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.
1. 1.2. Introduction - p. 5 2. 1.3. Mathematical Motivation - p. 14 3. 1.4. Nonextensive Statistical Mechanics, in Brief! - p. 16 4. 1.5. Entropy Maximization - p. 22 5. 1.6. Connection of Kappa Distributions With Nonextensive Statistical Mechanics - p. 29 6. 1.7. Structure of the Kappa Distribution - p. 37 7. 1.8. The Concept of Temperature - p. 41 8. 1.9. The Concept of the Kappa (or q) Index - p. 51 9. 1.10. Concluding Remarks - p. 62 10. 1.11. Science Questions for Future Research - p. 63 11. Chapter 2. Entropy Associated With Kappa Distributions - p. 65 12. 2.1. Summary - p. 66 13. 2.2. Introduction - p. 66 14. 2.3. The Role and Impact of Scale Parameters in the Entropic Formulation - p. 67 15. 2.4. Derivation of the Entropic Formula for Velocity Kappa Distributions - p. 78 16. 2.5. Entropy for Isothermal Transitions Between Stationary States - p. 87 17. 2.6. The Discrete Dynamics of Transitions Between Stationary States - p. 96 18. 2.7. Concluding Remarks - p. 103 19. 2.8. Science Questions for Future Research - p. 103 20. Chapter 3. Phase Space Kappa Distributions With Potential Energy - p. 105 21. 3.1. Summary - p. 106 22. 3.2. Introduction - p. 106 23. 3.3. The Hamiltonian Distribution - p. 110 24. 3.4. Normalization of the Phase Space Kappa Distribution - p. 113 25. 3.5. Marginal Distributions - p. 116 26. 3.6. Mean Kinetic Energy in the Presence of a Potential Energy - p. 120 27. 3.7. Degeneration of the Kappa Index in the Presence of a Potential Energy - p. 123 28. 3.8. Local Kappa Distribution - p. 134 29. 3.9. Negative Potentials - p. 142 30. 3.10. Gravitational Potentials - p. 164 31. 3.11. Potentials with Angular Dependence - p. 171 32. 3.12. Potentials Forming Anisotropic Distribution of Velocity - p. 174 33. 3.13. Concluding Remarks - p. 175 34. 3.14. Science Questions for Future Research - p. 176 35. Chapter 4. Formulae of Kappa Distributions: Toolbox - p. 177 36. 4.1. Summary - p. 185 37. 4.2. Introduction - p. 186 38. 4.3. Isotropic Distributions (Without Potential) (Livadiotis and McComas, 2009; 2011b) - p. 187 39. 4.4. Anisotropic Distributions (Without Potential) - p. 196 40. 4.5. Distributions With Potential - p. 217 41. 4.6. Multiparticle Distributions - p. 234 42. 4.7. Non-Euclidean–Normed Distributions (Livadiotis, 2007, 2008, 2012, 2016b) - p. 243 43. 4.8. Discrete Distributions (Tsallis et al., 1998) - p. 245 44. 4.9. Concluding Remarks - p. 245 45. 4.10. Science Questions for Future Research - p. 246 46. Part 2. Plasma Physics - p. 246 47. Chapter 5. Basic Plasma Parameters Described by Kappa Distributions - p. 249 48. 5.1. Summary - p. 250 49. 5.2. Introduction - p. 250 50. 5.3. Polytropes - p. 252 51. 5.4. Correlation Between Particle Energies - p. 267 52. 5.5. Debye Length in Equilibrium and Nonequilibrium Plasmas - p. 272 53. 5.6. Electrical Conductivity - p. 286 54. 5.7. Collision Frequency and Mean Free Path - p. 292 55. 5.8. Magnetization: The Curie Constant - p. 295 56. 5.9. Large-Scale Quantization Constant - p. 300 57. 5.10. Concluding Remarks - p. 310 58. 5.11. Science Questions for Future Research - p. 311 59. Chapter 6. Superstatistics: Superposition of Maxwell–Boltzmann Distributions - p. 313 60. Chapter 7. Linear Kinetic Waves in Plasmas Described by Kappa Distributions - p. 329 61. 7.1. Summary - p. 330 62. 7.2. Introduction - p. 330 63. 7.3. Plasma Dielectric Tensor and the Dispersion Relation - p. 336 64. 7.4. Kappa Velocity Distribution Plasma Waves at Parallel Propagation (ϑ=0) - p. 339 65. 7.5. Kappa Velocity Distribution Plasma Waves at Oblique Propagation (ϑ ≠ 0) - p. 359 66. 7.6. Concluding Remarks - p. 361 67. 7.7. Science Questions for Future Research - p. 361 68. Chapter 8. Nonlinear Wave–Particle Interaction and Electron Kappa Distribution - p. 363 69. 8.1. Summary - p. 363 70. 8.2. Introduction - p. 364 71. 8.3. Plasma Weak Turbulence Theory - p. 365 72. 8.4. Turbulent Quasiequilibrium and Kappa Electron Distribution - p. 389 73. 8.5. Concluding Remarks - p. 398 74. 8.6. Science Questions for Future Research - p. 398 75. Chapter 9. Solitary Waves in Plasmas Described by Kappa Distributions - p. 399 76. Part 3. Applications in Space Plasmas - p. 418 77. Chapter 10. Ion Distributions in Space Plasmas - p. 421 78. 10.1. Summary - p. 422 79. 10.2. Introduction - p. 422 80. 10.3. Formulations of Ion Kappa Distributions - p. 425 81. 10.4. Toward Antiequilibrium, the Farthest State From Thermal Equilibrium - p. 432 82. 10.5. Arrangement of the Stationary States - p. 439 83. 10.6. Interpreting the Observations - p. 444 84. 10.7. Concluding Remarks - p. 463 85. 10.8. Science Questions for Future Research - p. 463 86. Chapter 11. Electron Distributions in Space Plasmas - p. 465 87. Chapter 12. The Kappa-Shaped Particle Spectra in Planetary Magnetospheres - p. 481 88. Chapter 13. Kappa Distributions and the Solar Spectra: Theory and Observations - p. 523 89. Chapter 14. Importance of Kappa Distributions to Solar Radio Bursts - p. 549 90. Chapter 15. Common Spectrum of Particles Accelerated in the Heliosphere: Observations and a Mechanism - p. 569 91. Chapter 16. Formation of Kappa Distributions at Quasiperpendicular Shock Waves - p. 609 92. Chapter 17. Electron Kappa Distributions in Astrophysical Nebulae - p. 633 93. Appendix A: Abbreviations - p. 657 94. Appendix B. Main Symbols - p. 661 95. References - p. 665 96. Index - p. 711
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.
1. 1.2. Introduction - p. 5 2. 1.3. Mathematical Motivation - p. 14 3. 1.4. Nonextensive Statistical Mechanics, in Brief! - p. 16 4. 1.5. Entropy Maximization - p. 22 5. 1.6. Connection of Kappa Distributions With Nonextensive Statistical Mechanics - p. 29 6. 1.7. Structure of the Kappa Distribution - p. 37 7. 1.8. The Concept of Temperature - p. 41 8. 1.9. The Concept of the Kappa (or q) Index - p. 51 9. 1.10. Concluding Remarks - p. 62 10. 1.11. Science Questions for Future Research - p. 63 11. Chapter 2. Entropy Associated With Kappa Distributions - p. 65 12. 2.1. Summary - p. 66 13. 2.2. Introduction - p. 66 14. 2.3. The Role and Impact of Scale Parameters in the Entropic Formulation - p. 67 15. 2.4. Derivation of the Entropic Formula for Velocity Kappa Distributions - p. 78 16. 2.5. Entropy for Isothermal Transitions Between Stationary States - p. 87 17. 2.6. The Discrete Dynamics of Transitions Between Stationary States - p. 96 18. 2.7. Concluding Remarks - p. 103 19. 2.8. Science Questions for Future Research - p. 103 20. Chapter 3. Phase Space Kappa Distributions With Potential Energy - p. 105 21. 3.1. Summary - p. 106 22. 3.2. Introduction - p. 106 23. 3.3. The Hamiltonian Distribution - p. 110 24. 3.4. Normalization of the Phase Space Kappa Distribution - p. 113 25. 3.5. Marginal Distributions - p. 116 26. 3.6. Mean Kinetic Energy in the Presence of a Potential Energy - p. 120 27. 3.7. Degeneration of the Kappa Index in the Presence of a Potential Energy - p. 123 28. 3.8. Local Kappa Distribution - p. 134 29. 3.9. Negative Potentials - p. 142 30. 3.10. Gravitational Potentials - p. 164 31. 3.11. Potentials with Angular Dependence - p. 171 32. 3.12. Potentials Forming Anisotropic Distribution of Velocity - p. 174 33. 3.13. Concluding Remarks - p. 175 34. 3.14. Science Questions for Future Research - p. 176 35. Chapter 4. Formulae of Kappa Distributions: Toolbox - p. 177 36. 4.1. Summary - p. 185 37. 4.2. Introduction - p. 186 38. 4.3. Isotropic Distributions (Without Potential) (Livadiotis and McComas, 2009; 2011b) - p. 187 39. 4.4. Anisotropic Distributions (Without Potential) - p. 196 40. 4.5. Distributions With Potential - p. 217 41. 4.6. Multiparticle Distributions - p. 234 42. 4.7. Non-Euclidean–Normed Distributions (Livadiotis, 2007, 2008, 2012, 2016b) - p. 243 43. 4.8. Discrete Distributions (Tsallis et al., 1998) - p. 245 44. 4.9. Concluding Remarks - p. 245 45. 4.10. Science Questions for Future Research - p. 246 46. Part 2. Plasma Physics - p. 246 47. Chapter 5. Basic Plasma Parameters Described by Kappa Distributions - p. 249 48. 5.1. Summary - p. 250 49. 5.2. Introduction - p. 250 50. 5.3. Polytropes - p. 252 51. 5.4. Correlation Between Particle Energies - p. 267 52. 5.5. Debye Length in Equilibrium and Nonequilibrium Plasmas - p. 272 53. 5.6. Electrical Conductivity - p. 286 54. 5.7. Collision Frequency and Mean Free Path - p. 292 55. 5.8. Magnetization: The Curie Constant - p. 295 56. 5.9. Large-Scale Quantization Constant - p. 300 57. 5.10. Concluding Remarks - p. 310 58. 5.11. Science Questions for Future Research - p. 311 59. Chapter 6. Superstatistics: Superposition of Maxwell–Boltzmann Distributions - p. 313 60. Chapter 7. Linear Kinetic Waves in Plasmas Described by Kappa Distributions - p. 329 61. 7.1. Summary - p. 330 62. 7.2. Introduction - p. 330 63. 7.3. Plasma Dielectric Tensor and the Dispersion Relation - p. 336 64. 7.4. Kappa Velocity Distribution Plasma Waves at Parallel Propagation (ϑ=0) - p. 339 65. 7.5. Kappa Velocity Distribution Plasma Waves at Oblique Propagation (ϑ ≠ 0) - p. 359 66. 7.6. Concluding Remarks - p. 361 67. 7.7. Science Questions for Future Research - p. 361 68. Chapter 8. Nonlinear Wave–Particle Interaction and Electron Kappa Distribution - p. 363 69. 8.1. Summary - p. 363 70. 8.2. Introduction - p. 364 71. 8.3. Plasma Weak Turbulence Theory - p. 365 72. 8.4. Turbulent Quasiequilibrium and Kappa Electron Distribution - p. 389 73. 8.5. Concluding Remarks - p. 398 74. 8.6. Science Questions for Future Research - p. 398 75. Chapter 9. Solitary Waves in Plasmas Described by Kappa Distributions - p. 399 76. Part 3. Applications in Space Plasmas - p. 418 77. Chapter 10. Ion Distributions in Space Plasmas - p. 421 78. 10.1. Summary - p. 422 79. 10.2. Introduction - p. 422 80. 10.3. Formulations of Ion Kappa Distributions - p. 425 81. 10.4. Toward Antiequilibrium, the Farthest State From Thermal Equilibrium - p. 432 82. 10.5. Arrangement of the Stationary States - p. 439 83. 10.6. Interpreting the Observations - p. 444 84. 10.7. Concluding Remarks - p. 463 85. 10.8. Science Questions for Future Research - p. 463 86. Chapter 11. Electron Distributions in Space Plasmas - p. 465 87. Chapter 12. The Kappa-Shaped Particle Spectra in Planetary Magnetospheres - p. 481 88. Chapter 13. Kappa Distributions and the Solar Spectra: Theory and Observations - p. 523 89. Chapter 14. Importance of Kappa Distributions to Solar Radio Bursts - p. 549 90. Chapter 15. Common Spectrum of Particles Accelerated in the Heliosphere: Observations and a Mechanism - p. 569 91. Chapter 16. Formation of Kappa Distributions at Quasiperpendicular Shock Waves - p. 609 92. Chapter 17. Electron Kappa Distributions in Astrophysical Nebulae - p. 633 93. Appendix A: Abbreviations - p. 657 94. Appendix B. Main Symbols - p. 661 95. References - p. 665 96. Index - p. 711
