Entropy

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11 pages, 3606 KiB

Open AccessArticle

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 2465

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

Open AccessArticle

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 14 | Viewed by 4452

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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The first order kinetic energy moment <math display="inline"><semantics> <mrow> <msup> <mi>M</mi> <mn>1</mn> </msup> </mrow> </semantics></math> as a function of the kappa index κ, for five different plasma temperatures T. Full article ">Figure 2
Measurement sample for plasma with n = 20 cm−3, <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mn>0</mn> </msub> </mrow> </semantics></math> = 500 kms−1 towards Θ = 0° and Φ = 0°, T = 20 eV, and κ = 3, recorded by the top-hat electrostatic analyzer design we consider in this study. The left panel shows the registered number of counts Cout as a function of log10(E) and Θ integrated over Φ, while the right panel shows Cout as a function of log10(E) and Φ, integrated over Θ. Full article ">Figure 3
Histograms of the derived (top left) nout, (top right) u0,out, (bottom left) Tout, and (bottom right) κout, by the analysis of 1000 measurement samples considering plasma with n = 20 cm−3, <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mn>0</mn> </msub> </mrow> </semantics></math> = 500 kms−1 towards Θ = 0° and Φ = 0°, T = 20 eV, and κ = 3. In each panel, we show the mean value μ and the standard deviation σ of the derived moments, while the vertical blue dashed line indicates the corresponding input value. Full article ">Figure 4
The mean kappa index κout and its standard deviation σκ,out as functions of the energy moment order we use to analyze the data-set. Full article ">Figure 5
(Top left) The occurrence of <math display="inline"><semantics> <mrow> <msubsup> <mi>M</mi> <mrow> <mi>out</mi> </mrow> <mn>1</mn> </msubsup> </mrow> </semantics></math> and (lower right) Tout, as derived from the analysis of 1000 samples of plasma with n = 20 cm−3, <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mn>0</mn> </msub> </mrow> </semantics></math> = 500 kms−1 towards Θ = 0° and Φ = 0°, T = 20 eV, and κ = 3. (Top right) Solutions of κout as a function of Tout and <math display="inline"><semantics> <mrow> <msubsup> <mi>M</mi> <mrow> <mi>out</mi> </mrow> <mn>1</mn> </msubsup> </mrow> </semantics></math> according to Equation (12). On each panel, the blue lines indicate the input parameters and the black lines the derived parameters in our example. Full article ">Figure 6
Solutions of κout as a function of Tout and <math display="inline"><semantics> <mrow> <msubsup> <mi>M</mi> <mrow> <mi>out</mi> </mrow> <mn>1</mn> </msubsup> </mrow> </semantics></math> according to Equation (12). The black circle indicates the average parameters as derived from the analysis of 1000 observation samples by our standard instrument model with field of view −22.5° < Θ < +22.5°, −45° < Φ < +45°, and angular resolution ΔΘ = 5° and ΔΦ = 6° respectively. The green circle indicates the average parameters as derived from the analysis of 1000 observation samples by an instrument with field of view −45° < Θ < +45°, −45° < Φ < +45°, and angular resolution ΔΘ = ΔΦ = 2.5°. The input plasma parameters are the same as in <a href="#sec3-entropy-22-00212" class="html-sec">Section 3</a> and are indicated by the blue circle. Full article ">

14 pages, 3812 KiB

Open AccessEditor’s ChoiceArticle

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 15 | Viewed by 4878

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

Open AccessArticle

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 - 7 Nov 2019

Cited by 7 | Viewed by 2935

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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Figure 1
Transformation of velocities under reflection. Full article ">Figure 2
The dependence of the smallest scales kappa-index, κ0(0), on the Reynolds number, according to a power fit of the flatness factor, Equation (28), and according to the specific fit of [<a href="#B5-entropy-21-01093" class="html-bibr">5</a>], for a range of Reynolds the best fit of [<a href="#B5-entropy-21-01093" class="html-bibr">5</a>] that have been covered by the recent direct numerical simulations (DNS). Full article ">Figure 3
Plot of the function g(κ0) defined by Equation (43). g becomes infinite as κ0 approaches the value 0.5. Full article ">Figure 4
The ratio g(κ0(0))B3(r)/(B2(r))3/2/S as it follows from DNS data [<a href="#B6-entropy-21-01093" class="html-bibr">6</a>] for Reynolds number Reλ= 1131 (continuous line) and Reλ= 732 (dashed line), and Equations (43) and (29). Full article ">Figure 5
The kappa-index κ0(r) as a function of r/η derived from DNS data [<a href="#B6-entropy-21-01093" class="html-bibr">6</a>] for Reynolds number Reλ= 1131 (continuous line) and Reλ= 732 (dashed line), and Equation (43). The kappa-index starts off with an estimated value κ0(0) ~ 1.4 and κ0(0) ~ 1.66, respectively, and increases without bound. The dotted lines indicate the expected behavior in the region where we do not have information from the DNS, through interpolation. Full article ">Figure 6
Continuous line: The model probability density function (PDF) of the normalized (longitudinal) velocity derivative for Reλ = 675, as it follows from Equations (61), (63), (29). Dashed line: The corresponding DNS data for Reλ = 675 [<a href="#B5-entropy-21-01093" class="html-bibr">5</a>]. Full article ">

17 pages, 5433 KiB

Open AccessArticle

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 19 | Viewed by 4744

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

Open AccessArticle

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 6 | Viewed by 4309

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

Open AccessArticle

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 - 2 Oct 2018

Cited by 34 | Viewed by 3442

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

Open AccessArticle

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 - 4 Jan 2018

Cited by 4 | Viewed by 5442

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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