Chaotic orbits in thermal-equilibrium beams: Existence and dynamical implications

Abstract

Phase mixing of chaotic orbits exponentially distributes these orbits through their accessible phase space. This phenomenon, commonly called ''chaotic mixing,'' stands in marked contrast to phase mixing of regular orbits which proceeds as a power law in time. It is operationally irreversible; hence, its associated e-folding time scale sets a condition on any process envisioned for emittance compensation. A key question is whether beams can support chaotic orbits, and if so, under what conditions? We numerically investigate the parameter space of three-dimensional thermal-equilibrium beams with space charge, confined by linear external focusing forces, to determine whether the associated potentials support chaotic orbits. We find that a large subset of the parameter space does support chaos and, in turn, chaotic mixing. Details and implications are enumerated.

Nonlinear δf Particle Simulations of Collective Effects in High-Intensity Bunched Beams

“…The nonintegrability is a result of lacking a third invariant of the dynamics, which is fundamentally due to the coupling between the transverse and longitudinal dynamics induced by the 2D nonlinear space-charge field. Previous studies on this subject can be found in References [9,10]. It is also clear from Fig.…”

“…Secondly, even in a thermal equilibrium with isotropic temperature, particles' trajectories on constant energy surfaces are non-integrable [9,10], which implies that it is impossible to perform an integration along unperturbed orbits to analytically calculate the linear eigenmodes. This paper is organized as follows.…”

Production of enhanced beam halos via collective modes and colored noise

“…Of course there are numerous ways to do this; one is to choose a distribution corresponding to a configuration of thermal equilibrium (TE) [12,13]. We construct a cylindrically symmetric TE configuration of test charges following a procedure recently used to devise spherically symmetric TE configurations [14]. The associated dimensionless Poisson equation is…”

On the viability of local criteria for chaos

“…In order to study the phase space of our system, we solve numerically the system governed by (13) and construct Poincaré's sections by using Henon's trick [16]. We could do it very accurately, with a cumulative error, measured by the constant Hamiltonian H, inferior to 10 −12 .…”