Kinetic turbulence in shining pair plasma: intermittent beaming and thermalization by radiative cooling

Dmitri A. Uzdensky; Gregory R. Werner; Mitchell C. Begelman; Vladimir Zhdankin

REVIEW 3 major objections 5 minor 1 cited by

Kinetic turbulence in shining pair plasma: intermittent beaming and thermalization by radiative cooling

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Strong inverse Compton cooling pins a turbulent pair plasma to a quasi-thermal state and concentrates its nonthermal particles into intermittent beams.

desk verdict The thermalization result is solid and directly measured; the beaming result is a suggestive but unresolved extrapolation, and the Fokker-Planck fit is basically a consistency check, not a validation. read the letter →

arxiv 1908.08032 v2 pith:3A2GRY6F submitted 2019-08-21 astro-ph.HE physics.plasm-ph

Vladimir Zhdankin , Dmitri A. Uzdensky , Gregory R. Werner , Mitchell C. Begelman This is my paper · ORCID

classification astro-ph.HEphysics.plasm-ph

keywords relativisticplasmaturbulenceradiativecoolinginverseComptonscatteringMaxwell-Jüttnerdistributionparticleaccelerationmagneticreconnectionkineticbeamingparticle-in-cellsimulation

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper asks what a plasma does when radiative cooling is strong enough to balance the energy pumped in by turbulence. Using particle-in-cell simulations of a relativistic electron-positron plasma with external inverse Compton drag, it tries to establish that the statistical steady state is quasi-thermal: the global particle energy distribution hugs a Maxwell-Jüttner shape, and the nonthermal tail extends only about a factor of two in energy, across magnetizations from $\sigma\approx 0.04$ to $\sigma\approx 11$. In that picture, strong radiative cooling quenches the power-law particle acceleration that appears in the same turbulence without cooling. The part with the clearest astrophysical payoff is that at high magnetization the remaining nonthermal particles form intermittent narrow beams in the global momentum distribution, aligned with current sheets, which could show up as rapid flares from blazar jets and other compact sources.

What carries the argument

The load-bearing mechanism is the inverse Compton radiation backreaction force, $\mathbf{F}_{\rm IC} = -(4/3)\sigma_T U_{\rm ph}\gamma^2\,\mathbf{v}/c$, added to each particle's Lorentz force; because it scales as $\gamma^2$, it selectively drains the high-energy tail and fixes a statistical steady state whose mean temperature follows from balancing injected power with radiated power. The interpretation then rides on two analytic constructions: a Maxwell-Jüttner reference distribution with a thermal/nonthermal decomposition, and a steady-state Fokker-Planck equation with advection and diffusion coefficients chosen linear and quadratic in momentum, whose closed-form solution is fitted to the simulation spectra. The momentum-anisotropy diagnostic $f(\theta,\phi|\gamma)$ converts the same particle data into maps whose tails quantify intermittent beams and their current-sheet correlation.

What would settle it

Run the $\sigma\approx 3.4$ radiative turbulence at $L/2\pi\rho_e\approx 80$–$100$ and measure the PDF of the global momentum anisotropy for $\gamma>1600$: if the beam tail narrows toward the low-magnetization case instead of preserving the roughly 40–100 times average enhancements seen at $L/2\pi\rho_e\approx 39$, the kinetic-beam explanation of rapid astrophysical flares is contradicted.

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Extended reading notes

Core claim

Across the parameter scan of driven relativistic pair-plasma turbulence with external inverse Compton cooling, the time-averaged particle energy distributions are quasi-thermal (Maxwell-Jüttner) in every simulation, with a nonthermal population that extends the tail by a factor of about two in energy rather than forming a power law; the nonthermal energy fraction grows with magnetization, reaching about 50% of the energy and 30% of the particles at $\sigma\approx 11$. The near-thermal shape is reproduced by a steady-state Fokker-Planck solution in which stochastic (second-order Fermi or gyroresonant) diffusion is balanced by radiation reaction. At $\sigma\gtrsim 3$, the nonthermal particles appear as highly variable, narrow beams in the global momentum anisotropy, most often perpendicular to the mean field, with enhancements of roughly 40 to 100 times the average and lifetimes near the eddy turnover time; the beam particles are spatially concentrated at current sheets and gain energy mostly from the perpendicular electric field, consistent with localized magnetic reconnection. The conclusion drawn is that strong radiative cooling quenches nonthermal acceleration in continuously driven turbulence, so broad power laws seen in astrophysical spectra may require weak cooling or a superposition of quasi-thermal regions, while swift flares could still come from beams.

Load-bearing premise

The flare connection assumes that the intermittent beams seen in the 256-cubed and 512-cubed boxes survive at astrophysically large system sizes; the paper's own size scan (Section 4.6, Fig. 26) shows the high-energy anisotropy PDF narrowing by roughly a factor of two when $L/2\pi\rho_e$ is doubled, so extrapolation to real jet scales is untested.

Editorial extensions

If this is right

A continuously driven, strongly cooled turbulent pair plasma cannot sustain the extended nonthermal power-law tails that non-radiative turbulence produces; observed blazar broadband spectra then require either weaker cooling, transient injection followed by cooling, or a superposition of quasi-thermal zones with a range of temperatures.
At high magnetization, high-energy particles are beamed into narrow solid angles on timescales of about $0.1\,L/v_A$, so a single beam sweeping across an observer's line of sight can produce a flare orders of magnitude above the average emission without requiring the whole source to vary.
Because the beam particles sit at current sheets and are energized mainly by the perpendicular electric field, the paper identifies localized magnetic reconnection as the kinetic beaming mechanism in the high-magnetization regime.
The nonthermal fraction rises with magnetization, from a few percent at $\sigma\approx 0.04$ toward roughly half the energy at $\sigma\approx 11$, so even a thermalized, radiatively cooled plasma can emit conspicuous nonthermal radiation if its magnetization is high.
The steady-state temperature is set by the external photon energy density, so observations that constrain the temperature and photon bath can be used to infer the turbulent injection efficiency.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The system-size trend in Section 4.6, where the beams weaken when the box grows, suggests the flare interpretation may depend on whether astrophysical emission is dominated by the global momentum distribution or by localized sub-regions; if only the largest coherent reconnection layers contribute, global anisotropy might fade while localized flares remain, a distinction the present diagnostics can
Directly measuring the Fokker-Planck advection and diffusion coefficients from tracked particles, rather than hand-chosen fits, would turn the well-fit claim into a quantitative test, since the paper notes the fit parameters carry considerable freedom.
External inverse Compton cooling is spatially uniform, whereas synchrotron cooling is strongest where the magnetic field is strong, so one can predict that synchrotron radiative turbulence would localize cooling in current sheets even more sharply and perhaps increase the beam-to-average contrast; that is a testable extension for future simulations.
If each beam is tied to a reconnection site, the relation between beam direction and the local guide-field orientation predicts flare polarization swings that could be searched for in time-resolved blazar polarimetry.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Read this one. It is the first PIC study of driven turbulence in relativistic pair plasma with radiation backreaction, and the central finding is likely to survive scrutiny: strong external IC cooling makes the steady-state particle distributions quasi-thermal, with only a modest nonthermal tail, across sigma from 0.04 to 11. That overturns the default expectation from non-radiative simulations of broad power-law tails, and it is based on direct measurement of f(gamma), with system-size and resolution checks. The energetics and the equilibrium temperature argument (Eq. 7) are reasonable; eta_inj = 1.4 is calibrated to these runs, but it is close to the value measured independently from tracked particles, so it is not circular. Credit also for the energy-transfer spectrum and tracked-particle diagnostics.

The soft spots are real but not fatal. The Fokker-Planck fits (Table 2, Fig. 17) have three free parameters chosen by hand, with the authors openly calling it a proof of concept. So the 'well fit' claim is a consistency check, not an independent validation of stochastic acceleration theory. That does not touch the main thermalization result, which is just the shape of f(gamma). The bigger issue is the beaming. The high-sigma momentum-anisotropy beams are striking, and the correlation with current sheets is worth reporting. But the paper's own size scan shows the anisotropy PDF narrows by a factor of two from L/2pi rho_e = 18 to 39 at sigma = 3.4, and the conclusion concedes beaming weakens with system size. There is no high-sigma run at L/2pi rho_e > 60; the one large run is sigma ~ 0.9. So the flare implications rest on an untested asymptotic limit. The authors say so explicitly, which is honest, but it means the beaming claims are an extrapolation, not an established result.

Who gets value: anyone working on particle acceleration in relativistic turbulence, reconnection, blazar flares, or GRB radiative models. It deserves a serious referee—the thermalization result alone earns that. I would not desk-reject it; I would send it out with a request that the beaming section be rewritten to state plainly that the size dependence is unresolved, and ideally add a 768^3 high-sigma run or a scaling argument. The FP section could also be trimmed or moved to an appendix, since it is not load-bearing. Net: a solid paper with two specific weaknesses, both self-disclosed.

Referee Report

3 major / 5 minor

Summary. The paper presents particle-in-cell simulations of externally driven relativistic pair-plasma turbulence with external inverse Compton cooling. It reports that, in the statistically steady state, the global particle energy distributions are quasi-thermal (Maxwell-Jüttner-like) across a magnetization scan 0.04 ≲ σ ≲ 11, with only a modest nonthermal tail that broadens the distribution by roughly a factor of two in energy. The authors interpret this as quenching of nonthermal acceleration by radiative cooling and show that the distributions can be fit by analytic Fokker-Planck solutions with radiative reaction. At high magnetization, the high-energy particles appear as intermittent, narrow beams in the global momentum anisotropy that are spatially correlated with current sheets, a phenomenon the authors suggest may explain rapid astrophysical flares. The paper also characterizes turbulence spectra, energetics, pressure anisotropy, and particle energization channels via tracked particles.

Significance. If the results hold, this is a valuable first self-consistent kinetic study of radiatively cooled driven turbulence in pair plasmas. The central thermalization claim is directly measured and robust across the magnetization scan and system-size checks, and it contrasts sharply with the extended power-law tails seen in non-radiative simulations. The paper also provides useful diagnostics of turbulence energetics and tracks particles to identify energization by perpendicular electric fields. The main qualifications are that the intermittent-beaming result is established only at moderate system sizes and weakens with size, and that the Fokker-Planck fits are explicitly provisional, so the headline flare implications and the stochastic-acceleration interpretation are weaker than the thermalization result. These concerns do not undermine the directly measured quasi-thermal distributions, which are the paper's strongest contribution.

major comments (3)

[§4.6, Fig. 26, §6(v)] The beaming claim, which is central to the abstract and to the flare implications in §5.2, depends on an untested asymptotic limit. The size scan at ⟨σ⟩=3.4 in Fig. 26 shows that the high-energy momentum-anisotropy PDF narrows substantially when L/2πρe increases from 18.0 to 39.1, and §6(v) concedes that "the beaming becomes weaker as the system size is increased... raising questions about beaming statistics in the limit of large system size." Since no high-magnetization run at L/2πρe ≳ 60 is presented (the 768³ run is at ⟨σ⟩≈0.9), the paper does not establish that intermittent beaming persists at scales relevant to astrophysical flares. I ask for either an additional high-σ run at larger system size or a substantial reframing of the flare conclusions as scale-dependent and not yet demonstrated.
[§4.2, Table 2, Appendix A] The claim that the quasi-thermal distributions are "well fit by analytic models" of stochastic particle acceleration is not an independent test of that model. In Eq. (A8), the solution is fit with Γ0=Γ2 and γ0=300 imposed, Γ2 is set by Eq. (A11), and Γh and Γa are chosen by hand, with the paper itself admitting "considerable amount of freedom" (§4.2). Thus the agreement demonstrates only that the functional form is flexible enough to accommodate the measured distributions, not that the physical transport coefficients take the assumed forms. Since conclusion (iv) and the abstract present this as supporting the stochastic-acceleration interpretation, the authors should either measure the Fokker-Planck coefficients directly from tracked-particle data (extending the analysis in §4.8) or explicitly downgrade this point to a proof of concept in the abstract, §4.2, and §6.
[§3.2, Fig. 6, Eq. (7)] The comparison between the measured mean particle energy and the equilibrium temperature predicted by Eq. (7) is weakened because the coefficient η_inj=1.4 is chosen to make the prediction agree with the measured values. This does not affect the directly observed quasi-thermal distributions, but it means that Eq. (7) should be described as a calibrated estimate rather than a parameter-free prediction. The text currently states that "Eq. 7 provides an accurate prediction for the steady-state temperature," which overstates the test performed.

minor comments (5)

[General/typographical] There are several typographical and wording errors, including "resemblence" (§3.1), "consistute" in the caption of Table 1, and the phrase "kinetic turbulence" in the abstract.
[§4.7 and Fig. 27] The spatial correlation between high-energy particle density and current sheets is asserted on the basis of visual overlays; a quantitative correlation or conditional-average statistic would make the claim more convincing.
[Figs. 22 and 26] The anisotropy PDF comparisons in Fig. 22 are not controlled for system size, since L/2πρe differs among the σ cases; the authors should note this confounding or show that the qualitative trend persists when size is held fixed.
[§3.5, Fig. 11] The negative energy transfer at high wavenumbers is acknowledged as likely numerical; the authors should state more explicitly that this implies the kinetic-range dissipation signatures in Fig. 11 are only tentative.
[§5.1] The blazar spectral model uses η_inj∼0.1 in Eq. (17), while the simulations measure η_inj≈1.4–2.1; since this is an order-of-magnitude discrepancy, the estimate should be presented as illustrative only.

Circularity Check

2 steps flagged · score 2.0 of 10

Core results are direct PIC measurements; only minor fitted-parameter checks are dressed as predictions, and the Fokker-Planck fit is a fit by construction.

fitted input called prediction [Section 3.2 (Equilibrium temperature), around Fig. 6 and Eq. 7]
"We compare ⟨γ⟩ to the equilibrium values predicted from the initial parameters, γss,0, and predicted from the time-averaged parameters, γss, both calculated from Eq. 7 with ηinj = 1.4 chosen to get good agreement between the simulation measurement and the predictions. This value of ηinj is close to the injection efficiency measured in our previous non-radiative simulations (Zhdankin et al. 2018a). Hence, we conclude that Eq. 7 provides an accurate prediction for the steady-state temperature."

The coefficient ηinj is adjusted (ηinj = 1.4) specifically to make Eq. 7 agree with the measured steady-state temperatures, and the same measured data are then used to conclude that Eq. 7 'provides an accurate prediction.' The agreement is therefore partly enforced by the fitted efficiency rather than being an independent prediction. This is a consistency check, not a derivation of the thermalization result, and the quasi-thermal distributions themselves are directly measured, so the circularity is minor and non-load-bearing.
fitted input called prediction [Section 4.2 and Appendix A, Eq. A8, Table 2, Eq. A11]
"After these restrictions, the resulting steady-state solution (Eq. A8) that we fit to has three free parameters (Γ2, Γa, and Γh) which may vary with ⟨σ⟩ and, in principle, L/2π⟨ρe⟩. ... Given the admittedly considerable amount of freedom in fitting to the Fokker-Planck solution and interpreting the parameters, we limit this work to a proof of concept."

The Fokker-Planck solution (Eq. A8) is matched to the measured f(γ) using three free parameters (plus γ0) chosen by hand, so the statement that the distributions are 'well fit' by the model is partly a statement about the flexibility of the fit rather than a validated prediction. The claimed scaling of the diffusion coefficient with second-order Fermi acceleration is also partly imposed, since Γ2 ∝ c/vA is assumed before fitting (Eq. A11). The paper is transparent about the freedom, and the central thermalization claim rests on the measured distributions rather than on the fit, so this is a partial, non-load-bearing circularity.

full rationale

The paper's headline results—quasi-thermal steady-state particle distributions, quenching of the nonthermal power-law tail, and intermittent high-energy beams at high magnetization—are directly measured outputs of PIC simulations, not derived quantities. The comparisons to non-radiative simulations are legitimate baseline comparisons to the authors' own prior work, and the beaming system-size caveat is explicitly acknowledged rather than hidden. The only steps that reduce to their own inputs are (1) the equilibrium-temperature check, where ηinj = 1.4 is chosen to force agreement and then called a prediction, and (2) the Fokker-Planck fits, where three free parameters are adjusted to match the measured distributions and the fit is then presented as support for the stochastic-acceleration interpretation. Both are transparently flagged in the paper and neither is load-bearing for the central measured conclusions. Therefore the overall circularity is low: score 2.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The paper rests on standard PIC methodology, the Thomson-limit IC drag formula, and an assumed Fokker-Planck structure for interpreting the distributions. The main measured result (quasi-thermal distributions) does not depend on the FP ansatz, but the 'well fit' claim and the physical interpretation do. The free parameters are the injection efficiency η_inj (fitted to match the predicted and measured temperature) and the three FP fit parameters Γ2, Γa, Γh (chosen by hand per simulation). No new entities are introduced.

free parameters (6)

η_inj (injection efficiency) = 1.4 (§3.2); 2.1 (§4.8 for σ=3.4)
Chosen to make the analytic steady-state temperature prediction (Eq. 7) agree with the measured mean particle energy; it is a global factor multiplying the assumed energy injection rate, not independently measured.
Γ2 (= Γ0) Fokker-Planck diffusion rate = 1.8-3.0 (Table 2)
Fitted so that the steady-state Fokker-Planck solution (Eq. A8) matches the measured energy distributions; constrained to scale as c/v_A (Eq. A11) but the normalization is free.
Γa (first-order acceleration/advection rate) = -20.0 to -3.5 (Table 2)
Hand-chosen per simulation to fit the distribution; the paper admits the parameters are not independent and the fit has considerable freedom.
Γh (energy-independent advection/heating rate) = -5.0 to -1.0 (Table 2)
Hand-chosen per simulation to fit the distribution; part of the underdetermined Fokker-Planck fit.
γ0 (reference energy for FP fit) = 300 (γ_ss = 225, Θ = 75)
Set to approximately the mean particle energy rather than fitted; acts as the fourth FP scale parameter the paper acknowledges.
Θ_ss (target steady-state temperature) = 75 (γ = 225)
Chosen for all runs to set the cooling strength; the thermalization result is demonstrated in this strong-cooling regime, and weaker cooling would change the outcome.

assumptions (5)

domain assumption The IC radiation backreaction on each particle is given by the Landau-Lifshitz drag F_IC = -(4/3)σ_T U_ph γ² v/c (Eq. 2), valid in the Thomson regime for ultra-relativistic particles and an isotropic photon field.
Invoked at the start of §2.1 to define the cooling physics; if Klein-Nishina corrections or anisotropic photon fields matter, the steady state and distribution shapes would change.
domain assumption The turbulence can be represented in a periodic cube with an external driving current J_ext that maintains a statistical steady state; the injected energy is assumed to cascade and dissipate on the Alfvén crossing time (Eq. 4).
This is the standard driving setup from the authors' prior work (Zhdankin et al. 2018a) and underlies the equilibrium estimate; it assumes the driving couples with efficiency η_inj.
domain assumption The particle kinetics are described by an isotropic Fokker-Planck equation with advection linear in momentum and diffusion quadratic in momentum (Eq. A4/A7).
The paper notes the validity and form of the Fokker-Planck equation for turbulence is not established (citing Isliker et al. 2017); this is an ansatz for interpreting the distributions.
domain assumption The plasma is optically thin, so radiated IC photons leave the domain and do not reheat the plasma.
Stated in the abstract and §2; required for the statistical steady state, but self-consistent radiation transport (including absorption and re-emission) is not modeled.
standard math Standard Maxwell-Jüttner statistics and the ultra-relativistic relation Θ = γ/3 for the thermal population.
Used throughout to relate temperature to mean energy and to define the thermal fit component.

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Pith. "Pith review of Kinetic turbulence in shining pair plasma: intermittent beaming and thermalization by radiative cooling." pith.science (2026). https://pith.science/paper/3A2GRY6F

@misc{pith2026190808032,
  author       = {Pith},
  title        = {Pith review of: Kinetic turbulence in shining pair plasma: intermittent beaming and thermalization by radiative cooling},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3A2GRY6F}},
  note         = {Machine review of arXiv:1908.08032}
}

abstract

High-energy astrophysical systems frequently contain collisionless relativistic plasmas that are heated by turbulent cascades and cooled by emission of radiation. Understanding the nature of this radiative turbulence is a frontier of extreme plasma astrophysics. In this paper, we use particle-in-cell simulations to study the effects of external inverse Compton radiation on turbulence driven in an optically thin, relativistic pair plasma. We focus on the statistical steady state (where injected energy is balanced by radiated energy) and perform a parameter scan spanning from low magnetization to high magnetization ($0.04 \lesssim \sigma \lesssim 11$). We demonstrate that the global particle energy distributions are quasi-thermal in all simulations, with only a modest population of nonthermal energetic particles (extending the tail by a factor of $\sim 2$). This indicates that nonthermal particle acceleration (observed in similar non-radiative simulations) is quenched by strong radiative cooling. The quasi-thermal energy distributions are well fit by analytic models in which stochastic particle acceleration (due to, e.g., second-order Fermi mechanism or gyroresonant interactions) is balanced by the radiation reaction force. Despite the efficient thermalization of the plasma, nonthermal energetic particles do make a conspicuous appearance in the anisotropy of the global momentum distribution as highly variable, intermittent beams (for high magnetization cases). The beamed high-energy particles are spatially coincident with intermittent current sheets, suggesting that localized magnetic reconnection may be a mechanism for kinetic beaming. This beaming phenomenon may explain rapid flares observed in various astrophysical systems (such as blazar jets, the Crab nebula, and Sagittarius A*).

Figures

Figures reproduced from arXiv: 1908.08032 by the authors.

**Figure 1.** Surface plot of the emissivity proxy nγ2 avg for the 7683 , hσi = 0.9 simulation. value, also seen in our previous non-radiative simulations (Zhdankin et al. 2018a) and in non-relativistic kinetic turbulence (e.g., Roytershteyn et al. 2015). These structures are correlated with high densities, consistent with local pressure equilibrium. Although they bear some resemblence to plasmoids resulting from the tearing in… view at source ↗

**Figure 2.** Current density component along the mean field, Jz (normalized to rms value Jz,rms), for hσi = 0.2 (top) and hσi = 3.4 (bottom) in the 5123 simulations. tuning Uph to get fixed Θss), hγi does not vary significantly with hσi, showing only a slight increase with hσi. We compare hγi to the equilibrium values predicted from the initial parameters, γss,0, and predicted from the time-averaged parameters, γss, both calcu… view at source ↗

**Figure 5.** [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗

Figures from the paper (22 more)

**Figure 6.** Figure 6: Top panel: Evolution of the mean particle energy γ for a set of 2563 simulations with hσi ≈ 1 and different initial temperatures, Θ0 ∈ {Θss/4, Θss, 4Θss}, demonstrating that all these cases are approach the specified steady state. Bottom panel: The time-averaged mean …

**Figure 7.** Figure 7: Evolution of various energies at hσi = 0.2 (top), hσi = 0.8 (center), and at hσi = 11.0 (bottom), taken from 3843 simulations. Turbulent magnetic energy (red), electric energy (blue), internal energy (magenta), and bulk kinetic energy (green) are shown, all normalized…

**Figure 8.** Figure 8: Partitioning of the overall energy into time-averaged internal energy (magenta), turbulent magnetic energy (red), bulk kinetic energy (green), and electric energy (blue), versus magnetization hσi. expected for subsonic MHD turbulence; this becomes drastically shallow…

**Figure 9.** Figure 9: Power spectra for magnetic fluctuations Emag(k⊥) (top), electric fluctuations Eelec(k⊥) (center), and density fluctuations En(k⊥) (bottom), compensated by k 5/3 ⊥ , for varying system size and fixed magnetization hσi ∼ 1. Inertial-range fits (prior to compensation) o…

**Figure 11.** Figure 11: Top panel: The energy transfer spectrum D(k) in a 3843 simulation with hσi ∼ 1, with positive values (red) and negative values (blue) both shown, and a k−1 power-law fit (green, dashed). Bottom panel: The normalized energy transfer rate, ΓD(k) = (L/vA)D(k)/EEM(k), whe…

**Figure 10.** Figure 10: Power spectra for magnetic fluctuations Emag(k⊥) (top), electric fluctuations Eelec(k⊥) (center), and density fluctuations En(k⊥) (bottom), compensated by k 5/3 ⊥ , for fixed lattice size (5123 ) and varying magnetization hσi ∈ {0.2, 0.9, 3.4}. Fits from [PITH_FULL_…

**Figure 12.** Figure 12: Top panel: probability distribution for the magnetic field magnitude B for the 3843 simulations with hσi ∈ {0.04, 0.8, 11}. Bottom panel: probability distribution for the fluctuating magnetic field components δBz (dashed) and Bx (solid) for the same simulations. pan…

**Figure 14.** Figure 14: 2D probability distribution of pressure anisotropy ratio P⊥/P|| versus β|| (for the 3843 series). Contours are shown at five values for each case: {1/2, 1/4, 1/8, 1/16, 1/32} of the maximum value of the hσi = 0.04 distribution. Thresholds for the firehose instabilit…

**Figure 15.** Figure 15: Top panel: Time-averaged particle energy distributions hfi(γ) for varying hσi (for the 3843 series). For reference, a Maxwell-Juttner distribution with the same mean energy is also ¨ shown (black, dashed). Bottom panel: similar distributions for varying system size (…

**Figure 16.** Figure 16: Top panel: Time-averaged thermal (dashed) and nonthermal (solid) components of the particle energy distribution f(γ), for the 3843 simulation series. Center panel: Nonthermal particle energy fraction Enth/Etot (blue) and number fraction Nnth/Ntot (red) versus hσi for …

**Figure 18.** Figure 18 [PITH_FULL_IMAGE:figures/full_fig_p015_18.png]

**Figure 17.** Figure 17: Top panel: Steady-state particle energy distribution f(γ) for the 5123 , hσi = 0.2 case (blue) with fit from the FokkerPlanck model (black, dashed). Center panel: similar for the 7683 , hσi = 0.9 case. Bottom panel: similar for the 5123 , σ = 3.4 case. we revisit in …

**Figure 20.** Figure 20: Momentum anisotropy distributions for low-energy particles, f(θ, φ|γ < 800), at nine different times for the 5123 , hσi = 3.4 case [PITH_FULL_IMAGE:figures/full_fig_p016_20.png]

**Figure 21.** Figure 21: Momentum anisotropy distributions for high-energy particles, f(θ, φ|γ > 1600), at nine different times for the 5123 , hσi = 3.4 case. lar structure, organized in an ensemble of narrow beams. These beams have random orientations, but are most commonly directed perpend…

**Figure 22.** Figure 22: The probability distribution function for the momentum anisotropy distribution f(θ, φ|γ) bin values, for low-energy particles (γ < 800, dashed) and high-energy particles (γ > 1600, solid), in 3843 simulations with σ ∈ {0.04, 0.8, 11} [PITH_FULL_IMAGE:figures/full_fi…

**Figure 23.** Figure 23: Momentum anisotropy distributions averaged over the azimuthal angle φ for low-energy particles, f(θ|γ < 800) (dashed), and for high-energy particles, f(θ|γ > 1600) (solid). Simulations with varying hσi from the 3843 series are shown in different colors. the low magnet…

**Figure 24.** Figure 24: Momentum anisotropy distributions for high-energy particles, f(θ, φ + π|γ > 1600), over a duration of 0.24L/vA covering the fiducial beaming event at θ = 122◦ and φ = 174◦, from the 5123 , hσi = 3.4 case. Note that the coordinate system has been shifted by 180◦ in φ f…

**Figure 26.** Figure 26: The probability distribution function for the momentum anisotropy distribution f(θ, φ|γ) bin values, for low-energy particles (γ < 800, blue) and high-energy particles (γ > 1600, red), in hσi = 3.4 simulations with two different system sizes, L/2πhρei = 18.0 (2563 ; …

**Figure 25.** Figure 25: The evolution of the momentum anisotropy distribution f(θ, φ|γ) versus time in two different energy bands (γ < 800, blue; γ > 1600, red), for the 5123 , hσi = 3.4 case, taken in two different directions. Top panel: θ = 122◦ and φ = 174◦, coinciding with the beam from…

**Figure 27.** Figure 27: The cell-averaged particle energy γavg (top), heating rate proxy E·J (center), and current density J (bottom), all with overlaid contours of the high-energy particle number density, nhe (with green contours at 4 and 8 times nhe). All quantities are averaged in the z d…

**Figure 29.** Figure 29: The different contributions to the particle energy gain ∆γ, similar to [PITH_FULL_IMAGE:figures/full_fig_p020_29.png]

**Figure 30.** Figure 30: The fraction of overall particle energy gain from perpendicular (rather than parallel) electric fields versus hσi, for the 3843 simulation series. MNRAS 000, 1–26 (2020) [PITH_FULL_IMAGE:figures/full_fig_p020_30.png]

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astro-ph.HE 2019-09 conditional novelty 6.0 of 10

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