Introducing Some Basic Concepts

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# Introducing Some Basic Concepts - PowerPoint PPT Presentation

Introducing Some Basic Concepts. 4.14.09. Linear Theories of Waves. (Vanishingly) small perturbations Particle orbits are not affected by waves. Dispersion relation is independent of wave energy.

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Presentation Transcript
Linear Theories of Waves
• (Vanishingly) small perturbations
• Particle orbits are not affected by waves.
• Dispersion relation is independent of wave energy.
• But linear theory may actually describe some conceptually nonlinear processes. The best example is the mode coupling process.
• Nonlinear process and linear theory.
Nonlinear or Quasilinear Theories of Waves
• Explicit appearance of wave energy in the theory.
• Physically nonlinear but mathematically linear
• Both physically and mathematically nonlinear
Ensemble of Systems

A group of similar systems but suitably randomized so that statistical study is meaningful.

Concepts of Random Quantities
• Theoretically each physical quantity in a many-particle system consists of two parts: the ensemble averaged value and a fluctuating part.
• By definition the fluctuating part is random and its ensemble averaged value vanishes.
Statistic Approaches to Plasma Physics
• BBGKY hierarchy
• Prigogine and Balecu scheme
• Klimontovich formalism

which introduces a totally new approach in statistical theory of plasma physics.

Random Density Function
• A random density function is defined as follows

where and denote the position and momentum of a given particle.

Phase-Space Continuity Equation
• The density function satisfies
• Here the microscopic fields yield
Field Equations

In addition to the kinetic equation we also need the Maxwell equations

So Far…
• The theory is completely formal.
• Practically not useful
• A statistical treatment is needed.
Phase Space Probability Density
• The ensemble averaged value is what we know as the distribution function
• We may also define
Microscopic Field and Fluctuations
• Ensemble averaged microscopic field
• We define
Ensemble Averaging of the Klimontovich Equations
• If we neglect fluctuations completely, it is obtained
These are the Vlasov equations
• If electromagnetic fields are neglected, the equations reduce to
Linearization Scheme
• For practical reason we introduce

and assume so that the equations can be linearized. The result is

We use linearized Vlasov equations for:
• Derivation of dispersion relations
• Discussion of propagating modes
• Study of plasma instabilities
First order fluctuating quantity

If we neglect the terms involving products of fluctuating quantities, we obtain

An Important Conclusion
• When an unperturbed distribution function describes an unstable state, it means that both ensemble-averaged perturbation and microscopic fluctuating field would grow with time.
• In general an instability is more important for the latter because it leads to the origin of the turbulence.
Fluctuating Fields
• Consisting two components
• One can propagate in plasmas
• The other cannot
Significance of Kinetic Instabilities
• A kinetic instability usually excites a spectrum of fluctuating fields whereas a reactive instability often amplifies coherent waves.
• Therefore in general plasma turbulence is attributed to kinetic instabilities.