Langevin simulation provides an effective way to study collisional effects in
beams by reducing the six-dimensional Fokker-Planck equation to a group of
stochastic ordinary differential equations. These resulting equations usually
have multiplicative noise since the diffusion coefficients in these equations
are functions of position and time. Conventional algorithms, e.g. Euler and
Heun, give only first order convergence of moments in a finite time interval.
In this paper, a stochastic leap-frog algorithm for the numerical integration
of Langevin stochastic differential equations with multiplicative noise is
proposed and tested. The algorithm has a second-order convergence of moments in
a finite time interval and requires the sampling of only one uniformly
distributed random variable per time step. As an example, we apply the new
algorithm to the study of a mechanical oscillator with multiplicative noise.Comment: 3 pages, 4 figures, to submit to XX International LINAC conferenc

Habib, Salman

Qiang, Ji

English

arXiv

QIANG, J.

HABIB, S.

UNT Digital Library

L4-”F?-00-(?o “ 3$ ~ 1i.TITLE:AUTHOR(S):SUBMIVED TO:L(QENATIONALAllaLABORInncl$s:A TORYA SECOND-ORDER STOCHASTIC LEAP-FROG ALGORITHMFOR LANGEVIN SIMULATIONJi QiangSalman HabibLINAC 2000 ConferenceMonterey, CaliforniaAugust 21-25,2000LANSCE-1T-8Los Alamos National Laboratory, an affirmative action/equal opportunity employer, is operated by the University of California for the U.S. Department of Energyunder contract W-7405-ENG-36. By acceptance of this article, the publisher recognizes that the U.S. Government retains a nonexclusive, royatty-free license topublish or reproduce the published form of this contribution, or to allow others to do so, for U.S. Government purposes. The Los Alamos National Laboratoryrequests that the publisher identify this article as work performed under the auspices of the U.S. Department of Energy.Form No. 636 R5ST 2629 10/91DISCLAIMERThis report was prepared as an account of work sponsoredby an agency of the United States Government. Neitherthe United States Government nor any agency thereof, norany of their employees, make any warranty, express orimplied, or assumes any legal liability or responsibility forthe accuracy, completeness, or usefulness of anyinformation, apparatus, product, or process disclosed, orrepresents that its use would not infringe privately ownedrights. Reference herein to any specific commercialproduct, process, or service by trade name, trademark,manufacturer, or otherwise does not necessarily constituteor imply its endorsement, recommendation, or favoring bythe United States Government or any agency thereof. Theviews and opinions of authors’ expressed herein do notnecessarily state or reflect those of the United StatesGovernment or any agency thereof.DISCLAIMERPortions of this document may be illegiblein electronic image products. Images areproduced from the best available originaldocument.r ,8A Second-Order Stochastic Leap-Frog Algorithm for Langevin SimulationJi Qiang and Salman EIabib, LANL, Los Alamos, NM 87545, USAAbstractLangevin simulation provides an effective way to study col-Iisional effects in beams by reducing the six-dimensionalFokker-Planck equation to a group of stochastic ordinarydifferential equations. These resulting equations usuallyhave multiplicative noise since the diffusion coeftlcients inthese equations are functions of position and time. Con-ventional algorithms, e.g. Euler and Heun, give only firstorder convergence of moments in a finite time interval. Inthis paper, a stochastic leap-frog algorithm for the numeri-cal integration of Langevin stochastic differential equationswith multiplicative noise is proposed and tested. The al-gorithm has a second-order convergence of momenta in afinite time interval and requires the sampling of only oneuniformly distributed random variable per time step. Asan example, we apply the new algorithm to the study of amechanic oscillator with multiplicative noise.1 INTRODUCTIONMultiple Coulomb scattering of charged particles, alsocalled intra-beam scattering, has importasit applications inaccelerator operation. It causes a diffusion process of par-ticles and leads to an increase of beam size and emittance.This results in a fast decay of the quality of beam and re-duces the beam lifetime when the size of beam is largeenough to hit the aperture [11.An appropriate way to study the multiple Coulomb scat-tering is to solve the Fokker-Pkmck equations for the dis-tribution function in six-dimensional phase space. Never-theless, the Fokker-Planck equations are very expensive tosolve numerically even for dynamical systems possessingonly a very modest number of degrees of freedom. Trunca-tion schemes or closures have had some success in extract-ing the behavior of low-order moments, but the systemat-ic of these approximations remains to be elucidated. Onthe other hand, the Fokker-Planck equations can be solvedusing equivalent Langevin simulation, which reduces thesix-dimensional partial differential equations into a groupof stochastic ordinary differential equations. Compared tothe Fokker-Pkmck equation, stochastic differential equa-tions are not difficult to solve, and with the advent of mod-em supercomputers, it is possible to run very large num-bers of realizations in order to compute low-ordermomentsaccurately. In general, the noise in these stochastic ordi-nary differential equations are multiplicative instead of ad-ditive since the dynamic friction coefficient and diffusioncoefficient in the Fokker-Planck equations depend on thespatiat position. An effective numerical algorithm to inte-grate the stochastic differential equation with multiplicativenoise will significantly improve the efficiency of large scale13ECLangevin simulation.The stochastic leap-frog algorithms in the Langevins @ulation are given in Section II. Numerical tests of this algo-rithms is presented in Section HI. A physical applicationof the algorithm to the multiplicative-noise mechanic os-cillator is given in Section IV. The conclusions are drawnin Section V.2 STOCHASTIC LEAP-FROGALGORITHMIn the Langevin simulation, the stochastic particIe equa-tions of motion that follow from the Fokker-Planck equa-tion are (Cf. Ref. [2])r’ = v, (1)V( = : – Vv + fir(t), (2)where F is the force includlng both the external force andthe self-generated mean field space charge force, m is themass of particle, v is friction coefficient, D is the diffusioncoefficien~ and I’(t) are Gaussian random variables with(r,(t)) = o, (3)(r,(qri(t’)) s qt - t’). (4)In the case not too far from thermodynamic equilibrium,the friction coefficient is given as4@n(r)Z4e4 in (A)v= 3m2 (T(r) /m)3J2(5)and the diffusion coefficient D is D = vW’/m [3]. Here,n(r) is the density of particle, Z’(r) is the temperature of ofbeam, Z is the charge number of particle, e is the charge ofelectron, A is the Coulomb logarithm, and k is the Boltz-mann constant. For above case, noise terms enter onlyin the dynamical equations for the particle momenta. InEqn. (6) below, the indices are single-particle phase-spacecoordinate indices; the convention used here is that the oddindices correspond to mornent~ and the even indices to thespatial coordinate. In the case of three dimensions, the dy-namical equations then take the general formxl = F~(zl , z~, X3,Z4,Z5,Z13) + a~~ (Z2,Z4, SLl)gl(t)X2 = F2 (XI)x3 = F~(~~,~2,X3,~4,~~,~~) +~33(*2,~4,~~)(~(~)k4 = F4(@x5 = F5(z1, zz, x3, z4, z5,zG) + ~~~(~z,~*,~G)&j(t)26 = F6 ($5) (6)1+ ,#In the dynamical equations for the momenta, the first term 2.t9 vliliamaLSawTcul~ !-+--on the right hand side is a systematic drift term which in- 2.i8eludes the effects due to external forces and damping. The 2.17second term is stochastic in nature and describes a noise 2.16force which, in general, is a fimction of position, The noise 2.15f(t) is first assumed to be Gaussian and white as defined .Q ,,,~by Eqns. (3)-(4). The stochastic leap-frog algorithm for the V ,,,,Eqns. (6) is written as2.i2:/%i(h) = ~i(h) + Si(h) (7) 2.tf1 /’”‘“p---’”The deterministic contribution D~(h) can be obtained us- 2.030 :, :, :, :4 :, ,,ing the deterministic leap-frog algorithm. Here, the deter- hministic contribution ~i (h) and the stochastic conrnbution~i(h) of the above recursion formula for one-step integra- Figure 1: Zero damping convergence test. (X2(t)) at t = 6tion are found to be as a function of step size with white Gaussian noise. Solidfii(h) =~i(h) =Si(h) =~i(h) =—*q =lines represent quadratic fits to the data points (diamonds).—*”—*-* -* -* -*)“E~(0) + hF~(Zl,Zz>Zs>Za,Zs,ZG ,{i= 1,3, 5} 3 NUMERICAL TESTS3;. The abovealgorithmwas” tested on a one-dimensional‘*)]; stochastic ha&onic oscillatorwith a simple form of the+~hFi [~i--l + hFi-1 (fi~, @, z; , ~~, ~:, Zfj{i= 2,4,6)uw%W@) + jFi,.uwh3/2W@)+ ~uii,jFjh3~2Wd (h)+:Fi,klakbaI/h2 tii(h)tii(h);{i=l,3,5; j=2,4,6; k,l=l,3,5}$Fitiojjh3/2wj(h){i= 1,2:3, 4,5,6}multiplicative noise. The equations of motion were$ = F1(P, ~) + ~(~)~(t)X=p (12)where F1(y, z) = –Tp – q2z and cr(z) = –m. Thestochastic leapfrog integrator for this case is given byEqns. (8) (white noise) with the substitutions xl = p,Z2 = z.As a first tes~ we computed (&’) as a fimction of timestep size. To begin, we took the case of zero darnping con-stant (-y = O),where (X2) can be determined analytically.The curve in Fig. 1 shows (Z2) at t = 6.0 as a fimctionof time step size with white Gaussian noise. Here, the pa-rameters q and CYare set to 1.0 and 0.1. The analyticallydetermined vtie of (S2) at t = 6.0 is 2.095222. The,fi, quadratic convergence of the stochastic leap-frog algorithm~~)is clearly seen in the numerical results. We also verifiedwhere Wi (h) is a series of random numbers with the mo-ments(iii(h)) = ((~i(h))3) = ((~i(h))5) = O (9)((l?i(h))2) = 1, ((fii(h))’) = 3 (lo)This can not only be achieved by choosing true Gaussianrandom numbers, but also by using the sequence of randomnumbers following{-/3 R < 1/6Wi(h) = 1/6 < R < 5/6 (11)3, 5/6 ~ Rwhere R is a uniformly dkributed random number on theinterval (O,1). TMs trick significantly reduces the computa-tional cost in generating random numbers.that the auadratic convemence is mescnt fornonzero damD-ing (T ~ 0.1). At t = 1~.0, and-with all other paramete-mas above, the convergence of (Z2) as a function of time stepis shown by the curve in Fig. 2. As a comparison againstthe conventional Heun’s algorithm [5], we computed ($2)as a function oft using 100,000 numerical realizations fora particle starting from (0.0, 1.5) in the (x, p) phase space.The results along with the analytical solution and a numer-ical solution using Heun’s algorithm are given in Fig. 3.Parameters used were h = 0.1, q = 1.0, and a = 0.1. Theadvantage inaccuracy of the stochastic leap-frog algorithmover Heun’s algorithm is clearly displayed, both in termsof error amplitude and lack of a systematic drift.4 APPLICA~ONIn this section, we apply our algorithm to studying the ap-proach to thermal equilibrium of an oscillator with multi-0.515W4iletdwwilhw —051o.m5 -0.50.4950.49 -.40 0.485 -0.4a -0.4750.47 -0.465 -0.460 0.1 02 0.3 0.4 0.5 0.6kFigure 2: Finite damping (~ = 0.1) convergence test.(Z2(t)} at t = 12 as a function of step size with whiteGaussian noise. Solid lines represent quadratic fits to thedata flOiIKS (diamonds).A‘xvErro~ Leapfrog-20 100 200t200 400 500F@re 3: Comparing stochastic leap-frog and the Heun al-gorithm: (Z2(t)) as a function of t. Errors are given relativeto the exact solution.plicative noise. The governing equations areP= –w~z– kJ2p– lmx& (t)where the diffusion coefficients D = AkT, A is the cou-pling constan~ and W. is the oscillator angular frequencywithout damping. In Fig. 4, we display the time evolu-tion of the average energy with multiplicative noise fromthe simulations and the approximate analytical calcula-tions [6]. The analytic approximation resulting from theapplication of the energy-envelope method is seen to be inreasonable agreement with the numerical simulations forkl!’ = 4.5. The slightly higher equilibrium rate from theanalytical calculation is due to the truncation in the energyenvelope equation using the (@(t)) = 2(13(t))2 relationwhich yields an upper bound on the rate of equilibration ofthe average energy [6].43,fA=3w2.s215i(,,;j0 !W 2WI 300 400 Ym600700EatFigure 4 Temporal evolution of the scaled average energy(1?(t)) with multiplicative noise from numerical simulationand analytical approximation.5 CONCLUSIONSWe have presented a stochastic leap-frog algorithm forLangevin simulation with multiplicative noise. Thismethod has the advantages of retaining the symplecticproperty in the deterministic limi~ ease of implementa-tion, and second-order convergence of moments for mul-tiplicative noise. Sampling a uniform distribution insteadof a Gaussian distribution helps to significantly reduce thecomputational cost. A comparison with the conventionalHeun’s algorithm highlights the gain in accuracy due to thenew method. Finally, we have applied the stochastic leap-frog algorithm to a nonlinearly mechanic oscillator systemto investigate the the nature of the relaxation process.6 ACKNOWLEDGMENTSWe acknowledge helpful dkcussions with Grant Lythe andRobert Ryne. Partial support for this work came fromthe DOE Grand ChalIenge in, Computational Accelera-tor Physics. Numerical simulations were performed onthe SGI 0rigin2000 systems at ACL at Los Alamos Na-tional Laboratory, and on the C.ray T3E at the NERSC atLawrence Berkeley National Laboratory.[1][2][3][4][5][6]7 REFERENCESA. Piwinski, Pmt. 9th Int. Conf. on High Energy Accelera-tors, Standord, 1974 (SLAC, Stanford, 1974) p. 405.H. Risken, The Fokker-PlanckEquation: MethodsofSolutionundApplicatwns(Springer, New York 1989).M. E. Jones, D. S. Lemons, R. J. Masonj V. A. Thomas, andD. Wlnske, J. Comput. Phys. 123, 169 (1996).R. Zwanzig, J. Stat. Phys. 9,215 (1973).A. Greiner, W. Strittmatter, and J. Honerkamp, J. Stat. Phys.51,94(1988).K. Lindenberg and V. Seshadri, Physics 109A,483 (1981).

A Second-Order Stochastic Leap-Frog Algorithm for Langevin Simulation

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