Мagnetized particle diffusion in a random electric field with jumping phase
Diffusion of charged particles across a constant magnetic field caused by random varying electric field is considered. The electric field is assumed to be a superposition of waves with a fixed frequency, various wave vectors and randomly jumping phase. The dependence of a diffusion coefficient on a...
Gespeichert in:
| Datum: | 2018 |
|---|---|
| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2018
|
| Schriftenreihe: | Вопросы атомной науки и техники |
| Schlagworte: | |
| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/147448 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Мagnetized particle diffusion in a random electric field with jumping phase / V.I. Zasenko, O.М. Cherniak // Вопросы атомной науки и техники. — 2018. — № 4. — С. 264-267. — Бібліогр.: 8 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
nasplib_isofts_kiev_ua-123456789-147448 |
|---|---|
| record_format |
dspace |
| spelling |
nasplib_isofts_kiev_ua-123456789-1474482025-02-23T17:18:26Z Мagnetized particle diffusion in a random electric field with jumping phase Дифузія замагнічених частинок у випадковому електричному полі зі стрибками фази Диффузия замагниченных частиц в случайном электрическом поле с прыжками фазы Zasenko, V.I. Cherniak, O.М. Нелинейные процессы Diffusion of charged particles across a constant magnetic field caused by random varying electric field is considered. The electric field is assumed to be a superposition of waves with a fixed frequency, various wave vectors and randomly jumping phase. The dependence of a diffusion coefficient on a field correlation time is calculated. The effect of particle trapping which is more pronounced for a field with a low frequency of phase jumps is taken into account. It is shown that the results of the statistical approach are consistent with the direct numerical simulation. Розглянуто дифузію заряджених частинок поперек постійного магнітного поля, обумовлену випадковим змінним електричним полем. Вважається, що електричне поле є суперпозицією хвиль з фіксованою частотою, різними хвильовими векторами та випадковими стрибками фази. Розрахована залежність коефіцієнта дифузії від часу кореляції поля. Враховано ефект захоплення частинок, який є більш вираженим для полів з малою частотою стрибків фази. Показано, що результати статистичного підходу узгоджуються з прямим числовим моделюванням. Рассмотрена диффузия заряженных частиц поперек постоянного магнитного поля, обусловленная случайным переменным электрическим полем. Предполагается, что электрическое поле представляет собой суперпозицию волн с фиксированной частотой, различными волновыми векторами и случайными прыжками фазы. Рассчитана зависимость коэффициента диффузии от времени корреляции поля. Учтен эффект захвата частиц, наиболее выраженный для полей с малой частотой скачков фазы. Показано, что результаты статистического подхода согласуются с прямым числовым моделированием. The work is supported by the Program on Plasma Physics, Controlled Fusion and Plasma Technology of the National Academy of Sciences of Ukraine, project No П-7/6300-2018. 2018 Article Мagnetized particle diffusion in a random electric field with jumping phase / V.I. Zasenko, O.М. Cherniak // Вопросы атомной науки и техники. — 2018. — № 4. — С. 264-267. — Бібліогр.: 8 назв. — англ. 1562-6016 PACS: 52.65.Cc https://nasplib.isofts.kiev.ua/handle/123456789/147448 en Вопросы атомной науки и техники application/pdf Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| topic |
Нелинейные процессы Нелинейные процессы |
| spellingShingle |
Нелинейные процессы Нелинейные процессы Zasenko, V.I. Cherniak, O.М. Мagnetized particle diffusion in a random electric field with jumping phase Вопросы атомной науки и техники |
| description |
Diffusion of charged particles across a constant magnetic field caused by random varying electric field is considered. The electric field is assumed to be a superposition of waves with a fixed frequency, various wave vectors and
randomly jumping phase. The dependence of a diffusion coefficient on a field correlation time is calculated. The
effect of particle trapping which is more pronounced for a field with a low frequency of phase jumps is taken into
account. It is shown that the results of the statistical approach are consistent with the direct numerical simulation. |
| format |
Article |
| author |
Zasenko, V.I. Cherniak, O.М. |
| author_facet |
Zasenko, V.I. Cherniak, O.М. |
| author_sort |
Zasenko, V.I. |
| title |
Мagnetized particle diffusion in a random electric field with jumping phase |
| title_short |
Мagnetized particle diffusion in a random electric field with jumping phase |
| title_full |
Мagnetized particle diffusion in a random electric field with jumping phase |
| title_fullStr |
Мagnetized particle diffusion in a random electric field with jumping phase |
| title_full_unstemmed |
Мagnetized particle diffusion in a random electric field with jumping phase |
| title_sort |
мagnetized particle diffusion in a random electric field with jumping phase |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2018 |
| topic_facet |
Нелинейные процессы |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/147448 |
| citation_txt |
Мagnetized particle diffusion in a random electric field with jumping phase / V.I. Zasenko, O.М. Cherniak // Вопросы атомной науки и техники. — 2018. — № 4. — С. 264-267. — Бібліогр.: 8 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT zasenkovi magnetizedparticlediffusioninarandomelectricfieldwithjumpingphase AT cherniakom magnetizedparticlediffusioninarandomelectricfieldwithjumpingphase AT zasenkovi difuzíâzamagníčenihčastinokuvipadkovomuelektričnomupolízístribkamifazi AT cherniakom difuzíâzamagníčenihčastinokuvipadkovomuelektričnomupolízístribkamifazi AT zasenkovi diffuziâzamagničennyhčasticvslučajnomélektričeskompolespryžkamifazy AT cherniakom diffuziâzamagničennyhčasticvslučajnomélektričeskompolespryžkamifazy |
| first_indexed |
2025-11-24T02:39:43Z |
| last_indexed |
2025-11-24T02:39:43Z |
| _version_ |
1849637715187859456 |
| fulltext |
ISSN 1562-6016. ВАНТ. 2018. №4(116) 264
MAGNETIZED PARTICLE DIFFUSION
IN A RANDOM ELECTRIC FIELD WITH JUMPING PHASE
V.I. Zasenko, O.М. Cherniak
Bogolyubov Institute for Theoretical Physics, Kyiv, Ukraine
E-mail: zasenko@bitp.kiev.ua
Diffusion of charged particles across a constant magnetic field caused by random varying electric field is consid-
ered. The electric field is assumed to be a superposition of waves with a fixed frequency, various wave vectors and
randomly jumping phase. The dependence of a diffusion coefficient on a field correlation time is calculated. The
effect of particle trapping which is more pronounced for a field with a low frequency of phase jumps is taken into
account. It is shown that the results of the statistical approach are consistent with the direct numerical simulation.
PACS: 52.65.Cc
INTRODUCTION
In recent times, there is still considerable interest in
waves with stochastic jumps of phases [1], the genera-
tion of which was implemented experimentally. It is due
to their ability to penetrate into overdense plasma [2, 3],
to maintain a low pressure discharge [4, 5], to heat
plasmas.
Previously, we considered the heating and longitudi-
nal diffusion of particles in the electric field of waves
with jumping phase [6]. It has been shown that phase
jumps not only increase the intensity of resonance parti-
cle heating but also involve nonresonance particles in this
process from a wide interval of their initial velocities.
In this paper an attention is focused on the transverse
diffusion of magnetized particles under the influence of
random electric fields with jumping phases. We are in-
terested in the dependence of particle diffusivity on a
frequency of phase jumps. The frequency of jumps de-
termines the decay rate of the Lagrangian correlation
function of an electric field, and thus affects its integral
characteristic such as a diffusion coefficient. It can be
expected that the mean square displacement of particles
in a field with jumping phases after initial stage be-
comes linear in time, i.e. diffusion resembles the classi-
cal process. By reducing the frequency of phase jumps,
all the more influential is the particle trapping, and diffu-
sion become increasingly different from the classical
regime. To study the dependence of a particle transverse
diffusion coefficient on a correlation time which is de-
termined by a frequency of phase jumps, we generalize
our approach formulated earlier for frozen (constant in
time) random fields [7, 8] on varying random fields with
stochastically jumping phase.
1. EQUATION OF MOTION
AND ELECTRIC FIELD
WITH RANDOMLY JUMPING PHASE
Two components of particle drift velocity yxv , (zero
Larmor radius approximation) in a plane perpendicular
to a magnetic field under action of an electric field is
governed by the equations
( , ), ( , ),x yt v
y
v t
x
j j
¶ ¶
= - =
¶ ¶
r r (1)
where j is proportional to the potential of the electric
field with a coefficient dependent on a mass and charge
of a particle and a magnitude of constant magnetic field.
An electric field depends on two coordinate
, ) ( ( cos , sin )x y r rr r= =r
in a plane transverse to magnetic field and time t. If the
electric field is random, the motion of particles has the
character of wandering, and the characteristic of a rate
of particle excursion from a starting point is the diffu-
sion coefficient. Such diffusion resembles classical
wandering only at small correlation times of a random
field ( , )tj r , or coming over to dimensionless charac-
teristics, at Kubo numbers much smaller than unity. For
large Kubo numbers (i.e. long correlation times), parti-
cles in the course of the movement react on a local elec-
tric field profile. In the limit of a constant electric field
and zero Larmor radiuses their movement becomes fi-
nite when almost all particles move along closed trajec-
tories. Then the strongest effect is a particle trapping by
an electric field. The particle trapping changes the char-
acter of diffusion from classical to anomalous at least on
some temporal interval; here particle mean square dis-
placement deviates from linear law.
Previously, we considered the problem of a random
frozen electric field, in which the effect of particle trap-
ping is most pronounced. The statistically isotropic
stream function was taken as a superposition of
kN Nf´ harmonics with random phases ,i ja a
1 1
( ) ( , )
cos( cos( ) ),
k
NN
i
i j
i j i j
r
k r
f
j j r j
f r a a
= =
´
´ -
= =
- -
åår
(2)
where max( / )i kk k N i= × , and (2 / )j N jff p= × .
The total intensity is distributed over a partial harmonics
by the Gaussian law
2
2 2
0 2
2
exp i
i
kk
k k
e
j j
p
æ ö÷ç ÷ç ÷= -ç ÷ç ÷D ç ÷ç Dè ø
with a spectrum width kD and the interval between
neighboring wave numbers ke
max / 2.5k kD = , max / kk k Ne = .
Statistical description of particle spread in frozen field
(2) proposed in [7, 8] gave us the key characteristic such
as the Lagrangian correlation function of velocities which
was shown to be in agreement with direct simulation.
In this paper we consider an electric field varying in
time. The time dependence is twofold. First, we assume
the field is time harmonic with a frequency ω. Second, a
ISSN 1562-6016. ВАНТ. 2018. №4(116)
265
variation of field is also introduced through stochastic
jumps of the random phase ( )tb .
Let us consider low frequency electric field with ω
much less than a cyclotron frequency when a polariza-
tion drift may be neglected. Then particle drift motion is
governed the same Eq.(1), and for j we take:
1 1
( , , )
cos( ( )cos( ) ).
k
NN
i
i j
i j i j
r t
t k tr
f
j r j
a bw f r a
= =
=
´ - - ,, ,
´åå (3)
The set of random phases i ja a, remains con-
stant for each field realization, while the common phase
( )tb stochastically jumps in a course of particle mo-
tion. The phase ( )tb is changed abruptly by an arbi-
trary value with the frequency of a jump f and the prob-
ability p . It can be shown that the relation between a
correlation time of the field t and parameters of the
phase jumps is following
1
ln(1 )f p
t =-
-
. (4)
Depending on values of f and p , the correlation
time can vary from zero to infinity.
2. NUMERICAL SIMULATION
Orbits of 10 4 particles moving in a field with sto-
chastically jumping phases and governed by Eqs.(1), (3)
were calculated. After statistical averaging of simulation
data, a temporal dependence of a mean square dis-
placement for the various combinations of parameters p
and f was obtained, and a diffusion coefficients corre-
sponding to these parameters were calculated. For as-
ymptotic diffusion coefficient was taken its value at
instant several times greater than the field correlation
time.
3. STATISTICAL EQUATION
Previously, we have proposed a statistical descrip-
tion of particles [7] moving in accordance with the Eqs
(1) in a random field (2). Then it was refined with in-
volvement of a subensemble concept [8].
For each partial particle subensembles, marked by
the value of stream function there is the relation be-
tween the mean square displacement 2rá ñ and the diffu-
sion coefficient ( )D t :
2
0
2 ( ) .
t
r D t dtá ñ = ò (5)
In turn the diffusion coefficient ( )D t and the Lagrangi-
an correlation function ( )L
V t obey the Taylor relation:
0
( ) ( ) .
t
LD t V dt t= ∫ (6)
The Lagrangian correlation function ( )L
V t is un-
known, and it should be found from the Eulerian corre-
lation function ( )E
V t defined in the laboratory coordi-
nate system
2( ) ( ) ( ) ( ) ( ) .yE x x yV r v v v v= á ñ,á ñr 0 r 0 (7)
For given velocity field the Eulerian correlation func-
tion can be obtained by averaging over a statistical en-
semble. The Eulerian correlation function of velocities
(1) corresponding to the stream function (2) is found
explicitly by averaging over an ensemble of random
phases.
The most important step was to establish a relation
between Eulerian and Lagrangian correlation functions
2( ) ( ( ) )L EV t V r t= á ñ . (8)
The system of equations (5)-(8) is closed and can be
solved by numerical methods. The efficiency of such a
closure is verified by agreement between solutions
2( )r tá ñ , ( )D t , ( )L
V t and results of direct simulation.
Transition from constant fields with fixed random
phases (2) to time harmonic fields (3) with jumping
phase ( )tb introduce additional time dependence to the
Lagrangian correlation function
( ) ( )cos( )exp( / ).L LV t V t t tw t® - (9)
Then the diffusion of particles in a field with jumps
of phase is governed by equations of the same form as
(5)-(8), but with account for the additional dependence
of the Lagrangian correlation function on time (9).
4. RESULTS
In this section we compare numerical solutions of
statistical equations (5) - (9) with results obtained from
direct simulation of particle motion governed by
Eqs. (1), (3). In Figs. 1-3 results for particular choice of
jumping phase ( )tb that occur with frequency 1f =
and probability 0.03p = are given. According to
Eq.(4) the correlation time corresponding to this param-
eters of phase jumps is 32.83t = . In Fig. 4 results col-
lected from a set of calculation for various correlation
times are presented.
It is shown in Fig. 1 that the decay of the correlation
function cos (0)cos ( )tb b< > for stochastically jump-
ing phase ( )tb generated in numerical simulation is
exponential. This validates the form of the correlation
function ( )L
V t (9) that is taken in statistical equations.
Fig. 1. Exponential decay of the correlation function
cos (0)cos ( )tb b< > , 32.83t =
In Fig. 2 the Lagrangian correlation function of drift
velocity ( )L
V t found as a solution of statistical equation
is compared with one obtained in numerical simulation.
ISSN 1562-6016. ВАНТ. 2018. №4(116) 266
Fig. 2. Lagrangian velocity correlation function ( )L
V t
for 32.83t = ; statistical model (line) vs numerical
simulation (dots)
Double integration of the Lagrangian velocity corre-
lation function over time in statistical description gives
dispersion – particle mean square excursion from an
initial point. Particle dispersion obtained in statistical
model is compared with direct simulation in Fig. 3.
Fig. 3. Particle mean square displacement
for 32.83t = ; statistical model (line) vs numerical
simulation (dots)
Beyond the initial interval, the dispersion of parti-
cles becomes linear over time, and its time derivative
that is proportional to the running diffusion coefficient
becomes constant. Thus a linear dispersion at times la-
ger than field correlation time is characterized by a con-
stant asymptotic diffusion coefficient.
The asymptotic diffusion coefficient dependence
from inverse field correlation time 1 / t is shown in
Fig. 4. Solutions of the analytical model are compared
with results of direct numerical simulation in a wide
range of electric field correlation time
Fig. 4. Asymptotic diffusion coefficient for various
electric field correlation times; statistical model (line)
vs numerical simulation (dots)
CONCLUSIONS
While the diffusion of particles in fields with small
correlation time was studied in details, the development
of statistical descriptions for large correlation times,
when the particle trapping effect is not negligible, faced
difficulties. We have proposed a method of statistical
equation closure to overcome them. Earlier the method
was applied to study diffusion of magnetized particles in
random frozen electric field.
Here it is used to examine diffusion of particles
across a magnetic field undergoing time-harmonic elec-
tric field with jumping phase. Jumps of phase cause the
exponential decay of field correlation function. A corre-
lation time is determined by a frequency and probability
of phase jumps.
Main statistical characteristic of particle ensembles
such as the Lagrangian correlation function of particle
drift velocity, running and asymptotic diffusion coeffi-
cients, and mean square particle displacement were
found as solutions of statistical equations and as well in
direct numerical simulations. It is shown that solutions
of statistical equations agree with results obtained in
direct numerical simulations in wide interval of correla-
tion times. The statistical approach accounts for particle
trapping effect pronounced for low frequency electric
field and low frequency phase jumps.
ACKNOWLEDGMENTS
The work is supported by the Program on Plasma
Physics, Controlled Fusion and Plasma Technology of
the National Academy of Sciences of Ukraine, project
No П-7/6300-2018.
REFERENCES
1. Ya.B. Fainberg. Plasma electronics and plasma ac-
celeration of charged particles // Plasma Physics Re-
ports. 2000, v. 26, p. 335-343.
2. V.I. Karas’, V.D. Levchenko. Penetration of micro-
wave with a stochastic jumping phase into overdense
plasmas and electron collisionless heating by it //
Problems of Atomic Science and Technology. Series
“Plasma Electronics and New Acceleration Meth-
ods”. 2003, № 4, p. 133-136.
3. V.I. Karas’, Ya.B. Fainberg, A.F. Alisov, et al. In-
teraction of microwave radiation undergoing sto-
chastic phase jumps with plasmas or gases // Plasma
Physics Reports. 2005, v. 31, p. 748-760.
4. A.F. Alisov, A.M. Artamoshkin, et al. Low-pressure
discharge induced by microwave radiation with a
stochastically jumping phase // Plasma Physics Re-
ports. 2010, v. 36, p. 736-749.
5. A.F. Alisov, A.M. Artamoshkin, S.A. Berdin, et al.
Low Pressure Discharge Induced by Microwave Ra-
diation with a Stochastically Jumping Phase // Plas-
ma Physics Reports. 2010, v. 36, № 8, p. 736-749.
6. V.I. Zasenko, А.G. Zagorodny, O.M. Cherniak. Par-
ticle diffusion in a wave with randomly jumping
phase // Problems of Atomic Science and Technolo-
gy. Series “Plasma Physics”. 2015, № 1, p. 62-64.
7. V.I. Zasenko, А.G. Zagorodny, O.M. Chernyak.
Diffusion in a frozen random velocity field // Ukr. J.
Phys. 2011, v. 56, № 10, p. 1007-1010.
ISSN 1562-6016. ВАНТ. 2018. №4(116)
267
8. V.I. Zasenko, O.M. Chernyak, А.G. Zagorodny.
Subesemble concept in 2d magnetized particle
transport model // Problems of Atomic Science and
Technology. Series “Plasma Physics”. 2016, № 6,
p. 77-80.
Article received 06.06.2018
ДИФФУЗИЯ ЗАМАГНИЧЕННЫХ ЧАСТИЦ
В СЛУЧАЙНОМ ЭЛЕКТРИЧЕСКОМ ПОЛЕ С ПРЫЖКАМИ ФАЗЫ
В.И. Засенко, А.Н. Черняк
Рассмотрена диффузия заряженных частиц поперек постоянного магнитного поля, обусловленная слу-
чайным переменным электрическим полем. Предполагается, что электрическое поле представляет собой
суперпозицию волн с фиксированной частотой, различными волновыми векторами и случайными прыжками
фазы. Рассчитана зависимость коэффициента диффузии от времени корреляции поля. Учтен эффект захвата
частиц, наиболее выраженный для полей с малой частотой скачков фазы. Показано, что результаты стати-
стического подхода согласуются с прямым числовым моделированием.
ДИФУЗІЯ ЗАМАГНІЧЕНИХ ЧАСТИНОК
У ВИПАДКОВОМУ ЕЛЕКТРИЧНОМУ ПОЛІ ЗІ СТРИБКАМИ ФАЗИ
В.І. Засенко, О.М. Черняк
Розглянуто дифузію заряджених частинок поперек постійного магнітного поля, обумовлену випадковим
змінним електричним полем. Вважається, що електричне поле є суперпозицією хвиль з фіксованою часто-
тою, різними хвильовими векторами та випадковими стрибками фази. Розрахована залежність коефіцієнта
дифузії від часу кореляції поля. Враховано ефект захоплення частинок, який є більш вираженим для полів з
малою частотою стрибків фази. Показано, що результати статистичного підходу узгоджуються з прямим
числовим моделюванням.
MAGNETIZED PARTICLE DIFFUSION IN A RANDOM ELECTRIC FIELD WITH JUMPING PHASE
Introduction
|