1. How to lock or limit a free ballistic expansion of Energetic Particles?

2. What could limit the free collisionless expansion? Dominant Scenario: -Instabilities / to create the “waves (as scatterers)”; -Feedback of growing fluctuations on particles due to the particles – waves interaction; -Non-Linear saturation of instability / build up of saturation spectrum;

3. Add Fermi Acceleration

4. Implementation of such scenario -Instability: Velocity Anisotropy (“Cyclotron Instability” of Alfven waves); -Feedback on particles: Quasilinear Theory of Particles/Cyclotron Waves Interaction; -Non-Linear saturation: Strong MHD / Alfven Waves/ Turbulence;

6. Connection of Quasilinear Theory to KAM-Theory: From Planetary Resonances to Plasma

7. x V

8. V X ADD MORE WAVES

11. Width of resonance Distance between resonances vs.

12. much less than This limit corresponds to KAM (Kolmogoroff- Arnold-Mozer) case. KAM-Theorem : As applied to our case of Charged Particle – Wave Packet Interaction – “Particle preserves its orbit “

13. greater than That means - overlapping of neighboring resonances Repercussions: -”collectivization” of particles between neighboring waves; -particles moving from one resonance to another – “random walk”? And if yes - what is Diffusion Coefficient ?(in velocity space)

14. V X dV/dt =

15. D= Repercussions: Quasilinear Theory, Plateau Formation, Beam + Plasma Instability Saturation etc.

16. General Conclusions Kolmogoroff: Application of KAM theory to the Dynamics of Planetary System Plasma case: Application to the Dynamics of Charged Particles more applications: Waves-Particles interaction at Cyclotron Resonance Magnetic Surfaces Splitting? (Trieste, 1966) Advection in Fluids (+20 years)

17. Quasilinear Diffusion. The simplified approach to such diffusion is equivalent to a truncation of quasilinear velocity space diffusion similar to tau-approximation form of collision integral in kinetic equation. Further simplifications: -Plasma pressure is much greater than Magnetic field pressure; -Bulk of plasma particles out of resonance with “waves” (even in strong turbulence definition); -CR particle density too small to produce competitive nonlinear effects by themselves and do not affect waves nonlinear saturation process.

18. Add Fermi Acceleration

19. Truncate Quasilinear Eqn

20. Nonlinear Saturation Conjecture

21. MHD + Expanding Cloud of Energetic Particles + “Return Current MHD Waves modified:

22. Net effect of Instabilities Both types of Instability together: