Lévy walks for turbulence: A numerical study

We present a numerical study of enhanced diffusion, for which the mean-squared displacement follows asymptotically r ²(t) t , > 1. We simulate continuous time random walks with waiting-time distributions which couple the spatial and temporal parameters; this gives rise to Lévy-walks. Our results confirm the theoretically predicted long-time behavior and demonstrate its temporal regime of validity. Furthermore, the simulations document the appearance of (parameter-dependent) transitions between regular and enhanced diffusion regimes.     

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.     

Discrete random walk models for symmetric Lévy–Feller diffusion processes

We propose a variety of models of random walk, discrete in space and time, suitable for simulating stable random variables of arbitrary index α (0<α⩽2), in the symmetric case. We show that by properly scaled transition to vanishing space and time steps our random walk models converge to the corresponding continuous Markovian stochastic processes which we refer to as Lévy–Feller diffusion processes.     

Lévy flights in the Landau-Teller model of molecular collisions

We consider the Landau-Teller model, which is a prototype for the exchanges of energy, in molecular collisions, between internal degrees of freedom and those of the center of mass. We show that the statistics of the energy exchanges computed through the dynamics over a finite time is of the Lévy type for high enough frequencies of the internal motions, while it reduces to the familiar Gaussian one in the limit of low frequencies. The relevance for the definition of the times of relaxation to equilibrium is also pointed out.     

Multiplicative Lévy noise in bistable systems

Stochastic motion in a bistable, periodically modulated potential is discussed. Thesystem is stimulated by a white noise increments of which have a symmetric stable Lévydistribution. The noise is multiplicative: its intensity depends on the process variablelike |x|^?θ . The Stratonovich and Itôinterpretations of the stochastic integral are taken into account. The mean first passagetime is calculated as a function of θ for different values of thestability index α and size of the barrier. Dependence of the outputamplitude on the noise intensity reveals a pattern typical for the stochastic resonance.Properties of the resonance as a function of α, θ andsize of the barrier are discussed. Both height and position of the peak strongly dependson θ and on a specific interpretation of the stochastic integral.     

Phase Diagram in Stored-Energy-Driven Lévy Flight

Phase diagram based on the mean square displacement (MSD) and the distribution of diffusion coefficients of the time-averaged MSD for the stored-energy-driven Lévy flight (SEDLF) is presented. In the SEDLF, a random walker cannot move while storing energy, and it jumps by the stored energy. The SEDLF shows a whole spectrum of anomalous diffusions including subdiffusion and superdiffusion, depending on the coupling parameter between storing time (trapping time) and stored energy. This stochastic process can be investigated analytically with the aid of renewal theory. Here, we consider two different renewal processes, i.e., ordinary renewal process and equilibrium renewal process, when the mean trapping time does not diverge. We analytically show the phase diagram according to the coupling parameter and the power exponent in the trapping-time distribution. In particular, we find that distributional behavior of time-averaged MSD intrinsically appears in superdiffusive as well as normal diffusive regime even when the mean trapping time does not diverge.     

Dynamical robustness of Lévy search strategies

We study the role of dynamical constraints in the general problem of finding the best statistical strategy for random searching when the targets can be detected only in the limited vicinity of the searcher. We find that the optimal search strategy depends strongly on the delay time tau during which a previously visited site becomes unavailable. We also find that the optimal search strategy is always described for large step lengths l by a power-law distribution P(l) approximately l(-mu), with 1相似文献   

Modelling čerenkov radiation of relativistic particles in crystals

From the aspects of classical mechanics and electrodynamics, an analysis has been performed of the possible influence of the kind of charged particle trajectory on the erenkov radiation spectrum in a crystal. Results of the analytical computation are compared with the data of a computer experiment. It is shown that the influence of the particle trajectory on the erenkov radiation spectrum is insignificant in the optical frequency band. The expected effect is possible when utilizing crystals with a superlattice and by observation of radiation in the xray frequency range.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 62–67, February, 1988.The authors are grateful to S. A. Vorob'ev for supporting the research and to Yu. L. Pivovarov for stimulating and useful discussions.     

Time characteristics of Lévy flights in a steep potential well

Using the method previously developed for ordinary Brownian diffusion, we derive a new formula to calculate the correlation time of stationary Lévy flights in a steep potential well. For the symmetric quartic potential, we obtain the exact expression of the correlation time of steady-state Lévy flights with index α = 1. The correlation time of stationary Lévy flights decreases with an increasing noise intensity and steepness of potential well.     

Lévy Flights in a Steep Potential Well

Lévy flights in steeper than harmonic potentials have been shown to exhibit finite variance and a critical time at which a bifurcation from an initial monomodal to a terminal bimodal distribution occurs (Chechkin et al., Phys. Rev. E 67:010102(R) (2003)). In this paper, we present a detailed study of Lévy flights in potentials of the type U(x)∝|x| ^c with c>2. Apart from the bifurcation into bimodality, we find the interesting result that for c>4 a trimodal transient exists due to the temporal overlap between the decay of the central peak around the initial δ-condition and the building up of the two emerging side-peaks, which are characteristic for the stationary state. Thus, for certain system parameters there exists a transient trimodal distribution of the Lévy flight. These properties of Lévy flights in external potentials of the power-law type can be represented by certain phase diagrams. We also present details about the proof of multimodality and the numerical procedures to establish the probability distribution of the process.     

Structure of Shocks in Burgers Turbulence with Lévy Noise Initial Data

We study the structure of the shocks for the inviscid Burgers equation in dimension 1 when the initial velocity is given by Lévy noise, or equivalently when the initial potential is a two-sided Lévy process ψ ₀. When ψ ₀ is abrupt in the sense of Vigon or has bounded variation with lim?sup_|h|↓0 h ^?2 ψ ₀(h)=∞, we prove that the set of points with zero velocity is regenerative, and that in the latter case this set is equal to the set of Lagrangian regular points, which is non-empty. When ψ ₀ is abrupt we show that the shock structure is discrete. When ψ ₀ is eroded we show that there are no rarefaction intervals.