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Theoretical Aspects of Kappa Distributions
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Theoretical Aspects of Kappa Distributions

A special issue of Entropy (ISSN 1099-4300). This special issue belongs to the section "Statistical Physics".

Deadline for manuscript submissions: closed (31 December 2019) | Viewed by 28291

Special Issue Editor

Dr. George Livadiotis

E-Mail Website1 Website2
Guest Editor
Space Science and Engineering, Southwest Research Institute, San Antonio, TX 78238, USA
Interests: statistical physics & thermodynamics; mathematical physics; plasma & space physics; nonlinear dynamics & complexity

Special Issue Information

Dear Colleagues,

Classical particle systems reside at thermal equilibrium with their velocity distribution function, stabilized into a Maxwell distribution. On the other hand, collisionless and correlated particle systems, such as space plasmas, are characterized by a non-Maxwellian behavior, typically described by the so-called kappa distributions. Empirical kappa distributions have become increasingly widespread across space and plasma physics. However, a breakthrough in the field came with the connection of kappa distributions with the solid background of non-extensive statistical mechanics. Understanding the statistical background and origin of kappa distributions was a cornerstone of further theoretical developments, for example, among many others: the physical meaning of thermal parameters, e.g., temperature and kappa index; the N-particle description of kappa distributions; the generalization to phase-space kappa distribution of a Hamiltonian with non-zero potential; the entropy associated with kappa distributions.

In this Special Issue, we welcome papers reporting on the progress of the theory of kappa distributions. The subjects may include, but are not limited to, the following three broad areas:

A.    Statistical background:

-    Connection of kappa distributions with Non-extensive statistical mechanics;
-    Superstatistics and formulation of kappa distributions;
-    Superposition on kappa indices.

B.    Formulation:

-     Multi-particle distributions;
-     Distributions in the presence of potential energy;
-     Anisotropy of velocity space;
-     Relativistic distributions;
-     Further generalization in Lp norms;
-     Discrete formalism.

C.    Properties:

-     Concept of temperature for stationary states out of thermal equilibrium;
-     Physical meaning of kappa and its connection to particle correlations;
-     Higher statistical moments;
-     Parameter estimation methods;
-     Rankine–Hugoniot conditions for shocks in particle systems described by kappa distributions;
-     Polytropic relations and connection with the theory of kappa distributions;
-     Entropic formulations associated with kappa distributions;
-     Information measures and kappa distributions.

Dr. George  Livadiotis
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Entropy is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Published Papers (8 papers)

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Research

11 pages, 3606 KiB  
Article
Characteristics of Nonthermal Dupree Diffusion on Space-Charge Wave in a Kappa Distribution Plasma Column with Turbulent Diffusion
by Myoung-Jae Lee and Young-Dae Jung
Entropy 2020, 22(2), 257; https://doi.org/10.3390/e22020257 - 24 Feb 2020
Cited by 2 | Viewed by 2000
Abstract
The nonthermal diffusion effects on the dispersion equations of ion-acoustic space-charge wave (SCW) in a nonthermal plasma column composed of nonthermal turbulent electrons and cold ions are investigated based on the analysis of normal modes and the separation of variables. It is found [...] Read more.
The nonthermal diffusion effects on the dispersion equations of ion-acoustic space-charge wave (SCW) in a nonthermal plasma column composed of nonthermal turbulent electrons and cold ions are investigated based on the analysis of normal modes and the separation of variables. It is found that the real portion of the wave frequency of the SCW in a Maxwellian plasma is greater than that in a nonthermal plasma. It is also found that the magnitude of the damping rate of the SCW decreases with an increase of the spectral index of the nonthermal plasma. It is also shown that the magnitude of the scaled damping rate increases with an increase of the Dupree diffusion coefficient. Moreover, the influence of the nonthermal character of the nonthermal plasma on the damping rate is found to be more significant in turbulent plasmas with higher diffusion coefficient. The variations of the wave frequency and the growth rate due to the characteristics of nonthermal diffusion are also discussed. Full article
(This article belongs to the Special Issue Theoretical Aspects of Kappa Distributions)
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11 pages, 2279 KiB  
Article
On the Determination of Kappa Distribution Functions from Space Plasma Observations
by Georgios Nicolaou, George Livadiotis and Robert T. Wicks
Entropy 2020, 22(2), 212; https://doi.org/10.3390/e22020212 - 13 Feb 2020
Cited by 9 | Viewed by 3525
Abstract
The velocities of space plasma particles, often follow kappa distribution functions. The kappa index, which labels and governs these distributions, is an important parameter in understanding the plasma dynamics. Space science missions often carry plasma instruments on board which observe the plasma particles [...] Read more.
The velocities of space plasma particles, often follow kappa distribution functions. The kappa index, which labels and governs these distributions, is an important parameter in understanding the plasma dynamics. Space science missions often carry plasma instruments on board which observe the plasma particles and construct their velocity distribution functions. A proper analysis of the velocity distribution functions derives the plasma bulk parameters, such as the plasma density, speed, temperature, and kappa index. Commonly, the plasma bulk density, velocity, and temperature are determined from the velocity moments of the observed distribution function. Interestingly, recent studies demonstrated the calculation of the kappa index from the speed (kinetic energy) moments of the distribution function. Such a novel calculation could be very useful in future analyses and applications. This study examines the accuracy of the specific method using synthetic plasma proton observations by a typical electrostatic analyzer. We analyze the modeled observations in order to derive the plasma bulk parameters, which we compare with the parameters we used to model the observations in the first place. Through this comparison, we quantify the systematic and statistical errors in the derived moments, and we discuss their possible sources. Full article
(This article belongs to the Special Issue Theoretical Aspects of Kappa Distributions)
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14 pages, 3812 KiB  
Article
Determining the Bulk Parameters of Plasma Electrons from Pitch-Angle Distribution Measurements
by Georgios Nicolaou, Robert Wicks, George Livadiotis, Daniel Verscharen, Christopher Owen and Dhiren Kataria
Entropy 2020, 22(1), 103; https://doi.org/10.3390/e22010103 - 16 Jan 2020
Cited by 13 | Viewed by 4165
Abstract
Electrostatic analysers measure the flux of plasma particles in velocity space and determine their velocity distribution function. There are occasions when science objectives require high time-resolution measurements, and the instrument operates in short measurement cycles, sampling only a portion of the velocity distribution [...] Read more.
Electrostatic analysers measure the flux of plasma particles in velocity space and determine their velocity distribution function. There are occasions when science objectives require high time-resolution measurements, and the instrument operates in short measurement cycles, sampling only a portion of the velocity distribution function. One such high-resolution measurement strategy consists of sampling the two-dimensional pitch-angle distributions of the plasma particles, which describes the velocities of the particles with respect to the local magnetic field direction. Here, we investigate the accuracy of plasma bulk parameters from such high-resolution measurements. We simulate electron observations from the Solar Wind Analyser’s (SWA) Electron Analyser System (EAS) on board Solar Orbiter. We show that fitting analysis of the synthetic datasets determines the plasma temperature and kappa index of the distribution within 10% of their actual values, even at large heliocentric distances where the expected solar wind flux is very low. Interestingly, we show that although measurement points with zero counts are not statistically significant, they provide information about the particle distribution function which becomes important when the particle flux is low. We also examine the convergence of the fitting algorithm for expected plasma conditions and discuss the sources of statistical and systematic uncertainties. Full article
(This article belongs to the Special Issue Theoretical Aspects of Kappa Distributions)
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20 pages, 1875 KiB  
Article
Kappa Distributions and Isotropic Turbulence
by Elias Gravanis, Evangelos Akylas, Constantinos Panagiotou and George Livadiotis
Entropy 2019, 21(11), 1093; https://doi.org/10.3390/e21111093 - 07 Nov 2019
Cited by 6 | Viewed by 2420
Abstract
In this work, the two-point probability density function (PDF) for the velocity field of isotropic turbulence is modeled using the kappa distribution and the concept of superstatistics. The PDF consists of a symmetric and an anti-symmetric part, whose symmetry properties follow from the [...] Read more.
In this work, the two-point probability density function (PDF) for the velocity field of isotropic turbulence is modeled using the kappa distribution and the concept of superstatistics. The PDF consists of a symmetric and an anti-symmetric part, whose symmetry properties follow from the reflection symmetry of isotropic turbulence, and the associated non-trivial conditions are established. The symmetric part is modeled by the kappa distribution. The anti-symmetric part, constructed in the context of superstatistics, is a novel function whose simplest form (called “the minimal model”) is solely dictated by the symmetry conditions. We obtain that the ensemble of eddies of size up to a given length r has a temperature parameter given by the second order structure function and a kappa-index related to the second and the third order structure functions. The latter relationship depends on the inverse temperature parameter (gamma) distribution of the superstatistics and it is not specific to the minimal model. Comparison with data from direct numerical simulations (DNS) of turbulence shows that our model is applicable within the dissipation subrange of scales. Also, the derived PDF of the velocity gradient shows excellent agreement with the DNS in six orders of magnitude. Future developments, in the context of superstatistics, are also discussed. Full article
(This article belongs to the Special Issue Theoretical Aspects of Kappa Distributions)
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17 pages, 5433 KiB  
Article
On the Calculation of the Effective Polytropic Index in Space Plasmas
by Georgios Nicolaou, George Livadiotis and Robert T. Wicks
Entropy 2019, 21(10), 997; https://doi.org/10.3390/e21100997 - 12 Oct 2019
Cited by 11 | Viewed by 4004
Abstract
The polytropic index of space plasmas is typically determined from the relationship between the measured plasma density and temperature. In this study, we quantify the errors in the determination of the polytropic index, due to uncertainty in the analyzed measurements. We model the [...] Read more.
The polytropic index of space plasmas is typically determined from the relationship between the measured plasma density and temperature. In this study, we quantify the errors in the determination of the polytropic index, due to uncertainty in the analyzed measurements. We model the plasma density and temperature measurements for a certain polytropic index, and then, we apply the standard analysis to derive the polytropic index. We explore the accuracy of the derived polytropic index for a range of uncertainties in the modeled density and temperature and repeat for various polytropic indices. Our analysis shows that the uncertainties in the plasma density introduce a systematic error in the determination of the polytropic index which can lead to artificial isothermal relations, while the uncertainties in the plasma temperature increase the statistical error of the calculated polytropic index value. We analyze Wind spacecraft observations of the solar wind protons and we derive the polytropic index in selected intervals over 2002. The derived polytropic index is affected by the plasma measurement uncertainties, in a similar way as predicted by our model. Finally, we suggest a new data-analysis approach, based on a physical constraint, that reduces the amount of erroneous derivations. Full article
(This article belongs to the Special Issue Theoretical Aspects of Kappa Distributions)
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45 pages, 8763 KiB  
Article
Non-Extensive Statistical Analysis of Energetic Particle Flux Enhancements Caused by the Interplanetary Coronal Mass Ejection-Heliospheric Current Sheet Interaction
by Evgenios G. Pavlos, Olga E. Malandraki, Olga V. Khabarova, Leonidas P. Karakatsanis, George P. Pavlos and George Livadiotis
Entropy 2019, 21(7), 648; https://doi.org/10.3390/e21070648 - 30 Jun 2019
Cited by 5 | Viewed by 3729
Abstract
In this study we use theoretical concepts and computational-diagnostic tools of Tsallis non-extensive statistical theory (Tsallis q-triplet: q s e n ,   q r e l ,   q s t a t ), complemented by other known tools of nonlinear dynamics [...] Read more.
In this study we use theoretical concepts and computational-diagnostic tools of Tsallis non-extensive statistical theory (Tsallis q-triplet: q s e n ,   q r e l ,   q s t a t ), complemented by other known tools of nonlinear dynamics such as Correlation Dimension and surrogate data, Hurst exponent, Flatness coefficient, and p-modeling of multifractality, in order to describe and understand Small-scale Magnetic Islands (SMIs) structures observed in Solar Wind (SW) with a typical size of ~0.01–0.001 AU at 1 AU. Specifically, we analyze ~0.5 MeV energetic ion time-intensity and magnetic field profiles observed by the STEREO A spacecraft during a rare, widely discussed event. Our analysis clearly reveals the non-extensive character of SW space plasmas during the periods of SMIs events, as well as significant physical complex phenomena in accordance with nonlinear dynamics and complexity theory. As our analysis also shows, a non-equilibrium phase transition parallel with self-organization processes, including the reduction of dimensionality and development of long-range correlations in connection with anomalous diffusion and fractional acceleration processes can be observed during SMIs events. Full article
(This article belongs to the Special Issue Theoretical Aspects of Kappa Distributions)
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16 pages, 1430 KiB  
Article
Tsallis Entropy Index q and the Complexity Measure of Seismicity in Natural Time under Time Reversal before the M9 Tohoku Earthquake in 2011
by Panayiotis A. Varotsos, Nicholas V. Sarlis and Efthimios S. Skordas
Entropy 2018, 20(10), 757; https://doi.org/10.3390/e20100757 - 02 Oct 2018
Cited by 30 | Viewed by 2871
Abstract
The observed earthquake scaling laws indicate the existence of phenomena closely associated with the proximity of the system to a critical point. Taking this view that earthquakes are critical phenomena (dynamic phase transitions), here we investigate whether in this case the Lifshitz–Slyozov–Wagner (LSW) [...] Read more.
The observed earthquake scaling laws indicate the existence of phenomena closely associated with the proximity of the system to a critical point. Taking this view that earthquakes are critical phenomena (dynamic phase transitions), here we investigate whether in this case the Lifshitz–Slyozov–Wagner (LSW) theory for phase transitions showing that the characteristic size of the minority phase droplets grows with time as t 1 / 3 is applicable. To achieve this goal, we analyzed the Japanese seismic data in a new time domain termed natural time and find that an LSW behavior is actually obeyed by a precursory change of seismicity and in particular by the fluctuations of the entropy change of seismicity under time reversal before the Tohoku earthquake of magnitude 9.0 that occurred on 11 March 2011 in Japan. Furthermore, the Tsallis entropic index q is found to exhibit a precursory increase. Full article
(This article belongs to the Special Issue Theoretical Aspects of Kappa Distributions)
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21 pages, 19549 KiB  
Article
High Density Nodes in the Chaotic Region of 1D Discrete Maps
by George Livadiotis
Entropy 2018, 20(1), 24; https://doi.org/10.3390/e20010024 - 04 Jan 2018
Cited by 4 | Viewed by 4955
Abstract
We report on the definition and characteristics of nodes in the chaotic region of bifurcation diagrams in the case of 1D mono-parametrical and S-unimodal maps, using as guiding example the logistic map. We examine the arrangement of critical curves, the identification and arrangement [...] Read more.
We report on the definition and characteristics of nodes in the chaotic region of bifurcation diagrams in the case of 1D mono-parametrical and S-unimodal maps, using as guiding example the logistic map. We examine the arrangement of critical curves, the identification and arrangement of nodes, and the connection between the periodic windows and nodes in the chaotic zone. We finally present several characteristic features of nodes, which involve their convergence and entropy. Full article
(This article belongs to the Special Issue Theoretical Aspects of Kappa Distributions)
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