Stochastic collective dynamics of charged--particle beams in the stability regime

TL;DR

This paper models the collective transverse dynamics of charged-particle beams using stochastic processes, linking classical stochastic mechanics to quantum-like equations, and proposes control methods for beam focusing and oscillations.

Contribution

It introduces a stochastic variational framework for beam dynamics, deriving a Schrödinger-like equation with a novel emittance-based constant, connecting to quantum-like approaches.

Findings

Diffusion coefficient scales as λ_c√N, representing effective emittance.

Hydrodynamic equations can be recast as a Schrödinger equation.

Control strategies for beam focusing and oscillations are proposed.

Abstract

We introduce a description of the collective transverse dynamics of charged (proton) beams in the stability regime by suitable classical stochastic fluctuations. In this scheme, the collective beam dynamics is described by time--reversal invariant diffusion processes deduced by stochastic variational principles (Nelson processes). By general arguments, we show that the diffusion coefficient, expressed in units of length, is given by $\lambda_c\sqrt{N}$, where $N$ is the number of particles in the beam and $\lambda_c$ the Compton wavelength of a single constituent. This diffusion coefficient represents an effective unit of beam emittance. The hydrodynamic equations of the stochastic dynamics can be easily recast in the form of a Schr\"odinger equation, with the unit of emittance replacing the Planck action constant. This fact provides a natural connection to the so--called ``quantum--like approaches'' to beam dynamics. The transition probabilities associated to Nelson processes can be exploited to model evolutions suitable to control the transverse beam dynamics. In particular we show how to control, in the quadrupole approximation to the beam--field interaction, both the focusing and the transverse oscillations of the beam, either together or independently.