A Collisional-Energy-Cascade Model for Nonthermal Velocity Distributions of Neutral Atoms in Plasmas Keisuke Fujii1

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A Collisional-Energy-Cascade Model

for Nonthermal Velocity Distributions of Neutral Atoms in Plasmas

Keisuke Fujii1, ∗

1Oak Ridge National Laboratory, Oak Ridge, TN 37831-6169, United States of America

(Dated: October 14, 2022)

Nonthermal velocity distributions with much greater tails than the Maxwellian have been observed

for radical atoms in plasmas for a long time. Historically, such velocity distributions have been

modeled by a two-temperature Maxwell distribution. In this paper, I propose a model based on

collisional energy cascade, which has been studied in the ﬁeld of granular materials. In the collisional

energy cascade, a particle ensemble undergoes energy input at the high-energy region, entropy

production by elastic collisions among particles, and energy dissipation. For radical atoms, energy

input may be caused by the Franck-Condon energy of molecular dissociation or charge-exchange

collision with hot ions, and the input energy is eventually dissipated by collisions with the walls. I

show that the steady-state velocity distribution in the collisional energy cascade is approximated

by the generalized Mittag-Leﬄer distribution, which is a one-parameter extension of the Maxwell

distribution. This parameter indicates the degree of the nonthermality and is related to the relative

importance of energy dissipation over entropy production. This model is compared with a direct

molecular dynamics simulation for a simpliﬁed gaseous system with energy input and dissipative

wall collisions, as well as some experimentally observed velocity distributions of light radicals in

plasmas.

I. INTRODUCTION

Nonthermal velocity distributions have been observed

for neutral atoms in plasmas for a long time [1–5]. Spec-

tral proﬁles with much larger wings than a Maxwellian

have been frequently observed. While this is typically

most apparent in hydrogen atomic emission lines, as

their Doppler broadening is easily observed with a high-

resolution spectrometer, similar nonthermal velocity dis-

tributions have been reported for other atoms [6]. The

origin of such a nonthermal velocity distribution has been

attributed to generation processes of energetic atoms,

such as Franck-Condon energy obtained through molec-

ular dissociation, and charge-exchange collision with hot

ions [6–11]. Many groups have empirically approximated

these non-thermal energy distributions by a sum of two

(or more) Maxwell distributions [2, 4, 6]. However, the

two-temperature model does not consider the relaxation

of energetic atoms. Furthermore, this model does not

have a direct connection to a physical quantity, and thus

it is diﬃcult to extract knowledge from the observed non-

thermal velocity distribution. Some Monte-Carlo simula-

tions also have reproduced the observed non-thermal ve-

locity distribution [11–13], but it is not always applicable

as all the physics quantities should be known beforehand

for the system of interest.

In this paper, I propose to model such nonthermal

velocity distributions of neutral atoms focusing more

on the energy dissipation / relaxation processes than

the energy input processes. In particular, the applica-

tion of the collisional-energy-cascade model is proposed,

which has been studied in the ﬁeld of granular gaseous.

∗fujiik@ornl.gov

This model has been originally proposed by Ben-Naim

et al. [14–16], where they consider the three essential

properties in the system; 1. a heat source in the high-

energy limit, 2. energy dissipation, 3. elastic collision

among particles. They point out that, under this condi-

tion, the steady-state velocity distribution has a power-

law tail in the high-velocity region. In an accompa-

nying paper of this work [17], it is pointed out that

this steady-state kinetic-energy distribution can be repre-

sented by the generalized Mittag-Leﬄer (GML) distribu-

tion, which is a one-parameter extension of the Maxwell

distribution. Although the GML distribution has no an-

alytic representation except for a few special cases, its

Laplace transform can be simply written by LfGML (s) =

R∞

0fGML(E)e−sE dE = [1+2D−1(hEiαs)α]−D/2α. Here,

hEiα>0 is the energy scale (see Ref.[17] for the relation

to the fractional-calculus-extension of the mean energy)

and Dis the spatial dimension of the system. 0 < α ≤1

is the stability parameter related to the relative impor-

tance of the dissipation process, and α= 1 corresponds

to the thermal system, where the GML distribution re-

duces to the Maxwell distribution. Thus, αcan be seen

as a dimensionless parameter representing the degree of

nonthermality.

For radical atoms in plasmas, the heat source may be

caused by the Franck-Condon energy of molecular dis-

sociation or by charge-exchange collision with hot ions,

while this input energy is eventually dissipated by the

wall collisions. Elastic collision among atoms may ran-

domize the kinetic energy. This similarity suggests the

applicability of this collisional-energy-cascade model to

the velocity distribution of radical atoms in plasmas,

which is the purpose of this paper. This paper is or-

ganized as follows. In section II, the kinetic theory of

gaseous particles with energy dissipation and its connec-

tion to GML distribution [17] is brieﬂy summarized. In

arXiv:2210.06938v1 [physics.plasm-ph] 13 Oct 2022

section III, the direct molecular simulation for a simpli-

ﬁed situation will be presented. In this simulation, the

energy source, energy dissipation, and elastic collisions

are taken into account. The velocity distribution of parti-

cles is directly compared with the theoretical prediction.

In section IV, several previous measurements for the ve-

locity distribution of atoms in plasmas will be presented

and compared with a GML distribution.

II. THEORY

In this section, a brief summary of the theory lead-

ing the GML distribution [17] is shown. Also, a numeri-

cal computation of the GML distribution, as well as the

velocity distribution corresponding to the GML energy

distribution is described.

A. Derivation of GML energy distribution for

Maxwell gases

Consider an isotropic and spatially uniform ensem-

ble of particles undergoing elastic collisions (i.e., no en-

ergy dissipation at this point) in D-dimensional space.

A Maxwell-type inter-particle interaction is assumed for

now. Particle ensembles with other interactions will be

discussed in subsection II C. With a Maxwell interaction,

the kinetic energies of two colliding particles, E1and

E2, can be thought of as random samples from the en-

ergy distribution f(E). For many collision systems, the

post-collision energy E0

1can be written by the following

form [17],

1←xE1+yE2,(1)

where x, y ∈[0,1] are random numbers following the

probability distribution p(x, y), which are determined by

the collision geometry, such as the scattering angle and

the relation between the relative and center-of-mass ve-

locities. In steady state, E0

1should also follow f(E).

Several forms of p(x, y) have been proposed. The sim-

plest example of valid p(x, y) is the so-called diﬀuse col-

lision [18, 19], where after the elastic collision the two

energies will be completely randomized, i.e., no memory

eﬀect of pre-collision energies,

p(x, y) = Bx

2,D

2δ(x−y),(2)

where B(x|a, b) = xa−1(1 −x)b−1/B(a, b) is beta distri-

bution with beta function B(a, b) = R1

0xa−1(1 −x)b−1dx

and δ(t) is Dirac’s delta function. The p-qmodel [18, 20],

which takes the memory eﬀect into account, as well as its

linear superposition also gives a valid p(x, y) [17].

The steady-state solution of Eq. (1) can be written

in the following form with the Laplace transform of the

energy distribution Lf(s)≡R∞

0f(E)e−sE dE,

Lf(s) = ZLf(xs)Lf(ys)p(x, y)dx dy, (3)

With any valid p(x, y), the steady-state distribution

converges to a Maxwell distribution Lf(s) = [1 +

2D−1hEis]−D/2according to Boltzmann’s H-theorem.

Here, hEiis the mean kinetic energy of the system.

Additionally consider a system with energy-

dissipation. It is assumed that, by this dissipation

process, a particle loses its kinetic energy by the fraction

of 1 −e−∆(with ∆ ≥0). The energy transfer to the

surrounding walls can be considered as this dissipation

process, but another process can be also considered. A

similar recursive relation with this dissipation can be

constructed as follows

1←(e−∆E1,with probability ξ

xE1+yE2,with probability 1 −ξ,(4)

where ξis the rate of this dissipation process relative

to the elastic collision. The Laplace representation of

Eq. (4) at the steady state is

Lf(s) = ξLf(e−∆s)

+ (1 −ξ)ZLf(xs)Lf(ys)p(x, y)dx dy. (5)

Here, it is implicitly assumed that constant energy in-

jection exists in the high-energy limit so that the system

will eventually arrive at a nontrivial steady state [14].

Consider the ﬁrst two orders of Lf(s). From the

normalization condition Lf(0) = 1, it can be written

that Lf(s)≈1−(hEiαs)αin the small-|s|region, with

0< α ≤1. Here, hEiαis the energy scale of the

distribution, which is related to the fractional-calculus-

extension of the mean energy [17]. Note that this corre-

sponds to an assumption of f(E) in the large-Eregion,

i.e., either f(E)≈E−α−1/(hEiα)αΓ(1 −α) if α < 1, or

f(E)≈exp(−E/hEiα)/hEiαif α= 1. By substituting

this into Eq. (5), we obtain

1 = (1 −ξ)Z(xα+yα)p(x, y)dx dy +ξe−α∆.(6)

Note that the symmetry of the elastic collision leads

p(x, y) = p(1 −y, 1−x), which results in R(x+

y)p(x, y)dx dy = 1 [17]. This indicates that α= 1 is the

necessary and suﬃcient condition for the non-dissipative

system, i.e., ∆ = 0 or ξ= 0. With a ﬁnite energy dissi-

pation, α < 1.

At the large-slimit, Lf(s) asymptotically behaves

≈2D−1(hEiαs)−D/2at the steady state and with ∆ 

1 [17]. By combining with its lowest order approxima-

tion Lf(s)≈1−(hEiαs)α, we ﬁnd that the generalized

Mittag-Leﬄer (GML) distribution [21–23],

Lf(s)≈1 + 2

DhEiαsα−D/2α

,(7)

FIG. 1. The GML distribution with several values of αwith

hEiα= 1. The power-law tail E−α−1/Γ(1 −α) is shown by

dotted lines. Appropriate oﬀsets are introduced for clarity.

is the simplest approximation of the steady-state solution

of Eq. (4). The GML distribution naturally reduces to

the Maxwell distribution at α→1.

B. Numerical evaluation of GML distribution

Although the GML distribution has no analytical

forms except for few special cases, some eﬃcient nu-

merical computation methods have been proposed [21–

23]. The GML distribution, fGML(E|α, D, hEiα) =

L−1(1 + 2(hEiαs)α/D)−D/2αcan be written as a mix-

ture representation of the exponential function;

fGML(E|α, D, hEiα) = 1

πhEiαD

2

×Z∞

exp −yD

2

αE

hEiαsin πD

2Fα(y)

(y2α+ 2yαcos(πα) + 1)D/4αdy, (8)

where Fα(y) is deﬁned as follows,

Fα(y) = 1 −1

πα cot−1cot(πα) + yα

sin(πα).(9)

Observe that Eq. (8) is the form of the weighted integra-

tion, R∞

0exp(−cE)w(c)dc, where cis the scale and w(c)

is its weight (weight can be negative in this case).

Figure 1 shows the GML distribution for several values

of αwith D= 3 and hEiα= 1. Here, the function val-

ues of the GML distribution are computed by integrating

Eq. (8) numerically. As expected from the small-|s|de-

pendence, it has a power-law tail, E−α−1/Γ(1 −α). The

dotted lines in the ﬁgure are these power-law functions.

The GML distribution approaches to this power-law tail

in the large-Eregion.

The GML distribution approximates the distribution

of the total kinetic energy. With a spectroscopic mea-

surement, only the velocity distribution along a particu-

lar axis is observed. In D= 3 dimensional space, this

FIG. 2. Velocity distributions corresponding to GML energy

distributions. D= 3 and λ= 1/3 are used. Distributions

with several values of αare shown with vertical oﬀsets for

clarity. The distribution with α= 1 is a Gaussian. The

distribution with α.1 is similar to a Gaussian in the central

part but has much larger wings. With a smaller value of α,

the wing fraction is larger. (a) is a semi-log plot and (b) is

a log-log plot. The dotted lines in (b) shows the power-law

distribution ∝ |v|−2α−λ−1.

velocity distribution can be evaluated by substituting

E=m(v2

x+v2

y+v2

z)/2 and integrate it over vxand

vyby taking the statistical weight of free space into ac-

count, where vx,vy. and vzare the velocity components

in the three dimensional space and mis the mass of the

particle. Since only the term depending on Ein Eq. (8)

is exp(−cE) with c=yD

2

α/hEiα, the integration of

this term is suﬃcient.

Consider the energy distribution g(E|c) = cexp(−cE)

in 3-dimensional space. As the statistical weight of the

space is √2mE in the energy domain, the distribution of

vzis

g(vz|c) = cZ∞

−∞

exp(−cE)1

√2mE dvxdvy

2rc

2mΓ1

2, c v2

2m,(10)

where Γ(s, x) = R∞

xts−1e−tdt is the lower incomplete

gamma function. By substituting Eq. (10) into Eq. (8),

i.e., by replacing the term exp −y(D/2)1/α E/hEiαin

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ACollisional-Energy-CascadeModelforNonthermalVelocityDistributionsofNeutralAtomsinPlasmasKeisukeFujii1,1OakRidgeNationalLaboratory,OakRidge,TN37831-6169,UnitedStatesofAmerica(Dated:October14,2022)NonthermalvelocitydistributionswithmuchgreatertailsthantheMaxwellianhavebeenobservedforradicalatomsinpl...

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A Collisional-Energy-Cascade Model for Nonthermal Velocity Distributions of Neutral Atoms in Plasmas Keisuke Fujii1

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