Ok, closer to what I should actually be working on...
We have an electron beam with a variety of interactions. At high energy in a storage ring, mainly you get a Gaussian due to damping and diffusion processes from synchrotron radiation. The self interaction mainly manifests as a beam lifetime- scattered particles are lost in what's known as the Touschek lifetime. Also, if there's an aperture close enough to this Gaussian, then particles are lost through this diffusion process in what's known as the Quantum lifetime.
Suppose we have a non-Gaussian beam. How did it get this way? What does this say about the lifetime?
We can treat the synchrotron radiation effect on the distribution via the Fokker-Planck equation. Can other noise processes on the beam also be treated via the FP equation? What would a non-Gaussian distribution do to the emitted synchrotron radiation?
Finally, does anyone care? We use this radiation for all sorts of experiments. Which experiments care about lifetime? Which care about beam size? Which care about the coherence of the radiation?
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