G. Györgyi

1/f Noise and Extreme Value Statistics

We study finite-size scaling of the roughness of signals in systems displaying Gaussian 1/f power spectra. It is found that one of the extreme value distributions, the Fisher-Tippett-Gumbel (FTG) distribution, emerges as the scaling function when boundary conditions are periodic. We provide a realistic example of periodic 1/f noise, and demonstrate by simulations that the FTG distribution is a good approximation for the case of nonperiodic boundary conditions as well. Experiments on voltage fluctuations in GaAs films are analyzed and excellent agreement is found with the theory.

Submicron Plasticity: Yield Stress, Dislocation Avalanches, and Velocity Distribution

The existence of a well-defined yield stress, where a macroscopic crystal begins to plastically flow, has been a basic observation in materials science. In contrast with macroscopic samples, in microcrystals the strain accumulates in random bursts, which makes controlled plastic formation difficult. Here we study by 2D and 3D simulations the plastic deformation of submicron objects under increasing stress. We show that, while the stress-strain relation of individual samples exhibits jumps, its average and mean deviation still specify a well-defined critical stress. The statistical background of this phenomenon is analyzed through the velocity distribution of dislocations, revealing a universal cubic decay and the appearance of a shoulder due to dislocation avalanches.

Kepler’s equation, Fock variables, Bacry’s generators and Dirac brackets

SummaryA formulation of the Kepler problem, manifestly invariant with respect to theSO4 andSO3,1 groups, respectively, is given in terms of the Fock variables and their canonical conjugates; one is led to introduce a new time parameter, proportional to the eccentric anomaly. A transformation of the dynamical variables performed in order to get back the standard timet leads in a natural way to Bacry’s generators. A manifestlySO4,2 symmetric formulation of the problem is given. The concept of the Dirac bracket is used to establish a connection with the usual three-dimensional description.RiassuntoSi dà una formulazione del problema di Keplero manifestamente invariante rispetto ai gruppiSO4 eSO3,1, in termini delle rispettive variabili di Fock e dei loro coniugati canonici; si è indotti ad introdurre un nuovo parametro temporale, proporzionale alla anomalia eccentrica. Una trasformazione delle variabili dinamiche eseguita per riavere il tempo standardt conduce naturalmente ai generatori di Bacry. Si dà una formulazione del problema manifestamente simmetrica rispetto aSO4,2. Si usa il concetto di parentesi di Dirac per stabilire un collegamento con la comune descrizione tridimensionale.РезюмеВ терминах переменных Фока и их канонических сопряженных приводится формулировка проблемы Кеплера, явно инвариантная относительно группSO4 иSO3.1, соответственно; эта формулировка ведёт к введению нового временного параметра, пропорционального эксцентрической аномалии. Преобразование динамических переменных, выполненное для того, чтобы вернуться к стандартномы времениt, приводит естественным образом к генераторам Ъэкри. В явном виде представленаSO4.2 симметрическая формулировка проблемы. Концепция скобок Дирака используется для установления связи с обычным трехмерным описанием.

Properties of fully developed chaos in one-dimensional maps

We consider single-humped symmetric one-dimensional maps generating fully developed chaotic iterations specified by the property that on the attractor the mapping is everywhere two to one. To calculate the probability distribution function, and in turn the Lyapunov exponent and the correlation function, a perturbation expansion is developed for the invariant measure. Besides deriving some general results, we treat several examples in detail and compare our theoretical results with recent numerical ones. Furthermore, a necessary condition is deduced for the probability distribution function to be symmetric and an effective functional iteration method for the measure is introduced for numerical purposes.

Dynamics of coarse grained dislocation densities from an effective free energy

A continuum description of the time evolution of an ensemble of parallel straight dislocations has recently been derived from the equations of motion of individual dislocations. The predictions of the continuum model were compared to the results of discrete dislocation dynamics (DDD) simulations for several different boundary conditions. It was found that it is able to reproduce all the features of the dislocation ensembles obtained by DDD simulations. The continuum model, however, is systematically established only for single slip. Due to the complicated structure of the equations extending the derivation procedure for multiple slip is not straightforward. In this paper an alternative approach based on a thermodynamics-like principle is proposed to derive continuum equations for single slip. An effective free energy is introduced even for zero physical temperature, which yields equilibrium conditions giving rise to Debye-like screening; furthermore, it generates dynamical equations along the lines of phase field theory. It is shown that this leads essentially to the same evolution equations as obtained earlier. In addition, it seems that this framework is extendable to multiple slip as well.

Finite-size scaling in extreme statistics

We study the deviations from the limit distributions in extreme value statistics arising due to the finite size (FS) of data sets. A renormalization method is introduced for the case of independent, identically distributed (iid) variables, showing that the iid universality classes are subdivided according to the exponent of the FS convergence, which determines the leading order FS shape correction function as well. It is found that, for the correlated systems of subcritical percolation and 1/f;(alpha) stationary (alpha<1) noise, the iid shape correction compares favorably to simulations. Furthermore, for the strongly correlated regime (alpha>1) of 1/f;(alpha) noise, the shape correction is obtained in terms of the limit distribution itself.

Roughness distributions for 1/f alpha signals.

The probability density function (PDF) of the roughness, i.e., of the temporal variance, of 1/f(alpha) noise signals is studied. Our starting point is the generalization of the model of Gaussian, time periodic, 1/f noise, discussed in our recent Letter [Phys. Rev. Lett. 87, 240601 (2001)], to arbitrary power law. We investigate three main scaling regions (alpha < or = 1/2, 1/2 < alpha < or = 1, and 1< alpha), distinguished by the scaling of the cumulants in terms of the microscopic scale and the total length of the period. Various analytical representations of the PDF allow for a precise numerical evaluation of the scaling function of the PDF for any alpha. A simulation of the periodic process makes it possible to study also nonperiodic, thus experimentally more relevant, signals on relatively short intervals embedded in the full period. We find that for alpha < or = 1/2 the scaled PDFs in both the periodic and the nonperiodic cases are Gaussian, but for alpha > 1/2 they differ from the Gaussian and from each other. Both deviations increase with growing alpha. That conclusion, based on numerics, is reinforced by analytic results for alpha = 2 and alpha-->infinity, in the latter limit the scaling function of the PDF being finite for periodic signals, but developing a singularity for the aperiodic ones. Finally, an overview is given for the scaling of cumulants of the roughness and the various scaling regions in arbitrary dimensions. We suggest that our theoretical and numerical results open a different perspective on the data analysis of 1/f(alpha) processes.

Statistical properties of chaos demonstrated in a class of one‐dimensional maps

One-dimensional maps with complete grammar are investigated in both permanent and transient chaotic cases. The discussion focuses on statistical characteristics such as Lyapunov exponent, generalized entropies and dimensions, free energies, and their finite size corrections. Our approach is based on the eigenvalue problem of generalized Frobenius-Perron operators, which are treated numerically as well as by perturbative and other analytical methods. The examples include the universal chaos function relevant near the period doubling threshold. Special emphasis is put on the entropies and their decay rates because of their invariance under the most general class of coordinate changes. Phase-transition-like phenomena at the border state of chaos due to intermittency and super instability are presented.

Maximal height statistics for 1/fα signals

Numerical and analytical results are presented for the maximal relative height distribution of stationary periodic Gaussian signals (one-dimensional interfaces) displaying a 1/f(alpha) power spectrum. For 0<or=alpha<1 (regime of decaying correlations), we observe that the mathematically established limiting distribution (Fisher-Tippett-Gumbel distribution) is approached extremely slowly as the sample size increases. The convergence is rapid for alpha>1 (regime of strong correlations) and a highly accurate picture gallery of distribution functions can be constructed numerically. Analytical results can be obtained in the limit alpha-->infinity and, for large alpha, by perturbation expansion. Furthermore, using path integral techniques we derive a trace formula for the distribution function, valid for alpha=2n even integer. From the latter we extract the small argument asymptote of the distribution function whose analytic continuation to arbitrary alpha>1 is found to be in agreement with simulations. Comparison of the extreme and roughness statistics of the interfaces reveals similarities in both the small and large argument asymptotes of the distribution functions.

Statistics of extremal intensities for Gaussian interfaces

The extremal Fourier intensities are studied for stationary Edwards-Wilkinson-type, Gaussian, interfaces with power-law dispersion. We calculate the probability distribution of the maximal intensity and find that, generically, it does not coincide with the distribution of the integrated power spectrum (i.e., roughness of the surface), nor does it obey any of the known extreme statistics limit distributions. The Fisher-Tippett-Gumbel limit distribution is, however, recovered in three cases: (i) in the nondispersive (white noise) limit, (ii) for high dimensions, and (iii) when only short-wavelength modes are kept. In the last two cases the limit distribution emerges in nonconventional scenarios.