Evy Distribution Research Articles

Wave transmission and its universal fluctuations in one-dimensional systems with Lévy-like disorder: Schrödinger, Klein-Gordon, and Dirac equations.

We investigate the propagation of waves in one-dimensional systems with Lévy-type disorder. We perform a complete analysis of nonrelativistic and relativistic wave transmission submitted to potential barriers whose width, separation, or both follow Lévy distributions characterized by an exponent 0<α<1. For the first two cases, where one of the parameters is fixed, nonrelativistic and relativistic waves present anomalous localization, 〈T〉∝L^{-α}. However, for the latter case, in which both parameters follow a Lévy distribution, nonrelativistic and relativistic waves present a transition between anomalous and standard localization as the incidence energy increases relative to the barrier height. Moreover, we obtain the localization diagram delimiting anomalous and standard localization regimes, in terms of incidence angle and energy. Finally, we verify that transmission fluctuations, characterized by its standard deviation, are universal, independent of barrier architecture, wave equationtype, incidence energy, and angle, further extending earlier studies on electronic localization.

Breaking of time-reversal symmetry for a particle in a parabolic potential that is subjected to Lévy noise: Theory and an application to solar flare data.

The noise in nonequilibrium systems commonly contains more outliers as compared to equilibrium systems and is often best described with a Lévy distribution. Many systems in which there are fluctuations around a steady-state throughput can be modeled as a Lévy-noise-subjected particle in a parabolic potential. We consider an overdamped particle in a parabolic potential that is subjected to noise. Microscopic reversibility and time-reversal symmetry apply if the particle is subject to Gaussian distributed noise, but are violated if the noise is Lévy. A parameter to detect this violation is formulated. We, furthermore, develop an understanding for how the time-reversal asymmetry depends on the time Δt between the sample points and on the stability index, α, of the Lévy noise. With solar flare data it is shown how the time-reversal asymmetry parameter of a signal can be used to obtain the α of the underlying noise.

Eigenstate entanglement between quantum chaotic subsystems: Universal transitions and power laws in the entanglement spectrum

We derive universal entanglement entropy and Schmidt eigenvalue behaviors for the eigenstates of two quantum chaotic systems coupled with a weak interaction. The progression from a lack of entanglement in the noninteracting limit to the entanglement expected of fully randomized states in the opposite limit is governed by the single scaling transition parameter, $\Lambda$. The behaviors apply equally well to few- and many-body systems, e.g.\ interacting particles in quantum dots, spin chains, coupled quantum maps, and Floquet systems as long as their subsystems are quantum chaotic, and not localized in some manner. To calculate the generalized moments of the Schmidt eigenvalues in the perturbative regime, a regularized theory is applied, whose leading order behaviors depend on $\sqrt{\Lambda}$. The marginal case of the $1/2$ moment, which is related to the distance to closest maximally entangled state, is an exception having a $\sqrt{\Lambda}\ln \Lambda$ leading order and a logarithmic dependence on subsystem size. A recursive embedding of the regularized perturbation theory gives a simple exponential behavior for the von Neumann entropy and the Havrda-Charv{\' a}t-Tsallis entropies for increasing interaction strength, demonstrating a universal transition to nearly maximal entanglement. Moreover, the full probability densities of the Schmidt eigenvalues, i.e.\ the entanglement spectrum, show a transition from power laws and L\'evy distribution in the weakly interacting regime to random matrix results for the strongly interacting regime. The predicted behaviors are tested on a pair of weakly interacting kicked rotors, which follow the universal behaviors extremely well.

Wave scattering from two-dimensional self-affine Dirichlet and Neumann surfaces and its application to the retrieval of self-affine parameters

Wave scattering from two-dimensional self-affine Dirichlet and Neumann surfaces is studied for the purpose of using the intensity scattered from them to obtain the Hurst exponent and topothesy that characterize the self-affine roughness. By the use of the Kirchhoff approximation a closed form mathematical expression for the angular dependence of the mean differential reflection coefficient is derived under the assumption that the surface is illuminated by a plane incident wave. It is shown that this quantity can be expressed in terms of the isotropic, bivariate ($\alpha$-stable) L\'evy distribution of a stability parameter that is two times the Hurst exponent of the underlying surface. Features of the expression for the mean differential reflection coefficient are discussed, and its predictions compare favorably over large regions of parameter space to results obtained from rigorous computer simulations based on equations of scattering theory. It is demonstrated how the Hurst exponent and the topothesy of the self-affine surface can be inferred from scattering data it produces. Finally several possible scattering configurations are discussed that allow for an efficient extraction of these self-affine parameters.

Centrality Dependent Lévy-Stable Two-Pion Bose-Einstein Correlations in \( {\sqrt{s_{NN}}} \) = 200 GeV Au+Au Collisions at the PHENIX Experiment

Investigation of femtoscopic correlation functions in relativistic heavy ion reactions is an important tool to access the space-time structure of particle production in the sQGP. The shape of the correlation functions is often assumed to be Gaussian, but a detailed analysis reveals that the statistically correct assumption could be the so-called L\'evy distribution. The detailed analysis of correlation functions in various systems may shed light on the location of the critical endpoint on the QCD phase diagram. It could also reveal if there is partially coherent pion production or could indicate the possible in-medium mass modification of the $\eta'$ meson due to the (partial) restoration of the $U_A(1)$ axial symmetry. In this paper we present the status of the centrality dependent measurements of two-pion L\'evy Bose-Einstein correlation functions in Au+Au collisions at PHENIX.

Coherent wave transmission in quasi-one-dimensional systems with Lévy disorder.

We study the random fluctuations of the transmission in disordered quasi-one-dimensional systems such as disordered waveguides and/or quantum wires whose random configurations of disorder are characterized by density distributions with a long tail known as Lévy distributions. The presence of Lévy disorder leads to large fluctuations of the transmission and anomalous localization, in relation to the standard exponential localization (Anderson localization). We calculate the complete distribution of the transmission fluctuations for a different number of transmission channels in the presence and absence of time-reversal symmetry. Significant differences in the transmission statistics between disordered systems with Anderson and anomalous localizations are revealed. The theoretical predictions are independently confirmed by tight-binding numerical simulations.

Extreme-value statistics of intensities in a cw-pumped random fiber laser

We report on the extreme-value statistics of output intensities in a one-dimensional cw-pumped erbium-doped random fiber laser, with a strongly scattering disordered medium consisting of randomly spaced Bragg gratings. The experimental findings from the analysis of a large number of emission spectra are well described by the Gumbel distribution below and above the laser threshold, whereas the Fr\'echet distribution, typical of strongly fluctuating extreme events with heavy power-law probability tails, provides a nice support to the data near the threshold. We establish a close connection, relying on theoretical arguments, between the reported extreme-value statistics and the shifts in the statistics of intensity fluctuations, from the Gaussian to the L\'evy distribution at the threshold and back to the Gaussian well above threshold.

Efficient nonparametric inference for discretely observed compound Poisson processes

A compound Poisson process whose parameters are all unknown is observed at finitely many equispaced times. Nonparametric estimators of the jump and L\'evy distributions are proposed and functional central limit theorems using the uniform norm are proved for both under mild conditions. The limiting Gaussian processes are identified and efficiency of the estimators is established. Kernel estimators for the mass function, the intensity and the drift are also proposed, their asymptotic properties including efficiency are analysed, and joint asymptotic normality is shown. Inference tools such as confidence regions and tests are briefly discussed.

Rescuing the nonjet (NJ) azimuth quadrupole from the flow narrative

According to the flow narrative commonly applied to high-energy nuclear collisions a cylindrical-quadrupole component of 1D azimuth angular correlations is conventionally denoted by quantity $v_2$ and interpreted to represent elliptic flow. Jet angular correlations may also contribute to $v_2$ data as "nonflow" depending on the method used to calculate $v_2$, but 2D graphical methods are available to insure accurate separation. The nonjet (NJ) quadrupole has various properties inconsistent with a flow interpretation, including the observation that NJ quadrupole centrality variation in A-A collisions has no relation to strongly-varying jet modification ("jet quenching") in those collisions commonly attributed to jet interaction with a flowing dense medium. In this presentation I describe isolation of quadrupole spectra from pt-differential $v_2(p_t)$ data from the RHIC and LHC. I demonstrate that quadrupole spectra have characteristics very different from the single-particle spectra for most hadrons, that quadrupole spectra indicate a common boosted hadron source for a small minority of hadrons that "carry" the NJ quadrupole structure, that the narrow source-boost distribution is characteristic of an expanding thin cylindrical shell (strongly contradicting hydro descriptions), and that in the boost frame a single universal quadrupole spectrum (L\'evy distribution) on transverse mass $m_t$ accurately describes data for several hadron species scaled according to their statistical-model abundances. The quadrupole spectrum shape changes very little from RHIC to LHC energies. Taken in combination those characteristics strongly suggest a unique {\em nonflow} (and nonjet) QCD mechanism for the NJ quadrupole conventionally represented by $v_2$.

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

We study the spatial distribution of minority carriers 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. The experiment provides a realization of the "L\'evy flight" random walk of holes, in which the L\'evy distribution index gamma is controlled by the temperature. We show that the variation \gamma(T) is close to that predicted earlier on the basis of the assumed quasi-equilibrium (van Roosbroek-Shockley) intrinsic emission spectrum, \gamma=1-\Delta /kT, where \Delta(T) is the Urbach tailing parameter of the absorption spectra. The decreasing \gamma at lower temperatures results in a giant enhancement in the spread of holes -- over distances exceeding 1 cm from the region of photo-excitation.

Recognition of stable distribution with Lévy indexαclose to 2

We address the problem of recognizing α-stable Lévy distribution with Lévy index close to 2 from experimental data. We are interested in the case when the sample size of available data is not large, thus the power law asymptotics of the distribution is not clearly detectable, and the shape of the empirical probability density function is close to a Gaussian. We propose a testing procedure combining a simple visual test based on empirical fourth moment with the Anderson-Darling and Jarque-Bera statistical tests and we check the efficiency of the method on simulated data. Furthermore, we apply our method to the analysis of turbulent plasma density and potential fluctuations measured in the stellarator-type fusion device and demonstrate that the phenomenon of the L-H transition from low confinement, L mode, to a high confinement, H mode, which occurs in this device is accompanied by the transition from Lévy to Gaussian fluctuation statistics.

Multiplicative Lévy noise in bistable systems

Stochastic motion in a bistable, periodically modulated potential is discussed. The system is stimulated by a white noise increments of which have a symmetric stable L\'evy distribution. The noise is multiplicative: its intensity depends on the process variable like |x|^{-\theta}. The Stratonovich and It\^o interpretations of the stochastic integral are taken into account. The mean first passage time is calculated as a function of \theta for different values of the stability index \alpha and size of the barrier. Dependence of the output amplitude on the noise intensity reveals a pattern typical for the stochastic resonance. Properties of the resonance as a function of \alpha, \theta\ and size of the barrier are discussed. Both height and position of the peak strongly depends on \theta\ and on a specific interpretation of the stochastic integral.

SqueezedK+K−correlations in high energy heavy ion collisions

The hot and dense medium formed in high energy heavy ion collisions may modify some hadronic properties. In particular, if hadron masses are shifted in-medium, it was demonstrated that this could lead to back-to-back squeezed correlations (BBC) of particle-antiparticle pairs. Although well-established theoretically, the squeezed correlations have not yet been discovered experimentally. A method has been suggested for the empirical search of this effect, which was previously illustrated for $\ensuremath{\phi}\ensuremath{\phi}$ pairs. We apply here the formalism and the suggested method to the case of ${K}^{+}{K}^{\ensuremath{-}}$ pairs, since they may be easier to identify experimentally. The time distribution of the emission process plays a crucial role in the survival of the BBC's. We analyze the cases where the emission is supposed to occur suddenly or via a Lorentzian distribution, and compare with the case of a L\'evy distribution in time. Effects of squeezing on the correlation function of identical particles are also analyzed.

On the validity of the local diffusive paradigm in turbulent plasma transport

A systematic, constructive and self-consistent procedure to quantify nonlocal, nondiffusive action at a distance in plasma turbulence is exposed and applied to turbulent heat fluxes computed from the state-of-the-art full- f, flux-driven gyrokinetic GYSELA and XGC1 codes. A striking commonality is found: heat transport below a dynamically selected mesoscale has the structure of a Lévy distribution, is strongly nonlocal, nondiffusive, scale-free, and avalanche mediated; at larger scales, we report the observation of a self-organized flow structure which we call the " E × B staircase" after its planetary analog.

Multiplicative Lévy processes: Itô versus Stratonovich interpretation

Langevin equation with a multiplicative stochastic force is considered. That force is uncorrelated, it has the Lévy distribution and the power-law intensity. The Fokker-Planck equations, which correspond both to the Itô and Stratonovich interpretation, are presented. They are solved for the case without drift and for the harmonic oscillator potential. The variance is evaluated; it is always infinite for the Itô case whereas for the Stratonovich one it can be finite and rise with time slower that linearly, which indicates subdiffusion. Analytical results are compared with numerical simulations.

Nonalgebraic length dependence of transmission through a chain of barriers with a Lévy spacing distribution

The recent realization of a ``L\'evy glass'' (a three-dimensional optical material with a L\'evy distribution of scattering lengths) has motivated us to analyze its one-dimensional analog: A linear chain of barriers with independent spacings $s$ that are L\'evy distributed: $p(s)\ensuremath{\propto}{s}^{\ensuremath{-}1\ensuremath{-}\ensuremath{\alpha}}$ for $s\ensuremath{\rightarrow}\ensuremath{\infty}$. The average spacing diverges for $0&lt;\ensuremath{\alpha}&lt;1$. A random walk along such a sparse chain is not a L\'evy walk because of the strong correlations of subsequent step sizes. We calculate all moments of conductance (or transmission), in the regime of incoherent sequential tunneling through the barriers. The average transmission from one barrier to a point at a distance $L$ scales as ${L}^{\ensuremath{-}\ensuremath{\alpha}}\text{ }\text{ln}\text{ }L$ for $0&lt;\ensuremath{\alpha}&lt;1$. The corresponding electronic shot noise has a Fano factor ($\ensuremath{\propto}$ average noise power/average conductance) that approaches 1/3 very slowly, with $1/\text{ln}\text{ }L$ corrections.

Fractional dynamics in the Lévy quantum kicked rotor

We investigate the quantum kicked rotor in resonance subjected to momentum measurements with a Lévy waiting-time distribution. We find that the system has a sub-ballistic behavior. We obtain an analytical expression for the exponent of the power law of the variance as a function of the characteristic parameter of the Lévy distribution. We also connect the anomalous diffusion found with a fractional dynamics.

Mapping of the Roughness Exponent for the Fuse Model for Fracture

The roughness exponent for fracture surfaces in the fuse model has been thought to be universal for narrow threshold distributions and has been important in the numerical studies of fracture roughness. We show that the fuse model gives a disorder dependent roughness exponent for narrow disorders when the lattice is influencing the fracture growth. When the influence of the lattice disappears, the local roughness exponent approaches zeta(local)=0.65+/-0.03 for distribution with a tail toward small thresholds, but with large jumps in the profiles giving corrections to scaling on small scales. For very broad disorders the distribution of jumps becomes a Lévy distribution and the Lévy characteristics contribute to the local roughness exponent.

Accuracy of roughness exponent measurement methods

We test methods for measuring and characterizing rough profiles with emphasis on measurements of the self-affine roughness exponent, and describe a simple test to separate between roughness exponents originating from long range correlations in the signs of the profile, and roughness exponents originating from Lévy distributions of jumps. Based on tests on profiles with known roughness exponents we find that the power spectrum density analysis and the averaged wavelet coefficients method give the best estimates for roughness exponents in the range 0.1-0.9. The error bars are found to be less than 0.03 for profile lengths larger than 256, and there is no systematic bias in the estimates. We present quantitative estimates of the error bars and the systematic error and their dependence on the value of the roughness exponent and the profile length. We also quantify how power-law noise can modify the measured roughness exponent for measurement methods different from the power spectrum density analysis and the second order correlation function method.

Crossover between Lévy and Gaussian regimes in first-passage processes

We propose an approach to the problem of the first-passage time. Our method is applicable not only to the Wiener process but also to the non-Gaussian Lévy flights or to more complicated stochastic processes whose distributions are stable. To show the usefulness of the method, we particularly focus on the first-passage time problems in the truncated Lévy flights (the so-called KoBoL processes from Koponen, Boyarchenko, and Levendorskii), in which the arbitrarily large tail of the Lévy distribution is cut off. We find that the asymptotic scaling law of the first-passage time t distribution changes from t(-(alpha+1)/alpha)-law (non-Gaussian Lévy regime) to t(-32)-law (Gaussian regime) at the crossover point. This result means that an ultraslow convergence from the non-Gaussian Lévy regime to the Gaussian regime is observed not only in the distribution of the real time step for the truncated Lévy flight but also in the first-passage time distribution of the flight. The nature of the crossover in the scaling laws and the scaling relation on the crossover point with respect to the effective cutoff length of the Lévy distribution are discussed.