Chaotic scattering and transmission fluctuations

We discuss the phenomenon of chaotic scattering and its application in the study of transmission of electrons in mesoscopic devices as well as the transmission of microwaves through junctions. We show that the fact that the ray optics (classical dynamics) is chaotic, implies fluctuations in the observed transmission coefficients, whose statistics is determined by the theory of random matrices. We also show how the classical distribution functions which reflect the chaotic nature of the classical dynamics, determine the dependence of the correlations observed in the fluctuating transmission coefficients on external parameters. The time domain properties of chaotic scattering systems are also examined, and are shown to depend on the chaotic nature of the classical dynamics, together with a wave mechanical enhancement in time reversal invariant systems. Finally, we study the role of absorption and discuss its effects on the transmission fluctuations and their statistics.     

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.     

Effect of electron interaction on statistics of conductance oscillations in open quantum dots: does the dephasing time saturate?

We perform self-consistent quantum transport calculations in open quantum dots taking into account the effect of electron interaction. We demonstrate that, in the regime of the ultralow temperatures 2 pi kappa BT < or = delta (delta being the mean-level spacing), the electron interaction strongly smears the conductance oscillations and thus significantly affects their statistics. Our calculations are in good quantitative agreement with the observed ultralow temperature statistics of Huibers et al. [Phys. Rev. Lett. 81, 1917 (1998)]. Our findings question a conventional interpretation of the ultralow temperature saturation of the coherence time in open dots which is based on the noninteracting theories, where the agreement with the experiment is achieved by introducing additional phenomenological channels of dephasing.     

Dead time corrections to photon counting statistics II: Quantum theory

The recent formulation of the quantum theory of photodetection, based on the quantum theory of continuous measurements, is extended to the case of a (nonideal) detector which has non-zero dead time. A general result is proven which expresses the dead time modified counting statistics in terms of the counting statistics of anassociated ideal detector. As an illustration, the dead time corrections to the counting statistics of a single-mode free field are worked out, and these corrections are shown to be identical in form to the dead time corrections for a classical optical field of constant intensity.     

Interoccurrence times in the Bak-Tang-Wiesenfeld sandpile model: a comparison with the observed statistics of solar flares

A sequence of bursts observed in an intermittent time series may be caused by a single avalanche, even though these bursts appear as distinct events when noise and/or instrument resolution impose a detection threshold. In the Bak-Tang-Wiesenfeld sandpile, the statistics of quiet times between bursts switches from Poissonian to scale invariant on raising the threshold for detecting instantaneous activity, since each zero-threshold avalanche breaks into a hierarchy of correlated bursts. Calibrating the model with the time resolution of GOES data, qualitative agreement with the interoccurrence time statistics of solar flares at different intensity thresholds is found.     

Multiscale reconstruction of time series

A new method is proposed which allows a reconstruction of time series based on higher order multiscale statistics given by a hierarchical process. This method is able to model the time series not only on a specific scale but for a range of scales. It is possible to generate complete new time series, or to model the next steps for a given sequence of data. The method itself is based on the joint probability density which can be extracted directly from given data, thus no estimation of parameters is necessary. The results of this approach are shown for a real world dataset, namely for turbulence. The unconditional and conditional probability densities of the original and reconstructed time series are compared and the ability to reproduce both is demonstrated. Therefore in the case of Markov properties the method proposed here is able to generate artificial time series with correct n-point statistics.     

The interaction of an atom with a sub-Poissonian single field mode

The resonant interaction of a single atom with a single quantized field mode characterized by sub-Poissonian statistics is compared with its interaction with a coherent field mode. Two models are considered: the Jaynes-Cummings model (JCM) and the Raman-coupled model (RCM). Quadrature squeezing variances and photon number uncertainties are evaluated. In the JCM, sub-Poissonian field statistics lead to longer collapse times and more complete revivals than Poissonian statistics. In the RCM, which is characterized by periodic collapses and revivals, the collapse time is longer for sub-Poissonian fields.This work was supported in part by the UK Science and Engineering Research Council.     

Electron energy level statistics in graphene quantum dots

Motivated by recent experimental observations of size quantization of electron energy levels in graphene quantum dots [7] we investigate the level statistics in the simplest tight-binding model for different dot shapes by computer simulation. The results are in a reasonable agreement with the experiment which confirms qualitatively interpretation of observed level statistics in terms of “Dirac billiards” without taking into account many-body effects. It is shown that edge effects are in general sufficient to produce the observed level distribution and that even strong bulk disorder does not change the results drastically.

     

Electron energy level statistics in graphene quantum dots

Motivated by recent experimental observations of size quantization of electron energy levels in graphene quantum dots [7] we investigate the level statistics in the simplest tight-binding model for different dot shapes by computer simulation. The results are in a reasonable agreement with the experiment which confirms qualitatively interpretation of observed level statistics in terms of “Dirac billiards” without taking into account many-body effects. It is shown that edge effects are in general sufficient to produce the observed level distribution and that even strong bulk disorder does not change the results drastically. The article is published in the original.     

Characteristics of level-spacing statistics in chaotic graphene billiards

A fundamental result in nonrelativistic quantum nonlinear dynamics is that the spectral statistics of quantum systems that possess no geometric symmetry, but whose classical dynamics are chaotic, are described by those of the Gaussian orthogonal ensemble (GOE) or the Gaussian unitary ensemble (GUE), in the presence or absence of time-reversal symmetry, respectively. For massless spin-half particles such as neutrinos in relativistic quantum mechanics in a chaotic billiard, the seminal work of Berry and Mondragon established the GUE nature of the level-spacing statistics, due to the combination of the chirality of Dirac particles and the confinement, which breaks the time-reversal symmetry. A question is whether the GOE or the GUE statistics can be observed in experimentally accessible, relativistic quantum systems. We demonstrate, using graphene confinements in which the quasiparticle motions are governed by the Dirac equation in the low-energy regime, that the level-spacing statistics are persistently those of GOE random matrices. We present extensive numerical evidence obtained from the tight-binding approach and a physical explanation for the GOE statistics. We also find that the presence of a weak magnetic field switches the statistics to those of GUE. For a strong magnetic field, Landau levels become influential, causing the level-spacing distribution to deviate markedly from the random-matrix predictions. Issues addressed also include the effects of a number of realistic factors on level-spacing statistics such as next nearest-neighbor interactions, different lattice orientations, enhanced hopping energy for atoms on the boundary, and staggered potential due to graphene-substrate interactions.     

Donor ordering and the statistics of quantum dot level spacings and widths from full 3D self-consistent electronic structure

Using full 3D self-consistent electronic structure calculations of small (electron numberN 100) lateral quantum dots formed on GaAs–AlGaAs HEMT devices we calculate the statistics of level spacings Δε_pand tunneling coefficients Γ_pbetween leads and confined states of the dot. We employ random and ordered donor layer charge distributions, the latter generated through Monte Carlo variable range hopping simulations, as well as a homogeneous (jellium) ionic charge distribution, and examine the effects on these statistics.It has recently been argued that the statistics of the level spacings and widths follow from random matrix theory when the Hamiltonian is described by the Gaussian orthogonal ensemble (GOE) for zero magnetic fieldB, and by the Gaussian unitary ensemble (GUE) forBsufficiently large to break time reversal symmetry. Specifically it is argued that when the dot wave functions are expanded in an arbitrary basis the expansion coefficients, according to the postulate of Porter and Thomas, are uniformly distributed in Hilbert space.In our calculation we obtain statistics of level spacings and widths by generating many configurations of disordered and ordered donor charge. This corresponds to the experimental situation of thermal cycling of the device. We find that a pronounced transition occurs in the level spacing statistics between the completely disordered donor layer ensemble, which seems to be well described by random matrix theory, and the ordered ensemble which is dominated by secular variations in the coefficients. In particular, a shell structure in the levels, which results from approximate parabolicity in the self-consistent confining potential, is observed. This, and the effects of symmetry under inversion and azimuthal symmetry, are speculated to undermine level repulsion and result in Poisson statistics for the levels here at the band edge.Finally we find that distortions in the dot shape are markedly less significant in varying the widths (and level spacings) than calculations based on a hard wall potential for the dot predict. This suggests that the notion of invariant atomic structure for sufficiently small dots is not invalidated by the randomness inherent in donor positions and shape distortion but, on the contrary, a systematic study of dot structure is possible.     

A prediction theory of random level distribution based on a generalized difference type of Fokker-Planck equation and its application to environmental noise and vibration,I: Theoretical study

For generalized discrete random signals, of arbitrary correlations among arbitrarily chosen samples, and also arbitrary distribution form, the short time prediction problem, in terms of the transition probability distribution, is theoretically considered, first for discrete time interval sampling. A general expression is derived from which any signal statistics, e.g., the average, the variance, the 90% range value, and so on, can be predicted. This general expression is equivalent to the well-known Fokker-Planck equation, with continuous time sampling, in the special case of a Markovian process. Explicit algorithms for estimating moment statistics of arbitrary order are derived, by introducing the generalized difference equation of Fokker-Planck type for the probability distribution function.     

Statistical mechanics of Arakawa’s discretizations

The results of statistical analysis of simulation data obtained from long time integrations of geophysical fluid models greatly depend on the conservation properties of the numerical discretization used. This is illustrated for quasi-geostrophic flow with topographic forcing, for which a well established statistical mechanics exists. Statistical mechanical theories are constructed for the discrete dynamical systems arising from three discretizations due to Arakawa [Arakawa, Computational design for long-term numerical integration of the equations of fluid motion: two-dimensional incompressible flow. Part I. J. Comput. Phys. 1 (1966) 119–143] which conserve energy, enstrophy or both. Numerical experiments with conservative and projected time integrators show that the statistical theories accurately explain the differences observed in statistics derived from the discretizations.     

Poissonian bursts in e-mail correspondence

Recent work has shown that the distribution of inter-event times for e-mail communication exhibits a heavy tail which is statistically consistent with a cascading Poisson process. In this work we extend this analysis to higher-order statistics, using the Fano and Allan factors to quantify the extent to which the empirical data are more correlated — bursty — than a Poisson process. Our analysis demonstrates that the correlations in the empirical data are indistinguishable from those of randomly reordered time series, illustrating that any correlations in the data are not due to the precise ordering of events. We further find that correlations in synthetic time series generated from a cascading Poisson process agree quite well with the correlations observed in the empirical data. Finally, we rescale the empirical time series to confirm that e-mail correspondence is no more correlated than expected from a suitably chosen Poisson process.     

Statistical mechanics of Arakawa’s discretizations

The results of statistical analysis of simulation data obtained from long time integrations of geophysical fluid models greatly depend on the conservation properties of the numerical discretization used. This is illustrated for quasi-geostrophic flow with topographic forcing, for which a well established statistical mechanics exists. Statistical mechanical theories are constructed for the discrete dynamical systems arising from three discretizations due to Arakawa [Arakawa, Computational design for long-term numerical integration of the equations of fluid motion: two-dimensional incompressible flow. Part I. J. Comput. Phys. 1 (1966) 119–143] which conserve energy, enstrophy or both. Numerical experiments with conservative and projected time integrators show that the statistical theories accurately explain the differences observed in statistics derived from the discretizations.     

Effective temperature of self-similar time series: Analytical and numerical developments

Within both slightly non-extensive statistics and a related numerical model, a picture is elaborated to treat self-similar time series as a thermodynamic system. Thermodynamic-type characteristics relevant to temperature, pressure, entropy, internal and free energies are introduced and tested. Predictability conditions of time series analysis are discussed on the basis of Van der Waals model. Maximal magnitude for time interval and minimal resolution scale of the value under consideration are found and analyzed in details. The statistics developed is shown to be governed by effective temperature being an exponential measure of the fractal dimension of the time series. Testing of the analytical consideration is based on numerical scheme of non-extensive random walk. A statistical scheme is introduced to present the numerical model as a grand canonical ensemble for which entropy and internal energy are calculated as functions of particle number. Effective temperature is found numerically to show that its value is reduced to averaged energy per one degree of freedom.     

Methods for detrending success metrics to account for inflationary and deflationary factors*

Time-dependent economic, technological, and social factors can artificially inflate or deflate quantitative measures for career success. Here we develop and test a statistical method for normalizing career success metrics across time dependent factors. In particular, this method addresses the long standing question: how do we compare the career achievements of professional athletes from different historical eras? Developing an objective approach will be of particular importance over the next decade as major league baseball (MLB) players from the “steroids era” become eligible for Hall of Fame induction. Some experts are calling for asterisks (*) to be placed next to the career statistics of athletes found guilty of using performance enhancing drugs (PED). Here we address this issue, as well as the general problem of comparing statistics from distinct eras, by detrending the seasonal statistics of professional baseball players. We detrend player statistics by normalizing achievements to seasonal averages, which accounts for changes in relative player ability resulting from a range of factors. Our methods are general, and can be extended to various arenas of competition where time-dependent factors play a key role. For five statistical categories, we compare the probability density function (pdf) of detrended career statistics to the pdf of raw career statistics calculated for all player careers in the 90-year period 1920–2009. We find that the functional form of these pdfs is stationary under detrending. This stationarity implies that the statistical regularity observed in the right-skewed distributions for longevity and success in professional sports arises from both the wide range of intrinsic talent among athletes and the underlying nature of competition. We fit the pdfs for career success by the Gamma distribution in order to calculate objective benchmarks based on extreme statistics which can be used for the identification of extraordinary careers.     

Superdiffusion and viscoelastic vortex flows in a two-dimensional complex plasma

Viscoelastic vortical fluid motion in a strongly coupled particle system has been observed experimentally. Optical tracking of particle motion in a complex plasma monolayer reveals high grain mobility and large scale vortex flows coexistent with partial preservation of the global hexagonal lattice structure. The transport of particles is superdiffusive and ascribed to Lévy statistics on short time scales and to memory effects on the longer scales influenced by cooperative motion. At these longer time scales, the transport is governed by vortex flows covering a wide spectrum of temporal and spatial scales.     

Numerical experiments on billiards

We investigate decay properties of correlation functions in a class of chaotic billiards. First we consider the statistics of Poincaré recurrences (induced by a partition of the billiard): the results are in agreement with theoretical bounds by Bunimovich, Sinai, and Bleher, and are consistent with a purely exponential decay of correlations out of marginality. We then turn to the analysis of the velocity-velocity correlation function: except for intermittent situations, the decay is purely exponential, and the decay rates scale in a simple way with the (uniform) curvature of the dispersing arcs. A power-law decay is instead observed when the system is equivalent to an infinite-horizon Lorentz gas. Comments are given on the behaviour of other types of correlation functions, whose decay, during the observed time scale, appears slower than exponential.     

Dye laser model with pump and quantum fluctuations; colored noise

A single mode dye laser model with two stochastically fluctuating forces representing pump and quantum fluctuations is discussed. In extension to a previous paper we investigate the different influences ofcolored pump noise andwhite quantum fluctuations on the laser light statistics. The corresponding two-variable Fokker-Planck equation is solved by means ofmatrix continued fractions. A comparison to the model with two white noise forces is also included. We focus especially on the intensity correlation time enhancement observed for low noise strengths which seems to be the result of a competition between (multiplicative) pump noise and (additive) quantum noise. An increase in correlation time of the colored pump noise causes the intensity correlation time enhancement to disappear.