Conceptual foundation of the Fokker-Planck approach to space-charge effects Page: 3 of 6
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where AV(x, u) is a phase-space volume element centered on the coordinates
(x, u) which is large compared to the size of the turbulent fluctuations but
small compared to the size of the beam. In what follows, the "bar" is a
signature of quantities calculated from the coarse-grained distribution f.
After resolving the Klimontovich distribution into two components, f and
fluctuations about f, and averaging Liouville's equation, we are left with a
"collision" term involving the fluctuations. Working with a coarse-grained
distribution function is tantamount to neglecting nonlinear coupling between
fluctuations in the particle distribution and fluctuations in the electromagnetic
field. This approach results in the reduction of Liouville's equation to an
equation of the Fokker-Planck type (6):
(8t + u.-V,+KV U)f=Vu -(Ff) + Vs (D -Vuf) ,(3)
where K is the net acceleration of a particle in the comoving frame found
from the potentials 4'f and 4, associated with the external focusing force and
coarse-grained internal space-charge force, respectively, i.e.,
K = QM--1Vr(4fr + o) ; (4)
the vector F and tensor D are coefficients of friction and diffusion, respectively,
and Q and M are the charge and mass, respectively, of the beam particles.
According to Poisson's equation, , is determined from the coarse-grained
density, which is in turn determined from f:
Vpi,(x,t)= - Jduf(xu,t)., (5)
n
in which eo is the permittivity of free space.
If the coarse-grained beam is regarded to he uniform so that it can be
Fourier transformed using the periodic boundary conditions of a homogeneous
cube of volume V, then the transport coefficients are
1 2irQ2 k 6(wk - k - u)
F =-- (6)
V Meo tvk2z (kw)
and
1 2rQ2 EC kk 6(k - k u)
D M o W , wkk2 E(k,wk)
In these expressions, Ek is the energy contained in h fluctuation with
wavevector k and angular frequency wk, and in this qjuasnliiiear formulation
it evolves in the manner
2E T +rMwkJ
t 2E + k (kk) du f(u)b(wk - k 'iI . (8)
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Bohn, C.L. Conceptual foundation of the Fokker-Planck approach to space-charge effects, report, July 1, 1996; Newport News, Virginia. (https://digital.library.unt.edu/ark:/67531/metadc670019/m1/3/: accessed April 25, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.