Levy-Student Distributions for Halos in Accelerator Beams

Abstract

We describe the transverse beam distribution in particle accelerators within the controlled, stochastic dynamical scheme of the Stochastic Mechanics (SM) which produces time reversal invariant diffusion processes. This leads to a linearized theory summarized in a Shchrödinger--like (\Sl) equation. The space charge effects have been introduced in a recent paper~\cite{prstab} by coupling this \Sl equation with the Maxwell equations. We analyze the space charge effects to understand how the dynamics produces the actual beam distributions, and in particular we show how the stationary, self--consistent solutions are related to the (external, and space--charge) potentials both when we suppose that the external field is harmonic (\emph{constant focusing}), and when we \emph{a priori} prescribe the shape of the stationary solution. We then proceed to discuss a few new ideas~\cite{epac04} by introducing the generalized Student distributions, namely non--Gaussian, Lévy \emph{infinitely divisible} (but not \emph{stable}) distributions. We will discuss this idea from two different standpoints: (a) first by supposing that the stationary distribution of our (Wiener powered) SM model is a Student distribution; (b) by supposing that our model is based on a (non--Gaussian) Lévy process whose increments are Student distributed. We show that in the case (a) the longer tails of the power decay of the Student laws, and in the case (b) the discontinuities of the Lévy--Student process can well account for the rare escape of particles from the beam core, and hence for the formation of a halo in intense beams.

revtex4, 18 pages, 12 figures

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• Stochastic effects in real and simulated charged particle beams
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• Inference of Planck action constant by a classical fluctuative postulate holding for stable microscopic and macroscopic dynamical systems

Cited by in corpus (4)

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• Mixtures in non stable Levy processes
• Lévy-Schrödinger wave packets
• Markov processes and generalized Schroedinger equations