1. Introducing Some Basic Concepts 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.