Amplitude growth due to random, correlated kicks
Abstract
Historically, stochastic processes, such as gas scattering or stochastic cooling, have been treated by the FokkerPlanck equation. In this approach, usually considered for one dimension only, the equation can be considered as a continuity equation for a variable which would be a constant of the motion in the absence of the stochastic process, for example, the action variable, I = epsilon/2x for betatron oscillations, where epsilon is the area of the CourantSnyder ellipse, or energy in the case of unbunched beams, or the action variable for phase oscillations in case the beam is bunched. A flux, Phi, including diffusive terms can be defined, usually to second order: Phi = M sub 1 F(I) + M(2) deltaF deltaI +. M(1) and M(2) are the expectation values of deltaI and (deltaI) sq due to the individual stochastic kicks over some period of time, long enough that the variance of these quantities is sufficiently small. Then the FokkerPlanck equation is just deltaF deltaI + delta Phi deltaI = 0. In many cases those where the beam distribution has already achieved its final shape, it is sufficient to find the rate of increase of (I) by taking simple averages over the FokkerPlanck equation. At the time this work was begun, there was good knowledge of the second moment for general stochastic processes due to stochastic cooling theory, but the form of the first moment was known only for extremely wideband processes. The purposes of this note are to derive an expression relating the expected single particle amplitude growth to the noise autocorrelation function and to obtain, thereby, the form of M(1) for narrow band processes.
 Publication:

Presented at the 13th Particle Accelerator Conference
 Pub Date:
 March 1989
 Bibcode:
 1989paac.conf.....M
 Keywords:

 Betatrons;
 Magnets;
 Particle Acceleration;
 Pulse Generators;
 Electrical Impedance;
 Equations Of Motion;
 FokkerPlanck Equation;
 Green'S Functions;
 Stochastic Processes;
 Fluid Mechanics and Heat Transfer