The process of numerical thermalization in particle-in-cell (PIC) simulations has been studied extensively. It is analogous to Coulomb collisions in real plasmas, causing particle velocity distributions (VDFs) to evolve toward a Maxwellian as macroparticles experience polarization drag and resonantly interact with the fluctuation spectrum. This paper presents a practical tutorial on the effects of numerical thermalization in 2D PIC applications. Scenarios of interest include simulations, which must be run for many thousands of plasma periods and contain a population of cold electrons that leave the simulation space very slowly. This is particularly relevant to many low-temperature plasma discharges and materials processing applications. We present numerical drag and diffusion coefficients and their associated timescales for a variety of grid resolutions, discussing the circumstances under which the electron VDF is modified by numerical thermalization. Though the effects described here have been known for many decades, direct comparison of analytically derived, velocity-dependent numerical relaxation timescales to those of other relevant processes has not often been applied in practice due to complications that arise in calculating thermalization rates in 1D simulations. Using these comparisons, we estimate the impact of numerical thermalization in several examples of low-temperature plasma applications including capacitively coupled plasma discharges, inductively coupled plasma discharges, beam plasmas, and hollow cathode discharges. Finally, we discuss possible strategies for mitigating numerical relaxation effects in 2D PIC simulations.

1.
Z.
Donko
,
A.
Derzsi
,
M.
Vass
,
B.
Horvath
,
S.
Wilczek
,
B.
Hartmann
, and
P.
Hartmann
, “
eduPIC: An introductory particle based code for radio-frequency plasma simulation
,”
Plasma Sources Sci. Technol.
30
,
095017
(
2021
).
2.
S.
Rauf
,
D.
Sydorenko
,
S.
Jubin
,
W.
Villafana
,
S.
Ethier
,
A.
Khrabrov
, and
I.
Kaganovich
, “
Particle-in-cell modeling of electron beam generated plasma
,”
Plasma Sources Sci. Technol.
32
(
5
),
055009
(
2023
).
3.
M. M.
Turner
,
A.
Derzsi
,
Z.
Donko
,
D.
Eremin
,
S. J.
Kelly
,
T.
Lafleur
, and
T.
Mussenbrock
, “
Simulation benchmarks for low-pressure plasmas: Capacitive discharges
,”
Phys. Plasmas
20
,
013507
(
2013
).
4.
M.
Vass
,
P.
Palla
, and
P.
Hartman
, “
Revisiting the numerical stability/accuracy conditions of explicit PIC/MCC simulations of low-temperature gas discharges
,”
Plasma Sources Sci. Technol.
31
,
064001
(
2022
).
5.
D.-Q.
Wen
,
J.
Krek
,
J. T.
Gudmundsson
,
E.
Kawamura
,
M. A.
Lieberman
, and
J. P.
Verboncoeur
, “
Particle-in-cell simulations with fluid metastable atoms in capacitive argon discharges: Electron elastic scattering and plasma density profile transition
,”
IEEE Trans. Plasma Sci.
50
(
9
),
2548
2557
(
2022
).
6.
O. C.
Eldridge
and
M.
Feix
, “
Numerical experiments with a plasma model
,”
Phys. Fluids
6
,
398
(
1963
).
7.
J.-B.
Fouvry
,
B.
Bar-Or
, and
P.-H.
Chavanis
, “
Kinetic theory of one-dimensional homogeneous long-range interacting systems sourced by 1/N2 effects
,”
Phys. Rev. E
100
,
052142
(
2019
).
8.
R. W.
Hockney
, “
Measurements of collision and heating times in a two-dimensional thermal computer plasma
,”
J. Comput. Phys.
8
,
19
44
(
1971
).
9.
D.
Montgomery
and
C. W.
Nielson
, “
Thermal relaxation in one- and two-dimensional plasma models
,”
Phys. Fluids
13
,
1405
(
1970
).
10.
D.
Sydorenko
,
A.
Khrabrov
,
W.
Villafana
,
S.
Ethier
, and
S.
Janhunen
, see https://github.com/PrincetonUniversity/EDIPIC-2D for “
EDIPIC-2D online repository
” (
2022
).
11.
S. P.
Gary
,
Y.
Zhao
,
R. S.
Hughes
,
J.
Wang
, and
T. N.
Parashar
, “
Species entropies in the kinetic range of collisionless plasma turbulence: Particle-in-cell simulations
,”
Astrophys. J.
859
,
110
(
2018
).
12.
C.
Birdsall
and
A.
Langdon
,
Plasma Physics via Computer Simulation
(
Taylor and Francis
,
2004
).
13.
M.
Touati
,
R.
Codur
,
F.
Tsung
,
V. K.
Decyk
,
W. B.
Mori
, and
L. O.
Silva
, “
Kinetic theory of particle-in-cell simulation plasma and the ensemble averaging technique
,”
Plasma Phys. Controlled Fusion
64
,
115014
(
2022
).
14.
M. M.
Turner
, “
Kinetic properties of particle-in-cell simulations compromised by Monte Carlo collisions
,”
Phys. Plasmas
13
,
033506
(
2006
).
15.
P. Y.
Lai
,
T. Y.
Lin
,
Y. R.
Lin-Liu
, and
S. H.
Chen
, “
Numerical thermalization in particle-in-cell simulations with Monte-Carlo collisions
,”
Phys. Plasmas
21
,
122111
(
2014
).
16.
P. Y.
Lai
,
L.
Chen
,
Y. R.
Lin-Liu
, and
S. H.
Chen
, “
Study of discrete-particle effects in a one-dimensional plasma simulation with the Krook type collision model
,”
Phys. Plasmas
22
,
092127
(
2015
).
17.
H.
Okuda
and
C. K.
Birdsall
, “
Collisions in a plasma of finite-size particles
,”
Phys. Fluids
13
,
2123
(
1970
).
18.
B.
Abraham-Shrauner
, “
Test particle in a two-dimensional plasma
,”
Physica
43
,
95
104
(
1969
).
19.
M. A.
Reynolds
,
B. D.
Fried
, and
G. J.
Morales
, “
Velocity-space drag and diffusion in a model, two-dimensional plasma
,”
Phys. Plasmas
4
(
5
),
1286
1296
(
1997
).
20.
J. M.
Dawson
, “
Thermal relaxation in a one-species, one-dimensional plasma
,”
Phys. Fluids
7
,
419
(
1964
).
21.
J.
Hsu
,
G.
Joyce
, and
D.
Montgomery
, “
Thermal relaxation of a two-dimensional plasma in a d.c. magnetic field. Part 2. Numerical simulation
,”
J. Plasma Phys.
12
,
27
31
(
1974
).
22.
I.
Jechart
,
T.
Katsouleas
, and
J.
Dawson
, “
Anomalous thermal relaxation of a two-dimensional magnetized plasma
,”
Phys. Fluids
30
(
1
),
65
(
1987
).
23.
J.
Virtamo
and
H.
Tuomisto
, “
Verification of a simple collision operator for one-dimensional plasma by simulation experiments
,”
Phys. Fluids
22
,
172
(
1979
).
24.
J.
Dawson
, “
One-dimensional plasma model
,”
Phys. Fluids
5
,
445
(
1962
).
25.
J.-Y.
Hsu
,
D.
Montgomery
, and
G.
Joyce
, “
Thermal relaxation of a two-dimensional plasma in a d.c. magnetic field. Part 1. Theory
,”
J. Plasma Phys.
12
,
21
26
(
1974
).
26.
N.
Gatsonis
and
A.
Spirkin
, “
A three-dimensional electrostatic particle-in-cell methodology on unstructured Delaunay-Voronoi grids
,”
J. Comput. Phys.
228
,
3742
3761
(
2009
).
27.
S.
Averkin
and
N.
Gatsonis
, “
A parallel electrostatic Particle-in-Cell method on unstructured tetrahedral grids for large-scale bounded collisionless plasma simulations
,”
J. Comput. Phys.
363
,
178
199
(
2018
).
28.
A. T.
Powis
and
I. D.
Kaganovich
, “
Accuracy of the explicit energy-conserving particle-in-cell method for under-resolved simulations of capacitively coupled plasma discharges
,” arXiv:2308.13092 (
2023
).
29.
H.
Sun
,
S.
Banarjee
,
S.
Sharma
,
A. T.
Powis
,
A. V.
Khrabrov
,
D.
Sydorenko
,
J.
Chen
, and
I. D.
Kaganovich
, “
Direct implicit and explicit energy-conserving particle-in-cell methods for modeling of capacitively-coupled plasma devices
,”
Phys. Plasmas
30
,
103509
(
2023
).
30.
S. G.
Walton
,
D. R.
Boris
,
S. C.
Hernandez
,
E. H.
Lock
,
T. B.
Petrova
,
G. M.
Petrov
,
A. V.
Jagtiani
,
S. U.
Engelmann
,
H.
Miyazoe
, and
E. A.
Joseph
, “
Electron beam generated plasmas: Characteristics and etching of silicon nitride
,”
Microelectron. Eng.
168
,
89
96
(
2017
).
31.
S. G.
Walton
,
D. R.
Boris
,
S. G.
Rosenberg
,
H.
Miyazoe
,
E. A.
Joseph
, and
S. U.
Engelmann
, “
Etching with electron beam-generated plasmas: Selectivity versus ion energy in silicon-based films
,”
J. Vac. Sci. Technol., A
39
,
033002
(
2021
).
32.
K.
Nanbu
, “
Theory of cumulative small-angle collisions in plasmas
,”
Phys. Rev. E
55
,
4642
4652
(
1997
).
33.
T.
Charoy
,
J.-P.
Boeuf
,
A.
Bourdon
,
J.
Carlsson
,
P.
Chabert
,
B.
Cuenot
,
D.
Eremin
,
L.
Garrigues
,
K.
Hara
,
I. D.
Kaganovich
,
A. T.
Powis
,
A.
Smolyakov
,
D.
Sydorenko
,
A.
Tavant
,
O.
Vermorel
, and
W.
Villafana
, “
2D axial-azimuthal particle-in-cell benchmark for low-temperature partially magnetized plasmas
,”
Plasma Sources Sci. Technol.
28
,
105010
(
2019
).
34.
V. A.
Godyak
,
R. B.
Piejak
, and
B. M.
Alexandrovich
, “
Probe diagnostics of non-Maxwellian plasmas
,”
J. Appl. Phys.
73
,
3657
3663
(
1993
).
35.
Y.
He
,
Y.-M.
Lim
,
J.-H.
Lee
,
J.-H.
Kim
,
M.-Y.
Lee
, and
C.-W.
Chung
, “
Effect of parallel resonance on the electron energy distribution function in a 60 MHz capacitively coupled plasma
,”
Plasma Sci. Technol.
25
,
045401
(
2023
).
36.
I. D.
Kaganovich
and
L. D.
Tsendin
, “
The space-time-averaging procedure and modeling of the RF discharge, Part II: Model of collisional low-pressure RF discharge
,”
IEEE Trans. Plasma Sci.
20
(
2
),
66
75
(
1992
).
37.
S. V.
Berezhnoi
,
I. D.
Kaganovich
, and
L. D.
Tsendin
, “
Generation of cold electrons in a low-pressure RF capacitive discharge as an analogue of a thermal explosion
,”
Plasma Phys. Rep.
24
,
556
563
(
1998
).
38.
K.
Yee
, “
Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media
,”
IEEE Trans. Antennas Propag.
14
(
3
),
302
307
(
1966
).
39.
J.
Villasenor
and
O.
Buneman
, “
Rigorous charge conservation for local electromagnetic field solvers
,”
Comput. Phys. Commun.
69
,
306
316
(
1992
).
40.
PlasmaPy Community,
PlasmaPy, Version 2023.5.1
(
Zenodo
,
2023
).
You do not currently have access to this content.