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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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Introducing Some BasicConcepts 4.14.09
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
Considering Density Fluctuation • Since we know • And
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.