The role of higher moments on the distribution of particles in the space of impulses at cyclotron resonances
The results of the analysis of the dynamics of charged particles under conditions of cyclotron resonances in the field of an intense electromagnetic wave are presented. Particular attention is paid to regimes with dynamic chaos. It is shown that there are two qualitatively different regimes. The app...
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Buts, V.A. Kuzmin, V.V. 2023-12-11T11:48:09Z 2023-12-11T11:48:09Z 2023 The role of higher moments on the distribution of particles in the space of impulses at cyclotron resonances / V.A. Buts, V.V. Kuzmin // Problems of Atomic Science and Technology. — 2023. — № 4. — С. 16-20. — Бібліогр.: 9 назв. — англ. 1562-6016 PACS: 05.45.-a, 05.10.Gg DOI: https://doi.org/10.46813/2023-146-016 https://nasplib.isofts.kiev.ua/handle/123456789/196166 The results of the analysis of the dynamics of charged particles under conditions of cyclotron resonances in the field of an intense electromagnetic wave are presented. Particular attention is paid to regimes with dynamic chaos. It is shown that there are two qualitatively different regimes. The appearance of the first one is due to the overlap of nonlinear cyclotron resonances. The second mode is related to intermittency. The moments and spectra of each of these regimes are determined. It is shown that with an increase in the intensity of an external electromagnetic wave, the first regime appears at the beginning and only then the second regime appears. A characteristic feature of the second regime is intermittency. Steps appear on the time dynamics of pulses in the second mode. It is shown that the spectra in the second mode are narrower than in the first mode. A characteristic feature of the second regime (the regime with intermittency) is the fact that the higher moments turn out to be larger than the lower moments. In the first regime, the highest moments decrease rapidly. To find the particle momentum distribution function, the generalized Fokker-Planck equation was used. Solutions of this equation are written out for some important cases. Викладено результати аналізу динаміки заряджених частинок в умовах циклотронних резонансів у полі інтенсивної електромагнітної хвилі. Особливу увагу приділено режимам з динамічним хаосом. Показано, що існує два якісно різні режими. Виникнення першого обумовлено перекриттям нелінійних циклотронних резонансів. Другий режим пов'язаний із перемежуванням. Визначено моменти та спектри кожного з цих режимів. Показано, що зі збільшенням напруженості зовнішньої електромагнітної хвилі на початку з’являється перший режим і потім другий. Характерною рисою другого режиму є перемежування. На залежності імпульсів від часу у другому режимі з’являються сходинки. Показано, що спектри у другому режимі вужчі, ніж у першому. Характерною особливістю другого режиму (режиму з перемежуванням) є той факт, що вищі моменти виявляються більшими, ніж нижчі. У першому режимі вищі моменти швидко зменшуються. Для знаходження функції розподілу частинок за імпульсами було використано узагальнене рівняння Фоккера-Планка. Для деяких важливих випадків виписано рішення цього рівняння. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Problems of Atomic Science and Technology Relativistic and nonrelativistic plasma electronics The role of higher moments on the distribution of particles in the space of impulses at cyclotron resonances Роль вищих моментів у розподілі частинок у просторі імпульсів при циклотронних резонансах Article published earlier |
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| title |
The role of higher moments on the distribution of particles in the space of impulses at cyclotron resonances |
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The role of higher moments on the distribution of particles in the space of impulses at cyclotron resonances Buts, V.A. Kuzmin, V.V. Relativistic and nonrelativistic plasma electronics |
| title_short |
The role of higher moments on the distribution of particles in the space of impulses at cyclotron resonances |
| title_full |
The role of higher moments on the distribution of particles in the space of impulses at cyclotron resonances |
| title_fullStr |
The role of higher moments on the distribution of particles in the space of impulses at cyclotron resonances |
| title_full_unstemmed |
The role of higher moments on the distribution of particles in the space of impulses at cyclotron resonances |
| title_sort |
role of higher moments on the distribution of particles in the space of impulses at cyclotron resonances |
| author |
Buts, V.A. Kuzmin, V.V. |
| author_facet |
Buts, V.A. Kuzmin, V.V. |
| topic |
Relativistic and nonrelativistic plasma electronics |
| topic_facet |
Relativistic and nonrelativistic plasma electronics |
| publishDate |
2023 |
| language |
English |
| container_title |
Problems of Atomic Science and Technology |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| format |
Article |
| title_alt |
Роль вищих моментів у розподілі частинок у просторі імпульсів при циклотронних резонансах |
| description |
The results of the analysis of the dynamics of charged particles under conditions of cyclotron resonances in the field of an intense electromagnetic wave are presented. Particular attention is paid to regimes with dynamic chaos. It is shown that there are two qualitatively different regimes. The appearance of the first one is due to the overlap of nonlinear cyclotron resonances. The second mode is related to intermittency. The moments and spectra of each of these regimes are determined. It is shown that with an increase in the intensity of an external electromagnetic wave, the first regime appears at the beginning and only then the second regime appears. A characteristic feature of the second regime is intermittency. Steps appear on the time dynamics of pulses in the second mode. It is shown that the spectra in the second mode are narrower than in the first mode. A characteristic feature of the second regime (the regime with intermittency) is the fact that the higher moments turn out to be larger than the lower moments. In the first regime, the highest moments decrease rapidly. To find the particle momentum distribution function, the generalized Fokker-Planck equation was used. Solutions of this equation are written out for some important cases.
Викладено результати аналізу динаміки заряджених частинок в умовах циклотронних резонансів у полі інтенсивної електромагнітної хвилі. Особливу увагу приділено режимам з динамічним хаосом. Показано, що існує два якісно різні режими. Виникнення першого обумовлено перекриттям нелінійних циклотронних резонансів. Другий режим пов'язаний із перемежуванням. Визначено моменти та спектри кожного з цих режимів. Показано, що зі збільшенням напруженості зовнішньої електромагнітної хвилі на початку з’являється перший режим і потім другий. Характерною рисою другого режиму є перемежування. На залежності імпульсів від часу у другому режимі з’являються сходинки. Показано, що спектри у другому режимі вужчі, ніж у першому. Характерною особливістю другого режиму (режиму з перемежуванням) є той факт, що вищі моменти виявляються більшими, ніж нижчі. У першому режимі вищі моменти швидко зменшуються. Для знаходження функції розподілу частинок за імпульсами було використано узагальнене рівняння Фоккера-Планка. Для деяких важливих випадків виписано рішення цього рівняння.
|
| issn |
1562-6016 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/196166 |
| citation_txt |
The role of higher moments on the distribution of particles in the space of impulses at cyclotron resonances / V.A. Buts, V.V. Kuzmin // Problems of Atomic Science and Technology. — 2023. — № 4. — С. 16-20. — Бібліогр.: 9 назв. — англ. |
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16 ISSN 1562-6016. Problems of Atomic Science and Technology. 2023. № 4(146)
https://doi.org/10.46813/2023-146-016
THE ROLE OF HIGHER MOMENTS ON THE DISTRIBUTION
OF PARTICLES IN THE SPACE OF IMPULSES
AT CYCLOTRON RESONANCES
V.A. Buts
1,2
, V.V. Kuzmin
1
1
National Science Center “Kharkov Institute of Physics and Technology”, Kharkiv, Ukraine;
2
Institute of Radio Astronomy of NAS of Ukraine, Kharkiv, Ukraine
E-mail: vbuts1225@gmail.com
The results of the analysis of the dynamics of charged particles under conditions of cyclotron resonances in the
field of an intense electromagnetic wave are presented. Particular attention is paid to regimes with dynamic chaos. It
is shown that there are two qualitatively different regimes. The appearance of the first one is due to the overlap of
nonlinear cyclotron resonances. The second mode is related to intermittency. The moments and spectra of each of
these regimes are determined. It is shown that with an increase in the intensity of an external electromagnetic wave,
the first regime appears at the beginning and only then the second regime appears. A characteristic feature of the
second regime is intermittency. Steps appear on the time dynamics of pulses in the second mode. It is shown that the
spectra in the second mode are narrower than in the first mode. A characteristic feature of the second regime (the
regime with intermittency) is the fact that the higher moments turn out to be larger than the lower moments. In the
first regime, the highest moments decrease rapidly. To find the particle momentum distribution function, the gener-
alized Fokker-Planck equation was used. Solutions of this equation are written out for some important cases.
PACS: 05.45.-a, 05.10.Gg
INTRODUCTION
Moments of random processes are one of the im-
portant characteristics of these processes. The first and
second moments (mean and variance) are of the greatest
importance. Higher moments play a secondary role and
are rarely used. In addition, in most cases, the moments
decrease rapidly as their number increases. However,
there are random processes, which are called intermit-
tent regimes, and are characterized by the fact that their
higher moments exceed the lower ones. Processes with
intermittency are distinguished by the fact that rare, but
very intense bursts appear against the background of
some moderate dynamics [1, 2]. In some cases, these
surges can pose a significant danger. It was found in [3]
that at cyclotron resonances at sufficiently high field
strengths of electromagnetic waves, regimes with inter-
mittency can also arise. Cyclotron resonances are wide-
ly used. In particular, they are used in thermonuclear
fusion facilities for plasma heating. Therefore, the study
of regimes with intermittency is of both general scien-
tific and practical interest. The first step in studying the
influence of higher moments on the development of
random processes can be the Fokker-Planck (FP) equa-
tion. However, the usual FP equation contains only the
first and second moments. In [3], a generalization of the
FP equation was written for the case when the influence
of moments with any number is taken into account. Be-
low we will consider some solutions to this equation.
The work consists of an introduction, three parts and
a conclusion. In the first part, the problem statement is
formulated, and the main system of equations is written
out. The second part presents the results of the analysis
of particle dynamics at cyclotron resonances. Two
qualitatively different modes of particle dynamics are
described. Spectra of particle dynamics in these regimes
are determined. In the third part, the generalized FP
equations are written out. This new equation takes into
account all higher moments. Note that in the usual ex-
pressions of the FP equation, only the first and second
moments are taken into account. Some analysis of this
equation is given, in particular, the results of the sta-
tionary regime are given. In conclusion, the most im-
portant results are formulated.
1. STATEMENT OF THE PROBLEM
AND BASIC EQUATIONS
Consider a charged particle that moves in an exter-
nal constant magnetic field directed along the axis z, and
in the field of a plane electromagnetic wave, which in
the general case has the following components:
Re( exp( )),
Re exp( )
E i t i
c
i t i
E α kr
H kE kr
(1)
where , ,x y zi α is wave polarization vector.
Without limiting of generality, we can choose a co-
ordinate system in which the wave vector of the wave
has only two components
xk and
zk . It is also conven-
ient to use the following dimensionless dependent and
independent variables:
/ mcp p , t ,
c
r r .
The equations of motion in these variables will have
the form:
1 Re Rei iHd
e e
d
p kp k
ε ph ε p , (2)
d
d
r p
v , 1
d
d
kp ,
where 0/ Hh H ; /H eH mc ; 0ε α ;
0 0( / )eE mc ; kr ; k unit vector in the
direction of the wave propagation; 2 1 2(1 )p – di-
mensionless particle energy (measured in units mc
2
);
p particle momentum.
,
ISSN 1562-6016. Problems of Atomic Science and Technology. 2023. № 4(146) 17
The system of vector equations (2) can be fully ana-
lyzed only by numerical methods. However, many im-
portant features of charged particle dynamics can be
discovered using new variables. We will use, similarly
to [4 - 6], the following variables
cosxp p , sinyp p ,
zp p , 2 2
x yp p p ,
sin
H
p
x
, cos
H
p
y
. (3)
For new variables, the system of equations (2) can
be reduced to the form:
0 1 cos( )z z n n
dp
k v J
d
, (4)
0 cosz
z n n
dp
k v J
d
, (5)
0 cosn n
d
v J
d
, (6)
n z xk z k n . (7)
where ( )n nJ J ( ) ,n nJ dJ d
/x Hk p .
Details of obtaining system (4)-(7) can be found in
[3, 5].
2. MODES WITH DYNAMIC CHAOS
Below we will consider two chaotic regimes. In the
first mode, chaos arises as a result of overlapping of
homoclinic trajectories (overlapping of non-linear cy-
clotron resonances). With an increase in the field
strength, a second regime arises the regime with in-
termittency.
The conditions for the emergence of the first regime
are formulated, for example, in [4, 5, 7], and can be
written in the form:
,
2/ 1 zk , 2/ 1H zk , (8)
where nonlinear resonance width, distance
between cyclotron resonances.
The regime with intermittency arises at sufficiently
high field strengths of the wave. It is characterized by
steps in the dependence of particle momenta on time.
Such a regime appears as a result of solving the Adler
equation, which describes the dynamics of particles at
high field strengths. This mode is described in detail in
[6].
2.1. OVERLAPPING OF NONLINEAR
CYCLOTRON RESONANCES
An analytical analysis of chaotic regimes in the case
of overlapping cyclotron resonances was carried out in
[4 - 6]. Such a regime is almost always observed at a
wave strength parameter (nonlinearity parameter) of
0.26 (
0 0.26 ). Below we will present some results of
a numerical analysis of this regime. For definiteness, we
choose the following parameters: 0 0.26 ,
0.878zk , 0.987H .
It can be shown that, for these parameters, the non-
linear cyclotron resonances are covered. In addition, we
choose the following initial conditions: (0) (0) 0x y ,
(0) / 2z , (0) (0) (0) 0.01x y zP P P .
Below, in Figs. 1 and 2, the dependence of the longi-
tudinal momentum of the particle on time and the spec-
trum are presented. It can be seen that the spectrum of
particle dynamics is wide.
Fig. 1. Longitudinal impulse, before the appearance
of steps,
0 0.26
Fig. 2. Spectrum before the appearance
of steps,
0 0.26
In the considered case, the moments have the fol-
lowing values:
even
0 1 ,
2 0.07 , 3
4 1.6 10 , 5
6 1.8 10 ;
odd 5
1 1.7 10 ,
3 0.001 , 5
5 5.2 10 ,
7
7 7.5 10 .
It can be seen that the magnitudes of the moments
rapidly decrease with increasing their number.
2.2. MODES WITH INTERMITTED
With an increase in the field strength of the wave, a
regime with intermittency arises. The results of a de-
tailed study of this mode are given in [6]. Figs. 3 and 4
present the results of a numerical analysis of this mode
for the value of the wave strength parameter
0 1.2 .
The initial conditions are the same as in Figs. 1 and 2.
Fig. 3. Longitudinal pulse in intermittent mode, 0 1.2
18 ISSN 1562-6016. Problems of Atomic Science and Technology. 2023. № 4(146)
Fig. 3 shows the steps characteristic of the intermit-
tent mode on the time dependence of the longitudinal
pulse. However, the most interesting result is shown in
Fig. 4. This figure shows a narrow spectrum of particle
dynamics. This result is, to some extent, unexpected.
More common is the broadening of the particle dynam-
ics spectrum with increasing wave field strength.
Fig. 4. Spectrum in intermittent mode,
0 1.2
This feature of the spectrum in the regime with in-
termittency is due to the fact that the particle dynamics
at the steps themselves is regular. Randomness in this
case is due only to the appearance of particle jumps
from one stage to another stage. These jumps are ran-
dom.
In this case, the moments have the following values:
even
0 1 ,
2 82.5 , 3
4 4 10 , 5
6 1.2 10 ;
odd
1 0.001 ,
3 405 , 4
5 2 10 , 5
7 5.3 10 .
The main tendency in moments is that the greater the
field strength of the wave, and thus the more pro-
nounced the steps, the higher the moments are greater
and the greater their magnitude.
3. ROLE OF MOMENTS IN PARTICLE
DYNAMICS
Moments in the theory of random processes play a
significant role. In particular, if the process is ergodic
[8, 9], then the values averaged over the ensemble can
be replaced by averages over time. This, in turn, makes
it possible to use the results of a single-particle analysis
of particle motion to find the distribution function of an
ensemble of particles. This can be done using the FP
equation. Indeed, in this equation, as is known, the main
parameters are the first and second moments. However,
as shown in [3], at cyclotron resonances, regimes appear
whose characteristic feature is the fact that the higher
moments turn out to be larger than the lower moments.
In the same work, the generalized FP equation is pre-
sented, in which higher moments are taken into account.
It should also be noted that in the FP equation (and in
the generalized equation) the moment values are divided
by the factorial of the moment number.
!
m
m
m
m
pn n
m p
, 2 ; 1,2,3...m j j . (9)
To find out the role of higher moments, it is enough
for us to analyze the solutions of equation (9) taking
into account only the second and fourth moments:
2 4
2 2
2 4
n n n
p p
. (10)
If the parameter is small ( 1 ), then the solu-
tion of equation (10) can be sought in the form of a se-
ries in this parameter:
2
0 1 2 ....n n n n (11)
Substituting this series into equation (10), we will
find equations for finding the terms of this series. For
example, to find the second term, you can get the fol-
lowing sequence:
2
0 0
0 2
ˆ 0
n n
Ln
p
;
4
0
1 4
ˆ n
Ln
p
;
4
0
1 4
ˆ
ˆ ˆ 0
Ln
LLn
p
since
0
ˆ 0Ln , then and
1
ˆ 0Ln . The equations for the
other terms of the series (11) will have an analogous
form. Finally, the series (11) can be written in the form
of a series of geometric progression:
2
0 0( , ) 1 ... ( , ) / 1n p t n n p t . (12)
This expression shows that the solutions of the FP
equation are stable with respect to the influence of small
higher moments.
Below we will use the results obtained in [3] and
write equation (9) up to the 6th moment, which already
becomes sufficient to take into account the influence of
growing higher moments. We also take into account
only even moments and rename p x .
Then equation (9) takes the form:
2 4 6
2 4 6
n n n n
x x x
. (13)
The resulting equation is quite complex for both
analytical and numerical analysis. Therefore, we con-
sider the stationary case, for which the solution of
Eq. (13) takes the form:
0
1
( ) exp[ ( )]exp[ ( )]
2
n x ik x x F k dk
, (14)
where
2 4 6( )F k k k k .
Below are the results of numerical studies of expres-
sion (14) in the stationary case. The difference in taking
into account the highest moments, namely
4 and
6 ,
is shown in the graphs below, the red curve describes
only the second moment. The blue curve was obtained
taking into account additionally the fourth and sixth
moments. The calculation was carried out for different
values of higher moments. Fig. 5 shows the distribution
function when the magnitudes of the higher moments
divided by the factorial of the moment number are in-
significant (order 10
-4
…10
-5
). In this case, their contri-
bution is also insignificant.
Fig. 6 shows the distribution function for the case
when the higher moments are still less than the lower
ones. In this case, the values of the higher moments
divided by the factorial of the moment number are also
not significant, and their value is several orders of mag-
nitude smaller than the value of the lower moments. It
can be noted that within the limits of the change in the
field strength of the wave from 0 0.01 (see Fig. 5) to
0 0.26 (see Fig. 6) there is an insignificant but
ISSN 1562-6016. Problems of Atomic Science and Technology. 2023. № 4(146) 19
smooth increase in the higher moments, which leads to a
smooth broadening of the distribution function.
Fig. 5. Distribution function, for the case when the
highest moments are less than the lowest moments,
0 0.01
Fig. 6. Distribution function, for the case when the
highest moments are less than the lowest moments,
0 0.26
In the future, a slight change in the field strength
(
0 0.261 ) qualitatively changes the particle dynamics
the higher moments become larger than the lower ones
(Fig. 7). So the second moment
2 , divided by the facto-
rial of the moment number, becomes
2 12 ,
4 43 ,
6 84 . This leads to a sharp broadening of the distribu-
tion function. It is also worth noting that at the same mo-
ment there are significant changes in the dynamics of
particles, namely the appearance of a stepwise character
of the longitudinal momentum of particles.
Fig. 7. Distribution function, for the case
when the highest moments are greater
than the lowest moments, 0 0.261
It should be noted that for the correct display of the
distribution function, it is necessary to take the values of
the moments calculated for a specific implementation.
Fig. 8 shows the plots of the distribution function for the
moments calculated for a specific implementation, the
blue curve (the moments are equal to
2 0.021 ,
4
4 1.1 10 , 7
6 2.7 10 ) and arbitrary, red curve
(
2 0.03 ,
4 0.004 , 4
6 3 10 ). It can be seen
that an arbitrary choice of moments leads to a non-
physical result areas with negative particle densities
appear.
Fig. 8. Distribution function
CONCLUSIONS
1. The most interesting, unexpected and important
result is that as the intensity of the electromagnetic
wave increases, the particle dynamics changes qualita-
tively. These changes are characterized by a new regime
of dynamic chaos. From chaos, which was determined
by the overlap of nonlinear cyclotron resonances, chaos
becomes intermittent. With such a transition, the wave
amplitude increased, but the width of the particle dy-
namics spectrum narrowed significantly. However, at
the same time, the moments began to increase. Moreo-
ver, the higher moments become larger than the lower
moments. Let us try to explain such, at first glance, con-
tradictory characteristics of the regime with intermitten-
cy. In [4], see also Fig. 3, it is shown that as a result of
the phase synchronization of the wave and the particle,
steps appear on the time dependence of the pulses. The
dynamics of particles on the steps themselves is regular.
Randomness occurs only at moments of jumps. These
jumps are random in both magnitude and direction. In
general, the dynamics are more regular. Therefore, the
spectrum becomes much narrower (see Fig. 4). Now
consider the features of the moments. Suppose we have
some function ( )x . Her moment with number n will
be determined by the formula
n
nm x x . Aver-
aging is carried out over the ensemble. However, if the
system under study is ergodic, then averaging can be
carried out over time. Looking at Fig. 3, we see that the
mean function has x jumps are moderated. In the
function itself, the magnitude of the jumps in most cases
is much larger than the average values. Therefore, the
value x x more than one. As a consequence, each
next moment will be greater than the previous one.
2. Let's answer the main question of the article: an
increase in higher moments leads to a more rapid broad-
ening of the particle distribution function. Note that
jumps in particle momentum can be initiators of runa-
way electrons.
20 ISSN 1562-6016. Problems of Atomic Science and Technology. 2023. № 4(146)
3. We also note that the magnitudes of the moments
must be determined from the actual particle dynamics.
An attempt to change these values leads to erroneous
results the distribution function can become negative
(see Fig. 8). This result may be useful as a diagnostic
test.
REFERENCES
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53.1515.
Article received 06.06.2023
РОЛЬ ВИЩИХ МОМЕНТІВ У РОЗПОДІЛІ ЧАСТИНОК У ПРОСТОРІ ІМПУЛЬСІВ
ПРИ ЦИКЛОТРОННИХ РЕЗОНАНСАХ
В.О. Буц, В.В. Кузьмін
Викладено результати аналізу динаміки заряджених частинок в умовах циклотронних резонансів у полі
інтенсивної електромагнітної хвилі. Особливу увагу приділено режимам з динамічним хаосом. Показано, що
існує два якісно різні режими. Виникнення першого обумовлено перекриттям нелінійних циклотронних ре-
зонансів. Другий режим пов'язаний із перемежуванням. Визначено моменти та спектри кожного з цих ре-
жимів. Показано, що зі збільшенням напруженості зовнішньої електромагнітної хвилі на початку з'являється
перший режим і потім другий. Характерною рисою другого режиму є перемежування. На залежності імпу-
льсів від часу у другому режимі з'являються сходинки. Показано, що спектри у другому режимі вужчі, ніж у
першому. Характерною особливістю другого режиму (режиму з перемежуванням) є той факт, що вищі мо-
менти виявляються більшими, ніж нижчі. У першому режимі вищі моменти швидко зменшуються. Для зна-
ходження функції розподілу частинок за імпульсами було використано узагальнене рівняння Фоккера-
Планка. Для деяких важливих випадків виписано рішення цього рівняння.
https://www.researchgate.net/profile/Vyacheslav-Buts?_sg%5B0%5D=sYsx5OnwmSoc92A6XJj4hkO9PcQojY3dr3WtVO87iwoMToHjpqzByD-SvgozsvqzVyj0CA0._-rTiAG4g7ZYnMQ4u_eU3rwgNoRqq3C4IB6Lkjo4LaDD9ld3qD7i5pZS7zkfMwrOqWsKLDvWqGodoKWn0WdElw&_sg%5B1%5D=8ARfIH5kPhx6HB9As4sjk0HYA7R8v5k-5MBUg1lr9X3BA9fQKJzolVRzlnIR2uIdm-Z9Ccw.KsJQXiy_0li_1cAPnmYjZl92FufTS7Rv5s7FZzgFiV4ZqJ0GF1DlP9k2j2CKpH5eBy4DdQlxcz8m4ufmjyXqpA
https://www.researchgate.net/scientific-contributions/PI-Kolykhalov-2014106897?_sg%5B0%5D=sYsx5OnwmSoc92A6XJj4hkO9PcQojY3dr3WtVO87iwoMToHjpqzByD-SvgozsvqzVyj0CA0._-rTiAG4g7ZYnMQ4u_eU3rwgNoRqq3C4IB6Lkjo4LaDD9ld3qD7i5pZS7zkfMwrOqWsKLDvWqGodoKWn0WdElw&_sg%5B1%5D=8ARfIH5kPhx6HB9As4sjk0HYA7R8v5k-5MBUg1lr9X3BA9fQKJzolVRzlnIR2uIdm-Z9Ccw.KsJQXiy_0li_1cAPnmYjZl92FufTS7Rv5s7FZzgFiV4ZqJ0GF1DlP9k2j2CKpH5eBy4DdQlxcz8m4ufmjyXqpA
https://www.researchgate.net/scientific-contributions/S-S-Moiseev-2202617884?_sg%5B0%5D=sYsx5OnwmSoc92A6XJj4hkO9PcQojY3dr3WtVO87iwoMToHjpqzByD-SvgozsvqzVyj0CA0._-rTiAG4g7ZYnMQ4u_eU3rwgNoRqq3C4IB6Lkjo4LaDD9ld3qD7i5pZS7zkfMwrOqWsKLDvWqGodoKWn0WdElw&_sg%5B1%5D=8ARfIH5kPhx6HB9As4sjk0HYA7R8v5k-5MBUg1lr9X3BA9fQKJzolVRzlnIR2uIdm-Z9Ccw.KsJQXiy_0li_1cAPnmYjZl92FufTS7Rv5s7FZzgFiV4ZqJ0GF1DlP9k2j2CKpH5eBy4DdQlxcz8m4ufmjyXqpA
https://www.researchgate.net/scientific-contributions/V-G-Pungin-10761185?_sg%5B0%5D=sYsx5OnwmSoc92A6XJj4hkO9PcQojY3dr3WtVO87iwoMToHjpqzByD-SvgozsvqzVyj0CA0._-rTiAG4g7ZYnMQ4u_eU3rwgNoRqq3C4IB6Lkjo4LaDD9ld3qD7i5pZS7zkfMwrOqWsKLDvWqGodoKWn0WdElw&_sg%5B1%5D=8ARfIH5kPhx6HB9As4sjk0HYA7R8v5k-5MBUg1lr9X3BA9fQKJzolVRzlnIR2uIdm-Z9Ccw.KsJQXiy_0li_1cAPnmYjZl92FufTS7Rv5s7FZzgFiV4ZqJ0GF1DlP9k2j2CKpH5eBy4DdQlxcz8m4ufmjyXqpA
https://link.springer.com/book/10.1007/978-1-4615-1829-7
https://link.springer.com/book/10.1007/978-1-4615-1829-7
http://dx.doi.org/10.1007/978-1-4615-1829-7_44
https://archive.org/details/introductiontoer0000walt
https://en.wikipedia.org/wiki/Springer_Science%2BBusiness_Media
https://link.aps.org/doi/10.1103/PhysRevLett.53.1515
https://link.aps.org/doi/10.1103/PhysRevLett.53.1515
https://link.aps.org/doi/10.1103/PhysRevLett.53.1515
https://en.wikipedia.org/wiki/Bibcode_(identifier)
https://ui.adsabs.harvard.edu/abs/1984PhRvL..53.1515H
https://en.wikipedia.org/wiki/Doi_(identifier)
https://doi.org/10.1103%2FPhysRevLett.53.1515
https://doi.org/10.1103%2FPhysRevLett.53.1515
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