Cauchy flights in confining potentials

We analyze confining mechanisms for Lévy flights evolving under an influence of external potentials. Given a stationary probability density function (pdf), we address the reverse engineering problem: design a jump-type stochastic process whose target pdf (eventually asymptotic) equals the preselected one. To this end, dynamically distinct jump-type processes can be employed. We demonstrate that one “targeted stochasticity” scenario involves Langevin systems with a symmetric stable noise. Another derives from the Lévy-Schrödinger semigroup dynamics (closely linked with topologically induced super-diffusions), which has no standard Langevin representation. For computational and visualization purposes, the Cauchy driver is employed to exemplify our considerations.     

Stretched-Gaussian asymptotics of the truncated Lévy flights for the diffusion in nonhomogeneous media

The Lévy, jumping process, defined in terms of the jumping size distribution and the waiting time distribution, is considered. The jumping rate depends on the process value. The fractional diffusion equation, which contains the variable diffusion coefficient, is solved in the diffusion limit. That solution resolves itself to the stretched Gaussian when the order parameter μ→2. The truncation of the Lévy flights, in the exponential and power-law form, is introduced and the corresponding random walk process is simulated by the Monte Carlo method. The stretched Gaussian tails are found in both cases. The time which is needed to reach the limiting distribution strongly depends on the jumping rate parameter. When the cutoff function falls slowly, the tail of the distribution appears to be algebraic.     

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   

On the first passage time and leapover properties of Lévy motions

We investigate two coupled properties of Lévy stable random motions: the first passage times (FPTs) and the first passage leapovers (FPLs). While, in general, the FPT problem has been studied quite extensively, the FPL problem has hardly attracted any attention. Considering a particle that starts at the origin and performs random jumps with independent increments chosen from a Lévy stable probability law λ_α,β(x), the FPT measures how long it takes the particle to arrive at or cross a target. The FPL addresses a different question: given that the first passage jump crosses the target, then how far does it get beyond the target? These two properties are investigated for three subclasses of Lévy stable motions: (i) symmetric Lévy motions characterized by Lévy index α(0<α<2) and skewness parameter β=0, (ii) one-sided Lévy motions with 0<α<1, β=1, and (iii) two-sided skewed Lévy motions, the extreme case, 1<α<2, β=−1.     

Can spontaneous cell movements be modelled as Lévy walks?

Spontaneous cell movement is a random motion that takes place in the absence of external guiding stimuli. The spontaneous movements of HaCaT and NHDF cells (cells of the epidermis) are well represented as continuous Markovian processes driven by multiplicative noise [D. Selmeczi, S. Mosler, P.H. Hagedorn, N.B. Larsen, H. Flyvbjerg, Biophysical Journal 89 (2005) 912]. Model components are, however, ad hoc as they are inspired by fits to experimental data. As a consequence, model agreement with experimental data does not add much to our understanding of spontaneous movements of these cells beyond demonstrating that they can be modelled phenomenologically. Here it is noted that a slight re-parameterization and re-interpretation of the driving noise leads to the model of Lubashevsky et al. (2009) [I. Lubashevsky, R. Friedrich, A. Heuer, Physical Review E 79 (2009) 011110] that realises Lévy walks as Markovian stochastic processes. This brings forth new biological insight as Lévy walks are advantageous when searching in the absence of external stimuli and without knowledge of the target distribution, as may be the case with cells of the epidermis that form new tissue by locating and then attaching on to one another. The Hänggi-Klimontovich interpretation of the driving noise in the model of Lubashevsky et al. (2009) and Cauchy distributions of predicted velocities do, however, appear problematic, even unphysical. Here it is shown that these are perceived rather than actual difficulties. Intermittent stop-start motions of the kind displayed by some cells and protozoan are found to underlie the formulation of the model of Lubashevsky et al. (2009) and the velocities of starved Dictyostelium discoideum (a unicellular organism) are found to be Cauchy distributed to a good approximation. It is therefore suggested that the model of Lubashevsky et al. (2009) can describe the spontaneous movements of some cells, and that some cells have spontaneous movement patterns that can be approximated by Lévy walks, as first proposed by Schuster and Levandowsky (1996) [F.L. Schuster, M. Levandowsky, Journal of Eukaryotic Microbiology 43 (1996) 150].     

Adaptive Lévy walks can outperform composite Brownian walks in non-destructive random searching scenarios

Recently it has been found that composite Brownian walk searches are more efficient than any Lévy walk when searching is non-destructive and when the Lévy walks are not responsive to conditions found in the search. Here a new class of adaptive Lévy walk searches is presented that encompasses composite Brownian walks as a special case. In these new models, bouts of Lévy walk searching alternate with bouts of more intensive Brownian walk searching. Switching from extensive to intensive searching is prompted by the detection of a target. And here, switching back to extensive searching arises if a target is not located after travelling a distance equal to the ‘giving-up distance’. It is found that adaptive Lévy walks outperform composite Brownian walks when searching for sparsely distributed resources. Consequently there is stronger selection pressures for Lévy processes when resources are sparsely distributed within unpredictable environments. The findings reconcile Lévy walk search theory with the ubiquity of two modes of searching by predators and with their switching search mode immediately after finding a prey.     

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.     

Lévy-type complex diameter modulation in semiconductor nanowire growth

We have observed a non-periodic diameter modulation in SiC nanowire growth. The modulation was tentatively attributed to a stick-slip motion in a self-organized vapor-liquid-solid (VLS) growth, and was characterized by methods of time-series analysis. From the power-law decay behavior of a fat tail of the probability distribution of increments with the exponent 2.1±0.2, the modulation was interpreted as a Lévy flight. The degree of determinism is also discussed.     

Lévy processes and Schrödinger equation

We analyze the extension of the well known relation between Brownian motion and the Schrödinger equation to the family of the Lévy processes. We consider a Lévy-Schrödinger equation where the usual kinetic energy operator-the Laplacian-is generalized by means of a selfadjoint, pseudodifferential operator whose symbol is the logarithmic characteristic of an infinitely divisible law. The Lévy-Khintchin formula shows then how to write down this operator in an integro-differential form. When the underlying Lévy process is stable we recover as a particular case the fractional Schrödinger equation. A few examples are finally given and we find that there are physically relevant models-such as a form of the relativistic Schrödinger equation-that are in the domain of the non stable Lévy-Schrödinger equations.     

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.     

A Cautionary note on modeling with fractional Lévy flights

Temporal scaling and infinite variance are two stylized features often seen together in times series of complex systems. We find that because of their infinite moments samples from fractional Lévy flights produce bi-linear scaling functions which may be incorrectly attributed as evidence of multifractality. We argue that it is unnecessary to consider truncated fractional Lévy flights which are inherently problematic.     

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.     

Resonant regions of Josephson junction equation in case of large damping

The dynamics of Josephson junction equation in case of damping α>2 is investigated numerically. In this case the second-order system can be asymptotically reduced in the large to a one-dimensional circle map. We study the parametric dependence of the resonances of this system and plot the resonant regions in two-dimensional parameter space. The periodic variation of the widths of harmonic regions with increase of the periodic driving force is observed. In the limit of infinite damping, we study a first order system through suitable re-scaling and the same property is observed. We conjecture this may caused by the competition between the periodic potential and the periodic external driving in these systems.     

Cascade-like and scaling behavior of wind velocity increments in the atmospheric surface layer

By using a large amount of data collected in the atmospheric surface layer, we analyze the probability density functions (PDFs), the probability of return and the moments of wind velocity increments. Results show that the PDFs change from the non-Gaussian long-tailed distributions to Gaussian with the increase of time scales. This is similar to what has been observed and interpreted as an indication of cascade in the fully developed homogeneous and isotropic turbulence. Besides, both the probability of return and the moments are found to be scaling with time scales. We then compare above results with the truncated Lévy flights and the log-normal PDF model. It is found that although both models show the cascade-like behavior in the PDFs and the scaling behavior in the probability of return and the moments under some conditions, they are not good enough for quantitatively describing the random process of wind velocity increments.     

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.     

Exact solutions for a diffusion equation with a nonlinear external force

This work is devoted to investigate the solutions of the one-dimensional diffusion equation by taking the nonlinear external force F(x,t;ρ)=−k(t)x+K/x+κx|x|^α−1η[ρ(x,t)] into account. Our investigation is first performed by considering the case α=0 and η=1, which results in a Burgers like equation with a spatial and time dependent external force. After, we consider the case α≠0 and η=α+1 and show that the solution found may be expressed in terms of the q-exponential functions present in the Tsallis formalism. In addition, we also discuss the stationary solution for α and η arbitraries.     

Optimal transport in a ratchet coupled to a modulated environment: The role of Levy walks

We consider a Brownian particle in a ratchet potential coupled to a modulated environment and subjected to an external oscillating force. The modulated environment is modelled by a finite number N of uncoupled harmonic oscillators. Superdiffusive motion and Levy walks (anomalous random walks) are observed for any N and for low values of the external amplitude F. The coexistence of left and right running states enhances the power α from the time dependence of the mean square displacement (MSD). It is shown that α is twice the average of the power of the separated left and right MSDs. Normal random walks are obtained by increasing F. We show that the maximal mobility of particles along the periodic structure occurs just before superdiffusive motion disappears and Levy walks are transformed into normal random walks.     

Stochastic resonance, reverse-resonance, and resonant activation induced by a multi-state noise

In this paper, we investigate the periodic response for a linear system driven by a multiplicative multi-state noise (which is composed of the multiplication of two dichotomous noises) to an input temporal oscillatory signal, and the escape of Brownian particles over the fluctuating potential barrier for a system with a piece-wise linear potential and driven by an additive multi-state noise (which is also composed of the multiplication of two dichotomous noises). For the first system, we get the stochastic resonance phenomenon for the amplitude of the periodic response vs. the two dichotomous noise strengths, and the phenomenon of reverse-resonance for the amplitude of the periodic response vs. k, which represents the asymmetry degree of the dichotomous noises. For the second system, we obtain the resonant activation phenomenon, for which the mean first passage time of the Brownian particles over the fluctuating potential barrier shows a minimum as the function of the transition rates of the multi-state noise.