Introducing Some Basic Concepts - PowerPoint PPT Presentation

4.14.09

• (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.

• Explicit appearance of wave energy in the theory.
• Physically nonlinear but mathematically linear
• Both physically and mathematically nonlinear

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

• 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.

• BBGKY hierarchy
• Prigogine and Balecu scheme
• Klimontovich formalism

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

• A random density function is defined as follows

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

• The density function satisfies
• Here the microscopic fields yield

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

• The theory is completely formal.
• Practically not useful
• A statistical treatment is needed.

• The ensemble averaged value is what we know as the distribution function
• We may also define

• Ensemble averaged microscopic field
• We define

• If we neglect fluctuations completely, it is obtained

• If electromagnetic fields are neglected, the equations reduce to

• For practical reason we introduce

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

• Derivation of dispersion relations
• Discussion of propagating modes
• Study of plasma instabilities

• Since we know
• And

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

• 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.

• Consisting two components
• One can propagate in plasmas
• The other cannot

• 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.