Amplitude growth due to random, correlated kicks

Conference · Wed Mar 01 00:00:00 EST 1989

OSTI ID:6233988

Michelotti, L; Mills, F

Historically, stochastic processes, such as gas scattering or stochastic cooling, have been treated by the Fokker-Planck 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/2..pi.. for betatron oscillations, where epsilon is the area of the Courant-Snyder 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/sub 2/par. deltaF/par. deltaI + /hor ellipsis/. M/sub 1/ and M/sub 2/ are the expectation values of deltaI and (deltaI)/sup 2/ due to the individual stochastic kicks over some period of time, long enough that the variance of these quantities is sufficiently small. Then the Fokker-Planck equation is just par. deltaF/par. deltaI + par. delta/Phi//par. 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 by taking simple averages over the Fokker-Planck 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/sub 1/ for narrow band processes. 4 refs.

• Research Organization: Fermi National Accelerator Lab. (FNAL), Batavia, IL (United States)
• DOE Contract Number: AC02-76CH03000
• OSTI ID: 6233988
• Report Number(s): FNAL/C-89/93; CONF-890335-86; ON: DE89011504
• Resource Relation: Conference: 13. particle accelerator conference, Chicago, IL, USA, 20 Mar 1989; Other Information: Paper copy only, copy does not permit microfiche production
• Country of Publication: United States
• Language: English

Related Subjects

43 PARTICLE ACCELERATORS
BEAM DYNAMICS
STOCHASTIC PROCESSES
BEAM EMITTANCE
BETATRON OSCILLATIONS
CORRELATION FUNCTIONS
ELECTRIC IMPEDANCE
EQUATIONS OF MOTION
FOKKER-PLANCK EQUATION
GREEN FUNCTION
KICKER MAGNETS
PHASE SPACE
DIFFERENTIAL EQUATIONS
ELECTRICAL EQUIPMENT
ELECTROMAGNETS
EQUATIONS
EQUIPMENT
FUNCTIONS
IMPEDANCE
MAGNETS
MATHEMATICAL SPACE
OSCILLATIONS
PARTIAL DIFFERENTIAL EQUATIONS
SPACE
430200* - Particle Accelerators- Beam Dynamics
Field Calculations
& Ion Optics