First-principles Fermi acceleration in magnetized turbulence Martin Lemoine Institut dAstrophysique de Paris

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First-principles Fermi acceleration in magnetized turbulence

Martin Lemoine

Institut d’Astrophysique de Paris,

CNRS – Sorbonne Universit´e,

98 bis boulevard Arago, F-75014 Paris, France

(Dated: October 4, 2022)

This work provides a concrete implementation of E. Fermi’s model of particle acceleration in

magnetohydrodynamic (MHD) turbulence, connecting the rate of energization to the gradients of

the velocity of magnetic ﬁeld lines, which it characterizes within a multifractal picture of turbulence

intermittency. It then derives a transport equation in momentum space for the distribution function.

This description is shown to be substantiated by a large-scale numerical simulation of strong MHD

turbulence. The present, general framework can be used to model particle acceleration in a variety

of environments.

Introduction– Particle energization through scatterings

oﬀ inhomogeneous, random moving structures is a uni-

versal process [1,2], which has stirred considerable in-

terest in various branches of physics: primarily astro-

physics, with applications ranging from solar ﬂares [3] to

more remote phenomena involving plasmas in extreme

conditions, e.g. [4], but also statistical plasma physics [5]

or high energy density physics [6]. Remarkably, the two

papers of E. Fermi [1,2] represent the ﬁrst concrete sce-

narios for the origin of non-thermal particles in the Uni-

verse. While the literature has placed signiﬁcant empha-

sis on acceleration at shock fronts, numerical experiments

have demonstrated that stochastic acceleration can be ef-

ﬁcient [7–9], notably so at large turbulent Alfv´en veloc-

ity [10–14], in the sense that it produces extended, hard

powerlaw distributions of suprathermal particles. Be-

sides, the stochastic Fermi process assuredly plays a role

in the vicinity of shock fronts [15–17], just as it seemingly

controls part of the energization in reconnection environ-

ments [18,19].

While the overall scenario is commonly pictured as

originally formulated by E. Fermi – a sequence of discrete,

point-like interactions between a particle and inﬁnitely

massive, perfectly conducting plasma clouds – its imple-

mentation in a realistic turbulent context has remained

a challenge [20–22], to the extent that phenomenological

applications rely on a Fokker-Planck model parameter-

ized by a momentum diﬀusion coeﬃcient.

The present Letter proposes a novel approach to this

problem and formulates a transport equation to describe

the evolution of the distribution function in momentum

space. It is ﬁrst shown that particle momenta obey a

continuous-time random walk (CTRW), whose random

force scales as the gradients of the velocity of magnetic

ﬁeld lines, coarse-grained on a scale comparable to the

particle gyroradius rg≡pc/eB (pmomentum, B=|B|

with Bmagnetic ﬁeld). A key observation is that those

gradients are subject to intermittency on small length

scales. Hence, the random forces are neither Gaussian,

nor white noise in time and consequently, the random

walk deviates from Brownian motion, just as the trans-

port equation, which is derived here from known prop-

erties of CTRW, diﬀers from Fokker-Planck. This equa-

tion is characterized by the statistics of velocity gradi-

ents, which are captured via a multifractal description of

turbulence intermittency. This framework is eventually

shown to reproduce the time- and momentum-dependent

Green functions obtained by tracking a large number

of test particles in a large-scale MHD simulation. The

present formalism thus provides a successful implemen-

tation of stochastic Fermi acceleration in realistic, colli-

sionless MHD turbulence.

A (continuous-time) random walk picture– To evalu-

ate energy gains/losses in the original Fermi model, it

proves convenient to boost to the scattering center frame

where the motional electric ﬁeld Evanishes. The gen-

eralization of that model to a continuous random ﬂow

similarly tracks the particle momentum in the instan-

taneous (here, non-inertial) frame R/

Ein which Evan-

ishes [23,24], which, in ideal MHD, moves at velocity

vE=cE×B/B2. In that frame, momentum gains or

losses scale in direct proportion to the (lab frame) spatio-

temporal gradients of the velocity ﬁeld vE, as expressed

by Γacc, Γkand Γ⊥below. In detail, the momentum pof

particles with gyroradius rg`c(`ccoherence length of

the turbulence) evolves as

˙p=pΓacc + Γk+ Γ⊥,(1)

with Γacc =−v−1µb·∂tvE(vparticle velocity; µ=

p·b/p pitch-angle cosine with respect to the magnetic

ﬁeld direction b=B/B); Γk=−µ2b·(b·∇)vEand

Γ⊥=−1−µ2[∇·vE−b·(b·∇)vE]/2. For sim-

plicity, the present work focuses on the sub-relativistic

limit vEc. Equation (1) – more precisely, its rela-

tivistic limit – has been shown to account for the bulk of

acceleration in numerical simulations of collisionless tur-

bulence [25], putting the present model on solid footing.

It generalizes the two contributions originally identiﬁed

by E. Fermi: interactions with moving magnetic mirrors

are captured by Γ⊥, while orbits in dynamic, curved mag-

netic ﬁeld lines are represented by Γk; the remaining term

Γacc describes the eﬀective gravity force associated to ac-

celeration/deceleration of the ﬁeld lines; of second order

in vE/c, it is subdominant in the sub-relativistic limit,

unless vvA.

arXiv:2210.01038v1 [astro-ph.HE] 3 Oct 2022

All quantities in Eq. (1) are understood to be coarse-

grained on wavenumber scale kg∼r−1

g(length scales lg∼

2π rg), where rg=pc/eB denotes the particle gyroradius

in the lab frame. This procedure ﬁlters out wavenumbers

k > kg, whose contribution averages out over a gyro-

orbit, to retain the larger scales that shape the velocity

structures responsible for acceleration (in accord with the

original Fermi picture).

Henceforth, Eq. (1) is simpliﬁed to the symbolic ˙p=

pΓlg, Γlgrepresenting an aggregate (random) force ex-

erted by electromagnetic ﬁelds, coarse-grained on scale

lg; order of unity factors related to µare thus omitted;

we also consider relativistic particles (v∼c) to ease the

discussion. For technical details concerning the model-

ing of this random process, see [Supp. Mat. A]. Sepa-

rate now ﬂuctuations from the mean, Γl=hΓli+δΓl,

the average carrying over the statistical realizations of

the turbulent ﬂow: hΓlicharacterizes systematic heat-

ing, while the random δΓlrepresents the diﬀusive part.

If δΓlwere Gaussian distributed, and its time correlation

function that of white noise, the process would describe

Brownian motion, in one-to-one correspondence with a

Fokker-Planck equation for the distribution function [26].

As anticipated above, however, those random forces are

neither Gaussian in amplitude, nor white noise in time:

at small scales, they develop large powerlaw tails as a re-

sult of intermittency, while at large scales, the coherence

time of the random force &lg/c cannot be regarded as

inﬁnitesimal.

To obtain the transport equation, we ﬁrst observe that

the process ˙p=pΓlgcan be described as a CTRW: unlike

discretized Brownian motion, which operates at a ﬁxed

and uniform time step, the random walk is here deﬁned

by the joint probability φ(p|p0;t−t0) to jump from p0to

pin time ∆t=t−t0, with both ∆p=p−p0and ∆t

regarded as random variables. Expectations are ∆t∼

lg(p0)/c – thus, a function of p0– and ∆ ln p∼Γlg∆t.

We will assume ∆tto be exponentially distributed with

mean waiting time tp≡lg(p0)/c and ∆ ln pdistributed as

Γlglg/c [lg=lg(p0)], see [Supp. Mat. A] for methodology.

The random walk is then entirely deﬁned by the statistics

of the velocity gradients Γlg.

Statistics of momentum jumps– In turbulence theories,

such statistics are conveniently described within a mul-

tifractal analysis [27–29], which ascribes to each position

xa local scaling exponent h(x) for gradients on coarse-

graining scale l,viz.

Γl(x)∼Γ`c(x) (l/`c)h(x),(2)

and which describes the set of locations xwith index

h(x) as a fractal of dimension d(h). The statistics of Γl

are thus entirely captured by the probability distribution

function (p.d.f.) pΓ`cof Γ`cand by the spectrum d(h),

since the probability of being at xin a set with exponent

hon scale levolves as the volume ﬁlling fraction lD−d(h)

(Dnumber of spatial dimensions). The gradient Γ`c(x)

on the coherence scale `cis naturally modeled as a Gaus-

sian variable with standard deviation σc∼vA/`c, where

vAdenotes the Alfv´en velocity of the turbulent compo-

nent. The spectrum d(h) can take diﬀerent forms, the

simplest being log-normal [30], modern descriptions of

the statistics of Els¨asser ﬁelds in MHD turbulence rather

relying on log-Poisson models [31–34]. We use the former

log-normal form, as it provides a simple and satisfactory

description of the statistics of the gradients of vE; see

[Supp. Mat. B], which includes Refs. [35–38]. We thus

derive the p.d.f. pΓlof Γlas (using D= 3) [29]

pΓl∼ZdΓ`cpΓ`cZdh l3−d(h)δhΓl−Γ`c(l/`c)hi.

(3)

This p.d.f. remains to be properly normalized. In this

formulation, the gradient statistics pΓlon all scales re-

duce to a function of the main quantities vA,`cand the

few parameters characterizing d(h), which themselves de-

pend on the physical properties of the turbulence. This

oﬀers a ﬁrst-principles connection between the funda-

mental statistics of turbulence intermittency and the

physics of particle energization.

The transport equation– The CTRW is exactly equivalent

to the following kinetic equation for the volume averaged

distribution np(t)=4πp2f(p, t), where f(p, t) represents

the angle-averaged distribution function [39,40]:

∂tnp(t) = Z+∞

dp0Zt

dt0ψ(p|p0;t−t0)np0(t0)

−ψ(p0|p;t−t0)np(t0).

(4)

The kernel ψ(p|p0;t−t0) diﬀers from the CTRW jump

distribution probability φ(p|p0;t−t0) introduced earlier,

yet the two are related as follows. Denoting with a tilde

symbol the Laplace transform in time, and νthe Laplace

variable conjugate to t−t0,

ψ(p|p0;ν) = ν˜

φ(p|p0;ν)

1−˜

φp0(ν),(5)

with the short-hand notation ˜

φp0(ν)≡

R+∞

0dp˜

φ(p|p0;ν), the subscript p0emphasizing the

dependence on p0. As discussed above, we characterize

the CTRW with a joint probability distribution of the

form:

φ(p|p0;t−t0) = ϕ(p|p0)e−(t−t0)/tp0

tp0

,(6)

recalling that tp0≡lg(p0)/c. Then,

ψ(p|p0;ν) = ϕ(p|p0)

tp0

,(7)

in which case the transport equation becomes local in

time [40],

∂tnp(t) = Z+∞

dp0ϕ(p|p0)

tp0

np0(t)−ϕ(p0|p)

np(t).

(8)

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First-principlesFermiaccelerationinmagnetizedturbulenceMartinLemoineInstitutd'AstrophysiquedeParis,CNRS{SorbonneUniversite,98bisboulevardArago,F-75014Paris,France(Dated:October4,2022)ThisworkprovidesaconcreteimplementationofE.Fermi'smodelofparticleaccelerationinmagnetohydrodynamic(MHD)turbulence,co...

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First-principles Fermi acceleration in magnetized turbulence Martin Lemoine Institut dAstrophysique de Paris

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