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From quantum to classical modeling of radiation reaction: A focus on stochasticity effects

F. Niel, C. Riconda, F. Amiranoff, R. Duclous, and M. Grech
Phys. Rev. E 97, 043209 – Published 25 April 2018
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  • INTRODUCTION
  • DYNAMICS OF A RADIATING ELECTRON IN…
  • RADIATION REACTION AND HIGH-ENERGY…
  • FROM QUANTUM TO CLASSICAL RADIATION…
  • TEMPORAL EVOLUTION OF INTEGRATED…
  • NUMERICAL ALGORITHMS
  • NUMERICAL RESULTS
  • CONCLUSION
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    Radiation reaction in the interaction of ultrarelativistic electrons with a strong external electromagnetic field is investigated using a kinetic approach in the nonlinear moderately quantum regime. Three complementary descriptions are discussed considering arbitrary geometries of interaction: a deterministic one relying on the quantum-corrected radiation reaction force in the Landau and Lifschitz (LL) form, a linear Boltzmann equation for the electron distribution function, and a Fokker-Planck (FP) expansion in the limit where the emitted photon energies are small with respect to that of the emitting electrons. The latter description is equivalent to a stochastic differential equation where the effect of the radiation reaction appears in the form of the deterministic term corresponding to the quantum-corrected LL friction force, and by a diffusion term accounting for the stochastic nature of photon emission. By studying the evolution of the energy moments of the electron distribution function with the three models, we are able to show that all three descriptions provide similar predictions on the temporal evolution of the average energy of an electron population in various physical situations of interest, even for large values of the quantum parameter χ. The FP and full linear Boltzmann descriptions also allow us to correctly describe the evolution of the energy variance (second-order moment) of the distribution function, while higher-order moments are in general correctly captured with the full linear Boltzmann description only. A general criterion for the limit of validity of each description is proposed, as well as a numerical scheme for the inclusion of the FP description in particle-in-cell codes. This work, not limited to the configuration of a monoenergetic electron beam colliding with a laser pulse, allows further insight into the relative importance of various effects of radiation reaction and in particular of the discrete and stochastic nature of high-energy photon emission and its back-reaction in the deformation of the particle distribution function.

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    • Received 12 July 2017
    • Revised 22 February 2018

    DOI:https://doi.org/10.1103/PhysRevE.97.043209

    ©2018 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Classical field theoryCosmic rays & astroparticlesDiffusionElectromagnetismGamma ray burstsGamma-ray generation in plasmasKinetic theoryLaser driven electron accelerationLaser wakefield accelerationLaser-plasma interactionsParticle acceleration in plasmasPlasma acceleration & new acceleration techniquesRadiation & particle generation in plasmasRelativistic multiple-particle dynamicsSpecial relativityStochastic processesX-ray generation in plasmas
    Plasma PhysicsAccelerators & BeamsGeneral PhysicsParticles & FieldsGravitation, Cosmology & AstrophysicsInterdisciplinary PhysicsStatistical Physics & Thermodynamics

    Authors & Affiliations

    F. Niel1,*, C. Riconda1, F. Amiranoff1, R. Duclous2, and M. Grech3,†

    • 1LULI, UPMC Université Paris 06: Sorbonne Universités, CNRS, École Polytechnique, CEA, Université Paris-Saclay, F-75252 Paris cedex 05, France
    • 2CEA, DAM, DIF, F-91297 Arpajon, France
    • 3LULI, CNRS, École Polytechnique, CEA, Université Paris-Saclay, UPMC Université Paris 06: Sorbonne Universités, F-91128 Palaiseau cedex, France

    • *fabien.niel@polytechnique.edu
    • †mickael.grech@polytechnique.edu

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    Vol. 97, Iss. 4 — April 2018

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    • Figure 1
      Figure 1

      (a) Dependence of g(χ) on the electron quantum parameter χ leading to a reduction of the emitted power due to quantum effects. (b) Quantum emissivity G(χ/χγ)/χ2 and (c) its classical limit as a function of χ and χγ/χ=γγ/γ. Dashed lines in panels (b) and (c) show χγ≃0.435χ2 for which the classical limit of G(χ,χγ) is maximum.

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    • Figure 2
      Figure 2

      Dependence on χ of the functions an(χ)/n! for n=1 to 4, in black, blue, green, and red (respectively). The left vertical line at χ=χcl=1×10−3 indicates the threshold of the classical to the intermediate regime and the right vertical line at χ=χqu=2.5×10−1 the limit of the intermediate to the quantum regime (see explicit definitions in the text).

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    • Figure 3
      Figure 3

      Dependence on 〈χ〉 and σ̂γ of Er(〈χ〉,σ̂γ) which represents the relative difference between dt〈γ〉MC and dt〈γ〉cLL. The curves Er=10−3,10−2, and 10−1 are plotted in white lines.

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    • Figure 4
      Figure 4

      Dependence on 〈χ〉 of Er(〈χ〉,σγthr), which represents the relative difference between dt〈γ〉MC and dt〈γ〉cLL using σ̂γthr(〈χ〉). The green and blue crosses represent the value of [〈γ〉MC(theat)−〈γ〉cLL(theat)]/〈γ〉cLL(theat) for the plane-wave field and the constant, uniform magnetic field (respectively) and for χ0=10−2,10−1, and 1, theat being the time at which σγ stops to increase.

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    • Figure 5
      Figure 5

      Dependence on 〈χ〉 and σ̂γ of ϕ(〈χ〉,σ̂γ). When ϕ>1, the electron population is predicted to experience energy spreading (heating), while it is predicted to cool down when ϕ<1. The black dashed line corresponds to ϕ=1 and represents the threshold σ̂γthr(〈χ〉) between the regions of energy spreading (heating) and cooling. The green dashed line shows the first-order expansion in 〈χ〉 of the previous equation and corresponds to the prediction of Ref. [42]. The black lines represent the trajectories σ̂γ(〈χ〉) for the interaction of an ultrarelativistic electron bunch with different constant, uniform magnetic fields corresponding to (a) χ0=10−2, (b) χ0=10−1, and (c) χ0=1 (the corresponding simulations are discussed in Sec. 7). The green and blue crosses represent the value of σ̂γ extracted from the simulations considering the plane-wave field and the constant, uniform magnetic field (respectively) for χ0=10−2,10−1, and 1.

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    • Figure 6
      Figure 6

      Dependence on 〈χ〉 and σ̂γ of ψ(〈χ〉,σ̂γ,μ̂3=0). When ψ>1, the electron population is predicted to acquire an asymmetry toward high energies (μ3 increases, positive skewness) while it is predicted to acquire an asymmetry toward low energies (μ3 decreases, negative skewness) when ψ<1. The black dashed line corresponds to ψ=1 and represents the threshold σ̂γ,0lim(〈χ〉) [Eq. (83)]. The blue and green crosses represent the value of σ̂γ when μ3=0 in the interaction of an ultrarelativistic electron bunch with different constant, uniform magnetic field and plane waves (respectively) and initial quantum parameter (a) χ0=10−2, (b) χ0=10−1, and (c) χ0=1. Vertical crosses correspond to the initial values of σ̂γ and χ. Diagonal crosses correspond to the values of σ̂γ and χ when μ3 goes back to 0 (see the sixth panels of Figs. 9 and 12). These simulations are discussed in Sec. 7.

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    • Figure 7
      Figure 7

      Dependence of σ̂γlim [Eq (84)] on 〈χ〉 for different values of μ̂3. The red line represents σ̂γlim for the case μ3̂=0.2 used in Sec. 7c. The black solid line corresponds to μ̂3=0 while the black dotted line corresponds to σ̂γthr [Eq. (74)]. The green crosses correspond to the three initial situations considered in Sec. 7c considering a broad Maxwell-Jüttner energy distribution.

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    • Figure 8
      Figure 8

      Domain of validity of the different approaches. (a) In the case of an initially symmetric electron energy distribution; as given by Eqs. (85) and (86) for a symmetric distribution function (μ̂3=0). (b) In the case of an initially asymmetric distribution function; as given by Eqs. (87) and (88). Here presented for μ̂3≃0.2 corresponding to the Maxwell-Jüttner distribution presented in Sec. 7c.

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    • Figure 9
      Figure 9

      Simulations of an ultrarelativistic electron beam in a constant, uniform magnetic field for (a) χ0=10−3, (b) χ0=10−2, (c) χ0=10−1, and (d) χ0=1. The first three panels of each row show the electron distribution functions from the Monte Carlo simulations (MC, first panels), stochastic (Fokker-Planck) simulations (FP, second panels), and quantum-corrected deterministic simulations (cLL, third panels). The fourth panel shows the difference in the prediction of the mean electron energy in between the MC simulation and the deterministic (black line) and FP (red line) simulations. The two last panels (in each row) correspond to the moments of order 2 (energy variance) and 3 for the MC (blue line), FP (red line), and deterministic (black line), simulations.

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    • Figure 10
      Figure 10

      Simulations of an ultrarelativistic electron beam in a constant, uniform magnetic field for (a) χ0=10−3, (b) χ0=10−2, (c) χ0=10−1, and (d) χ0=1 electron distribution functions at times t=0, t=tend/2, and t=tend (from right to left). The red lines correspond to FP simulations, the blue ones to MC simulations.

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    • Figure 11
      Figure 11

      Simulations of an ultrarelativistic electron beam in a constant, uniform magnetic field for (a) χ0=10−1, (b) χ0=1. This figure focuses on the early times of interaction (t≤theating during which the energy dispersion increases). The first column corresponds to MC simulations, the second to the FP ones. The last column shows snapshots of the electron distribution functions at different times t=0, t=theating/4, and t=theating/2 (from right to left). The red lines correspond to FP simulations, the blue ones to MC simulations.

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    • Figure 12
      Figure 12

      Simulations of an ultrarelativistic electron beam in a counterpropagating linearly polarized plane wave for (a) χ0=10−2, (b) χ0=10−1, and (c) χ0=1. The first three panels of each row show the electron distribution functions from the Monte Carlo simulations (MC, first panels), stochastic (Fokker-Planck) simulations (FP, second panels), and quantum-corrected deterministic simulations (cLL, third panels). The fourth panel shows the difference in the prediction of the mean electron energy in between the MC simulation and the deterministic (black line) and FP (red line) simulations. The two last panels (in each row) correspond to the moments of order 2 (energy variance) and 3 for the MC (blue line), FP (red line), and deterministic (black line) simulations.

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    • Figure 13
      Figure 13

      Simulations of an ultrarelativistic electron beam in a counterpropagating linearly polarized plane wave for (a) χ0=10−2, (b) χ0=10−1, and (c) χ0=1 electron distribution functions at times t=0, t=tend/2, and t=tend (from right to left). The red lines correspond to FP simulations, the blue ones to MC simulations.

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    • Figure 14
      Figure 14

      Simulations of an ultrarelativistic electron beam in a counterpropagating linearly polarized plane wave for (a) χ0=10−1, (b) χ0=1. This figure focuses on the early times of interaction (t≤theating during which the energy dispersion increases). The first column corresponds to MC simulations, the second to the FP ones. The last column shows snapshots of the electron distribution functions at different times t=0, t=theating/4, and t=theating/2 (from right to left). The red lines correspond to FP simulations, the blue ones to MC simulations.

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    • Figure 15
      Figure 15

      Simulations of an electron bunch with initially broad (Maxwell-Jüttner) energy distribution in a constant, uniform magnetic field for (a) χ0=10−3, (b) χ0=10−2, (c) χ0=10−1, and (d) χ0=1. The first three panels of each row show the electron distribution functions from the Monte Carlo simulations (MC, first panels), stochastic (Fokker-Planck) simulations (FP, second panels), and quantum-corrected deterministic simulations (cLL, third panels). The fourth panels show the difference in the prediction of the mean electron energy in between the MC simulation and the deterministic (black line) and FP (red line) simulations. The two last panels (in each row) correspond to the moments of order 2 (energy variance) and 3 for the MC (blue line), FP (red line), and deterministic (black line) simulations.

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