Equilibrium state of colliding electron beams
We study a nonlinear integral equation that is a necessary condition on the equilibrium phase-space distribution function of stored, colliding electron beams. It is analogous to the Haïssinski equation, being derived from Vlasov-Fokker-Planck theory, but is quite different in form. The equation is a...
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American Physical Society
2003-10-01
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Series: | Physical Review Special Topics. Accelerators and Beams |
Online Access: | http://doi.org/10.1103/PhysRevSTAB.6.104401 |
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doaj-c28b3a4998a543868d8b4a8d36d82d792020-11-24T21:05:18ZengAmerican Physical SocietyPhysical Review Special Topics. Accelerators and Beams1098-44022003-10-0161010440110.1103/PhysRevSTAB.6.104401Equilibrium state of colliding electron beamsR. L. WarnockJ. A. EllisonWe study a nonlinear integral equation that is a necessary condition on the equilibrium phase-space distribution function of stored, colliding electron beams. It is analogous to the Haïssinski equation, being derived from Vlasov-Fokker-Planck theory, but is quite different in form. The equation is analyzed for the case of the Chao-Ruth model of the beam-beam interaction in 1 degree of freedom, a so-called strong-strong model with nonlinear beam-beam force. We prove the existence of a unique solution, for sufficiently small beam current, by an application of the implicit function theorem. We have not yet proved that this solution is positive, as would be required to establish existence of an equilibrium. There is, however, numerical evidence of a positive solution. We expect that our analysis can be extended to more realistic models.http://doi.org/10.1103/PhysRevSTAB.6.104401 |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
R. L. Warnock J. A. Ellison |
spellingShingle |
R. L. Warnock J. A. Ellison Equilibrium state of colliding electron beams Physical Review Special Topics. Accelerators and Beams |
author_facet |
R. L. Warnock J. A. Ellison |
author_sort |
R. L. Warnock |
title |
Equilibrium state of colliding electron beams |
title_short |
Equilibrium state of colliding electron beams |
title_full |
Equilibrium state of colliding electron beams |
title_fullStr |
Equilibrium state of colliding electron beams |
title_full_unstemmed |
Equilibrium state of colliding electron beams |
title_sort |
equilibrium state of colliding electron beams |
publisher |
American Physical Society |
series |
Physical Review Special Topics. Accelerators and Beams |
issn |
1098-4402 |
publishDate |
2003-10-01 |
description |
We study a nonlinear integral equation that is a necessary condition on the equilibrium phase-space distribution function of stored, colliding electron beams. It is analogous to the Haïssinski equation, being derived from Vlasov-Fokker-Planck theory, but is quite different in form. The equation is analyzed for the case of the Chao-Ruth model of the beam-beam interaction in 1 degree of freedom, a so-called strong-strong model with nonlinear beam-beam force. We prove the existence of a unique solution, for sufficiently small beam current, by an application of the implicit function theorem. We have not yet proved that this solution is positive, as would be required to establish existence of an equilibrium. There is, however, numerical evidence of a positive solution. We expect that our analysis can be extended to more realistic models. |
url |
http://doi.org/10.1103/PhysRevSTAB.6.104401 |
work_keys_str_mv |
AT rlwarnock equilibriumstateofcollidingelectronbeams AT jaellison equilibriumstateofcollidingelectronbeams |
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