Analysis and Simulation for a Model of Electron Impact
Excitation/Deexcitation and Ionization/Recombination ˚
Bokai Yan:, Russel E. Caisch:, Farzin Barekat: and Jean-Luc Cambier;
April 10, 2015
Abstract
This paper describes a kinetic model and a corresponding Monte Carlo simulation method
for excitation/deexcitation and ionization/recombination by electron impact in a plasma free
of external elds. The atoms and ions in the plasma are represented by continuum densities
and the electrons by a particle distribution. A Boltzmann-type equation is formulated and a
corresponding H-theorem is formally derived. An ecient Monte Carlo method is developed
for an idealized analytic model of the excitation and ionization collision cross sections. To
accelerate the simulation, the reduced rejection method and binary search method are used to
overcome the singular rate in the recombination process. Numerical results are presented to
demonstrate the eciency of the method on spatially homogeneous problems. The evolution of
the electron distribution function and atomic states is studied, revealing the possibility under
certain circumstances of system relaxation towards stationary states that are not the equilibrium
states, a potential non-ergodic behavior.
Keywords.
Monte Carlo Method, Excitation-deexcitation, Ionization-recombination, Boltzmann
equation, H theorem, Singular reaction rates.
Distribution A: Approved for public release; distribution unlimited.
This research was supported by AFOSR and AFRL. The work of Barekat, Caisch and Yan was partially
supported by AFOSR grant FA9550-14-1-0283.
:
Mathematics Department, University of California at Los Angeles, Los Angeles, CA 90095-1555, USA.
[email protected], [email protected], [email protected]
;
Air Force Research Laboratory, Edwards AFB, CA 93524, USA. [email protected]
˚
1
Contents
1 Introduction
4
2 Formulation
5
2.1
The Distribution Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
2.2
Interaction Types and Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2.3
The Principle of Detailed Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
2.4
The Boltzmann Equation
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2.4.1
Ionization and Recombination . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
2.4.2
Excitation and Deexcitation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
3 Entropy and the H-Theorem
11
3.1
Entropy
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
3.2
Relation to H-theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
3.3
H-Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
3.3.1
Ionization and Recombination . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
3.3.2
Excitation and Deexcitation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
3.3.3
Full Kinetic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
4 Idealized Energetic Cross Sections
20
4.1
Ionization
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2
Recombination
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
4.3
Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
4.4
Deexcitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
5 Monte Carlo Particle Method
21
22
5.1
The Framework of Algorithm
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
5.2
Sample Electrons from a Singular Distribution . . . . . . . . . . . . . . . . . . . . . .
23
5.2.1
Acceptance/Rejection Method . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
5.2.2
Direct Search Method
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
5.2.3
Binary Search Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
5.2.4
Reduced Rejection Method
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
5.3
Large Scale Variation in the Rates
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
5.4
Particle/Continuum Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
5.5
A No Time Counter DSMC method
26
. . . . . . . . . . . . . . . . . . . . . . . . . . .
2
6 Computational Results
29
6.1
Equilibrium initial data
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
6.2
Non-equilibrium initial data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
6.3
The stationary states without ionization/recombination
. . . . . . . . . . . . . . . .
38
6.4
Comparison of dierent methods for sampling electrons in recombination . . . . . . .
38
6.5
The large scale variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
7 Conclusions and Future Work
42
Appendix A Proof of Lemma 2.1
45
Appendix B Proof of Lemma 3.6
46
Appendix C Proof of Proposition 3.1
47
Appendix D Proof of Lemma 3.8 and Theorem 3.9
49
Appendix E Algorithm for binary search method
52
E.1
Padding by zeros
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
E.2
Building a table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
E.3
Sampling a particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
E.4
Update the table when some
vk
changes
3
. . . . . . . . . . . . . . . . . . . . . . . . .
53
1
Introduction
Simulations of plasma dynamics can be computationally challenging due to the large number of
relevant physical processes and the wide range of space/time scales over which they operate. This is
particularly true when accounting for inelastic collisional processes such as excitation/deexcitation
and ionization/recombination in an atomic plasma, due to the large number of electronic states and
reaction pathways. The relative signicance of collisional interactions is determined by examining
the strengths of the various terms of the Boltzmann equation for the distribution function
Bt f ` v¨∇x f ` a¨∇v f “ Qpf q
where
Q
f:
(1.1)
eq being the equilibrium, Maxwellian
is the collision operator, with a stationary state f
distribution. The left-hand-side of (1.1) is the streaming operator in physical (x) and velocity (v)
apE, Bq is the acceleration due to the electric (E ) and magnetic (B ) elds. Thus, the
number Kn “ Lc {L0 , in which Lc is the mean free path between collisions and L0 is a
space, and
Knudsen
characteristic length scale of the gradient operator, determines the relative strength of collisions
versus advection. In the limit of small Knudsen number and weak elds, the energy distribution
function
f
for each of the particle species is close to an Maxwellian equilibrium, and the evolution
of the plasma can be described by uid-like equations, in which individual particle interactions do
not need to be resolved.
On the other hand, for
Kn Á 1
and/or large elds, the distribution
f
can be far from equilibrium, in which case particle representations and a Monte Carlo simulation
procedure are more appropriate. A variety of physical and numerical models have been used to solve
for the Boltzmann equation (mostly designed for transport studies) using uid approximations
(e.g. [1]), spherical harmonics expansions (e.g. [24]), or Monte-Carlo (e.g. [5, 6]).
In this work,
we do not intend to provide yet another analysis tool for non-equilibrium plasma, but examine
in detail the algorithms for a fully kinetic, Monte Carlo simulation of electron-induced inelastic
collisions for excitation, ionization, and their reverse processes (deexcitation and recombination) for
a dilute atomic plasma. In particular, we develop an ecient procedure for dealing with divergent
recombination rates at low energy, provide a proof of the H-theorem for inelastic processes, and
examine the properties of the stationary states.
For simplicity, we consider the simplest atomic
species, Hydrogen, using a Bohr model of the Atomic State Distribution Function (ASDF); this
is sucient to examine the eects of multiple scales in the energies exchanged, as well as timescales of reactions, while focusing on some mathematical and algorithmic aspects of the solution.
For the same reason, other physical processes such as radiative transitions, atomic (heavy-particle)
collisions, charge-exchange and elastic (Coulomb) collisions are also not considered. Note that we
are also solving only for the local inelastic collision operator, and other eects due to transport in
physical and velocity space, i.e. left-hand-side of (1.1), would be treated separately.
This paper consists of two parts. The rst part (Section 2 - 4) reviews the mathematical description
of excitation/deexcitation and ionization/recombination, in terms of the dierential cross sections
for each process. The modeling of the ionization cross sections by electron impact has been started
from the early quantum models in the 1920s (see for example a recent review [7]). In this work, as
a platform for development of numerical methods, our model uses the semi-classical cross section
(for example, see [8]) for the energetic cross section of ionization and an extension of this cross
section for excitation. The cross sections for the reverse processes are then found through detailed
balance. A Boltzmann equation is also formulated for the evolution of the electron distribution and
for the atom and ion densities. We then formulate and derive a version of Boltzmann's
H -Theorem
for excitation/deexcitation and ionization/recombination, originating from the classical denition
of physical entropy. Although not unexpected, this
new result, to the best of our knowledge.
4
H
function formulation for atomic plasma is a
The second part (Section 5 and 6) of this paper describes the development of a particle simulation
method for these kinetic processes. Ionization and recombination processes have been included in
the kinetic equations describing particles collisions in, for example, [911]. In those work the reaction
rates depend on the macroscopic number density and temperature, since they were computed by
assuming the electron distribution to be a Maxwellian.
For non-equilibrium conditions, (see for
example [12]), more realistic reaction rates depend on the electron velocity (or energy) rather than
temperature. The recombination rates are particularly dicult to treat, since they are singular as
the electron velocity goes to
0;
thus, an electron with lower energy can more easily be recombined,
see for example [13]. This singularity presents a new numerical challenge that we overcome in this
work, using either the Reduced Rejection method of [14] or a binary search method. We employ
here the framework of the Kinetic Monte Carlo (KMC) method ( [15, 16]), or equivalently the
Gillespie method [17].
kinetic processes.
Simulations are performed by random sampling from all of the relevant
We would also like to mention that a deterministic method to solve the same
system was developed as well [18].
The remainder of this paper is organized as follows. The formulation of the kinetic processes and
their rates and the resulting Botlzmann equations, along with some preliminaries, are presented
in Section 2.
The analogue of Boltzmann's
H -theorem
is formulated in Section 3.
The models
for the cross sections are described in Section 4, and the Monte Carlo simulation method, using
a particle representation for the electrons, is presented in Section 5. Computational results from
this simulation method are presented in Section 6. Finally, conclusions and future directions are
discussed in Section 7, while some technical details are presented in Appendices A-E.
2
Formulation
2.1 The Distribution Function
The plasma is assumed to consist of electrons (e), a single type of ion, of unit charge, and neutral
atoms in various electronic states. The atoms and ions are modeled as a continuum with uniform
density; the electrons are modeled through an energy distribution function, which is simulated using
a discrete set of particles. Throughout this paper, the temperature
that the Boltzmann constant
kB “ 1
Te
will have units of energy so
(i.e., it can be omitted).
The problems solved here will be spatially homogeneous (no spatial gradients and no transport)
and isotropic in velocity. Thus, we can describe the system in terms of a time-dependent electron
energy distribution function (EEDF) for the free electrons,
f pE, tq (E is the energy of the particle).
ne and temperature Te can be
The normalization is such that macroscopic electron number density
obtained from by taking the rst two moments:
ż8
ne “
3
ne Te “
2
f pEq dE,
(2.1a)
Ef pEq dE.
(2.1b)
ż08
0
Te denes the scale of the macroscopic energy of the electrons, without necessarily assuming an
f and the velocity
distribution function F (under the assumption that F is isotropic, i.e., F “ F pvq for v “ |v|) are
Here,
equilibrium, Maxwellian distribution. Note that the energy distribution function
related by
4π
f pEq “
me
since
E “ me v 2 {2
and
ˆ
dv “ 4πv 2 dv .
5
2E
me
˙1
2
F,
(2.2)
We use the following notation for the other variables.
kmax : the
total number of levels included in atoms,
nk ptq : the
number density of atoms at excitation level, k
˜
na : number
density of all neutral atoms
“ 1, . . . , kmax ,
¸
“
ÿ
nk
,
k
n` : number
density of ions p”
ne q ,
Lk : the
energy of atoms at the
L` : the
ionization energy of an unexcited atom,
gk : the
degeneracy of level
g` : the
degeneracy of an ion,
ge : the
k -th
excitation level,
k “ 1, . . . , kmax ,
k,
(spin) degeneracy of the electronp“
2q.
As mentioned earlier, we use the simple Bohr model for Hydrogen, such that:
Lk “ p1 ´ k ´2 qIH ,
where
IH
gk “ 2k 2 ,
L` “ IH “ L8 ,
g` “ 2,
(2.3)
is the Rydberg unit of energy (13.6eV). By denition, the atomic ground state, from which
k “ 1, i.e. L1 ” 0. Without transport of particles,
n` ” ne , while mass and energy conservation imply
number density of (bound and free) electrons and εtot the
all energy levels are measured, corresponds to
charge neutrality of the system is assured, i.e.
d
dt ntot
“
d
dt εtot
“ 0,
with
ntot
the total
total energy of the system dened by
ntot “ ne `na “ n` `na ,
ż
ÿ
εtot “ εe `εa `ε` “ Ef pEq dE `
nk Lk ` n` L` .
(2.4a)
(2.4b)
k
Note that momentum conservation is not used here, since it leads to terms of order of the ratio
of masses,
me {M` Æ 1{2000,
which are neglected. At equilibrium, the electron distribution is the
(isotropic) Maxwell-Boltzmann distribution
1?
f eq pEq “ ne 2pπTe3 q´ 2 E e´E{Te .
The equilibrium densities
neq
k
and
neq
`
(2.5)
for atomic level populations and ions are given by the Boltz-
mann and Saha distributions respectively [19]. The former describes the ratio of number densities
between two levels (e.g.
l, u),
and the latter is the ratio between electron and ion densities and that
1
of the atomic level from which ionization takes place
Boltzmann :
Saha :
where
h
neq
u
Blu pTe q “ eq “
nl
neq neq
e
Sk pTe q “ ` eq
nk
gu ´∆Elu {Te
e
,
gl
“
g`
Ze e´Bk {Te ,
gk
(2.6a)
(2.6b)
is the Planck constant, and
ˆ
Ze “ ge
2πme Te
h2
1
˙3{2
“ ge λ´3
e ,
(2.7)
We emphasize that only electron-impact collisions are examined in tho work. Therefore Te , the temperature of
the Maxwellian distribution of electrons (2.5) is also the governing temperature of Boltzmann and Saha equilibria,
when these are reached. Therefore, the complete system at equilibrium can be described by a temperature T , applied
to all plasma components. Furthermore, the equilibrium distributions are used here only to express thermodynamic
relations and verify the H-theorem; no assumption is made about the actual thermodynamic state of the plasma. For
example, partial equilibrium conditions can be initially created with dierent temperatures, such as in some of the
test cases described in later sections.
6
is the total electron partition function, product of a term for internal states (spin states,
one for the translational states. Here
λe
is the de Broglie wavelength of an electron (see [19] for
example). We have also dened the energy gap
∆Elu “ Lu´Ll
Zxa
and
c`
Bk “ L`´Lk .
ne “ n` ) to:
and the binding energy
These expressions are equivalent (under the assumption oc charge neutrality,
where
ge ” 2) and
neq
a
neq
“
gk e´Lk {Te ,
k
Zxa
(2.8a)
´L` {2Te
,
neq
` “c` e
(2.8b)
are:
Zxa “
kÿ
max
gk e´Lk {Te ,
(2.9a)
k“1
«
2g`
na
c` “
Zxa
Here
Zxa
ˆ
2πme Te
h2
˙ 3 ff1{2
2
.
(2.9b)
is the atomic partition function for internal states only (ground and excited electronic
states). We end this section with a lemma, which shows that specication of temperature is equivalent to specication of total energy; the proof is given in Appendix A for completeness.
Lemma 2.1. Given ntot ą 0 the total number of free and bound electrons, and εtot ą 0 the total
energy of the system, there are unique values of temperature Te and densities ne , na and nk , @k
satisfying
ne ` na “ ntot ,
ÿ eq
3
ne Te `
nk Lk ` neq
` L` “ εtot ,
2
k
eq
with neq
k and n` “ ne obtained using (2.6a)-(2.6b).
2.2 Interaction Types and Rates
We consider here a thermal (single-uid) plasma; because of the small electron/atom mass ratio,
the relative velocity and relative energy of the collision are well approximated by electron velocity
and kinetic energy. We also assume also that the dierential cross sections are independent of the
angular variables and only depend on the collision energy.
We consider four types of collisions,
ordered in pairs of forward and backward processes:
u by an electron of energy E0 with E0 ą
∆Elu leaves a post-collision electron with energy E1 “ E0 ´∆Elu . We write this process as
exc pE ; E q.
pl, E0 q Ñ pu, E1 q, with a cross section σlu
0
1
The reverse process is deexcitation from the upper-state u down to a lower-state l after collision
with an electron of energy E1 , leaving a post-collision electron with energy E0 “ E1 `∆Elu .
dex
We write this process as pu, E1 q Ñ pl, E0 q, with cross section σul pE1 ; E0 q.
Ionization from an atomic level k , by a free electron with energy E0 ą Bk , which generates
a scattered electron with energy of E1 “ E0 ´Et , a new electron with energy E2 “ Et ´Bk ,
and an ion. The transferred energy Et satises Bk ď Et ď E0 . We write this process as
pk, E0 , Et q Ñ pE1 , E2 q, with σkion pE0 ; E1 , E2 q.
The reverse process is three-body recombination, where a free electron (E1 ) and an ion recombine into an atomic state k , after collision with a second free electron (E2 ). This second
electron gains energy from the collision, with post-collisional energy E0 “ E2 `E1 `Bk . We
rec
write this process as pE1 , E2 q Ñ pk, E0 q and σk pE1 , E2 ; E0 q.
1. The excitation of a lower state
2.
3.
4.
l
to an upper-state
7
Explicit formulae for these cross sections are presented in Section 4. For each process, we can dene
a rate
r
per electron (more precisely, per electron pair for recombination):
rion pk, E0 , Et q “ nk vpE0 qσkion pE0 ; E1 , E2 q,
(2.10a)
rrec pk, E1 , E2 q “ n` vpE1 qvpE2 qσkrec pE1 , E2 ; E0 q,
exc pE ; E q,
rexc pl, u, E0 q “ nl vpE0 qσlu
0
1
dex
dex
r pu, l, E q “ n vpE qσ pE ; E q
1
where
vpEq “
a
2E{me
u
1
1
ul
(2.10b)
(2.10c)
(2.10d)
0
2
is the electron velocity. An integrated ionization rate per electron
ion pk, E q “
rtot
0
ż E0
is:
ion pk, E q.
rion pk, E0 , Et q dEt “ nk vpE0 qσtot
0
(2.11)
Bk
The total, macroscopic rates are (in units of (volume
Rion “
Rrec “
ÿż 8
Rdex “
´1 ):
time)
ion
rtot pk, E0 qf pE0 q dE0 ,
k Bk
ÿ ż 8ż 8
(2.12a)
rrec pk, E1 , E2 qf pE1 qf pE2 q dE1 dE2 ,
k
Rexc “
¨
0
ÿÿ
0
ż8
(2.12b)
rexc pl, u, E0 qf pE0 q dE0 ,
(2.12c)
l uąl ∆Elu
ÿÿż8
rdex pu, l, E1 qf pE1 q dE1
(2.12d)
u lău 0
2.3 The Principle of Detailed Balance
The principle of detailed balance determines the relations between the rates of dierent processes [19], by asserting that, at thermal equilibrium, each reaction
by its inverse
q Ñ p.
In other words,
rpÑq “ rqÑp
for all pairs
p Ñ q is exactly counterbalanced
of states p and q . Hereafter, to
simply expressions and unless specied otherwise, we will use the short notations
vp “ vpEp q,
and
fp “ f pEp q,
for
p “ 0, 1, 2
(2.13)
σ ion “ σkion pE0 ; E1 , E2 q,
(2.14a)
σ rec “ σkrec pE1 , E2 ; E0 q,
σ exc “ σ exc pE ; E q,
0
lu
(2.14b)
(2.14c)
1
dex pE ; E q,
σ dex “ σul
1
0
(2.14d)
For the ionization and recombination processes, the principle of detailed balance states that:
eq
ion “ neq f eq f eq v v σ rec .
neq
` 1 2 1 2
k f0 v0 σ
eq
eq and
Note that nk , f
(2.15)
neq
` are temperature dependent, describing the statistical properties of the
reactants (average over initial states), while the microscopic quantities
vp , σ ion
and
σ rec
are indepen-
dent of temperature. Using the Saha equilibrium relation (2.6b), we obtain the Fowler-Nordheim
relation between microscopic variables [19]:
gk E0 σ ion “
16πme
g` E1 E2 σ rec .
h3
2
(2.16)
To lighten the formulations, we slightly abuse the notation by using the same symbols σ for cross-sections (in
units of m´2 ) and dierential cross sections (m´2 /eV). The appropriate identication and choice of units can be easily
obtained by counting the number of energy integration variables in the macroscopic rates.
8
Similarly, for the excitation and deexcitation processes, detailed balance implies:
eq
exc “ neq f eq v σ dex .
neq
u 1 1
l f0 v0 σ
(2.17)
Using the equilibrium Boltzmann relation (2.6a), we then obtain a similar relationship between
microscopic variables, the Klein-Rosseland relation
[19]:
gl E0 σ exc “ gu E1 σ dex .
(2.18)
The fact that the cross-sections for forward and reverse processes are not independent is to be
expected from invariance under time-reversal of the microscopic, quantum interactions. Thus, eqs.
(2.16,2.18) are more fundamental than (2.15,2.17) and in turn, can be used to relate the rates of
forward and backward processes, independently of thermodynamic equilibrium conditions.
2.4 The Boltzmann Equation
We now formulate the Boltzmann-type equations for the electron energy distribution
atom density function
nk ptq, for inelastic collisions only.
f pE, tq and the
We present the results here and go through
the details for ionization/recombination processes in Section 2.4.1 and excitation/deexcitation processes in Section 2.4.2. Since we are only examining here the collisional processes, we can restrict
ourselves to a homogeneous system with no electro-magnetic elds, so that the streaming operators
on the left-side of (1.1) vanish. Thus, the reduced Boltzmann equations for ionization / recombination (IR) and excitation / deexcitation (ED) are:
Bt f pEq “ QeIR ` QeED ,
Bt nk “ QkIR ` QkED ,
Q`
IR
Bt n` “
`
(2.19)
Q`
ED ,
with the ionization/recombination collision operators
QeIR
“
ÿij
`
˘
pδ1 `δ2 ´δ0 q nk f0 v0 σ ion ´ n` f1 f2 v1 v2 σ rec dE0 dEt ,
k
QkIR
ij
`
“´
Q`
IR “
ÿij
˘
nk f0 v0 σ ion ´ n` f1 f2 v1 v2 σ rec dE0 dEt ,
(2.20)
pnk f0 v0 σ ion ´ n` f1 f2 v1 v2 σ rec q dE0 dEt ,
k
and the excitation/deexcitation collision operators
QeED
“
ÿÿż
´
¯
pδ1 ´δ0 q nl f0 v0 σ exc ´ nu f1 v1 σ dex dE0 ,
u lău
QkED “
ÿÿ
pδuk ´δlk q
ż ´
¯
nl f0 v0 σ exc ´ nu f1 v1 σ dex dE0 ,
(2.21)
u lău
Q`
ED
Here
E1
and
E2
“ 0.
in the ionization/recombination equations (2.20) are dened by
E1 “ E0 ´Et ,
and
E1
E2 “ Et ´Bk ,
(2.22)
in the excitation/deexcitation equations (2.21) is dened by
E1 “ E0 ´∆Elu .
k
In QED ,
δij “ 1
is the Kronecker delta function (“
1
if
i “ j, 0
(2.23)
otherwise) and we used the short
notations (2.13), (2.14) and
δp “ δpE ´Ep q,
9
p “ 0, 1, 2,
(2.24)
2.4.1 Ionization and Recombination
Considering ionization and recombination from and to a single level
k
only, the kinetic equation for
electrons includes a source term:
ż
e
f0 v0 σkion pE0 ; E1 , E2 qδpE1 ´ Eq dEt dE0
ż
` nk f0 v0 σkion pE0 ; E1 , E2 qδpE2 ´ Eq dEt dE0
ż
´ nk f0 v0 σkion pE0 ; E1 , E2 qδpE0 ´ Eq dEt dE0
ż
´ n` f1 f2 v1 v2 σkrec pE1 , E2 ; E0 qδpE1 ´ Eq dE1 dE2
ż
´ n` f1 f2 v1 v2 σkrec pE1 , E2 ; E0 qδpE2 ´ Eq dE1 dE2
ż
` n` f1 f2 v1 v2 σkrec pE1 , E2 ; E0 qδpE0 ´ Eq dE1 dE2 .
Bt f pEq|k “ QIR |k “ nk
The rst three lines describe respectively the gain/gain/loss of electrons during an ionization process,
the next three lines describe the loss/loss/gain of electrons during a recombination process, and the
integration takes place over the initial states. These expressions implicitly contain the relationships
between the electron energies, i.e.
dE1 dE2 .
(2.22) from energy conservation, and such that
dE0 dEt “
Summing over all excitation levels, we obtain:
e
Bt f pEq “ QIR “
kÿ
max ij
`
˘
pδ1 `δ2 ´δ0 q nk f0 v0 σ ion ´ n` f1 f2 v1 v2 σ rec dE0 dEt ,
(2.25)
k“1
with
δp
dened in (2.24). Similarly, the equation for the atomic state is:
Bt nk “ QkIR “
ij
n` f1 f2 v1 v2 σkrec pE1 , E2 ; E0 q dEt dE0
ij
´
nk f0 v0 σkion pE0 ; E1 , E2 q dEt dE0
ij
`
˘
nk f0 v0 σ ion ´ n` f1 f2 v1 v2 σ rec dE0 dEt .
“´
(2.26)
The rst term describes the gain of atomic level density in the recombination process, while the
second term describes the loss of neutral density in the
k th level due to the ionization.
The equation
for the ions is
ż
`
Bt n` “ QIR “ Bt
f pEq dE “ ´
ÿ
Bt nk
k
“
ÿij
pnk f0 v0 σ ion ´ n` f1 f2 v1 v2 σ rec q dEt dE0 .
k
10
(2.27)
2.4.2 Excitation and Deexcitation
Consider now an excitation and deexcitation process between two levels (l, u) only.
The kinetic
equation for the electrons is
ż
e
Bt f pEqulu “ QED ulu “
exc
pE0 ; E1 qδpE1 ´ Eq dE1
nl f pE0 qvpE0 qσlu
ż
exc
´ nl f pE0 qvpE0 qσlu
pE0 ; E1 qδpE0 ´ Eq dE0
ż
dex
pE1 ; E0 qδpE0 ´ Eq dE0
` nu f pE1 qvpE1 qσul
ż
dex
´ nu f pE1 qvpE1 qσul
pE1 ; E0 qδpE1 ´ Eq dE1 .
(2.28)
The rst two lines correspond respectively to the gain/loss of electrons in the excitation processes,
while the next two lines correspond to the gain/loss of electrons in the deexcitation processes.
pl, uq and
using short notations again, we obtain the compact form:
´
¯
ÿÿż
pδ1 ´ δ0 q nl f0 v0 σ exc ´ nu f1 v1 σ dex dE0 .
“
(2.29)
Summing over all pairs of levels
Bt f pEq “ QeED
u lău
Here
E0
and
E1
k
Bt nk “QED “
are related by (2.23). Similarly the equation for the density function of atoms is
ÿż
lăk
`
ÿż
ÿż
exc
nl f pE0 qvpE0 qσkl
pE0 ; E1 q dE0 ´
uąk
uąk
“
ÿż
dex
nu f pE1 qvpE1 qσku
pE1 ; E0 q dE1 ´
exc
nk f pE0 qvpE0 qσuk
pE0 ; E1 q dE0
dex
nk f pE1 qvpE1 qσlk
pE1 ; E0 q dE1
(2.30)
lăk
¯
ÿÿż ´
nl f0 v0 σ exc ´ nu f1 v1 σ dex pδuk ´ δlk q dE0 .
u lău
The rst two tems correspond to the gain and loss of atoms at
k -th
level in excitation processes,
and the next two terms corresponds to the deexcitation processes. In the last line, the summation
is over all pairs of
3
pl, uq
with
lău
by introducing the Kronecker delta function.
Entropy and the H-Theorem
3.1 Entropy
Np particles
Sp “ kB log Wp ,
The Boltzmann entropy of a macroscopic system of
where
Wp
of type
p
is dened as
is the number of independent micro-states available and compatible with a given total
energy of the system. The total phase space available to the system can be divided into unit cells,
such that the number of possible arrangements of
quantum states in the
nth
Np pnq
particles distributed into
Gp
available
cell is
N
Wp pnq “
We have assumed here a
classical
Gp p
.
Np !
approximation (Gp
" Np ),
since the studied conditions are such
that no quantum statistical eects need to be considered. The
Np ! factor is a correction for the fact
that particles in the same unit phase space are indistinguishable. The number of particles being
variable, the total number of possible arrangements for a given macro state (in units of
Sp “ log
ź
Wp pnq “
n
ÿ
n
11
log
N
Gp p
Np !
,
kB “ 1 )
is
(3.1)
in which the dependence of
N log N ´N ,
Gp
Np
and
on
n
is implicit. Using the Stirling approximation
logpN !q «
the entropy (3.1) becomes
Sp “ ´
ř
”
ı
Np
N
log
´
1
.
n p
Gp
(3.2)
For a very large number of cells (continuous distributions), we can approximate the discrete summation by an integral, yielding
ż
Sp “ ´
„
ȷ
Np pEq
Np pEq log
´1 dE.
Gp pEq
We introduce now the denition of the occupation number
variable.
ζp pEq “
ζp ,
(3.3)
a temperature-dependent, intensive
Np pEq
.
Gp pEq
For electrons, consider a small (or innitesimal) volume
(3.4)
d3 v d3 p
in phase space. The number of
possible micro-states in this phase space cell is
Ge ” ge
where
ge “ 2
d3 x d3 p
m3e 3 3
“
g
d x d v,
e
h3
h3
is the spin degeneracy, and the remainder is the elementary volume in phase space.
Assuming that the distributions are spatially homogeneous, we can replace
volume 1. After a change of variable from velocity
v to energy Ee
d3 x
by a spatial cell of
(assuming isotropic distributions),
one has
«
Ge ” ge
in which
get
4πm2e
h3
ˆ
2Ee
me
˙1{2 ff
.
dEe “ ge get dEe ,
is the energy-dependent degeneracy per unit volume due to the
(3.5)
translational
modes
only. Similarly, the number of particles in a unit spatial volume with energy in an interval of size
dEe
can be written as
Ne pEe q “ F pvq d3 v “ f pEe q dEe
The occupation number (3.4) becomes
ζe pEe q “
It will be useful in the following to dene
ş
translational modes only (
x
φpEqdE ” 1),
Ne pEe q
f pEe q
“
.
Ge pEe q
ge get
φpEe q
(3.6)
as a normalized distribution function for the
and separate the occupation number into a component
for internal modes only (ζ ) and one for translational modes; thus, for electrons,
ζex “ ne {ge .
From
(3.3), we can write down the entropy for the free electrons:
˙
ˆ
ż
ż
f pEe q
´1
dEe
Se “ ´ f pEe q plog ζe ´1q dEe “ ´ f pEe q log
ge get
ˆ
˙
ż
ζex φpEe q
“ ´ne φpEe q log
´1 .
get
For atoms in a specied atomic state
«
Gk “ g k
where
gk
k,
the same denition applies.
4πMa2
h3
ˆ
2Ea
Ma
˙1{2 ff
dEa “ gk gat dEa ,
Ea , Ma are respectively the kinetic energy and mass of the
Nk “ fk pEa q dEa ; note that since the atomic mass is the same for any atomic state k ,
use the same index (a) to describe the kinetic energy of the atom. Combining with (3.5),
is now the level degeneracy,
atom, and
we can
(3.7)
12
and performing a similar split between internal and translational modes, the occupation number for
atoms at level
k
is
Nk
nk φpEa q
φpEa q
“ ζkx
,
“
Gk
gk gat
gat
ζk “
(3.8)
and the entropy for the atoms has a form similar to (3.7), i.e.
Sa “
ÿ
Sk
ˆ
˙
ż
ζ x φpEa q
´1 .
Sk “ ´nk φpEa q log k t
ga
with
k
(3.9)
Similarly, the occupation number for ions is
ζ` “
x φpE q
ζ`
n` φpE` q
a
“
,
t
t
g` g`
g`
(3.10)
with the translational degeneracy
2
t “ 4πM`
g`
3
ˆ
h
2E`
M`
˙1{2
.
Because of the small electron/atom mass ratio, the kinetic energy of the atom is approximately
constant during collisions, i.e.
Ea
M` « Ma
and
E` « Ea .
Therefore
t « gt
g`
a
and again, we can use
to describe the kinetic energy of the ions. The entropy for the ions is then:
ż
S` “ ´n`
˙
x φpE q
ζ`
a
φpEa q log
´1 .
gat
ˆ
(3.11)
The total entropy of the system is the sum of that of its constituents,
S “ Se ` Sa ` S` ,
with
Se , Sa
S0 ,
constant
and
S`
dened by (3.7,3.9,3.11).
(3.12)
Note that the entropy is always dened up to a
which is arbitrary, but commonly chosen such that
SpT “ 0q “ 0.
3.2 Relation to H-theorem
It is traditional to dene a
H
function
3
to describe the constrained evolution of the system towards
H function includes the equilibrium distributions,
*
ż "
f pEq
eq
´ pf pEq ´ f pEqq dE
H“
f pEq log eq
f pEq
*
ÿ"
nk
eq
`
nk log eq ´ pnk ´ nk q
nk
k
ni
` ni log eq ´ pni ´ neq
i q.
ni
equilibrium. The denition of a
This expression is commonly referred to as the
relative
librium. We can also write it as the sum of partial
(3.13)
entropy, since it is identically zero at equi-
H -function
for each plasma component, which
will be dened below. We can now make the correspondence with the physical entropy denitions
(3.7,3.9,3.11). Using (2.5), (2.7) and (3.6), we can express the two distribution functions for the
electrons as
f pEe q “ ge get ζe pEe q,
ge get ´Ee {T
f eq pEe q “ neq
e
.
e
Ze pT q
3
(3.14a)
(3.14b)
In most cases, this function is equivalent to the negative entropy, and therefore is minimized at equilibrium.
13
where
T
is the equilibrium temperature of the complete thermodynamic system (electrons, ions
and atomic states).
Alternatively, we can write the equilibrium distribution for the normalized
distribution function of translational modes only as
gpt
φ pEp q “ t e´Ep {T
Zp
eq
Ztp “
with
this expression being valid for all particles (e.g.
ˆ
2πMp T
h2
˙3
Mp “ Me , Ma , M` ).
2
,
Let us now compute the
following expressions. First we remark that
ζp “
np φp
np e´Ep {T φp
“
.
gp gpt
gp Ztp φeq
p
(3.15)
It follows that
ż
ż
np φp log ζp dEp “ np φp log
where
φp
xEp y
np
´ np
.
eq dEp ` np log
t
gp Zp
T
φp
xEp y is the average thermal energy of particle p.
(3.16)
Since the collisions do not track the exchange
of momentum, the velocity distribution functions for the atoms and ions remain the equilibrium,
Maxwellian distributions; thus, for any atom in state
k , φk Ñ φeq
k ,
and the rst term on the
right-hand-side of (3.16) vanishes. We also take notice of the following relation, using (2.8a)
nk
nk neq
a ´Lk {T
“
e
.
eq
t
gk Zk
nk Za
Therefore,
nk log
(3.17)
Lk
nk
nk
neq
a
“
n
log
´nk ,
`n
log
k
k
eq
t
gk Zk
Za
T
nk
(3.18)
Za “ Zxa Zta is the complete partition function. We also remark the following,
neutrality (ne ” n` ) and the Saha equation (2.6b),
„ eq eq eq ȷ
n` ne na
ne
ne
n`
n`
ne log
“ ne log eq ` n` log eq ` ne log
` n` log
t
t
t
ge Ze
g` Z`
ne
n`
g` Ze neq
a Za
neq
ne
n`
a
“ ne log eq ` n` log eq ` n` log
.
Za
ne
n`
where
using charge
(3.19)
Combining these expressions, we arrive at the following
ż
ÿ
φe
n`
nk
nk log eq ` n` log eq
eq dEe `
φe
nk
n`
k
„
ȷ
eq
na
εtot
` ntot log
´
´ ne ´na ´n` .
Za
T
following H -functions
*
ż"
f pEq
He “
f pEq log eq
´ pf pEq ´ f eq pEqq dE,
f pEq
nk
Hk “ nk log eq ´ pnk ´ neq
k q,
nk
n`
H` “ n` log eq ´ pn` ´ neq
` q.
n`
ř
H “ He ` k Hk `H` is just the denition (3.13). It is
p´Sq “ne φe log
Let us now dene the
The sum of all terms,
(3.20)
(3.21)
then easy to see from
(3.20) that
p´Sq “ He `
ÿ
k
ȷ
neq
εtot
a
Hk ` H` ` ntot 1`log
` neq
.
e ´
Za
T
loooooooooooooooooomoooooooooooooooooon
„
CpT q
14
(3.22)
The last term involves a combination of conserved quantities (ntot , εtot ) and variables evaluated
at equilibrium only.
Since, according to lemma 2.1 there is a unique equilibrium state for the
given conserved quantities, the last term in (3.22) is indeed a constant
CpT q
as indicated, for given
initial conditions. This constant can be absorbed into the denition of the arbitrary
S0
mentioned
earlier, and the expression (3.22) is then formally equivalent to the negative of the system entropy.
Therefore, the minimum of
H
is also the point of maximum entropy, i.e. the equilibrium solution.
We point out that the equilibrium temperature
closed system; it is equal to
Te
T
is a unique and xed value for the complete,
when the velocity distribution function for electrons has reached
its equilibrium state along with the atomic and ionized states, but in general
Te
is a function of
time; for example, one can initialize the system with a Maxwellian distribution at a temperature
Te pt “ 0q,
which is dierent from the nal, equilibrium value
T.
As the electrons start to exchange
energy via the inelastic collisions with the atoms, and the distribution function will relax towards the
equilibrium. The
eective
4 T ptq Ñ T
e
temperature
under which the system can actually
reach
as the system nears equilibrium; the conditions
the exact equilibrium are discussed further below.
3.3 H-Theorems
In this section we show the H-Theorems for the Boltzmann equations (2.19, 2.20, 2.21), based on
the relative entropy. We rst summarize the results on moments conservations, while the proof is
left in Appendix C.
Proposition 3.1 (Conservation of density and energy.).
(a) Suppose tf, nk , n` u is the solution to the ionization/recombination kinetic equations, i.e.
(2.19, 2.20) with QeED “ QkED “ Q`ED “ 0. Then the total density and total energy are
conserved, i.e.,
d
d
ntot “
εtot “ 0,
dt
dt
where ntot and εtot are dened by (2.4a) and (2.4b).
(b) Suppose tf, nk , n` u is the solution to the excitation/deexcitation kinetic equations, i.e. (2.19)
(2.21) with QeIR “ QkIR “ Q`IR “ 0. Then the densities for each species and the total energy are
conserved, i.e.,
ż
d
dt
f pEq dE “
d
d ÿ
nk “
εtot “ 0,
dt k
dt
where εtot is dened by (2.4b).
(c) Suppose tf, nk , n` u is the solution to the ionization/recombination and excitation/deexcitation
kinetic equations (2.19, 2.20, 2.21). Then the total density and total energy are conserved,
d
d
ntot “
εtot “ 0.
dt
dt
Now we give the stationary states:
Proposition 3.2 (Stationary states).
8
• If the ionization and recombination processes are included, the stationary solutions tf 8 , n8
k , n` u
are given by
eq
eq
f 8 “ f eq , n8
n8
for k “ 1, . . . , kmax .
(3.23)
` “ n` ,
k “ nk ,
eq
eq
eq
Here f , nk and n` are the equilibria with temperature Te determined by the total number
density ntot and the total energy εtot .
4
Dened as average thermal energy, without implying a Maxwellian distribution.
15
•
If only the excitation and deexcitation processes are included, i.e. without the ionization and
recombination processes, let ∆E be the greatest common divisor of ∆Elu 's, i.e.
∆E “ max t∆E ą 0 : ∆Elu is an integer multiplication of ∆E , for every l, uu .
(3.24)
8
Then the stationary solutions tf 8 , n8
k , n` u are given by
eq
8
eq
8
0
f “ f pE; Te qηpEq, n` “ n` , n8
for k “ 1, . . . , kmax ,
(3.25)
k “ nk pTe q,
where ηpEq is a periodic function in E ř
with period ∆E , dened by
j
ηpEq “ ř
f pE ` j∆E, t “ 0q
f eq pE ` j∆E; Te q
j
.
(3.26)
with f pE, t “ 0q the initial electron distribution and Te the temperature of the stationary state
determined
ż
ż from energy conservation
Ef eq pE; Te qηpEq dE `
ÿ
neq
k pTe qLk “
Ef pE, t “ 0q dE `
k
ÿ
n0k Lk .
(3.27)
k
where n0` is the initial ion density and neq
k pT q is the k -level equilibrium atom density (2.8a) at
temperature Te . The summation in (3.26) is over all integers j “ jpEq, such that E`j∆E ě 0.
Remark 3.3.
•
The minimum amount of energy change for one electron in the excitation-deexcitation process
is
$
,
&
.
ÿ
∆E “ min ∆E ą 0 : ∆E “
mlu ∆Elu ,
mlu PZ %
l,u
∆E
ηpEq.
which is exactly the period
periodicity of the function
dened in (3.24).
Here
Z
This is the mechanism generating the
denotes the set of all positive (corresponding to
excitation) and negative (corresponding to deexcitation) integers.
•
The non-equilibrium stationary state (3.25) depends on the periodic function
∆E .
Every energy dierence
∆Elu
must be an integer multiple of
∆E .
ηpEq
of period
It follows that the
non-equilibrium stationary state exists if and only if the ratio of any two energy dierences
∆Elu
and
∆El1 u1
∆Elu
is a rational number. This is true for the Bohr model, since every
IH .
0 very fast as the number of excitation levels increase.
is a
rational multiple of the ionization energy
•
The period
0, η
∆E
decreases to
As
∆E Ñ
becomes a constant function and the stationary solutions are given by the Boltzmann
distribution (2.5) and (2.8a), with temperature determined by energy conservation.
The proposition 3.2 is a direct result of the following H-theorem on the relative entropy. We dene:
Denition 3.4
.
(Relative entropy)
Dene the relative entropy of the system
H “ He ` Ha ` H` ,
where
He , Ha “
ř
k
Hk
and
H`
(3.28)
are the relative entropy of electrons, atoms and ions respectively
ż
f
f log 8 dE ´ ne ` n8
e ,
f
˙
ÿˆ
nk
Ha “
nk log 8 ´ nk ` n8
,
k
n
k
k
n`
H` “ n` log 8 ´ n` ` n8
`,
n`
ş
8
“ f 8 dE ; tf 8 , n8
k , n` u are the the
He “
where
ne “
ş
f dE
and
n8
e
Proposition 3.2.
16
(3.29)
stationary solutions given in
ř
p nk `n` q
Note that
is constant and
ne “ n` ,
so that the total relative entropy (3.28) can be
rewritten as
ż
H“
f log
˙
ÿˆ
nk
f
n`
8
dE
`
n
log
`
n
´
n
` n` log 8 .
k
k
k
8
f8
n
n
`
k
k
(3.30)
The H-theorem can be presented as follows:
Theorem 3.5 (H-theorem). One always has He ě 0, Ha ě 0 and H` ě 0 in (3.29), hence H ě 0
in (3.28). If one further assumes the solution f to the kinetic system (2.19)(2.20)(2.21) is smooth,
then for the entropy dened by (3.28)(3.29), one has dtd H ď 0. The equality holds if and only if
H “ 0, which is equivalent to the fact that f pEq, nk and n` are the stationary solutions given
in Proposition 3.2. This result is also valid for the solely ionization/recombination process (with
`
e
k
QeED “ QkED “ Q`
ED “ 0) or the excitation/deexcitation process (with QIR “ QIR “ QIR “ 0).
The non-negativity of
He , Ha
and
H`
comes from the inequality
y log xy ě py ´ xq,
for
x, y ą 0.
We
prove the decay of the relative entropy for dierent processes in the following sections.
3.3.1 Ionization and Recombination
We rst prove the H-theorem for the ionization/recombination process, i.e., the equations (2.19)
(2.20) with
QeED “ QkED “ Q`
ED “ 0.
For any smooth function
ż
Bt
ϕ “ ϕpEq,
To start, we derive the weak form of the Boltzmann equations.
one has for electrons,
ż
ϕ QeIR dE
ÿij
`
˘
“
pϕ1 ` ϕ2 ´ ϕ0 q nk f0 v0 σ ion ´ n` f1 f2 v1 v2 σ rec dE0 dEt
ϕf dE “
k
“
ÿij
ˆ
pϕ1 ` ϕ2 ´ ϕ0 q
k
nk f0
n` f1 f2
eq eq ´ eq eq eq
nk f0
n` f1 f2
˙
(3.31)
eq
ion
neq
dE0 dEt ,
k f 0 v0 σ
using the detailed balance relation (2.15). Similarly, for atoms and ions,
˙
n` f1 f2
nk f 0
eq eq
ion
´
dE0 dEt ,
eq
eq eq eq nk f0 v0 σ
neq
f
n
f
f
` 1
2
k 0
˙
ÿ ij ˆ nk f 0
n` f1 f2
eq eq
ion
“
´
dE0 dEt .
eq eq
eq eq eq nk f0 v0 σ
n
f
n
f
f
`
0
1
2
k
k
Bt nk “ QkIR “ ´
Bt n` “ Q`
IR
ij ˆ
(3.32)
(3.33)
The following lemma, proven in Appendix B, will be used in the derivation of the H-theorem.
eq
Lemma 3.6. Suppose f pEq is a smooth function and f eq , tneq
k u, n` are the equilibrium functions
dened by (2.5)-(2.8b), sharing the same total number density and total energy with f , tnk u and
n` . If
nk f0
n` f1 f2
eq “ eq eq eq ,
neq
f
n
` f1 f2
k 0
for every E0 and Et , then
f “ f eq ,
nk “ neq
k ,
n` “ neq
`.
Now we are ready to give the H-theorem for the ionization/recombination process,
17
(3.34)
Theorem 3.7 (H-theorems for ionization/recombination). Assume the solution f to the ionization/recombination kinetic equations ((2.19) (2.20) with QeED “ QkED “ Q`ED “ 0) is smooth. Then
for the entropy H dened by (3.28)(3.29) with the stationary solution (3.23), one has dtd H ď 0. The
equality holds if and only if H “ 0, which is equivalent to say f pEq, nk and n` are the equilibrium
values.
Proof.
Utilizing the weak forms (3.31)-(3.33), a straightforward computation gives
ż
ÿ
f
nk
n`
dE
`
Bt nk log eq ` Bt n` log eq ` 2Bt ntot
eq
f
nk
n`
k
ˆ
ˆ
˙
ˆ
˙˙
ij
ÿ
nk f0
n` f1 f2
log
“´
´ log
eq eq
eq eq eq
n
f
n
` f1 f2
k 0
k
ˆ
˙
nk f0
n` f1 f2
eq eq
ion
¨
dE0 dEt
eq ´ eq eq eq nk f0 v0 σ
neq
f
n
f
f
`
1
2
k 0
Bt H “
Bt f log
ď 0.
The equality holds if and only if
nk f0
n` f1 f2
eq “ eq eq eq ,
neq
f
n
` f1 f2
k 0
for every
E0
and
Et .
Lemma 3.6 then implies
f “ f eq ,
which is equivalent to
nk “ neq
k ,
n` “ neq
`,
H “ 0.
3.3.2 Excitation and Deexcitation
We then prove the H-theorem for the excitation/deexcitation process, i.e., the equations (2.19)
(2.21) with
QeIR “ QkIR “ Q`
IR “ 0.
Again we derive the weak form of the Boltzmann equations.
ϕ “ ϕpEq, one has for electrons
ij
ϕf dE “
ϕ QeED dE
´
¯
ÿż
“
pϕ1 ´ ϕ0 q nl f0 v0 σ exc ´ nu f1 v1 σ dex dE0
For any smooth function
ż
Bt
lău
“
ÿż
lău
ˆ
pϕ1 ´ ϕ0 q
nu f1
nl f0
eq eq ´ eq eq
nl f0
nu f1
˙
(3.35)
eq
exc
neq
dE0 ,
l f0 v0 σ
using the detailed balance relation (2.17). Similarly for atoms
Bt nk “ QkED “
˙
ÿ ż ˆ nl f0
nu f1
eq eq
exc
dE0 .
´
eq eq
eq eq pδuk ´ δlk qnl f0 v0 σ
n
f
n
f
u
0
1
l
lău
Note that for any sequence
Bt
ÿ
tϕk u,
˙
ÿ ż ˆ nl f0
nu f 1 ÿ
eq
exc
n k ϕk “
ϕk pδuk ´ δlk qneq
dE0
eq eq ´ eq eq
l f0 v0 σ
n
f
n
f
u
0
1
l
lău
k
˙
ÿ ż ˆ nl f0
nu f 1
eq eq
exc
“
´
dE0 .
eq eq
eq eq pϕu ´ ϕl qnl f0 v0 σ
n
f
n
f
u
0
1
l
lău
(3.36)
With only excitation/deexcitation, the equilibria
f eq
and
neq
k
dened in Section 2.1 are not the only
stationary solutions. The proof of this uses the following results.
18
(3.37)
Lemma 3.8. (i). Suppose f is a solution to the excitation/deexcitation kinetic equations (2.19)
(2.21) with initial value f pE, t “ 0q. Then
d ÿ
f pE ` j∆E, tq “ 0,
dt j
for any E ě 0,
(3.38)
where the summation is over all integers j such that pE ` j∆Eq ě 0.
(ii). If f is in the form f pEq “ f eq pE; Te qηpE; Te q, for some temperature Te and some periodic
function ηpE; Te q in E with the period ∆E , then η is given by (3.26).
(iii). There is a unique value of Te satisfying the energy conservation (3.27).
The detailed proof is given in Appendix D. The equation (3.38) describes the fact that an electron
can only gain or lose an energy of integer multiplication of
included. (ii) gives the formula of
η , as a result of (3.38).
∆E
with only excitation/deexcitation
(iii) shows the uniqueness of the tempera-
ture at stationary state, as an analogy to Lemma 2.1 where the ionization/recombination processes
are included.
Theorem 3.9. Assume the period ∆E dened in (3.24) exists. Suppose f pEq is a smooth function
such that
nu f1
nl f0
eq eq “ eq eq
nl f 0
nu f1
(3.39)
eq
holds for every l, u and E0 , in which f eq “ f eq pE; Te q, tneq
k u “ tnk pTe qu are of the form (2.5) and
(2.8a)(2.8b) for the temperature Te determined from (3.27). Then
f pEq “ f eq pE; Te qηpEq,
nk “ neq
k pTe q,
(3.40)
where ηpEq is the periodic function (3.26), with period ∆E .
A detailed proof is given in Appendix D.
Now we are ready to give the H-theorem for the excitation/deexcitation process,
Theorem 3.10 (H-theorems for excitation/deexcitation). Assume the solution f to the excitation/deexcitation kinetic equations ((2.19) (2.21) with QeIR “ QkIR “ Q`IR “ 0) is smooth. Then for
the entropy H dened by (3.28)(3.29) with the stationary solution (3.25), one has dtd H ď 0. The
equality holds if and only if H “ 0, which is equivalent to say f and tnk u are given by (3.40).
Proof.
Note that with only excitation and deexcitation, one has
ÿ
pnk ´ n8
k q “ n` log
k
n`
“ 0.
n8
`
The entropy becomes
ż
H“
˙
ÿˆ
f pEq
nk
f pEq log eq
dE `
nk log eq
.
f pE; T qηpEq
nk pT q
k
19
Utilizing the weak forms (3.35)(3.36), a straightforward computation gives
ż
ż
ÿ
nk
Bt H “ Bt f log eq dE ` Bt nk log eq ` Bt f dE ` Bt nk
f η
nk
k
k
ˆ
˙˙ ˆ
˙
ÿ ż ˆ ˆ nl f0 ˙
nu f1
nl f0
nu f1
eq eq
exc
dE0
´ log
“´
log
eq eq
eq eq
eq eq ´ eq eq nl f0 v0 σ
η
η
n
f
n
f
n
f
n
f
u
u
0
1
0
1
0
1
l
l
lău
ˆ
˙˙ ˆ
˙
ÿ ż ˆ ˆ nl f0 ˙
nu f1
nl f0
nu f1
eq eq
exc
log
“´
´ log
dE0
eq eq
eq eq
eq eq ´ eq eq nl f0 v0 σ
n
f
n
f
n
f
n
f
u 1
u 1
0
0
l
l
lău
f
ÿ
ď 0.
This inequality used
ηpE1 q “ ηpE0 ´∆Elu q “ ηpE0 q,
since
∆Elu
l, u
H “ 0.
for every
to
∆E , the period of η .
nl f0
nu f1
eq eq “ eq eq ,
nl f0
nu f 1
is an integer multiplication of
and
E0 .
By Theorem 3.9 and 3.8,
f
and
tnk u
The equality holds if and only if
are given by (3.40), which is equivalent
The stationary solution is unique for any specied initial data as a result of this theorem.
3.3.3 Full Kinetic Equations
Finally consider the full kinetic equations including all four processes (2.19)(2.20)(2.21).
We are
ready to give:
Proof of Theorem 3.5.
The proof of the non-negativity of
H
is the same as the one in Theorem 3.7.
Then it is straightforward to obtain
˙˙
n` f1 f2
´ log
Bt H “ ´
log
eq eq
neq
` f1 f2
k
ˆ
˙
nk f 0
n` f1 f2
eq eq
ion
¨
´
dE0 dEt
eq
eq eq eq nk f0 v0 σ
neq
f
n
f
f
` 1
2
k 0
ˆ
˙˙ ˆ
˙
ÿ ż ˆ ˆ nl f0 ˙
nu f1
nu f1
nl f0
eq eq
exc
´
log
´ log
dE0
eq eq
eq eq
eq eq ´ eq eq nl f0 v0 σ
n
f
n
f
n
f
n
f
u
u
0
1
0
1
l
l
lău
ÿij ˆ
ˆ
nk f0
eq
neq
k f0
˙
ˆ
ď 0.
The equality holds if and only if both (3.34) and (3.39) hold. Then by (3.34) and Lemma 3.6,
f “ f eq ,
nk “ neq
k ,
which is consistent with (3.39) with the periodic function
4
n` “ neq
`,
η “ 1.
Idealized Energetic Cross Sections
This section describes semi-classical cross sections, which have explicit analytic formulas.
20
4.1 Ionization
For ionization events
pk, E0 , Et q Ñ pE1 , E2 q, experimental measurements and numerical calculations
of the dierential cross section have been performed (e.g., [20], [21]) but the results are rather
complicated. As a simpler alternative, the semi-classical cross section [8] is used. This is sucient
for demonstrating the Monte-Carlo procedure, while highlighting some key diculties due to the
scaling of the cross-sections with energy an issue which remains for any physically realistic crosssection model. In the semi-classical approximation, we have the dierential cross section
2
σkion pE0 ; E1 , E2 q “ 4πa20 IH
1 1
1B ďE ďE .
E0 Et2 k t 0
(4.1)
1 ” 1 if the conditional argument is satised, ” 0 otherwise. The total ionization cross section
given initial atomic energy level k , and electron energy E0 , is
ż E0
ion
σtot pk, E0 q “
σkion pE0 ; E1 , E2 q dEt
Bk
(4.2)
ˆ
˙
1
1
2 2 1
1E0 ąBk .
“ 4πa0 IH
´
E0 Bk
E0
where
for a
“
E0 from (2.11) is
ion
nk vpE0 qσtot pk, E0 q
“
2
4πa20 IH
The corresponding ionization rate per electron
ion
rtot
pk, E0 q
c
In simulation of ionization from level
the energy transfer
Et
k,
2 nk
?
me E0
ˆ
1
1
´
Bk
E0
˙
1E0 ąBk .
(4.3)
E0 is sampled according to (4.2), then
Bk ă Et ă E0 according to the probability
the incident energy
is chosen from the energy interval
density
pion pEt q “
σkion pE0 ; E1 , E2 q
E0 Bk 1
1B ďE ďE .
“
ion
E0 ´ Bk Et2 k t 0
σtot pk, E0 q
Note that this expression diverges rapidly when the excess energy
(4.4)
E0 ´Bk Ñ 0.
4.2 Recombination
The cross section for a recombination event
using
pE1 , E2 q Ñ pk, E0 q
is obtained from the relation (2.16),
Et “ E2 ` Bk ,
1
gk E0 ion
σ pE0 ; E1 , E2 q
´3
16πme h g` E1 E2 k
2
4πa20 IH
1
gk 1
“
.
´3
16πme h g` E1 E2 pE2 ` Bk q2
σkrec pE1 , E2 ; E0 q “
E1 , E2 from (2.10) is
rec
n` vpE1 qvpE2 qσk pE1 , E2 ; E0 q
2
a20 IH
gk
n
1
? `
?
.
2
´3
2me h
E1 E2 g` pE2 ` Bk q2
(4.5)
The corresponding recombination rate per pair of electrons
r
rec
pk, E1 , E2 q “
“
Both cross-sections are divergent as
E1 Ñ 0
allowed range of energy transferred, i.e. near
E2 Ñ 0, conditions
threshold (Et » Bk ) or
or
21
(4.6)
obtained at either end of the
near maximum (Et
» E0 ).
4.3 Excitation
pl, E0 q Ñ pu, E1 q, the semi-classical
ˆ
˙
1
1
1
exc
2 2
σlu pE0 ; E1 q “ p4πa0 IH qp3ful q
´
1E0 ą∆Elu ,
E0 ∆Elu E0
For the energetic cross section of excitation
cross section is [8]
(4.7)
where the absorption oscillator strength is
ful
“
32
1
1
? 5 3 ´2
.
3π 3 l u pl ´ u´2 q3
The corresponding excitation rate per electron
c
2
rexc pl, u, E0 q “ 12πa20 IH
ful
(4.8)
E0 from (2.10) is
ˆ
˙
2 nl
1
1
?
1E0 ą∆Elu .
´
me E0 ∆Elu E0
(4.9)
4.4 Deexcitation
For the cross section of the deexcitation
pu, E1 q Ñ pl, E0 q,
the Klein-Rosseland relation (2.18) is
used:
gl E0 exc
σ pE0 ; E1 q
gu E1 lu
ˆ
˙
gl 1
1
1
2 2
“ 12πa0 IH ful
´
.
gu E1 ∆Elu E1 `∆Elu
dex
σul
pE1 ; E0 q “
E1 from (2.10) is
ˆ
˙
2
gl 1
1
1
nu ?
´
.
me gu E1 ∆Elu E1 `∆Elu
(4.10)
The corresponding deexcitation rate per electron
c
2
rdex pu, l, E1 q “ 12πa20 IH
ful
5
(4.11)
Monte Carlo Particle Method
Using the rates detailed in the previous sections, we now formulate a Monte Carlo simulation
method for the system (2.19)-(2.20), which describes the ionization, recombination, excitation and
deexcitation interactions between electrons and atoms/ions in plasma dynamics. The simulation is
performed using a method like that of Direct Simulation Monte Carlo (DSMC) [22] designed for
binary elastic collisions. Coulomb collisions and other physical processes are not included here, but
will be added in future work.
There are several diculties in simulation of excitation/deexcitation and ionization/recombination:
1.
There are many processes to be simulated, including
pkmax pkmax ´ 1qq
excitation/deexcitation
processes and (2kmax ) ionization/recombination processes. One needs to pick the processes
and sample corresponding electrons with the right rates.
2.
The recombination rate depends in a singular way on the electron energies, so that ecient and
3.
The rates for dierent interactions are widely varying, so that it is dicult to eciently include
correct sampling of the electrons for recombination events is dicult.
all of the interactions.
In this section we mainly focus on dealing with the rst two diculties.
The third involves the
multiscale properties which will be investigated in future work.
5.1 The Framework of Algorithm
We employ a Kinetic Monte Carlo (KMC) method to deal with the rst diculty. The method can
be summarized as
22
Step 1.
Find upper bounds for the total rates of each of the four collision types, denoted as
with the superscript
R̃ “
Step 2.
ř
j “ exc, dex, ion, rec.
r̃j
The total rate of all types of collisions is given by
j
j r̃ .
j
Choose collision type
with rate
r̃j {R̃.
Update the time with
t “ t ` ∆t, where ∆t “ 1{R̃,
so that there is only one virtual collision in one time step.
Step 3.
For the selected collision type, choose the participating particles, including electrons and
atom levels before and/or after the collision, with the rate used in computing the upper bound
r̃j .
Step 4.
Denote this rate to be
r̃Pj ,
with the subscript
P
representing the particles.
For the selected collision type and particles, compute the actual rate
j
j
Accept and perform the collision with probability rP {r̃P .
rPj
of this collision.
5.2 Sample Electrons from a Singular Distribution
In step 3 of the framework, electrons for excitation, deexcitation and ionization processes are uniformly chosen from the numerical particles. For the recombination process, the actual rate
uniformly chosen
rPj
scales
´1{2 , if two electrons with energy E and E are
like pE1 E2 q
from the numerical
1
2
rec extremely large and the acceptance/rejection procedure
particles. This makes the upper bound r̃
in step 4 very inecient. This is the second diculty mentioned above.
To avoid this singularity, we choose electrons
´1{2 as
scales like E
E Ñ 0.
E
with rate
ppEq,
in which the distribution
As described in the following sections, the two electrons
E1
and
ppEq
E2 in
the recombination are chosen with probabilities
1
“? ,
E
1 ÿ
gl
p2 pEq “ ?
.
E l pE ` Bl q2
p1 pEq
(5.1)
Now consider the following model problem. Suppose we want to simulate some reaction process over
a xed time interval
r0, tmax s,
with the reaction rate
R.
Here
R
such as (2.12). For each collision one needs to choose an electron
Ej
with
tEj , j “ 1, . . . , Ne u. Then the total number of collisions
Rtmax {Ne “ Rtmax Ne {ne “ OpNe q, where ne is the physical density of electron, Ne is
number of numerical particles and Ne is the eective number.
from
Ne
pvolume ¨ timeq´1 ,
probability pj “ ppEj q
is in the unit of
numerical electrons with energy
needed is
the
We propose four dierent methods here for this model problem and give a numerical comparison in
Section 6.4.
5.2.1 Acceptance/Rejection Method
For this method, rst compute
pmax “ max pj ,
(5.2)
1ďjďNe
then uniformly choose an electron
is accepted. The value
pmax
Ej
and accept it with rate
pj
pmax . Keep sampling until one electron
can be updated after each collision.
This method is used for the excitation, deexcitation and ionization. But it is very inecient for
the recombination due to the singularity as
is at equilibrium. At equilibrium for small
E ă E˚
E Ñ 0. To see this, consider the case
E˚ , using equation (2.5), the fraction
when the system
of electrons with
is
P pE ă E˚ q “ C
ż E˚ ?
3{2
Ee´E{Te dE “ OpE˚ q,
0
23
(5.3)
so that
ˆ
P
˙
min Ej ď E˚
1ďjďNe
Hence the minimum of
E
is
´2{3
OpNe
q
3{2
“ 1 ´ p1 ´ P pE ă E˚ qqNe “ OpNe E˚ q.
and
rAR “
To simulate over
r0, tmax s,
pmax “
?1
E
1{3
“ OpNe q.
The acceptance rate is
1
“ OpNe´1{3 q.
pmax
the total cost is
CAR “ O
ˆ
Ne
rAR
˙
“ OpNe4{3 q.
5.2.2 Direct Search Method
A direct search method is to sample a number
r
#
j “ min ξ, s.t.
ξ
ÿ
pi ě r
i“1
Here the summation
s“
r0, tmax s
Ne
ÿ
and look for
pi .
(5.4)
i“1
řNe
i“1 pi can be updated after each collision. In this method the singularity
is not a problem. However the searching for
over
r0, 1s
+
uniformly from
j
takes
OpNe q
operations. The total cost to simulate
is
Cdirect “ OpNe2 q.
5.2.3 Binary Search Method
j in (5.4). Divide
r0, Ne s into two equal size subdivisions and rst locate which division Ej is in. If this idea is applied
iteratively to each subdivision, the total cost can be reduced to Oplog2 Ne q. This is the widely used
ř
binary search method. For this method the sum s “
pi in each subdivision can be updated after
each collision. A total number of Oplog2 Ne q updates is need for each collision. For completeness a
detailed algorithm is given in Appendix E. The total cost to simulate over r0, tmax s is
Cbinary “ OpNe log2 Ne q.
The direct search method can be accelerated by a binary search method to nd
5.2.4 Reduced Rejection Method
The recently developed reduced rejection method [14] can be applied, by combining with the
Marsaglia table method [23]. See Section 6 in [14] for a detailed description.
There are two sources of computation costs.
collisions.
Then there are
method is applied.
OpKq
Suppose the Marsaglia table is re-created after
K
particles in the L-region, where a regular acceptance/rejection
According to Section 5.2.1, the cost of sampling one electron is
the L-region. Hence the total cost of sampling
OpNe q
OpK 1{3 q
in
1{3 q. A total number of
particles is OpNe K
OpNe {Kq Marsaglia tables are
since the cost of creating one table is OpNe q, the resulting
` Nre-created,
˘
e
total cost in this part is O
N
. Therefore the total cost of the reduced rejection method to
e
K
simulate over r0, tmax s is
ˆ
˙
Ne2
1{3
CRR “ O Ne K `
“ OpNe5{4 q,
K
if
3{4
K “ Ne
.
Note that to minimize the total cost, we need the costs in the two sources to be
approximately equal, i.e.,
Ne K 1{3 «
Ne2
K . In practice, we use a simple technique to make the cost of
re-creating Marsaglia tables and the cost of sampling from L-regions approximately equal:
24
•
•
Step 1. Create the Marsaglia table. Record the computation time as
Step 2. Perform collisions. Keep track of the total time
tsample
ttable .
used in sampling particles for
recombination. When
tsample ą ttable ,
set
tsample “ 0
and go to step 1.
5.3 Large Scale Variation in the Rates
This section illustrates the large variation in the rates of dierent processes. Ecient multiscale
algorithms will be investigated in future works.
The total rate of excitation from level
R
exc
ż
pl, uq “
l
to level
u
at equilibrium is
rexc pl, u, E0 qf eq pE0 q dE0 “ 3ful nl Cpne , Te qψ
Cpne , Te q is a
˙
1 1 ´y
´
e dy.
x y
where ful is the absolute oscillatory strength (4.8),
ż8ˆ
ψpxq “
x
ˆ
∆Elu
Te
function of
ne
˙
,
and
Te ,
and
One can easily check
ˆ ˙
1
,
ψpxq “ O
x
as
x Ñ 0.
For large l, the value of ful is maximized at
∆Elu
is minimized at the same
Therefore, since
u “ l ` 1, for which
4
? l.
fl`1,l «
3π 3
post-collisional level u “ l ` 1,
2
∆El,l`1 « 3 IH .
l
nl “ Opl2 q,
Rexc pl, l ` 1q “ nl fl`1,l ¨ O
ˆ
1
∆El,l`1
˙
Similarly, at equilibrium the total rate of ionization from level
Rion plq “ nl ¨ O
ˆ
1
Bl
˙
l
“ Opl6 q.
(5.5)
is
“ nl ¨ Opl2 q “ Opl4 q.
(5.6)
(5.5) and (5.6) indicate that there are two dierent scales in the excitation/ionization problem.
First, the scales of excitation and ionization vary dramatically as the atom level
4
of ionization increases as Opl q and the rate of excitation increases as
for typical values of
l increases. The rate
104 „ 106
Opl6 q, a range of
kmax “ 10.
Second, the rate of excitation is much larger than that of ionization for large excitation level, with
ratio
Rexc pl, l ` 1q
“ Opl2 q.
Rion plq
Some numerical examples are presented in Section 6.5.
25
5.4 Particle/Continuum Representation
The plasma is described by a discrete set of particles for the electron diestribution
continuum description for the atoms
nk ptq and ions n` ptq.
f pE, tq and a
f pE, tq
The electron distribution function
is represented by the following discrete distribution
f pE, tq “ Ne
Ne
ÿ
δpE ´ Ej ptqq.
(5.7)
j“1
The number
units of
f
Ne “ Ne ptq
of electrons is not constant because of ionization and recombination. The
are (number of particles) per energy per volume, and the constant
Ne
is the eective
number which represents the number of physical particles per numerical particle per volume.
5.5 A No Time Counter DSMC method
This section describes in detail a Direct Simulation Monte Carlo method without time counter. The
ntot
and εtot
method follows the framework illustrated in Section 5.1. It preserves the total electron density
(including the free and bound electrons) and the total energy
εtot
, due to the fact that
ntot
are preserved in each collision. Moreover, since the algorithm satises detailed balance, if initially
the system is in an equilibrium state, then it is always in equilibrium. This is numerically tested in
Section 6.1.
Step 1 : Decide the rates of virtual collisions.
First nd upper bounds on the total rates of collisions of each type. The upper bounds need to
be not too large, while at the same time easy to compute. The actual total rates of collisions
in each type can be derived by taking the summation of the rates (4.3), (4.6), (4.9), (4.11)
over all electrons and atom levels:
rexc “ Ne
ÿÿÿ
rexc pl, u, Ek q
l uąl k
c
ÿ ÿ ÿ nl
2
? ful
Ne
“
me
Ek
l uąl k
ÿÿÿ
rdex “ Ne
rdex pu, l, Ek q
2
12πa20 IH
ˆ
1
1
´
∆Elu Ek
˙
1Ek ą∆Elu ,
u lău k
c
“
2
12πa20 IH
rion “ Ne
ÿÿ
l
ion pl, E q
rtot
k
k
c
2
“ 4πa20 IH
2
rrec “ Ne
?
ÿ ÿ ÿ nu g l
2
ful Ek
Ne
,
me
gu ∆Elu p∆Elu ` Ek q
u lău k
ÿÿ
ÿ ÿ nl
2
?
Ne
me
Ek
l k
ˆ
1
1
´
Bl Ek
˙
1Ek ąBl ,
rrec pl, Ei , Ej q
i‰j l
¸
ÿ
ÿ 1 ÿ
gl
gl
1
a
“
´
Ei l pEi ` Bl q2
Ei Ej l pEi ` Bl q2
i,j
i
˜
˜
¸
¸
ÿ 1 ÿ
ÿ 1 ÿ
ÿ
g
a20 n`
1
g
l
l
a
?
“
I2 N 2
´
.
4m2e h´3 g` H e j
Ei l pEi ` Bl q2
Ei l pEi ` Bl q2
Ej i
i
a20 n`
I2 N 2
4m2e h´3 g` H e
˜
ÿ
(5.8)
26
Note that these are numerical approximations of (2.12). They have the following upper bounds:
2
˜ “ 12πa20 IH
rexc ď rexc
˜ “ 12πa2 I 2
rdex ď rdex
0 H
c
ÿ ÿ 2
2
ful
?
Ne Ne nl
,
me
3 3 p∆Elu q3{2
l
uąl
c
ÿ
ÿ gl
2
ful
Ne Ne nu
,
me
g 2p∆Elu q3{2
u
lău u
c
ÿ 2
2
nl
?
Ne Ne
,
3{2
me
pB
q
3
3
l
l
˜
¸
2
ÿ
ÿ 1 ÿ
1
g
a
n
2
`
l
0
˜ “
a
?
rrec ď rrec
I2 N 2
,
4m2e h´3 g` me H e j
Ei l pEi ` Bl q2
Ej i
˜ “
rion ď rion
using the inequalities
2
4πa20 IH
2
?
a´3{2 , for
3 3
x´1{2 p1{a ´ 1{xq ď
x ě a,
and
?
2 x
apa`xq
ď a´3{2 .
Then
˜ ` rion
˜ ` rrec
˜ ` rdex
˜
R̃ “ rexc
is an upper bound on the total rate of all collisions.
Evaluation of
R̃ requires total computational cost of Oppkmax q2 ` Ne q at beginning,
with
kmax
the number of excitation levels. Since only one or two electrons and one or two atom levels
are updated after each collision, it is easy to update
R̃
in each step.
Step 2 : Update time and choose collision type.
For a time step
∆t, there are R̃∆t{Ne
virtual collisions. For the space homogeneous problem,
one can take
∆t “
Ne
.
R̃
This eliminates the time error. An equivalent choice is to take
Ne
lnp1{uq,
R̃
r0, 1s. This has
∆t “
with
u
a number uniformly sampled from
been widely used in the simulation
of chemical reactions since Gillespie's work [17]
Then update the time
t “ t ` ∆t.
r̃j
,
R̃
The collision type is chosen with probability
for
j P texc, dex, ion, recu.
Step 3 and 4 : Perform one virtual collision
We combined the steps 3 and 4 in Section 5.1. For the dierent collision types, the algorithms
are slightly dierent.
•
For excitation:
Step 3.1 Choose a free electron E0 uniformly.
Step 3.2 Choose a prior-collisional level l and post-collisional level u ą l with rate
ř
l
by the following method: Denote
nl ful 3{2
p∆Elu q
ř
ful
nl
,
uąl p∆E q3{2
lu
alu “
ful
p∆Elu q3{2
and
bl “
ř
uąl
alu . talu u
and
tbl u
are pre-computed. Then apply the direct search method described in Section 5.2.2)
to choose rst
Opkmax q.
l
nb
with rate ř l l
l
nl bl , then
27
u
with rate
alu
bl . The total cost in this step is
Step 3.3
Sample a number
q
r0, 1s. If
¯
´ E10 1E0 ą∆Elu
uniformly from
qď
?1
E0
´
1
∆Elu
2
1
?
3 3 p∆Elu q3{2
,
accept the collision and perform the excitation
pl, E0 q Ñ pu, E1 “ E0 ´ ∆Elu q.
Otherwise reject it.
•
For deexcitation:
Step 3.1 Choose a free electron E1 uniformly.
Step 3.2 Choose a prior-collisional level u and post-collisional level l ă u with rate
nu ggul 2p∆Eful q3{2
lu
,
ř
ř
gl
ful
n
u
u
lău gu 2p∆E q3{2
lu
by a method similar to Step 3.2 in the case of excitation.
Step 3.3
Sample a number
q
r0, 1s.
uniformly from
?
E1
∆Elu `E1
1
2p∆Elu q1{2
qď
If
,
accept the collision and perform the deexcitation
pu, E1 q Ñ pl, E0 “ E1 ` ∆Elu q.
Otherwise reject it.
•
For ionization:
Step 3.1 Choose a free electron E0 uniformly.
Step 3.2 Choose a prior-collisional level l with rate
nl
pBl q3{2
nl
l pBl q3{2
ř
Step 3.3
Sample a number
q
.
r0, 1s. If
¯
´ E10 1E0 ąBl
uniformly from
qď
?1
E0
´
1
Bl
2
1
?
3 3 pBl q3{2
Et
E0 Bk 1
1B ďE ďE .
E0 ´ Bk Et2 k t 0
accept the collision and sample the transfer energy
,
according to
and perform the ionization
pl, E0 q Ñ pE1 “ E0 ´ Et , E2 “ Et ´ Bl q.
Otherwise reject it.
•
For recombination:
Step 3.1 Pick up one free electron with with
?1
Ej
ř 1 ,
?
j
Ej
and denote it by
E1 .
Pick up a second electron with rate
?1
Ei
ř ´
i
28
gl
l pEi `Bl q2
ř
?1
Ei
gl
l pEi `Bl q2
ř
¯,
and denote it by
E2 .
As described in Section 5.2, this step can be accomplished
by several dierent methods. In the numerical section part, we apply four dierent
methods and compare their eciencies.
Step 3.2
If the two electrons are the same, reject the collision. Otherwise, choose the
l
post-collisional level
with rate
gl
pE2 `Bl q2
ř
gl
l pE2 `Bl q2 .
Step 3.3
Perform the recombination,
pE1 , E2 q Ñ pl, E0 “ E1 ` E2 ` Bl q.
6
Computational Results
This section presents computational results from the simulation method described above. 5 atomic
levels are included, unless specied. All the results except Section 6.4 are obtained using the binary
search method in recombination. However all the other methods described in Section 5.2 give the
same results.
6.1 Equilibrium initial data
The rst results are for a distribution of electrons, neutral atoms and ions that are in the equilibrium
Te “ 4.31 eV and number
100, 000 computational particles are used
distribution given by (2.5), (2.8a) and (2.8b), with initial temperature of
density
ntot “ 1028 m´3
for all (free and bound) electrons.
to represent the electron distribution function.
The model for the neutral atoms consists of ve
levels.
The left of Figure 1 shows the time evolution of the atom density
right shows the electron energy distribution function
f pEq
nk ,
for dierent level
k;
the
at various times. The plot on the top
is for a simulation that includes excitation/deexcitation alone with no ionization/recombination;
while that on the bottom is for ionization/recombination with no excitation/deexcitation. In this
simulation the time step is
∆t “ 6.5´11
for the ionization/recombination.
ps for the excitation/deexcitation, and
∆t “ 4.5´10
8
The simulation is performed for about 10 time steps.
ps
These
results show that the simulation method does preserve the equilibrium distribution as required.
The uctuations in the results are due to the statistical error from a nite number of computational
particles.
6.2 Non-equilibrium initial data
The second simulation results, shown in Figure 2 - 7, are for initial data for which all of the
neutral atoms are distributed at the ground level, with total number density
na “ 5 ˆ 1027 m´3 .
The electrons and ions are initially in an equilibrium distribution given by (2.5) and (2.8b), with
initial temperature of
Te “ 50000K “ 4.31
eV and number density
ne “ 5 ˆ 1027 m´3 . 1, 000, 000
computational particles are used to represent the electron distribution function. The model for the
neutral atoms consists of ve levels.
In Figures 2, 3, 4, only excitation/deexcitation processes are included. The plots in the left column
of Figure 2 show the density of the ve dierent atomic levels (blue bars) at various times, demonstrating convergence to equilibrium (white bars) given by (2.8a). The plots in the right column show
the density distribution of free electrons (blue lines) at various times, demonstrating convergence
to equilibrium (red dotted lines) given by (2.5). Note that the atomic level population inversion
29
27
44
x 10
k=5
16
x 10
t = 0.002825 ps
t = 0.005682 ps
t = 0.008539 ps
t = 0.011395 ps
t = 0.014254 ps
t = 0.017111 ps
t = 0.019968 ps
14
k=1
EEDF / m−3 J−1
12
k=4
k=3
10
8
6
4
k=2
0.005
2
0.01
time / ps
0.015
0
0.02
27
0
0.2
0.4
0.6
0.8
1
energy / IH
1.2
1.4
1.6
1.8
2
44
x 10
16
k=5
x 10
t = 0.002423 ps
t = 0.005053 ps
t = 0.007686 ps
t = 0.010319 ps
t = 0.012945 ps
t = 0.015571 ps
t = 0.018201 ps
14
k=1
EEDF / m−3 J−1
12
k=4
k=3
10
8
6
4
k=2
2
0.005
0.01
time / ps
0.015
0.02
0
0
0.2
0.4
0.6
0.8
1
energy / IH
1.2
1.4
1.6
1.8
2
Figure 1: Atomic (left) and electron (right) density function at various times starting from equilibrium, with excitation/deexcitation only (top), and ionization/recombination only (bottom) 100000
samples for free electrons.
30
is expected, because the level degeneracy grows faster than the exponential drop o due to
∆E
(except for between level 1 & 2) since the higher levels are close together.
Figure 3 shows the time evolution of the electron temperature (top)
ş8
εe
2 0 Ef pEq dE
ş
“
Te “
p3{2qne
3 08 f pEq dE
and the normalized relative entropy (bottom)
H“
where
H
H
ntot
Teeq “ 27959K determined from the
Te decays to Teeq as time evolves, due
is dened by (3.13), with the equilibrium temperature
density and energy conservation. The electron temperature
to the fact that the excitations from ground level to excited levels dominate in the evolution. The
normalized relative entropy function (negative entropy) is monotonically decreasing to 0, except for
uctuations due to the nite number of sampled particles which scales like
Figure 4 shows the time evolution of various ratios. In the top we show
type of virtual collision; in the middle
collisions; in the bottom
?
1{ N .
r̃j
, the fraction of each
R̃
rj
, the ratio of each type of accepted collision over all virtual
R̃
rj
, the acceptance rate of each type. Here
r̃j
j P texc, dexu.
In the middle
gure, the two ratios become equal as time evolves, meaning that the two processes get balanced.
In the bottom gure, the acceptance rates of each virtual collision are above 50%, showing the high
eciency of our method.
consecutive collisions.
Each point in these gures is computed by taking average over 50000
Note that compared to the entropy decay in Figure 3, these ratios reach
steady states in a much shorter time. This is because the collisions involving high levels (in this
example, level 4 and 5) have much larger rates, and hence dominate in computing these ratios. The
contribution from the low levels is very small. As the high levels can reach equilibrium in a very
short time, these ratios reach steady states very fast. On the other hand, the contribution from the
low levels in computing the entropy is not negligible, leading to the slow decay of the entropy in
Figure 3.
Note that with only excitation/deexcitation, the stationary state of the electron distribution diers
from the equilibrium by a periodic function
the period of
η
η,
according to (3.40). However with 5 excitation levels
is
∆E “
1
IH
32 42 52
“
IH
,
3600
which is too small to be observed in this experiment. The relaxation towards this stationary state
does not preclude the generation of physical entropy, as shown by Figure 3, since according to the
discussion of Section 3.2, the evolution of the system can be described by either form of entropy,
but it does imply that by allowing only transitions with nite energy gaps, the nal value of
does not correspond to the absolute minimum (maximum of physical entropy).
H
This eect was
also observed in a non-stochastic solver [18], and is not an artifact of our numerical approach, but
rather a physical feature of the system, indicating breakdown of ergodicity for system with only
excitation/deexcitation.
In Figure 5, 6, 7, similarly results are obtained with only ionization and recombination. In contrast
to the previous case of excitation/deexcitation, there is a continuum of energy that can be exchanged
with the free electrons. Therefore in this case the stationary state of the electron distribution is
exactly the equilibrium state.
Figure 6 shows the time evolution of the degree of ionization (top), the electron temperature (middle)
and the normalized relative entropy (bottom), with
31
H
dened by (3.13).
More recombination
atom distribution / m−3
27
t = 0.00 ps
8
x 10
45
8
actual
equilibrium
6
4
2
2
1
2
3
Levels
4
0
5
27
t = 0.02 ps
8
8
6
6
4
4
2
2
1
2
3
Levels
4
0
5
27
t = 0.50 ps
8
Figure 2:
0
0.5
1
Energy / IH
1.5
2
0.5
1
Energy / IH
1.5
2
0.5
1
Energy / IH
1.5
2
x 10
0
45
x 10
8
6
6
4
4
2
2
0
actual
equilibrium
45
x 10
0
electron distribution / m−3J −1
6
4
0
x 10
1
2
3
Levels
4
0
5
x 10
0
Snapshots of time evolution of atom density (left) and electron distribution (right),
compared to the equilibrium (white bars and red dotted lines), starting from all neutral atoms at
the ground level. Only excitation/deexcitation processes are included.
collisions happen than the ionization, leading to the decay of the degree of ionization. In the early
time, the electron temperature increases due to the fact that electron with lower energy is easied
combined with an ion. This can also be observed in the second snapshot of EEDF in Figure 5. As
the number of low energy electrons decreases, the high energy electrons start to dominate. More
high energy electrons are combined with ions, leading to the decrease of the electron temperature.
The relative entropy is monotonically decreasing to
0,
except for the statistical uctuations. Note
that compared to the excitation/deexcitation collisions (see Figure 3), the ionization/recombination
collisions take longer time to reach equilibrium. This is due to the fact that the collision rates for
excitation/deexcitation are larger than those of ionization/recombination (see Figure 13 for an
example).
Again Figure 7 shows the acceptance rate of each virtual ionization collision is above
acceptance rate of each virtual recombination collision is almost
50%.
The
100%, since we are using the binary
search method for the recombination process and the only rejected collisions in our algorithm (Step
3.2) are those with two identical incident electrons.
32
4
electron temperature / K
5
x 10
4.5
4
3.5
3
temperature at equilibrium
2.5
0
0.01
0.02
0.03
0.04
0.05
time / ps
relative entropy
0.06
0.07
0.08
0.09
0.1
0
0.01
0.02
0.03
0.04
0.05
time / ps
0.06
0.07
0.08
0.09
0.1
0
10
−1
10
−2
10
−3
10
−4
10
Figure 3: Time evolution of the electron temperature (top) and the normalized relative entropy
H
(bottom), starting from all neutral atoms at the ground level.
processes are included.
33
Only excitation/deexcitation
ratio of virtual collisions
1
excitation
deexcitation
0.5
0
0
0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004 0.0045 0.005
time / ps
ratio of actual collisions
0
0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004 0.0045 0.005
time / ps
acceptance rate
0
0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004 0.0045 0.005
time / ps
1
0.5
0
1
0.5
0
Figure 4: Time evolution of the fraction of each type of virtual collision (top), the ratio of each type
of accepted collision over all virtual collisions (middle) and acceptance rate (bottom), starting from
all neutral atoms at the ground level. Only excitation/deexcitation processes are included.
34
−3
27
t = 0.00 ps
8
45
atom distribution / m
x 10
4
actual
equilibrium
6
2
2
1
1
2
3
Levels
4
0
5
t = 0.02 ps
4
6
3
4
2
2
1
1
2
3
Levels
4
0
5
t = 0.50 ps
Figure 5:
1
Energy / I
1.5
2
1
Energy / I
1.5
2
1
Energy / IH
1.5
2
x 10
0
0.5
45
x 10
4
6
3
4
2
2
1
0
0.5
H
27
8
0
45
x 10
0
actual
equilibrium
H
27
8
−3 −1
electron distribution / m J
3
4
0
x 10
1
2
3
Levels
4
0
5
x 10
0
0.5
Snapshots of time evolution of atom density (left) and electron distribution (right),
compared to the equilibrium (white bars and red dotted lines), starting from all neutral atoms at
the ground level. Only ionization/recombination processes are included.
35
0.5
0.45
degree of ionization
0.4
0.35
0.3
0.25
0.2
0
0.01
0.02
0.03
0.04
0.05
time / ps
0.06
0.07
0.08
0.09
0.1
0.01
0.02
0.03
0.04
0.05
time / ps
0.06
0.07
0.08
0.09
0.1
0.3
0.35
0.4
0.45
0.5
4
x 10
9.5
9
electron temperature / K
8.5
8
7.5
7
6.5
6
5.5
5
4.5
0
relative entropy
0
10
−1
10
−2
10
−3
10
−4
10
−5
10
0
0.05
0.1
0.15
0.2
0.25
time / ps
Figure 6: Time evolution of the degree of ionization (top), the electron temperature (middle) and
the normalized relative entropy
H
(bottom), starting from all neutral atoms at the ground level.
Only ionization/recombination processes are included.
36
ratio of virtual collisions
ionization
recombination
1
0.5
0
0
0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016 0.0018 0.002
time / ps
ratio of actual collisions
0
0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016 0.0018 0.002
time / ps
acceptance rate
0
0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016 0.0018 0.002
time / ps
1
0.5
0
1
0.5
0
Figure 7: Time evolution of the fraction of each type of virtual collision (top), the ratio of each type
of accepted collision over all virtual collisions (middle) and acceptance rate (bottom), starting from
all neutral atoms at the ground level. Only ionization/recombination processes are included.
37
6.3 The stationary states without ionization/recombination
In this section we compute the stationary states with only excitation/deexcitation processes. The
model includes only two atomic levels. The period of the stationary function
Initially the free electrons are
ηpEq
in (3.40) is
3
∆E “ ∆E12 “ L2 ´ L1 “ IH .
4
placed near E “ ∆E with distribution
2
f pE, 0q “ ?
with
σ0 “
IH
10 and
ne “ 0.9ntot ,
where the total density distribution is
The time evolution is shown in Figure 8.
period of
´ pE´∆Eq
ne
2
2σ0
e
,
2πσ0
(6.1)
ntot “ 1028 m´3 .
The stationary electron distribution clearly shows the
3
4 IH . Figure 9 shows the time evolution of the normalized entropy
H“
H
ntot , where
H
is
the entropy dened by (3.28)(3.29) with the stationary solutions given by (3.25). This veries the
convergence to the theoretical result.
Furthermore, one can see from Figure 9 that the relaxation is much slower than the one in Figure
3. This is due to the fact that the collision between the electron and the higher atomic level has
a larger rate. In Figure 3, ve atomic levels are included; while in Figure 9 only two atomic levels
included.
We have also solved a similar problem with three atomic levels and observed the period of
1
36 IH in
the stationary solution.
In the next we start with the same initial data, but with all the processes (excitation/deexcitation
and ionization/recombination) included. The time evolutions of the atomic distribution, with only
two levels, and the electron distribution are shown in Figure 10. With the ionization/recombination
f
odicity in electron distribution f
processes included, the EEDF
decays to the Maxwell-Boltzmann distribution. However the perican still be observed at the early time of evolution, due to the fact
that the collision rates of excitation/deexcitation is higher than those of ionization/recombination
(see Figure 13 for an example).
6.4 Comparison of dierent methods for sampling electrons in recombination
We apply the four methods described in Section 5.2 to the excitation/ionization problem and make
a comparison.
The rst test problem is the simulation of a system in equilibrium. Initially all atoms are ionized
and free electrons are given by a Maxwellian.
ntot “ 1028 m´3 and the temperature of free electron Te “ 4.31
´3
eV. The degree of ionization stays around 38.04%. The simulation is performed until time t “ 10
6
ps, which requires around 2 ˆ 10 time steps for Ne “ 100000.
We take the total electron density
We solve this problem with numbers of simulation particles for electrons to be
Ne “ 781, 1562, 3150, 6250, 12500, 25000, 50000, 100000, 200000, 400000, 800000
respectively for the initial free electrons. We track the time spent on the processes related to the
sampling of electrons for recombination collisions. For reduced rejection method this includes the
rebuilding of the Marsaglia tables and the sampling process. For the binary search method, this
includes the updating of the summation tables and the sampling process. For direct search method,
only the sampling process are timed.
38
atom distribution / m−3
26
t = 0.00 ps
8
x 10
2
6
1.5
4
1
2
0.5
0
1
electron distribution / m−3J −1
46
actual
stationary
0
2
x 10
actual
stationary
0
0.5
1
1.5
2
Energy / IH
2.5
3
3.5
1
1.5
2
Energy / IH
2.5
3
3.5
1
1.5
2
Energy / IH
2.5
3
3.5
Levels
26
t = 0.02 ps
8
46
x 10
2
6
1.5
4
1
2
0.5
0
1
0
2
x 10
0
0.5
Levels
26
t = 0.52 ps
8
46
x 10
2
6
1.5
4
1
2
0.5
0
1
0
2
x 10
0
0.5
Levels
Figure 8: Snapshots of time evolution of atom density (left) and electron distribution (right), compared to the stationary states (white bars and red dashed lines), starting from the initial data (6.1).
Only excitation/deexcitation processes are included.
200000
samples are used for free electrons.
relative entropy
0
10
−1
10
−2
10
−3
10
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
time / ps
Figure 9: Time evolution of the normalized relative entropy
H
dened by (3.28)(3.29), with the
initial data (6.1). Only excitation/deexcitation processes are included.
free electrons.
39
200000
samples are used for
27
t = 0.000 ps
2
x 10
atom distribution / m−3
46
2
actual
stationary
1.5
1
0.5
0.5
1
electron distribution / m−3J −1
actual
stationary
1.5
1
0
x 10
0
2
0
1
2
Energy / IH
3
1
2
Energy / IH
3
1
2
Energy / IH
3
Levels
27
t = 0.002 ps
2
46
x 10
2
1.5
1.5
1
1
0.5
0.5
0
1
0
2
x 10
0
Levels
27
t = 0.010 ps
2
46
x 10
2
1.5
1.5
1
1
0.5
0.5
0
1
0
2
Levels
Figure 10:
x 10
0
Snapshots of time evolution of atom density (left) and electron distribution (right),
compared to the stationary states (white bars and red dashed lines), starting from the initial data
(6.1). All processes are included.
1000000
samples are used for free electrons.
40
time spent on sampling for recombination
5
time spent on sampling for recombination
4
10
10
regular rejection
direct search
reduced rejection
binary search
4
10
3
10
regular rejection
direct search
reduced rejection
binary search
3
10
slope = 1.35
slope = 1.85
2
10
slope = 1.90
slope = 1.18
10
cputime / s
cputime / s
slope = 1.27
slope = 1.24
2
1
10
1
10
slope = 1.11
0
10
slope = 0.96
0
−1
10
10
−1
−2
10
10
−2
−3
10
3
4
10
5
10
10
number of initial samples
10
6
10
3
4
10
5
6
10
10
number of initial samples
10
Figure 11: The time cost for recombination collisions when various numbers of samples are used for
the initial free electron distribution. Left: the system stays in equilibrium for all the time. Right:
the equilibration process with all atoms ionized at beginning.
The results are shown in the left of Figure 11. As expected the direct search method needs
computations.
It is getting extremely slow when more than
106
samples are used.
OpNe2 q
The accep-
1.35 q computation, which is very close to the theoretical results.
tance/rejection method needs OpNe
However, due to the large upper bound caused by the singularity of collision rate as
E Ñ 0,
the
acceptance rate in the acceptance/rejection method is very small, leading to ineciency of the acceptance/rejection method. The reduced rejection method is faster than the acceptance/rejection
OpNe1.25 q computation. The cost for binary search method is OpNe q, which agrees
with the theoretical result OpNe log Ne q. Note that the binary method is about 10 times faster than
method and needs
the reduced rejection method, although these two methods have almost the same convergence order.
A similar test is applied on the simulation of a equilibration process.
electron density
ntot “
Again we take the total
1028 m´3 and the temperature of free electron
Te “ 4.31 eV. In this case the
44.50%. We simulate the decaying process until t “ 10´3 ps,
steps for Ne “ 100000. At the end time, the degree of ionization is
degree of ionization at equilibrium is
which requires around
around
106
time
48.18%, indicating that the system is close to the equilibrium. The simulation is performed
t “ 10´3 ps, which requires around 2 ˆ 106 time steps for Ne “ 100000. The results are
until time
shown in the right of Figure 11. We observe similar eciencies as in the equilibrium test.
6.5 The large scale variation
This section numerically checks the large scale variation in the rates illustrated in Section 5.3.
Figure 12 shows the rates for excitation and deexcitation at equilibrium. Here 10 atomic levels are
Ñ lq gives the rate of excitation or deexcitation (for k ă l or k ą l respectively)
l. The rates with k “ l are not dened and left blank in the gure. For each
6
level, the entries closest to the diagonal have the largest rates. These rates vary over a range of 10
exc
dex
for dierent levels. These rates are obtained by computing each terms in r
and r
in (5.8).
included. ratepk
from level
k
to level
Figure 13 shows the total rates from/to each level. The blue circle shows the total rate of excitation/deexcitation from a given level. The red cross shows the total rate of excitation/deexcitation
to a given level. The black square shows the total rate of ionization from a given level. The magenta
plus shows the total rate of recombination to a given level. As predicted in Section 5.3, the rates
41
log10 rate ( k −> l )
23
atom level l after collision
10
9
22
8
21
7
20
6
19
5
18
4
17
3
2
16
1
15
1
2
3
4
5
6
7
atom level k before collision
8
9
10
Figure 12: The total rates for excitation/deexcitation at equilibrium. The rates with
k“l
are not
dened and left blank in the gure.
of excitation/deexcitation vary over
dierent levels.
For high levels like
106 while those of ionization/recombination vary over 104 for
k “ 9, 10, the rates of excitation/deexcitation are 100 times
larger than those of ionization/recombination.
Note that the total excitation/deexciation rate to/from level 10 is smaller that that of level 9. This
is due to the fact that we only include 10 levels. If the more levels are included, we will have the
opposite relation.
7
Conclusions and Future Work
This paper presents an idealized model for excitation/deexcitation and ionization/recombination
by electron impact in plasma. The energetic cross sections for the model are all given by analytic
formulas and they are consistent with detailed balance. A Boltzmann type equation for the energy
distribution function is then formulated and an
H -theorem
is derived.
A conservative and equilibrium-preserving Direct Simulation Monte Carlo method is also developed
for this system. The rates of recombination has a singularity as the energy of electron goes to
0.
To overcome this diculty, four dierent methods are proposed and compared, among which the
binary search method and the reduced rejection method give the highest eciency.
A simulation method suers from two diculties: First, there are a large number of kinetic processes
(i.e., excitation and deexcitation between any two atomic levels) with widely varying rates. Fair
representation of all of these processes can require very small time steps. Second, random uctuations can exhaust the nite number of simulation particles in some state. In subsequent work, we
plan to develop coarse-graining and a multi-scale simulation strategy, for example the level grouping
42
24
10
22
total rates
10
20
10
excitation/deexcitation from this level
excitation/deexcitation to this level
ionization from this level
recombination to this level
18
10
16
10
1
2
3
4
5
6
7
8
9
10
levels
Figure 13: The total rates for all collisions at equilibrium.
method developed in [24], to reduce the eect of these two computational bottlenecks.
The model and method here has been developed under a number of simplifying assumptions: The
energy levels come from the Bohr model of hydrogen; only Hydrogen atoms and ions are included;
Coulomb collisions have been neglected.
Generalization to more realistic models will be further
investigated.
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44
Appendix A
Proof of Lemma 2.1
In Appendix A-D we provide the proof of several lemmas and propositions given in Section 3. For
The reader's convenience, we restate these results in the appendices.
The proof of Lemma 2.1 is given in this appendix.
Lemma 2.1.
Given
ntot ą 0
εtot ą 0 the total
ne , na and nk , @k
the total number of free and bound electrons, and
energy of the system, there are unique values of temperature
Te
and densities
satisfying
ne ` na “ ntot ,
ÿ eq
3
ne Te `
nk Lk ` neq
` L` “ εtot ,
2
k
with
neq
k
Proof.
neq
` “ ne
and
obtained using (2.6a)-(2.6b).
For ease of notation, the sux
eq
is ignored, with the understanding that all quantities
computed here are at thermodynamic equilibrium. For a given
with temperature
Te
ntot ,
the total energy at equilibrium
is
ÿ
3
εpTe q “ ne Te `
nk Lk ` n` L` ,
2
k
with
nk
and
n`
given by (2.8a)(2.8b). We only need to show
ε
is monotonic in
Te .
Charge neutrality gives
ntot ´ na “ ne “ n` “ n1{2
a γpTe q,
with
γpTe q “ CTe3{4 pZxa q´1{2 e´L` {p2Te q ,
where
´
˘ ¯1{2
`
e 3{2
C “ 2g` 2πm
h2
is a constant. The function
γ
can be shown to be monotonic in
ř
´Lk {Te
dγ
3 γ
L` γ
γ
k gk Lk e
ř
“
`
´
.
´Lk {Te
dTe
4 Te
Te 2Te 2Te2
k gk e
We dene by
x¨y
the averaging over the atomic partition function
Zxa ,
Te :
(A.1)
i.e. for
ϕ “ pϕ1 , . . . , ϕkmax q,
dene
ř
´Lk {Te
ÿ nk
k ϕk gk e
xϕy “ ř
“
ϕk .
´Lk {Te
na
k gk e
k
The last term in (A.1) can be recognized as
xLy.
Since
L` ą Lk , @k
majorant for this expression
„
ȷ
dγ
γ 3 L` xLy
3γ
“
`
´
ě
ą 0.
dTe
2Te 2
Te
Te
4Te
The solution for the neutral atom density is then
na1{2 “ ´
γ 1a 2
`
γ `4ntot ,
2 2
na “ ´a
4n2tot
γ 2 `4ntot ` γ
45
¯2 ,
(A.2)
and
γ ě 0,
we can nd a
with the following variation
dna
´2na
“a
2
dγ
γ ` 4ntot ` γ
1` a
¸
γ
ă 0.
γ 2 ` 4ntot
Te increases, and since
dne
ne monotonically increases with Te , i.e. dTe ě 0.
Therefore,
na
˜
monotonically decreases as
ne “ ntot ´ na ,
this implies that
The variation of the energy can be computed as follows
dε
3
“ ne `
dTe
2
ˆ
3
Te ` L`
2
˙
dne ÿ
dnk
` Lk
.
dTe
dTe
k
(A.3)
The last term is not a-priori trivial; we expect the population of excited states to increase with
(note that the ground state, being depleted, does not contribute to the sum since
L1 ” 0),
Te
but it is
entirely possible that the rate of changes of some levels increase while others decrease, especially as
ionization proceeds preferentially from the upper states. Let us consider this variation,
dnk
dna x ´1 ´Lk {Te nk x ´1 ÿ
nk Lk
“
pZa q gk e
´ 2 pZa q
gk Lk e´Lk {Te `
.
dTe
dTe
Te
Te2
k
Identifying again the averaging over the atomic partition function
Zxa ,
we obtain
nk dna
nk
dnk
“
` 2 pLk ´ xLyq .
dTe
na dTe
Te
Inserting into (A.3),
¸
ÿ nk Lk dne ÿ nk `
˘
3
Te ` L` ´
`
L2k ´ Lk xLy
2
2
na
dTe
Te
k
k
ˆ
˙
˘
3
dne
na `
3
Te ` L` ´ xLy
` 2 xL2 y ´ xLy2 .
“ ne `
2
2
dTe Te
3
dε
“ ne `
dTe
2
˜
The second term is nonnegative since
L` ą Lk
for any
k and
dne
dTe
ě 0.
The third term is nonnegative
due to Cauchy-Schwarz inequality. Therefore at equilibrium, the total energy is also monotonically
increasing as a function of
Te .
This completes the proof of Lemma 2.1.
Appendix B
Proof of Lemma 3.6
This appendix gives the proof of Lemma 3.6.
Lemma 3.6.
Suppose
f pEq
is a smooth function and
eq
f eq , tneq
k u, n`
are the equilibrium functions
dened by (2.5)-(2.8b), sharing the same total number density and total energy with
n` .
f , tnk u
and
If
nk f0
n` f1 f2
eq eq “ eq eq eq ,
nk f0
n` f 1 f 2
for every
E0
and
Et ,
then
f “ f eq ,
Proof.
nk “ neq
k ,
n` “ neq
`.
Denote
Ck “
nk
,
neq
k
C` “
46
n`
,
neq
`
f
f˜ “ eq .
f
(B.1)
Then
Ck f˜pE0 q “ C` f˜pE0 ´ Et qf˜pEt ´ Bk q,
for any
k , E0 Et
satisfying
Bk ď Et ď E0 .
Et on both sides to obtain
1
˜
˜
´f pE0 ´ Et qf pEt ´ Bk q ` f˜pE0 ´ Et qf˜1 pEt ´ Bk q “ 0,
Take the derivative with respect to
f˜1 pE0 ´ Et q
f˜1 pEt ´ Bk q
“
.
f˜pE0 ´ Et q
f˜pEt ´ Bk q
Denote
ϕ “ log f˜,
then
ϕ1 pE0 ´ Et q “ ϕ1 pEt ´ Bk q.
This formula is valid for any
where both
α
and
β
E0 ,
therefore
ϕ1 is a constant,
f˜ “ eαE`β ,
so that
f˜ has
the form
are constants. Then
Ck “ C` e´αBk `β “ C` e´αL` `β eαLk .
Denote
c0 “ C` e´αL` `β ,
then
Ck “ c0 eαLk ,
This means that
f , nk
and
n`
in (B.1) are at equilibrium with temperature
ˆ
where
Te
is the temperature of
C` “ c0 eαL` ´β .
f eq .
1
´α
Te
˙´1
,
From Lemma 2.1, the equilibrium temperature is unique, hence
α “ 0.
Then the conservation of number density gives
f˜ ” 1,
β“0
and
Ck “ 1,
c0 “ 1,
so that
C` “ 1,
which nishes the proof.
Appendix C
Proof of Proposition 3.1
Proposition 3.1.
(a) Suppose
tf, nk , n` u is the solution to the ionization/recombination kinetic
QeED “ QkED “ Q`
ED “ 0. Then the total density and total
equations, i.e. (2.19, 2.20) with
energy are conserved, i.e.,
d
d
ntot “
εtot “ 0,
dt
dt
ntot and εtot are dened by (2.4a) and (2.4b).
Suppose tf, nk , n` u is the solution to the excitation/deexcitation kinetic equations, i.e. (2.19)
`
e
k
(2.21) with QIR “ QIR “ QIR “ 0. Then the densities for each species and the total energy are
where
(b)
conserved, i.e.,
d
dt
where
εtot
ż
f pEq dE “
d ÿ
d
nk “
εtot “ 0,
dt k
dt
is dened by (2.4b).
47
(c) Suppose
tf, nk , n` u is the solution to the ionization/recombination and excitation/deexcitation
kinetic equations (2.19, 2.20, 2.21). Then the total density and total energy are conserved,
d
d
ntot “
εtot “ 0.
dt
dt
Proof.
For case (a), where only the ionization/recombinaiton processes are included, take
ϕ“1
in
(3.31) to obtain
˙
ÿ ij ˆ nk f0
n` f1 f2
eq eq
ion
Bt ne “
dE0 dEt .
eq eq ´ eq eq eq nk f0 v0 σ
n
f
n
f
f
`
0
1
2
k
k
Combining this with (3.32) leads to the conservation of total density of electrons
ÿ d
d
d
ntot “
ne `
nk “ 0.
dt
dt
dt
Now take
ϕ“E
in (3.31) to obtain
ż
Ef pEq dE “
Bt
ÿij
ˆ
pE1 ` E2 ´ E0 q
k
nk f0
n` f1 f2
eq eq ´ eq eq eq
nk f0
n` f1 f2
˙
eq
ion
neq
dE0 dEt .
k f0 v0 σ
Combining this with (3.32) and (3.33) leads to
ż
ÿ
d
d
d
d
εtot “
Ef pEq dE ` Lk nk ` Lion n`
dt
dt
dt
dt
k
ˆ
˙
ij
ÿ
nk f0
n` f 1 f 2
eq eq
ion
“
pE1 `E2 ´E0 ´Lk `Lion q
dE0 dEt .
eq eq ´ eq eq eq nk f0 v0 σ
n
f
n
f
f
`
0
1
2
k
k
Noting that
d
dt εtot
E1 ` E2 ` Bk “ E0
from energy conservation at the microscopic level, which yields
“ 0.
For case (b), where only the excitation/deexcitataiton processes are included, take
ϕ“1
and (3.37) to obtain
d
dt
Now take
ż
f pEq dE “
d ÿ
nk “ 0.
dt k
ϕ “ E in (3.35) to obtain
ˆ
˙
ż
ÿż
nl f0
nu f1
eq eq
exc
Bt Ef pEq dE “
pE1 ´ E0 q
dE0 .
eq eq ´ eq eq nl f0 v0 σ
n
f
n
f
u
0
1
l
lău
ϕk “ Lk in (3.37) gives
˙
ż ˆ
ÿ
ÿ
d
nl f 0
nu f1
eq eq
exc
´
nk Lk “
dE0 .
eq eq
eq eq pLu ´ Ll qnl f0 v0 σ
dt k
n
f
n
f
u
0
1
l
lău
As for the energy of atoms, taking
Therefore,
ż
ÿ
d
d
d
d
εtot “
Ef pEq dE ` Lk nk ` Lion n`
dt
dt
dt
dt
k
˙
ˆ
ÿż
nu f 1
nl f0
eq eq
exc
“
pE1 ´ E0 ` Lu ´ Ll q
dE0
eq eq ´ eq eq nl f0 v0 σ
n
f
n
f
u
0
1
l
lău
“ 0.
The last equal sign comes from microscopic energy conservation,
48
E1 “ E0 ´∆Elu .
in (3.35)
For case (c), where all processes are included, the conservations of moments come from the fact that
the IR operators and the ED operators conserve the moments separately.
Appendix D
Proof of Lemma 3.8 and Theorem 3.9
This appendix provides the proof of Lemma 3.8 and Theorem 3.9.
Lemma 3.8.
f is a solution to the excitation/deexcitation
f pE, t “ 0q. Then
d ÿ
f pE ` j∆E, tq “ 0, for any E ě 0,
dt j
(i). Suppose
(2.21) with initial value
where the summation is over all integers
(ii).
f
If
function
such that
pE ` j∆Eq ě 0.
f pEq “ f eq pE; Te qηpE; Te q, for some temperature Te
E with the period ∆E , then η is given by (3.26).
is in the form
ηpE; Te q
in
(iii). There is a unique value of
Proof.
j
kinetic equations (2.19)
(i). For any
Ẽ ą 0,
Te
and some periodic
satisfying the energy conservation (3.27).
take
ϕpEq “
8
ÿ
δpE ´ Ẽ ´ j∆Eq
(D.1)
j“0
in (3.35) to obtain (3.38), using the fact that
ϕpE1 q “ ϕpE0 ´ ∆Elu q “ ϕpE0 q,
since
ϕ
(ii). If
is a periodic function with period
f
∆E
∆Elu
and
is an integer multiplication of
∆E .
is in the form of (3.40), multiply both sides of (3.40) by (D.1) and integrate over
tE ą 0u
to obtain
ż ÿ
δpE ´ Ẽ ´ j∆Eqf pEq dE “
ż ÿ
j
δpE ´ Ẽ ´ j∆Eqf eq pEqηpEq dE,
j
ÿ
f pẼ ` j∆Eq “
ÿ
f eq pẼ ` j∆EqηpẼq.
j
j
Therefore,
ř
ηpẼq “ ř
j
j
ř
f pẼ ` j∆Eq
f eq pẼ ` j∆Eq
“
f pẼ ` j∆E, t “ 0q
.
ř eq
j f pẼ ` j∆Eq
j
(D.2)
The last equality comes from (3.38).
na and ne are constant in time. The total energy
(
eq
nk pTe q at temperature Te is
(iii). With only excitation/deexcitation included,
for
f“
f eq pE; T
e qηpE; Te q and
ż
εpTe q “
␣
Ef eq pE; Te qηpE; Te q dE `
ÿ
neq
k pTe qLk ` n` L` .
k
For a given total energy
ε
is monotonic in
εtot ,
to prove
εpTe q “ εtot
Te .
49
has a unique solution
Te ,
we only need to show
First notice that
where
x¨y
d
dTe pn` L` q
“ 0 and from the proof of Lemma 2.1,
ÿ
ÿ
˘
d
d eq
na `
neq
L
“
Lk
nk “ 2 xL2 y ´ xLy2 ě 0,
k
k
dTe k
dTe
Te
k
is dened in (A.2).
Next one can check
1
d eq
f “ f eq
dTe
Te
ˆ
E 3
´
T
2
˙
,
and
ÿ d
´η
d
η “ ř eq
f eq pE ` j∆Eq
dTe
dT
f
pE
`
j∆Eq
e
j
j
ˆ
˙
ÿ
´η{Te
E ` j∆E 3
eq
“ ř eq
´
f pE ` j∆Eq
.
Te
2
j f pE ` j∆Eq j
Hence, for
f “ f eq pE; Te qηpE; Te q, after simplication,
d
d eq
d
f “η
f ` f eq
η
dTe
dTe
dTe
ř
eq
f eq η∆E
j jf pE ` j∆Eq
ř
,
“´
¨
eq
Te2
j f pE ` j∆Eq
where the summation is over all integers
r¨s
j “ jpEq
E ` j∆E ě 0,
such that
i.e.
jě´
“
E
∆E , with
‰
representing the integer part.
Then for the electron energy
ż
εe pTe q “
one has
d
∆E
εe “ ´ 2
dTe
Te
Here the
For any
ż
ř
j
Ef pEqηpEq ř
eq
Ef eq pE; Te qηpE; Te q dE,
jf eq pE ` j∆Eq
dE
f eq pE ` j∆Eq
ř
ż
eq
∆E ÿ pj0 `1q∆E
j jf pE ` j∆Eq
eq
ř
dE
“´ 2
Ef pEqηpEq
eq
Te j j0 ∆E
j f pE ` j∆Eq
0
ř
ż
eq
∆E ∆E ÿ
j jf pE ` j0 ∆E ` j∆Eq
eq
ř
“´ 2
dE.
pE ` j0 ∆Eqf pE ` j0 ∆EqηpEq
eq
Te 0
j f pE ` j∆Eq
j0
”
ı
“ E ‰
E`j0 ∆E
summation in the numerator is over all integers j ě ´
“ ´j0 ´ ∆E
.
∆E
E P r0, ∆Eq,
j
denote
ÿ
Aj0 pEq “
jf eq pE ` j0 ∆E ` j∆Eq
jě´j0 ´rE{∆Es
ÿ
“
pj 1 ´ j0 qf eq pE ` j 1 ∆Eq
j 1 ě´rE{∆Es
ÿ
“
ÿ
j 1 f eq pE ` j 1 ∆Eq ´ j0
j 1 ě´rE{∆Es
f eq pE ` j 1 ∆Eq.
j 1 ě´rE{∆Es
Let
«ř
j0˚
“
j0˚ pEq
“
j 1 ě´rE{∆Es j
ř
1 f eq pE
j 1 ě´rE{∆Es f
50
eq pE
` j 1 ∆Eq
` j 1 ∆Eq
ff
,
then
Aj0 pEq ď 0,
for
j0 ď j0˚ ,
and
Aj0 pEq ą 0,
for
j0 ą j0˚ .
Therefore
ř
ż
eq
d
∆E ∆E ÿ
j jf pE ` j0 ∆E ` j∆Eq
˚
eq
ř eq
εe ě ´ 2
pE ` j0 ∆Eqf pE ` j0 ∆EqηpEq
dE
dTe
Te 0
j f pE ` j∆Eq
j0
ż ∆E
ÿ d
pE ` j0˚ ∆Eq
“
f pE ` j0 ∆Eq dE
dT
e
0
j0
ż ∆E
d ÿ
“
pE ` j0˚ ∆Eq
f pE ` j0 ∆Eq dE
dTe j
0
0
ż ∆E
ÿ
d
f pE ` j0 ∆E, t “ 0q dE.
“
pE ` j0˚ ∆Eq
dT
e j
0
0
d ř
d
We have used (3.38) in the last step. Note that
j0 f pE ` j0 ∆E, t “ 0q “ 0, one has dTe εe “ 0.
dTe
Therefore
d
d
d ÿ eq
d
ε“
εe `
nk L k `
pn` L` q ě 0.
dTe
dTe
dTe k
dTe
This ends the proof.
Theorem 3.9.
every
l, u.
Assume there is
Suppose
f pEq
∆E ą 0,
such that
∆Elu
is an integer multiplication of
∆E ,
for
is a smooth function such that
nu f1
nl f0
eq eq “ eq eq
nl f 0
nu f1
holds for every
l, u
and
E0 ,
in which
(2.8a)(2.8b) for the temperature
Te
eq
f eq “ f eq pE; Te q, tneq
k u “ tnk pTe qu
are of the form (2.5) and
determined from (3.27). Then
f pEq “ f eq pE; Te qηpEq,
nk “ neq
k pTe q,
where
ηpEq
Proof.
is the periodic function (3.26), with period
∆E .
Denote
Ck “
nk
,
neq
k
f
f˜ “ eq .
f
Then
Cl f˜pE0 q “ Cu f˜pE0 ´ ∆Elu q,
for any
l, u
and
E0 .
E0 on both
1
1
˜
˜
Cl f pE0 q “ Cu f pE0 ´ ∆Elu q,
Take the derivative with respect to
sides to obtain
f˜1 pE0 q
f˜1 pE0 ´ ∆Elu q
“
.
f˜pE0 q
f˜pE0 ´ ∆Elu q
Denote
ψ “ log f˜,
then
ψ 1 pE0 q “ ψ 1 pE0 ´ ∆Elu q.
l, u and E0 . Therefore ψ 1 is a periodic function
l and u, the period of ψ 1 is ∆E dened in (3.24).
This formula is valid for any
Since this is true for every
51
with period
∆Elu .
Then
ψpE0 q “ ψpE0 ´ ∆Eq ` C ,
for some constant
C.
Let
α “ C{∆E ,
one has
ψpE0 q ´ αE0 “ ψpE0 ´ ∆Eq ´ αpE0 ´ ∆Eq.
Denote
ηpEq “ exppψpEq ´ αEq,
then
η
is a periodic function with period
∆E
and
f pEq
is given by
f pEq “ f pE; Te qf˜pEq “ f pE; Te q exppψpEqq “ eαE f eq pE; Te qηpEq.
eq
eq
(D.3)
Moreover, from
Cl f˜pE0 q “ Cu f˜pE0 ´ ∆Elu q
one has
f˜pE0 ´ ∆Elu q
eαLl
Cl
1
“
“ α∆E “ αLu .
lu
Cu
e
e
f˜pE0 q
Choose a constant
c0
such that
C1 “ c0 eαL1 ,
nk “
then
Ck neq
k pTe q
Ck “ c0 eαLk
“
for any level
k.
Hence
c0 eαLk neq
k pTe q.
p1{Te ´ αq´1 . From
Lemma 3.8 (iii), the temperature is uniquely determined from the energy conservation, hence α “ 0.
Combined with (D.3),
f pEq
and
nk
are in the form of (3.40) with temperature
This completes the proof.
Appendix E
Algorithm for binary search method
This appendix provides the detailed algorithm for the application of binary search method in the
recombination problem.
E.1 Padding by zeros
For a given set
tvk , k “ 1, . . . , N u, rst enlarge its size to N1 by padding zeros.
N1 ě N . The method is easier to formulate for N1 ,
minimum power of 2 satisfying
Here
N1
is the
a power of two,
and padding by zeros also reserves some slots for possible new electrons generated in the ionization
collisions. In the following we assume
N
is a power of 2 and
N1 “ N ,
without loss of generality.
E.2 Building a table
Let
L “ log2 N .
Build a table to store the pre-computed summations,
L´l
j2ÿ
sl,j “
vk ,
with
0 ď l ď L, 1 ď j ď 2l .
k“pj´1q2L´l `1
Note that
sL,j “ vj ,
there is no need to store
1 ` 2 ` ¨¨¨ ` 2
Moreover, the table
tsl,j u
sL,j . The total memory used for storing this table is
L´1
L
“ 2 ´ 1 “ N ´ 1.
can be computed backwardly
sl,j “ sl`1,2j´1 ` sl`1,2j .
Hence we only need to carry out
N
summations.
52
E.3 Sampling a particle
Here we describe the algorithm for sampling.
r uniformly from
ř p0, 1q
k “ 1, q “ s0,1 “ j vj
for l “ 1, . . . , L do
q1 Ð sl,2k´1
p Ð qq1
if r ď p then
k Ð 2k ´ 1
q Ð q1
r Ð pr
Sample
Let
else
k Ð 2k
q Ð q ´ q1
r´p
r Ð 1´p
end if
end for
The variable
k
gives the solution for (5.4). The computational cost is
OpLq “ Oplog2 N q.
E.4 Update the table when some vk changes
If
vk
changes, there is exactly one entry in
a total number of
L
tsl,j u to be updated, for each l.
k“
L
ÿ
βl 2l ,
l“0
one only needs to update
sl,αl :“ sl,αl ` vknew ´ vkold ,
where
α0 “ 1
l “ 0, . . . , L,
and
αl “ 2αl´1 ` βL`1´l ´ 1.
This cost is also
Hence we need to update
entries in the table. More precisely, if we write the binary expansion of
Oplog2 N q.
53
k,