Integral Equation for the Equilibrium State of Colliding Electron Beams | Zendy

Robert Warnock | Zendy

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Integral Equation for the Equilibrium State of Colliding Electron Beams

Author(s) -

Robert Warnock

Publication year - 2002

Language(s) - English

Resource type - Reports

DOI - 10.2172/808677

Subject(s) - fokker–planck equation , vlasov equation , physics , thermodynamic equilibrium , beam (structure) , nonlinear system , electron , state (computer science) , equation of state , phase (matter) , classical mechanics , argument (complex analysis) , distribution (mathematics) , statistical physics , mathematics , differential equation , mathematical analysis , quantum mechanics , chemistry , biochemistry , algorithm , optics

We study a nonlinear integral equation for the equilbrium phase distribution of stored colliding electron beams. It is analogous to the Ha¨ õssinski equation, being derived from Vlasov-Fokker-Planck theory, but is quite different in form. We prove existence of a unique solution, thus the existence of a unique equilibrium state, for sufficiently small current. This is done for the Chao-Ruth model of the beam-beam in- teraction in one degree of freedom. We expect no difficulty in generalizing the argument to more realistic models.

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