Creating kappa-like distributions from a Galton board

Leitner, M.; Leubner, M. P.; Vörös, Z.

doi:10.1016/j.physa.2010.11.044

Cited by 8 publications

(9 citation statements)

References 31 publications

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“…We believe it is safer to say that the low values of κ index at the scale τ are associated with the structures repeatedly generated by turbulence introducing correlations to observed fluctuations. We mention that correlations in a generalized Galton board lead to peaked and fat‐tailed kappa‐like distributions, while in the absence of correlations Gaussian distributions are obtained [ Leitner et al , ]. Figure and the fitting results in Table show that the κ and σ parameters are significantly different for the yearly and for each conditional PDFs.…”

Section: Discussionmentioning

confidence: 89%

“…The correlation length associated with solar wind turbulence is about 0.015 AU, while the correlation length of large‐scale velocity fluctuations is roughly 0.25 AU [ Podesta et al , ]. Simple simulations of statistical distributions demonstrate the formation of kappa‐like distributions from Gaussian distributions, when interaction terms and long‐range correlations are included [ Leitner et al , ]. We expect that the long‐tailed PDFs or the change of the κ parameter in the solar wind also reflects the occurrence of processes exhibiting long‐range correlations in the one‐point unfiltered data sets.…”

Section: Probability Distribution Functions For Space Plasmasmentioning

confidence: 99%

“…The nonextensive entropy introduces interactions between subsystems (e.g., particles) [ Leubner , ], and its extremization under certain physically relevant constraints leads to kappa distribution [ Tsallis et al , ; Livadiotis and McComas , ]. A simple physical model based on the generalized Galton board clearly demonstrates the formation of fat‐tailed kappa‐like distributions from Gaussian distributions, when interaction terms and memory effects are switched on [ Leitner et al , ]. The peaked, fat‐tailed data distributions corresponding to turbulence in the solar wind are well modeled by the kappa distribution [ Leubner and Vörös , , ]:

P_{κ} (x, μ, σ, κ) = \frac{1}{σ \sqrt{κπ}} \frac{Γ (κ)}{Γ (κ - 1 / 2)} {((), 1 + \frac{{(x - μ)}^{2}}{σ^{2} κ})}^{- κ}, - \infty < x < \infty .

…”

Section: Probability Distribution Functions For Space Plasmasmentioning

confidence: 99%

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Probability density functions for the variable solar wind near the solar cycle minimum

et al. 2015

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Unconditional and conditional statistics are used for studying the histograms of magnetic field multiscale fluctuations in the solar wind near the solar cycle minimum in 2008. The unconditional statistics involves the magnetic data during the whole year in 2008. The conditional statistics involves the magnetic field time series split into concatenated subsets of data according to a threshold in dynamic pressure. The threshold separates fast‐stream leading edge compressional and trailing edge uncompressional fluctuations. The histograms obtained from these data sets are associated with both multiscale (B) and small‐scale (δB) magnetic fluctuations, the latter corresponding to time‐delayed differences. It is shown here that, by keeping flexibility but avoiding the unnecessary redundancy in modeling, the histograms can be effectively described by a limited set of theoretical probability distribution functions (PDFs), such as the normal, lognormal, kappa, and log‐kappa functions. In a statistical sense the model PDFs correspond to additive and multiplicative processes exhibiting correlations. It is demonstrated here that the skewed small‐scale histograms inherent in turbulent cascades are better described by the skewed log‐kappa than by the symmetric kappa model. Nevertheless, the observed skewness is rather small, resulting in potential difficulties of estimation of the third‐order moments. This paper also investigates the dependence of the statistical convergence of PDF model parameters, goodness of fit, and skewness on the data sample size. It is shown that the minimum lengths of data intervals required for the robust estimation of parameters is scale, process, and model dependent.

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Section: Discussionmentioning

confidence: 89%

Section: Probability Distribution Functions For Space Plasmasmentioning

confidence: 99%

P_{κ} (x, μ, σ, κ) = \frac{1}{σ \sqrt{κπ}} \frac{Γ (κ)}{Γ (κ - 1 / 2)} {((), 1 + \frac{{(x - μ)}^{2}}{σ^{2} κ})}^{- κ}, - \infty < x < \infty .

…”

Section: Probability Distribution Functions For Space Plasmasmentioning

confidence: 99%

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Probability density functions for the variable solar wind near the solar cycle minimum

et al. 2015

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“…After innumerable nail collisions, it was piled up at the bottom. The final slot that a single ball is falling into is unintentional, but the majority of balls eventually fall into the center slots, with a tiny number of balls falling into slots on both sides [22][23][24][25]. Similar to balls moving in nails, bubbles in the spaces of soil particles eventually overflow the water in large quantities from the central area.…”

Section: Resultsmentioning

confidence: 99%

A 3D DEM Model for Air Sparging Technology in the Saturated Granular Soils

Liu

et al. 2022

Front. Phys.

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This work provides a three-dimensional discrete element simulation (DEM) model to study the air sparging technology. The simulations have taken into account the multi-phases of bubble (gas) - fluid (water) - soil (solid) particles. Bubbles are treated as discrete individual particles, with buoyancy and drag forces applied to bubbles and soil particles. The trajectory of each discrete bubble particle can be tracked using the discrete element model. It is found that the diffusion of the whole bubble is inverted conical though the motion behavior of a single bubble particle is random. Furthermore, the distribution of the radius of influence (ROI) is not uniform. The bubbles become more concentrated as in the center of the inverted cone. The number of bubbles dissipated from the water surface is normally distributed. The DEM simulation is a novel approach to studying air sparging technology that can provide us a deeper insight into bubble migration at the microscopic level.

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“…This has its roots in the Boltzmann-Gibbs statistical mechanics, which is in turn based on the assumption that the collisions in the studied plasma are sufficiently frequent to attain the thermal equilibrium. However, the assumption of the Maxwellian distribution has been challenged in the past decades in a wide variety of astrophysical environments, ranging from laboratory plasmas to planetary magnetospheres, the solar wind, and even galaxies (e.g., Dalla & Ljepojevic 1997;Maksimovic et al 1997a,b;Viñas et al 2000;Petkaki et al 2003;Mauk et al 2004;Leubner 2004aLeubner ,b, 2005Maksimovic et al 2005;Ryu et al 2007Ryu et al , 2009Gaelzer et al 2008;Nieves-Chinchilla & Viñas 2008;Dialynas et al 2009;Prokhorov 2009;Prokhorov et al 2009;Le Chat et al 2009;Pierrard 2009;Pierrard & Lazar 2010;Livadiotis & McComas 2010;Lu et al 2010Lu et al , 2011Leitner et al 2011).…”

Section: Introductionmentioning

confidence: 99%

The non-Maxwellian continuum in the X-ray, UV, and radio range

Dudík

Kašparová

Dzifčáková

et al. 2012

A&A

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Aims. We investigate the X-ray, UV, and also the radio continuum arising from plasmas with a non-Maxwellian distribution of electron energies. The two investigated types of distributions are the κ-and n-distributions. Methods. We derived analytical expressions for the non-Maxwellian bremsstrahlung and free-bound continuum spectra. The spectra were calculated using available cross-sections. Then we compared the bremsstrahlung spectra arising from the different bremsstrahlung cross-sections that are routinely used in solar physics.Results. The behavior of the bremsstrahlung spectra for the non-Maxwellian distributions is highly dependent on the assumed type of the distribution. At flare temperatures and hard X-ray energies, the bremsstrahlung is greatly increased for κ-distributions and exhibits a strong high-energy tail. With decreasing κ, the maximum of the bremsstrahlung spectrum decreases and moves to higher wavelengths. In contrast, the maximum of the spectra for n-distributions increases with increasing n, and the spectrum then falls off very steeply with decreasing wavelength. In the millimeter radio range, the non-Maxwellian bremsstrahlung spectra are almost parallel to the thermal bremsstrahlung. Therefore, the non-Maxwellian distributions cannot be detected by off-limb observations made by the ALMA instrument. The free-bound continua are also highly dependent on the assumed type of the distribution. For n-distributions, the ionization edges disappear and a smooth continuum spectrum is formed for n 5. Opposite behavior occurs for κ-distributions where the ionization edges are in general significantly enhanced, with details depending on κ and T through the ionization equilibrium. We investigated how the non-Maxwellian κ-distributions can be determined from the observations of the continuum and conclude that one can sample the low-energy part of the distribution from the continuum.

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Creating kappa-like distributions from a Galton board

Cited by 8 publications

References 31 publications

Probability density functions for the variable solar wind near the solar cycle minimum

Probability density functions for the variable solar wind near the solar cycle minimum

A 3D DEM Model for Air Sparging Technology in the Saturated Granular Soils

The non-Maxwellian continuum in the X-ray, UV, and radio range

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