Effects of asymmetric Lévy noise in parameter-induced aperiodic stochastic resonance

We study properties of parameter-induced aperiodic stochastic resonance in the presence of asymmetric Lévy noise. The system performance is characterized by the bit error rate. Investigations are based on the numerical solution of the space-fractional Fokker-Planck equation and Monte Carlo simulations. After choosing the optimal detection threshold, it is shown that the skewness parameter β has very limited influence on the system performance. Under the same conditions, the system performance is slightly reduced with the increasing β. The stability index α has the dominant effects on the system performance. The lower value of α leads to the better system performance.     

Temperature controlled Lévy flights of minority carriers in photoexcited bulk n-InP

We study by photoluminescence the spatial distribution of minority carriers (holes) arising from their anomalous photon-assisted diffusion upon photo-excitation at an edge of n-InP slab for temperatures ranging from 300 K to 78 K. Giant enhancement in the spread of holes — over distances exceeding 1 cm from the excitation edge — is seen at lower temperatures. We show that the experiment provides a realization of the “Lévy flight” random walk of holes, in which the Lévy distribution index γ   is controlled by the temperature. The variation γ(T)$\gamma (T)$ is close to that predicted earlier, γ=1−Δ/kT$\gamma =1-\Delta /kT$, where Δ(T)$\Delta (T)$ is the Urbach tailing parameter of the absorption spectra. This theoretical prediction is based on the assumption of a quasi-equilibrium intrinsic emission spectrum in the form due to van Roosbroeck and Shockley.     

Balancing the competing demands of harvesting and safety from predation: Lévy walk searches outperform composite Brownian walk searches but only when foraging under the risk of predation

Some foragers have movement patterns that can be approximated by Lévy walks whilst others may be better represented as composite Brownian walks. Many attempts have been made to interpret these movement patterns in terms of optimal searching strategies for the location of randomly and sparsely distributed targets. Here it is shown that the relative merits of Lévy walk and composite Brownian walk searches are sensitively dependent upon target encounter dynamics which set the initial conditions for an extensive search. It is suggested these initial conditions are determined, at least in part, by the competing demands of harvesting and safety from predation. In accordance with observations, it is shown that Lévy walks are expected in tritrophic systems and where intraguild predation operates. Composite Brownian walks, on the other hand, are found to be advantageous when the risk of predation is low. Despite having fundamentally different properties, Lévy walks and composite Brownian walks can therefore compete a priori as possible models of animal movements. Throughout, attention is focused on searching for randomly and sparsely distributed resources that are not depleted or rejected once located but instead remain targets for future searches. We re-evaluate and overturn the widely held belief that in numerical simulations this ‘non-destructive’ searching scenario can faithfully and consistently represent destructive searching for patchily distributed resources, i.e. for resources that tend to occur in clusters rather than in isolation.     

Measuring the self-similarity exponent in Lévy stable processes of financial time series

Geometric method-based procedures, which will be called GM algorithms herein, were introduced in [M.A. Sánchez Granero, J.E. Trinidad Segovia, J. García Pérez, Some comments on Hurst exponent and the long memory processes on capital markets, Phys. A 387 (2008) 5543-5551], to efficiently calculate the self-similarity exponent of a time series. In that paper, the authors showed empirically that these algorithms, based on a geometrical approach, are more accurate than the classical algorithms, especially with short length time series. The authors checked that GM algorithms are good when working with (fractional) Brownian motions. Moreover, in [J.E. Trinidad Segovia, M. Fernández-Martínez, M.A. Sánchez-Granero, A note on geometric method-based procedures to calculate the Hurst exponent, Phys. A 391 (2012) 2209-2214], a mathematical background for the validity of such procedures to estimate the self-similarity index of any random process with stationary and self-affine increments was provided. In particular, they proved theoretically that GM algorithms are also valid to explore long-memory in (fractional) Lévy stable motions.     

Inertial effects, mass separation and rectification power in Lévy ratchets

We study the dynamics of particles in an external anisotropic periodic potential under the influence of additive white Lévy noise, in a general not overdamped situation. Different quantities characterizing directionality, coherence and dispersion are analyzed as functions of the mass and other systems parameters. We show that, while the current decreases monotonously with the stability index of the Lévy noise, there exists a particular intermediate value of such parameter (slightly dependent on the mass) that minimizes the time required to form a coherent particle package advancing in the preferred direction. Moreover, we show the possibility of observing mass separation. This means that particles of different masses may advance in opposite directions when influenced by the same ratchet potential and the same Lévy noise. Finally, we show that the ratio of the advanced distance to the total distance travelled constitutes a relevant measure for the rectification power, useful not only for Lévy ratchets but also for general ratchets systems. In particular, we find that it behaves quite similar to the rectification efficiency for standard models of rocking and flashing ratchets found in the literature.     

Fluctuations in a lévy flight gas

We consider the density fluctuations of an ideal Brownian gas of particles performing Lévy flìghts characterized by the indexf. We find that the fluctuations scale as N(t) t^H, where the Hurst exponentH locks onto the universal value 1/4 for Lévy flights with a finite root-mean-square range (f>2). For Lévy flights with a finite mean range but infinite root-mean-square range (1相似文献   

Some insights in superdiffusive transport

In a wide range of systems, the relaxation in response to an initial pulse has been experimentally found to follow a nonlinear relationship for the mean squared displacement, of the kind 〈x²(t)〉∝t^α$〈x^2(t)〉\propto t^{\alpha }$, where α$\alpha$ may be greater or smaller than 1. Such phenomena have been described under the generic term of anomalous diffusion. “Lévy flights” stochastic processes lead to superdiffusive behaviour (1<α<2)$(1<\alpha <2)$ and have been recently proposed to model—among the others—the subsurface contaminant spread in highly heterogeneous media under the effects of water flow. In this paper, within the continuous-time random walk (CTRW) approach to anomalous diffusion, we compare the analytical solution of the approximated fractional diffusion equation (FDE) with the Monte Carlo one, obtained by simulating the superdiffusive behaviour of an ensemble of particle in a medium. We show that the two are neatly different as the process approaches the standard diffusive behaviour. We argue that this is due to a truncation in the Fourier space expansion introduced by the FDE approach. We propose a second-order correction to this expansion and numerically solve the CTRW model under this hypothesis: the accuracy of the results thus obtained is validated through Monte Carlo simulation over all the superdiffusive range. The same kind of discrepancy is shown to occur also in the derivation of the fractional moments of the distribution: analogous corrections are proposed and validated through the Monte Carlo approach.     

Human memory retrieval as Lévy foraging

When people attempt to recall as many words as possible from a specific category (e.g., animal names) their retrievals occur sporadically over an extended temporal period. Retrievals decline as recall progresses, but short retrieval bursts can occur even after tens of minutes of performing the task. To date, efforts to gain insight into the nature of retrieval from this fundamental phenomenon of semantic memory have focused primarily upon the exponential growth rate of cumulative recall. Here we focus upon the time intervals between retrievals. We expected and found that, for each participant in our experiment, these intervals conformed to a Lévy distribution suggesting that the Lévy flight dynamics that characterize foraging behavior may also characterize retrieval from semantic memory. The closer the exponent on the inverse square power-law distribution of retrieval intervals approximated the optimal foraging value of 2, the more efficient was the retrieval. At an abstract dynamical level, foraging for particular foods in one's niche and searching for particular words in one's memory must be similar processes if particular foods and particular words are randomly and sparsely located in their respective spaces at sites that are not known a priori. We discuss whether Lévy dynamics imply that memory processes, like foraging, are optimized in an ecological way.     

Fluctuation-driven directed transport in the presence of Lévy flights

The role of Lévy flights on fluctuation-driven transport in time independent periodic potentials with broken spatial symmetry is studied. Two complementary approaches are followed. The first one is based on a generalized Langevin model describing overdamped dynamics in a ratchet type external potential driven by Lévy white noise with stability index α in the range 1<α<2. The second approach is based on the space fractional Fokker-Planck equation describing the corresponding probability density function (PDF) of particle displacements. It is observed that, even in the absence of an external tilting force or a bias in the noise, the Lévy flights drive the system out of the thermodynamic equilibrium and generate an up-hill current (i.e., a current in the direction of the steeper side of the asymmetric potential). For small values of the noise intensity there is an optimal value of α yielding the maximum current. The direction and magnitude of the current can be manipulated by changing the Lévy noise asymmetry and the potential asymmetry. For a sharply localized initial condition, the PDF of staying at the minimum of the potential exhibits scaling behavior in time with an exponent bigger than the −1/α exponent corresponding to the force free case.     

Power tails of index distributions in chinese stock market

The power α$\alpha$ of the Lévy tails of stock market fluctuations discovered in recent years are generally believed to be universal. We show that for the Chinese stock market this is not true, the powers depending strongly on anomalous daily index changes short before market closure, and weakly on the opening data.     

Fractional motions

Brownian motion is the archetypal model for random transport processes in science and engineering. Brownian motion displays neither wild fluctuations (the “Noah effect”), nor long-range correlations (the “Joseph effect”). The quintessential model for processes displaying the Noah effect is Lévy motion, the quintessential model for processes displaying the Joseph effect is fractional Brownian motion, and the prototypical model for processes displaying both the Noah and Joseph effects is fractional Lévy motion. In this paper we review these four random-motion models–henceforth termed “fractional motions” –via a unified physical setting that is based on Langevin’s equation, the Einstein–Smoluchowski paradigm, and stochastic scaling limits. The unified setting explains the universal macroscopic emergence of fractional motions, and predicts–according to microscopic-level details–which of the four fractional motions will emerge on the macroscopic level. The statistical properties of fractional motions are classified and parametrized by two exponents—a “Noah exponent” governing their fluctuations, and a “Joseph exponent” governing their dispersions and correlations. This self-contained review provides a concise and cohesive introduction to fractional motions.     

Lévy flights in inhomogeneous environments

We study the long time asymptotics of probability density functions (pdfs) of Lévy flights in confining potentials that originate from inhomogeneities of the environment in which the flights take place. To this end we employ two model patterns of dynamical behavior: Langevin-driven and (Lévy-Schrödinger) semigroup-driven dynamics. It turns out that the semigroup modeling provides much stronger confining properties than the standard Langevin one. For computational and visualization purposes our observations are exemplified for the Cauchy driver and its response to external polynomial potentials (referring to Lévy oscillators), with respect to both dynamical mechanisms. We discuss the links of the Lévy semigroup motion scenario with that of random searches in spatially inhomogeneous media.     

Heavy-tailed targets and (ab)normal asymptotics in diffusive motion

We show that, under suitable confinement conditions, the ordinary Fokker-Planck equation may generate non-Gaussian heavy-tailed probability density functions (pdfs) (like, for example, Cauchy or more general Lévy stable distributions) in its long-time asymptotics. In fact, all heavy-tailed pdfs known in the literature can be obtained this way. For the underlying diffusion-type processes, our main focus is on their transient regimes and specifically the crossover features, when an initially infinite number of pdf moments decreases to a few or none at all. The time dependence of the variance (if in existence), ∼t^γ with 0<γ<2, may in principle be interpreted as a signature of subdiffusive, normal diffusive or superdiffusive behavior under confining conditions; the exponent γ is generically well defined in substantial periods of time. However, there is no indication of any universal time rate hierarchy, due to a proper choice of the driver and/or external potential.     

Generalized Langevin equation driven by Lévy processes: A probabilistic, numerical and time series based approach

Lévy processes have been widely used to model a large variety of stochastic processes under anomalous diffusion. In this note we show that Lévy processes play an important role in the study of the Generalized Langevin Equation (GLE). The solution to the GLE is proposed using stochastic integration in the sense of convergence in probability. Properties of the solution processes are obtained and numerical methods for stochastic integration are developed and applied to examples. Time series methods are applied to obtain estimation formulas for parameters related to the solution process. A Monte Carlo simulation study shows the estimation of the memory function parameter. We also estimate the stability index parameter when the noise is a Lévy process.     

Lévy-Driven Langevin Systems: Targeted Stochasticity

Langevin dynamics driven by random Wiener noise (white noise), and the resulting Fokker–Planck equation and Boltzmann equilibria are fundamental to the understanding of transport and relaxation. However, there is experimental and theoretical evidence that the use of the Gaussian Wiener noise as an underlying source of randomness in continuous time systems may not always be appropriate or justified. Rather, models incorporating general Lévy noises, should be adopted. In this work we study Langevin systems driven by general Lévy, rather than Wiener, noises. Various issues are addressed, including: (i) the evolution of the probability density function of the system's state; (ii) the system's steady state behavior; and, (iii) the attainability of equilibria of the Boltzmann type. Moreover, the issue of reverse engineering is introduced and investigated. Namely: how to design a Langevin system, subject to a given Lévy noise, that would yield a pre-specified target steady state behavior. Results are complemented with a multitude of examples of Lévy driven Langevin systems.     

Lévy models and scale invariance properties applied to Geophysics

In this work we study scale invariant functions and stochastic Lévy models and we apply them to geophysical data. We show that a pattern arises from the scale invariance property and Lévy flight models that may be used to estimate parameters related to some major event–major earthquakes.     

Modelling price dynamics: A hybrid truncated Lévy Flight–GARCH approach

ARCH and GARCH stochastic processes are widely used in finance and are generally accepted as good approximations when modelling the price dynamics with Gaussian conditional probability. It can be seen that certain aspects of the empirical data for asset price changes seems to more closely fit a Truncated Lévy Flight or GARCH model, but each with individual shortfalls. In this paper therefore, we combine the GARCH process with a conditional truncated Lévy distribution in order to build a hybrid model that most notably describes the price change and associated volatility probability density distributions and scaling behaviour over different time horizons.     

Stochastic bifurcation for a tumor–immune system with symmetric Lévy noise

In this paper, we investigate stochastic bifurcation for a tumor–immune system in the presence of a symmetric non-Gaussian Lévy noise. Stationary probability density functions will be numerically obtained to define stochastic bifurcation via the criteria of its qualitative change, and bifurcation diagram at parameter plane is presented to illustrate the bifurcation analysis versus noise intensity and stability index. The effects of both noise intensity and stability index on the average tumor population are also analyzed by simulation calculation. We find that stochastic dynamics induced by Gaussian and non-Gaussian Lévy noises are quite different.