# Kappa Distributions

## 1st Edition

### Theory and Applications in Plasmas

**Authors:**George Livadiotis

**eBook ISBN:**9780128046395

**Paperback ISBN:**9780128046388

**Imprint:**Elsevier

**Published Date:**21st April 2017

**Page Count:**738

## Description

*Kappa Distributions: Theory and Applications in Plasmas* presents the theoretical developments of kappa distributions, their applications in plasmas, and how they affect the underpinnings of our understanding of space and plasma physics, astrophysics, and statistical mechanics/thermodynamics. Separated into three major parts, the book covers theoretical methods, analytical methods in plasmas, and applications in space plasmas. The first part of the book focuses on basic aspects of the statistical theory of kappa distributions, beginning with their connection to the solid backgrounds of non-extensive statistical mechanics. The book then moves on to plasma physics, and is devoted to analytical methods related to kappa distributions on various basic plasma topics, spanning linear/nonlinear plasma waves, solitons, shockwaves, and dusty plasmas. The final part of the book deals with applications in space plasmas, focusing on applications of theoretical and analytical developments in space plasmas from the heliosphere and beyond, in other astrophysical plasmas.

*Kappa Distributions* is ideal for space, plasma, and statistical physicists; geophysicists, especially of the upper atmosphere; Earth and planetary scientists; and astrophysicists.

## Key Features

- Answers important questions, such as how plasma waves are affected by kappa distributions and how solar wind, magnetospheres, and other geophysical, space, and astrophysical plasmas can be modeled using kappa distributions
- Presents the features of kappa distributions in the context of plasmas, including how kappa indices, temperatures, and densities vary among the species populations in different plasmas
- Provides readers with the information they need to decide which specific formula of kappa distribution should be used for a certain occasion and system (toolbox)

## Readership

*Kappa Distributions* is ideal for space, plasma, and statistical physicists; geophysicists especially of the upper atmosphere, Earth and planetary scientists, and astrophysicists

## Table of Contents

**Part 1. Theory and Formalism**

Chapter 1. Statistical Background of Kappa Distributions: Connection With Nonextensive Statistical Mechanics

- 1.1. Summary
- 1.2. Introduction
- 1.3. Mathematical Motivation
- 1.4. Nonextensive Statistical Mechanics, in Brief!
- 1.5. Entropy Maximization
- 1.6. Connection of Kappa Distributions With Nonextensive Statistical Mechanics
- 1.7. Structure of the Kappa Distribution
- 1.8. The Concept of Temperature
- 1.9. The Concept of the Kappa (or q) Index
- 1.10. Concluding Remarks
- 1.11. Science Questions for Future Research

Chapter 2. Entropy Associated With Kappa Distributions

- 2.1. Summary
- 2.2. Introduction
- 2.3. The Role and Impact of Scale Parameters in the Entropic Formulation
- 2.4. Derivation of the Entropic Formula for Velocity Kappa Distributions
- 2.5. Entropy for Isothermal Transitions Between Stationary States
- 2.6. The Discrete Dynamics of Transitions Between Stationary States
- 2.7. Concluding Remarks
- 2.8. Science Questions for Future Research

Chapter 3. Phase Space Kappa Distributions With Potential Energy

- 3.1. Summary
- 3.2. Introduction
- 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
- 3.8. Local Kappa Distribution
- 3.9. Negative Potentials
- 3.10. Gravitational Potentials
- 3.11. Potentials with Angular Dependence
- 3.12. Potentials Forming Anisotropic Distribution of Velocity
- 3.13. Concluding Remarks
- 3.14. Science Questions for Future Research

Chapter 4. Formulae of Kappa Distributions: Toolbox

- 4.1. Summary
- 4.2. Introduction
- 4.3. Isotropic Distributions (Without Potential) (Livadiotis and McComas, 2009; 2011b)
- 4.4. Anisotropic Distributions (Without Potential)
- 4.5. Distributions With Potential
- 4.6. Multiparticle Distributions
- 4.7. Non-Euclidean–Normed Distributions (Livadiotis, 2007, 2008, 2012, 2016b)
- 4.8. Discrete Distributions (Tsallis et al., 1998)
- 4.9. Concluding Remarks
- 4.10. Science Questions for Future Research

**Part 2. Plasma Physics**

Chapter 5. Basic Plasma Parameters Described by Kappa Distributions

- 5.1. Summary
- 5.2. Introduction
- 5.3. Polytropes
- 5.4. Correlation Between Particle Energies
- 5.5. Debye Length in Equilibrium and Nonequilibrium Plasmas
- 5.6. Electrical Conductivity
- 5.7. Collision Frequency and Mean Free Path
- 5.8. Magnetization: The Curie Constant
- 5.9. Large-Scale Quantization Constant
- 5.10. Concluding Remarks
- 5.11. Science Questions for Future Research

Chapter 6. Superstatistics: Superposition of Maxwell–Boltzmann Distributions

- 6.1. Summary
- 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
- 6.9. Concluding Remarks
- 6.10. Science Questions for Future Research

Chapter 7. Linear Kinetic Waves in Plasmas Described by Kappa Distributions

- 7.1. Summary
- 7.2. Introduction
- 7.3. Plasma Dielectric Tensor and the Dispersion Relation
- 7.4. Kappa Velocity Distribution Plasma Waves at Parallel Propagation (ϑ=0)
- 7.5. Kappa Velocity Distribution Plasma Waves at Oblique Propagation (ϑ ≠ 0)
- 7.6. Concluding Remarks
- 7.7. Science Questions for Future Research

Chapter 8. Nonlinear Wave–Particle Interaction and Electron Kappa Distribution

- 8.1. Summary
- 8.2. Introduction
- 8.3. Plasma Weak Turbulence Theory
- 8.4. Turbulent Quasiequilibrium and Kappa Electron Distribution
- 8.5. Concluding Remarks
- 8.6. Science Questions for Future Research

Chapter 9. Solitary Waves in Plasmas Described by Kappa Distributions

- 9.1. Summary
- 9.2. Introduction: Observations and Origin of Suprathermal Electrons
- 9.3. Model of Ion-Acoustic Solitons and Double Layers in Plasmas With Suprathermal Electrons
- 9.4. Model for Electron-Acoustic Solitons in Plasmas With Suprathermal Electrons
- 9.5. Concluding Remarks
- 9.6. Science Questions for Future Research

**Part 3. Applications in Space Plasmas**

Chapter 10. Ion Distributions in Space Plasmas

- 10.1. Summary
- 10.2. Introduction
- 10.3. Formulations of Ion Kappa Distributions
- 10.4. Toward Antiequilibrium, the Farthest State From Thermal Equilibrium
- 10.5. Arrangement of the Stationary States
- 10.6. Interpreting the Observations
- 10.7. Concluding Remarks
- 10.8. Science Questions for Future Research

Chapter 11. Electron Distributions in Space Plasmas

- 11.1. Summary
- 11.2. Introduction: Observations and Origins of Suprathermal Electrons
- 11.3. Coronal Heating by Velocity Filtration Due to Suprathermal Electrons
- 11.4. Heat Flux
- 11.5. Influence of Suprathermal Electrons on the Acceleration of Escaping Particles
- 11.6. Concluding Remarks
- 11.7. Science Questions for Future Research

Chapter 12. The Kappa-Shaped Particle Spectra in Planetary Magnetospheres

- 12.1. Summary
- 12.2. Introduction
- 12.3. Measuring and Interpreting the Kappa Distribution in Space Plasmas
- 12.4. Kappa Distribution in the Magnetospheres of the Gas Giant Planets
- 12.5. Kappa Distribution in the Magnetospheres of the Terrestrial Planets
- 12.6. Are Kappa Distributions Useful for Magnetospheric Research?
- 12.7. Concluding Remarks
- 12.8. Science Questions for Future Research

Chapter 13. Kappa Distributions and the Solar Spectra: Theory and Observations

- 13.1. Summary
- 13.2. Introduction
- 13.3. Synthetic Line and Continuum Intensities
- 13.4. Plasma Diagnostics From Emission Line Spectra
- 13.5. Differential Emission Measures for Kappa Distributions
- 13.6. Concluding Remarks
- 13.7. Science Questions for Future Research

Chapter 14. Importance of Kappa Distributions to Solar Radio Bursts

- 14.1. Summary
- 14.2. Introduction
- 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
- 14.6. Concluding Remarks
- 14.7. Science Questions for Future Research

Chapter 15. Common Spectrum of Particles Accelerated in the Heliosphere: Observations and a Mechanism

- 15.1. Summary
- 15.2. Introduction
- 15.3. Observations
- 15.4. Acceleration Mechanism That Yields the Common Spectrum
- 15.5. Applications of the Pump Acceleration Mechanism
- 15.6. Concluding Remarks
- 15.7. Science Questions for Future Research

Chapter 16. Formation of Kappa Distributions at Quasiperpendicular Shock Waves

- 16.1. Summary
- 16.2. Introduction
- 16.3. Upstream Distributions and Their Transmission Through Quasiperpendicular Shocks
- 16.4. Velocity Distribution Function Downstream of a Quasiperpendicular Shock
- 16.5. Simulations
- 16.6. Observational Tests
- 16.7. Dissipation and Particle Acceleration at Quasiperpendicular Shocks
- 16.8. Concluding Remarks
- 16.9. Science Questions for Future Research

Chapter 17. Electron Kappa Distributions in Astrophysical Nebulae

- 17.1. Summary
- 17.2. Introduction
- 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
- 17.7. Atomic Energy Levels and Kappa Distribution
- 17.8. Diagnostics for the Kappa Index
- 17.9. Modeling of Photoionized Nebulae
- 17.10. Other Applications of Kappa Distributions in Astrophysical Nebulae
- 17.11. Alternative Explanations of Abundance Discrepancy
- 17.12. Concluding Remarks
- 17.13. Science Questions for Future Research

Appendix A: Abbreviations

Appendix B. Main Symbols

## Details

- No. of pages:
- 738

- Language:
- English

- Copyright:
- © Elsevier 2017

- Published:
- 21st April 2017

- Imprint:
- Elsevier

- eBook ISBN:
- 9780128046395

- Paperback ISBN:
- 9780128046388

## About the Author

### George Livadiotis

Dr. George Livadiotis is a Senior Research Scientist in Southwest Research Institute. He is a leading expert on the field of kappa distributions and its statistical background, the framework of non-extensive statistical mechanics. Among other theoretical achievements, he developed (i) the connection of kappa distributions with non-extensive statistical mechanics, (ii) the formula of entropy that is related to the kappa distributions, (iii) the generalization of kappa distribution to describe the whole Hamiltonian of particles, the kinetic and potential energy, (iv) the different types of consistent formulae of kappa distributions, (v) the shock Rankine–Hugoniot conditions for kappa distributions. Among other applications, he used kappa distributions to describe the proton populations in many space plasmas in the heliosphere and the heliosheath.

### Affiliations and Expertise

Senior Research Scientist, Southwest Research Institute, USA