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Register a new userThe relationship between extremum statistics and universal fluctuations
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The normalized probability density function (PDF) of global measures of a large class of highly correlated systems has previously been demonstrated to fall on a single non-Gaussian universal curve. We derive the functional form of the global PDF in terms of the source PDF of the individual events in the system. A single parameter distinguishes the global PDF and is related to the exponent of the source PDF. When normalized, the global PDF is shown to be insensitive to this parameter and importantly we obtain the previously demonstrated universality from an uncorrelated Gaussian source PDF. The second and third moments of the global PDF are more sensitive, providing a powerful tool to probe the degree of complexity of physical systems.
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Recent work has suggested that in highly correlated systems, such as sandpiles, turbulent fluids, ignited trees in forest fires and magnetization in a ferromagnet close to a critical point, the probability distribution of a global quantity (i.e. total energy dissipation, magnetization and so forth) that has been normalized to the first two moments follows a specific non Gaussian curve. This curve follows a form suggested by extremum statistics, which is specified by a single parameter a (a=1 corresponds to the Fisher-Tippett Type I (Gumbel) distribution.) Here, we present a framework for testing for extremal statistics in a global observable. In any given system, we wish to obtain a in order to distinguish between the different Fisher-Tippett asymptotes, and to compare with the above work. The normalizations of the extremal curves are obtained as a function of a. We find that for realistic ranges of data, the various extremal distributions when normalized to the first two moments are difficult to distinguish. In addition, the convergence to the limiting extremal distributions for finite datasets is both slow and varies with the asymptote. However, when the third moment is expressed as a function of a this is found to be a more sensitive method.
We investigate the general property of the energy fluctuation for the canonical ensemble in Tsallis statistics and the ensemble equivalence. By taking the ideal gas and the non-interacting harmonic oscillators as examples, we show that, when the particle number N is large enough, the relative fluctuation of the energy is proportional to 1/N in the new statistics, instead of square root of 1/N in Boltzmann-Gibbs statistics. Thus the equivalence between the microcanonical and the canonical ensemble still holds in Tsallis statistics.
Subdiffusive transport in tilted washboard potentials is studied within the fractional Fokker-Planck equation approach, using the associated continuous time random walk (CTRW) framework. The scaled subvelocity is shown to obey a universal law, assuming the form of a stationary Levy-stable distribution. The latter is defined by the index of subdiffusion alpha and the mean subvelocity only, but interestingly depends neither on the bias strength nor on the specific form of the potential. These scaled, universal subvelocity fluctuations emerge due to the weak ergodicity breaking and are vanishing in the limit of normal diffusion. The results of the analytical heuristic theory are corroborated by Monte Carlo simulations of the underlying CTRW.
Radar data collected at an experimental facility arranged on purpose suggest that the footprint of atmospheric turbulence might be encoded in the radar signal statistics. Radar data probability distributions are calculated and nicely fitted by a one parameter family of generalized Gumbel (GG) distributions, G(a). A relation between the wind strength and the measured shape parameter a is obtained. Strong wind fluctuations return pronounced asymmetric leptokurtic profiles, while Gaussian profiles are eventually recovered as the wind fluctuations decrease. Besides stressing the crucial impact of air turbulence for radar applications, we also confirm the adequacy of G(a) statistics for highly correlated complex systems.
The relationship between irreversibility field, Hirr, and crystallinity of MgB2 bulks including carbon substituted samples was studied. The Hirr was found to increase with an increase of FWHM of MgB2 (110) peak, which corresponds to distortion of honeycomb boron sheet, and their universal correlation was discovered even including carbon substituted samples. Excellent Jc characteristics under high magnetic fields were observed in samples with large FWHM of (110) due to the enhanced intraband scattering and strengthened grain boundary flux pinning. The relationship between crystallinity and Hirr can explain the large variation of Hirr for MgB2 bulks, tapes, single crystals and thin films.
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