The square of the horizontal projected (rms) beam emittance is expressed as the sum of four nonnegative contributions, each described using the slice moments of the beam and possessing a natural interpretation in terms of the geometrical properties of the beam in the six-dimensional phase space. The mathematical formalism describing the relationships between projected beam quantities and slice beam quantities is reviewed. The results may be used to reconstruct the moments and emittances of the beam from the moments of its subpopulations, as well as to isolate and better understand a variety of slice and interslice dynamical contributions to the projected beam emittance growth.

This article explores the long-time behavior of the bounded orbits associated with an ensemble of initial conditions in a nondegenerate integrable Hamiltonian system. Such systems are inherently nonlinear and subject to highly regular phase space filamentation that can drive the ensemble of orbits toward a stationary state. Describing the statistical ensemble by a probability density on a neighborhood of a family of invariant tori, it is proved that the probability density describing the ensemble at time t converges weakly to an invariant density as t → ∞. More generally, we provide sufficient conditions for convergence to equilibrium of a multiphase system in action-angle form. These ideas are applied to an illustrative exactly soluble example. This work is relevant for understanding the statistical mechanics of integrable and near-integrable Hamiltonian systems.

An alternative representation for the vector potential of the nonlinear magnetic insert for the Integrable Optics Test Accelerator (IOTA), first described in Sec. V.A. of \cite{Danilov}, is determined from first principles using standard complex variable methods. In particular, it is shown that the coupled system consisting of the 2D Laplace equation and the Bertrand-Darboux equation is equivalent to a single ordinary differential equation in the complex plane, and a simple solution is constructed. The results are consistent with \cite{Danilov}, and this concise representation provides computational advantages for particle tracking through the nonlinear insert by avoiding numerical errors caused by small denominators that appear when evaluating transverse derivatives of the vector potential near the midplane. A similar representation is provided for the spatial dependence of the two invariants of motion.

In this paper, we propose a new adaptive unified differential evolution algorithmfor single-objective global optimization. Instead of the multiple mutation strate-gies proposed in conventional differential evolution algorithms, this algorithmemploys a single equation unifying multiple strategies into one expression. It hasthe virtue of mathematical simplicity and also provides users the flexibility forbroader exploration of the space of mutation operators. By making all controlparameters in the proposed algorithm self-adaptively evolve during the processof optimization, it frees the application users from the burden of choosing appro-priate control parameters and also improves the performance of the algorithm.In numerical tests using thirteen basic unimodal and multimodal functions, theproposed adaptive unified algorithm shows promising performance in compari-son to several conventional differential evolution algorithms.

When a charged-particle bunch in a storage ring is kicked to a large transverse offset, the time series describing the dynamics of the bunch centroid is determined both by the lattice focusing and by the Fourier transform of the 1D density profile of the bunch projected along the angle of the kick. In the presence of nonlinear focusing, we show that this fact can be exploited to enable 2D phase space reconstruction of the bunch (computational tomography) based only on turn-by-turn beam position monitor data. We demonstrate various tomography methods based on this principle, including machine learning methods, and discuss their advantages and disadvantages, and measure of reliability. We also mention a possible extension to 4D phase space computational tomography.

The determination of dynamic aperture in storage rings and colliders is a numerically intensive procedure. When realistic space-charge forces come into consideration, the numerical load becomes even heavier. Furthermore, dynamic aperture estimation using chaos indicators like frequency map analysis (FMA) raises reliability issues when the dynamical system has a time-dependent perturbation like the space-charge force. In this article, we apply a rapidly converging chaos indicator called reversibility error method (REM) to study the space-charge contribution to the dynamic aperture of the integrable optics test accelerator (IOTA) storage ring at a small value of the space charge tune shift. The strength of REM is addressed through examples, including a particle-core model of halo formation. We also develop a toy model of the IOTA lattice to further reduce the computing time required to estimate the dynamic aperture, and we compare this model with a realistic space-charge simulation for verification.

Measures of discrepancy between probability distributions (statistical distance) are widely used in the fields of artificial intelligence and machine learning. We describe how certain measures of statistical distance can be implemented as numerical diagnostics for simulations involving charged-particle beams. Related measures of statistical dependence are also described. The resulting diagnostics provide sensitive measures of dynamical processes important for beams in nonlinear or high-intensity systems, which are otherwise difficult to characterize. The focus is on kernel-based methods such as maximum mean discrepancy, which have a well-developed mathematical foundation and reasonable computational complexity. Several benchmark problems and examples involving intense beams are discussed. While the focus is on charged-particle beams, these methods may also be applied to other many-body systems such as plasmas or gravitational systems.

A numerical method is described for producing stationary solutions of the Vlasov-Poisson system describing a relativistic charged-particle beam in a constant focusing accelerator channel, confined transversely by a general (linear or nonlinear) focusing potential. The method utilizes a variant of the spectral Galerkin algorithm to solve a nonlinear partial differential equation (PDE) in two degrees of freedom for the beam space charge potential in equilibrium. Numerical convergence with an increasing number of computed spectral modes is investigated for several benchmark problems. Preservation of the stationary phase space density is verified using a strongly nonlinear focusing channel based on the Integrable Optics Test Accelerator at Fermi National Accelerator Laboratory.

The Integrable Optics Test Accelerator (IOTA) is a novel storage ring at Fermi National Accelerator Laboratory designed (in part) to investigate the dynamics of beams in the presence of highly nonlinear transverse focusing fields that generate integrable single-particle motion with a large spread in the intrinsic betatron tunes. We describe how contemporary geometrical methods from the theory of integrable Hamiltonian systems may be used to locate all critical separatrixlike structures in the 4D transverse phase space, and to construct a complete analysis of the dynamical bifurcations of the system. Application of these techniques results in a global picture of the nominal on-energy transverse dynamics, revealing a rich diversity of accessible dynamical behavior. Similar techniques may be applied to future facilities that exploit the concept of nonlinear integrable optics.

A numerical method to design nonlinear double- and multibend achromat (DBA and MBA) lattices with approximate invariants of motion is investigated. The search for such nonlinear lattices is motivated by Fermilab's Integrable Optics Test Accelerator, whose design is based on an integrable Hamiltonian system with two invariants of motion. While it may not be possible to design an achromatic lattice for a dedicated synchrotron light source storage ring with one or more exact invariants of motion, it is possible to tune the sextupoles and octupoles in existing double- and multibend achromat lattices to produce approximate invariants. In our procedure, the lattice is tuned while minimizing the turn-by-turn fluctuations of the Courant-Snyder actions Jx and Jy at several distinct amplitudes, while simultaneously minimizing diffusion of the on-energy betatron tunes. The resulting lattices share some important features with integrable ones, such as a large dynamic aperture, trajectories confined to invariant tori, robustness to resonances and errors, and a large amplitude-dependent tune spread. Compared to the nominal National Synchrotron Light Source-II lattice, the single- and multibunch instability thresholds are increased and the bunch-by-bunch feedback gain can be reduced.