MacLachlan, J A

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APPLICATION OF SCALING PROPERTIES OFTHE VLASOV AND FOKKER-PLANCK EQUATIONS TOIMPROVED MACROPARTICLE MODELSJ. A. MacLachlanFermi National Accelerator Laboratory, Box 500, Batavia IL 60510-0500∗1 INTRODUCTIONThere are two important partial differential equations(PDE’s) which describe the evolution of phase space den-sity distributions (denoted ψ) for beams in cyclic accelera-tors and storage rings. The Vlasov equation∂ψ∂t+ τ˙∂ψ∂ε+ ε˙∂ψ∂τ= 0 (1)applies when the forces are derivable from a potential. Itis written here just for the longitudinal degree of freedomin conjugate time τ and energy ε variables. The Fokker-Planck equation applies when there are incoherent coolingor diffusive processes to be taken into account:∂ψ∂t= − ∂∂E[C(E)ψ −D(E) ∂ψ∂E]. (2)Here ψ(E; t) is the energy distribution of a coasting beam.It is often easier in practical applications to follow the evo-lution of phase space density by tracking a representativedistribution of macroparticles using the equations of mo-tion. There are scaling relations derived from the PDE’swhich can make dramatic improvements in the efficiencyor practical scope of macroparticle models. Scaling rulesreported previously are recapitulated below. Their useful-ness in speeding up macroparticle model calculations is dis-cussed with most attention to the simpler case of time scal-ing in the Vlasov equation.Andre Gerasimov found scaling for the Fokker-Planckequation for longitudinal stochastic cooling permittingmacroparticle solutions correct even for macroparticlenumber six or seven orders of magnitude less than the beampopulation.[1] These ideas were used by the author in somecooling simulations described briefly at the end of this note.Their success encouraged the search for an appropriate scal-ing for speeding up large scale longitudinal dynamics mod-eling for synchrotrons, allowing time steps longer than abeam turn without misrepresenting the dynamics. The re-sult was more powerful than anticipated, being for a broadrange of cases the fourth power of the time scaling constantrather than the first power one might expect.∗Operated by the Universities Research Association under contractnumber DE-AC02-76CH03000 with the U. S. Department of Energy2 TIME SCALING IN THE VLASOVEQUATIONThe objective for scaling the Vlasov equation (eq. 1) isto find variables in which the evolution of the density func-tion is accelerated but not otherwise changed. In the ab-sence of a multiparticle potential it is obvious from inspec-tion of the equations of motion that multiplying the poten-tial and the phase-slip-per-time-step by a constant acceler-ates the change in the distributionproportionally. This scal-ing constant is hereafter denoted by λ. Because the Hamil-tonianH is the generating function for infinitesimal canon-ical transformations of coordinates from their value at timet to time t+dt, the Hamiltonian for the system that evolvesλ times as fast is simplyH ′(t′) = λH(t) for t′ = λt; ε′ = ε , (3)from which one can get particle equations of motion foran accelerated tracking. Note that writing the eq. 1 in thescaled system∂ψ′∂t′+∂ψ′∂τ ′dτ ′dt′+∂ψ′∂ε′dε′dt′= 0 (4)and substituting according to eq. 3 leads directly toλ∂ψ∂t+ λ∂ψ∂τdτdt+ λ∂ψ∂εdεdt= 0 . (5)Thus, the invariance of the Vlasov equation under scaling isalmost trivial. This derivation of time scaling, which startsfrom the existence of the multiparticle Hamiltonian, com-plements the previously reported derivation from the parti-cle equations of motion.[2] Somewhat more subtle is howthe scaled time will show up in the potential; any terms re-lating to frequencies, including rf potentials and frequencydomain expressions for impedance, will be scaled.Multiparticle tracking using the equations of motionfrom the scaled Hamiltonian will go faster by the factor λbecause the clock runs faster by that factor. However, it isthrough the scaling of frequencies and a statistical observa-tion that the scaling gets most of its power, another threepowers of λ. Scaling any frequency f with the time scalinggives f ′ = λf . Writing the potential in frequency domain,one can see that it takes λ−1 as many harmonics of the beamcirculation frequency to span the domain; the sampling is,of course, sparser by that same factor. If the potential issmooth enough that the reduced sampling can be accepted,the scaling will not compromise fidelity of the calculated0-7803-7191-7/01/$10.00 ©2001 IEEE. 3180Proceedings of the 2001 Particle Accelerator Conference, Chicagodistribution. In fact, for the space charge term, whose fre-quency domain form is linear, there is no loss of fidelityfrom reduced sampling.[2] Fewer harmonics in the poten-tial mean fewer bins needed in calculating fourier compo-nents of the beam current. It has been shown rigorously forthe space charge force in time domain[3] and heuristicallyfor smooth potentials in frequency domain[4] that when thebin number is reduced by λ−1, the macroparticle numbercan be reduced byλ−3 for the same statistical accuracy (nu-merical noise) in the distribution. Therefore, in suitable cir-cumstances λ may be considerably larger than one, and atracking simulation can be accelerated by λ4.Besides sampling deficiency, there is another way inwhich the fidelity of a scaled macroparticle model can becompromised; viz., the expanded time step can be too largea fraction of a characteristic period in the motion like, espe-cially, the synchrotron period. One must observe a limit onscaling λνc << 1, where νc is the synchrotron tune or thetune of some parameter variation. Usually a factor ten foreach < sign is acceptable.3 HYBRID MACROPARTICLE MODELSVLASOV + FOKKER-PLANCKGerasimov’s scaling rules[1] for the Fokker-Planckequation eq. 2 can be expressed in terms of their effect onthe gain G at each harmonic  of the beam circulation fre-quency, the cooling power P at each harmonic, the beamcirculation frequency as a function of energy ω(E), thescaled time t, and the scaled number of particles N . Thereare three rules which are called here number scaling, band-width scaling, and gain scaling. The scale factors are de-noted kN , kB, and kG respectively; ω◦ is the beam circula-tion angular frequency at the central orbit energy E◦. Therules are expressed in the following table:Scaled Scaling RuleResult Number Bandwidth GainG′ kNG GkB· kGGP ′ kNP P/kB kGPω′ − ω◦ ω − ω◦ kB(ω − ω◦) kG(ω − ω◦)t′ t/kN kBt t/kGN ′ N/kN N NThere are limits on the size of the scale factors which in-volve number of harmonics and particle statistics; they areakin to those discussed above for time scaling in the Vlasovequation. The reader is referred to ref. [1] for these limitsand the relations between tabulated quantities and the cool-ing coefficient C(E) and diffusion coefficient D(E) of eq.2. For the macroparticle approach, the tabulated quantitiesare the ones of interest.Consider a case where the beam particles are acted on byan rf system, for example, as well as a cooling system. Ifthe Fokker-Planck scaling rules can be used to give an ade-quate macroparticle model with a time step appropriate forthe phase motion at a macroparticle number low enough tobe practical, one can build a combined macroparticle modelby applying both the map for the equations of motion andthe Fokker-Planck operator for the cooling on each itera-tion. Notice it is also possible to scale the map for largertime step as discussed above. In the case of the hybridmodel, however, the macroparticle number will be deter-mined by the Fokker-Planck scaling so that the Schottkynoise is correctly scaled. This technique has been used tostudy rf stacking onto a cooled stack in the Fermilab Recy-cler storage ring. It was meaningful to simulate minutes ofreal time of cooling/stackingwith fewer than 104 macropar-ticles; computing times were a few hours.4 CONCLUSIONSNumerical simulations of cooling processes over min-utes or hours of real time are usually carried out using di-rect solution of the Fokker-Planck equation. However, byusing scaling rules derived from that equation, it is possi-ble to use macroparticle representations of the beam distri-bution. Besides having applications for cooling alone, themacroparticle approach allows combining the cooling pro-cess with other dynamical processes which are representedby area-preserving maps. A time-scaling rule derived fromthe Vlasov equation can be used to adjust the time step ofa map-based dynamics calculation to one more suitable forcombining with a macroparticle Fokker-Planck calculation.The time scaling for the Vlasov equation is also useful forsubstantially more rapid calculations when a macroparti-cle model of a conservative multiparticle system requiresa large number of macroparticles to faithfully produce thecollective potential or when the model must simulate a longtime period.5 REFERENCES[1] A. Gerasimov, “Macroparticle Simulation for StochasticCooling”, FERMILAB-Pub-94/376 (1994)[2] J. A. MacLachlan and Z. Nazario, “Scaling for fastermacroparticle simulation in longitudinal multiparticledynamics”, Phys. Rev. ST Acc. Beams vol. 3, 114401 (2000)[3] J. Wei, “Longitudinal Dynamics of the Non-AdiabaticRegime on Alternating Gradient Synchrotrons”, PhDdissertation SUNY at Stony Brook (1990)[4] J. A. MacLachlan, ”Particle Tracking in E-ϕ Space for Syn-chrotron Design and Diagnosis ”, FERMILAB-Conf-92/333(1992)3181Proceedings of the 2001 Particle Accelerator Conference, Chicago

Application of Scaling Properties of the Vlasov and Fokker-Planck Equations to Improved Macroparticle Models

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