the process terminating upon the first encounter between walkers of different groups. We show that the probability that the process survives up to time t decays as t-α at late times. We compute α up to the second order in ε expansion, where ε=σ-d, σ is the Lévy exponent, and d is the spatial dimension. For d=σ, we find the exponent of the logarithmic decay exactly. Theoretical values of the exponents are confirmed by numerical simulations. Our results indicate that walkers with smaller values of σ survive longer and are therefore more effective at avoiding each other.     

Lévy噪声与周期力共同驱动下的螺旋波动力学

研究加性Lévy噪声与周期外力对FitzHugh-Nagumo可激系统中螺旋波动力学行为的影响.螺旋波波头的运动随外力周期在一定范围内呈规则变化,该规则变化可用相应的傅立叶谱理解,维持该规则变化的是锁频行为.Lévy噪声序列中包含着小概率的大尺度噪声,螺旋波波头运动改变主要来自于它们的影响,本文指出Lévy噪声对波头运动的影响也依赖于外力周期的取值.在适当的参数取值下,Lévy噪声的存在也能导致螺旋波的消失,这为螺旋波的控制、消除提供了一种方法.分析了系统周期与外力周期的锁定行为,给出了不同噪声强度下的Arnold舌,指出随机共振行为的存在.     

Continuous Markovian model for Lévy random walks with superdiffusive and superballistic regimes

We consider a previously devised model describing Lévy random walks [I. Lubashevsky, R. Friedrich, A. Heuer, Phys. Rev. E 79, 011110 (2009); I. Lubashevsky, R. Friedrich, A. Heuer, Phys. Rev. E 80, 031148 (2009)]. It is demonstrated numerically that the given model describes Lévy random walks with superdiffusive, ballistic, as well as superballistic dynamics. Previously only the superdiffusive regime has been analyzed. In this model the walker velocity is governed by a nonlinear Langevin equation. Analyzing the crossover from small to large time scales we find the time scales on which the velocity correlations decay and the walker motion essentially exhibits Lévy statistics. Our analysis is based on the analysis of the geometric means of walker displacements and allows us to tackle probability density functions with power-law tails and, correspondingly, divergent moments.     

Infinite average lifetime of an unstable bright state in the green fluorescent protein

The time evolution of the fluorescence intensity emitted by well-defined ensembles of green fluorescent proteins has been studied by using a standard confocal microscope. In contrast with previous results obtained in single-molecule experiments, the photobleaching of the ensemble is well described by a model based on Lévy statistics. By assuming the presence of thermally activated barriers, this simple model allows us to obtain information about their height distribution.     

Statistical aging and nonergodicity in the fluorescence of single nanocrystals

The relation between single particle and ensemble measurements is addressed for semiconductor CdSe nanocrystals. We record their fluorescence at the single molecule level and analyze their emission intermittency, which is governed by unusual random processes known as Lévy statistics. We report the observation of statistical aging and ergodicity breaking, both related to the occurrence of Lévy statistics. Our results show that the behavior of ensemble quantities, such as the total fluorescence of an ensemble of nanocrystals, can differ from the time-averaged individual quantities, and must be interpreted with care.     

Nonergodicity of blinking nanocrystals and other Lévy-walk processes

We investigate the nonergodic properties of blinking nanocrystals modeled by a Lévy-walk stochastic process. Using a nonergodic mean field approach we calculate the distribution functions of the time averaged intensity correlation function. We show that these distributions are not delta peaked on the ensemble average correlation function values; instead they are W or U shaped. Beyond blinking nanocrystals our results describe ergodicity breaking in systems modeled by Lévy walks , for example, certain types of chaotic maps and spin dynamics to name a few.     

Lévy flights and superdiffusion in the context of biological encounters and random searches

We review the general problem of random searches in the context of biological encounters. We analyze deterministic and stochastic aspects of searching in general and address the destructive and nondestructive cases specifically. We discuss the concepts of Lévy walks as adaptive strategies and explore possible examples. We also review Lévy searches in other media and spaces, including lattices and networks as opposed to continuous environments. We analyze empirical evidence supporting the Lévy flight foraging hypothesis, as well as the more general idea of superdiffusive foraging. We compare these hypothesis with alternative theories of random searches. Finally, we comment on several issues relevant to the practical application of models of Lévy and superdiffusive strategies to the general question of biological foraging.     

Optimizing the encounter rate in biological interactions: Lévy versus Brownian strategies

An important application involving two-species reaction-diffusion systems relates to the problem of finding the best statistical strategy for optimizing the encounter rate between organisms. We investigate the general problem of how the encounter rate depends on whether organisms move in Lévy or Brownian random walks. By simulating a limiting generalized searcher-target model (e.g., predator-prey, mating partner, pollinator-flower), we find that Lévy walks confer a significant advantage for increasing encounter rates when the searcher is larger or moves rapidly relative to the target, and when the target density is low.     

Nonexponential decoherence and momentum subdiffusion in a quantum lévy kicked rotator

We investigate decoherence in the quantum kicked rotator (modeling cold atoms in a pulsed optical field) subjected to noise with power-law tail waiting-time distributions of variable exponent (Lévy noise). We demonstrate the existence of a regime of nonexponential decoherence where the notion of a decoherence rate is ill defined. In this regime, dynamical localization is never fully destroyed, indicating that the dynamics of the quantum system never reaches the classical limit. We show that this leads to quantum subdiffusion of the momentum, which should be observable in an experiment.     

Synchronization of oscillating reactions in an extended fluid system

We present experiments on the synchronization of a dynamical, chemical process in an extended, flowing, fluid system. The oscillatory Belousov-Zhabotinsky chemical reaction is the process studied, and the flow is an annular chain of counterrotating vortices. Azimuthal motion of the vortices is controlled externally, enabling us to vary the type of transport. We find that oscillations of the Belousov-Zhabotinsky reaction synchronize throughout the extended fluid system only if transport in the flow is superdiffusive, with tracers in the flow undergoing rapid, distant jumps called Lévy flights.     

Mapping of the roughness exponent for the fuse model for fracture

The roughness exponent for fracture surfaces in the fuse model has been thought to be universal for narrow threshold distributions and has been important in the numerical studies of fracture roughness. We show that the fuse model gives a disorder dependent roughness exponent for narrow disorders when the lattice is influencing the fracture growth. When the influence of the lattice disappears, the local roughness exponent approaches zeta(local)=0.65+/-0.03 for distribution with a tail toward small thresholds, but with large jumps in the profiles giving corrections to scaling on small scales. For very broad disorders the distribution of jumps becomes a Lévy distribution and the Lévy characteristics contribute to the local roughness exponent.     

Optical electron transitions in a two-dimensional disordered semiconductor

The frequency dependence of the light absorption coefficient associated with electron transitions between states of strong and weak localization is studied.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 11, pp. 22–24, November, 1982.One of us (V. L. Bonch-Bruevich) is grateful to Dr. M. Pepper for sending a preprint.     

A Monte Carlo Study of the Fractal Property inside Jets Produced in High Energy e+e－ Collisions

The anomalous scaling property inside jets produced in e⁺e^－ Collisions at s91.2GeV is studied using Monte Carlo method. A new variable r is applied, which has very good anomalous scaling property. The relation between the scaling indices of different orders is investigated. The Lévy index for the hadronic system inside jets is obtained, which turns out to be μ=2.498±0.025, very close to the same index of hadron-hadron collisions. On the contrary, the Lévy index of e⁺e^－ collisions is 1.701±0.043, being less than 2. Since the multi-hadron production in e⁺e^－ collisions is dominated by hard gluon emission or hard parton branching, while the parton fragmentation into jet, like the multi-production in hadron-hadron collisions, is a soft process, the above results suggest that the value of Lévy index may provide another signal for the qualitative difference between hard and soft processes.     

Leapover lengths and first passage time statistics for Lévy flights

Exact results for the first passage time and leapover statistics of symmetric and one-sided Lévy flights (LFs) are derived. LFs with a stable index alpha are shown to have leapover lengths that are asymptotically power law distributed with an index alpha for one-sided LFs and, surprisingly, with an index alpha/2 for symmetric LFs. The first passage time distribution scales like a power law with an index 1/2 as required by the Sparre-Andersen theorem for symmetric LFs, whereas one-sided LFs have a narrow distribution of first passage times. The exact analytic results are confirmed by extensive simulations.     

Random walks in two-dimensional glass-like environments

The asymptotic behavior of a diffusive particle under the influence of an environment and of a dynamical feedback coupling is considered in the framework of a two-dimensional numerical simulation. The feedback is controlled by a memory term of strength λ. A sufficient negative memory term (λ<0) offers a superdiffusive behavior with logarithmic corrections to conventional diffusion, whereas a positive feedback coupling (λ>0) is related to a weak subdiffusion or leads to localization of the particle. The numerical simulations are in agreement with results of a renormalization group approach and of the mode-coupling theory.     

Bounds on the fine structure constantvariability from Fe ii absorption lines in QSO spectra

The dispersal of individuals of a species is the key driving force of various spatiotemporal phenomena which occur on geographical scales. It can synchronise populations of interacting species, stabilise them, and diversify gene pools [1-3]. The geographic spread of human infectious diseases such as influenza, measles and the recent severe acute respiratory syndrome (SARS) is essentially promoted by human travel which occurs on many length scales and is sustained by a variety of means of transportation [4-8]. In the light of increasing international trade, intensified human traffic, and an imminent influenza A pandemic the knowledge of dynamical and statistical properties of human dispersal is of fundamental importance and acute [7,9,10]. A quantitative statistical theory for human travel and concomitant reliable forecasts would substantially improve and extend existing prevention strategies. Despite its crucial role, a quantitative assessment of human dispersal remains elusive and the opinion that humans disperse diffusively still prevails in many models [11]. In this chapter I will report on a recently developed technique which permits a solid and quantitative assessment of human dispersal on geographical scales [11]. The key idea is to infer the statistical properties of human travel by analysing the geographic circulation of individual bank notes for which comprehensive datasets are collected at the online bill-tracking website www.wheresgeorge.com. The analysis shows that the distribution of travelling distances decays as a power law, indicating that the movement of bank notes is reminiscent of superdiffusive, scale free random walks known as Lèvy flights [13]. Secondly, the probability of remaining in a small, spatially confined region for a time T is dominated by heavy tails which attenuate superdiffusive dispersal. I will show that the dispersal of bank notes can be described on many spatiotemporal scales by a two parameter continuous time random walk (CTRW) model to a surprising accuracy. To this end, I will provide a brief introduction to continuous time random walk theory [14] and will show that human dispersal is an ambivalent, effectively superdiffusive process.     

Coherent backscattering and localization in a self-attracting random walk model

Intensity propagation of waves in dilute 2D and 3D disordered systems is well described by a random walk path-model. In strongly scattering media, however, this model is not quite correct because of interference effects like coherent backscattering. In this letter, coherent backscattering is taken into account by a modified, self-attracting random walk. Straightforward simulations of this model essentially reproduce the results of current theories on “non-classical” transport behavior, i.e. Anderson localization in 1D and 2D for any amount of disorder and a phase transition from weak to strong localization in 3D. However, in the strongly scattering regime corrections are necessary to account for the finite number of light modes due to their non-vanishing lateral extention. Within our model this correction leads to the observation that strong localization does not take place. Received 17 September 2001