Power-law Distribution Function Research Articles

Multiscale dynamics and robust critical scaling in a continuum current sheet model.

We analyze the self-organized critical behavior of a continuum running avalanche model. We demonstrate that over local interaction scales, the model behavior is affected by low-dimensional chaotic dynamics that plays the role of the primary noise source. With the help of scale-free avalanches, the uncertainty associated with chaos is distributed over a variety of intermediate scales and thus gives rise to spatiotemporal fluctuations that are characterized by power-law distribution functions. We show that globally, the continuum model displays structurally stable critical scaling that can be observed in a finite region in the control parameter space. In this region, the system exhibits a power-law critical divergence of the integrated response function over a broad range of dissipation rates. The observed behavior involves a remarkably stable spatial configuration. We explain the robust features of the model by the adjustable dynamics of its global loading-unloading cycle, which allows maintaining the long-term stationary state without affecting the intrinsic avalanche dynamics.

Single molecule waiting time distribution functions in quantum processes

The statistics of single molecule blinking events often reveal underlying quantum mechanisms. The golden rule rate expression for quantum transitions is shown to be the inverse of the mean waiting time. The distribution function for the waiting time is related to the density of states such that simple power-law distribution functions can be predicted based on the functional form for the density of states. Explicit formulas are derived for waiting time distribution functions in three kinetic processes: Quantum tunneling, intersystem conversion, and nonstationary electron transfer.

ITAL]HUBBLE SPACE TELESCOPE[/ITAL][ITAL]Hubble Space Telescope[/ITAL] Observations of Dust and Star-forming Regions in the Ocular Galaxy IC 2163 and Its Spiral Companion NGC 2207

Hubble Space Telescope observations in U, B, V, and I passbands of the interacting spiral galaxies IC 2163 and NGC 2207 are used to measure extinctions in the cloud and intercloud regions and ages and luminosities of the star-forming regions. The extinction in the part of NGC 2207 seen in projection against IC 2163 was determined by using the method of White & Keel. The extinctions there and elsewhere were also determined from radiative transfer models of the magnitude differences between clouds and their surroundings. The intercloud extinction in V band ranges from 0.5 to 1 mag on the line of sight, and the cloud extinction ranges from 1 to 2 mag. The measured star-forming regions in these galaxies have a power-law relation between size and luminosity and a power-law luminosity distribution function. These power laws are consistent with a fractal dimension for the star formation that is the same as that for interstellar gas, D ~ 2.2, extending over scales ranging from 20 to 1000 pc. Fifteen compact massive star clusters that are analogous to super–star clusters found in starburst regions are in the spiral arms of NGC 2207. Nothing is peculiar about these regions except for a high H I velocity dispersion (~50 km s-1). Two more super–star clusters are in the tidally compressed oval of IC 2163. These clusters have masses ranging from ~10^4 to 2 × 10^5 M⊙ and ages of a few times 10^6 yr.

Spectral characteristics of urban aerosols and their association with relative humidity

Multi-spectral extinction measurements made with co-located high-spectral resolution radiometer (spectroradiometer) and sunphotometer at the Indian Institute of Tropical Meteorology (IITM), Pune (18°32′N, 73°5′E, 559 m AMSL) on 188 cloud-free days between March 1993 and May 1995 are presented. The observed aerosol optical size spectrum was approximatd by a composite power-law distribution function. The seasonal mean size distributions, inferred from both the radiometers, exhibit a power-law type distribution with different exponents changing at an intermediate size. The mean Junge size exponents ( ν 1 and ν 2 ) and switching radius ( r 0) obtained with spectroradiometer are found to be in good agreement with those obtained with sunphotometer. The relationships among aerosol optical depths and derived size distributions, and meteorological parameters (height-integrated) are discussed.

Ac Response of Heterogeneous Materials: A Numerical Study

ABSTRACTThe ac and dc electrical properties of composite materials are studied using hierarchical lattices. First we show that the hierarchical model can correctly account for the main scaling properties of critical percolative structures. Then we study the effect of potential disorder by assuming that the microscopic conductances are distributed according to a power law distribution function. We find that in the limit of strong disorder, the predictions are in qualitative agreement with reported experimental measurements.

One-dimensional Vlasov simulations of Langmuir solitons

Interactions of Langmuir solitons with plasma are simulated using a one-dimensional electrostatic Vlasov code, including both electrons and ions. The Zakharov’s solutions are used as initial conditions. The wave is found to transfer energy into electrons during the heating process due to the interaction of resonant electrons with high-frequency electric fields. For Te≫Ti, the temporal evolution of the electron distribution function shows different heating behaviors depending on whether the electrons are interacting with a single soliton or many solitons. In single-soliton interactions, the final electron distribution function approaches an exponential form, f(v)∝exp(−6v/vte). However, for multisoliton interactions, the final stage of the heated electrons establishes a power-law distribution function, f(v)∝v−4, which agrees with that of Gorev and Kingsep [Sov. Phys. JETP 39, 1008 (1974)]. Simulation results for soliton motion show strong ion Landau damping, such that the solitons dissipated quickly when Ti=Te. Analyses indicate that nonlinear damping of Langmuir solitons by thermal ions for vg≤vti plays the most important role in slowing down the moving solitons. A double-hump structure was also found in the near-sonic soliton motion.

Phase Behavior of Linear/Branched Polymer Blends

The phase behavior of a linear/branched polymer blend is predicted within the framework of the Flory-Huggins theory. The linear polymer is treated as monodisperse and the branched polymer as polydisperse with power law statistics, O(N) ∞ N -(τ-1) , cut off at some upper degree of polymerization N 2 . The latter is dependent on the reacted fraction, α, and the functionality, f, of the functional groups of the branched polymer. We calculate analytically the spinodal curves for various α up to the gel point and find unusual behavior of the critical point. In particular, it is found to be very sensitive to the exponent of the power law distribution function, e.g., whether τ takes the classical value of 2.5 or the percolation value of 2.20. Example cloud point curves and coexistence curves have been calculated numerically, again for various α, and a picture of the evolution of the phase diagram during cross-linking has been constructed. The coexistence curves are surprisingly steep as χ is varied over a large range ; this is particularly evident for the parameters of the linear-rich phase. Hence it is found that although the distribution of the branched polymer in the linear-rich phase varies considerably with increasing χ, the ratio of linear and branched polymer volume fractions within this phase does not. From our results we are able to predict, at least qualitatively, a secondary phase separation that is known to occur in such blends.

Ion-acoustic waves in a relativistic plasma for various particle velocity distributions

It is established that longitudinal ion-acoustic modes can exist only in the one-dimensional case in a relativistic plasma with an arbitrary particle velocity distribution. The spectrum of the given wave mode is calculated on the basis of Vlasov's kinetic theory for an arbitrary particle velocity distribution function. In particular, the dispersion and logarithmic decrement are found for power-law and Maxwell distribution functions.

Space and Time Distribution of Hard X-Ray Emission in a Loop at the Beginning of a Flare

AbstractUsing a new ID hybrid model of the electron bombardment in flare loops, we study not only the evolution of densities, plasma velocities and temperatures in the loop, but also the temporal and spatial evolution of hard X-ray emission. In the present paper a continuous bombardment by electrons isotropically accelerated at the top of flare loop with a power-law injection distribution function is considered. The computations include the effects of the return-current that reduces significantly the depth of the chromospheric layer which is evaporated. The present modelling is made with superthermal electron parameters corresponding to the classical resistivity regime for an input energy flux of superthermal electrons of 109erg cm−2s−1. It was found that due to the electron bombardment the two chromospheric evaporation waves are generated at both feet of the loop and they propagate up to the top, where they collide and cause temporary density and hard X-ray enhancements.

Optimal design of a flocculator

An approach is presented for the least cost design of a flocculator with respect to flocculation time and velocity gradient, in order to obtain a desired effluent condition, which is measured using the negative slope of power law distribution function (β). The model proposed is a non-linear programming problem involving a non-linear constraint, which in turn comprises a verified simulation model. The case study demonstrates the existence of a relationship between design criterion and design parameters. It also shows that the optimal solution is sensitive to influent particle size distribution (PSD) and particle contact efficiency (α).

Concentration, Clustering and Stretching in Homogeneous Turbulence

Incompressible Navier-Stokes equation is numerically simulated for investigating the characteristics of the energy-dissipation ε field and the vorticity-square ω2 field of homogeneous isotropic turbulence. The concentration of ω2 into a volume is higher than that of ε. The sizes of the concentrating regions (clusters) of ω2 have almost the same power-law distribution function as that of ε. The vorticity stretching in regions of large ω2 or ε has two directions of expansion and one direction of contraction. On the other hand, in the regions of small ω2, two directions show expansion and contraction respectively, but the remaining third direction is inactive, where a two-dimensional stretching appears to occur.

Concentration, Clustering and Stretching in Homogeneous Turbulence

Incompressible Navier-Stokes equation is numerically simulated for investigating the characteristics of the energy-dissipation ε field and the vorticity-square ω2 field of homogeneous isotropic turbulence. The concentration of ω2 into a volume is higher than that of ε. The sizes of the concentrating regions (clusters) of ω2 have almost the same power-law distribution function as that of ε. The vorticity stretching in regions of large ω2 or ε has two directions of expansion and one direction of contraction. On the other hand, in the regions of small ω2, two directions show expansion and contraction respectively, but the remaining third direction is inactive, where a two-dimensional stretching appears to occur.

対流圏工一ロゾルの放射効果の航空機観測

The size distribution and the complex refractive index of tropospheric aerosols have been retrieved from the simultaneous observations of solar radiation and aerosols described in Part I of this study (Asano and Shiobara, 1989). Three sets of observations over the Tsukuba area have been analyzed, for which the aerosol optical thicknesses were available from ground-based spectral attenuation measurements of direct solar radiation.The bimodal size distribution for tropospheric aerosols was approximated by a composite powerlaw distribution function. The parameters of the distribution function were determined by fitting the function to the columnar size distributions inverted from photometrically measured spectral optical thicknesses. Estimation was made of wavelength-mean effective values of the complex refractive index m*=mr-mii which optimally simulate the observed solar fluxes and heating rates for the total solar spectral region. In the simulated calculations, the observed profiles of aerosols and water vapor were considered. The single scattering properties were computed from Mie theory for the determined size distribution and for a reasonably assumed real part of the index of refraction of mr=1.52. The optimal value of mi=0.03×0.02 was estimated for the imaginary part of the index of refraction of the tropospheric aerosols.

Void fraction, bubble velocity and bubble size in two-phase flow

Experiments were performed in atmosphereic vertical air-water flows, for void fractions between 0.25 and 0.75 (cross-sectional averages) and superficial liquid velocities of 1.3, 1.7 and 2.1 m/s. Local values of void fraction and bubble velocity as well as the bubble diameter were measured by means of a resistivity probe technique. Reliable values were obtained for the local void fraction over the entire range 0 ≤ α ≤ 1. The void fraction profiles appeared to have a local maximum at the pipe center, local maxima close to the wall were obviously absent. The resistivity probes are shown to measure the velocity of the interface between the conducting and nonconducting phases, which equals the gas velocity only for low void fractions. The measured data for void fraction and bubble velocity were correlated by means of power law distribution functions, with exponents given by a function of the cross-sectionally averaged void fraction. The Sauter mean diameters for the bubble size spectra found, agree reasonably well with diameters predicted by a theoretical model based on the energy dissipation in the flow.

Kinetics of phase transitions: Theory of Ostwald ripening

A new derivation is presented for the last stage of phase separation in the kinetics of a first order phase transition, precipitation, where Ostwald ripening is the dominant mechanism. We use a time scaling technique and derive the power law time dependence and distribution function for the size of the particles of the new phase. Equations are derived for the correction terms to the distribution function and power laws. The derivation clarifies and corrects prior work.

X-ray bremsstrahlung measurements on an ECR-discharge in a magnetic mirror

The hot electron component of a microwave heated electron cyclotron resonance (ECR)-discharge is investigated. The plasma formed from different noble gases is contained in a simple magnetic mirror configuration. Improved X-ray diagnostic methods and an improved numerical deconvolution scheme are used to measure the volume bremsstrahlung spectrum emitted by the high energy electron and to derive their energy distribution function. For kinetic energies below 350 keV the electron energy distribution obeys a power law, if a cold plasma component (kTe<100 eV) is present. The density of electrons with energies above 10 keV turns out to be smaller than 1014 m-3. These results, together with the observed dependence of the power-law distribution function both on the pressure and on the nuclear charge number Z of the neutral gas, provide new insights into the electron acceleration and loss mechanisms of an ECR-discharge.

The velocity distribution of interstellar gas observed in strong UV absorption lines

Observations of three strong interstellar UV absorption lines of N I (1199 A), N II (1083 A), and Si III (1206 A) in 47 stars of widely varying distance and a variety of spectral types are analyzed to obtain a velocity distribution function for the interstellar gas. A technique based on the maximum and minimum velocities observed along a line of sight is adopted because of heavy line blending, and results are discussed for both power-law and exponential distribution functions. The expected distribution of radiative-phase supernova remnants (SNRs) in the interstellar medium is calculated as a function of SNR birthrate and of the interstellar density in which they evolve. The results are combined with observed distance estimates, and it is shown that an interstellar density in excess of 0.1 per cu cm would be required to keep the SNRs sufficiently confined so that their cross sections are consistent with the observed number of components. The alternative possibility is considered that SNRs do not enter the radiative phase before escaping from the Galaxy or colliding with neighboring remnants.

Comparison of flare bremsstrahlung resulting from energetic thermal and nonthermal electrons

We have investigated the behavior of the X-ray bremsstrahlung spectra resulting from two distinct types of electron distribution functions impinging on a target atmosphere during flare activity. A power-law distribution function is compared with two double-peaked Maxwellians. The results of these calculations show that it would be difficult to rule out multithermal interpretations for the emitted high-energy X-rays.

Lossy radial diffusion of relativistic Jovian electrons

The radial diffusion equation with synchrotron losses is solved by the Laplace transform method for near equatorially mirroring relativistic electrons. The evolution of a power law distribution function is found, and the characteristics of synchrotron burn-off are stated in terms of explicit parameters for an arbitrary diffusion coefficient of the form DLL = D0Lα. The peaking of the 10.4-cm volume emissivity from Jupiter at L ≃ 1.8 provides an estimate D0 ≃ 9 × 10−11 s−1 in the radiation belts; this value is suggested as the appropriate modification for an equatorial field strength of 4.2 G of the Birmingham et al. (1974) result. Nonsynchrotron losses are included phenomenologically, and from the phase space densities reported by McIlwain and Fillius (1975) the particle lifetime is estimated as τ ∼ 6 × 108L2−α s. Asymptotic forms for the distribution in the strong synchrotron loss regime are provided.