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Extra info for Collisionless Relaxation in Beam-Plasma Systems [thesis]
The dynamical CME predictions (3), theoretical estimates ( ) (given by Eqs. 40)) and PIC results (4) for an initially gaussian distribution are plotted in Fig. 14. The dynamical CME results are obtained by the numerical integration until stationary values of x and v are achieved. It can be seen that the dynamical CME results o er an improvement in the accuracy compared to the analytical estimates of Sec. 3. The numerical and analytical predictions for the asymptotic rms values of an initially uniform beam are presented in Fig.
We now derive the predictions for the rms beam properties reached after the relaxation process is complete. Unnormalized, physical variables are used in this section. 21) Under the assumption that phase mixing results in a coarse-grained distribution function with the symmetry f (x vx ) = f (;x vx ) = f (x ;vx), all moments hxp vxq i with p and q odd vanish in the asymptotic state. Thus, the stationary forms of Eqs. 19) and Eq. 21) are satis ed. The remaining Eq. 22) where h:::ieq stands for the average evaluated using the asymptotic coarse-grained distribution.
23) has been dictated by the desire to predict the evolution of rms beam properties these are expressed through only low order 36 coe cients akl . It can be easily seen that the damping is weak for low order coe cients and strong for akl with large values of k + l. The action of the damping term smoothes the otherwise abrupt transition to akl = 0 at k + l > jmax . Note that no relaxation is introduced for the lowest order coe cients, a20 , a11 and a02 , which correspond to the bulk beam properties x, v and hxvx i.
Collisionless Relaxation in Beam-Plasma Systems [thesis]