Chapter
1.5.2 Continuous Description
1.6 Connection of Kappa Distributions With Nonextensive Statistical Mechanics
1.6.2 Historical Comments
1.7 Structure of the Kappa Distribution
1.7.1 The Base of the Kappa Distribution
1.7.2 The Exponent of the Kappa Distribution
1.8 The Concept of Temperature
1.8.1 The Definition of Temperature Out of Equilibrium and the Concept of Physical Temperature
1.8.2 Mean Kinetic Energy Defines Temperature
1.8.3 Misleading Considerations About Temperature
1.8.3.1 The Misleading Temperature-Like Parameter Tκ
1.8.3.1.1 Misinterpretation
1.8.3.2 The Misleading Dependence of the Temperature on the Kappa Index
1.8.3.2.1 Misinterpretation
1.8.3.3 The Misleading “Nonequilibrium Temperature”
1.8.3.3.1 Misinterpretation
1.8.3.4 The Misleading “Equilibrium Temperature”
1.8.3.4.1 Misinterpretation
1.8.3.5 The Most Frequent Speed
1.8.3.5.1 Misinterpretation
1.8.3.6 The Divergent Temperature at Antiequilibrium
1.8.3.6.1 Misinterpretation
1.8.3.7 The Thermal Pressure
1.8.3.7.1 Misinterpretation
1.8.3.8.1 Misinterpretation
1.8.4 Relativity Principle for Statistical Mechanics
1.9 The Concept of the Kappa (or q) Index
1.9.2 Dependence of the Kappa Index on the Number of Correlated Particles: Introduction of the Invariant Kappa Index κ0
1.9.3 Formulation of the N-Particle Kappa Distributions
1.9.4 Negative Kappa Index
1.9.5 Misleading Considerations About the Kappa (or q) Index
1.9.5.1 Kappa Index Sets an Upper Limit on the Total Number of Particles
1.9.5.2 Correlation: Independent of the Total Number of Particles
1.9.5.3 The Problem of Divergence
1.11 Science Questions for Future Research
2 - Entropy Associated With Kappa Distributions
2.3 The Role and Impact of Scale Parameters in the Entropic Formulation
2.3.2 The Length Scale, σr
2.3.3 The Speed Scale, σu
2.4 Derivation of the Entropic Formula for Velocity Kappa Distributions
2.4.2.2 Thermodynamic Limits
2.5 Entropy for Isothermal Transitions Between Stationary States
2.5.3 Spontaneous Entropic Procedures
2.6 The Discrete Dynamics of Transitions Between Stationary States
2.6.2 The Discrete Map of Stationary States Transitions
2.6.3 Numerical Application of the Discrete Transitions of Stationary States
2.6.4 The Five Stages of Stationary State Transitions
2.6.4.1 The Intermediate Transitions Toward Antiequilibrium
2.6.4.2 The Isentropic Switching and Its Importance
2.6.4.3 The Double Role of Antiequilibrium
2.6.4.4 The Final Transitions Toward Equilibrium
2.6.4.5 Back to Far-Equilibrium Region
2.8 Science Questions for Future Research
3 -
Phase Space Kappa Distributions With Potential Energy
3.3 The Hamiltonian Distribution
3.4 Normalization of the Phase Space Kappa Distribution
3.5 Marginal Distributions
3.6 Mean Kinetic Energy in the Presence of a Potential Energy
3.7 Degeneration of the Kappa Index in the Presence of a Potential Energy
3.7.2 1-D Linear Gravitational Potential
3.7.3 Positive Attractive Power Law Potential (Oscillator Type)
3.8 Local Kappa Distribution
3.9.2 The “Negative” Kappa Distribution
3.9.3 Formulation of Phase Space Distributions
3.9.3.1 Positive Potential Φ(r→)﹥0
3.9.3.1.1 Positive Potential: Positive Kappa Index, κ0 ﹥ 0
3.9.3.1.2 Positive Potential: Negative Kappa Index, κ0<0
3.9.3.2 Negative Potential, Φ(r→)<0
3.9.3.2.1 Negative Potential: Positive Kappa Index, κ0﹥0
3.9.3.2.2 Negative Potential: Negative Kappa Index, κ0<0
3.9.4 Negative Attractive Power Law Potential (Gravitational Type)
3.9.5 Potentials With Positive and Negative Values Acquiring Stable/Unstable Equilibrium Points
3.10 Gravitational Potentials
3.10.1 Linear Gravitational Potential: The Barometric Formula
3.10.2 Spherical Gravitational Potential
3.10.3 Virial Theorem and Jeans ‘Radius’
3.11 Potentials with Angular Dependence
3.12 Potentials Forming Anisotropic Distribution of Velocity
3.14 Science Questions for Future Research
4 - Formulae of Kappa Distributions: Toolbox
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 κ0
4.3.1.1.1 In Terms of the Velocity u→
4.3.1.1.2 In Terms of the Kinetic Energy εK=12m(u→−u→b)2
4.3.1.1.3 In Terms of Normalized Kinetic Energy ξ≡εK/(κ0kBT)
4.3.1.2 Distributions, for dK Degrees of Freedom, Using the 3-D Kappa Index κ3= κ0+32
4.3.1.2.1 Distribution in Terms of the Velocity u→
4.3.1.2.2 In Terms of the Kinetic Energy εK
4.3.1.3 Distributions, for dK Degrees of Freedom, Using the dK-Dimensional Kappa Index κ= κ0+12dK
4.3.1.3.1 In Terms of the Velocity u→
4.3.1.3.2 In Terms of the Kinetic Energy εK
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 κ0
4.3.2.1.1 In Terms of the Kinetic Energy EK=12mu→2 and the Polar Angle, ϑ, Set Between u→ and u→b
4.3.2.1.2 In Terms of the Kinetic Energy EK
4.3.2.1.3 In Terms of the Polar Angle ϑ
4.3.2.2 Distributions, for dK=3 Degrees of Freedom, Using the Invariant Kappa Index κ0
4.3.2.2.1 In Terms of the Kinetic Energy EK=12mu→2 and the Polar Angle ϑ
4.3.2.2.2 In Terms of the Kinetic Energy EK
4.3.2.2.3 In Terms of the Speed u
4.3.2.2.4 In Terms of the Polar Angle ϑ (Compare With Eq. 4.10b)
4.3.2.3 Distributions, for dK=3 Degrees of Freedom, Using the 3-D Kappa Index κ=κ0+32
4.3.2.3.1 In Terms of the Kinetic Energy EK=12mu→2 and the Polar Angle ϑ
4.3.2.3.2 In Terms of the Kinetic Energy EK
4.3.2.3.3 In Terms of the Speed u
4.3.2.3.4 In Terms of the Polar Angle ϑ
4.3.3 Negative, Multidimensional Kappa Distribution (Livadiotis, 2015b)
4.3.3.1 Distributions Using the Invariant Kappa Index κ0
4.3.3.1.1 In Terms of the Velocity u→
4.3.3.1.2 In Terms of the Kinetic Energy εK
4.3.3.2 Distributions Using the Kappa Index κ=κ0 −12dK
4.3.3.2.1 In Terms of the Velocity u→
4.3.3.2.2 In Terms of the Kinetic Energy εK
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 ÷ (1−c)
4.3.4.1.2 Positive to Negative Proportion is 1:1
4.3.4.1.3 Positive to Negative Proportion is 1:1, and Common Kappa Index (Leubner and Voros, 2005)
4.3.4.2 Linear Superposition Given a Density of Kappa Indices D(κ0)
4.3.4.3 Linear Superposition in Terms of the Kinetic Energy εK with Components of Different Temperature
4.3.4.4 Nonlinear Superposition of Positive and Negative Distributions in Terms of the Kinetic Energy εK
4.4 Anisotropic Distributions (Without Potential)
4.4.1 Correlated Degrees of Freedom
4.4.1.1 Distributions, for dK Degrees of Freedom, Using the Invariant Kappa Index κ0
4.4.1.1.1 In Terms of the Velocity u→
4.4.1.1.2 In Terms of the Kinetic Energy per Degree of Freedom εKi=12m(ui−ubi)2 for i: 1, …, dK
4.4.1.1.3 The Temperature is Given by
4.4.1.2 Distribution in Terms of the Velocity u→ for dK Degrees of Freedom, Using the dK-Dimensional Kappa Index κ=κ0 +12dK
4.4.1.3 Distribution in Terms of the Velocity u→ for dK Degrees of Freedom, Using the 3-D Kappa Index κ3=κ0 +12 (dK − 3)
4.4.1.4 Distributions, for dK=3 Degrees of Freedom, Using the Invariant Kappa Index κ0
4.4.1.4.1 In Terms of the Velocity u→
4.4.1.4.2 In Terms of the Kinetic Energy Per Degree of Freedom εKi=12m(ui−ubi)2 for i: x, y, z
4.4.1.5 Distribution, for dK=3 Degrees of Freedom, Using the 3-D Kappa Index κ=κ0 +32
4.4.1.5.1 In Terms of the Velocity u→
4.4.1.5.2 In Terms of the Kinetic Energy per Degree of Freedom εKi for i: x, y, z
4.4.1.5.3 The Temperature is Given by
4.4.2 Correlation Between the Projection at a Certain Direction and the Perpendicular Plane
4.4.2.1 Distributions, for dK=3 Degrees of Freedom, Using the Invariant Kappa Index κ0
4.4.2.1.1 In Terms of the Velocity u→=(u‖,u→⊥)
4.4.2.1.2 In Terms of the Kinetic Energy, εK‖=12m(u‖−ub‖)2 εK⊥=12m(u→⊥−u→b⊥)2
4.4.2.2 Distribution in Terms of the Velocity u→=(u‖,u→⊥) for dK = 3 Degrees of Freedom, Using the 3-D Kappa Index κ=κ0 +32
4.4.2.3 The Temperature is Given by
4.4.3 Self-Correlated Degrees of Freedom
4.4.3.1 Distributions, for dK Degrees of Freedom, Using the Invariant Kappa Index κ0
4.4.3.1.1 In Terms of the Velocity u→
4.4.3.1.2 In Terms of the Kinetic Energy per Degree of Freedom εKi=12m(ui−ubi)2 for i: 1, …, dK
4.4.3.1.3 The Temperature is Given by
4.4.3.2 Distribution in Terms of the Velocity, u→, for dK Degrees of Freedom, Using the dK-Dimensional Kappa Index κ=κ0+ 12dK
4.4.3.3 Distribution in Terms of the Velocity, u→, for dK Degrees of Freedom, Using the 3-D Kappa Index κ=κ0 +32
4.4.3.4 Distributions, for dK=3 Degrees of Freedom, Using the Invariant Kappa Index κ0
4.4.3.4.1 In Terms of the Velocity u→
4.4.3.4.2 In Terms of the Kinetic Energy per Degree of Freedom εKi=12m(ui−ubi)2 for i: x, y, z
4.4.3.5 Distribution, for dK=3 Degrees of Freedom, Using the 3-D Kappa Index κ=κ0 +32
4.4.3.5.1 In Terms of the Velocity u→
4.4.3.5.2 In Terms of the Kinetic Energy per Degree of Freedom εKi for i: x, y, z
4.4.3.5.3 The Temperature is Given by
4.4.4 Self-Correlated Projections at a Direction and Perpendicular Plane
4.4.4.1 Distributions, for dK=3 Degrees of Freedom, Using the Invariant Kappa Index κ0
4.4.4.1.1 In Terms of the Velocity u→=(u‖,u→⊥)
4.4.4.1.2 In Terms of the Kinetic Energy, εK‖=12m(u‖−ub‖)2 εK⊥=12m(u→⊥−u→b⊥)2
4.4.4.2 Distribution in Terms of the Velocity u→=(u‖,u→⊥) for dK=3 Degrees of Freedom, Using the 3-D Kappa Index κ=κ0 +32
4.4.4.3 The Temperature is Given by
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 κ0
4.4.5.1.1 In Terms of the Velocity u→
4.4.5.1.2 In Terms of the Kinetic Energy per Degree of Freedom εKi=12m(ui−ubi)2 for i: 1, …, dK
4.4.5.1.3 The Temperature is Given by
4.4.5.2 Distribution in Terms of the Velocity u→ for dK Degrees of Freedom, Using the 1-D Kappa Index κi=κ0i+12
4.4.5.3 Distribution in Terms of the Velocity u→ for dK Degrees of Freedom, Using the 3-D Kappa Index κ=κ0 +32
4.4.6 Self-Correlated Projections at a Direction and Perpendicular Plane With Different Kappa
4.4.6.1 Distributions, for dK=3 Degrees of Freedom, Using the Invariant Kappa Index κ0
4.4.6.1.1 In Terms of the Velocity u→=(u‖,u→⊥)
4.4.6.1.2 In Terms of the Kinetic Energy, εK‖=12mu‖−ub‖2, εK⊥=12m(u→⊥−u→b⊥)2 (also see Eq. (4.54))
4.4.6.2 Distribution in Terms of the Velocity u→=(u‖,u→⊥) for dK=3 Degrees of Freedom, Using the Kappa Indices κ‖=κ‖0+12,κ⊥=κ⊥0+1
4.4.6.3 The Temperature is Given by
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 o ...
4.4.7.1.1 In Terms of the Velocity u→
4.4.7.1.2 In Terms of the Kinetic Energy εKi=12m(u→−u→b)2|(fi) for i: 1, …, M
4.4.7.1.3 The Total Degrees of Freedom dK Are Given by
4.4.7.2 Distributions, for M Correlated Groups, Each With fi Degrees of Freedom, Self-Correlated With Kappa Index κ0i and Temperatu ...
4.4.7.2.1 In Terms of the Velocity u→
4.4.7.2.2 In Terms of the Kinetic Energy εKi=12m(u→−u→b)2|(fi)
4.4.7.2.3 The Total Degrees of Freedom dK Are Given by
4.4.7.2.4 The Temperature is Given by
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, κ0i, a ...
4.4.8.1.1 In Terms of the Velocity u→
4.4.8.1.2 In Terms of the Kinetic Energy εKi=12m(u→−u→b)2|(fi)
4.4.8.2 Distributions, for M Correlated Groups, Each With fi Degrees of Freedom, Self-Correlated With Kappa Index, κi, and Intercor ...
4.4.8.2.1 In Terms of the Velocity u→
4.4.8.2.2 In Terms of the Kinetic Energy εKi=12m(u→−u→b)2|(fi)
4.4.8.2.3 Internal Energy
4.4.8.2.4 Degeneration of the Kappa Index
4.4.8.2.5 Nonlinear Superposition
4.4.8.3 Distributions, for M Correlated Groups, Each With fi Degrees of Freedom, Self-Correlated With Invariant Kappa Index, κ0i=κ0 ...
4.4.8.3.1 In Terms of the Velocity u→
4.4.8.3.2 In Terms of the Kinetic Energy εKi=12m(u→−u→b)2|(fi)
4.4.8.4 Distributions, for dK=3 Correlated Groups, Each With fi=1 Degrees of Freedom, Self-Correlated With Kappa Index, κ0i, and In ...
4.4.8.4.1 In Terms of the Velocity u→
4.4.8.4.2 In Terms of the Kinetic Energy εKi=12m(ui−ubi)2 for i: x, y, z
4.4.8.4.3 Correlated Degrees in Uncorrelated Groups κ0int→∞
4.4.8.4.4 Correlated Degrees in Equally Correlated Groups κ0int→κ0
4.4.8.5 Distributions, for Two Correlated Groups, With f1=1 and f2=2 Degrees of Freedom, Self-Correlated With Kappa Index κ0i, and ...
4.4.8.5.1 Distribution in Terms of the Velocity u→=(u‖,u→⊥)
4.4.8.5.2 Distribution in Terms of the Kinetic Energy, εK‖=12m(u‖−ub‖)2 and εK⊥=12m(u→⊥−u→b⊥)2
4.4.8.5.3 Correlated Degrees in Equally Correlated Groups κ0int→κ0 (See 4.4.2.1.2)
4.4.8.5.4 Correlated Degrees in Uncorrelated Groups κ0int→∞ (See 4.4.4.1.2)
4.5 Distributions With Potential
4.5.1 General Hamiltonian Distribution
4.5.1.1 Phase Space Distribution (Livadiotis et al., 2012; Livadiotis and McComas, 2013a, 2014a)
4.5.1.2 Hamiltonian Function
4.5.1.3 Hamiltonian Degrees of Freedom Summing Up the Kinetic and Potential Degrees of Freedom
4.5.2 Positive Attractive Potential Φ(r→)﹥0
4.5.2.1 Phase Space Distributions
4.5.2.1.1 In Terms of the Velocity u→
4.5.2.1.2 In Terms of the Kinetic Energy εK (compare With Eq. (4.2a))
4.5.2.2 Positional Distribution Function
4.5.2.3 Potential Degrees of Freedom
4.5.3 Negative Attractive Potential Φ(r→)<0
4.5.3.1 Phase Space Distributions
4.5.3.1.1 In Terms of the Velocity u→
4.5.3.1.2 In Terms of the Kinetic Energy εK
4.5.3.2 Positional Distribution Function
4.5.3.3 Potential Degrees of Freedom
4.5.4 Small Positive/Negative Attractive/Repulsive Potential |Φ(r→)/(κ0kBT)|<<1 Defined in a Finite Volume r→∈V
4.5.4.1 Phase Space Distribution
4.5.4.1.1 In Terms of the Velocity u→
4.5.4.1.2 In Terms of the Kinetic Energy εK
4.5.4.2 Positional Distribution Function
4.5.4.3 Potential Degrees of Freedom
4.5.5 Equivalent Local Distribution
4.5.5.1 Phase Space Distribution With Potential Φ(r→)
4.5.5.2 Equivalent Local Distribution With No Potential Energy
4.5.6 Positive Power Law Central Potential (Oscillation Type) Φ(r)= 1bkrb (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
4.5.6.1.2 In Terms of the Position Distance r and the Kinetic Energy εK
4.5.6.2 Potential Degrees of Freedom
4.5.6.3 Positional Distribution
4.5.6.3.1 In Terms of the Position Vector r→
4.5.6.3.2 In Terms of the Position Distance r
4.5.6.3.3 In Terms of the Potential Energy Φ
4.5.7 Negative Power Law Central Potential (Gravitational Type) Φ(r)=−1bkr−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
4.5.7.1.2 In Terms of the Position Distance r and the Kinetic Energy εK
4.5.7.2 Potential Degrees of Freedom
4.5.7.3 Positional Distribution
4.5.7.3.1 In Terms of the Position Vector r→
4.5.7.3.2 In Terms of the Position Distance r
4.5.7.3.3 In Terms of the Potential Energy Φ
4.5.8 Properties for Φ(r)= ±1bkr±b
4.5.8.2 Degeneration of the Kappa Index
4.5.8.3.3 Thermal Pressure
4.5.8.5 Statistical Moments
4.5.8.5.1 Kinetic Moments (Livadiotis, 2014a)
4.5.8.5.2 Potential Moments (Based on Eqs. 4.106a, 4.111)
4.5.9 Marginal and Conditional Distributions (Livadiotis, 2015b)
4.5.9.1 Marginal Distributions
4.5.9.2 Conditional Distributions
4.5.10 Angular Potentials Φ(ϑ,ϕ)
4.5.10.1 Phase Space Distribution
4.5.10.1.1 In Terms of the Angular Dependence (ϑ, ϕ) of the (dr=3)-dimensional Position Vector r→ and the (dK)-Dimensional Velocity u→
4.5.10.1.2 In Terms of the Position Vector r→ and the Kinetic Energy εΚ
4.5.10.2 Potential Degrees of Freedom
4.5.10.3 Kappa Index Degeneration
4.5.11 Magnetization Potential Φ(ϑ)∝−cosϑ (Livadiotis, 2015b, 2015c, 2016a)
4.5.11.1 Potential Energy
4.5.11.2 Phase Space Distribution, in Terms of the Positional Polar Angle ϑ and the Kinetic Energy εK, for 1 Positional and dK Kinet ...
4.5.11.3 Distribution of the Kinetic Energy εK
4.5.11.4 Distribution of the Polar Angle cosϑ
4.5.11.4.3 Average 〈cosϑ〉 is Given by
4.5.11.5 Degeneration of the Kappa Index
4.6 Multiparticle Distributions
4.6.1 Standard N-Particle (N·d)–Dimensional 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 ...
4.6.1.2 Distributions, in Terms of the Kinetic Energy, εK(n)=12m× (u→(n)−u→b)2, for N Particles, dK Degrees of Freedom per Particle ...
4.6.2 Negative N-Particle (N·d)–Dimensional 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 ...
4.6.2.2 Distributions, in Terms of the Kinetic Energy, εΚ(n), for N Particles, dK Degrees of Freedom per Particle, Using the Invari ...
4.6.3 N-Particle (N·d)–Dimensional 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 κ0
4.6.3.2 Hamiltonian Function
4.6.3.3 Hamiltonian Degrees of Freedom Summing up the Kinetic and Potential Degrees of Freedom
4.6.4 Standard N-Particle (N·d)–Dimensional 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 κ0
4.6.4.1.1 In Clusters of NC Uncorrelated Particles
4.6.4.1.2 In Clusters of NC Correlated Particles; See (4.6.1.1)
4.6.4.1.3 In Clusters of NC Correlated Particles, But the Correlation Among the Clusters (κ0int) Differs From That Among the Particle ...
4.6.4.2 Distributions, in Terms of the Kinetic Energies {εK(n)}n=1N With dK Degrees of Freedom per Particle, Using the Invariant Ka ...
4.6.4.2.1 In Clusters of NC Uncorrelated Particles
4.6.4.2.2 In Clusters of NC Correlated Particles; See (4.6.1.2)
4.6.4.2.3 In Clusters of NC Correlated Particles, But the Correlation Among the Clusters (κ0int) Differs From That Among the Particle ...
4.6.4.3 Phase Space Distributions {H(n)= H(r→(n),u→(n))}n=1NC With d Degrees of Freedom per Particle, Summing dK Kinetic and dΦ Pot ...
4.6.4.3.1 In Clusters of NC Uncorrelated Particles
4.6.4.3.2 In Clusters of NC Correlated Particles
4.6.4.3.3 In Clusters of NC Correlated Particles, But the Correlation Among the Clusters (κ0int) Differs From That Among the Particle ...
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→(1), u→(2), ⋯, u→(N), With dK Degrees of Freedo ...
4.6.5.1.1 N Uncorrelated Species
4.6.5.1.2 N Correlated Species (Compare With Eqs. 4.1a, 4.136a)
4.6.5.1.3 N Correlated Species, But the Correlation Among the Species (κ0int) Differs From That Among the Particles (κ0)
4.6.5.2 Distributions, in Terms of the Kinetic Energies, {εK(n)}n=1N, With dK Degrees of Freedom per Particle, Using the Invariant ...
4.6.5.2.1 N Uncorrelated Species
4.6.5.2.2 N Correlated Species
4.6.5.2.3 N Correlated Species, But the Correlation Among the Species (κ0int) Differs From That Among the Particles (κ0)
4.6.5.3 Phase Space Distributions of N Species {H(n)= H(r→(n), u→(n))}n=1N With d Degrees of Freedom per Particle, Summing dK Kinet ...
4.6.5.3.1 N Uncorrelated Species
4.6.5.3.2 N Correlated Species
4.6.5.3.3 N Correlated Species, But the Correlation Among the Species (κ0int) Differs From That Among the Particles (κ0)
4.7 Non-Euclidean–Normed Distributions (Livadiotis, 2007, 2008, 2012, 2016b)
4.7.1 Standard dK–Dimensional Kappa Distribution of Velocity, u→
4.7.2 Standard dK–Dimensional Kappa Distribution of Kinetic Energy, εK
4.8 Discrete Distributions (Tsallis et al., 1998)
4.8.1 Distribution of Energy
4.10 Science Questions for Future Research
5 - Basic Plasma Parameters Described by Kappa Distributions
5.3.1 Simple Polytropes and Their Characteristic Exponent: The Polytropic Index
5.3.2 Generalized Polytropic Relations
5.3.3 Connection With the Kappa Index
5.3.3.1 Thermal Equilibrium, κ0→∞
5.3.3.2 Out of Thermal Equilibrium, κ0<∞
5.3.4 Complicated Relations Between Polytropic and Kappa Indices
5.4 Correlation Between Particle Energies
5.5 Debye Length in Equilibrium and Nonequilibrium Plasmas
5.5.2 Poisson Equation for the Electrostatic Potential
5.5.3 Symmetric Poisson Equation and Solutions
5.5.3.1 1-D Potential: Planar Charge Perturbation
5.5.3.2 2-D Potential: Linear Charge Perturbation
5.5.3.3 3-D Potential: Point Charge Perturbation
5.5.4 The Case of Large Potential Energy
5.5.5 Main Interpretations
5.6 Electrical Conductivity
5.7 Collision Frequency and Mean Free Path
5.8 Magnetization: The Curie Constant
5.9 Large-Scale Quantization Constant
5.9.1 The Role of Correlations
5.9.2 The Smallest Phase Space Parcel in Plasmas
5.9.3 Estimation of ℏ∗ for Space Plasmas
5.9.4 Application: Missing Plasma Parameters
5.11 Science Questions for Future Research
6 - Superstatistics: Superposition of Maxwell–Boltzmann Distributions
6.2 Introduction: Dynamical Creation of Kappa Distributions
6.3 Timescale Separation in Nonequilibrium Situations
6.4 Typical Universality Classes for f(β)
6.5 Asymptotic Behavior for Large Energies
6.6 Universality for Not Too Large Energies ε
6.7 From Measured Time Series to Superstatistics
6.8 Some Examples of Applications
6.8.1 Classical Lagrangian Turbulence: Acceleration Statistics
6.8.3 Kappa Distributions in High-Energy Scattering Processes
6.10 Science Questions for Future Research
7 - Linear Kinetic Waves in Plasmas Described by Kappa Distributions
7.3 Plasma Dielectric Tensor and the Dispersion Relation
7.4 Kappa Velocity Distribution Plasma Waves at Parallel Propagation (ϑ=0)
7.4.2 Low-Frequency Electromagnetic Alfvén Cyclotron Plasma Waves
7.4.3 Low-Frequency Instabilities
7.4.3.1 Alfvén Cyclotron Instability
7.4.3.2 Proton Firehose Instability
7.4.4 High-Frequency Electromagnetic Whistler Cyclotron Plasma Waves
7.4.5 High-Frequency Instabilities
7.4.5.1 Whistler Cyclotron Instability
7.4.5.2 Electron Firehose Instability
7.4.6 High-Frequency Electromagnetic Langmuir Plasma Waves
7.4.7 Low-Frequency Electromagnetic Ion-Acoustic Plasma Waves
7.5 Kappa Velocity Distribution Plasma Waves at Oblique Propagation (ϑ ≠ 0)
7.7 Science Questions for Future Research
8 - Nonlinear Wave–Particle Interaction and Electron Kappa Distribution
8.3 Plasma Weak Turbulence Theory
8.3.1 General Formulation
8.3.2 Weak Turbulence Theory for Plasmas Described by Kappa Distributions
8.3.2.1 Induced and Spontaneous Emissions
8.3.2.2 Induced and Spontaneous Decay Processes
8.3.2.3 Induced and Spontaneous Scattering Processes
8.4 Turbulent Quasiequilibrium and Kappa Electron Distribution
8.4.1 Steady-State Particle Distribution
8.4.2 Steady-State Langmuir Turbulence
8.4.2.1 Balance of Induced and Spontaneous Emission
8.4.2.2 Absence of Steady-State Decay Processes
8.4.2.3 Balance of Induced and Spontaneous Scattering
8.6 Science Questions for Future Research
9 - Solitary Waves in Plasmas Described by Kappa Distributions
9.2 Introduction: Observations and Origin of Suprathermal Electrons
9.3 Model of Ion-Acoustic Solitons and Double Layers in Plasmas With Suprathermal Electrons
9.3.1 Two-Component Plasmas: Cold Protons and Suprathermal Electrons
9.3.2 Three-Component Plasmas: Protons, Heavier Ions, and Suprathermal Electrons
9.3.3 Two-Component Magnetized Plasmas: Protons and Suprathermal Electrons
9.4 Model for Electron-Acoustic Solitons in Plasmas With Suprathermal Electrons
9.6 Science Questions for Future Research
3 - Applications in Space Plasmas
10 - Ion Distributions in Space Plasmas
10.3 Formulations of Ion Kappa Distributions
10.3.1 Standard Kappa Distribution of Velocities
10.3.2 “Negative” Kappa Distribution of Velocities
10.3.3 Superposition of Kappa Distribution of Velocities
10.3.4 Anisotropic Kappa Distribution of Velocities
10.3.5 Phase Space Marginal Kappa Distribution of Velocities
10.4 Toward Antiequilibrium, the Farthest State From Thermal Equilibrium
10.5 Arrangement of the Stationary States
10.5.1 A Measure of the Thermodynamic Distance From Thermal Equilibrium
10.5.2 A Generalized Measure
10.5.3 The Kappa Spectrum
10.6 Interpreting the Observations
10.6.1 Observations and Measurements of Kappa Indices in the Heliosphere
10.6.2 Observations in the Inner Heliosheath
10.6.2.1 Far-Equilibrium Inner Heliosheath
10.6.2.2 Anticorrelation Between Density and Temperature
10.6.2.3 Isobaric Process
10.6.3 Near Versus Far Equilibrium
10.6.4 The Role of PickUp Ions in the Transitions of Kappa Distributions
10.8 Science Questions for Future Research
11 - Electron Distributions in Space Plasmas
11.2 Introduction: Observations and Origins of Suprathermal Electrons
11.3 Coronal Heating by Velocity Filtration Due to Suprathermal Electrons
11.5 Influence of Suprathermal Electrons on the Acceleration of Escaping Particles
11.7 Science Questions for Future Research
12 - The Kappa-Shaped Particle Spectra in Planetary Magnetospheres
12.3 Measuring and Interpreting the Kappa Distribution in Space Plasmas
12.4 Kappa Distribution in the Magnetospheres of the Gas Giant Planets
12.4.1.2 Energetic Ion Spectra
12.4.1.3 Particle Energization Processes and Anisotropies
12.4.1.4 Energetic Particle Moments
12.4.1.5 Plasma Sources and Aurora
12.4.2.4 Electron Spectra
12.4.3.2 Plasma Energy Spectra
12.4.3.3 Energetic Ion Spectra
12.5 Kappa Distribution in the Magnetospheres of the Terrestrial Planets
12.5.1.2 “Quiet” Plasma Sheet Spectra
12.5.1.3 “Disturbed” Plasma Sheet Spectra
12.5.1.4 Middle Magnetosphere and Substorms
12.5.1.5 High-Latitude Spectra
12.5.1.6 Magnetosheath Spectra
12.5.2.2 Energetic Electron Bursts
12.5.2.3 MESSENGER Measurements
12.6 Are Kappa Distributions Useful for Magnetospheric Research?
12.8 Science Questions for Future Research
13 - Kappa Distributions and the Solar Spectra: Theory and Observations
13.3 Synthetic Line and Continuum Intensities
13.3.1 Ionization and Recombination Rates: Behavior of the Ionization Equilibrium
13.3.5 Synthetic Spectra and the Atmospheric Imaging Assembly Response to Emissions
13.4 Plasma Diagnostics From Emission Line Spectra
13.4.1 Density Diagnostics
13.4.2 Single-Ion Diagnostics of the κ-index
13.4.3 Diagnostics Involving Ionization Equilibrium
13.4.4 Diagnostics From Transition Region Lines: Si III
13.5 Differential Emission Measures for Kappa Distributions
13.7 Science Questions for Future Research
14 - Importance of Kappa Distributions to Solar Radio Bursts
14.3 Qualitative Aspects for the Generation and Damping of Plasma Waves and Radio Emissions
14.4 Type III Bursts, Electron Beams, and Langmuir Waves
14.5 Type II Bursts, Shocks, and Electron Reflections
14.7 Science Questions for Future Research
15 - Common Spectrum of Particles Accelerated in the Heliosphere: Observations and a Mechanism
15.3.1 Common Spectrum in the Inner Heliosphere
15.3.1.1 August 12, 2001, Local Acceleration Event
15.3.1.2 October 25, 2001, Local Acceleration Event
15.3.2 Common Spectrum of Anomalous Cosmic Rays Accelerated in the Nose of the Heliosheath
15.3.3 Common Spectrum in the Fast, Polar Coronal Holes Solar Wind
15.4 Acceleration Mechanism That Yields the Common Spectrum
15.4.1 Conditions in Which the Acceleration Must Operate
15.4.2 Illustration of How the Acceleration Mechanism Works
15.4.3 Parker Equation: Describing the Behavior of Accelerated Particles
15.4.4 Why Is the Spectral Index in the Velocity Space Equal to −5?
15.4.5 Deriving an Equation for the Time Evolution of the Common Spectrum
15.4.6 Comparing Solutions for the Time Evolution of the Common Spectrum With Observations
15.4.7 Why Is Pump Acceleration the Dominant Acceleration Mechanism?
15.4.8 Subtleties Associated With Solutions for Pump Acceleration
15.4.9 Source Particle Spectrum
15.5 Applications of the Pump Acceleration Mechanism
15.5.1 Acceleration of Energetic Particles at Shocks
15.5.1.1 Model for the Shock
15.5.1.2 Spectrum at the Shock
15.5.1.3 Role of Diffusive Shock Acceleration
15.5.2 Acceleration in the Solar Corona
15.5.2.1 Choice of the Diffusion Coefficient
15.5.2.2 Impulsive Solar Energetic Particle Events
15.7 Science Questions for Future Research
16 - Formation of Kappa Distributions at Quasiperpendicular Shock Waves
16.3 Upstream Distributions and Their Transmission Through Quasiperpendicular Shocks
16.4 Velocity Distribution Function Downstream of a Quasiperpendicular Shock
16.7 Dissipation and Particle Acceleration at Quasiperpendicular Shocks
16.9 Science Questions for Future Research
17 - Electron Kappa Distributions in Astrophysical Nebulae
17.2.3 Kappa Distributions
17.3 Are Energy Kappa Distributions Present in Astrophysical Nebulae?
17.4 Ionization Structures in an HII Region
17.5 Magnetic Structures in HII Regions
17.6 Nebular Spectral Lines
17.6.1 Collisional Excitation
17.6.2 Recombination Lines
17.6.3 Abundance Discrepancy Problem
17.7 Atomic Energy Levels and Kappa Distribution
17.7.1 Calculating Electron Temperatures From Collisionally Excited Lines
17.7.2 Explaining the Optical Recombination Line/Collisonally Excited Line Abundance Discrepancy
17.7.3 Discrepancies in Collisionally Excited Line Temperatures
17.8 Diagnostics for the Kappa Index
17.9 Modeling of Photoionized Nebulae
17.9.2 Complexities in HII Region Structures
17.10 Other Applications of Kappa Distributions in Astrophysical Nebulae
17.11 Alternative Explanations of Abundance Discrepancy
17.13 Science Questions for Future Research
Kappa Distributions
：Theory and Applications in Plasmas
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