Levy walk is a fundamental model with applications ranging from quantum physics to paths of animal foraging. Taking animal foraging as an example, a natural idea that comes to one's mind is to introduce the multiple internal states for dealing with the dependence of the PDF of waiting time on the energy of the animal and richness of the food at a particular location, etc; the framework can also be used to model the moving trajectories of smart animals without returning to the directions or locations which they come from immediately. After building the Levy walk model with multiple internal states and deriving the governing equation of the distribution of the positions of the particles, some applications are discussed with specific transition matrices. The type of diffusion for non-immediately-repeating Lévy walk is uncovered, and the distribution and average of first passage time are numerically simulated.
We refute here the concernes raised in the Comment of our letter. This reply states clearly the validity range of our results and shows that the optimality of inverse-square Levy walks at the basis of the Levy flight foraging hypothesis is generically unfounded. We also give the precise set of conditions for which inverse-levy square Levy walks turn to be optimal, conditions which are unlikely to be verified biologically.
Aging phenomena have been observed in numerous physical systems. Many statistical quantities depend on the aging time $t_a$ for aging anomalous diffusion processes. This paper pays more attention to how an external force field affects the aging Lévy walk. Based on the Langevin picture of Lévy walk and generalized Green-Kubo formula, we investigate the quantities which include the ensemble- and time-averaged mean-squared displacements in both weak aging $t_a\ll t$ and strong aging $t_a\gg t$ cases, and compare them to the quantities in the absence of any force field. Two typical force fields, c…
Lévy walkers inside spherical shells with absorbing boundaries: Towards settling the optimal Lévy walk strategy for random searches
Federal University of Rio Grande do Norte（北里奥格兰德联邦大学）；Harvard-Smithsonian Center for Astrophysics（哈佛-史密森天体物理中心）；Universidade Federal de Pernambuco（伯南布哥联邦大学）；National Institute of Science and Technology of Complex Systems, Federal University of Rio Grande do Norte（复杂系统科学与技术国家研究所，北里奥格兰德联邦大学）
We investigate the nonergodicity of the generalized Lévy walk introduced by Shlesinger et al. [Phys. Rev. Lett. 58, 1100 (1987)] with respect to the squared displacements. We present detailed analytical derivations of our previous findings outlined in a recent Letter [Phys. Rev. Lett. 120, 104501 (2018)], give profound interpretations, and especially emphasize three surprising results: First, we find that the mean-squared displacements can diverge for a certain range of parameter values. Second, we show that an ensemble of trajectories can spread subdiffusively, whereas individual time-average…
This article considers the statistical properties of Lévy walks possessing a regular long-term linear scaling of the mean square displacement with time, for which the conditions of the classical Central Limit Theorem apply. Notwithstanding this property, their higher-order moments display anomalous scaling properties, whenever the statistics of the transition times possesses power-law tails. This phenomenon is perfectly consistent with the classical Central Limit Theorem, as it involves the convergence properties towards the normal distribution. It is closely related to the property that the h…
A stochastic process with movement, return, and rest phases is considered in this paper. For the movement phase, the particles move following the dynamics of Gaussian process or ballistic type of Lévy walk, and the time of each movement is random. For the return phase, the particles will move back to the origin with a constant velocity or acceleration or under the action of a harmonic force after each movement, so that this phase can also be treated as a non-instantaneous resetting. After each return, a rest with a random time at the origin follows. The asymptotic behaviors of the mean squared…
The spectral theory of random walks on networks of arbitrary topology can be readily extended to study random walks and Lévy flights subject to resetting on these structures. When a discrete-time process is stochastically brought back from time to time to its starting node, the mean search time needed to reach another node of the network may be significantly decreased. In other cases, however, resetting is detrimental to search. Using the eigenvalues and eigenvectors of the transition matrix defining the process without resetting, we derive a general criterion for finite networks that establis…
We study Markovian continuous-time random walk models for Lévy flights and we show an example in which the convergence to stable densities is not guaranteed when jumps follow a bi-modal power-law distribution that is equal to zero in zero. The significance of this result is two-fold: i) with regard to the probabilistic derivation of the fractional diffusion equation and also ii) with regard to the concept of site fidelity in the framework of Lévy-like motion for wild animals.
Continuous Markovian model for Levy random walks with superdiffusive and superballistic regimes
我们考虑先前设计的描述Levy随机游动的模型（Phys. Rev E 79，011110; 80，031148，（2009））。数值结果表明，该模型描述了Levy随机游动与超扩散，弹道，以及超弹道动力学。以前只有超扩散制度进行了分析。在这个模型中，步行者的速度是由一个非线性朗之万方程。分析从小到大的时间尺度的交叉，我们发现的时间尺度上的速度相关性衰减和步行运动基本上表现出Levy统计。我们的分析是基于分析的几何平均沃克位移，并允许我们处理概率密度函数的幂律尾巴，相应地，发散的时刻。
Lévy walk is a popular and more `physical' model to describe the phenomena of superdiffusion, because of its finite velocity. The movements of particles are under the influences of external potentials almost at anytime and anywhere. In this paper, we establish a Langevin system coupled with a subordinator to describe the Lévy walk in the time-dependent periodic force field. The effects of external force are detected and carefully analyzed, including nonzero first moment (even though the force is periodic), adding an additional dispersion on the particle position, the consistent influence on th…
We study the ballistic Lévy walk stemming from an infinite mean traveling time between collision events. Our study focuses on the density of spreading particles all starting from a common origin, which is limited by a `light' cone $-v_0 t<x<v_0 t$. In particular we study this density close to its maximum in the vicinity of the `light' cone. The spreading density follows the Lamperti-arcsine law describing typical fluctuations far from the `light' cone. However this law blows up in the vicinity of the `light' cone horizon which is nonphysical, in the sense that any finite time observation will …
We perform a thorough analysis of the survival probability of symmetric random walks with stochastic resetting, defined as the probability for the walker not to cross the origin up to time $n$. For continuous symmetric distributions of step lengths with either finite (random walks) or infinite variance (Lévy flights), this probability can be expressed in terms of the survival probability of the walk without resetting, given by Sparre Andersen theory. It is therefore universal, i.e., independent of the step length distribution. We analyze this survival probability at depth, deriving both exact …
The time that waves spend inside 1D random media with the possibility of performing Lévy walks is experimentally and theoretically studied. The dynamics of quantum and classical wave diffusion has been investigated in canonical disordered systems via the delay time. We show that a wide class of disorder--Lévy disorder--leads to strong random fluctuations of the delay time; nevertheless, some statistical properties such as the tail of the distribution and the average of the delay time are insensitive to Lévy walks. Our results reveal a universal character of wave propagation that goes beyond st…
Superdiffusion arises when complicated, correlated and noisy motion at the microscopic scale conspires to yield peculiar dynamics at the macroscopic scale. It ubiquitously appears in a variety of scenarios, spanning a broad range of scientific disciplines. The approach of superdiffusive systems towards their long-time, asymptotic behavior was recently studied using the Lévy walk of order $1<β<2$, revealing a universal transition at the critical $β_{c}=3/2$. Here, we investigate the origin of this transition and identify two crucial ingredients: a finite velocity which couples the walker's posi…
Lévy walks are a particular type of continuous-time random walks which results in a super-diffusive spreading of an initially localized packet. The original one-dimensional model has a simple schematization that is based on starting a new unidirectional motion event either in the positive or in the negative direction. We consider two-dimensional generalization of Lévy walks in the form of the so-called XY-model. It describes a particle moving with a constant velocity along one of the four basic directions and randomly switching between them when starting a new motion event. We address the ball…
A Lévy random medium, in a given space, is a random point process where the distances between points, a.k.a. targets, are long-tailed. Random walks visiting the targets of a Lévy random medium have been used to model many (physical, ecological, social) phenomena that exhibit superdiffusion as the result of interactions between an agent and a sparse, complex environment. In this note we consider the simplest non-trivial Lévy random medium, a sequence of points in the real line with i.i.d. long-tailed distances between consecutive targets. A popular example of a continuous-time random walk in th…
Truncated Levy Random Walks and Generalized Cauchy Processes
提出了截断Levy随机游动的连续马尔可夫模型。它概括了先前由Lubashevsky等人Phys.Rev.E79，011110（2009）; 80，031148（2009），Eur. J. B 78，207（2010），其允许在怀疑粒子运动中的非线性摩擦和取决于粒子速度的噪声强度的饱和。这两种效应都有自己的理由被认为是，并单独产生截断Levy随机游动的文件中所示。采用1.5阶强随机Runge-Kutta方法数值求解了控制颗粒运动的非线性Langevin方程，并利用所得数值数据计算了一定时间间隔内颗粒位移的几何平均值，构造了其分布函数。它表明，几何平均值的时间依赖性包括三个片段，随着时间尺度的增加，可以被归类为弹道制度，列维型制度（超弹道，准弹道，或superdiffusive一个），和布朗粒子的标准运动。对于中间Levy型部分，粒子位移的分布被发现是广义Cauchy形式的截止。此外，随机游动的性质主要取决于摩擦系数和噪声强度的某一比值，而不是它们各自的特性。
We introduce a method to exactly generate bridge trajectories for discrete-time random walks, with arbitrary jump distributions, that are constrained to initially start at the origin and return to the origin after a fixed time. The method is based on an effective jump distribution that implicitly accounts for the bridge constraint. It is illustrated on various jump distributions and is shown to be very efficient in practice. In addition, we show how to generalize the method to other types of constrained random walks such as generalized bridges, excursions, and meanders.
It is widely accepted that inverse square Lévy walks are optimal search strategies because they maximize the encounter rate with sparse, randomly distributed, replenishable targets when the search restarts in the vicinity of the previously visited target, which becomes revisitable again with high probability, i.e., non-destructive foraging [Nature 401, 911 (1999)]. The precise conditions for the validity of this Lévy flight foraging hypothesis (LFH) have been widely described in the literature [Phys. Life Rev. 14, 94 (2015)]. Nevertheless, three objecting claims to the LFH have been raised rec…
