Formulae of Kappa Distributions

CHAPTER 4 Formulae of Kappa Distributions: Toolbox G. Livadiotis Southwest Research Institute, San Antonio, TX, United States Chapter Outline 4.1 Su...

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CHAPTER 4

Formulae of Kappa Distributions: Toolbox G. Livadiotis Southwest Research Institute, San Antonio, TX, United States

Chapter Outline 4.1 Summary 185 4.2 Introduction 186 4.3 Isotropic Distributions (Without Potential) 187 4.3.1 Standard (Positive) Multidimensional Kappa Distributions 187 4.3.1.1 Distributions, for dK Degrees of Freedom, Using the Invariant Kappa Index k0 187 4.3.1.2 Distributions, for dK Degrees of Freedom, Using the 3-D Kappa Index k3 ¼ k0 þ 32 188 4.3.1.3 Distributions, for dK Degrees of Freedom, Using the dKDimensional Kappa Index k ¼ k0 þ 12dK 188

4.3.2 Standard (Positive) Multidimensional Kappa Distributions in an

Kappa Distributions. http://dx.doi.org/10.1016/B978-0-12-804638-8.00004-8 Copyright 2017 Elsevier Inc. All rights reserved.

Inertial Reference Frame 189 4.3.2.1 Distributions, for dK Degrees of Freedom, Using the Invariant Kappa Index k0 189 4.3.2.2 Distributions, for dK ¼ 3 Degrees of Freedom, Using the Invariant Kappa Index k0 190 4.3.2.3 Distributions, for dK ¼ 3 Degrees of Freedom, Using the 3-D Kappa Index k ¼ k0 þ 32 192

4.3.3 Negative, Multidimensional Kappa Distribution 193 4.3.3.1 Distributions Using the

177

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PART 1 Theory and Formalism

Invariant Kappa Index k0 193 4.3.3.2 Distributions Using the Kappa Index k ¼ k0 12dK 193

4.3.4 Superposition of Multidimensional Kappa Distributions 194 4.3.4.1 Linear Superposition of Positive and Negative Distributions in Terms of the Kinetic Energy εK 194 4.3.4.2 Linear Superposition Given a Density of Kappa Indices D(k0) 195 4.3.4.3 Linear Superposition in Terms of the Kinetic Energy εK with Components of Different Temperature 196 4.3.4.4 Nonlinear Superposition of Positive and Negative Distributions in Terms of the Kinetic Energy εK 196

4.4 Anisotropic Distributions (Without Potential) 196

4.4.1 Correlated Degrees of Freedom 196 4.4.1.1 Distributions, for dK Degrees of Freedom, Using the Invariant Kappa Index k0 196 4.4.1.2 Distribution in Terms of the / Velocity u for dK Degrees of Freedom, Using the dKDimensional Kappa Index k ¼ k0 þ 12dK 197 4.4.1.3 Distribution in Terms of the / Velocity u for dK Degrees of Freedom, Using the 3-D Kappa Index k3 ¼ k0 þ 12 (dK 3) 198 4.4.1.4 Distributions, for dK ¼ 3 Degrees of Freedom, Using the Invariant Kappa Index k0 198 4.4.1.5 Distribution, for dK ¼ 3 Degrees of Freedom, Using the 3-D Kappa Index k ¼ k0 þ 32 199

4.4.2 Correlation Between the Projection at a Certain

Formulae of Kappa Distributions: Toolbox CHAPTER 4

Direction and the Perpendicular Plane 200 4.4.2.1 Distributions, for dK ¼ 3 Degrees of Freedom, Using the Invariant Kappa Index k0 200 4.4.2.2 Distribution in Terms of the Velocity / / u ¼ uk ; u t for dK ¼ 3 Degrees of Freedom, Using the 3-D Kappa Index k ¼ k0 þ 32 201 4.4.2.3 The Temperature is Given by 201

4.4.3 Self-Correlated Degrees of Freedom 201 4.4.3.1 Distributions, for dK Degrees of Freedom, Using the Invariant Kappa Index k0 201 4.4.3.2 Distribution in Terms of the /

Velocity, u , for dK Degrees of Freedom, Using the dKDimensional Kappa Index k ¼ k0 þ 12dK 202 4.4.3.3 Distribution in Terms of the / Velocity, u , for

dK Degrees of Freedom, Using the 3-D Kappa Index k ¼ k0 þ 32 202 4.4.3.4 Distributions, for dK ¼ 3 Degrees of Freedom, Using the Invariant Kappa Index k0 202 4.4.3.5 Distribution, for dK ¼ 3 Degrees of Freedom, Using the 3-D Kappa Index k ¼ k0 þ 3 203 2

4.4.4 Self-Correlated Projections at a Direction and Perpendicular Plane 204 4.4.4.1 Distributions, for dK ¼ 3 Degrees of Freedom, Using the Invariant Kappa Index k0 204 4.4.4.2 Distribution in Terms of the Velocity / / u ¼ uk ; u t for dK ¼ 3 Degrees of Freedom, Using the 3-D Kappa Index k ¼ k0 þ 3 205 2 4.4.4.3 The Temperature is Given by 205

179

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PART 1 Theory and Formalism

4.4.5 Self-Correlated Degrees of Freedom With Different Kappa 205 4.4.5.1 Distributions, for dK Degrees of Freedom, Using the Invariant Kappa Index k0 205 4.4.5.2 Distribution in Terms of the / Velocity u for dK Degrees of Freedom, Using the 1-D Kappa Index ki ¼ k0i þ 12 206 4.4.5.3 Distribution in Terms of the / Velocity u for dK Degrees of Freedom, Using the 3-D Kappa Index k ¼ k0 þ 32 206

4.4.6 Self-Correlated Projections at a Direction and Perpendicular Plane With Different Kappa 207 4.4.6.1 Distributions, for dK ¼ 3 Degrees of Freedom, Using the Invariant Kappa Index k0 207 4.4.6.2 Distribution in Terms of the Velocity / / u ¼ uk ; u t

for dK ¼ 3 Degrees of Freedom, Using the Kappa Indices 207 4.4.6.3 The Temperature is Given by 208

4.4.7 Self-Correlated Projections of Different Dimensionality and Kappa 208 4.4.7.1 Distributions, for dK Degrees of Freedom of Mixed Correlation, i.e., i: 1, ., M Uncorrelated Groups of Correlated Degrees of Freedom, With Each Group Having Different Degrees of Freedom fi, Using Different (Invariant) Kappa Index, k0i, but Common Temperature T or q 208 4.4.7.2 Distributions, for M Correlated Groups, Each With fi Degrees of Freedom, Self-Correlated With Kappa Index k0i and Temperature qi 209

Formulae of Kappa Distributions: Toolbox CHAPTER 4

4.4.8 Different Self-Correlation and Intercorrelation Between Degrees of Freedom 210 4.4.8.1 Distributions, for M Correlated Groups, Each With fi Degrees of Freedom, Self-Correlated With Invariant Kappa Index, k0i, and Intercorrelated With Invariant Kappa Index kint 0i 210 4.4.8.2 Distributions, for M Correlated Groups, Each With fi Degrees of Freedom, Self-Correlated With Kappa Index, ki, and Intercorrelated With Kappa Index kint 211 i 4.4.8.3 Distributions, for M Correlated Groups, Each With fi Degrees of Freedom, Self-Correlated With Invariant Kappa Index, k0i ¼ k0, and Intercorrelated With Invariant Kappa Index kint 0 213 4.4.8.4 Distributions, for dK ¼ 3 Correlated Groups, Each

With fi ¼ 1 Degrees of Freedom, SelfCorrelated With Kappa Index, k0i, and Intercorrelated With Kappa Index kint 213 0 4.4.8.5 Distributions, for Two Correlated Groups, With f1 ¼ 1 and f2 ¼ 2 Degrees of Freedom, Self-Correlated With Kappa Index k0i, and Intercorrelated With Kappa Index kint 216 0

4.5 Distributions With Potential 217 4.5.1 General Hamiltonian Distribution 218 4.5.1.1 Phase Space Distribution 218 4.5.1.2 Hamiltonian Function 218 4.5.1.3 Hamiltonian Degrees of Freedom Summing Up the Kinetic and Potential Degrees of Freedom 218

4.5.2 Positive Attractive / Potential F r > 0 218 4.5.2.1 Phase Space Distributions 218

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PART 1 Theory and Formalism

4.5.2.2 Positional Distribution Function 219 4.5.2.3 Potential Degrees of Freedom 219

4.5.3 Negative Attractive / Potential F r < 0 220 4.5.3.1 Phase Space Distributions 220 4.5.3.2 Positional Distribution Function 221 4.5.3.3 Potential Degrees of Freedom 221

4.5.4 Small Positive/Negative Attractive/Repulsive Potential / jF r = k0 k B T j << 1 Defined in a Finite /

Volume r ˛V

221

4.5.4.1 Phase Space Distribution 221 4.5.4.2 Positional Distribution Function 222 4.5.4.3 Potential Degrees of Freedom 222

4.5.5 Equivalent Local Distribution 222 4.5.5.1 Phase Space Distribution With Potential / 223 F r 4.5.5.2 Equivalent Local Distribution With No Potential Energy 223

4.5.6 Positive Power Law Central Potential (Oscillation Type) Fðr Þ ¼ 1b kr b 224 4.5.6.1 Phase Space Distributions 224 4.5.6.2 Potential Degrees of Freedom 224 4.5.6.3 Positional Distribution 225

4.5.7 Negative Power Law Central Potential (Gravitational Type) Fðr Þ ¼ 1b kr b 226 4.5.7.1 Phase Space Distributions 226 4.5.7.2 Potential Degrees of Freedom 226 4.5.7.3 Positional Distribution 227

4.5.8 Properties for Fðr Þ ¼ 1 kr b 228 b 4.5.8.1 Parameters 228 4.5.8.2 Degeneration of the Kappa Index 228 4.5.8.3 Local Parameters 228 4.5.8.4 Polytropic Index 229 4.5.8.5 Statistical Moments 229

4.5.9 Marginal and Conditional Distributions 230

Formulae of Kappa Distributions: Toolbox CHAPTER 4

4.5.9.1 Marginal Distributions 230 4.5.9.2 Conditional Distributions 230

4.5.10 Angular Potentials F(w,4) 231 4.5.10.1 Phase Space Distribution 231 4.5.10.2 Potential Degrees of Freedom 232 4.5.10.3 Kappa Index Degeneration 232

4.5.11 Magnetization Potential F(w) f cosw 232 4.5.11.1 Potential Energy 232 4.5.11.2 Phase Space Distribution, in Terms of the Positional Polar Angle w and the Kinetic Energy εK, for 1 Positional and dK Kinetic Degrees of Freedom 232 4.5.11.3 Distribution of the Kinetic Energy εK 232 4.5.11.4 Distribution of the Polar Angle cosw 233 4.5.11.5 Degeneration of the Kappa Index 233

4.6 Multiparticle Distributions

234

4.6.1 Standard N-Particle (N$d) eDimensional Kappa Distributions 234 4.6.1.1 Distributions, in Terms of the / Velocity, u , for N Particles, dK Degrees of Freedom per Particle, Using the Invariant Kappa Index k0 234 4.6.1.2 Distributions, in Terms of the Kinetic Energy, εKðnÞ ¼ 12 m / / 2 u ðnÞ u b , for N Particles, dK Degrees of Freedom per Particle, Using the Invariant Kappa Index k0 234

4.6.2 Negative N-Particle (N$d) eDimensional Kappa Distribution 235 4.6.2.1 Distributions, in Terms of the / Velocity, u , for N Particles, dK Degrees of Freedom per Particle, Using the Invariant Kappa Index k0 235 4.6.2.2 Distributions, in Terms of the Kinetic Energy, εK(n), for N Particles, dK

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PART 1 Theory and Formalism

Degrees of Freedom per Particle, Using the Invariant Kappa Index k0 235

4.6.3 N-Particle (N$d) eDimensional Kappa Distributions With Potential 235 4.6.3.1 Phase Space Distribution, for N Particles, dK Degrees of Freedom per Particle, Using the Invariant Kappa Index k0 235 4.6.3.2 Hamiltonian Function 236 4.6.3.3 Hamiltonian Degrees of Freedom Summing up the Kinetic and Potential Degrees of Freedom 236

4.6.4 Standard N-Particle (N$d) eDimensional Kappa Distributions 236 4.6.4.1 Distributions, in Terms of the / Velocity u With dK Degrees of Freedom per Particle, Using the Invariant Kappa Index k0 236 4.6.4.2 Distributions, in Terms of the Kinetic Energies

feKðnÞ gNn¼1 With dK Degrees of Freedom per Particle, Using the Invariant Kappa Index k0 237 4.6.4.3 Phase Space Distributions r ðnÞ ; fHðnÞ ¼ Hð~ c ~ uðnÞ ÞgNn¼1 With

d Degrees of Freedom per Particle, Summing dK Kinetic and dF Potential Degrees of Freedom, Using the Invariant Kappa Index k0 238

4.6.5 Multispecies Distributions

240

4.6.5.1 Distributions, in Terms of the Velocity, of N Different Particle / Species, u ð1Þ ; / / u ð2Þ ; /; u ðN Þ , With dK Degrees of Freedom per Particle, Using the Invariant Kappa Index, k0 240 4.6.5.2 Distributions, in Terms of the Kinetic Energies, feKðnÞ gNn¼1 , With dK Degrees of Freedom per

Formulae of Kappa Distributions: Toolbox CHAPTER 4

Particle, Using the Invariant Kappa Index, k0 241 4.6.5.3 Phase Space Distributions of N Species r ðnÞ ; fHðnÞ ¼ Hð~ ~ uðnÞ ÞgNn¼1 With d Degrees of Freedom per Particle, Summing dK Kinetic and dF Potential Degrees of Freedom, Using the Invariant Kappa Index, k0 242

4.7 Non-Euclideane Normed Distributions 243 4.7.1 Standard dKeDimensional Kappa Distribution of Velocity, / u 243 4.7.2 Standard dKe Dimensional Kappa Distribution of Kinetic Energy, εK 244 4.7.3 Argument x* 244 4.8 Discrete Distributions 245 4.8.1 Distribution of Energy 245 4.8.2 Partition Function 245 4.8.3 Internal Energy 245 4.9 Concluding Remarks 245 4.10 Science Questions for Future Research 246

4.1 Summary The chapter presents all the formulae of kappa distributions necessary to describe particle populations out of thermal equilibrium in plasmas and beyond. In summary, we first provide the isotropic and anisotropic distributions in the absence of a potential energy, while then we continue with the formulae of kappa distributions in the presence of a potential energy; we present the formulation of multiparticle and multispecies kappa distributions, and the generalized kappa distributions based on non-Euclidean norms or under the discrete description of energy. The presented formulations and their guidelines constitute the most updated “toolbox” of useful and statistically well-grounded equations for future analyses that seek to apply kappa distributions in data analysis, simulations, modeling, theory, and other work in space, geophysical, laboratory, and other plasmas, or related particle systems. Science Question: What are the various kappa distribution formulae that describe particle populations out of thermal equilibrium? Keywords: Angular; Anisotropic; Correlation; Degrees of freedom; Distributions: phase space; Hamiltonian; Isotropic; Kinetic energy; Multiparticle; Multispecies; One-particle; Positional; Speed; Velocity.

185

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PART 1 Theory and Formalism

4.2 Introduction This chapter provides the kappa distribution formulae, which are the basic and frequently used tools for describing the statistics of systems out of thermal equilibrium. Useful information about the properties and the parameters of these formulae is also given. There are several categories and subcategories of distributions. For instance, multiparticle distributions can be reduced to oneparticle distributions, which are more convenient to handle, but less accurate for describing the statistics of particle systems, such as plasmas. Similarly, the distributions of Hamiltonian are more general, but they can be reduced to distributions of velocities for cases where the potential energy can be ignored. The provided toolbox should be used for future space and plasma physics analyses that seek to apply kappa distributions in data analyses, simulations, modeling, or other theoretical work. Using these equations guarantees results that remain firmly grounded on the foundation of nonextensive statistical mechanics. The chapter is structured as follows: first, we start with the isotropic distributions in the absence of a potential energy. Section 4.3 presents the standard (positive) multidimensional kappa distributions in the plasma flow (comoving) reference frame or any other inertial reference frame (in which the plasma flows with constant velocity). Next, the negative, multidimensional kappa distribution is presented; (as we will see, the nomenclature of positive and negative corresponds to the values of the kappa indices). Section 4.4 presents the anisotropic distributions, again, in the absence of a potential energy. The following important subcategories are described according to the degrees of freedom per particle: correlated degrees of freedom, correlation between the projection at a certain direction and the perpendicular plane, self-correlated degrees of freedom, self-correlated projections at a direction and perpendicular plane, self-correlated degrees of freedom with different kappa indices, self-correlated projections at a direction and perpendicular plane with different kappa, self-correlated projections of different dimensionality and kappa, and different self-correlation and intercorrelation between degrees of freedom. Section 4.5 presents the formulae of kappa distributions in the presence of a potential energy. In particular, we describe the subcategories: Hamiltonian kappa distribution, in general; in the presence of positive or negative attractive potentials, or of any small positive/negative attractive/repulsive potentials defined in a finite volume, ! r ˛V; reduction of the distribution of the kinetic and potential energy to the equivalent local distribution of solely the kinetic energy; kappa distributions for positive or negative power law central attractive potentials, that is, of oscillation or gravitational types, respectively; properties of the distributions with these potentials (local density, temperature, thermal pressure, polytropic index); marginal and conditional distributions; and distributions with angular potentials, e.g., magnetization potential. Section 4.6 provides the formulae of (positive or negative) multiparticle kappa distributions, and of multispecies kappa distributions, both in the presence or not of a potential energy. Section 4.7 provides the formulation of generalized Lp kappa distributions, which are based on nonEuclidean Lp-norms. Section 4.8 provides briefly the formulation of discrete kappa distributions (distribution of energy, partition function, internal energy).

Formulae of Kappa Distributions: Toolbox CHAPTER 4

Finally, the concluding remarks are given in Section 4.9, while three general science questions for future analyses are posed in Section 4.10.

4.3 Isotropic Distributions (Without Potential) (Livadiotis and McComas, 2009; 2011b) 4.3.1 Standard (Positive) Multidimensional Kappa Distributions 4.3.1.1 Distributions, for dK Degrees of Freedom, Using the Invariant Kappa Index k0

4.3.1.1.1 In Terms of the Velocity ! u Pð! u ; k0 ; qÞ ¼

pk0 q

1 #k0 11dK G k0 þ 1 þ dK " 2 1 ð! u ! u b Þ2 2 $ 1þ $ . k0 Gðk0 þ 1Þ q2

1 2 2 dK

(4.1a) Normalization Z N Pð! u ; k0 ; qÞd! u ¼ 1; N

where q ¼

(4.1b)

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2kB T=m.

4.3.1.1.2 In Terms of the Kinetic Energy εK ¼ PðεK ; k0 ; TÞ ¼

2 1 mð! u ! u bÞ 2

1 k0 11dK 2 1 ðk k TÞ2dK 1 ε d 1 0 B $ 1 þ $ K εK2 K . dK k0 kB T ; k0 þ 1 B 2

Normalization Z N PðεK ; k0 ; TÞdεK ¼ 1;

(4.2a)

(4.2b)

0

where the Beta function is defined by Bðx; yÞhGðxÞ$GðyÞ=Gðx þ yÞ.

4.3.1.1.3 In Terms of Normalized Kinetic Energy xhεK =ðk0 kB T Þ Pðx; k0 ; TÞ ¼ Fðx; 2k0 þ 2; f Þ.

(4.3a)

Normalization Z N Pðx; k0 ; TÞdx ¼ 1;

(4.3b)

0

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PART 1 Theory and Formalism

where the F-distribution is defined by Fðx; m; nÞ ¼

mþn n 1 m n $ x21 $ ð1 þ xÞ 2 . B ; 2 2

(4.3c)

4.3.1.2 Distributions, for dK Degrees of Freedom, Using the 3-D Kappa Index k3 ¼ k0 þ 32

4.3.1.2.1 Distribution in Terms of the Velocity ! u 1 3 2 2dK 1 dK G k3 þ q k3 2 2 2 Pð! u ; k3 ; qÞ ¼ 1 G k3 2 2 3k3 112 ðdK 3Þ 2 ! ! 1 ð u u Þ b 5 41 þ . $ 2 3 q k3 2

(4.4)

4.3.1.2.2 In Terms of the Kinetic Energy εK

1dK 0 1k3 112 ðdK 3Þ 2 3 kB T k3 1 1 εK C 2 d 1 $B εK2 K . $ A @1 þ 3 kB T dK 1 k3 ; k3 B 2 2 2 (4.5)

PðεK ; k3 ; TÞ ¼

4.3.1.3 Distributions, for dK Degrees of Freedom, Using the dK-Dimensional Kappa Index k ¼ k0 þ 12dK

4.3.1.3.1 In Terms of the Velocity ! u Pð! u ; k; qÞ ¼

1 d K 2 2 dK #k1 Gðk þ 1Þ " q 1 ð! u ! u b Þ2 2 . $ $ 1þ dK dK q2 k G kþ1 2 2

k

(4.6)

Formulae of Kappa Distributions: Toolbox CHAPTER 4 4.3.1.3.2 In Terms of the Kinetic Energy εK

1dK 0 1k1 2 dK k kB T 1 1 εK C 2 d 1 $B PðεK ; k; TÞ ¼ εK2 K . $ A @1 þ dK kB T dK dK k ;k þ 1 B 2 2 2

(4.7)

Comment: For the (dK ¼ 3)-dimensional distribution, the two descriptions, using the 3-D kappa index: k3 ¼ k0 þ 32, or the dK-dimensional kappa index: k ¼ k0 þ 12dK , obviously, coincide.

4.3.2 Standard (Positive) Multidimensional Kappa Distributions in an Inertial Reference Frame (Livadiotis and McComas, 2013a) 4.3.2.1 Distributions, for dK Degrees of Freedom, Using the Invariant Kappa Index k0 2 4.3.2.1.1 In Terms of the Kinetic Energy EK ¼ 12 m! u and the Polar Angle, w, Set Between ! u and ! ub

ðk0 kB TÞ2dK 1 dK 1 dK ; k0 þ 1 B ; B 2 2 2 !k0 11dK pﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃ 2 1 EK þ EK;b 2 EK;b EK cos w 1þ $ k0 kB T 1 1 ðd 3Þ d 1 1 cos2 w 2 K . EK2 K 1

PðEK ; cos w; k0 ; TÞ ¼

(4.8a) Normalization Z

1

1

Z

N

PðEK ; cos w; k0 ; TÞdEK d cos w ¼ 1;

(4.8b)

0

2 where EK;b h12 m! u b.

4.3.2.1.2 In Terms of the Kinetic Energy EK 1 1 ðk0 kB TÞ2dK 1 EK þ EK;b k0 12dK 12dK 1 PðEK ; k0 ; TÞ ¼ EK $ 1þ $ 1 dK 1 dK k0 kB T ; k0 þ 1 B ; B 2 2 2 ! pﬃﬃﬃﬃﬃﬃﬃﬃpﬃﬃﬃﬃﬃﬃ 2 EK;b EK dK dK 3 g x ¼ ; k ¼ k0 þ ; l ¼ ; 2 k0 kB T þ EK þ EK;b 2

(4.9a)

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PART 1 Theory and Formalism

where Z gðx; k; lÞh

1

1

l ð1 x$xÞk1 1 x2 dx.

(4.9b)

Normalization Z N PðEK ; k0 ; TÞdEK ¼ 1:

(4.9c)

0

4.3.2.1.3 In Terms of the Polar Angle w 1 ðd 3Þ

k0 1 2 1 cos2 w 2 K 1 2 $ 1 þ $xb 1 cos w Pðcos w; k0 ; xb Þ ¼ 1 dK 1 dK k0 ; k0 þ 1 B ; B 2 2 2

1 1 f h cos w; k0 xb ; k0 þ dK ; dK ; 2 (4.10a) Z

Nh

1 þ ðy xÞ2

f ðx; k; dÞh

ik1

0

1 EK;b . yd1 dy; hðx; yÞhx$ y þ 1 x2 2 ; xb h kB T (4.10b)

Normalization Z 1 Pðcos w; k0 ; xb Þd cos w ¼ 1:

(4.10c)

1

4.3.2.2 Distributions, for dK ¼ 3 Degrees of Freedom, Using the Invariant Kappa Index k0

4.3.2.2.1 In Terms of the Kinetic Energy EK ¼ and the Polar Angle w

PðEK ; cos w; k0 ; TÞ ¼ ðk0 kB TÞ2 3

2 1 m! u 2

5 G k0 þ 2 pﬃﬃﬃﬃ pGðk0 þ 1Þ

!k0 5 pﬃﬃﬃﬃﬃﬃﬃﬃpﬃﬃﬃﬃﬃﬃ 2 1 1 EK þ EK;b 2 EK;b EK cos w 1þ $ E2K . k0 kB T (4.11)

Formulae of Kappa Distributions: Toolbox CHAPTER 4

4.3.2.2.2 In Terms of the Kinetic Energy EK 82 3 3 3 pﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃ 2 k0 2 > G k0 þ < 1 E E 2 1 6 K K;b 7 $ PðEK ; k0 ; TÞ ¼ pk0 kB T EK;b 2 41 þ 5 Gðk0 þ 1Þ 2 > k k T 0 B : 9 3 3 pﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃ 2 k0 2 > > = EK þ EK;b 7 6 . 41 þ 5 > k0 kB T > ; 2

(4.12)

4.3.2.2.3 In Terms of the Speed u 3 G k0 þ 1 2 P ðu; k0 ; qÞ ¼ pk0 q2 u2b 2 Gðk0 þ 1Þ 8" 9 3 3 # " # < 2 k0 2 2 k0 2 = ðu u b Þ ðu þ u b Þ 1þ $u. 1þ : ; k0 q2 k0 q2

(4.13a)

Normalization Z

N

Pðu; k0 ; qÞdu ¼ 1:

(4.13b)

0

4.3.2.2.4 In Terms of the Polar Angle w (Compare With Eq. 4.10b) 5

2G k0 þ k0 1 1 2 2 P ðcos w; k0 ; xb Þ ¼ pﬃﬃﬃ $ 1 þ $xb 1 cos w k0 pGðk0 þ 1Þ

3 ; 3 . ; k f h cos w; k0 x1 þ 0 b 2

(4.14)

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PART 1 Theory and Formalism 4.3.2.3 Distributions, for dK ¼ 3 Degrees of Freedom, Using the 3 3-D Kappa Index k ¼ k0 þ 2 2 4.3.2.3.1 In Terms of the Kinetic Energy EK ¼ 12 m! u and the

Polar Angle w

3 2 3 Gðk þ 1Þ PðEK ; cos w; k; TÞ ¼ k kB T pﬃﬃﬃﬃ 1 2 pG k 2 0 1k1 pﬃﬃﬃﬃﬃﬃﬃﬃpﬃﬃﬃﬃﬃﬃ 1 1 EK þ EK;b 2 EK;b EK cos wC B @1 þ E2K . $ A 3 kB T k 2 (4.15)

4.3.2.3.2 In Terms of the Kinetic Energy EK 82 3 pﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃ 2 k >

1 < 2 EK EK;b 7 3 GðkÞ 1 6 $ PðEK ; k; TÞ ¼ p k kB T EK;b 41 þ 5 > 1 3 2 2: G k kB T k 2 2 3k 9 2 > pﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃ 2 > EK þ EK;b 7 = 6 . 41 þ 5 > 3 > ; kB T k 2 (4.16)

4.3.2.3.3 In Terms of the Speed u 82 3k > >

1 <6 3 2 2 2 GðkÞ ðu ub Þ2 7 $ 6 7 1 þ Pðu; k; qÞ ¼ p k q ub 4 1 > 3 25 2 > : G k q k 2 2 3k 9 2 > = 2 7 > 6 ðu þ u Þ b 7 $u. 6 1 þ 4 3 25 > > ; q k 2

(4.17)

Formulae of Kappa Distributions: Toolbox CHAPTER 4

4.3.2.3.4 In Terms of the Polar Angle w 2

3kþ1

7 2Gðk þ 1Þ 6 1 $41 þ P ðcos w; k; xb Þ ¼ $x 1 cos2 w 5 3 b pﬃﬃﬃ 1 k pG k 2 2

3 1 f h cos w; k x ; k; 3 . 2 b

2

(4.18)

4.3.3 Negative, Multidimensional Kappa Distribution (Livadiotis, 2015b) 4.3.3.1 Distributions Using the Invariant Kappa Index k0

4.3.3.1.1 In Terms of the Velocity ! u

1 # " 2 k0 12dK ! ! Gðk Þ 1 ð u u Þ 0 b $ 1 $ Pð! u ; k0 ; qÞ ¼ pk0 q $ . dK k0 q2 þ G k0 2 (4.19a) where the symbol “þ” denotes the cut-off condition (also see Eq. 1.13 in Chapter 1; Livadiotis and McComas, 2009; Tsallis, 2009b):

1 2 2 dK

xþ ¼ x for x 0 and xþ ¼ 0 for x < 0:

(4.19b)

4.3.3.1.2 In Terms of the Kinetic Energy εK

P ðεK ; k0 ; T Þ ¼

1 k0 11dK 2 1 ðk k T Þ2dK 1 ε d 1 0 B $ 1 $ K εK2 K . dK dK k 0 kB T þ ; k0 B 2 2

(4.20)

4.3.3.2 Distributions Using the Kappa Index k ¼ k0 12dK

4.3.3.2.1 In Terms of the Velocity ! u

2 dK 1 G kþ 6 d K 2 2 dK 2 $6 Pð! u ; k; qÞ ¼ p k þ $ q 41 2 GðkÞ

3k1 27 ! ! 1 ð u ubÞ 7 $ 5 . dK q2 kþ 2 þ (4.21)

193

194

PART 1 Theory and Formalism 4.3.3.2.2 In Terms of the Kinetic Energy εK

1dK 0 1k1 2 dK kþ kB T 1 1 εK C 2 dK 1 B $ @1 P ðεK ; k0 ; T Þ ¼ . $ A εK2 d dK K kB T kþ ;k B 2 þ 2

(4.22)

4.3.4 Superposition of Multidimensional Kappa Distributions 4.3.4.1 Linear Superposition of Positive and Negative Distributions in Terms of the Kinetic Energy εK

4.3.4.1.1 Positive to Negative Proportion is c O (1 c) þ P εK ; kþ 0 ; k0 ; T; c ¼ c$P εK ; k0 ; T þ ð1 cÞ$P εK ; k0 ; T ; 0 c 1;

εK ; kþ 0;T

where P distributions, i.e.,

is the positive and P εK ; k 0;T

(4.23a) is the negative kappa

ðkB TÞ2dK 12dK 1 $εK P εK ; kþ 0 ; k0 ; T; c ¼ dK G 2 2 1 dK þ 2 dK þ 1 þ G k0 þ 1 þ k0 6 1 εK k0 12dK 2 6 4c$ 1 þ þ$ k kB T G kþ þ 1 1

0

0

þ ð1 cÞ$

12dK k G k0 0

dK G k 0 2

1

1 εK $ k 0 kB T

k0 11dK 2

þ

3 7 7. 5 (4.23b)

4.3.4.1.2 Positive to Negative Proportion is 1:1 1 þ P εK ; kþ 0 ; k0 ; T ¼ $ P εK ; k0 ; T þ P εK ; k0 ; T ; or 2

(4.24a)

Formulae of Kappa Distributions: Toolbox CHAPTER 4

P

εK ; k þ 0 ; k0 ; T

ðkB TÞ2dK 12dK 1 $εK ¼ dK 2G 2 2 1 dK 2dK þ 1 þ G k þ 1 þ kþ 0 6 0 1 εK k0 12dK 2 6 1 þ þ$ 4 k kB T G kþ þ 1 1

0

0

3 1 1 d k 2 K G k 1 εK k0 12dK 7 0 7. þ 0 1 $ 5 dK k0 kB T þ G k0 2

(4.24b)

4.3.4.1.3 Positive to Negative Proportion is 1:1, and Common Kappa Index (Leubner and Voros, 2005) 1 PðεK ; k0 ; TÞ ¼ $½PðεK ; k0 ; TÞ þ PðεK ; k0 ; TÞ; or 2 ðk0 kB TÞ2dK 12dK 1 $εK dK 2G 2 2 dK 1 6G k0 þ 1 þ 2 1 εK k0 12dK 6 6 1þ $ 4 Gðk0 þ 1Þ k0 kB T

(4.25a)

1

PðεK ; k0 ; TÞ ¼

(4.25b)

3 k0 11dK 2 7 Gðk0 Þ 1 ε 7. 1 $ K þ 5 dK k 0 kB T þ G k0 2

4.3.4.2 Linear Superposition Given a Density of Kappa Indices D(k0) PðεK ; TÞ ¼ c

X X þ D kþ D k 0 P εK ; k0 ; T . 0 P εK ; k0 ; T þ ð1 cÞ kþ 0

k 0

(4.26)

195

196

PART 1 Theory and Formalism 4.3.4.3 Linear Superposition in Terms of the Kinetic Energy εK with Components of Different Temperature

PðεK ; Ts Þ ¼ c

X X þ þ þ ð1 cÞ D kþ D k 0 P εK ; k0 ; T k0 ; 0 P εK ; k0 ; T k0 k 0

kþ 0

(4.27a) Ts ¼ c

X X þ D kþ D k 0 $T k0 ¼ hTi. 0 $T k0 þ ð1 cÞ

(4.27b)

k 0

kþ 0

4.3.4.4 Nonlinear Superposition of Positive and Negative Distributions in Terms of the Kinetic Energy εK 8 < X $f P εK ; kþ PðεK ; TÞ ¼ f1 c D kþ ;T 0 0 : þ k0

9 = X ; þ ð1 cÞ D k ; k ; T $f P ε K 0 0 ; k

(4.28)

0

where f is some monotonic function (Livadiotis and McComas, 2013a).

4.4 Anisotropic Distributions (Without Potential) The reader may take a look to the references (Pierrard and Lazar, 2010; Lazar et al., 2012; Livadiotis, 2015a).

4.4.1 Correlated Degrees of Freedom 4.4.1.1 Distributions, for dK Degrees of Freedom, Using the Invariant Kappa Index k0

4.4.1.1.1 In Terms of the Velocity ! u

dK G k0 þ 1 þ 1 2 K P ! u ; k0 ; fqi gdi¼1 ¼ ðpk0 Þ2dK $ Gðk0 þ 1Þ " #k0 112dK dK dK 2 Y X 1 ð u u Þ i bi q1 $ ; i $ 1þ 2 k q 0 i i¼1 i¼1 where qi ¼

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2kB Ti =m, for i: 1, ., dK.

(4.29)

Formulae of Kappa Distributions: Toolbox CHAPTER 4

4.4.1.1.2 In Terms of the Kinetic Energy per Degree of Freedom εKi ¼ 12 mðui ubi Þ2 for i: 1, ., dK dK G k þ 1 þ 0 1 2 K K P fεKi gdi¼1 ; k0 ; fTi gdi¼1 ¼ ðpk0 Þ2dK $ Gðk0 þ 1Þ dK Y

dK 1 X εKi ðkB Ti Þ $ 1 þ $ k0 i¼1 kB Ti i¼1 12

!k0 112dK $

dK Y i¼1

1

εKi2 .

(4.30a)

Normalization Z 0

N

Z /

N

0

K K P fεKi gdi¼1 ; k0 ; fTi gdi¼1 dεK1 /dεKdK ¼ 1:

(4.30b)

4.4.1.1.3 The Temperature is Given by

T ¼ hTi i ¼

dK 1 X Ti . dK i ¼ 1

(4.31)

4.4.1.2 Distribution in Terms of the Velocity ! u for dK Degrees of Freedom, Using the dK-Dimensional Kappa Index k ¼ k0 þ 12dK 1 d K 2 dK Gðk þ 1Þ dK ! P u ; k; fqi gi¼1 ¼ p k $ dK 2 G k þ1 2 2 3k1 d d 2 K K Y X ðui ubi Þ 7 1 6 q1 . $ 5 i $41 þ dK q2i i¼1 i¼1 k 2

(4.32)

197

198

PART 1 Theory and Formalism 4.4.1.3 Distribution in Terms of the Velocity ! u for dK Degrees of Freedom, Using the 3-D Kappa Index k3 ¼ k0 þ 12 (dK 3) dK 3 1

G k3 þ 1 þ 3 2 dK 2 K P ! u ; k3 ; fqi gdi¼1 $ ¼ p k3 3 2 G k3 þ 1 2 2 3k3 11 ðdK 3Þ 2 d d 2 K K Y X ðui ubi Þ 7 1 1 6 qi $41 þ . $ 5 3 q2i i¼1 k3 i¼1 2 (4.33)

4.4.1.4 Distributions, for dK ¼ 3 Degrees of Freedom, Using the Invariant Kappa Index k0

4.4.1.4.1 In Terms of the Velocity ! u

5 G k0 þ 3 2 $q1 q1 q1 P ! u ; k0 ; qx ; qy ; qz ¼ ðpk0 Þ2 $ Gðk0 þ 1Þ x y z ( " #)k0 52 2 uy uby 1 ðux ubx Þ2 ðuz ubz Þ2 1þ $ þ þ . k0 q2x q2y q2z (4.34)

4.4.1.4.2 In Terms of the Kinetic Energy Per Degree of Freedom εKi ¼ 12 mðui ubi Þ2 for i: x, y, z 5 G k0 þ 1 3 2 $ ðkB Tx Þ kB Ty ðkB Tz Þ 2 P εKx ; εKy ; εKz ; k0 ; Tx ; Ty ; Tz ¼ ðpk0 Þ2 $ Gðk0 þ 1Þ k0 5 2 εKy 1 εKx εKz 1 1 1 1þ $ þ þ εKx2 εKy2 εKz2 . k0 kB Tx kB Ty kB Tz (4.35a)

Formulae of Kappa Distributions: Toolbox CHAPTER 4

Normalization Z NZ NZ N PðεKx ; εKy ; εKz ; k0 ; Tx ; Ty ; Tz ÞdεKx dεKy dεKz ¼ 1: 0

0

(4.35b)

0

4.4.1.5 Distribution, for dK ¼ 3 Degrees of Freedom, Using the 3-D Kappa Index k ¼ k0 þ32

4.4.1.5.1 In Terms of the Velocity ! u

3 3 2 Gðk þ 1Þ 1 1 1 $qx qy qz P ! u ; k; qx ; qy ; qz ¼ p k $ 1 2 G k 2 9 #>k1 ðuy uby Þ 1 ðux ubx Þ ðuz ubz Þ = 1þ þ þ . $ 3 > > q2x q2y q2z ; : k 2 (4.36) 8 > <

"

2

2

2

4.4.1.5.2 In Terms of the Kinetic Energy per Degree of Freedom εKi for i: x, y, z 3 1 3 2 Gðk þ 1Þ $½ðkB Tx ÞðkB Ty ÞðkB Tz Þ2 PðεKx ; εKy ; εKz ; k; Tx ; Ty ; Tz Þ ¼ p k $ 1 2 G k 2 3k1 2 εKy 1 εKx εKz 7 1 1 1 6 41 þ þ þ εKx2 εKy2 εKz2 . $ 5 3 kB Tx kB Ty kB Tz k 2 (4.37)

4.4.1.5.3 The Temperature is Given by T ¼ ðTx þ Ty þ Tz Þ=3.

(4.38)

199

200

PART 1 Theory and Formalism

4.4.2 Correlation Between the Projection at a Certain Direction and the Perpendicular Plane This formalism is most frequently used for describing plasma populations in the presence of an ambient magnetic field.

4.4.2.1 Distributions, for dK ¼ 3 Degrees of Freedom, Using the Invariant Kappa Index k0

4.4.2.1.1 In Terms of the Velocity ! u ¼

uk ; ! ut

5 G k þ 0 3 2 $q1 q2 P ! u ; k0 ; qk ; qt ¼ ðpk0 Þ2 $ Gðk0 þ 1Þ k t 8 2 39k0 52 2 ! 2 > > ! < u u bk u t u bt 7= 1 6 k 1 þ $4 þ . 5 > > k0 q2k q2t ; : (4.39a)

Normalization Z N P ! u ; k 0 ; q k ; q t d! u ¼ 1:

(4.39b)

N

4.4.2.1.2 In Terms of the Kinetic Energy, εKk ¼ 2 εKt ¼ 12 mð! ut ! u bt Þ

P εKk ; εKt ; k0 ; Tk ; Tt

0

2 uk ubk

1 32 1 k0 2 $ kB Tk ðkB Tt Þ1 ¼ 2 3 B ; k0 þ 1 2 " !#k0 5 2 εKk 1 εKt 1 1þ $ þ εKk2 . k0 kB Tk kB Tt

Normalization Z NZ N P εKk ; εKt ; k0 ; Tk ; Tt dεKk dεKt ¼ 1. 0

1m 2

(4.40a)

(4.40b)

Formulae of Kappa Distributions: Toolbox CHAPTER 4 4.4.2.2 Distribution in Terms of the Velocity ! u ¼ uk ; ! ut for dK ¼ 3 Degrees of Freedom, Using the 3-D Kappa Index k ¼ k0 þ32 3 3 2 Gðk þ 1Þ 1 2 ! $qk qt P u ; k; qk ; qt ¼ p k $ 1 2 G k 2 8 2 39k1 2 ! 2 > > ! < u u bk u t u bt 7= 1 6 k 1þ þ . $4 5 3 > > q2k q2t ; : k 2 (4.41)

4.4.2.3 The Temperature is Given by T ¼

Tk þ 2Tt

. 3.

(4.42)

4.4.3 Self-Correlated Degrees of Freedom 4.4.3.1 Distributions, for dK Degrees of Freedom, Using the Invariant Kappa Index k0

4.4.3.1.1 In Terms of the Velocity ! u dK Y K P ! u ; k0 ; fqi gdi¼1 P ðui ; k0 ; qi Þ ¼ i¼1

3 2 3 dK G k0 þ 6 1 2 7 7 ¼ ðpk0 Þ2dK $6 4 Gð k 0 þ 1 Þ 5

(4.43)

8 9 3 " # dK < 2 k0 2 = Y 1 ð u u Þ i bi . q1 1þ $ i 2 : ; k q 0 i i¼1

4.4.3.1.2 In Terms of the Kinetic Energy per Degree of Freedom εKi ¼ 12 mðui ubi Þ2 for i: 1, ., dK 3 2 3 dK G k0 þ dK Y 6 1 2 7 K K 7 P fεKi gdi¼1 ; k0 ; fTi gdi¼1 PðεKi ; k0 ; Ti Þ ¼ ðpk0 Þ2dK $6 ¼ 4 Gðk0 þ 1Þ 5 i¼1

dK Y i¼1

ðkB Ti Þ

12

3

dK Y 1 εKi k0 2 12 1þ $ $εKi . k 0 kB T i i¼1

(4.44)

201

202

PART 1 Theory and Formalism

4.4.3.1.3 The Temperature is Given by

T ¼ hTi i ¼

dK 1 X Ti . dK i ¼ 1

(4.45)

4.4.3.2 Distribution in Terms of the Velocity, ! u , for dK Degrees of Freedom, Using the dK-Dimensional Kappa Index k ¼ k0 þ 12dK 2 3 dK dK 1 7 1 6G k þ 1 7 dK 2dK 6 2 dK ! 7 6 P u ; k; fqi gi¼1 ¼ p k $6 7 d 2 5 4 G k K þ1 2 8 2 3kþ1 ðdK 3Þ 9 2 > > > > dK < = Y 1 ðui ubi Þ2 7 1 6 . $ qi 41 þ 5 2 dK > > qi > i¼1> ; : k 2

(4.46)

4.4.3.3 Distribution in Terms of the Velocity, ! u , for dK Degrees of Freedom, Using the 3-D Kappa Index k ¼ k0 þ 32 2

3 dK

1 7 3 2 dK 6 6 Gðk3 Þ 7 K P ! u ; k3 ; fqi gdi¼1 $6 ¼ p k3 7 1 5 4 2 G k3 2 8 2 3k3 9 > > d < 2 K Y 1 ðui ubi Þ 7 = 6 . q1 1 þ $ 4 5 3 > > i q2i ; i¼1: k3 2

(4.47)

4.4.3.4 Distributions, for dK ¼ 3 Degrees of Freedom, Using the Invariant Kappa Index k0

4.4.3.4.1 In Terms of the Velocity ! u

3 3 " 2 #k0 32 G k0 þ ! 1 ux ubx 2 32 1 1 1 P u ; k0 ; qx ; qy ; qz ¼ ðpk0 Þ $ $qx qy qz $ 1 þ $ k0 q2x Gðk0 þ 1Þ3 " 2 #k0 32 " 2 #k0 32 1 uy uby 1 uz ubz 1þ $ $ 1þ $ . k0 k0 q2y q2z (4.48)

Formulae of Kappa Distributions: Toolbox CHAPTER 4

4.4.3.4.2 In Terms of the Kinetic Energy per Degree of Freedom εKi ¼ 12 mðui ubi Þ2 for i: x, y, z 3 3 G k0 þ 3 1 2 PðεKx ; εKy ; εKz ; k0 ; Tx ; Ty ; Tz Þ ¼ ðpk0 Þ2 $ $½ðkB Tx ÞðkB Ty ÞðkB Tz Þ2 3 Gðk0 þ 1Þ !k0 3 !k0 3 2 2 1 εKx 1 εKy 1þ $ 1þ $ k0 kB Tx k0 kB Ty !k0 3 2 1 εKz 1 1 1 1þ $ εKx2 εKy2 εKz2 . k0 kB Tz

(4.49)

4.4.3.5 Distribution, for dK ¼ 3 Degrees of Freedom, Using the 3-D Kappa Index k ¼ k0 þ32

4.4.3.5.1 In Terms of the Velocity ! u

" #32 ! 3 GðkÞ3 1 1 1 P u ; k; qx ; qy ; qz ¼ p k $ $qx qy qz 2 1 3 G k 2 3k 2 2 2 3k 2 uy uby 7 6 ux ubx 7 6 1 7 $6 1 þ 1 $ 7 6 $ 5 5 41 þ 4 2 3 3 qx q2y k k 2 2 3k 2 2 6 uz ubz 7 1 7 . 6 41 þ $ 5 2 3 q z k 2 (4.50)

4.4.3.5.2 In Terms of the Kinetic Energy per Degree of Freedom εKi for i: x, y, z

3 3 2 GðkÞ3 1 PðεKx ; εKy ; εKz ; k; Tx ; Ty ; Tz Þ ¼ p k $ 3 $½ðkB Tx ÞðkB Ty ÞðkB Tz Þ 2 2 1 G k 2 1k 0 1k 0 B @1 þ 0

εKy C εKx C B 1 1þ $ A @ 3 kB Tx 3 kB Ty A k k 2 2 1 1

$

k

εKz C 12 12 12 B @1 þ ε ε ε . $ 3 kB Tz A Kx Ky Kz k 2 1

(4.51)

203

204

PART 1 Theory and Formalism

4.4.3.5.3 The Temperature is Given by T ¼ ðTx þ Ty þ Tz Þ=3.

(4.52)

Comment: The exponent of each component is ek instead of ek 1. This is because each of the direction is described by a 1-D kappa distribution. As it is mentioned, the kappa index depends on the degrees of freedom, i.e., kðdK Þ ¼ k0 þ 12dK ; hence, kðdK Þ ¼ k þ 12 ðdK 3Þ, where we symbolize with k the standard dK-dimensional kappa index and with k3 the 3-D kappa index. Thus, the exponent of the dK-dimensional kappa distribution is k(dK) 1, which, in terms of k3, becomes k3 1 12 ðdK 3Þ. For dK ¼ 1, the exponent becomes k3, while for dK ¼ 2, it becomes k3 12.

4.4.4 Self-Correlated Projections at a Direction and Perpendicular Plane This formalism may be used in cases of very strong ambient magnetic fields. It has been introduced in Lazar and Poedts (2014) and called “product-bi-kappa distributions” (e.g., Lazar and Poedts, 2014; dos Santos et al., 2015).

4.4.4.1 Distributions, for dK ¼ 3 Degrees of Freedom, Using the Invariant Kappa Index k0

4.4.4.1.1 In Terms of the Velocity ! u ¼

uk ; ! ut

3 G k0 þ ðk0 þ 1Þ 3 2 2 $q1 P ! u ; k0 ; qk ; qt ¼ ðpk0 Þ2 $ k qt Gðk0 þ 1Þ 2 2 3k0 32 u u k bk 7 1 6 41 þ $ 5 2 k0 qk

(4.53)

"

# ! ! 2 k0 2 1 u t u bt 1þ $ . k0 q2t

4.4.4.1.2 In Terms of the Kinetic Energy, εKk ¼ 2 εKt ¼ 12 mð! ut ! u bt Þ

1m 2

2 uk ubk

3 1 ðk0 þ 1Þ$k0 2 2 $ kB Tk P εKk ; εKt; k0 ; Tk ; Tt ¼ ðkB Tt Þ1 1 B ; k0 þ 1 2 " #k0 3

2 1 εKk 1 εKt k0 2 12 1þ $ $ 1þ $ εKk . k0 kB Tk k0 kB Tt

(4.54)

Formulae of Kappa Distributions: Toolbox CHAPTER 4 4.4.4.2 Distribution in Terms of the Velocity ! u ¼

uk ; ! ut for dK ¼ 3

Degrees of Freedom, Using the 3-D Kappa Index k ¼ k0 þ32 1 3 Gð k Þ k

3 2 2 2 $q1 P ! u ; k; qk ; qt ¼ p k $ k qt 1 2 G k 2 2 2 3k u u k bk 1 6 7 41 þ $ 5 2 3 q k k 2 1 2 3 ! 2 k2 ! u t u bt 7 1 6 41 þ . $ 5 3 q2t k 2

(4.55)

4.4.4.3 The Temperature is Given by T ¼

Tk þ 2Tt

. 3.

(4.56)

Comment: The exponent of the 1-D kappa distribution (left-hand side) is k, and the exponent of the 2-D kappa distribution (right-hand side) is k 12.

4.4.5 Self-Correlated Degrees of Freedom With Different Kappa 4.4.5.1 Distributions, for dK Degrees of Freedom, Using the Invariant Kappa Index k0

4.4.5.1.1 In Terms of the Velocity ! u dK Y K K P ! u ; fk0i gdi¼1 ; fqi gdi¼1 Pðui ; k0i ; qi Þ ¼ i¼1

8 3 > > G k þ 0i dK < Y 1 2 pk0i q2i 2 $ ¼ > Gðk þ 1Þ 0i i¼1> : "

9 #k0i 3 > 2> = 1 ðui ubi Þ2 . 1þ $ > k0i q2i > ;

(4.57)

205

206

PART 1 Theory and Formalism

4.4.5.1.2 In Terms of the Kinetic Energy per Degree of Freedom εKi ¼ 12 mðui ubi Þ2 for i: 1, ., dK dK Y K K K P fεKi gdi¼1 ; fk0i gdi¼1 ; fTi gdi¼1 P ðεKi ; k0i ; Ti Þ ¼ i¼1

8 > > dK > < Y

3 2 3 dK G k0i þ 6 1 2 7 7 ðpk0i Þ2dK $6 ¼ 4 > Gðk0i þ 1Þ 5 i¼1> > : 1 1 εKi ðkB Ti Þ2 $ 1 þ $ k0i kB Ti

k0i 32

1

9 > > > =

$εKi2 . > > > ; (4.58)

4.4.5.1.3 The Temperature is Given by T ¼ hTi i ¼

dK 1 X Ti . dK i ¼1

(4.59)

4.4.5.2 Distribution in Terms of the Velocity ! u for dK Degrees of Freedom, Using the 1-D Kappa Index ki ¼ k0i þ 12 8 > > 1 dK > < Y 1 2 2 Gð k i þ 1 Þ dK dK ! p ki P u ; fki gi¼1 ; fqi gi¼1 ¼ $ q > 1 2 i i¼1 > > þ G k i : 2 9 2 3ki 1 > > > = 1 ðui ubi Þ2 7 6 . (4.60) 41 þ 5 1 > q2i > ki > ; 2 4.4.5.3 Distribution in Terms of the Velocity ! u for dK Degrees of Freedom, Using the 3-D Kappa Index k ¼ k0 þ 32

K K P ! u ; fk3i gdi¼1 ; fqi gdi¼1

8

dK > < Y

1 3 2 2 Gðk3i Þ p k3i qi $ ¼ > 1 2 i¼1: G k3i 2 2 3k3i 9 > 1 ðui ubi Þ2 7 = 6 $ 41 þ . 5 3 > q2i ; k3i 2

(4.61)

Formulae of Kappa Distributions: Toolbox CHAPTER 4

4.4.6 Self-Correlated Projections at a Direction and Perpendicular Plane With Different Kappa 4.4.6.1 Distributions, for dK ¼ 3 Degrees of Freedom, Using the Invariant Kappa Index k0

4.4.6.1.1 In Terms of the Velocity ! u ¼

uk ; ! ut

12 3 P ! u ; kk0 ; kt0 ; qk ; qt ¼ p2 1 þ k1 t0 $kk0

3 G kk0 þ 2 2 $q1 k qt G kk0 þ 1

2

2 3kk0 32 1 uk ubk 7 6 41 þ $ 5 kk0 q2k " 1þ

1 kt0

$

! 2 #kt0 2 ut! u bt q2t

4.4.6.1.2 In Terms of the Kinetic Energy, εKk ¼ 2 ut ! u bt Þ (also see Eq. (4.54)) εKt ¼ 12 mð!

.

(4.62)

1m 2

uk ubk

2

;

12 1 1 þ k1 t0 $kk0 2 $ kB Tk P εKk ; εKt ; kk0 ; kt0 ; Tk ; Tt ¼ ðkB Tt Þ1 1 B ; kk0 þ 1 2 " #kk0 3

2 1 εKk 1 εKt kt0 2 12 1þ $ $ 1þ $ εKk . kk0 kB Tk kt0 kB Tt (4.63)

4.4.6.2 Distribution in Terms of the Velocity ! u ¼ uk ; ! u t for dK ¼ 3 Degrees of Freedom, Using the Kappa Indices kk ¼ kk0 þ 12; kt ¼ kt0 þ 1 1 G k þ 1 2 k 3 kt 1 2 $q1 $ kk P ! u ; kk ; kt ; qk ; qt ¼ p2 k qt 1 2 kt 1 G kk þ 2 2 2 3kk 1 uk ubk 7 1 6 41 þ $ 5 1 q2k kk 2 " ! 2 #kt 1 ut! u bt 1 $ 1þ . (4.64) kt 1 q2t

207

208

PART 1 Theory and Formalism

4.4.6.3 The Temperature is Given by T ¼

Tk þ 2Tt

. 3.

(4.65)

4.4.7 Self-Correlated Projections of Different Dimensionality and Kappa 4.4.7.1 Distributions, for dK Degrees of Freedom of Mixed Correlation, i.e., i: 1, ., M Uncorrelated Groups of Correlated Degrees of Freedom, With Each Group Having Different Degrees of Freedom fi, Using Different (Invariant) Kappa Index, k0i, but Common Temperature T or q

4.4.7.1.1 In Terms of the Velocity ! u 8 > > < M > Y

fi G k0i þ 1 þ 1 2 2 2fi pk P ! u ; fk0i gM ; q ¼ q $ 0i i¼1 > G k ð þ 1 Þ 0i i¼1 > > : 9 ! ! 2 3k0i 112fi > > > u ub = 1 ðfi Þ7 6 ; 41 þ $ 5 > k0i q2 > > ; 2

(4.66a)

where we set the magnitude of the vector ! u ! u b as ð! u ! u b Þ2 ¼

dK X

ðuj ubj Þ2 ¼

M X ð! u ! u b Þ2 ðf Þ

and ð! u ! u b Þ2

ðfi Þ

¼

jbef þfi X

ðuj ubj Þ2 ; jbef

j ¼ jbef þ1

(4.66b)

i

i¼1

j¼1

8 1 > < iP fi0 if i > 0 ; ¼ i0 ¼ 1 > : 0 if i ¼ 0 .

4.4.7.1.2 In Terms of the Kinetic Energy εKi ¼ for i: 1, ., M 2

1 mð! u 2

2 ! u bÞ

ðfi Þ

3 k0i 11fi 2 1 1 ε f 1 7 $ 1 þ $ Ki ε2Kii 7 5. k0i kB T

M 6 Y 0i kB TÞ M 6 ðk P fεKi gM i¼1 ; fk0i gi¼1 ; T ¼ 4 fi i¼1 B ; k0i þ 1 2

12fi

(4.67)

Formulae of Kappa Distributions: Toolbox CHAPTER 4

4.4.7.1.3 The Total Degrees of Freedom dK Are Given by dK ¼

M X

fi .

(4.68)

i¼1

4.4.7.2 Distributions, for M Correlated Groups, Each With fi Degrees of Freedom, Self-Correlated With Kappa Index k0i and Temperature qi

4.4.7.2.1 In Terms of the Velocity ! u 8 > > < M > Y

fi G k0i þ 1 þ 1 2 M 2 2fi pk P ! u ; fk0i gM ; q q $ ¼ f g i 0i i i¼1 i¼1 > Gðk0i þ 1Þ i¼1 > > : 9 ! ! 2 3k0i 112fi > > > u ub = 1 ðfi Þ7 6 . 41 þ $ 5 > k0i q2i > > ; 2

(4.69)

4.4.7.2.2 In Terms of the Kinetic Energy εKi ¼

1 mð! u 2

2 ! u bÞ

ðfi Þ

2

3

1 M 6 1fi Y 1 εKi k0i 12fi 12fi 1 7 6 ðk 7 0i kB Ti Þ 2 M M M P fεKi gi¼1 ; fk0i gi¼1 ; fTi gi¼1 ¼ εKi 7. $ 1þ $ 6 f 4 5 k k T 0i B i i¼1 B i ; k0i þ 1 2

(4.70)

4.4.7.2.3 The Total Degrees of Freedom dK Are Given by dK ¼

M X

fi .

(4.71)

i¼1

4.4.7.2.4 The Temperature is Given by T ¼ hTi i ¼

M 1 X fi Ti . dK i¼1

(4.72)

209

210

PART 1 Theory and Formalism

4.4.8 Different Self-Correlation and Intercorrelation Between Degrees of Freedom 4.4.8.1 Distributions, for M Correlated Groups, Each With fi Degrees of Freedom, Self-Correlated With Invariant Kappa Index, k0i, and Intercorrelated With Invariant Kappa Index kint 0i

4.4.8.1.1 In Terms of the Velocity ! u

int M 1 M int P ! u ;fk0i gM i¼1 ; k0i i¼1 ; k0 ; fqi gi¼1 ¼ Z int 1 8 39k0 12dK 2 > fi > > > k0i þ1þ 2 > > > 7> 6 > > 0 1 > f > > > 6 > > kint þ1þ 2i 7 0i f f > > 7 6 i i 2 ! ! 2C > > = < M kint 7 6 B q ð u u Þ X b i 0i þ ðfi Þ 1 7 6 B C 2 2 1 $ $61 B1 þ . 7 C 2 d 6 fi > 7> @ A qi > > i¼1 kint þ K 6 > > 7 þ k 0i > 0 > > 7> 2 6 2 > > > 5> 4 > > > > > > ; : (4.73a)

The partition function is given by the normalization: 8 > > > > > > > > > > <

int 1 39k0 12dK > f k0i þ1þ 2i > > 7> 6 > 1 0 fi 7> 6 > int > k þ1þ f i 0i f 2 > 7 6 ! ! 2 2 i > Z N ð u u b Þ qi C M kint þ 7= 6 B X 0i 1 ðfi Þ 2 C 7 6 B 2 Z ¼ d! u. $ $6 1 B 1 þ 1 C 7 2 dK 6 f > A 7 @ i q N > int > > i i ¼1 k > > 7 6 þ þ k 0i > 0 > > 7> 2 6 2 > > > 5> 4 > > > > > > : ;

2

(4.73b)

Comment: The kappa index kint 0 characterizes overall the intercorrelation. While it may be used as an independent parameter, it must be aligned int to the following conditions: (1) if kint 0i ¼ 0 for every i, then k 0 ¼ 0; int int int (2) if k0i /N for any i, then k 0 / N; (3) if k 0i ¼ constant for any i, then int kint 0i ¼ k 0.

Formulae of Kappa Distributions: Toolbox CHAPTER 4 4.4.8.1.2 In Terms of the Kinetic Energy εKi ¼

1 mð! u 2

2 ! u bÞ

ðfi Þ

M int M M int P fεKi gM i¼1 ; fk0i gi¼1 ; k0i i¼1 ; k0 ; fTi gi¼1 1 int 8 39k0 12dK 2 > > f > > i > > > 6 0 > > 1 k0i þ1þ 2 7> > > f > > 7 6 i > > kint fi 6 fi > 0i þ1þ 2 7> int > > M M = < B 7 6 C k k þ ε T X 0i B i Ki 1 Y 12fi 1 1 7 6 B C 2 2 ¼ $ εKi $ 1 . $ $61 B1 þ 7 C d 6 @ fi > 7> A Z i ¼1 kB Ti > > i ¼1 kint þ K 6 > > 7 k þ 0i > 0 > > 7> 2 6 2 > > > 5> 4 > > > > > > ; :

(4.74a) The partition function is given by the normalization: Z N Z N M 1 Y int 1 f 1 2 i Z ¼ / f/gk0 12dK εKi dεK1 /dεKM 0

0

i¼1

1 8 2 39 int 0 1 k þ1þ fi >k0 12dK > 0i fi fi > 2 > int > > M k < 6 εKi kB Ti int fi 7= X 0i þ 1 B C k0i þ1þ 2 7 2 $6 2 where f/g ¼ 1 . $ 6 1 @1 þ 7 A dK 4 f > 5> k T i B i > > int i¼1 k > > þ k þ 0i ; : 0 2 2

(4.74b)

4.4.8.2 Distributions, for M Correlated Groups, Each With fi Degrees of Freedom, Self-Correlated With Kappa Index, ki, and Intercorrelated With Kappa Index kint i

4.4.8.2.1 In Terms of the Velocity ! u int M P ! u ; fki gM ; kint ; fqi gM i¼1 ; ki i¼1 i¼1

int 8 99k 1 8 ki þ1 > > > > 2 3kint þ1 > > >> > > > > > > > > > fi 2 i > 2 > > > > ! ! > > > > q ð u u Þ M int < < = = 6 7 b i X ki 1 1 2 7 ðfi Þ 6 ¼ $ 1 $ 1 $ . 1 þ 6 7 2 int > > > 4 5 Z > ki k > qi > > > i ¼1 > > > > > > > > > > > > > > > > ;> > : : ;

(Partition functions as above.)

(4.75)

211

212

PART 1 Theory and Formalism 4.4.8.2.2 In Terms of the Kinetic Energy εKi ¼

1 mð! u 2

2 ! u bÞ

ðfi Þ

M int M M int P fεKi gM ; ; k ; k ; g g fk fT i i i i¼1 i¼1 i¼1 i¼1 39kint 1 > 1 kinti þ1 > > k þ1 > 6 7> i f > i > 6 7 M M B C k ε T int Xk 6 Ki B i 7= 1 Y 12fi 1 1 B C i 2 6 1 B1 þ $ 7 εKi $ 1 $ . ¼ $ C 7> > @ A Z i ¼1 ki kB Ti kint 6 > 6 7> i ¼1 > > > 4 5> > > > > > > ; : 8 > > > > > > > <

2

0

(4.76)

4.4.8.2.3 Internal Energy Ui ¼

1 2 1 fi kB Ti ; Ui ¼ fi q2i . 2 m 2

(4.77)

4.4.8.2.4 Degeneration of the Kappa Index 1 1 1 int ki ¼ k0i þ fi ; kint ¼ kint ¼ kint i 0i þ fi ; k 0 þ dK . 2 2 2

(4.78)

4.4.8.2.5 Nonlinear Superposition Eq. (4.76) can be written as 8 <

9 =

int1 k þ1 ¼ kint $ 1 ZP fεKi gM i¼1 ; :

M X

8 > <

kint i $>1 ½Zi PðεKi Þ : i ¼1

int1 k þ1 i

9 > = > ;

;

(4.79a)

or using the ordinary distributions and their relation to the escort distributions, 1 1 P kþ1 fpk (Chapter 1; Livadiotis and McComas, 2009) 9 8 > > < 1 1

kint = M kint int int e e k $ 1 Z p fεKi gi¼1 ¼ ki $ 1 Zi pðεKi Þ i . > ; : > ; : i ¼1 8 <

9 =

M X

(4.79b)

Comment: This is a special (4.28), where the distribution on the P case of Eq. int kappa indices is Dðk0 Þ ¼ M i ¼ 1 d k0 k0i , while the monotonic function f is given by the deformed logarithm function (Chapter 1): 1 (4.79c) fðxÞ ¼ lnk ðxÞhk$ 1 xk ; with x ¼ Z$P fεKi gM i¼1 .

Formulae of Kappa Distributions: Toolbox CHAPTER 4

4.4.8.3 Distributions, for M Correlated Groups, Each With fi Degrees of Freedom, Self-Correlated With Invariant Kappa Index, k0i ¼ k0, and Intercorrelated With Invariant Kappa Index kint 0

4.4.8.3.1 In Terms of the Velocity ! u M P ! u ; k0 ; kint 0 ; fqi gi¼1 8 > > > > > > > fi > > M kint X 0 þ 1 < 2 ¼ $ 1 dK Z > > int i ¼1 k > > 0 þ > 2 > > > > :

2 0 6 6 2 ! ! 6 ð u u Þ B b 6 1 ðfi Þ B $ $6 1 1 þ B 6 fi @ q2i 6 k0 þ 6 2 4

int 1 39k0 12dK

fi > > > 1 k0 þ1þ 2 7> > fi 7> > fi 2 kint 0 þ1þ 2 7> > qi C 7=

2 C C A

7 7> 7> > 7> > 5> > > > ;

.

(4.80)

4.4.8.3.2 In Terms of the Kinetic Energy εKi ¼

2 1 mð! u ! u bÞ 2 ðf Þ i

M int P fεKi gM i¼1 ; k0 ; k0 ; fTi gi¼1 int 1 8 39k0 12dK 2 > > fi > > > > > 6 > > 1 k0 þ1þ 2 7> 0 > fi 7> > > 6 int > k þ1þ fi 6 fi > > 7> 0 2 int > > M M = < 7 C 6 B k k þ ε T X 0 B i Ki 1 Y 12fi 1 1 7 C 6 B 2 2 $ ¼ $ $61 B1 þ εKi $ 1 . 7 C d 6 fi > 7> A @ Z i ¼1 kB Ti > > i ¼1 kint þ K 6 > > 7 þ k 0 > > 0 > 7> 2 6 2 > > > 5> 4 > > > > > > ; : (4.81)

4.4.8.4 Distributions, for dK ¼ 3 Correlated Groups, Each With fi ¼ 1 Degrees of Freedom, Self-Correlated With Kappa Index, k0i, and Intercorrelated With Kappa Index kint 0

4.4.8.4.1 In Terms of the Velocity ! u

3 P ! u ; fk0i g3i¼1 ; kint 0 ; fqi gi¼1 int 5 8 8 99k0 2 > > 3 > > >> > > > 80 9 k0i þ 2 > > > > > > > >> 1 3 > > > int > > > > > > k0 þ 2 > 1 > > > > > > " # > > > > > > int 3 < = < < = = 2 k þ X B C 0 1 k 1 ðu u Þ 0i i bi 2 B C 1 @ $ 1þ $ ¼ $ 1 . $ A 3 1 > > > > > Z > k0i q2i > > > > > kint þ i ¼1 > > k0i þ > > > > > 0 : > > > > ; > > > > 2 2 > > > > > > > > > > > > > > : ; ; : (4.82)

213

214

PART 1 Theory and Formalism

The partition function is given by the normalization: int 5 8 99k0 2 > 3 > >> > 80 9 k0i þ 2 > > > >> > > 1 3> > > > int > > > > > þ k 0 1 > > > > > # " 2 > > > > > = < < = = 3 2 kint þ X B C 0 k 1 ðu u Þ 0i C i bi 2 B 1 1 @ $ $ 1 þ Z ¼ $ d! u. 2 3 1A > > > > > k q 0i int N > > > > > > > i i ¼1 > k þ k þ > > > > > 0i 0 > : > > > ; > > > > 2 2 > > > > > > > > > > > > > > : ; ; :

8 > > > > > > > > Z N> <

(4.83)

4.4.8.4.2 In Terms of the Kinetic Energy εKi ¼ for i: x, y, z

1 mðu i 2

ubi Þ2

3 P fεKi g3i¼1 ; fk0i g3i¼1 ; kint 0 ; fTi gi¼1 int 5 8 8 99k0 2 > > > > >> > > k0i þ 3 > > > > >> > > > > > > > 20 3 int 23 > > > > > 1 > > > > þ k > > > 0 2> 1 > > > > int > > > > = < < = 3 3 6 7 k þ X Y B C 0 1 k 1 ε 12 6 7 0i Ki 2 B C 1 6@ εKi $ 1 . $ $ 1þ $ ¼ $ 7 A 3 1 > > > > 4 Z i ¼1 k0i kB Ti 5 > > > > kint þ i ¼1 > k0i þ > > > 0 > > > > > > > > 2 2 > > > > > > > > > > > > > > > > > > > > > > : ; ; : (4.84)

The partition function is given by the normalization: NZ

Z Z ¼ 0

NZ

0 3 Y

i¼1

0

99kint 5 8 1 20 3 3 1 k0i þ > 0 2 > > > > 2 = => < 3 þ X 3 k 1 ε Ki 7 kint þ 6B 0i C 2$ 0 2 1 1 4@ $ 1þ $ A 5 3 1 > > > > k0i kB Ti > > i ¼ 1> kint k0i þ ; : ;> : 0 þ 2 2

8 > < N>

kint 0

1

εKi2 dεK1 dεK2 dεK3 . (4.85)

Formulae of Kappa Distributions: Toolbox CHAPTER 4 4.4.8.4.3 Correlated Degrees in Uncorrelated Groups kint 0 /N Using the approximations: 8 > > > > > > > > > > <

int 5 8 99k0 2 > 3 > > > k0i þ >> > > > > > 3 int 23 > 20 > >> > 1 > > k0 þ 2 > > > > 1 > > > > > > 3 < = = 7 6 kint þ X C B 0 k 1 ε 6B 0i C Ki 7 2 $ $ $ 1 þ 1 6@ 1 7 3 1A > > > > 4 k0i kB Ti 5 > > > i ¼ 1> kint k0i þ > > > > 0 þ > > > > > > > > 2 2 > > > > > > > > > > > > > > > > : ; > > : ;

8 > > > > > > <

5 8 99kint 0 2 3k0i þ32 >> 20 > > 1 > > > > > > > > > > > 7 3 6B = C 1 k 1 ε 6B 0i C Ki 7 z 1þ $ $ln $ 1 þ 7 6 3 > 1A > > > 4@ k0i kB Ti 5 > > > > i¼1 kint k0i þ > > > > 0 þ > > > > > > > 2 2 : ;> > > : ;

8 8 99 > 3k0i þ32 >> 20 > > > 1 > > > > > > > > > > > > > > > > 7 3 6B > > > 4 k0i kB Ti 5 > > > > i¼1 k0i þ > > > > > > > > > > > 2 : ;> > > : ;

we find P

3 fεKi g3i¼1 ; fk0i g3i¼1 ; kint 0 /N; fTi gi¼1

( ) 3 3 Y 1 εKi k0i 2 12 1þ $ $εKi ; f k0i kB Ti i¼1

(4.86)

coinciding, thus, with Eq. (4.58) in 4.4.5.1.2.

4.4.8.4.4 Correlated Degrees in Equally Correlated Groups kint 0 /k0 P3 εKi 1 3 3 3 int P fεKi gi¼1 ; fk0i /k0 gi¼1 ; k0 /k0 ; fTi gi¼1 f 1 þ $ i ¼ 1 k0 kB Ti

!k0 5 2

$

3 Y i¼1

1

εKi2 ;

(4.87) coinciding, thus, with Eq. (4.30a) in 4.4.1.1.2.

215

216

PART 1 Theory and Formalism 4.4.8.5 Distributions, for Two Correlated Groups, With f1 ¼ 1 and f2 ¼ 2 Degrees of Freedom, Self-Correlated With Kappa Index k0i, and Intercorrelated With Kappa Index kint 0 This is the generalization of the cases of “correlation between the projection at a certain direction and the perpendicular plane” and “self-correlated projections at a direction and perpendicular plane,” in Subsections 4.4.2 and 4.4.4 (or 4.4.6), respectively.

uk ; ! ut

4.4.8.5.1 Distribution in Terms of the Velocity ! u ¼

8 3 > k0 þ > > > 2 3 int 23 > 2 1 > k þ > int 1 uk ubk q2k 0 2 1 q2k > k0 þ > k0 > > 2 > > :

þ

kint 0 þ kint 0

1

k0 þ2

" $ 1þ

1 $ k0 þ 1

# kint þ2 ! 2 0 ut! u bt q2t q2t

1

5 9kint 0 2 > > > > > > > =

> > > > > > > ;

.

(4.88a)

Normalization 8 3 > k0 þ > 2 > > 2 3 > 3 2 1 1 2 kint þ > int Z N> 0 2 > k0 þ > k0 > > 2 > > :

þ

kint 0 þ kint 0

1

" $ 1þ

k0 þ2

5 9kint 0 2 > > > > > =

1 $ k0 þ 1

# kint þ2 ! 2 0 ut! u bt q2t q2t

1

> > > > > ;

d! u.

(4.88b)

Formulae of Kappa Distributions: Toolbox CHAPTER 4

4.4.8.5.2 Distribution in Terms of the Kinetic Energy, 2 2 ut ! u bt Þ εKk ¼ 12 m uk ubk and εKt ¼ 12 mð! 2

3

0 1 k0 þ 2 6 1 1 kint þ 3 int 6 k0 þ B εKk kB Tk C 0 2 6 1 1 2$B1 þ 2 C 6 P εKk ; εKt ; k0 ; kint $ 0 ; Tk ; Tt ¼ $6 @ A 1 kB Tk Z 6 kint 0 k0 þ 4 2 5 3kint 0 2

0 þ2 kint 1 εKt kB Tt kint 0 0 þ1 $ $ 1 þ int kB Tt k0 þ 1 k0 k þ2

þ

7 7 7 17 7 7 5 (4.89a)

Normalization 2

3

1 kint0 þ 23 0 1 k þ Z NZ N 6kint þ 1 εKk kB Tk 0 2 1 6 0 C B 2 Z ¼ $ 6 int 2$@1 þ A 1 kB Tk 0 0 4 k0 k0 þ 2 kint þ þ 0 int k0

1

$ 1þ

1 εKt kB Tt $ k0 þ 1 kB Tt

kint0 þ2

k þ2 0

5 3kint 0 2

7 15

1

εKk2 dεKk dεKt (4.89b)

4.4.8.5.3 Correlated Degrees in Equally Correlated Groups kint 0 /k0 (See 4.4.2.1.2) " !#k0 5 2 εKk 1 εKt 1 P εKk ; εKt ; k0 ; Tk ; Tt f 1 þ $ þ εKk2 . k0 kB Tk kB Tt

(4.89c)

4.4.8.5.4 Correlated Degrees in Uncorrelated Groups kint 0 /N (See 4.4.4.1.2)

P εKk ; εKt ; k0 ; Tk ; Tt

"

1 εKk f 1þ $ k0 kB Tk

#k0 3

2 1 εKt k0 2 12 $ 1þ $ εKk . k0 kB Tt

(4.89d)

4.5 Distributions With Potential The kappa distributions in the presence of a potential energy have been first systematically studied in Livadiotis (2015b). Special cases can be found in Livadiotis

1

εKk2 .

217

218

PART 1 Theory and Formalism

(2015c, 2016a). General Hamiltonian distributions formulae can be found also in Livadiotis et al. (2012) and Livadiotis and McComas (2013a, 2014a).

4.5.1 General Hamiltonian Distribution 4.5.1.1 Phase Space Distribution (Livadiotis et al., 2012; Livadiotis and McComas, 2013a, 2014a) Pð! r ;! u ; k0 ; TÞ ¼

8
N

N

Z

N

N

91 1

1 Hð! r ;! u Þ k0 12 d ! != 1þ $ dr du ; k0 kB T

1

1 Hð! r ;! u Þ k0 12 d 1þ $ . k0 kB T

Normalization Z N Z N Pð! r ;! u ; k0 ; TÞd! r d! u ¼ 1: N

N

(4.90a)

(4.90b)

4.5.1.2 Hamiltonian Function Hð! r ;! u Þ ¼ εK ð ! u Þ þ Fð! r Þ.

(4.90c)

4.5.1.3 Hamiltonian Degrees of Freedom Summing Up the Kinetic and Potential Degrees of Freedom r ;! u Þi r Þi 1 u Þi hFð! 1 1 hHð! hεK ð! dh ¼ þ ¼ dK þ dF . kB T kB T 2 kB T 2 2

(4.90d)

4.5.2 Positive Attractive Potential F ! r >0 4.5.2.1 Phase Space Distributions

4.5.2.1.1 In Terms of the Velocity ! u

dK dF G k0 þ 1 þ þ ! 1 2 2 1d $qdK P ! r ; u ; k0 ; q ¼ p2dK k0 2 K $ dF G k0 þ 1 þ 2 8 91 > > 3k0 112dF 2 > > > > Z < N = ! 1 Fð r Þ ! 5 4 1þ $ dr > > k0 kB T N > > > > : ; 1 1 39k0 12dK 2dF 2 > 2 ! u ! ub 1 Fð! r Þ5 = 1 þ $4 þ . > k0 kB T > q2 ; :

8 > <

(4.91a)

Formulae of Kappa Distributions: Toolbox CHAPTER 4

Normalization Z N Z N Pð! r ;! u ; k0 ; qÞd! r d! u ¼ 1: N

N

(4.91b)

4.5.2.1.2 In Terms of the Kinetic Energy εK (compare With Eq. (4.2a)) P ! r ; εK ; k0 ; T ¼

ðk k T Þ2dK 0 B dK dF ; k0 þ 1 þ B 2 2 91 8 3k0 112dF 2 > > > > = > k 0 kB T > > ; : N 1

3k0 112dK 12dF ! 1 1 ε þ Fð r Þ d 1 K 5 41 þ $ εK2 K . k0 kB T 2

Normalization Z NZ N Pð! r ; εK ; k0 ; TÞd! r dεK ¼ 1: 0

N

(4.92a)

(4.92b)

4.5.2.2 Positional Distribution Function

Pð! r ; k0 ; TÞ ¼

8
N

N

91 1

k0 11 dF

= ! 2 1 Fð r Þ 1 Fð! r Þ k0 12 dF d! r $ 1þ $ . 1þ $ ; k0 kB T k0 kB T (4.93a)

Normalization Z N P ! r ; k 0 ; T d! r ¼ 1. N

(4.93b)

4.5.2.3 Potential Degrees of Freedom r Þi 1 hFð! dF h . kB T 2

(4.94)

219

220

PART 1 Theory and Formalism 4.5.3 Negative Attractive Potential F ! r <0 4.5.3.1 Phase Space Distributions

4.5.3.1.1 In Terms of the Velocity ! u

dK dF G k0 þ 1 þ ! 1 2 2 1d $qdK P ! r ; u ; k0 ; T ¼ p2dK k0 2 K $ dF G k0 þ 1 2 91 8 3 2 k0 1þ12dF > > = < Z N Fð! r Þ 4 15 d! r > > ; : N k0 kB T þ

9k0 112dK þ12dF 8 2 ! ! 2 3 = < 1 Fð! r Þ u ub 51 $4 . ; :k0 kB T q2 þ

(4.95)

4.5.3.1.2 In Terms of the Kinetic Energy εK P ! r ; εK ; k 0 ; T ¼

ðk k T Þ2dK 0 B dK dF ; k0 þ 1 B 2 2 91 8 3k0 1þ12dF 2 > > = < Z N Fð! r Þ ! 5 4 1 dr > > ; : N k0 kB T 1

þ

3k0 112dK þ12dF 2 r Þ εK Fð! 15 4 . k0 kB T

(4.96)

þ

4.5.3.1.3 Restrictions dF dK . N < Fð! r Þ k0 kB T or k0 kB T Fð! r Þ N; 0 < k0 < 2 2

(4.97)

Formulae of Kappa Distributions: Toolbox CHAPTER 4

4.5.3.2 Positional Distribution Function 91 2 3k0 1þdF 3k0 1þdF 2 ! 2 2 > = Fð! rÞ Fð r Þ ! ! 5 5 4 4 1 1 Pð r ; k0 ; TÞ ¼ dr $ . > > k 0 kB T ; : N k0 kB T 8 >
N

þ

þ

(4.98) See: xþ ¼ x for x 0 and xþ ¼ 0 for x < 0 (See Eq. (1.13) in Chapter 1).

4.5.3.3 Potential Degrees of Freedom 1 hjFð! r Þji dF ¼ . 2 kB T

(4.99)

4.5.4 Small Attractive/Repulsive Positive/Negative r ðk0 kB T Þ << 1 Defined in a Finite Potential F ! Volume ! r ˛V 4.5.4.1 Phase Space Distribution

4.5.4.1.1 In Terms of the Velocity ! u

dK dF G k0 þ 1 þ ! 1 2 2 1d $qdK P ! r ; u ; k0 ; T ¼ p2dK k0 2 K $ dF G k0 þ 1 2 1 91 ! 3k0 1H2dF = r Þ Fð 1 ! 5 41 $ dr ; : ! k 0 kB T r ˛V 8
2

2 k0 112dK H12dF 2 ! 39 ! ! = r Þ Fð u ub 1 5 1 þ $4 . : k0 kB T ; q2 8 <

(4.100)

221

222

PART 1 Theory and Formalism 4.5.4.1.2 In Terms of the Kinetic Energy εK P ! r ; εK ; k0 ; T ¼

ðk k T Þ2dK 0 B $ dK dF ; k0 þ 1 B 2 2 1 91 8 2 ! 3k0 1H2dF =
1 1 ! 3k0 12dK H2dF ε r Þ Fð K 1 1 d 1 5 41 þ $ εK2 K . kB T k0 2

(4.101)

4.5.4.2 Positional Distribution Function 91 >

k0 1H1 dF = ! 2 1 jFð r Þj Pð! r ; k0 ; TÞ ¼ d! r 1 $ > > k 0 kB T r ˛V ; : ! 8 >
! 3k0 1H2 dF Fð r Þ 1 5 41 $ . k0 kB T 2

1

(4.102)

Comment: This is similar to Eq. (4.93a) for F ! r ¼ F ! r (4.5.2.2), for positions spanning a finite volume ! r ˛V.

4.5.4.3 Potential Degrees of Freedom r Þji 1 hjFð! dF ¼ . kB T 2

(4.103)

4.5.5 Equivalent Local Distribution Reduction of the distribution function of the kinetic and potential energies to the simpler but equivalent local distribution of solely the kinetic energy.

Formulae of Kappa Distributions: Toolbox CHAPTER 4 4.5.5.1 Phase Space Distribution With Potential F ! r ðk0 kB T Þ2dK f ! r ; εK hN$P ! r ; εK ¼ N$ dK dF ; k0 þ 1 þ B 2 2 91 8 3k0 112dF 2 =
3k0 112dK 12dF ! 1 1 εK þ Fð r Þ5 d 1 41 þ $ εK2 K . k0 kB T 2

(4.104a)

Normalization to N number of particles Z

N

N

Z 0

N

f ð! r ; εK Þd! r dεK ¼ N$

Z

N

N

Z

N

Pð! r ; εK Þd! r dεK ¼ N.

(4.104b)

0

4.5.5.2 Equivalent Local Distribution With No Potential Energy

! ! dF ! f r ; εK ¼ n r $P εK ; F ¼ 0; e k0 ¼ k0 þ ; T r 2 h i12dK r k0 kB T ! ! e ¼ n r $ dK ;e k0 þ 1 B 2

k0 11dK 2 1 1 εK d 1 1þ $ εK2 K : ! e k0 kB Tð r Þ

(4.104c)

223

224

PART 1 Theory and Formalism

4.5.6 Positive Power Law Central Potential (Oscillation Type) FðrÞ ¼ 1b kr b (Livadiotis, 2015c) 4.5.6.1 Phase Space Distributions

4.5.6.1.1 In Terms of the Position Vector ! r and the Kinetic Energy εK dr dr b 1bdr 12dK b k0 ! 1 b $ðkB T Þ2dK r0dr P r ; εK ; k0 ; T ¼ $ dr d d d d K r K r 2 2p ; þ ; k0 þ 1 B B 2 b 2 b 8 2 k0 112dK 1bdr !!b 39 < = 1 r 1 εK dr d 1 5 þ 1 þ $4 εK2 K . : ; k0 kB T b r0 dr G 2

(4.105)

4.5.6.1.2 In Terms of the Position Distance r and the Kinetic Energy εK dr 1 dr b 1bdr 12dK b k0 $ðkB T Þ2dK r0dr b Pðr; εK ; k0 ; T Þ ¼ dK dr dK dr ; þ ; k0 þ 1 B B 2 b 2 b ( " #)k0 112dK 1bdr 1 1 εK dr r b d 1 þ 1þ $ εK2 K r dr 1 . k0 kB T b r0 (4.106a) Normalization Z NZ N Pðr; εK ; k0 ; TÞdrdεK ¼ 1; 0

(4.106b)

0

where dr is the position vector dimensionality.

4.5.6.2 Potential Degrees of Freedom r Þi 1 dr hFð! dF h ¼ . kB T 2 b

(4.107)

Formulae of Kappa Distributions: Toolbox CHAPTER 4

4.5.6.3 Positional Distribution

4.5.6.3.1 In Terms of the Position Vector ! r dr d dr dr b br dr " #k0 11b dr G b k0 j! rj b 2 b dr ! b $r0 $ 1 þ $ Pð r ; k0 ; r0 Þ ¼ $ . dr r0 k0 2p 2 B dr ; k þ 1 0 b (4.108)

4.5.6.3.2 In Terms of the Position Distance r dr d 3k0 11 dr 2 b dr b br dr b k0 b 7 6 r 7 b b $r0dr $6 Pðr; k0 ; r0 Þ ¼ r dr 1 . 41 þ k0 $ r0 5 dr ; k0 þ 1 B b

(4.109a)

Normalization Z

N

Pðr; k0 ; r0 Þdr ¼ 1:

(4.109b)

0

4.5.6.3.3 In Terms of the Potential Energy F k0 11dr b 1 dr $ 1 þ $R Rdr =b1 ; Rh $ðr=r0 Þb fFðr Þ. dr k0 b ; k0 þ 1 B b (4.110a) 1dr

P ðR; k0 Þ ¼

k0 b

Normalization Z N PðR; k0 ÞdR ¼ 1: 0

(4.110b)

225

226

PART 1 Theory and Formalism

4.5.7 Negative Power Law Central Potential (Gravitational Type) FðrÞ ¼ 1b kr b (Livadiotis, 2015b) 4.5.7.1 Phase Space Distributions

4.5.7.1.1 In Terms of the Position Vector ! r and the Kinetic Energy εK dr dr dr b 1bdr 12dK b k0 ! 1 2 b $ðkB T Þ2dK r0dr P r ; εK ; k 0 ; T ¼ $ dr d d d d 2p 2 B r k ; k þ 1 B K ; k K þ r 0 0 0 b 2 2 b 9k0 112dK þ1bdr 8 2 0 3 1b > > = <1 d 1 εK 7 d 1 6 r B r0 C $ 4 @ A εK2 K : 51 ! > > k b k T B ; : 0 r G

þ

(4.111)

4.5.7.1.2 In Terms of the Position Distance r and the Kinetic Energy εK dr dr b 1bdr 12dK b k0 1 b $ðkB T Þ2dK r0dr P ðr; εK ; k0 ; T Þ ¼ dr dK dr dK k0 ; k0 þ 1 B ; k0 B b 2 b 2

k0 11dK þ1dr 2 b 1 1 dr r 0 b εK d 1 $ εK2 K r dr 1 . 1 k0 b r kB T þ (4.112)

4.5.7.2 Potential Degrees of Freedom r Þji 1 dr hjFð! dF h ¼ . kB T 2 b

(4.113)

Formulae of Kappa Distributions: Toolbox CHAPTER 4

4.5.7.3 Positional Distribution

4.5.7.3.1 In Terms of the Position Vector ! r dr 1b 2 0 3k0 1þ1bdr b dr dr dr d r b G b k0 2 b B r0 C 7 $r0dr $6 r ¼ P ! r ; k 0 ; r 0 d! $ d! r; 4 b $@ A 15 dr d k 0 ! 2p 2 B r k ; k þ 1 r 0 0 b (4.114)

4.5.7.3.2 In Terms of the Position Distance r dr 2 3k0 1þ1bdr b dr dr dr b k0 b r0 b b 7 $r0dr $6 P ðr; k0 ; r0 Þdr ¼ 15 r dr 1 dr; 4b$ dr k0 r k0 ; k0 þ 1 B b (4.115)

4.5.7.3.3 In Terms of the Potential Energy F 1

d

k0b r

k0 1þ1dr

1 $ $R 1 k0

dr k0 ; k0 þ 1 B b dr R h $ðr=r0 Þb fFðr Þ. b

P ðR; k0 ÞdR ¼

b

Rdr =b1 dR; (4.116)

4.5.7.3.4 Restrictions 0 < k0 <

dr dK ; b 2dr =dK . b 2

(4.117)

227

228

PART 1 Theory and Formalism 4.5.8 Properties for FðrÞ ¼ 1b kr b 4.5.8.1 Parameters 1 1 b dr 1 k B b kB T C 1 1 hFi b b C ¼ B @ 1 A ; 2dF h kB T ¼ kB T r0 ¼ b dr . k b 0

D E r0 h r b

1b

(4.118)

4.5.8.2 Degeneration of the Kappa Index 1 kðdK Þ ¼ k0 þ dK ðno potentialÞ0kðdK ; dr Þ 2 1 1 1 ¼ k0 þ dK dr with potential FðrÞ ¼ kr b . 2 b b

(4.119)

4.5.8.3 Local Parameters

4.5.8.3.1 Density 2 dr b 3k0 1H1bdr r Hb 17 6k $ r 6 0 7 0 nðr Þ ¼ n0 $6 . 7 dr 4 5 Hb 1 k0

(4.120a)

4.5.8.3.2 Temperature 2 dr b 3 r Hb 17 6k $ r 6 7 T ðr Þ ¼ T 0 $ 6 0 d 0 7. r 4 5 Hb 1 k0

(4.120b)

Formulae of Kappa Distributions: Toolbox CHAPTER 4

4.5.8.3.3 Thermal Pressure 2 dr b 3k0 H1bdr r Hb 17 6k $ r 6 0 7 0 p ðr Þ ¼ p 0 $ 6 . 7 dr 4 5 Hb 1 k0

(4.120c)

4.5.8.4 Polytropic Index

nðr Þ T ðr Þ v 1 ¼ ; n ¼ H dr 1 k0 ; n0 T0 b

(4.120d)

where (n0, T0, p0): density, temperature, and thermal pressure at radius r ¼ r0, given by Eq. (4.118). (See Meyer-Vernet et al., 1995; Livadiotis, 2015b, 2015c, 2016c.) The polytropic relation (Eq. 4.120d) can be modified in cases where (n0, T0, p0) depend on the position (see Chapter 5, Section 5.3).

4.5.8.5 Statistical Moments

4.5.8.5.1 Kinetic Moments (Livadiotis, 2014a) Z N u ja Pð! r ;! u Þd! r d! u j! N N dK þ a a G G k þ 1 0 1 2 a 2 ; $ ¼ qa k20 $ dK Gðk0 þ 1Þ G 2

Z D E u ja ¼ j!

Z D 1 E a ε2K ¼

N

N

Z

N

Pð! r ; εK Þd! r dεK 0 N dK þ a a G G k þ 1 0 1 1 2 a 2 . $ ¼ ðkB TÞ2 a k20 dK Gðk0 þ 1Þ G 2 1

a

ε2K

(4.121a)

(4.121b)

229

230

PART 1 Theory and Formalism

4.5.8.5.2 Potential Moments (Based on Eqs. 4.106a, 4.111) r ja iF>0 ¼ hj!

Z

N

r ja j!

Z

N

Pð! r ;! u ; F > 0Þd! u d! r N N d þa a a G r G k0 þ 1 b 1 d b a r b ; $ ¼ r0a k0b $ dr Gðk0 þ 1Þ b G b Z

N

a Z ! r

N

! P ! r ; u ; F < 0 d! u d! r N N d a a G r þ 1 G k þ 1 þ 0 b 1 dr b a b . $ ¼ r0a k0 b $ dr þ a Gðk0 þ 1Þ b G þ1 b

r ja iF<0 ¼ hj!

(4.122a)

(4.122b)

4.5.9 Marginal and Conditional Distributions (Livadiotis, 2015b) 4.5.9.1 Marginal Distributions The marginal distributions provide the positional and velocity distributions, which are derived from integrating the phase space distribution of the Hamiltonian over the velocities and the positions, respectively, Pð! rÞ ¼

Z

N

N

! ! P ! r ; u d u and

Pð! uÞ ¼

Z

N

N

! ! P ! r; u dr.

(4.123)

4.5.9.2 Conditional Distributions The conditional distributions provide the positional and velocity distributions ! at a certain velocity ! u ¼ b or position ! r ¼ ! a ! (4.124a) r ;! u ¼ b r ¼ ! a ;! u ; P ! r ¼ P ! and P ! u ¼ P ! while a local probability distribution may be defined after the normalization: ! P ! r ;! u ¼ b ! P ! r ¼ ! a ;! u ! Pð r Þ ¼ R . ! ! and P u ¼ R N P! N !! u r ¼ ! a ;! u d! N N P r ; u ¼ b d r (4.124b)

Formulae of Kappa Distributions: Toolbox CHAPTER 4 4.5.10 Angular Potentials F(w,4) This may apply for positive or small negative potentials with jFðw; 4Þj < k0 kB T.

4.5.10.1 Phase Space Distribution

4.5.10.1.1 In Terms of the Angular Dependence (w, 4) of the (dr ¼ 3)-dimensional Position Vector ! r and the (dK)-Dimensional Velocity ! u

dK dF G k0 þ 1 þ 1 2 2 1d $qdK P w; 4; ! u ; k0 ; T ¼ p2dK k0 2 K $ dF G k0 þ 1 2 (Z )1 1

Z 2p 1 1 Fðw; 4Þ k0 1H2dF 1þ $ dcos wd4 k0 kB T 0 1 (

1 1þ $ k0

"

2 ! u ! ub q2

Fðw; 4Þ þ kB T

#)k0 112dK H12dF .

(4.125a) where the upper/lower sign corresponds to positive/negative potential. Normalization Z

N

Z

N

2p

Z

1

1

0

Pðw; 4; εK ; k0 ; TÞdcos wd4 d! u ¼ 1:

(4.125b)

4.5.10.1.2 In Terms of the Position Vector ! r and the Kinetic Energy εK ðk k T Þ2dK 0 B dK dF ; k0 þ 1 B 2 2 (Z )1 1

Z 2p 1 1 Fðw; 4Þ k0 1H2dF 1þ $ dcos wd4 k0 kB T 0 1 1

P ðw; 4; εK ; k0 ; T Þ ¼

1 1

1 εK þ Fðw; 4Þ k0 12dK H2dF 12dK 1 1þ $ εK . k0 kB T

(4.126a)

Normalization Z 0

N

Z

2p 0

Z

1

1

Pðw; 4; εK ; k0 ; TÞ dcos wd4 dεK ¼ 1.

(4.126b)

231

232

PART 1 Theory and Formalism

4.5.10.2 Potential Degrees of Freedom 1 jFðw; 4Þj dF ¼ . kB T 2

(4.127)

4.5.10.3 Kappa Index Degeneration 1 1 k ¼ k 0 þ dK dF . 2 2

(4.128)

4.5.11 Magnetization Potential F(w) f cosw (Livadiotis, 2015b, 2015c, 2016a) 4.5.11.1 Potential Energy ! FðwÞ ¼ ! m $ B ¼ mB cos w; b h mB=ðkB TÞ.

(4.129)

4.5.11.2 Phase Space Distribution, in Terms of the Positional Polar Angle w and the Kinetic Energy εK, for 1 Positional and dK Kinetic Degrees of Freedom 1 G k0 bhcos wi þ dK þ 1 ðk0 þ bÞk0 bhcos wi ð1 þ bhcos wi bÞ 2 Pðcos w; εK ; k0 ; bÞ ¼ 1 k þ1þ12dK bhcos wi 2Gðk0 bhcos wiÞG dK k00 2 " #k0 112dK þbhcos wi 1 1 εK d 1 12dK b cos w ðkB TÞ $ 1þ $ εK2 K . k0 kB T (4.130a) Normalization Z

N

0

Z

1

1

Pðcos w; εK ; k0 ; bÞd cos w dεK ¼ 1.

(4.130b)

4.5.11.3 Distribution of the Kinetic Energy εK

P ðεK ; k0 ; bÞ ¼

1 ðk0 kB T Þ2dK b k0 bhcos wi $ 1 þ 1 k0 B k0 bhcos wi; dK 2 ( 1

1 1 εK bkB T k0 2dK þbhcos wi b þ hcos wi 1 $ 1þ 2 k 0 kB T ) 1

1 εK þ bkB T k0 2dK þbhcos wi d 1 1þ (4.131) $εK2 K . k0 kB T

Formulae of Kappa Distributions: Toolbox CHAPTER 4

Comment: The distribution of the kinetic energy is a superposition of two kappa distributions, one with energy εK bkBT and one with energy εK þ bkBT.

4.5.11.4 Distribution of the Polar Angle cosw

4.5.11.4.1 Distribution

Pðcos w; k0 ; bÞ ¼

1 ðk0 bhcos wiÞðk0 þ bÞk0 bhcos wi ð1 þ bhcos wi bÞ 2 ðk0 b cos wÞk0 1þbhcos wi . (4.132a)

Normalization Z

1

1

Pðcos w; k0 ; bÞdcos w ¼ 1:

(4.132b)

4.5.11.4.2 Restrictions k0 > b.

(4.133)

4.5.11.4.3 Average hcos wi is Given by 1 þ bhcos wi þ b ¼ 1 þ bhcos wi b

k0 þ b k0 bhcos wi . k0 b

(4.134)

4.5.11.5 Degeneration of the Kappa Index 1 kðdK Þ ¼ k0 þ dK ðno potentialÞ 0 2 1 kðdK ; dF Þ ¼ k0 þ dK bhcos wi ðwith potentialÞ; 2 where 1 dF ¼ bhcos wi. 2

(4.135a)

(4.135b)

Comment: The application was first examined using the kappa index k0 given by kðdK Þ ¼ k0 þ 12dK (Livadiotis, 2015b; 2016a), and this kappa index is used in Chapter 5, Section 5.8. For instance, the reader may compare Eq. (5.159a) with Eq. (4.134), where the degeneration caused by the potential degrees of freedom is also considered, using k0 0k0 bhcos wi.

233

234

PART 1 Theory and Formalism

4.6 Multiparticle Distributions 4.6.1 Standard N-Particle (N$d)eDimensional Kappa Distributions (Livadiotis and McComas, 2011b)

4.6.1.1 Distributions, in Terms of the Velocity, ! u , for N Particles, dK Degrees of Freedom per Particle, Using the Invariant Kappa Index k0 dK G k0 þ 1 þ N n oN 1d N 2 P ! u ðnÞ ; k0 ; q ¼ pk0 q2 2 K $ n¼1 Gðk0 þ 1Þ 1 " # N 2 k0 12dK N 1 1 X ! ! 1þ $ 2 . u ðnÞ u b k0 q n ¼ 1

(4.136a)

Normalization Z N Z N n oN / P ! u ðnÞ ; k0 ; q d! u ð1Þ /d! u ðNÞ ¼ 1: N

N

n¼1

(4.136b)

4.6.1.2 Distributions, in Terms of the Kinetic Energy, ε K ðnÞ ¼ 12 m 2 ! u b , for N Particles, dK Degrees of Freedom per Particle, Using u ðnÞ ! the Invariant Kappa Index k0 dK G k0 þ 1 þ N n oN 1 2 P εKðnÞ ; k0 ; T ¼ ðk0 kB TÞ2dK N $ N n¼1 dK Gðk0 þ 1Þ$G 2 !k0 11dK N 2 N N Y 1 1 1 X d 1 2 K 1þ $ εKðnÞ $ εKðnÞ . k 0 kB T n ¼ 1 n¼1 (4.137a)

Normalization Z N Z N n oN / P εKðnÞ ; k0 ; T dεKð1Þ /dεKðNÞ ¼ 1: 0

0

n¼1

(4.137b)

Formulae of Kappa Distributions: Toolbox CHAPTER 4

4.6.2 Negative N-Particle (N$d)eDimensional Kappa Distribution

4.6.2.1 Distributions, in Terms of the Velocity, ! u , for N Particles, dK Degrees of Freedom per Particle, Using the Invariant Kappa Index k0 n P

! u ðnÞ

oN n¼1

; k0 ; q

1d N Gðk0 Þ $ ¼ pk0 q2 2 K $ dK G k0 N 2 1 " # N 2 k0 12dK N 1 1 X ! ! 1 $ 2 . u ðnÞ u b k0 q n ¼ 1 þ

(4.138)

4.6.2.2 Distributions, in Terms of the Kinetic Energy, εK(n), for N Particles, dK Degrees of Freedom per Particle, Using the Invariant Kappa Index k0 n P

εKðnÞ

oN n¼1

; k0 ; T

1 Gðk0 Þ ¼ ðk0 kB TÞ2dK N $ N dK dK G k0 N $G 2 2 !k0 11dK N 2 N N Y 1 1 1 X d 1 2 K 1 $ εKðnÞ $ εKðnÞ . k0 kB T n ¼ 1 n¼1

(4.139)

4.6.3 N-Particle (N$d )eDimensional Kappa Distributions With Potential (Livadiotis, 2015c) 4.6.3.1 Phase Space Distribution, for N Particles, dK Degrees of Freedom per Particle, Using the Invariant Kappa Index k0

n P

! r ðnÞ ; ! u ðnÞ

oN n¼1

; k0 ; T

8
Z

2

H 41 þ 1 $ ¼ / : N k0 N N

N

n

! r ðnÞ ; ! u ðnÞ kB T

oN 3k0 11 dN 2 n¼1

5

1 2 n oN 3k0 12 dN 91 ! ! H r ðnÞ ; u ðnÞ = 6 7 1 n¼1 7 6 ! ! ! ! d r ð1Þ d u ð1Þ /d r ðNÞ d u ðNÞ $61 þ $ : 7 ; 4 5 k0 kB T

(4.140a)

235

236

PART 1 Theory and Formalism

Normalization Z N Z N n oN / P ! r ðnÞ ; ! u ðnÞ ; k0 ; T d! r ð1Þ d! u ð1Þ /d! r ðNÞ d! u ðNÞ ¼ 1. N

n¼1

N

(4.140b)

4.6.3.2 Hamiltonian Function !

n oN H ! r ðnÞ ; ! u ðnÞ

n¼1

¼

N n 2 oN 1 X ! m ub þF ! r ðnÞ u ðnÞ ! . n¼1 2 n¼1

(4.140c)

4.6.3.3 Hamiltonian Degrees of Freedom Summing up the Kinetic and Potential Degrees of Freedom

1 1 dh $ 2 N

n oN H ! r ðnÞ ; ! u ðnÞ n¼1

kB T

N X

n oN F ! r ðnÞ

1 n¼1 $ N kB T n oN F ! r ðnÞ 1 1 1 1 n¼1 . ¼ dK þ dF ; dF h $ 2 2 2 N kB T (4.140d) ¼

1 $ N n¼1

εK ! u ðnÞ kB T

þ

4.6.4 Standard N-Particle (N$d )eDimensional Kappa Distributions

4.6.4.1 Distributions, in Terms of the Velocity ! u With dK Degrees of Freedom per Particle, Using the Invariant Kappa Index k0

4.6.4.1.1 In Clusters of NC Uncorrelated Particles n P

! u ðnÞ

oNC n¼1

; k0 ; q

¼

NC n oNC Y P ! u ðnÞ ; k0 ; q

n¼1

n¼1

dK 3NC 2 G k0 þ 1 þ 1d 2 5 ¼ 4 pk0 q2 2 K $ Gðk0 þ 1Þ 2 2 3k0 112dK ! ! NC u u Y b 7 ðnÞ 1 6 . 41 þ $ 5 2 k q 0 n¼1

(4.141a)

Formulae of Kappa Distributions: Toolbox CHAPTER 4

4.6.4.1.2 In Clusters of NC Correlated Particles; See (4.6.1.1) n P

! u ðnÞ

1 " # NC 2 k0 12dK NC 1 1 X ! ; k0 ; q f 1 þ $ 2 ub . u ðnÞ ! n¼1 k0 q n ¼ 1

oNC

(4.141b)

4.6.4.1.3 In Clusters of NC Correlated Particles, But the Correlation Among the Clusters (kint 0 ) Differs From That Among the Particles (k0) n P 8 > > > > > > > > <

! u ðnÞ

oN n¼1

; k0 ; kint 0 ;q

¼

1 Z 1 39kint 0 12dK NC

2

6 0 6 dK NC 6 þ X 6 1 2 $ 61 B $ 1 @1 þ 6 d N dK > K C > n ¼ 16 > k0 þ kint þ > 6 0 > 2 2 > 4 > > : kint 0

> > d k þ1þ 2K 7> > > 2 d 1 0 > 7 K 2 kint þ1þ dK 7> ! ! > 0 u ðnÞ u b q 2 7= C q2

2

7 7> 7> > 7> > 5> > > ;

A

:

(4.141c)

oN n 4.6.4.2 Distributions, in Terms of the Kinetic Energies εK ðnÞ

n¼1

With dK

Degrees of Freedom per Particle, Using the Invariant Kappa Index k0

4.6.4.2.1 In Clusters of NC Uncorrelated Particles n P

εKðnÞ

oNC n¼1

; k0 ; T

¼

NC Y P εK ðn Þ ; k 0 ; T

n¼1

2

3 NC 12dK

6 ðk0 kB T Þ 7 7 ¼ 6 4 dK 5 ; k0 þ 1 B 2

1 NC NC Y 1 1 εKðnÞ k0 12dK Y $ 1þ $ εKðnÞ 2dK 1 k k T 0 B n¼1 n¼1

(4.142a)

237

238

PART 1 Theory and Formalism

4.6.4.2.2 In Clusters of NC Correlated Particles; See (4.6.1.2) n P

εKðnÞ

oNC

NC 1 1 X ; k0 ; T f 1 þ $ ε n¼1 k0 kB T n ¼ 1 KðnÞ

!k0 112dK NC

NC Y n¼1

εKðnÞ 2dK 1 1

(4.142b)

4.6.4.2.3 In Clusters of NC Correlated Particles, But the Correlation Among the Clusters (kint 0 ) Differs From That Among the Particles (k0) 8 > dK > > n < o NC kint 0 þ 1 int 2 P εK ðn Þ ; k0 ; k0 ; T ¼ $ 1 dK N C n¼1 Z > > int > k0 þ : 2 1 9 int 1 k þ1þ dK 3>k0 12dK NC dK > 0 NC 6 = εKðnÞ kB T int 2dK 7> X 1 B C k0 þ1þ 2 7 2 6 1 1 þ $ $ @ A 4 5> dK kB T > n¼1 > k0 þ ; 2

2

NC Y

0

εKðnÞ 2dK 1 1

n¼1

(4.142c)

oNC n 4.6.4.3 Phase Space Distributions HðnÞ ¼ H ! r ðnÞ ; ! u ðnÞ d Degrees of Freedom per Particle, Summing dK Kinetic and Degrees of Freedom, Using the Invariant Kappa Index k0

With

n¼1 dF Potential

4.6.4.3.1 In Clusters of NC Uncorrelated Particles n P

HðnÞ

oNC n¼1

; k0 ; T

¼

NC Y P HðnÞ ; k0 ; T

n¼1

3k0 11 d 2 2 ! ! NC H r ; u Y ðnÞ ðnÞ 1 1 5 4 ¼ $ 1þ $ . Z n¼1 k0 kB T

(4.143a)

Formulae of Kappa Distributions: Toolbox CHAPTER 4

Normalization 3k 11 d 8 9 2 0 2 NC < Z N Z N = H ! r ðnÞ ; ! u ðnÞ Y 1 5 41 þ $ Z ¼ d! r ðnÞ d! u ðnÞ . : N N ; k0 kB T n¼1 (4.143b)

4.6.4.3.2 In Clusters of NC Correlated Particles n P

HðnÞ

o NC n¼1

; k0 ; T

1 " # NC k0 12 d$NC 1 1 1 X ¼ $ 1þ $ H ! r ðnÞ ; ! u ðnÞ . Z k0 kB T n ¼ 1

(4.144a) Normalization Z

Z

"

NC 1 1 X Z ¼ 1þ $ H ! r ðn Þ ; ! u ðn Þ k 0 kB T n ¼ 1 N N ! ! d r ð1Þ d u ð1Þ /d! r ð NC Þ d ! u ð NC Þ . N

N

#k0 112 d$NC (4.144b)

4.6.4.3.3 In Clusters of NC Correlated Particles, But the Correlation Among the Clusters kint Differs From That 0 Among the Particles (k0) n P

oNC

1 Z 1 8 8 2 3 k þ1þ d 99kint 0 12 d$NC d d > > > 0 ! ! 2> > > > > int N = < = < k0 þ H r ðnÞ ; u ðnÞ kB T int d C X 1 6 7 k0 þ1þ 2 2 $ 2 $ 1 1 41 þ : 5 d$N d > > > > T k C B int > > > > n ¼ 1 ; : ; : k0 þ k0 þ 2 2 (4.145a) HðnÞ

n¼1

; k0 ; kint 0 ;T

¼

239

240

PART 1 Theory and Formalism

Z Z ¼

Normalization Z N N / r ð1Þ d! u ð1Þ /d! r ðNC Þd! u ðNC Þ f/gd!

N

8 > > > > > > <

N

1 99kint 0 12 d$NC d > 2> > > 3 int d > > >> > k þ1þ > 2> = => T 0

8 > > > > > NC > < X

2 d d ! ! kB H r ; u ð n Þ ð n Þ 1 6 7 2 $ 2 $ 1 41 þ f/g ¼ 1 5 d$N d > > T k C B > n ¼ 1> > > kint k0 þ þ > > 0 > > 2 2 > > : : kint 0 þ

k0 þ1þ

> > > > > > > > > > > ; ;> (4.145b)

4.6.5 Multispecies Distributions (Livadiotis and McComas, 2014a) 4.6.5.1 Distributions, in Terms of the Velocity, of N Different Particle Species, ! u ð1Þ ; ! u ð2Þ ; /; ! u ðNÞ , With dK Degrees of Freedom per Particle, Using the Invariant Kappa Index, k0

4.6.5.1.1 N Uncorrelated Species

P

n N oN n oN n oN Y ! ¼ u ðnÞ ; k0ðnÞ ; qðnÞ P ! u ðnÞ ; k0ðnÞ ; qðnÞ n¼1

n¼1

n¼1

n¼1

2 3k0ðnÞ 112dK dK 2 ! ! N 1dK G k0ðnÞ þ 1 þ 2 u u Y b 7 ðnÞ 1 2 6 ¼ pk0ðnÞ q2ðnÞ $ $ . 41 þ 5 2 k qðnÞ 0ðnÞ G k0ðnÞ þ 1 n¼1 (4.146a)

4.6.5.1.2 N Correlated Species (Compare With Eqs. 4.1a, 4.136a) n P

! u ðnÞ

oN n¼1

n oN ; k0 ; qðnÞ

n¼1

dK G k0 þ 1 þ N 1 2 ¼ ðpk0 Þ2dK N $ Gðk0 þ 1Þ 2 2 3k0 112dK N ! ! N N u u Y b 7 ðnÞ 1 X 6 qðnÞ dK $41 þ $ . $ 5 2 k q 0 n¼1 ðnÞ n¼1 (4.146b)

.

Formulae of Kappa Distributions: Toolbox CHAPTER 4

Comment: The correlation among the species is the same with the correlation among the particles.

4.6.5.1.3 N Correlated Species, But the Correlation Among Differs From That Among the Particles (k0) the Species kint 0 n P

! u ðnÞ

oN n¼1

; k0 ; kint 0 ;

n qðnÞ

oN n¼1

8 > > > <

dK kint 0 þ 1 2 $ ¼ $ 1 d N Z > K > > kint : 0 þ 2

1 9 int 1 k þ1þ dK 3>k0 12dK N 2 d > 0 K ! N 6 = u ðn Þ ! u b q2ðnÞ int 2dK 7> X 1 C k0 þ1þ 2 7 2 61 B 1 þ . $ @ A 4 5 2 dK > qðnÞ > n¼1 > k0 þ ; 2

2

0

(4.146c)

oN n 4.6.5.2 Distributions, in Terms of the Kinetic Energies, εK ðnÞ , With n¼1

dK Degrees of Freedom per Particle, Using the Invariant Kappa Index, k0

4.6.5.2.1 N Uncorrelated Species n P

εKðnÞ

N Y

¼

n¼1

oN n¼1

N n oN n oN Y ; k0ðnÞ ; TðnÞ P εKðnÞ ; k0ðnÞ ; TðnÞ ¼ n¼1

n¼1

1dK 2

n¼1

k0ðnÞ kB TðnÞ 1 εKðnÞ $ 1 þ $ dK k0ðnÞ kB TðnÞ ; k0ðnÞ þ 1 B 2

!k0ðnÞ 11dK 2

N Y n¼1

εKðnÞ 2dK 1 . 1

(4.147a)

4.6.5.2.2 N Correlated Species n P

εKðnÞ

oN

n

N ε oN 1 X KðnÞ ; k0 ; TðnÞ f 1þ $ n¼1 n¼1 k0 n ¼ 1 kB TðnÞ

!k0 11dK N 2

N Y n¼1

εKðnÞ 2dK 1 . 1

(4.147b)

241

242

PART 1 Theory and Formalism

4.6.5.2.3 N Correlated Species, But the Correlation Among the Species kint Differs From That Among the Particles (k0) 0 8 > dK > > n oN n oN kint 0 þ 1 < int 2 ¼ $ 1 P εKðnÞ ; k0 ; k0 ; TðnÞ dK N n¼1 n¼1 Z > > int > k0 þ : 2 int 1 9 2 0 1 k þ1þ dK 3>k0 12dK N dK 0 2 > > N 6 N εKðnÞ kB TðnÞ int dK 7= X Y 1 1 C k0 þ1þ 2 7 2 61 B 1 þ εKðnÞ 2dK 1 . $ @ A 4 5 d > kB TðnÞ K > n¼1 n¼1 > k0 þ ; 2 (4.147c)

4.6.5.3 Phase Space Distributions of N Species n oN HðnÞ ¼ H ! r ðnÞ ; ! u ðnÞ With d Degrees of Freedom per Particle, n¼1

Summing dK Kinetic and dF Potential Degrees of Freedom, Using the Invariant Kappa Index, k0

4.6.5.3.1 N Uncorrelated Species n P

HðnÞ

oN n¼1

n oN n oN ; k0ðnÞ ; TðnÞ n¼1

n¼1

3k0ðnÞ 11 d 2 2 ! ! N N H r ; u Y Y ðnÞ ðnÞ 1 1 4 5 ¼ 1þ P HðnÞ ; k0ðnÞ ; TðnÞ ¼ $ $ . Z n¼1 k0ðnÞ kB TðnÞ n¼1 (4.148a)

4.6.5.3.2 N Correlated Species

n P

HðnÞ

oN n¼1

n

; k0 ; TðnÞ

oN n¼1

!

3k0 11 d$N 2 ! N H ! r ; u X ðnÞ ðnÞ 1 4 1 5 . ¼ $ 1þ $ Z k0 n ¼ 1 kB TðnÞ 2

(4.148b)

Formulae of Kappa Distributions: Toolbox CHAPTER 4

4.6.5.3.3 N Correlated Species, But the Correlation Among the Species kint Differs From That Among the Particles (k0) 0 n P

HðnÞ

oN n¼1

; k0 ; kint 0 ;

8 > > > <

d n oN kint 0 þ 1 2 TðnÞ ¼ $ 1 d$N n¼1 Z > > > kint : 0 þ 2

d 9 int 3 k þ1þ d 9>k0 12 d$N d > > 0 = => H ! r ðn Þ ; ! u ðnÞ kB TðnÞ int 2d > 1 6 7 k0 þ1þ 2 2 $ 1 41 þ : 5 d > > > kB TðnÞ > n ¼ 1> > ;> : k0 þ ; 2

8 > < N > X

2

(4.148c)

4.7 Non-EuclideaneNormed Distributions (Livadiotis, 2007, 2008, 2012, 2016b) 4.7.1 Standard dKeDimensional Kappa Distribution of Velocity, ! u ( ( ! ! 2 d p2 1 1 dK K p1 ð u u b Þ ! $ þ x * $ Pð u ; k0 ; q; pÞ ¼ $ 1 þ Z k0 $ðp 1Þ 2 2 q2 "

ð! u ! u b Þ 2 dK 2 q2

#))k0 11dK $ðp1Þ1 2

;

(4.149a)

with the high-energy approximation (e.g., see also Qureshi et al., 2003, 2014; Randol and Christian, 2014, 2016), 2

3k0 112dK $ðp1Þ1 ! ! 2ðp1Þ u u b 1 6 7 P ! u ; k0 ; q; p y $41 þ $ ; 5 2 ð p1 Þ Z k0 $ðp 1Þ q p1 x*

2 1 for ! u ! u b >> dK q2 . 2 Partition function:

Z

N(

Z ¼ N

"

(4.149b)

( ! ! 2 d p2 1 dK K p1 ð u u b Þ $ þ x * $ 1þ k0 $ðp 1Þ 2 2 q2

ð! u ! u b Þ2 dK 2 q2

#))k0 11dK $ðp1Þ1 2

d! u.

(4.149c)

243

244

PART 1 Theory and Formalism 4.7.2 Standard dKeDimensional Kappa Distribution of Kinetic Energy, εK PðεK ; k0 ; T; pÞ ¼

1 Z

k0 11dK $ðp1Þ1 2 1 1 dK dK p2 εK dK d 1 p1 εK $ þ x * $ 1þ εK2 K ; k0 $ðp 1Þ 2 kB T 2 kB T 2 (4.150a)

with the high-energy approximation. " #k0 112dK $ðp1Þ1 p1 1 x* 1 εK p1 d 1 $ εK2 K ; P ðεK ; k0 ; T; pÞy $ 1 þ Z k0 $ðp 1Þ kB T 1 for εK >> dK kB T. 2

(4.150b)

Partition function: Z ¼ ðkB T=x* Þ

12dK

Z $ 0

8 > < N>

2

1 dK p2 6dK 1þ $4 þ x x* > k0 $ðp 1Þ 2 2 > :

39k0 112dK $ðp1Þ1 > > 1 dK 7 = x x* x2dK 1 dx: 5 > 2 > ;

(4.150c)

4.7.3 Argument x* k0 11dK $ðp1Þ1 2 1 dK dK p2 dK $ þ x x* 1þ x x* k0 $ðp 1Þ 2 2 2 0 p2 dK dK 1dK 1 x x* dx ¼ 0; x x* x2 2 2

Z

N

(4.151) e.g., x* ¼ 1 for p ¼ 2 and any k0.

Formulae of Kappa Distributions: Toolbox CHAPTER 4

4.8 Discrete Distributions (Tsallis et al., 1998) 4.8.1 Distribution of Energy 3k1 W ε U ; k; T fε g i j 1 j¼1 5 $41 þ $ kB T k 2

Pi ðεi ; k; TÞ ¼

1 Z fεj gW j¼1

(4.152a)

4.8.2 Partition Function 3k1 2 W εi U fεj gW ; k; T X 1 j¼1 5 41 þ $ Z fεj gW j¼1 ; k; T ¼ k k T B i¼1

(4.152b)

4.8.3 Internal Energy

The internal energy U fεj gW ; k; T hhεi i is implicitly given by j¼1 W X i¼1

1 εi U 1þ $ k kB T

k1 $ðεi UÞ ¼ 0:

(4.152c)

4.9 Concluding Remarks The chapter provided all the formulae of kappa distributions necessary to describe particle populations out of thermal equilibrium in plasmas and beyond. In particular, it was shown: n

n

isotropic distributions, in the absence of a potential energy: the positive and negative multidimensional kappa distributions, in the plasma flow (comoving) or an inertial reference frame; anisotropic distributions, in the absence of a potential energy, provided for the cases: correlated degrees of freedom, correlation between the projection at a certain direction and the perpendicular plane, self-correlated degrees of freedom, self-correlated projections at a direction and perpendicular plane, self-correlated degrees of freedom with different kappa indices, self-correlated projections at a direction and perpendicular plane with different kappa, self-correlated projections of different dimensionality and kappa, different self-correlation, and intercorrelation between degrees of freedom;

245

246

PART 1 Theory and Formalism

n

n

n

n

formulae of kappa distributions in the presence of a potential energy: general Hamiltonian kappa distribution; in the presence of positive or negative attractive potentials; or, of any small positive/negative attractive/repulsive potentials defined in a finite volume; reduction of the distribution of the kinetic and potential energy to the equivalent local distribution of solely the kinetic energy; kappa distributions for positive (oscillation type) or negative (gravitational type) power law central potentials; properties of power law central potentials (local density, temperature, thermal pressure, polytropic index); marginal and conditional distributions; distributions with angular potentials (e.g., magnetization potential); formulae of (positive or negative) multiparticle kappa distributions in the presence or not of a potential energy and of multispecies kappa distributions; formulation of generalized Lp kappa distributions, which are based on non-Euclidean Lp-norms; and formulation of discrete kappa distributions (distribution of energy, partition function, internal energy).

These developments allow the researcher to express and work with the distributions of any particle populations out of thermal equilibrium in space, geophysical, laboratory, or other plasmas.

4.10 Science Questions for Future Research Future analyses and observations need to address the following questions: 1. What is the magnetization angular distribution in an inertial frame? 2. What is the temperature if particle and group correlations are different? 3. Can a superposition on k form a complete/orthogonal basis?

Formulae of Kappa Distributions

Formulae of Kappa Distributions

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