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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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linear theories of waves
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
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
Ensemble of Systems

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

concepts of random quantities
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
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
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
Phase-Space Continuity Equation
  • The density function satisfies
  • Here the microscopic fields yield
field equations
Field Equations

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

so far
So Far…
  • The theory is completely formal.
  • Practically not useful
  • A statistical treatment is needed.
phase space probability density
Phase Space Probability Density
  • The ensemble averaged value is what we know as the distribution function
  • We may also define
microscopic field and fluctuations
Microscopic Field and Fluctuations
  • Ensemble averaged microscopic field
  • We define
ensemble averaging of the klimontovich equations
Ensemble Averaging of the Klimontovich Equations
  • If we neglect fluctuations completely, it is obtained
these are the vlasov equations
These are the Vlasov equations
  • If electromagnetic fields are neglected, the equations reduce to
linearization scheme
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
We use linearized Vlasov equations for:
  • Derivation of dispersion relations
  • Discussion of propagating modes
  • Study of plasma instabilities
first order fluctuating quantity
First order fluctuating quantity

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

an important conclusion
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
Fluctuating Fields
  • Consisting two components
  • One can propagate in plasmas
  • The other cannot
significance of kinetic instabilities
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.