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Using kernel-based statistical distance to study the dynamics of charged particle beams in particle-based simulation codes

Chad E. Mitchell, Robert D. Ryne, and Kilean Hwang
Phys. Rev. E 106, 065302 – Published 8 December 2022
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  • INTRODUCTION
  • BEAM PHYSICS CONTEXT AND NOTATION
  • STATISTICAL DISTANCE
  • PROPERTIES OF MAXIMUM MEAN DISCREPANCY
  • HILBERT-SCHMIDT CORRELATION
  • APPLICATIONS TO IDEALIZED AND EXACTLY…
  • APPLICATION TO HIGH-INTENSITY BEAMS
  • CONCLUSIONS
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    Measures of discrepancy between probability distributions (statistical distance) are widely used in the fields of artificial intelligence and machine learning. We describe how certain measures of statistical distance can be implemented as numerical diagnostics for simulations involving charged-particle beams. Related measures of statistical dependence are also described. The resulting diagnostics provide sensitive measures of dynamical processes important for beams in nonlinear or high-intensity systems, which are otherwise difficult to characterize. The focus is on kernel-based methods such as maximum mean discrepancy, which have a well-developed mathematical foundation and reasonable computational complexity. Several benchmark problems and examples involving intense beams are discussed. While the focus is on charged-particle beams, these methods may also be applied to other many-body systems such as plasmas or gravitational systems.

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    • Received 8 April 2022
    • Revised 18 October 2022
    • Accepted 19 November 2022

    DOI:https://doi.org/10.1103/PhysRevE.106.065302

    ©2022 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Beam code development & simulation techniquesHigh intensity beam dynamicsNonlinear beam dynamicsRelativistic multiple-particle dynamics
    Accelerators & Beams

    Authors & Affiliations

    Chad E. Mitchell* and Robert D. Ryne

    • Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA

    Kilean Hwang

    • Facility for Rare Isotope Beams, Michigan State University, East Lansing, Michigan 48824, USA

    • *ChadMitchell@lbl.gov

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    Issue

    Vol. 106, Iss. 6 — December 2022

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    Images

    • Figure 1
      Figure 1

      (Upper) Difference between the result γk(fKV,fG) given in (47) and its numerical estimate obtained using (24) for varying number of particles n=m and number of frequency samples L. Statistical averaging was performed using 1000 distinct realizations of the sampled distributions fKV, fG, and Λ. Curves for increasing values of L approach the curve obtained using the estimate based on (21), shown in black. (Lower) The same quantities, shown for the case of two (4D) Gaussian distributions with the same second moments. In this case the rms error Δγrms is independent of the number of frequency samples L and is equal to γknoise (25).

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    • Figure 2
      Figure 2

      Hilbert-Schmidt correlation Rk between variables z and pz in the longitudinal phase space of a bunch with a quadratic correlation (50). (Inset) Sampled particles (104) for the case σp/aσz2=1, yielding the computed value Rk≈0.375.

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    • Figure 3
      Figure 3

      Dynamics of a 2D beam with n=104 particles sampled from a matched Gaussian distribution (52) evolving under iteration of the map (51). The quantity γk(ft,f0) is shown as a function of the iteration number t for four distinct random seeds, showing that the distribution remains stationary.

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    • Figure 4
      Figure 4

      Evolution of a distribution with n=105 particles sampled from (52) with initial offset q↦q+q0 under iteration of the map (51). The distance of the distribution to the predicted equilibrium (53) after t iterates is shown. (Red curve) Analytical prediction (54). (Black, inset) Absolute error in the numerical result obtained using (21). (Blue, inset) Absolute error in the numerical result obtained using (24) with L=103. (Red, dashed) Prediction (25) of the numerical noise level evaluated using the distribution feq.

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    • Figure 5
      Figure 5

      Dynamics of a beam with n=105 particles sampled from the density (58) evolving under iteration of the map (57). The quantity Rk(t) is shown as a function of the iteration number t, illustrating the decay of correlations due to mixing. The red curve shows the prediction (59), the black points are the results of simulation, and the black curve denotes the expected rms value due to noise (39). (Inset) Plot of initial particle coordinates (q(0),p(0)) sampled from the density (58).

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    • Figure 6
      Figure 6

      Initial phase space coordinates (q(0),p(0)) vs the final coordinate q(t) at t=1,2,3,10 for the particles used to generate Fig. 5, showing the visible correlations that are quantified by Rk(t) and their evolution over time.

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    • Figure 7
      Figure 7

      Test of stationarity for an unbunched (4D) beam with n=106 particles sampled from a thermal equilibrium distribution (62) propagating in a linear constant focusing channel. The quantity γk(ft,f0) is shown as a function of propagation distance t for four distinct random seeds.

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    • Figure 8
      Figure 8

      Collisionless relaxation of an unbunched (4D) beam with n=106 particles sampled from the distribution (65) in a linear constant focusing channel (63), where t denotes the number of linearized envelope periods. The MMD between the distribution on successive periods γk(ft,ft−1) decays to the level of noise as the beam relaxes to a stationary state. (Inset) The final particle distribution (one quadrant).

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    • Figure 9
      Figure 9

      Projections into the (x,px) plane of an initially rms-matched (but nonstationary) distribution (65) evolving in a linear constant focusing channel, shown at four distinct times. The particles shown are those used to produce Fig. 8. (Red) The complete simulated particle distribution. (Black) A subset of initial conditions, illustrating dynamical filamentation of the phase space.

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    • Figure 10
      Figure 10

      Matched (K-V) beam envelopes for a 10 A proton beam at 200 MeV in the FODO cell used in Sec. 7b. Red rectangle, focusing quadrupole; blue rectangle, defocusing quadrupole.

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    • Figure 11
      Figure 11

      Dynamics of an unbunched (4D) beam with n=106 particles sampled from the distribution (66) in the FODO channel shown in Fig. 10. The MMD between the distribution on successive lattice periods γk(ft,ft−1) decays to near (but remains slightly above) the level of noise (black line). The MMD to the initial distribution γk(ft,f0) is largely unchanged after the first 100 periods.

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    • Figure 12
      Figure 12

      Projections into the (x,px) plane of an initially rms-matched (but nonstationary) distribution (66) evolving in a periodic FODO channel, shown at four distinct lattice periods t. The particles shown are those used to produce Fig. 11. (Red) The complete simulated particle distribution. (Black) A subset of initial conditions, illustrating dynamical filamentation of the phase space.

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    • Figure 13
      Figure 13

      Dynamics of an unbunched (4D) beam with n=106 particles sampled from the distribution (66) in the FODO channel shown in Fig. 10. The correlation Rk(t) between the distribution after t periods and the initial distribution at t=0 is shown. Correlations appear to persist well above the level of the noise (black curve). (Inset) Plot of initial y vs final y after 1K periods, showing that the correlations are not easily visible in low-dimensional projections.

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