Inverse Power Law Behavior Research Articles

Diffusion entropy and waiting time statistics of hard-x-ray solar flares.

We show at work a technique of scaling detection based on evaluating the Shannon entropy of the diffusion process obtained by converting the time series under study into trajectories. This method, called diffusion entropy, affords information that cannot be derived from the direct evaluation of waiting times. We apply this method to the analysis of the distribution of time distance tau between two nearest-neighbor solar flares. This traditional part of the analysis is based on the direct evaluation of the distribution function psi(tau), or of the probability Psi(tau), that no time distance smaller than a given tau is found. We adopt the paradigm of the inverse power-law behavior, and we focus on the determination of the inverse power index mu, without ruling out different asymptotic properties that might be revealed, at larger scales, with the help of richer statistics. We then use the DE method, with three different walking rules, and we focus on the regime of transition to scaling. This regime of transition and the value of the scaling parameter itself, delta, depends on the walking rule adopted, a property of interest to shed light on the slow process of transition from dynamics to thermodynamics often occurring under anomalous statistical conditions. With the first two rules the transition regime occurs throughout a large time interval, and the information contained in the time series is transmitted, to a great extent, to it, as well as to the scaling regime. By using the third rule, on the contrary, the same information is essentially conveyed to the scaling regime, which, in fact, emerges very quickly after a fast transition process. We show that the DE method not only causes to emerge the long-range correlation with a given mu < 3, and so a basin of attraction different from the ordinary Gaussian one, but it also reveals the presence of memory effects induced by the time dependence of the solar flare rate. When this memory is annihilated by shuffling, the scaling parameter delta is shown to fit the theoretically expected function of mu. All this leads us to the compelling conclusion that mu = 2.138+/-0.01.

Fluorescence Intermittency in Single InP Quantum Dots

Fluorescence “blinking” kinetics of isolated colloidal InP quantum dots (QDs) are investigated via confocal single molecule microscopy. Analysis of fluorescence trajectories reveals inverse power law behavior (∝ 1/τm) in on/off time probability densities with mon ≈ 2.0(2) and moff ≈ 1.5(1). Such an inverse power law in both on/off times is inconsistent with a static distribution of electron/hole trapping sites and highlights the role of fluctuations in the QD nanoenvironment.

“On”/“off” fluorescence intermittency of single semiconductor quantum dots

Single molecule confocal microscopy is used to investigate the detailed kinetics of fluorescence intermittency in colloidal II–VI (CdSe) semiconductor quantum dots. Two distinct modes of behavior are observed corresponding to (i) sustained “on” episodes (τon) of rapid laser absorption/fluorescence cycling, followed by (ii) sustained “off” episodes (τoff) where essentially no light is emitted despite continuous laser excitation. Both on-time and off-time probability densities follow an inverse power law, P(τon/off)∝1/τon/offm, over more than seven decades in probability density and five decades in time. Such inverse power law behavior is an unambiguous signature of highly distributed kinetics with rates varying over 105-fold, in contrast with models for switching between “on” and “off” configurations of the system via single rate constant processes. The unprecedented dynamic range of the current data permits several kinetic models of fluorescence intermittency to be evaluated at the single molecule level and indicate the importance of fluctuations in the quantum dot environment.

Nonexponential “blinking” kinetics of single CdSe quantum dots: A universal power law behavior

Single molecule confocal microscopy is used to study fluorescence intermittency of individual ZnS overcoated CdSe quantum dots (QDs) excited at 488 nm. The confocal apparatus permits the distribution of “on” and “off” times (i.e., periods of sustained fluorescence emission and darkness) to be measured over an unprecedentedly large dynamic range (109) of probability densities, with nonexponential behavior in τoff over a 105 range in time scales. In dramatic contrast, these same τoff distributions in all QDs are described with remarkable simplicity over this 109-fold dynamic range by a simple inverse power law, i.e., P(τoff)∝1/τoff1+α. Such inverse power law behavior is a clear signature of distributed kinetics, such as predicted for (i) an exponential distribution of trap depths or (ii) a distribution of tunneling distances between QD core/interface states. This has important statistical implications for all previous studies of fluorescence intermittency in semiconductor QDs and may have broader implications for other systems such as single polymer molecules.

Two-state model for sub-exponential fluorescence

We outline a theory of fluorescence in systems where the decay of the fluorescence takes place more slowly than exponential, a behavior we refer to as `sub-exponential'. We develop a two-state model for the fluorescence where the two states are the trapped state and the fluorescing level. The sub-exponential behavior is associated with a distribution of detrapping rates. Asymptotic inverse power-law behavior of the fluorescence, i.e., t − δ , arises from a distribution that varies near the origin as the δ−2 power of the detrapping rate. Stretched exponential behavior, where the asymptotic time dependence of the fluorescence is dominated by the factor exp[−(t/τ) β], 0<β<1 , is associated with a steep but continuous decrease in the density of transition rates. A one-parameter thermal activation model is introduced in which the distribution of detrapping rates arises from a distribution of activation energies. Power-law behavior of the fluorescence is associated with an exponential decrease in the activation energy distribution at high energies, while stretched exponential dynamics reflects an even faster decrease. The thermal activation model is used to make predictions about the temperature dependence of the exponents characterizing the fluorescence decay. Possible experimental tests of the theory are discussed.

Late-time evolution of charged gravitational collapse and decay of charged scalar hair. II

We study analytically the initial value problem for a charged massless scalar-field on a Reissner-Nordstr\"om spacetime. Using the technique of spectral decomposition we extend recent results on this problem. Following the no-hair theorem we reveal the dynamical physical mechanism by which the charged hair is radiated away. We show that the charged perturbations decay according to an inverse power-law behaviour at future timelike infinity and along future null infinity. Along the future outer horizon we find an oscillatory inverse power-law relaxation of the charged fields. We find that a charged black hole becomes ``bald'' slower than a neutral one, due to the existence of charged perturbations. Our results are also important to the study of mass-inflation and the stability of Cauchy horizons during a dynamical gravitational collapse of charged matter in which a charged black-hole is formed.

ON THE UBIQUITY OF 1/f NOISE

A generic mechanism for the ubiquitous phenomenon of 1/f noise is reviewed. This mechanism arises in random processes expressible as a product of several random variables. Under mild conditions this product form leads to the log-normal distribution which we show straightforwardly generates 1/f noise. Thus, 1/f noise is tied directly to a probability limit distribution. A second mechanism involving scaling is introduced to provide a natural crossover from log-normal to inverse power-law behavior and generates 1/fα noise instead of pure 1/f noise. Examples of these distributions and the transitions between them are drawn from such diverse areas as economics, scientific productivity, bronchial structure and cardiac activity.

Electrons on one-dimensional quasi-periodic lattices

We study electronic states on one-dimensional quasi-periodic lattices numerically by using the tight binding Fibonacci model generated by the Fibonacci substitution rule. It is found that the electronic states are critical, that is, intermediate between the localized and extended states. This critical state is characterized by the branching rule of the energy spectra and the inverse power law behavior of the level spacings. The Harper model with an incommensurate cosine potential, and quasi-periodic lattice models generated by other substitution rules are examined. Phonon problem in the quasi-periodic lattice is also discussed.