Pith. sign in

REVIEW 3 major objections 5 minor 48 references

Ultra-long simulations of collisionless relativistic shocks in a front-comoving frame find an asymptotic steady downstream state whose parameters are independent of numerical realization, with Fermi acceleration stalling near ten times equi

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-02 02:43 UTC pith:3RHJCROV

load-bearing objection Genuinely new front-comoving setup reaches record durations and gives credible but not conclusive evidence for a downstream steady state; the reflecting wall is part of the system and wall-independence is tested at only two distances. the 3 major comments →

arxiv 2607.14235 v1 pith:3RHJCROV submitted 2026-07-15 astro-ph.HE physics.plasm-ph

Ultra-long simulations of collisionless relativistic shocks in front-comoving frame: evidence for a steady state and its properties

Mikhail Garasev , Evgeny Derishev This is my paper
classification astro-ph.HE physics.plasm-ph
keywords collisionless relativistic shocksparticle-in-cell simulationsfront-comoving framemoving-wall boundary conditionWeibel instabilitysteady stateFermi accelerationmagnetic field structure
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper asks whether the conventional Weibel-mediated picture of relativistic collisionless shocks has a steady-state solution, and answers yes for the downstream: across several very long runs it finds an asymptotic, time-independent downstream whose parameters depend only on the upstream temperature at injection, not on the numerical realization. This matters because if the steady state is physical, shock microphysics—magnetic field strength, coherence scale, particle spectrum—can be computed once and used predictively in radiation models of gamma-ray bursts and other relativistic sources. The paper also finds that Fermi-type acceleration stalls near ten times the equipartition energy, so no true power-law tail forms, and that the downstream magnetic field breaks into independently evolving soliton-like spots with roughly Lorentzian profiles. These results put pressure on the conventional model's ability to explain GRB afterglow observations and sharpen the contrast with the pair-balance model.

Core claim

We present ultra-long front-comoving simulations of unmagnetized electron-positron shocks, running beyond 100,000 plasma periods. The central claim is that, after a transient of a few light-crossing times, the downstream reaches a steady state: the magnetic energy density profile, transverse coherence length (about 60 c/omega_p), density contrast, and particle energy spectrum stop evolving, and this asymptotic state is the same across runs that vary upstream and downstream lengths, transverse size, and particle-per-cell count. The steady-state magnetization peaks near 0.03 at the front and decays as a power law downstream. Fermi-type acceleration is limited to about ten times the equipartiti

What carries the argument

The central mechanism is the front-comoving simulation frame with a moving-wall downstream boundary condition and continuous upstream injection. This keeps the simulation domain fixed in size, making computational cost grow linearly with time instead of quadratically, so runs can exceed 100,000 plasma periods while particles make several round trips between the shock front and the downstream boundary. The authors argue that fully established feedback between the front and the reflecting wall is required for the downstream to settle into a steady state, and they use the wall speed matched to the downstream bulk velocity as the control parameter.

Load-bearing premise

The load-bearing premise is that the fixed reflecting wall in the downstream—needed to keep the simulation box from growing—does not itself create the measured steady state, i.e., that the wall-bounded downstream behaves like the downstream of an unbounded shock.

What would settle it

Run a downstream-comoving simulation without any downstream wall past ~50,000 plasma periods and check whether the downstream magnetization, coherence length, and spectral cutoff keep evolving; if they do, the steady state is an artifact of the moving wall. A cheaper check is to keep the front-comoving setup but make the downstream several times longer and see whether the fitted steady-state parameters shift with wall distance.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The downstream microphysics of the shock is computable: magnetization at the front ~0.03, transverse coherence length ~60 c/omega_p, and the particle spectrum are time-independent once the steady state is reached.
  • Fermi acceleration in these shocks saturates at about ten times the equipartition energy; no true power-law tail forms, so afterglow models that rely on a nonthermal power-law electron distribution need a different energy input or a different shock model.
  • The downstream magnetic field organizes into independently evolving soliton-like spots with approximately Lorentzian radial profiles; the average magnetization decays as a power law with distance downstream, with most volume weakly magnetized.
  • The steady state is insensitive to changes in upstream and downstream box length, transverse size, and particle-per-cell count, but the far-upstream precursor keeps evolving and depends on upstream length, which the authors argue is of minor observational consequence because radiation comes mainly from the downstream.
  • Raising the upstream temperature to relativistic values increases magnetization, coherence lengths, and maximum accelerated energy, moving the simulation closer to conditions relevant to the pair-balance shock model.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • A downstream-comoving run long enough to reach the same nominal age would be the cleanest check of whether the wall-bounded steady state survives; if it does not, the reported state is a property of the wind-tunnel setup rather than of astrophysical shocks.
  • If Fermi acceleration truly caps near ten times equipartition with weak scattering, observed high-energy emission from GRB afterglows would require either additional acceleration in the decaying magnetic spots or support for the competing pair-balance model.
  • The Lorentzian magnetic-spot structure suggests a concrete observational signature: the synchrotron brightness and polarization maps of the downstream emission should trace isolated compact spots rather than a quasi-uniform turbulent field, testable in synthetic radiation-transfer post-processing.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper introduces a front-comoving 2D3V PIC setup for unmagnetized relativistic e± shocks, using continuous upstream injection and a moving-wall downstream boundary to keep the computational domain fixed. With runs extending to 1.05×10^5 ω_p^{-1} (several times longer than previous work) and variations in upstream/downstream lengths, transverse size, particle number, and upstream temperature, it claims strong evidence for an asymptotic steady state in the downstream, independent of numerical realization and depending only on the upstream temperature; that the upstream precursor continues to evolve and retains a dependence on the upstream box length; that Fermi-type acceleration saturates at roughly ten times equipartition without forming a true power-law tail; and that the downstream magnetic field consists of independent, soliton-like magnetic spots with an approximately Lorentzian profile.

Significance. If the steady-state claim is correct, the paper provides an answer to a long-standing question about the conventional Weibel-mediated shock model: there is a computable stationary downstream solution with ϵ_B ~ 0.03, λ_y ~ 60 c/ω_p, and a non-power-law particle spectrum. This bears directly on GRB afterglow modeling and on the competition between the conventional and pair-balance shock models. The paper's strengths are its unprecedentedly long runs, the explicit algorithm for the moving-wall boundary condition, and the cross-check of downstream properties across several numerical realizations. The main caveats are that the steady-state convergence metric is partly self-referential and that the independence of the downstream state from the reflecting wall is tested only over a factor of two in wall distance.

major comments (3)
  1. [Section III, ΔF_γ definition and Table II] The quantitative steady-state metric is circular: ΔF_γ(t) is defined with f_{γ,s} taken as the end-of-run spectrum, so the fitted relaxation time t_s measures how quickly each run approaches its own final state, not convergence to a run-independent attractor. The statement that “the system reaches a late-time stationary state” therefore rests partly on the assumption it is meant to prove. I request a non-self-referential diagnostic, e.g., differences between successive non-overlapping time windows, or a reference spectrum fixed a priori, with error bars and a criterion for 'no further evolution'.
  2. [Section IV, Table II and Section V] The wall-independence test is only Reference (L_ds=1600 c/ω_p) versus Short Downstream (L_ds=800 c/ω_p), with identical β_wall, injection scheme, and moving-wall boundary. Since Section IV states that reaching steady state 'requires fully established feedback between the shock front and the reflecting boundary,' and since the settling time t_s is shorter in Short Downstream (2800 vs 4000 ω_p^{-1}), the wall is dynamically involved in the relaxation. A factor-of-two change in L_ds does not rule out that the plateau is a property of the front–wall feedback loop rather than of the unbounded shock. I ask for a third downstream length (e.g., L_ds=3200 or 400 c/ω_p), or a downstream-comoving control run at comparable physical time, to demonstrate that the asymptotic values in a downstream slab are independent of wall location. Without this, the abstract's claim that the steady state depends on
  3. [Appendix A, Eqs. (A1)-(A6)] The statistical implementation of the moving wall is built on the assumption f'(p'_x)=f'(-p'_x) in the wall frame, and the reflection prescription in Eq. (A6) is designed to maintain that symmetry. The boundary condition thus actively enforces the isotropic downstream symmetry that characterizes the claimed steady state. This is not necessarily wrong—a physical downstream may be nearly isotropic—but it shifts the burden of proof: the downstream state could be co-produced by the symmetrizing mirror. I recommend an additional test with a different downstream boundary treatment (e.g., an absorbing boundary with re-injection from a prescribed downstream distribution, or a much longer downstream region) to show that ϵ_B, λ_y, and the particle spectrum do not depend on the mirror's presence.
minor comments (5)
  1. [Section IV] The sentence 'no matter whether we change the upstream length, the downstream length, the spatial resolution, or increase the width of the simulation box' claims invariance under spatial resolution changes, but Table I lists no run with a different cell size. Please either add such a run or remove the claim.
  2. [Section IV, accelerated-particle definitions] The lifetime-based definitions of accelerated particles (t_life > (L_up/c)(1+x/L_up) upstream; t_life > L_up/c + 4|x|/c downstream) are presented without a sensitivity study. Because the energy fraction in accelerated particles and the 'no power-law tail' conclusion depend on these thresholds, a brief test of threshold variations would strengthen the result.
  3. [Figure 4 caption and Section IV] Typos: 'trasversely' in the caption of Fig. 4; 'magnec energy density' in Section IV; 'the the shock's' in Section IV.
  4. [Appendix A and Figure 11] The sign conventions for β_w and β_x are stated only in passing ('both β_w and β_x are negative here'). A short explicit sign convention at the start of Appendix A would make the reflection formulas easier to check.
  5. [Section III] The relationship between w_B and ϵ_B is described as 'about 2 times greater,' but the exact frame transformation and density-jump factors are not written out. A precise definition would help readers compare with previous work.

Circularity Check

1 steps flagged

One self-referential steady-state diagnostic; wall-feedback limitation weakens external validity but is not itself a circular step.

specific steps
  1. self definitional [Section III, 'Long-term evolution and formation of the steady state' (definition of ΔFγ and its interpretation)]
    "To quantify the rate of transition to the steady state, we compute the time-dependent deviation ∆Fγ(t) = Z γ (ln fγ(t) − ln fγ,s)2dγ, where fγ(t) is the particle distribution function at time t and fγ,s is the final steady-state spectrum (end of simulation). ... The values ∆F(tk) at discrete times are then fitted with ∆Fk = C1 + C2 · exp (−tk/ts)."

    The convergence diagnostic is defined against the run's own end-of-run spectrum, which is labeled the 'final steady-state spectrum.' By construction ΔFγ(t_final)=0, so an exponential fit to ΔFγ will always describe relaxation to the run's final state, whether or not that state is physically time-independent. The reported settling time ts therefore partly encodes the definitional choice of reference spectrum, rather than independently establishing that the downstream particle spectrum has an asymptotic attractor.

full rationale

The central steady-state claim does not reduce to the self-referential ΔFγ metric alone: Figures 1, 2, 3, and 6 show overlapping time-averaged profiles of wB, λy, density contrast, and spectra, and the multi-run comparison (Reference, Short Downstream, Long Upstream, Noisy, Wide) provides independent evidence of reproducibility of the downstream state. The main external-validity threat is the moving-wall boundary: the paper states that reaching steady state 'requires fully established feedback between the shock front and the reflecting boundary in the downstream' (Section IV), and wall independence is tested only by varying Lds by a factor of two (800 vs 1600 c/ωp), so the unbounded-shock extrapolation is underdetermined. This is a limitation, not a formal circularity. The ΔFγ construction is a genuine but non-load-bearing self-definitional step; it affects the reported ts values but not the primary plateaus. No load-bearing self-citation, imported uniqueness theorem, or ansatz-by-citation was found; comparisons with [26] and [29] are external and testable.

Axiom & Free-Parameter Ledger

6 free parameters · 7 axioms · 0 invented entities

The ledger shows that the paper's quantitative claims are mostly descriptive fits to the runs themselves (power-law indices, relaxation times, lifetime exponents), while the interpretive claims rest on standard plasma-shock assumptions (Weibel mediation, 2D approx 3D) and on the boundary-condition premises of the front-comoving box — the most fragile of which (the wall co-producing the steady state) is stated in the authors' own words in Section IV. No new particles, forces, or conserved quantities are postulated; the 'soliton-like magnetic spots' are observed structures within the simulated field, whose only independent handle is the comparison to Groselj et al. 2024.

free parameters (6)
  • moving-wall speed beta_wall = 0.481 (T_up=1), 0.498 (T_up=0.1), 0.468 (T_up=5)
    Set to the downstream bulk velocity, which is not exactly the ideal jump-condition value ('small difference ... due to escaping particles and magnetic field'); each run's wall speed is a tuned input defining the downstream frame.
  • upstream magnetization growth index = -1.2 (Reference), -1 (Long Upstream)
    Power-law fit epsilon_B proportional to |x|^{-alpha} to simulation profiles (Section IV); index varies between runs and is quoted without uncertainty.
  • downstream magnetization decay index = -0.8 (all but Cold), -1 (Cold)
    Power-law fit epsilon_B proportional to |x|^{-0.8} to the downstream profile (Section IV); used in the summary as a finding.
  • magnetic-energy distribution exponent = 0.65 (full downstream)
    Fit w_B^2 f(w_B) proportional to w_B^{0.65}; compared to 0.5 expected for Lorentzian spots; the slope varies with segment length.
  • exponential lifetime exponents = -2e-4 omega_p and +0.18e-4 omega_p (Fig. 10)
    Exponential fits to dN/dt and average gamma vs particle lifetime; their ratio yields the formal f(gamma) proportional to gamma^{-11} claim.
  • spectrum relaxation time t_s = 2800-7900 omega_p^{-1} per run (Table II)
    Exponential fit Delta_F_k = C1 + C2 exp(-t_k/t_s) of the distance to the run's own final spectrum; sets the steady-state settling claim.
axioms (7)
  • domain assumption Weibel/filamentation instability alone can mediate unmagnetized relativistic shock formation
    Section I; the simulation is built on this conventional-model premise; the competing pair-balance model [5] is not simulated.
  • domain assumption 2D3V PIC results capture the essential physics of 3D relativistic pair shocks
    Section I ('qualitative agreement between 2D and 3D gives confidence'); Section V admits intrinsic uncertainty in the 2D3V non-thermal energy fraction.
  • ad hoc to paper The moving-wall frame is a fluid-comoving frame with symmetric distribution f'(p'_x) = f'(-p'_x)
    Appendix A, statistical implementation of the moving-wall BC; the reflection probability Eq. (A6) rests on this symmetry.
  • domain assumption A fixed box with steady injection and a moving reflecting wall represents an unbounded shock
    Section II and Section V; the authors concede the boundary biases the far-upstream precursor and note the steady state 'requires fully established feedback between the shock front and the reflecting boundary' (Section IV).
  • standard math The upstream distribution is Maxwell-Juttner at the injection boundary
    Appendix B sampler; standard thermal model, with no independent evidence for real astrophysical shock upstreams.
  • ad hoc to paper Accelerated particles are defined by lifetime thresholds (t_life > L_up/c (1 + x/L_up), etc.)
    Section IV; the 15% non-thermal energy fraction and anisotropy claims depend on these definitions.
  • ad hoc to paper End-of-run spectrum f_{gamma,s} is the steady-state reference
    Section III; the convergence measure Delta_F_gamma compares to the final spectrum of the same run, a self-referential steady-state definition.

pith-pipeline@v1.3.0-alltime-deepseek · 19546 in / 26522 out tokens · 277773 ms · 2026-08-02T02:43:24.686319+00:00 · methodology

0 comments
read the original abstract

We present a series of unprecedently long 2D3V PIC simulations of unmagnetized relativistic $e^{-}e^{+}$-pair shocks performed in a front-comoving frame. By implementing a moving-wall boundary condition in the downstream together with continuous injection at the upstream boundary, we maintain a fixed simulation domain size, opening the way to perform substantially longer simulations. Our longest runs extend beyond $100000\,\omega_p^{-1}$, exceeding the duration of the previously published simulations by a factor of several. Across a diverse set of simulations -- varying upstream/downstream lengths, transverse sizes, and particle-per-cell counts -- we find strong evidence that the shock approaches an asymptotic, time-independent state. In the downstream region, the steady state depends only on the upstream temperature at the injection boundary and does not depend on a particular numerical realization. The upstream precursor evolves slower and retains a dependence on the simulation's upstream length, that may be of minor observational consequence, since radiation from astrophysical shocks predominantly originates from the downstream region. We also find that Fermi-type acceleration is limited in energy and a true power-law tail never forms. Another important finding is that the downstream magnetic field has a soliton-like structure, where individual magnetic domains evolve independently, each comprising a compact, highly magnetized core embedded within an extended, weakly magnetized region. The magnetic-field distribution around the centers of these spots has approximately Lorentzian profile.

Figures

Figures reproduced from arXiv: 2607.14235 by Evgeny Derishev, Mikhail Garasev.

Figure 1
Figure 1. Figure 1: FIG. 1. Evolution of normalized magnetic energy density, [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Late-time profiles of the magnetic energy density [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Normalized out-of-plane magnetic field [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Evolution of magnetic field characteristics behind [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. The distribution of magnetic energy per logarithmic [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Comparison of the steady state for magnetic en [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: This distribution is highly anisotropic. The most [PITH_FULL_IMAGE:figures/full_fig_p011_9.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. The distribution of accelerated particles (see text for the definition) in the upstream region [PITH_FULL_IMAGE:figures/full_fig_p012_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Normalized number of particles per unit lifetime [PITH_FULL_IMAGE:figures/full_fig_p013_10.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Acceptance rate as a function of logarithm of the [PITH_FULL_IMAGE:figures/full_fig_p015_12.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

48 extracted references · 16 linked inside Pith

  1. [1]

    Note that both βw and βx are negative here

    Out of all particles leaving the simulation box, select only those whose velocity satisfies the condition βx < 2βw 1 + β2w , (A7) whereas other particles are discarded. Note that both βw and βx are negative here

  2. [2]

    For every selected particle, calculate the position at the end of the timestep according to Eq. (A4). The par- ticles that have xn+1 < 0 are discarded. Other particles are kept within the simulation box with new coordinates, and their energy and momentum is recalculated accord- ing to Eq. (A1). Another, statistical, implementation of the moving- wall boun...

  3. [3]

    (A7) and discard other particles

    Out of all particles leaving the simulation box, se- lect only those whose velocity satisfies the condition from Eq. (A7) and discard other particles

  4. [4]

    Every selected particle causes injection of the re- flected particle with probability given by Eq. (A6). En- ergy and momentum of the injected particle are given by Eq. (A1). Appendix B: Maxwell-Juttner distribution Let us introduce the particle’s dimensionless energy (Lorentz factor) γ = E/(mc2) and dimensionless mo- mentum p = P/(mc), note that p = p γ2...

  5. [5]

    Start with ϵ0 = p 1 + 11θ2 − 1 + 2θ

  6. [6]

    Calculate q = 3 1 − exp − ϵn 2θ

  7. [7]

    Calculate the next approximation ϵn+1 = ϵn + ϵn(ϵn + 2) − 2qθ(ϵn + 1) ϵn + 1 + q(2θ − ϵn − 1)

  8. [8]

    Go to step 2 if the desired precision is not achieved

  9. [9]

    Calculate A from Eq. (B6)

  10. [10]

    Using the value of ϵ3 for ϵc in Eq

    Multiply A by a factor (1 + δ) with small δ > 0 to allow for imprecise value of A. Using the value of ϵ3 for ϵc in Eq. (B6) gives rela- tive precision better than 10 −7 everywhere in the range ϵc ∈ (0, ∞). Alternatively, the normalization factor can be quickly estimated as A = 11.38 θ2 + 0.5534 θ 3/2 at the expense of lower acceptance rate. The acceptance...

  11. [11]

    Generate two independent random numbers, u and w, uniformly distributed in the range [0 , 1)

  12. [12]

    Calculate p = A u 1 − u 1/3 , which is distributed as -4 -2 0 2 4 0 .5 0 .5 5 0 .6 0 .6 5 0 .7 0 .7 5 0 .8 0 .8 5 FIG. 12. Acceptance rate as a function of logarithm of the dimensionless temperature. Solid line is for optimal value of the normalization factor A, dashed line – for approximate value of A (see text for explanations). f (p) = 3Ap2 (A + p3)2 ....

  13. [13]

    If w >exp 1 − p 1 + p2 θ ! 1 + p3 A 2 go to step 1

  14. [14]

    Return p, which is the momentum of a random particle with the Maxwell-J¨ uttner distribution. With the optimal choice of the normalization factor A, the acceptance rate varies from 0.803 in the non- relativistic limit ( θ → 0) to 0.528 in the ultrarelativistic limit (θ → ∞), reaching maximum of 0.841 at θ ≃ 0.18. It is often desirable to generate particle...

  15. [15]

    Generate the particle’s total momentum p from any isotropic (in the hydrodynamical rest frame) distribu- tion

  16. [16]

    Generate a random number u uniformly distributed in the range [0 , 1]

  17. [17]

    Calculate µ = p (1 − β0β)2 + 4uβ0β − 1 β0β , which is distributed as f (µ) = 1 + β0βµ

  18. [18]

    Generate another independent random number w uniformly distributed in the range [0 , 1)

  19. [19]

    Calculate px = p 1 − µ2 cos(2πw) p and py = p 1 − µ2 sin(2πw) p

  20. [20]

    V. A. Acciari et al. (MAGIC Collaboration), Nature (London) 575, 455 (2019)

    2019

  21. [21]

    Abdalla et al

    H. Abdalla et al. (H. E. S. S. Collaboration), Na- ture (London) 575, 464 (2019), arXiv:1911.08961 [astro- ph.HE]

    Pith/arXiv arXiv 2019

  22. [22]

    Abdalla et al.(H

    H. Abdalla et al.(H. E. S. S. Collaboration), Science372, 1081 (2021), arXiv:2106.02510 [astro-ph.HE]

    Pith/arXiv arXiv 2021

  23. [23]

    Derishev and T

    E. Derishev and T. Piran, Astrophys. J. 923, 135 (2021), arXiv:2106.12035 [astro-ph.HE]

    Pith/arXiv arXiv 2021

  24. [24]

    E. V. Derishev and T. Piran, MNRAS 460, 2036 (2016), arXiv:1512.04257 [astro-ph.HE]

    Pith/arXiv arXiv 2036

  25. [25]

    R. Z. Sagdeev, Reviews of Plasma Physics 4, 23 (1966)

    1966

  26. [26]

    E. S. Weibel, Physical Review Letters 2, 83 (1959)

    1959

  27. [27]

    M. V. Medvedev and A. Loeb, Astrophys. J. 526, 697 (1999)

    1999

  28. [28]

    Gruzinov, Astrophys

    A. Gruzinov, Astrophys. J. Lett. 563, L15 (2001), arXiv:astro-ph/0107106 [astro-ph]

    Pith/arXiv arXiv 2001

  29. [29]

    L. O. Silva, R. A. Fonseca, J. W. Tonge, J. M. Dawson, W. B. Mori, and M. V. Medvedev, Astrophys. J. Lett. 596, L121 (2003)

    2003

  30. [30]

    K. I. Nishikawa, P. Hardee, G. Richardson, R. Preece, H. Sol, and G. J. Fishman, Astrophys. J.595, 555 (2003), arXiv:astro-ph/0305091 [astro-ph]

    Pith/arXiv arXiv 2003

  31. [31]

    J. T. Frederiksen, C. B. Hededal, T. Haugbølle, and ˚A. Nordlund, Astrophys. J. Lett. 608, L13 (2004)

    2004

  32. [32]

    C. B. Hededal, T. Haugbølle, J. T. Frederiksen, and ˚A. Nordlund, Astrophys. J. Lett. 617, L107 (2004), arXiv:astro-ph/0408558 [astro-ph]

    Pith/arXiv arXiv 2004

  33. [33]

    K. I. Nishikawa, P. Hardee, G. Richardson, R. Preece, H. Sol, and G. J. Fishman, Astrophys. J.622, 927 (2005), arXiv:astro-ph/0409702 [astro-ph]

    Pith/arXiv arXiv 2005

  34. [34]

    Spitkovsky (AIP, 2005) pp

    A. Spitkovsky (AIP, 2005) pp. 345–350, arXiv:astro- ph/0603211 [astro-ph]

    arXiv 2005

  35. [35]

    T. N. Kato, Astrophys. J. 668, 974 (2007)

    2007

  36. [36]

    Spitkovsky, Astrophysical Journal Letters 673, L39 (2008)

    A. Spitkovsky, Astrophysical Journal Letters 673, L39 (2008)

    2008

  37. [37]

    Chang, A

    P. Chang, A. Spitkovsky, and J. Arons, Astrophys. J. 674, 378 (2008), arXiv:0704.3832 [astro-ph]

    Pith/arXiv arXiv 2008

  38. [38]

    Spitkovsky, Astrophysical Journal Letters 682, L5 (2008)

    A. Spitkovsky, Astrophysical Journal Letters 682, L5 (2008)

    2008

  39. [39]

    A. Bret, A. Stockem Novo, R. Narayan, C. Ruyer, M. E. Dieckmann, and L. O. Silva, Laser and Particle Beams 34, 362 (2016)

    2016

  40. [40]

    Sironi and A

    L. Sironi and A. Spitkovsky, Astrophys. J. 726, 75 (2011), arXiv:1009.0024 [astro-ph.HE]

    Pith/arXiv arXiv 2011

  41. [41]

    Sironi, U

    L. Sironi, U. Keshet, and M. Lemoine, Space Science Re- views 191, 519 (2015)

    2015

  42. [42]

    Sironi, A

    L. Sironi, A. Spitkovsky, and J. Arons, Astrophys. J.771, 54 (2013), arXiv:1301.5333 [astro-ph.HE]

    Pith/arXiv arXiv 2013

  43. [43]

    Ardaneh, D

    K. Ardaneh, D. Cai, K.-I. Nishikawa, and B. Lemb´ ege, Astrophys. J. 811, 57 (2015), arXiv:1507.05374 [astro- ph.HE]

    Pith/arXiv arXiv 2015

  44. [44]

    Keshet, B

    U. Keshet, B. Katz, A. Spitkovsky, and E. Waxman, Astrophys. J. Lett. 693, L127 (2009), arXiv:0802.3217 [astro-ph]

    Pith/arXiv arXiv 2009

  45. [45]

    Groˇ selj, L

    D. Groˇ selj, L. Sironi, and A. Spitkovsky, Astrophys. J. Lett. 963, L44 (2024), arXiv:2401.02392 [astro-ph.HE]

    Pith/arXiv arXiv 2024

  46. [46]

    T. D. Arber, K. Bennett, C. S. Brady, A. Lawrence- Douglas, M. G. Ramsay, N. J. Sircombe, P. Gillies, R. G. Evans, H. Schmitz, A. R. Bell, and C. P. Ridgers, Plasma Physics and Controlled Fusion 57, 113001 (2015)

    2015

  47. [47]

    Gruzinov, arXiv e-prints , arXiv:2501.17341 (2025), arXiv:2501.17341 [astro-ph.HE]

    A. Gruzinov, arXiv e-prints , arXiv:2501.17341 (2025), arXiv:2501.17341 [astro-ph.HE]

    Pith/arXiv arXiv 2025

  48. [48]

    Lemoine, L

    M. Lemoine, L. Gremillet, G. Pelletier, and A. Van- thieghem, Phys. Rev. Lett. 123, 035101 (2019), arXiv:1907.07595 [astro-ph.HE]

    Pith/arXiv arXiv 2019