Non Gaussian transverse distributions in a stochastic model for
Transcript
Non Gaussian transverse distributions in a stochastic model for
ISBN 92-9083-231-2 (Web version) ISBN 92-9083-232-0 (CD-ROM version) Copyright © 2004 by EPAC. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the written permission of EPAC, except as stated below. Single photocopies of single articles may be made for private study or research. Illustrations and short extracts from the text of individual contributions may be copied provided that the source is acknowledged and the permission of EPAC is obtained. Published by the European Physical Society Accelerators Group (EPS-AG) and CERN. CONFERENCE ORGANIZATION Organized by Paul Scherrer Institut (PSI), Villigen Under the Auspices of European Physical Society Supported by BESSY CCLRC (ISIS & ASTeC) CEA DSM CERN DESY ELETTRA ESRF FZ Jülich FZ Karlsruhe GSI Chairman Organizing Committee Chairman Scientific Programme Committee Chairman Local Organizing Committee EPAC Conference Coordinator INFN MSL Novartis PSI Roche Synchrotron Soleil Serono Swiss International Air Lines Syngenta LeonidRivkin, PSI Christopher Prior, ASTeC CCLRC RAL & Trinity College Oxford Hans-Arno Synal, ETH & PSI Christine Petit-Jean-Genaz, CERN ORGANIZING COMMITTEE C.Biscari, INFN-LNF, K. Blasche*, GSI, C.Bocchetta*, Sincrotrone Trieste, J. Bordas, LSB-IFAE, R.Brinkmann*, DESY, H.Danared*, MSL, J.-P.Delahaye*, CERN, J.-M.Lagniel*, CEA, E.Lee*, LBNL, A.Mosnier*, CEA, S.P.Møller*, ISA, W.Namkung, PAL, C.Pagani*, LASA-INFN, Ch. Petit-Jean-Genaz*, CERN, C. Prior*, ASTeC CCLRC RAL & Trinity College Oxford, F.Ruggiero*, CERN, Y.Shatunov, BINP, R.Siemann, SLAC, S.Smith*, ASTeC CLRC DL, H.-A.Synal*, ETH & PSI, U.VanRienen, U. Rostock, T. Weiland, TU-Darmstadt, A.Wrulich, EPFL & PSI * Member of the Scientific Programme Committee LOCAL ORGANIZING COMMITTEE A.Adelmann, PSI, M.Bugmann, PSI, J.Chrin, PSI, P.Dietrich, PSI, M.Heiniger, PSI, C.Petit-Jean-Genaz, CERN, J.Poole,CERN L.Rivkin, PSI, G.Schneider, Conventus, D.Vermeulen, PSI, A.Wrulich, EPFL and PSI SCIENTIFIC ADVISORY BOARD N. Angert, GSI, R.Assmann, CERN, J.Bergoz, Bergoz Inst., F.Bordry, CERN, P.Burrows, QMUL, J.Byrd, LBNL, Y.Cho, ANL, L.Dallin, CLS, D.Dinev, INRNE, R.Eichler, PSI, P.Elleaume, ESRF, M.Eriksson, MAX-lab, J.X.Fang, IHIP, Beijing, G.Fortuna, INFN-LNL, J.Hangst, IPA, Aarhus, H.Henke, TU- Berlin, N.Holtkamp, ORNL, A.Jackson, ALS, T.Katayama, INS, Tokyo, D.Krämer, BESSY, S.Kulinski, Soltan Inst., S.Y.Lee, IUCF, M.P.Level, Soleil, T.Linnecar, CERN, G. Luijckx, NIKHEF-K, P.Maccioni, CERCA, R.Maier, FZJ, I.Meshkov, JINR, Y.Mori, KEK, S.Nath, LANL, P.O’Shea, U. Maryland, S.Ozaki, BNL, R. Penco, Ansaldo, F. Perez, FZK-ISS, L.Picardi, ENEA, N. Pichoff, CEA, M.Poole, ASTeC CLRC DL, P.Raimondi, INFN-LNF, U. Ratzinger, IAP, T.Raubenheimer, SLAC, D.Reistad, TSL, G.Ryckewaert, U. Louvain, P.Schmüser, DESY, B.Sharkov, ITEP, Moscow, H.Tanaka, JASRI, S.Tazzari, U. Roma II, M.Tigner, U. Cornell, A.Tkatchenko, IPN, N.Vinokurov, BINP, R.Walker, Diamond, J. Wei, BNL, T.Weis, U. Dortmund, F.Willeke, DESY, Y. Yamazaki, J-PARC, K.Yokoya, KEK, C.Zhang, BEPC, F.Zimmermann, CERN EDITORIAL BOARD J.Chrin, PSI, Ch.Petit-Jean-Genaz, CERN, J.Poole, CERN, C.Prior, ASTeC CCLRC RAL & Trinity College Oxford, H.-A.Synal, ETH & PSI LIST OF PARTICIPANTS ABO-BAKR, Michael ABREU, Natalia ADAMS, Dean ADELMANN, Andreas ADOLPHSEN, Chris AGAPOV, Ilya AÏT ABDERRAHIM, Hamid ALEKSANDROV, Vladimir ALESINI, David ALEX, Juergen ALLEN, Curtis ANDERSON, Oscar ANDERSSON, Ake ANDLER, Guillermo ANGAL-KALININ, Deepa ANGERT, Norbert ANTHON, Lars ARDUINI, Gianluigi ARKAN, Tug ARNAUDON, Luca ARTOOS, Kurt ASSMANN, Ralph ATTAL, Maher AUDITORE, Lucrezia AUSLENDER, Vadim AUSSET, Patrick AYMAR, Robert BAGLIN, Vincent BAI, Mei BALEWSKI, Klaus BALTATEANU, Nicolae BANE, Karl BARON, Eric BARTH, Winfried BARTOLINI, Riccardo BARTOLOME, Maria Sonia BATHKE, Horst BAUER, Günter BAUER, Pierre BEAUVAIS, Pierre-Yves BECKER, Reinard BELLER, Peter BELLODI, Giulia BELLOMO, Giovani BENEDETTO, Elena BENEDIKT, Michael BERDEN, Giel BERG, J. Scott BERG, Russ BERGOZ, Julien BERNAUDIN, Pierre-Emmanuel BERNHARD, Axel BERRY, Stéphane BERTARELLI, Alessandro BERTONE, Caterina BERTRAND, Patrick BESSY LNLS CCLRC RAL Paul Scherrer Institut SLAC DESY SCK CEN JINR INFN-LNF Thales Broadcast & Multimedia AG Altair technologies, inc. 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Diamond Light Source LMU München BNL BESSY PAL POSTECH PAL POSTECH PAL POSTECH PAL POSTECH ANL CCLRC RAL KEK RIKEN/SPring-8 IAP ACCEL Instruments GmbH IAP PTB ESRF CERN BESSY SPring-8/JASRI JINR Alstom Magnets, Superconductors ITEP BINP Paul Scherrer Institut CERN [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] KOSCIELNIAK, Shane KOSTKA, Barbara KOVALENKO, Alexandre KRÄMER, Dieter KRASILNIKOV, Mikhail KREMPASKY, Juraj KRISCHEL, Detlef KUBLER, Kristina KUEHNEL, Kai-Uwe KUNZE, Marco KUO, Changhor KURODA, Ryunosuke KUROKAWA, Shin-ichi KUSHNIR, Volodymyr KUSKE, Peter KVARDAKOV, Vasily KWIATKOWSKI, Slawomir LA CHINA, Marco LAGNIEL, Jean-Michel LAMBERT, Guillaume LAMY, Thierry LARI, Luisella LAU, Thomas LAVERTY, Michael LAWSON, Wesley LEBEDEV, Valeri LEBLANC, Greg LEBRUN, Paul L. 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DESY SLAC Columbia University DESY CERN UKAEA INFN-LNL LAL - IFUSP BNL Paul Scherrer Institut TU Eindhoven GSI LANL CERN University of Abertay CERN Schulz-Electronic GmbH CERN Saes Getters Spa CERN INFN-LNF [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] MARCHAND, Patrick MARDOR, Israel MARHAUSER, Frank MARKS, Neil MARQUE, Sebastien MARTI, Zeus MARTIN, Ian MARTIN, Siegfried MASULLO, Maria Rosaria MATHIAS, Brandin MAURI, Marco MCGAHERN, William MEDDAHI, Malika MELIKYAN, Robert MERMINGA, Lia MERTENS, Volker MERZ, Wolfhard MESECK, Atoosa MESHKOV, Igor MESHKOV, Oleg MICHALCZYK, Pierre MIKHAYLOV, Vladimir MILARDI, Catia MINTY, Michiko MISHIN, Andrey MITSUHASHI, Toshiyuki MIURA, Sadao MOLODOZHENTSEV, Alexendre MONDELAERS, Wim MONTAG, Christoph MORGAN, Alun MORITZ, Gebhard MORRIS, William MOSER, Herbert O MOSNIER, Alban MOUAT, Mike MUELLER, Anke-Susanne MUELLER, Norbert MUELLER, Wolfgang MUÑOZ, Marc MURATORI, Bruno MURDOCH, Graeme MURPHY, James MUSTAFIN, Edil MUTO, Toshiya MYERS, Stephen MYTSYKOV, Andriy NADJI, Amor NAGAITSEV, Sergei NAGAOKA, Ryutaro NAKAMURA, Norio NAKAMURA, Takeshi NAKAYAMA, Masahiko NAM, Sang Hoon NEFF, Stephan NEMTANU, Monica NESMIYAN, Inna NEUMANN, Axel NEZHEVENKO, Oleg Synchrotron SOLEIL Soreq NRC BESSY CCLRC DL CERN CELLS Diamond Light Source Forschungszentrum Jülich INFN MAX-Lab, Lund University INFN-LASA BNL CERN YerPhI Jefferson Lab CERN DESY BESSY JINR BINP TPC JINR INFN-LNF DESY American Science and Engineering KEK Mitsubichi Heavy Industries, Ltd. 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Paul Scherrer Institut / EPFL BESSY RIKEN Japan Nuclear Cycle Dev. Inst. Osaka University PAL POSTECH Osaka University NASU/IOC Sincrotrone Trieste NSC KIPT CERN SINAP LBNL Uppsala University MAN CERN IAP FNAL INFN-LNF MIT TU Dresden IHEP MIT University of Texas at Austion [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] Proceedings of EPAC 2004, Lucerne, Switzerland NON GAUSSIAN TRANSVERSE DISTRIBUTIONS IN A STOCHASTIC MODEL FOR BEAM HALOS N. Cufaro Petroni, Bari University, INFN Sez. Bari, Italy S. De Martino, S. De Siena, and F. Illuminati, Salerno University, INFN Sez. Napoli, Italy Abstract We face, in a stochastic framework, the problem of the halo formation in high intensity beams of charged particles. We sum up our recent results on this argument, and in the final part of the paper we introduce non Gaussian distributions as good candidates to account for sensible escape of particles from the density core. have adapted our scheme to a preliminary study of the problem of halo formation and control, and we have described the collective, not deterministically controllable effects by diffusive (Gaussian) fluctuations. In the present paper we make the hypothesis that non–Gaussian processes, for example the family of the Student distributions, with longer tails and non-zero jump probability could provide a better understanding of the long–distance particle behavior. INTRODUCTION STOCHASTIC MODEL FOR HALOS In high intensity beams of charged particles, proposed in recent years for a wide variety of accelerator–related applications, it is very important to keep at low level the beam loss to the wall of the beam pipe, since even small fractional losses in a high–current machine can cause exceedingly high levels of radioactivation. However, numerical simulations, and a few experiments at high current proton linacs, suggest that an extended, low–density halo is formed around the beam: it is then necessary to obtain a more quantitative understanding of the physics of the halo, in order to control and reduce its undesirable effects. In recent papers [1, 2] we faced this problem in the framework of a stochastic approach to the collective beam dynamics. Charged particle beams are usually described either in terms of classical dynamical systems, i. e. by a collisionless plasma with the dynamics of a suitable phase space, or by a Fokker–Planck equation in the phase–space variables. In our alternative approach [3] we propose instead to describe the collective dynamics of the particle beams by a stochastic and time–reversal invariant description in configuration space. This picture is effective in the regime of stability, where a balance is realized between the energy dissipation and the external energy pumping. The sample paths of the representative particle are described by a diffusive stochastic differential equation ruled by the beam emittance, and the two coupled stochastic–hydrodynamic equation describing the dynamics of the beam density profile and the motion of its center are derived by a stochastic variational principle [4]. The two coupled equations are also equivalent to a unique linear equation of the form of a (mesoscopic) Schrödinger equation, with the universal Planck constant replaced by the emittance of the beam. Our approach, starting by an apparently different point of view, formally converges then to the so–called quantum– like approach to beam dynamics [5]. In the framework of our scheme is also possible to implement an active control for the beam dynamics [6], an aspect which could be useful in order to eliminate halos. In references [1, 2] we The coupled stochastic–hydrodynamic equations derived by the stochastic variational principle [4] and describing the particle beam dynamics in the stability regime [3] take the form , respectively, of a continuity equation: ∂t ρ = −∇ · (ρv) , (1) and of an evolution (Madelung) equation for the conservative dynamics √ ∇2 ρ m + V (r, t) = 0 , (2) ∂t S + v2 − 2mD2 √ 2 ρ where , if m is the mass of a particle, the drift velocity and the (Jacobi-Madelung) function S are connected by the gradient form mv = ∇S, and the diffusion coefficient is proportional to the beam emittance α: α = 2mD. Eqns. (1) and (2) describe the collective behavior of the beam at each instant of time through the evolution of both its particle density and its velocity field. Introducing the repre√ sentation ψ = ρ eiS/α , the two coupled real equations (1) and (2) are equivalent to a single complex Schrödinger equation for the function ψ, with the Planck action constant h̄ replaced by the unit of action α: iα∂t ψ = − α2 2 ∇ ψ+Vψ. 2m (3) In this “Schrödinger-like” (Sl) formulation the phenomenological “wave function” ψ carries the information on the dynamics of both the beam density and the beam velocity field, since the velocity field is determined via the gradient relation by the phase function S. Space–charge effects Among the various possible sources of halos, in [1] we have taken into account the spreading effects due to the presence of a space–charge distribution in the framework of our approach. We have implemented the Sl-equation (3) 2056 Proceedings of EPAC 2004, Lucerne, Switzerland 1 associated potential and of the associated velocity fields. The potential exhibits an extra relative maximum if compared with the harmonic one, while in correspondence the velocity field exhibits a “bump” with respect to the straight (linear) behavior. 0.8 0.6 0.4 s w2 +s/ Dynamical control of the halo 0.2 s 1 2 3 4 5 Figure 1: The radial distribution s w2 (s) compared with the distribution in absence of space charge (dashed line) for a strong space-charge strength . All quantities are dimensionless. in a cylindrically symmetric configuration and coupled it with the Maxwell equations for the electromagnetic field generated by the charge distribution. We have thus obtained two nonlinear coupled equations. Considering the stationary distribution of the particle density, we have obtained that the density profile associated to the presence of space–charge is sensibly spread with respect to the Gaussian profile obtained if space–charge is absent (see Fig. 1). The distribution exhibits a larger dispersion, and we have shown that, for sufficiently intense charge density, we can find up to 105 particles per meter beyond a distance of 10σ (with σ denoting the width of the Gaussian core), at variance with the zero particles in absence of charge effects. Stationary halo distributions The solutions discussed in the previous subsection have been obtained through a numerical approach, because the nonlinearly coupled equations do not allow for exact analytical solutions. In the same reference [1] we have then also employed a variational technique to obtain an approximate analytical version of the numerical solution for the density profile. In fact, an analytic expression for the density profile leads to analytical forms for the drift velocity and the generating potential, which in turn are needed if one wish to engineer a controlling mechanism to eliminate the halo. We have then optimized a reasonable test form for the density profile, and we have found that the variational solutions show the presence of a node in the density profile, and a corresponding singularity in the potential. A node can be the signature for the presence of a ring–shaped halo, well separated by the core; in fact, it is completely open the question if halo is due to long tails in the density distribution, or if a true halo must be separated from the core. On the other hand, we have obtained a tail by space–charge effects, which however are not the only possible source of halo. Then, in the spirit of a preliminary inquiry about both the interpretations, we have reversed our point of view; we have in fact explicitly introduced a realistic ring distribution, and we have used the dynamical equations to obtain information on the characteristics of the In reference [2] we have extended our analysis to the problem of the halo dynamics, considering both the the halo elimination and its reformation. This last problem cannot be accounted for in a pure Quantum–like (Ql) description since, because of the stochastic extremal principle, a change in the distribution produces a change in the dynamics. On the other hand the Ql dynamics leads to distributions which are stable attractors; thus, the halo, if scraped away, will be restored in characteristic times, but through a not extremal Fokker–Planck evolution. We have then accordingly modified our scheme: the velocity field changes slowly on the relaxation time scales, while the evolutions are ruled only by a Fokker–Planck equation until the balance has been restored. We got an estimate of the time τ needed by a non stationary, halo–free distribution to relax toward the stationary distribution with a halo when the dynamics is supposed frozen in the configuration that produces this halo: we have obtained τ ≈ 10−8 ÷10−7 sec [2]. Since this relaxation time depends on the parameters of our beam, a comparison with possibly measured phenomenological times could constitute a good check on the soundness of the model. We have also begun to analyze possible transitions from a beam with halo toward a halo–free one, putting an emphasis on the possible dynamics which allows this halo elimination. NON GAUSSIAN DISTRIBUTIONS Non gaussian stationary states Values of the maximum-to-RMS ratio as large as 10 or 12 have been observed in simulations and it is generally agreed that this is called halo (see [8] p. 286). Hence there are good reasons to try non gaussian transverse distributions with longer tails so that the probability of finding particles far away from the beam axis becomes larger. Generalized Student distributions Σ(ν; a2 ) Γ ν+1 aν 2 f (x) = √ ν > 0. (4) ν+1 , π Γ ν2 (a2 + x2 ) 2 are good candidates. They have finite variance σ2 only for ν > 2 and a2 = (ν − 2)σ 2 . Moreover Σ(ν, (ν − 2)σ 2 ) converges in distribution to N (0, σ 2 ) for ν → +∞. The radial control potential for a stationary cylindrical beam: 2 2 r σ α2 2 ν−2 + (ν + 10) 4 r (ν + 2) Vν (r) = 2 2mσ 2 [r2 + (ν − 2)σ 2 ] tends to be harmonic for ν → +∞, and is constant beyond a certain radius. This possibly points to some effect due to the presence of the conducting walls. 2057 Proceedings of EPAC 2004, Lucerne, Switzerland Lévy driven processes The trajectories of a process with a stationary Student Σ(ν, (ν − 2)σ 2 ) ground state can be simulated from the Gaussian driven SDE √ dX(t) = vν (X(t))dt + 2D dW (t) -100 -200 -300 with suitable velocity field. On the other hand a different kind of process can be generated by independent increments with Student law (4) through a SDE Figure 2: Student process with ν = 2.0 dX(t) = β(X(t))dt + γ dS(t) 125 where dS(t) is distributed according to Σ(ν; a2 ), and β(x) is a suitable velocity field. No complete theory is available for a Lévy driven stochastic mechanics. The Student distributions Σ(ν; a2 ) are infinitely divisible (Lévy distributions) [9]. Hence they are possible limit distributions of row-sums of r.v.’s, but different from the gaussian. As every infinitely divisible law, the Σ(ν; a2 ) are possible distributions of the increments of a Markov process with independent increments (Lévy process) [10]. Non-gaussian Lévy processes have moving discontinuities: only gaussian Lévy processes have almost every trajectory everywhere continuous [11]. The standard example of a non-gaussian Lévy process is the compound Poisson process. The rate of occurrence, and the height of the moving discontinuities are regulated by the Lévy function Lt (x) characteristic of a given process with independent increments [11]. However the Lévy function of a Student Σ(ν; a2 ) process is not explicitly known. Discontinuities in these processes can be used to describe the occasional escape of particles out from an otherwise well collimated beam in an accelerator. Moreover, by suitably tuning the parameters (a and ν) we can also keep the transverse distribution of the beam reasonably close to a gaussian. We can contrast the trajectories for both a Gauss and a Student process with comparable dispersions. In the Figures 2, 3 a few examples of the Student trajectories are produced. Since for fixed a the Student variance (which diverges for 0 < ν ≤ 2) is a decreasing function of ν, it is apparent that also the jump probability and their length will generally decrease with ν. The Figures for ν > 2 (finite, decreasing variance) are selected among the possible cases by excluding all the trajectories showing an almost gaussian behavior. For ν > 2 several trials are needed before getting a trajectory definitely drifting away from the beam core. In this model we suppose that a Student-Lévy noise will be active only inside the beam, while beyond a given distance from the core the process will be driven by a Gauss noise; moreover the velocity field attracts the trajectory back toward the core only in its neighborhood, while beyond a given distance the process is just a free diffusion. Remark that in this example the Gaussian trajectories would be essentially that of an Ornstein–Uhlenbeck process. 100 75 50 25 Figure 3: Student process with ν = 3.0 We conclude by remarking that in our opinion for a halo model the possibility of rare, long jumps in the trajectories is more relevant than just the modest increase in the variance produced by the Student noise with respect to the Gauss (almost everywhere continuous) noise. REFERENCES [1] N. Cufaro Petroni, S. De Martino, S. De Siena, and F. Illuminati, Phys. Rev. ST Accel. Beams 6, 034206 (2003). [2] N. Cufaro Petroni, S. De Martino, S. De Siena, and F. Illuminati, Int. Journal of Mod. Phys. B 18, 607 (2004). [3] N. Cufaro Petroni, S. De Martino, S. De Siena, and F. Illuminati, Phys. Rev. E 63, 016501 (2001) [4] F. Guerra and L. M. Morato, Phys. Rev. D 27, 1774 (1983); E. Nelson, Quantum Fluctuations (Princeton University Press, Princeton N. J., 1985). [5] R. Fedele, G. Miele and L. Palumbo, Phys. Lett. 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