Photoluminescence decay of silicon nanocrystals and Lévy stable distributions

We incorporate the tools of Lévy processes and distributions to describe the photoluminescence of silicon nanocrystals. The method relies on two novel features: first we use exact forms of one-sided Lévy distributions to get an excellent reproduction of experimental data. Then we show that the dynamics leading to photoluminescence decay can be modelled in terms of fractional Fokker–Planck equation.     

Bose-Einstein correlations for Lévy stable source distributions

The peak of the two-particle Bose-Einstein correlation functions has a very interesting structure. It is often believed to have a multivariate Gaussian form. We show here that for the class of stable distributions, characterized by the index of stability , the peak has a stretched exponential shape. The Gaussian form corresponds then to the special case of . We give examples for the Bose-Einstein correlation functions for univariate as well as multivariate stable distributions, and we check the model against two-particle correlation data.Received: 19 November 2003, Revised: 27 April 2004, Published online: 23 June 2004     

Exact and explicit probability densities for one-sided Lévy stable distributions

We study functions gα(x) which are one-sided, heavy-tailed Lévy stable probability distributions of index α, 0<α<1, of fundamental importance in random systems, for anomalous diffusion and fractional kinetics. We furnish exact and explicit expressions for gα(x), 0 ≤ x<∞, for all α=l/k<1, with k and l positive integers. We reproduce all the known results given by k ≤ 4 and present many new exact solutions for k > 4, all expressed in terms of known functions. This will allow a "fine-tuning" of α in order to adapt gα(x) to a given experimental situation.     

A relationship between asymmetric Lévy-stable distributions and the dielectric susceptibility

This paper, as a complement to the work of Montroll and Bendler, is concerned with the Lévy-stable distributions and their connection to the dielectric response of dipolar materials in the frequency domain. The necessary and sufficient condition for this connection is found. The presented probabilistic analysis is based on the mathematically correct representation of the meaning of the relaxation function of a system of dipoles and shows why the same form of a distribution of relaxation rates, namely, the completely asymmetric Lévy-stable distribution, should apply in all different relaxing systems. This is in contrast to the traditional definition of the relaxation function, expressed as a weighted average of exponential relaxation functions, which does not explain the universality of the dielectric relaxation law. It also follows from the present considerations that not only is the imaginary part () of the dielectric susceptibility directly related to the Lévy-stable distribution (as was found by Montroll and Bendler), but so is the real part(). As a consequence the relation()/()=cot(n/2) for> _p and 0<n<1, implied by experimental results, is obtained.     

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.     

Lévy model for interstellar scintillations

Observations of radio signals from distant pulsars provide a valuable tool for investigation of interstellar turbulence. The time shapes of the signals are the result of pulse broadening by the fluctuating electron density in the interstellar medium. While the scaling of the shapes with the signal frequency is well understood, the observed anomalous scaling with respect to the pulsar distance has remained a puzzle for more than 30 years. We propose a new model for interstellar electron density fluctuations, which explains the observed scaling relations. We suggest that these fluctuations obey Lévy statistics rather than Gaussian statistics, as assumed in previous treatments of interstellar scintillations.     

Fractional transport equations for Lévy stable processes

The influence functional method of Feynman and Vernon is used to obtain a quantum master equation for a system subjected to a Lévy stable random force. The corresponding classical transport equations for the Wigner function are then derived, both in the limits of weak and strong friction. These are fractional extensions of the Klein-Kramers and the Smoluchowski equations. It is shown that the fractional character acquired by the position in the Smoluchowski equation follows from the fractional character of the momentum in the Klein-Kramers equation. Connections among fractional transport equations recently proposed are clarified.     

Verhulst model with Lévy white noise excitation

The transient dynamics of the Verhulst model perturbed by arbitrary non-Gaussian white noise is investigated. Based on the infinitely divisible distribution of the Lévy process we study the nonlinear relaxation of the population density for three cases of white non-Gaussian noise: (i) shot noise; (ii) noise with a probability density of increments expressed in terms of Gamma function; and (iii) Cauchy stable noise. We obtain exact results for the probability distribution of the population density in all cases, and for Cauchy stable noise the exact expression of the nonlinear relaxation time is derived. Moreover starting from an initial delta function distribution, we find a transition induced by the multiplicative Lévy noise, from a trimodal probability distribution to a bimodal probability distribution in asymptotics. Finally we find a nonmonotonic behavior of the nonlinear relaxation time as a function of the Cauchy stable noise intensity.     

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.     

Langevin Picture of Lévy Walks and Their Extensions

In this paper we derive Langevin picture of Lévy walks. Applying recent advances in the theory of coupled continuous time random walks we find a limiting process of the properly scaled Lévy walk. Next, we introduce extensions of Levy walks, in which jump sizes are some functions of waiting times. We prove that under proper scaling conditions, such generalized Lévy walks converge in distribution to the appropriate limiting processes. We also derive the corresponding fractional diffusion equations and investigate behavior of the mean square displacements of the limiting processes, showing that different coupling functions lead to various types of anomalous diffusion.     

Lévy flights and Lévy-Schrödinger semigroups

We analyze two different confining mechanisms for Lévy flights in the presence of external potentials. One of them is due to a conservative force in the corresponding Langevin equation. Another is implemented by Lévy-Schrödinger semigroups which induce so-called topological Lévy processes (Lévy flights with locally modified jump rates in the master equation). Given a stationary probability function (pdf) associated with the Langevin-based fractional Fokker-Planck equation, we demonstrate that generically there exists a topological Lévy process with the same invariant pdf and in reverse.     

Universal record statistics of random walks and Lévy flights

It is shown that statistics of records for time series generated by random walks are independent of the details of the jump distribution, as long as the latter is continuous and symmetric. In N steps, the mean of the record distribution grows as the sqrt[4N/pi] while the standard deviation grows as sqrt[(2-4/pi)N], so the distribution is non-self-averaging. The mean shortest and longest duration records grow as sqrt[N/pi] and 0.626 508...N, respectively. The case of a discrete random walker is also studied, and similar asymptotic behavior is found.     

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.     

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   

Theory of decoherence due to scattering events and Lévy processes

A general connection between the characteristic function of a Lévy process and loss of coherence of the statistical operator describing the center of mass degrees of freedom of a quantum system interacting through momentum transfer events with an environment is established. The relationship with microphysical models and recent experiments is considered, focusing on the recently observed transition between a dynamics described by a compound Poisson process and a Gaussian process.