Stephan I. TZENOV CCLRC Daresbury Laboratory Accelerator Science and Technology Centre (ASTeC)

1. Simulation of a 3½-cell SRF gun with the GPT codeParticle-Core Model and the Method of the Frobenius-Perron Operator for the Study of the Long-Range Beam-Beam Interaction in the Modified Head-On Scheme of the ILC Stephan I. TZENOV CCLRC Daresbury Laboratory Accelerator Science and Technology Centre (ASTeC) LC-ABD Meeting, 25th September 2006, Durham

2. Contents of the Presentation • Introduction • The Frobenius-Perron Operator • Particle-Core Model for the Beam-Beam Interaction • Equilibrium Distributions from Jaynes’ Principle • Mapping the Centred Second Moments • Continuous Limit of the Particle-Core Model • Perspectives Stephan I. Tzenov

3. Introduction Recursive maps represent a useful and powerful tool to model and to facilitate the understanding of the physical processes taking place in complex nonlinear systems. • In particular: 1) they are widely used to study the various transition scenarios from regular to chaotic behaviour in nonlinear dynamical systems; 2) to simulate physical systems exhibiting anomalous diffusion; 3) or to analyze the underlying dynamics in time series with 1 / f noise in their power spectrum. • Iterative maps provide a convenient and effective method to investigate single-particle dynamics as well as collective effects in accelerators and storage rings. Stephan I. Tzenov

4. The Frobenius-Perron Operator • The extremely complicated behaviour of specific trajectories in chaotic systems strongly suggests a probabilistic approach to the dynamics. • The iterative map yields complete information of how the value of an individual phase-space point jumps around during successive iterations, so that one gets a good sense of the point dynamics but no sense of how iteration acts on densities with support on sets in phase space. • The latter gap is filled by the Frobenius-Perron operator, which provides a rule to determine how the evolution of densities over repeated iterations is accomplished. • Instead of tracing an individual trajectory in phase space, one employs a statistical mechanics approach by means of a distribution function of an ensemble of trajectories. • The Frobenius-Perron Operator of a phase-space density (distribution) function, which sometimes is called the Transfer Operator of that function or a phase-space density propagator, provides a tool for studying the dynamics of the iteration of the distribution function itself. Stephan I. Tzenov

5. Derivation of the Frobenius-Perron Operator The derivation is demonstrated on the Henon map: Alternative representation of the Henon map: The Frobenius-Perron Operator U can be calculated explicitly: Generalization for a generic symplectic map is straightforward. In action-angle variables the Frobenius-Perron operator is represented as: Stephan I. Tzenov

6. The Frobenius-Perron Operator Continued… The potentialcan be in general a functional (linear or nonlinear) of the distribution function, and depends on the coordinate only: A reduction of the Frobenius-Perron operator can be systematically performed for both the non-resonant and the resonant cases. As an example, the results for the Henon map are shown below. Non-resonant case. The distribution function can be represented as: where In the continuous limit the renormalized “amplitude” satisfies the equation: Stephan I. Tzenov

7. The Frobenius-Perron Operator Continued… Resonant case : The renormalized “amplitude” satisfies a Fokker-Planck equation of the form: where Stephan I. Tzenov

8. Particle-Core Model for the Beam-Beam Interaction The Frobenius-Perron operator for the one-degree-of freedom beam-beam interaction can be written as (k=1,2): Let us define the mean value of a dynamical variable G Performing a corresponding marginalization of the Frobenius-Perron equation it is straightforward to write the beam centroid map: In analogy, the map for the second moments can be obtained: Stephan I. Tzenov

9. Particle-Core Model Continued… The centred moments are defined according to: Using the map for the beam centroid it is straightforward to write: Stephan I. Tzenov

10. Equilibrium Distributions from Jaynes’ Principle Assuming that the two beams are in equilibrium, we can use the Jaynes’ principle of maximum information entropy to determine the distribution functions provided the first and second moments are known: In particular: Using the identity: and noting that only centred variables are involved in the corresponding calculations, we obtain: Stephan I. Tzenov

11. Mapping the Centred Second Moments Therefore the centred moments map becomes: and a similar map for the (3-k)-beam. It should be noted that the particle-core model combined with the Jaynes’ principle represents an EXACT CLOSURE A LA HYDRODYNAMIC TYPE. One continue further by including higher moments. Stephan I. Tzenov

12. Mapping the Centred Second Moments Continued… What information do we obtain from the mean-square beam dimension, momentum and correlation map? • Evolution of the Mean-Square Beam Dimension. • Evolution of the Mean-Square Beam Momentum. • Evolution of the Deviation-Momentum Correlation. • Evolution of the Average Beam Emittance defined according to: To treat the long-range beam-beam interaction one need to introduce a gradually increasing OFFSET: and repeat the above considerations, which lead us to the derivation of the map for the centred second moments. Stephan I. Tzenov

13. Continuous Limit of the Particle-Core Model The continuous limit of the particle-core model can be written as: The zero-order solutions are: In the case of a symmetric collider, to first order the amplitude A satisfies the equation: where K denotes the complete elliptic integral of the first kind. Stephan I. Tzenov

14. Continuous Limit of the Particle-Core Model Continued… Clearly the modulus of the complex amplitude is constant, while its phase linearly increases. Therefore, the expression implies the first-order nonlinear correction to the phase advance due to the beam-beam kick. Although the algebra involved becomes rather cumbersome, higher order approximations can be calculated explicitly. Further, the case of an asymmetric collider can be treated accordingly. Stephan I. Tzenov

15. Perspectives The approach presented here is useful to study a variety of problems: • Beam-beam collisions in circular and linear colliders can be treated to a high degree of accuracy, since only the core of the particle distribution contributes significantly to the luminosity. • Long-range beam-beam effects as well as beam-beam collisions at a crossing angle can be handled with a corresponding modification. • Nonlinear effects and crab cavity schemes can be easily incorporated. • Noise and dissipation due to synchrotron radiation can be also incorporated. • Stationary effects like equilibrium RMS beam size, emittance, flip-flop distribution can be studied as well. • Last, but not least the method can be used to study effects in space-charge dominated beams. Stephan I. Tzenov