Introducing Some Basic Concepts

1. Introducing Some BasicConcepts 4.14.09

2. 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.

3. Nonlinear Process Described by Linear Theory

4. Nonlinear or Quasilinear Theories of Waves • Explicit appearance of wave energy in the theory. • Physically nonlinear but mathematically linear • Both physically and mathematically nonlinear

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

6. 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.

7. 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.

8. Random Density Function • A random density function is defined as follows where and denote the position and momentum of a given particle.

9. Phase-Space Orbit

10. Phase-Space Continuity Equation • The density function satisfies • Here the microscopic fields yield

11. Field Equations In addition to the kinetic equation we also need the Maxwell equations

12. So Far… • The theory is completely formal. • Practically not useful • A statistical treatment is needed.

13. Phase Space Probability Density • The ensemble averaged value is what we know as the distribution function • We may also define

14. Microscopic Field and Fluctuations • Ensemble averaged microscopic field • We define

15. Ensemble Averaging of the Klimontovich Equations • If we neglect fluctuations completely, it is obtained

16. These are the Vlasov equations • If electromagnetic fields are neglected, the equations reduce to

17. Linearization Scheme • For practical reason we introduce and assume so that the equations can be linearized. The result is

18. We use linearized Vlasov equations for: • Derivation of dispersion relations • Discussion of propagating modes • Study of plasma instabilities

19. Considering Density Fluctuation • Since we know • And

20. First order fluctuating quantity If we neglect the terms involving products of fluctuating quantities, we obtain

21. Comparison of Two Sets of Equations

22. 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.

23. Fluctuating Fields • Consisting two components • One can propagate in plasmas • The other cannot

24. 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.