Fluctuations and Non-Linear Wave Interactions in Plasmas

List of Notations


1. Statistical Description of a Plasma


1.1. Microscopic Equations

The Microscopic Density

The Equation of Continuity and Maxwell's Equations


1.2. The Liouville Distribution

The Liouville Equation

Statistical Averaging


1.3. The One-particle Distribution Function and Correlation Functions

Sequence of Distribution Functions

Connection between One-particle and Many-particle Distribution Functions


1.4. The Bogoliubov-Born-Green-Kirkwood-Yvon Hierarchy


1.5. Self-consistent Field Approximation

The Vlasov Kinetic Equation

Effects of Electromagnetic Interactions between Particles


1.6. Influence of Binary Correlations and The Collision Term

The Kinetic Equation with Collisions

Coulomb Collisions

Correlation Weakening Principle and Irreversibility


2. Non-linear Electrodynamic Equations


2.1. The Non-linear Equation for the Potential Field in a Plasma

Solution of the Kinetic Equation

Relation between the Polarization and the Electric Field Strength


2.2. Longitudinal Dielectric Permittivity and Non-linear Electric Susceptibilities of a Plasma

General Relations

Symmetry Properties

The Long-Wave Limiting Case

Non-Linear Susceptibilities of a Cold Plasma


2.3. Multiple-time-scale Perturbation Analysis of the Non-linear Equation for the Longitudinal Field

Hierarchy of Times and Expansion in the Non-linearity Parameter Series

Linear Approximation

Quadratic Interaction and Three-wave Resonances

Cubic Interaction and Four-wave Resonances


2.4. The General Non-linear Equation for the Field in a Plasma

Maxwell'S Equations and the Non-linear Material Relation

The Non-linear Equation for the Electric Field in a Plasma


2.5. The Dielectric Permittivity Tensor and Non-linear Tensor Susceptibilities of a Plasma

General Formulas

Symmetries of the Dielectric Permittivity Tensor

Symmetries of the Non-Linear Susceptibilities

Non-Linear Tensor Susceptibilities of a Cold Plasma


2.6. Hierarchy of Multiple-Time-Scale Approximate Equations for the Electromagnetic Field in a Plasma

Basic Set of Equations

Linear Approximation

Three-wave Resonances

Four-wave Resonances


3. Electrodynamic Properties of a Magneto-active Plasma


3.1. Longitudinal Dielectric Permittivity and Non-linear Susceptibilities of a Plasma in the Presence of a Magnetic Field

Solution of the Kinetic Equation

The Dielectric Permittivity of an Equilibrium Plasma

The Dielectric Permittivity of a Plasma with an Anisotropic Velocity Distribution

Non-Linear Susceptibilities of a Cold Plasma


3.2. The Dielectric Permittivity Tensor of a Magneto-active Plasma

Solution of the Linearized General Kinetic Equation

The General Form of the Dielectric Permittivity Tensor

an Equilibrium Plasma

A Plasma with an Anisotropic Velocity Distribution

A Cold Magneto-active Plasma


3.3. Non-linear Tensor Susceptibilities of a Magneto-active Plasma

The General Case

A Cold Plasma


4. Waves in Plasmas


4.1. Eigenoscillations and Eigenwaves in Space-time Dispersive Media (Linear Approximation)

The Dispersion Equation

Wave Polarization

Density and Flux of Energy

Fluctuations and Averaging over Initial Phases


4.2. Waves in an Isotropic Plasma

Dispersion Relations for Longitudinal and Transverse Waves

Longitudinal Langmuir and Ion-sound Waves

Transverse Electromagnetic Waves


4.3. Waves in a Magneto-active Plasma

The Dispersion Equation for Waves in a Magneto-Plasma

Plasma Resonances

Refractive Indexes and Eigenwave Frequencies in a Cold Magneto-Plasma

Thermal Waves in a Magneto-Plasma


5. Non-Linear Wave Interactions


5.1. Non-resonant Wave Interaction and Plasma Echoes

Resonant and Non-resonant Wave Interactions

Plasma Echoes

Longitudinal Echo Oscillations

Echo Oscillations in the Case of Anti-Parallel Wave Vectors of the Perturbations

Three-pulse Echo Oscillations


5.2. Three-wave Resonant Coupling

Dynamic Equations for Interacting Waves

Conservation of Energy and Momentum of Interacting Waves

Resonant Three-wave Coupling

Decay Instability

Explosive Instability


5.3. Four-Wave Resonant Coupling

The Resonance Condition and Dynamic Wave Equations

Resonant Coupling of Four Waves with Fixed Phases

Non-linear Frequency Shift of Eigenmodes in a Cold Plasma

Effect of Electron Thermal Motion on Non-linear Frequency Shifts

Influence of the Ion Motion

Non-linear Eigen Frequency Shifts of Langmuir Waves 95


5.4. Parametric Resonance in a Plasma

Dispersion Equation for Waves in a Pump Wave Field

Induced Scattering

Parametric Instability


6. Fluctuations in Plasmas


6.1. Fluctuations in Spatially Homogeneous Stationary Systems

Space-Time Correlation Functions

Correlators for the Distribution Function Fluctuations


6.2. Fluctuations in a System of Non-interacting Charged Particles

Spectral Correlators for the Fluctuations of the Distribution Function

Spectral Distributions for the Fluctuations of Particle, Charge and Current Densities

Higher-order Correlation Functions

Spectral Distributions of Fluctuations in the Presence of an External Magnetic Field


6.3. Fluctuations in an Ensemble of Charged Particles Taking into Account Electromagnetic

Interaction between Particles (Linear Approximation)

Effect of Coulomb Interactions between Particles on the Fluctuations of the Distribution Function

Averaging Procedure and the Correlator for the Fluctuations of the Distribution Function

Spectral Distributions of Particle, Charge and Current Density Fluctuations in a Plasma

Effect of Electromagnetic Interactions bBetween Particles on Plasma Fluctuations

Electromagnetic Fluctuations in an Isotropic Plasma


6.4. Collective Fluctuations and the Effective Temperature (Potential Field)

Incoherent and Coherent (Collective) Fluctuations

Collective Fluctuations in a Non-equilibrium Plasma and the Effective Temperature

Critical Fluctuations near the Instability Threshold

Critical Fluctuations in a Plasma with a Directed Particle Motion

Critical Fluctuations in a Plasma with an Anisotropic Velocity Distribution


6.5. Collective Electromagnetic Fluctuations

The Spectral Distribution of Fluctuations and the Effective Temperature

High-frequency Fluctuations in a Magneto-active Plasma

Low-frequency Fluctuations in a Magneto-active Plasma


7. Effect of Non-linear Wave Interactions on Fluctuations in a Plasma


7.1. Equations for the Spectral Correlation Functions (Potential Field)

The Non-linear Equation for the Fluctuation Field

Correlation Functions

The Equation for the Quadratic Correlation Function

Equations for Ternary and Quaternary Correlation Functions


7.2. The Stationary Fluctuation Spectrum Taking into Account Non-linear Corrections

Iteration Process

Induced Enhancement of Fluctuations


7.3. Non-linear Frequency Shifts and Saturation of Fluctuations in a Non-equilibrium Plasma

Non-linear Eigenfrequency Shift

Saturation of the Non-equilibrium Fluctuation Level

Turbulent Fluctuations in a Beam-Plasma System

Non-linear Stabilization of the Hydrodynamic Beam-Plasma Instability

Turbulent Fluctuations in a Plasma with an Anisotropic Velocity Distribution


7.4. Spectral Correlation Functions for Electromagnetic Fluctuations

The Non-linear Equation for the Fluctuations of the Electromagnetic Field

Set of Equations for the Sequence of Correlation Functions

The Stationary Spectrum of Electromagnetic Fluctuations

Non-linear Frequency Shifts of Electromagnetic Eigenoscillations

Saturation of the Level of Non-equilibrium Electromagnetic Fluctuations


8. Fluctuation-Dissipation Theorem


8.1. Inversion of the Fluctuation-Dissipation Theorem

Relation between the Dielectric Permittivity of an Equilibrium Plasma and the Correlation

Function for Fluctuations in a System of Non-interacting Particles

Fluctuation-dissipation Theorem for a Non-equilibrium Plasma


8.2. Non-linear Fluctuation-Dissipation Relations

Relation between Non-linear Susceptibilities and Higher Order Correlation Functions for Plasma Fluctuations (Potential Case)

Spectral Correlation Functions

Sum Rules

The Non-linear Fluctuation-dissipation Theorem for Electromagnetic Fluctuations


9. Kinetic Equations


9.1. The Kinetic Equation for Particles

Spectral Distribution of Field Fluctuations and the Equation for the One-particle Function

The Balescu-Lenard Collision Term

Dynamic Friction and Diffusion Coefficients

Quasi-linear Approximation


9.2. The Kinetic Equation for Waves (Potential Field)

Time Evolution of the Spectral Distribution of the Fluctuation Field

Linear Approximation

The Kinetic Equation for Longitudinal Plasma Waves

Non-linear Landau Damping


9.3. The Kinetic Equation for Waves (General Case)

Time Evolution of the Spectral Correlation Function for the Electromagnetic Field

The Kinetic Equation for Electromagnetic Waves in a Plasma

Parametric Instability and Turbulence


10. Scattering and Radiation of Waves


10.1. Scattering and Transformation of Waves in a Plasma

Derivation of Scattering and Transformation Cross-Sections from the Kinetic Equation for Waves

Scattering and Transformation Cross-sections in an Isotropic Plasma


10.2. Scattering and Transformation of Transverse Waves

Scattering of Transverse Electromagnetic Waves in a Plasma

Spectral Distribution of Scattered Radiation

Transformation of Transverse Electromagnetic Waves into Longitudinal Langmuir and Ion-sound Waves


10.3. Scattering and Transformation of Longitudinal Waves

Scattering of Longitudinal Langmuir Waves

Transformation of Longitudinal Langmuir Waves into Longitudinal Ion-sound

Transformation of Longitudinal Langmuir Waves into Transverse Electromagnetic Waves


10.4. Radiation from a Plasma I

Plasma Bremsstrahlung

Spontaneous Emission by a Non-equilibrium Plasma


11. Wave Interaction in a Semi-bounded Plasma


11.1. Waves in a Semi-bounded Plasma

The Non-linear Equation for the Potential Field in a Semi-bounded Plasma

Dispersion Equations for Bulk and Surface Waves (Linear Approximation)

Surface Waves

The Non-linear Equation for the Field (General Case)


11.2. Non-linear Interaction of Surface Waves

Resonant Wave Interaction

Three-wave Surface Wave Decays

Non-linear Interactions of Bulk and Surface Waves


11.3. Fluctuations and Kinetic Equations for Surface Waves

Fluctuations

Kinetic Equation for Surface Waves

References

Index

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