Probability Distribution Of Fluctuations Research Articles

Retraction Note to: Review on Plasma Edge Analysis Using the Auto-Correlation and Probability Distributions of Fluctuations

Retraction Note to: Review on Plasma Edge Analysis Using the Auto-Correlation and Probability Distributions of Fluctuations

Density Parameters of the Probability Distribution of Fluctuations of the Apparent Center of Radiation from the Helicopter Propeller Bushing during its Radar Observation

The properties of angular noise of radar reflections from the rotor head are considered. On the basis of a multipoint geometric model of the rotor head, analytical expressions are obtained that allow calculating the parameters of the probability distribution density of angular noise for the azimuth planes and the elevation of the rotor head of a single-bladed screw at an undirected angle of the control sleeve. Correlation obtained for one blade rotor head is generalized to thease of a rotor head with an arbitrary number of blades. It is shown that the rotor head angular noise is a random process with periodically changing parameters. The theoreticalresults are confirmed by mathematical modeling.

Density Parameters of the Probability Distribution of Fluctuations of the Apparent Center of Radiation from the Helicopter Propeller Bushing during Its Radar Observation

Density Parameters of the Probability Distribution of Fluctuations of the Apparent Center of Radiation from the Helicopter Propeller Bushing during Its Radar Observation

On the Moments of Probability Distribution Function of Amplitude Scintillation in the Polar Region

AbstractThe distortions seen in the Global Navigation Satellite Systems radio signal caused by ionospheric irregularities in the polar region are examined through the lens of probability distributions of fluctuations in the recorded scintillation signals. The first four moments of the probability distribution function (PDF) of the amplitude scintillation are computed and analyzed for 106 events collected at Pond Inlet station [Magnetic Coordinates = (80.0°N, 2.6°E)]. As a starting point, the background variation level is investigated using similar computations carried out with about 400 30‐s window segments. The resulting distribution for the skewness (third moment) and the kurtosis (fourth moment) strongly indicate that the background fluctuations are nearly Gaussian; given the computed variances, the Rayleigh distribution hypothesis is ruled out. In the case where scintillation is present, we found that the skewness distribution nearly overlaps with the background while the excess kurtosis strongly indicates a departure from normality. More precisely, the PDF appear mostly symmetric and leptokurtic (positive kurtosis K) and exhibit, in some cases, significant flatness (K ∼ 1). These results reveal that the analyzed scintillations are statistically consistent with stationarity and therefore vanishing skewness. Moreover, the empirical moments are compared with those derived from probability densities referred in the literature regarding scintillations. We have found that none of the log‐normal, Nakagami, Rician and α–μ distributions account for the analyzed data. The generalized Gaussian distribution with a shape parameter 1 < β < 2 seems the most probable and the most adequate to account for the results.

Non-poissonian Distribution of Point Mutations in DNA.

In general, for chemical reactions occurring in systems, where fluctuations are not negligibly small, it is necessary to introduce a master equation for distribution of probability of fluctuations. It has been established that the monomolecular reactions of a type as A ↔ X are described by the master equation, which leads to a Poisson distribution with the variance equal to the average value N0. However, the consideration of the Löwdin mechanism as autocatalytic non-linear chemical reactions such as A + X ↔ 2X and the corresponding master equation lead to a non-Poissonian probability distribution of fluctuations. In the presented work, first-order autocatalysis has been applied to the Löwdin's mechanism of spontaneous mutations in DNA. Describing double proton transfers between complimentary nucleotide bases along the chain by first-order autocatalytic reactions, the corresponding master equation for protons in tautomeric states becomes non-linear, and at non-equilibrium conditions this leads to the non-Poissonian distribution of spontaneous mutations in DNA. It is also suggested that the accumulation of large fluctuations of successive cooperative concerted protons along the chain may produce higher non-linearities which could have a significant impact on some biochemical processes, occurring in DNA.

Thermally activated crack fronts propagating in pinning disorder: simultaneous brittle/creep behaviour depending on scale.

We study theoretically the propagation of a crack front in mode I along an interface in a disordered elastic medium, with a numerical model considering a thermally activated rheology, toughness disorder and long-range elastic interactions. This model reproduces not only the large-scale dynamics of the crack front position in fast or creep loading regimes, but also the small-scale self-affine behaviour of the front. Two different scaling laws are predicted for the front morphology, with a Hurst exponent of 0.5 at small scales and a logarithmic scaling law at large scales, consistently with experiments. The prefactor of these scaling laws is expressed as a function of the temperature, and of the quenched disorder characteristics. The cross-over between these regimes is expressed as a function of the quenched disorder amplitude, and is proportional to the average energy release rate, and to the inverse of temperature. This model captures as well the experimentally observed local velocity fluctuation probability distribution, with a high-velocity tail P(v)∼v -2.6 This feature is shown to arise when the quenched disorder is sufficiently large, whereas smaller toughness fluctuations lead to a lognormal-like velocity distribution. Overall, the system is shown to obey a scaling determined by two distinct mechanisms as a function of scale: namely, the large scales display fluctuations similar to an elastic line in an annealed noise excited as the average front travels through the pinning landscape, while small scales display a balance between thresholds in possible elastic forces and quenched disorder.This article is part of the theme issue 'Statistical physics of fracture and earthquakes'.

Quantitative approach to multifractality induced by correlations and broad distribution of data

We analyze quantitatively the effect of spurious multifractality induced by the presence of fat-tailed symmetric and asymmetric probability distributions of fluctuations in time series. In the presented approach different kinds of symmetric and asymmetric broad probability distributions of synthetic data are examined starting from Levy regime up to those with finite variance. We use nonextensive Tsallis statistics to construct all considered data in order to have good analytical description of frequencies of fluctuations in the whole range of their magnitude and simultaneously the full control over exponent of power-law decay for tails of probability distribution. The semi-analytical compact formulas are then provided to express the level of spurious multifractality generated by the presence of fat tails in terms of Tsallis parameter q̃ and the scaling exponent β of the asymptotic decay of cumulated probability density function (CDF). The results are presented in Hurst and Hölder languages — more often used in study of multifractal phenomena. According to the provided semi-analytical relations, it is argued how one can make a clear quantitative distinction for any real data between true multifractality caused by the presence of nonlinear correlations, spurious multifractality generated by fat-tailed shape of distributions — eventually with their asymmetry, and the correction due to linear autocorrelations in analyzed time series of finite length. In particular, the spurious multifractal effect of fat tails is found basic for proper quantitative estimation of all spurious multifractal effects. Examples from stock market data are presented to support these findings.

An Exactly Soluble Equation for the Stationary Probability Distribution in a Nonlinear System under the Influence of Two-telegraph Noise: Application to the Noise Reduction in a Raman Ring Laser

In this paper, we will consider a model of nonlinear system with random telegraph noises and a Raman ring laser by modeling the laser pump light by a pregaussian process and find an exactly soluble equations for the stationary probability distribution of fluctuations in this nonlinear system under the influence of two-telegraph noise. In consequence, we will obtain the so-called noise reduction in this system: the Stokes output of this laser tends to the stabilize under the influence of the broad-band two-telegraph pregaussian pump and compare this results with that obtained in our previous paper (Cao Long Van, Doan Quoc Khoa, Opt. Quant. Electron. 43, 137 (2012)) for the case of one telegraph noise.

Review on Plasma Edge Analysis Using the Auto-Correlation and Probability Distributions of Fluctuations

We have reviewed the plasma edge fluctuations analysis over a wide range of plasma and field parameters in the IR-T1 tokamak. Fluctuation levels and spectra were measured using two arrays of Langmuir probes on the edge region. Under almost all conditions the edge plasma was turbulently unstable, with a broadband fluctuation spectrum in the drift wave range of frequencies f = 10–1000 kHz. A stable state was observed only in the very cold, low-current discharge formed at unusually high neutral filling pressure. Otherwise, the relative fluctuation level as monitored by the ion saturation current was very high while the fluctuation power spectra were roughly invariant in shape. The relative fluctuation level was always highest near the wall and decreased monotonically toward the plasma centre.

Scaling metabolic rate fluctuations.

Complex ecological and economic systems show fluctuations in macroscopic quantities such as exchange rates, size of companies or populations that follow non-Gaussian tent-shaped probability distributions of growth rates with power-law decay, which suggests that fluctuations in complex systems may be governed by universal mechanisms, independent of particular details and idiosyncrasies. We propose here that metabolic rate within individual organisms may be considered as an example of an emergent property of a complex system and test the hypothesis that the probability distribution of fluctuations in the metabolic rate of individuals has a "universal" form regardless of body size or taxonomic affiliation. We examined data from 71 individuals belonging to 25 vertebrate species (birds, mammals, and lizards). We report three main results. First, for all these individuals and species, the distribution of metabolic rate fluctuations follows a tent-shaped distribution with power-law decay. Second, the standard deviation of metabolic rate fluctuations decays as a power-law function of both average metabolic rate and body mass, with exponents -0.352 and -1/4 respectively. Finally, we find that the distributions of metabolic rate fluctuations for different organisms can all be rescaled to a single parent distribution, supporting the existence of general principles underlying the structure and functioning of individual organisms.

Heat fluctuations in Brownian transducers

The heat fluctuation probability distribution function in Brownian transducers operating between two heat reservoirs is studied. We find, both analytically and numerically, that the recently proposed fluctuation theorem for heat exchange [C. Jarzynski and D. K. Wojcik, Phys. Rev. Lett. 92, 230602 (2004)] has to be applied carefully when the coupling mechanism between both baths is considered. We also conjecture how to extend such a relation when an external work is present.

Space–time thermodynamics of the glass transition

We consider the probability distribution for fluctuations in dynamical action and similar quantities related to dynamic heterogeneity. We argue that the so-called "glass transition" is a manifestation of low action tails in these distributions where the entropy of trajectory space is subextensive in time. These low action tails are a consequence of dynamic heterogeneity and an indication of phase coexistence in trajectory space. The glass transition, where the system falls out of equilibrium, is then an order-disorder phenomenon in space-time occurring at a temperature T(g), which is a weak function of measurement time. We illustrate our perspective ideas with facilitated lattice models and note how these ideas apply more generally.

Scaling and commonality in anomalous fluctuation statistics in models for turbulence and ferromagnetism

Recently, Portelli et al (2003) have semi-numerically obtained a functional form of the probability distribution of fluctuations in the total energy flow in a model for fluid turbulence. This follows earlier work suggesting that fluctuations in the total magnetization in the 2D X-Y model for a ferromagnet also follow this distribution. Here, starting from the scaling ansatz that is the basis of the turbulence model we analytically derive the functional form of this distribution and find its single control parameter that depends upon the scaling exponents and system size of the model. Our analysis allows us to identify this explicitly with that of the X-Y model, and suggest a possible generalization.

Constraining the LogN-LogSRelation of Far-Infrared Unresolved Sources Using aP(D)Analysis; Application to FIRBACK Maps Obtained by ISO

The FIRBACK survey is a deep imaging survey carried out with ISOPHOT on board ISO, at 170 μ m. The survey covers ≈4 sq deg in three high galactic latitude fields (FIRBACK South Marano, FIRBACK North 1 and FIRBACK/ELAIS North 2). In these fields, fluctuations due to unresolved extra-galactic sources were isolated for the first time [1]. We are using a P(D) (probability distribution of fluctuations) analysis to gain information about the N(≥S) curve of the unresolved sources which make up the background. Many planned surveys ( e.g. SIRTF, FIRST, PLANCK...) will pose the same analysis challenges and we hope that the outcome of this study will be useful in that context. As a preparation to the analysis of the FIRBACK maps we analyze simulated sky maps. We create histograms of number of pixels per flux bin. The histograms are binned such that there are enough pixels per bin to allow us to apply Gaussian estimates for the errors. We then calculate the expected number of pixels within the relevant flux ranges and evaluate the χ 2 statistic. We will use a maximum likelihood algorithm to find the parameters of N(≥S) .

Probability Distribution of Density Fluctuations in the Nonlinear Regime

We present a general procedure for obtaining the present density fluctuation probability distribution given the statistics of the initial conditions. The main difficulties faced with regard to this problem are those related to the non-linear evolution of the density fluctuations and those posed by the fact that the fields we are interested in are the result of filtering an underlying field with structure down to scales much smaller than that of filtering. The solution to the latter problem is discussed here in detail and the solution to the former is taken from a previous work. We have checked the procedure for values of the rms density fluctuation as large as 3/2 and several power spectra and found that it leads to results in excellent agreement with those obtained in numerical simulations. We also recover all available exact results from perturbation theory.

Onset of symmetric plasma turbulence

We analyze the electrostatic turbulence in a stationary toroidal magnetoplasma, created by radio-frequency waves and confined by a toroidal magnetic field. The increase of toroidal magnetic field leads to gradients in the mean plasma radial profiles and to the enhancement of electrostatic turbulence. Another consequence is the emergence of symmetry in the probability distributions of fluctuations and of the time in which they return to a specified reference interval of values. This symmetry as well as scaling laws reported here are typically observed in low-dimensional chaotic dynamical systems. Decreasing the magnetic field breaks the symmetry in the probability distribution; however, it preserves the correlation properties of a few average quantities.

Planck—Wheeler Quantum Foam as White Noise: Metric Diffusion and Congruence Focussing for Fluctuating Spacetime Geometry

It is expected that quantum effects endow spacetime with stochastic properties near the Planck scale as exemplified by random fluctuations of the metric, usually referred to as spacetime foam or geometrodynamics. In this paper, a methodology is presented for incorporating Planck scale stochastic effects and corrections into general relativity within the ADM formalism, by coupling the Riemann 3-metric to white noise. The ADM—Cauchy evolution of a Riemann 3-metric h ij (t) induced on spacelike hypersurface C(t) can be interpreted within pure general relativity as a smooth geodesic flow in superspace, whose points consist of equivalence classes of 3-metrics. Coupling white noise to h ij gives Langevin stochastic differential equations for the Cauchy evolution of h ij, which is now a Brownian motion or diffusion in superspace. A fluctuation h′ij away from h ij is considered to be related to h ij by elements of the diffeomorphism group diff(C). Hydrodynamical Fokker—Planck continuity equations are formulated describing the stochastic Cauchy evolution of h ij as a probability flow. The Cauchy invariant or equilibrium solution gives a stationary probability distribution of fluctuations peaked around the deterministic metric. By selecting a physically viable ansatz for the scale dependent diffusion coefficient, one reproduces the Wheeler uncertainty relation for the metric fluctuations of quantum geometrodynamics. Treating h ij as a random variable, a non-linear Raychaudhuri—Langevin equation is derived describing the “geometro-hydrodynamics” of a congruence of fluid or dust matter propagating on the stochastic spacetime. For an initially converging congruence θ′>0 at s′ the singularity θ=−∞ at future proper time s=3/|θ′|$, which is expected in general relativity, is now smeared out near the Planck scale. Proper time s can be extended indefinitely (s→∞) so that intrinsic metric fluctuations can restore geodesic completeness although the geodesics remain trapped for all time: although a singularity can be removed the collapsing matter still creates a black hole. A Fokker—Planck formulation also gives zero probability that θ→−∞ for s→∞. Essentially, the short distance stochastic corrections to the deterministic equations of general relativity can remove pathologies such as singularities, conjugate points and geodesic incompleteness.

Hierarchical Extraction Methods for Higher Order Correlation Information among Sound, Light and Electromagnetic Waves in a VDT Periphery Environment and Mutual Prediction of Each Fluctuation Probability Distribution

Hierarchical Extraction Methods for Higher Order Correlation Information among Sound, Light and Electromagnetic Waves in a VDT Periphery Environment and Mutual Prediction of Each Fluctuation Probability Distribution

Weak turbulence and structure evolution inN-body Hamiltonian systems with long-range force

The dynamics of a family of one-dimensional spatially periodic systems of $N$ classical particles interacting by a repulsive pair force is investigated. This force is the long-range part of the one-dimensional Coulomb interaction; the family includes the mean-field Hamiltonian rotator model. Initial conditions generating turbulent structures are considered. These structures are density holes in $(x,v)$ space that produce a non-Gaussian probability distribution of fluctuations of the particle distribution function $f(x,v,t)$. These density holes appear in a velocity domain where $f(x,v)$ has large derivative ${\ensuremath{\partial}}_{v}f$ as predicted by the kinetic theory of clumps in plasmas. Their evolution is shown to be controlled by the motion of the particles in the $(x,v)$ space domain swept by the separatrix associated with the longest-range coupling field components, which implies that their lifetime is proportional to the number $N$ of particles. The relaxation time of the velocity distribution function of tagged particles in the system (for various initial conditions) is also shown to be quite insensitive to the presence of turbulent structures and to spatial scales smaller than the Debye length.

Mechanism of the generation of black hole entropy in Sakharov’s induced gravity

The mechanism of the generation of Bekenstein-Hawking entropy ${S}^{\mathrm{BH}}$ of a black hole in the Sakharov's induced gravity is proposed. It is suggested that the ``physical'' degrees of freedom, which explain the entropy ${S}^{\mathrm{BH}},$ form only a finite subset of the standard Rindler-like modes defined outside the black hole horizon. The entropy ${S}_{R}$ of the Rindler modes, or entanglement entropy, is always ultraviolet divergent, while the entropy of the ``physical'' modes is finite and coincides in the induced gravity with ${S}^{\mathrm{BH}}.$ The two entropies ${S}^{\mathrm{BH}}$ and ${S}_{R}$ differ by a surface integral $Q$ interpreted as a Noether charge of nonminimally coupled scalar constituents of the model. We demonstrate that energy $E$ and Hamiltonian $H$ of the fields localized in a part of space-time, restricted by the Killing horizon \ensuremath{\Sigma}, differ by the quantity ${T}_{H}Q,$ where ${T}_{H}$ is the temperature of a black hole. The first law of black hole thermodynamics enables one to relate the probability distribution of fluctuations of the black hole mass, caused by the quantum fluctuations of the fields, to the probability distribution of ``physical'' modes over energy $E.$ The latter turns out to be different from the distribution of the Rindler modes. We show that the probability distribution of the ``physical'' degrees of freedom has a sharp peak at $E=0$ with the width proportional to the Planck mass. The logarithm of number of ``physical'' states at the peak coincides exactly with the black hole entropy ${S}^{\mathrm{BH}}.$ This enables us to argue that the energy distribution of the ``physical'' modes and distribution of the black hole mass are equivalent in induced gravity. Finally it is shown that the Noether charge $Q$ is related to the entropy of the low-frequency modes propagating in the vicinity of the bifurcation surface $\ensuremath{\Sigma}$ of the horizon. We find in particular an explicit representation of $Q$ in terms of an effective action of some two-dimensional quantum fields ``living'' on \ensuremath{\Sigma}.