Recurrence time statistics in chaotic dynamics. I. Discrete time maps

The dynamics of transitions between the cells of a finite-phase-space partition in a variety of systems giving rise to chaotic behavior is analyzed, with special emphasis on the statistics of recurrence times. In the case of one-dimensional piecewise Markow maps the recurrence problem is cast into a-renewal process. In the presence of intermittency, transitions between cells define a non-Markovian, non-renewal process reflected in the presence of power-law probability distributions and of divergent variances and mean values.     

Correction algorithm for finite sample statistics

Assume in a sample of size M one finds M_i representatives of species i with . The normalized frequency , based on the finite sample, may deviate considerably from the true probabilities p_i. We propose a method to infer rank-ordered true probabilities r_i from measured frequencies M_i. We show that the rank-ordered probabilities provide important informations on the system, e.g., the true number of species, the Shannon- and the Renyi-entropies.Received: 14 July 2003, Published online: 5 February 2004PACS: 02.50.-r Probability theory, stochastic processes, and statistics - 02.60.-x Numerical approximation and analysis - 07.05.Kf Data analysis: algorithms and implementation; data management     

Residence time statistics for N renewal processes

We present a study of residence time statistics for N renewal processes with a long tailed distribution of the waiting time. Such processes describe many nonequilibrium systems ranging from the intensity of N blinking quantum dots to the residence time of N Brownian particles. With numerical simulations and exact calculations, we show sharp transitions for a critical number of degrees of freedom N. In contrast to the expectation, the fluctuations in the limit of N→∞ are nontrivial. We briefly discuss how our approach can be used to detect nonergodic kinetics from the measurements of many blinking chromophores, without the need to reach the single molecule limit.     

Recurrence plot statistics and the effect of embedding

Recurrence plots provide a graphical representation of the recurrent patterns in a timeseries, the quantification of which is a relatively new field. Here we derive analytical expressions which relate the values of key statistics, notably determinism and entropy of line length distribution, to the correlation sum as a function of embedding dimension. These expressions are obtained by deriving the transformation which generates an embedded recurrence plot from an unembedded plot. A single unembedded recurrence plot thus provides the statistics of all possible embedded recurrence plots. If the correlation sum scales exponentially with embedding dimension, we show that these statistics are determined entirely by the exponent of the exponential. This explains the results of Iwanski and Bradley [J.S. Iwanski, E. Bradley, Recurrence plots of experimental data: to embed or not to embed? Chaos 8 (1998) 861–871] who found that certain recurrence plot statistics are apparently invariant to embedding dimension for certain low-dimensional systems. We also examine the relationship between the mutual information content of two timeseries and the common recurrent structure seen in their recurrence plots. This allows time-localized contributions to mutual information to be visualized. This technique is demonstrated using geomagnetic index data; we show that the AU and AL geomagnetic indices share half their information, and find the timescale on which mutual features appear.     

The effects of multifractality on the statistics of return intervals

We study the statistics of the return intervals in multifractal data sets with and without linear correlations. In the absence of linear correlations, we find that the nonlinear correlations inherent in multifractal data yield (i) a power-law decay of the autocorrelation function of the return intervals, (ii) a power-law increase of the conditional return period as function of the previous return interval, and (iii) a power-law decay of the probability density function of the return intervals. These features remain unchanged in the presence of linear long-term correlations. Deviations observed in the asymptotic behaviour are probably due to finite size effects. We compare our results with those obtained for uncorrelated and for monofractal long-term correlated data, and demonstrate significant differences. Applications can be found in studying the dynamics of several processes characterised by multifractality, such as turbulence, climate dynamics, heartbeat dynamics, stock market dynamics, and tele-traffic in large networks.     

Multifractal properties of return time statistics

The global statistics of the return times of a dynamical system can be described by a new spectrum of generalized dimensions. Comparison with the usual multifractal analysis of measures is presented, and the difference between the two corresponding sets of dimensions is established. Theoretical analysis and numerical examples of dynamical systems in the class of iterated functions are presented.     

Classical and quantum statistics as finite random processes

We show: (1) It is possible to produce the three familiar statistics without referring to the problem of distinguishability; (2) what really distinguishes elementary particles is the correlation existing among them; (3) correlations existing among quantum particles, positive for bosons and negative form fermions, are completely different in character.     

Twist and finite size effects for the source method

We analyze the finite size effects for the source method in pure lattice gauge theory at weak coupling. They are found to be strongly suppressed by twisting the boundary conditions, forSU(3) by typically an order of magnitude.     

First-passage time statistics of stochastic transcription process for time-dependent reaction rates

The European Physical Journal E - To theoretically understand force generation properties of actin filaments, many models consider growing filaments pushing against a movable obstacle or barrier....     

Infinite invariant density determines statistics of time averages for weak chaos

Weakly chaotic nonlinear maps with marginal fixed points have an infinite invariant measure. Time averages of integrable and nonintegrable observables remain random even in the long time limit. Temporal averages of integrable observables are described by the Aaronson-Darling-Kac theorem. We find the distribution of time averages of nonintegrable observables, for example, the time average position of the particle, x[over ˉ]. We show how this distribution is related to the infinite invariant density. We establish four identities between amplitude ratios controlling the statistics of the problem.     

The size of finite size effects in lattice gauge theories

If a quantum field is enclosed in a spatial box of finite volume, its mass spectrum depends on the box size L. For field theories in the continuum Lüscher has shown to all orders in perturbation theory that for large L this dependence is related to certain scattering amplitudes of the infinite volume theory. We derived the corresponding relations for lattice field theories. Assuming their validity for lattice gauge theory outside the perturbative region the magnitude of finite size effects on the spectrum is determined by a glueball coupling constant. This quantity is estimated by strong coupling methods.     

Effective classical potential for fluctuating field systems of finite size

We extend a previously proposed variational method in which quantum mechanical path integrals over fluctuating orbits are approximated by single integrals over an effective classical potential, to field systems of finite size and point the way for higher order corrections.     

Analysis of the time-reversal operator for scatterers of finite size

Recently, it was shown that the time-reversal operator for a single, small spherical scatterer could have up to four distinguishable eigenstates [Chambers and Gautesen, J. Acoust. Soc. Am. 109, 2616-2624 (2001)]. In this paper, that analysis is generalized for scatterers of arbitrary shape and larger size. It is shown that the time-reversal operator may have an indefinitely large number of distinguishable eigenstates, with the exact number depending on the nature of the scatterer and the geometry of the time-reversal mirror. In addition, the case of a multiple number of well-separated scatterers is investigated, with the result that the total spectrum is the direct combination of the eigenstates associated with each scatterer. As an example, the singular value spectrum of the time-reversal operator for a linear array is calculated explicitly for bubbles and hard rubber spheres of finite size. Both resonance peaks and apparent crossing points can be observed in the spectrum as the size of the scatterer increases.     

对于具有一定宽度的光束通过展宽器的色散误差的评价

利用光线追踪法计算了具有一定宽度的光束通过Martinez展宽器时的色散误差.结果表明,通常的无宽度理想光线的展宽器模型有足够的精确度.但是如果将光束反转后再送入展宽器,将得到更加完美的色散误差补偿 关键词： 飞秒激光放大器 展宽器 压缩器 色散补偿     

First passage time distributions for finite one-dimensional random walks

We present closed expressions for the characteristic function of the first passage time distribution for biased and unbiased random walks on finite chains and continuous segments with reflecting boundary conditions. Earlier results on mean first passage times for one-dimensional random walks emerge as special cases. The divergences that result as the boundary is moved out to infinity are exhibited explicitly. For a symmetric random walk on a line, the distribution is an elliptic theta function that goes over into the known Lévy distribution with exponent 1/2 as the boundary tends to ∞.     

Scale-free statistics of time interval between successive earthquakes

The statistical property of the calm times, i.e., time intervals between successive earthquakes with arbitrary values of magnitude, is studied by analyzing the seismic time series data in California and Japan. It is found that the calm times obey the Zipf–Mandelbrot power law, exhibiting a new scale-free nature of the earthquake phenomenon. Dependence of the exponent of the power-law distribution on threshold for magnitude is examined. As threshold increases, the tail of the distribution tends to become longer, showing difficulty in statistically estimating time intervals of earthquakes.     

Occupation time statistics in the quenched trap model

We investigate the distribution of the occupation time of a particle undergoing a random walk among random energy traps and in the presence of a deterministic potential field. When the distribution of energy traps is exponential with a width T(g), we find in thermal equilibrium a transition between Boltzmann statistics when T>T(g) to Lamperti statistics when T < T(g). We explain why our main results are valid for other models of quenched disorder, and discuss briefly implications on single particle experiments.     

Firing time statistics for driven neuron models: analytic expressions versus numerics

Analytical expressions are put forward to investigate the forced spiking activity of abstract neuron models such as the driven leaky integrate-and-fire model. The method is valid in a wide parameter regime beyond the restraining limits of weak driving (linear response) and/or weak noise. The novel approximation is based on a discrete state Markovian modeling of the full long-time dynamics with time-dependent rates. The scheme yields excellent agreement with numerical Langevin and Fokker-Planck simulations of the full nonstationary dynamics, not only for the first-passage time statistics, but also for the important interspike interval (residence time) distribution.     

First passage time statistics for some stochastic processes with superimposed shot noise

We present a method for finding statistical properties of the first passage time to exit an interval of general diffusion processes subject to random delta function impulses. Exact solutions are found for the mean first passage time for Brownian motion. Other special cases, detailed in the text, can also be solved in some generality.