Nonlinear density waves in the single-wave model

The single-wave model equations are transformed to an exact hydrodynamic closure by using a class of solutions to the Vlasov equation corresponding to the waterbag model. The warm fluid dynamic equations are then manipulated by means of the renormalization group method. As a result, amplitude equations for the slowly varying wave amplitudes are derived. Since the characteristic equation for waves has in general three roots, two cases are examined. If all the three roots of the characteristic equation are real, the amplitude equations for the eigenmodes represent a system of three coupled nonlinear equations. In the case where the dispersion equation possesses one real and two complex conjugate roots, the amplitude equations take the form of two coupled equations with complex coefficients. The analytical results are then compared to the exact system dynamics obtained by solving the hydrodynamic equations numerically.

Soliton gas in space-charge-dominated beams

New J. Phys., Vol. 6, p. 19 (2004)

Based on the Vlasov–Maxwell equations describing self-consistent nonlinear beam dynamics and collective processes, the evolution of an intense sheet beam propagating through a periodic focusing field has been studied. It has been shown that in the case of a beam with uniform phase-space density, the Vlasov–Maxwell equations can be replaced exactly by the hydrodynamic equations with a triple adiabatic pressure law coupled with the Maxwell equations. We further demonstrate that starting from the system of hydrodynamic and Maxwell equations, a set of coupled nonlinear Schrödinger equations for the slowly varying amplitudes of density waves can be derived. In the case where a parametric resonance between a certain mode of density waves and the external focusing occurs, the slow evolution of the resonant amplitudes in the cold-beam limit is shown to be governed by a system of coupled Gross–Pitaevskii equations. Properties of the nonlinear Schrödinger equation as well as properties of the Gross–Pitaevskii equation are discussed, together with soliton and condensate formation in intense particle beams.

Hamiltonian formalism for solving the Vlasov-Poisson equations and its applications to periodic focusing systems and coherent beam-beam interaction

A Hamiltonian approach to the solution of the Vlasov-Poisson equations has been developed. Based on a nonlinear canonical transformation, the rapidly oscillating terms in the original Hamiltonian are transformed away, yielding a new Hamiltonian that contains slowly varying terms only. The formalism has been applied to the dynamics of an intense beam propagating through a periodic focusing lattice, and to the coherent beam-beam interaction. A stationary solution to the transformed Vlasov equation has been obtained.

Whistleron gas in magnetized plasmas

The nonlinear dynamics of whistler waves in magnetized plasmas is studied. Since the plasmas and beam-plasma systems considered here are assumed to be weakly collisional, the point of reference for the analysis performed in the present paper is the system of hydrodynamic and field equations. The renormalization group method is applied to obtain dynamical equations for the slowly varying amplitudes of whistler waves. Further, it has been shown that the amplitudes of eigenmodes satisfy an infinite system of coupled nonlinear Schrödinger equations. In this sense, the whistler eigenmodes form a sort of a gas of interacting quasiparticles, while the slowly varying amplitudes can be considered as dynamical variables heralding the relevant information about the system. An important feature of the approach is that whistler waves do not perturb the initial uniform density of plasma electrons. The plasma response to the induced whistler waves consists in velocity redistribution which follows exactly the behavior of the whistlers. In addition, selection rules governing the nonlinear mode coupling have been derived, which represent another interesting peculiarity of the description presented here.

IRROTATIONAL MOMENTUM FLUCTUATIONS CONDITIONING THE QUANTUM NATURE OF PHYSICAL PROCESSES

Starting from a simple classical framework and employing some stochastic concepts, the basic ingredients of the quantum formalism are recovered. It has been shown that the traditional axiomatic structure of quantum mechanics can be rebuilt, so that the quantum mechanical framework resembles to a large extent that of the classical statistical mechanics and hydrodynamics. The main assumption used here is the existence of a random irrotational component in the classical momentum. Various basic elements of the quantum formalism (calculation of expectation values, the Heisenberg uncertainty
principle, the correspondence principle) are recovered by applying traditional techniques, borrowed from classical statistical mechanics.

Renormalization group reduction of non-integrable Hamiltonian systems

Based on the renormalization group (RG) method, a reduction of non-integrable multi-dimensional Hamiltonian systems has been performed. The evolution equations for the slowly varying part of the angle-averaged phase space density and for the amplitudes of the angular modes have been derived. It has been shown that these equations are precisely the RG equations. As an application of the approach developed, the modulational diffusion in a one-and-a-half-degree-of-freedom dynamical system has been studied in detail.

Hydrodynamic Covariant Symplectic Structure from Bilinear Hamiltonian Functions

PROGRESS IN PHYSICS, VOLUME 2, PAGE 92-100, JULY 2005

Starting from generic bilinear Hamiltonians, constructed by covariant vector, bivector or tensor fields, it is possible to derive a general symplectic structure which leads to holonomic and anholonomic formulations of Hamilton equations of motion directly related to a hydrodynamic picture. This feature is gauge free and it seems a deep link common to all interactions, electromagnetism and gravity included. This scheme could lead toward a full canonical quantization.

Kinetic derivation of the hydrodynamic equations for capillary fluids

Based on the generalized kinetic equation for the one-particle distribution function with a small source, the transition from the kinetic to the hydrodynamic description of many-particle systems is performed. The basic feature of this interesting technique to obtain the hydrodynamic limit is that the latter has been partially incorporated into the kinetic equation itself. The hydrodynamic equations for capillary fluids are derived from the characteristic function for the local moments of the distribution function. Fick’s law appears as a consequence of the transformation law for the hydrodynamic quantities under time inversion.

Simulation of the beam-beam effects in e+e- storage rings with a method of reduced region of mesh

A highly accurate self-consistent particle code to simulate the beam-beam collision in e+e- storage rings has been developed. It adopts a method of solving the Poisson equation with an open boundary. The method consists of two steps: assigning the potential on a finite boundary using Green's function and then solving the potential inside the boundary with a fast Poisson solver. Since the solution of Poisson's equation is unique, our solution is exactly the same as the one obtained by simply using Green's function. The method allows us to select a much smaller region of mesh and therefore increase the resolution of the solver. The better resolution makes more accurate the calculation of the dynamics in the core of the beams. The luminosity simulated with this method agrees quantitatively with the measurement for the PEP-II B Factory ring in the linear and nonlinear beam current regimes, demonstrating its predictive capability in detail.

Renormalization group reduction of the Hénon map and application to the transverse betatron motion in cyclic accelerators

The renormalization group (RG) method is applied to the study of discrete dynamical systems. As a particular example, the Hénon map is considered as being applied to describe the transverse betatron oscillations in a cyclic accelerator or storage ring possessing a FODO-cell structure with a single thin sextupole. A powerful RG method is developed that is valid correct to fourth order in the perturbation amplitude, and a technique for resolving the resonance structure of the Hénon map is also presented. This calculation represents an application of the RG method to the study of discrete dynamical systems in a unified manner capable of reducing the dynamics of the system both far from and close to resonances, thus preserving the symplectic symmetry of the original map.

Kinetic description of intense beam propagation through a periodic focusing field for uniform phase-space density

The Vlasov-Maxwell equations are used to investigate the nonlinear evolution of an intense sheet beam with distribution function fb(x,x′,s) propagating through a periodic focusing lattice κx(s+S)  =  κx(s), where S  =  const is the lattice period. The analysis considers the special class of distribution functions with uniform phase-space density fb(x,x′,s)  =  A  =  const inside of the simply connected boundary curves, x+′(x,s) and x-′(x,s), in the two-dimensional phase space (x,x′). Coupled nonlinear equations are derived describing the self-consistent evolution of the boundary curves, x+′(x,s) and x-′(x,s), and the self-field potential ψ(x,s)  =  ebφ(x,s)/γbmbβb2c2. The resulting model is shown to be exactly equivalent to a (truncated) warm-fluid description with zero heat flow and triple-adiabatic equation of state with scalar pressure Pb(x,s)  =  const[nb(x,s)]3. Such a fluid model is amenable to direct analysis by transforming to Lagrangian variables following the motion of a fluid element. Specific examples of periodically focused beam equilibria are presented, ranging from a finite-emittance beam in which the boundary curves in phase space (x,x′) correspond to a pulsating parallelogram, to a cold beam in which the number density of beam particles, nb(x,s), exhibits large-amplitude periodic oscillations. For the case of a sheet beam with uniform phase-space density, the present analysis clearly demonstrates the existence of periodically focused beam equilibria without the undesirable feature of an inverted population in phase space that is characteristic of the Kapchinskij-Vladimirskij beam distribution.

Dynamics of Particles in Non Scaling Fixed Field Alternating Gradient Accelerators

Non scaling Fixed-Field Alternating Gradient (FFAG accelerators have an unprecedented potential for muon acceleration, as well as for medical purposes based on carbon
and proton hadron therapy. They also represent a possible active element for an Accelerator Driven Subcritical Reactor (ADSR). Starting from first principle the Hamiltonian formalism for the description of the dynamics of particles in non-scaling FFAG machines has been developed. The stationary reference (closed) orbit has been found within the Hamiltonian framework. The dependence of the path length on the energy
deviation has been described in terms of higher order dispersion functions. The latter have been used subsequently to specify the longitudinal part of the Hamiltonian. It has been shown that higher order phase slip coecients should be taken into account to adequately describe the acceleration in non-scaling FFAG accelerators. A complete theory of the fast (serpentine) acceleration in non-scaling FFAGs has been developed. An example of the theory is presented for the parameters of the Electron Machine with Many Applications (EMMA), a prototype electron non-scaling FFAG to be hosted at
Daresbury Laboratory.

RANDOM BEAM PROPAGATION IN ACCELERATORS AND STORAGE RINGS

A kinetic equation for the joint probability distribution for fixed values of the classical action, momentum and density has been derived. Further the hydrodynamic equations of continuity and balance of momentum density have been
transformed into a Schroedinger-like equation, describing particle motion in an effective electro-magnetic field and an expression for the gauge potentials has been obtained.

Beam dynamics in ee storage rings and a stochastic Schrödinger-like equation

The longitudinal dynamics of electrons in e storage rings has been studied, when the radiation damping and quantum excitation of synchrotron radiation are taken into account. It has been shown that the electron beam propagates according to the law specified by a stochastic Schrödinger-like equation, in which the role of Planck's constant is played by an effective longitudinal thermal beam emittance.

MACROSCOPIC FLUID APPROACH TO THE COHERENT BEAM-BEAM INTERACTION

Building on the Radon transform of the Vlasov-Poisson equations, a macroscopic fluid model for the coherent beam-beam interaction has been developed. It is shown that the Vlasov equation, expressed in action-angle variables, can be reduced to a closed set of hydrodynamic (fluid) equations for the beam density and current velocity. The linearized one-dimensional equations have been analysed, and explicit expressions for the coherent beam-beam tuneshifts are presented.

Formation of Patterns and Coherent Structures in Charged Particle Beams

In the present paper we study the long wavelength and slow time scale behavior of a coasting beam in a resonator adopting a broad-band impedance model. Based on the renormalization group approach we derive a set of coupled evolution equations for the beam envelope distribution function
and the resonator voltage amplitude. The equation for the resonator voltage amplitude is further transformed into a generalized Ginzburg-Landau equation.

STOCHASTIC PROPERTIES OF THE FROBENIUS-PERRON OPERATOR

In the present paper the Renormalization Group (RG) method is adopted as a tool for a constructive analysis of the properties of the Frobenius-Perron Operator. The renormalization group reduction of a generic symplectic map in the case, where the unperturbed rotation frequency of the map is far from structural resonances driven by the kick perturbation has been performed in detail. It is further shown that if the unperturbed rotation frequency is close to a resonance, the reduced RG map of the Frobenius-Perron operator (or phase-space density propagator) is equivalent to a discrete Fokker-Planck equation for the renormalized distribution function. The RG method has been also applied to study the stochastic properties of the standard Chirikov-Taylor map.

Nonlinear Longitudinal Waves in High Energy Stored Beams

We solve the Vlasov equation for the longitudinal distribution function and find stationary wave patterns when the distribution in the energy error is Maxwellian. In the long wavelength limit a stability criterion for linear waves has been obtained and a Korteweg-de Vries- Burgers equation for the relevant hydrodynamic quantities has been derived.

SOLITARY WAVES IN AN INTENSE BEAM PROPAGATING THROUGH A SMOOTH FOCUSING FIELD

Based on the Vlasov-Maxwell equations describing the self-consistent nonlinear beam dynamics and collective processes, the evolution of an intense sheet beam propagating through a periodic focusing field has been studied. In an earlier paper [1] it has been shown that in the case of a beam with uniform phase space density the Vlasov-Maxwell equations can be replaced exactly by the macroscopic warm fluid-Maxwell equations with a triple adiabatic pressure law. In this paper we demonstrate that starting from the macroscopic fluid-Maxwell equations a nonlinear Schroedinger equation for the slowly varying wave amplitude (or a set of coupled nonlinear Schroedinger equations for the wave amplitudes in the case of multi-wave interactions) can be derived. Properties of the nonlinear Schroedinger equation are discussed, together with soliton formation in intense particle beams.

Report of the Working Group on Plasma Phenomena In Beams

«Charge Exchange Injection of Heavy Ions in Synchrotrons»

FERMILAB-Pub-98/275

Dynamics of Particles in Non Scaling FFAG Accelerators

Collision Integrals and the Generalized Kinetic Equation for Charged Particle Beams

Application of master equation technique for the study of nonlinear dynamics of particles in accelerators and storage rings

Nonlinear $\ delta f $ Method for Beam-Beam Simulation

Generation and Propagation of Nonlinear Waves in Travelling Wave Tubes

The method of effective potential for the stability analysis of isolated nonlinear resonances

Hydrodynamic approximation with self-diffusion for collision-free beams

Diffusion of particles due to periodic crossing of non-linear resonances

The Schroedinger-like Equation with Electro-Magnetic Potentials in the Framework of Stochastic Quantization Approach

Renormalization Group Approach to the Beam-Beam Interaction in Circular Colliders

Hamiltonian formalism for solving the Vlasov-Poisson equations and its application to the coherent beam-beam interaction

Formation of Patterns in Intense Hadron Beams. The Amplitude Equation Approach

Nonlinear Waves and Coherent Structures in the Quantum Single-Wave Model

On the United Kingdom, hydrodynamic and diffusion description of particle beam propagation

Simulation of the Beam-Beam E ects in e

Renormalization Group Reduction of the Frobenius-Perron Operator

Solitary Waves in Intense Charged Particle Beams

Mesoscopic Description Of Dynamical Systems: Hierarchy Of Recursive Equations For The Moments Of The Density Distribution With An Application To Charged …

Hydrodynamic Approach to the Free Electron Laser Instability