The peculiarities of particles transition through split separatrix
The dynamics of charged particle transition through a stochastic layer generated by a split separatrix is investigated on an example of mathematical pendulum motion with damping under acting of external periodic disturbance. It is shown that such layer has properties of an effective potential barrie...
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| Опубліковано в: : | Вопросы атомной науки и техники |
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| Дата: | 2005 |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2005
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| Цитувати: | The peculiarities of particles transition through split separatrix / V.A. Buts, A.P. Tolstoluzhsky // Вопросы атомной науки и техники. — 2005. — № 2. — С. 134-136. — Бібліогр.: 10 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860012577619181568 |
|---|---|
| author | Buts, V.A. Tolstoluzhsky, A.P. |
| author_facet | Buts, V.A. Tolstoluzhsky, A.P. |
| citation_txt | The peculiarities of particles transition through split separatrix / V.A. Buts, A.P. Tolstoluzhsky // Вопросы атомной науки и техники. — 2005. — № 2. — С. 134-136. — Бібліогр.: 10 назв. — англ. |
| collection | DSpace DC |
| container_title | Вопросы атомной науки и техники |
| description | The dynamics of charged particle transition through a stochastic layer generated by a split separatrix is investigated on an example of mathematical pendulum motion with damping under acting of external periodic disturbance. It is shown that such layer has properties of an effective potential barrier, which can prevent to transition of charged particles through it. The dwelling time of particle in such layer can be anomalous large.
На прикладі руху математичного маятника із загасанням під дією зовнішнього періодичного збурення досліджена динаміка проходження зарядженої частинки через стохастичний шар, утворений розщепленою сепаратрисою. Показано, що такий шар має властивості ефективного потенційного бар'єру, який може перешкоджати проходженню заряджених частинок через нього. Час знаходження частинки в такому шарі може бути аномально великим.
На примере движения математического маятника с затуханием под действием внешнего периодического возмущения исследована динамика прохождения заряженной частицы через стохастический слой, образованный расщепленной сепаратрисой. Показано, что такой слой обладает свойствами эффективного потенциального барьера, который может препятствовать прохождению заряженных частиц через него. Время нахождения частицы в таком слое может быть аномально большим.
|
| first_indexed | 2025-12-07T16:42:37Z |
| format | Article |
| fulltext |
THE PECULIARITIES OF PARTICLES TRANSITION
THROUGH SPLIT SEPARATRIX
V.A. Buts, A.P. Tolstoluzhsky
NSC Kharkov Institute of Physics and Technology,
Kharkov, Ukraine, e-mail: tolstoluzhsky@kipt.kharkov.ua
The dynamics of charged particle transition through a stochastic layer generated by a split separatrix is investigated on
an example of mathematical pendulum motion with damping under acting of external periodic disturbance. It is shown
that such layer has properties of an effective potential barrier, which can prevent to transition of charged particles
through it. The dwelling time of particle in such layer can be anomalous large.
PACS: 05.45.–a, 45.50.–j
1. INTRODUCTION
Under development of dynamic chaos in a Hamiltonian
system all phase space is broken to regions with random and
regular behavior. The investigation of boundaries of the
regions, which separate random behavior from regular,
demonstrates that these boundaries are fractal and have
property of "sticky". The result of which one can be the
apparent violation of second law of thermodynamics [1].
The interaction of a wave-particle type plays the main
role for beam problems. Most important point of such
interaction is the point of a qualitative change of particles
motion nature. At this in the phase space the trajectory of
particle passes through separatrix. In real situations the
separatrix is split, more often, while as generating the
stochastic layer. The nature of particle motion inside
region of split separatrix and outside it differs essentially.
In the present paper the features of charged particle
transition through a split separatrix are investigated
analytically and numerically on the model of mathematical
pendulum with damping and with external high frequency
disturbance.
Besides it is considered the case of excitation by the
beam of monochromatic wave at the case of Cherenkov’s
effect; and the features of beam particles’ dynamics are
analyzed at such excitation.
2. PROBLEM STATEMENT. THE BASIC
EQUATIONS
The problem about interaction of charged particles with
a field of electromagnetic waves is the primary goal in
accelerator theory and in plasma theory. So, at charged
particle capturing by the field of electromagnetic waves,
and also at escaping from the capture the particle should
pass separatrix region, which separates captured particles
from transient-time. The separatrix is, practically always,
split. For analysis of dynamics of particles motion, if
there is a split separatrix, it is convenient to use the model
of mathematical pendulum with damping ν , on which
the external periodic force of amplitude h acts
ẍν ẋsin x=h sin t , =ω/ω0 .
(1)
In absence of damping and disturbance the Hamiltonian
of mathematical pendulum is:
H 0 ẋ , x =1 /2 ẋ2−ω0
2cos x , where ω0 -
frequency of small oscillations. On a phase plane of such
pendulum the separatrix separates infinite invariant
curves for transient-time particles from closed curves for
captured particles.
The presence of external force ( h≠0 ) at ν =0 leads
to
splitting of separatrix and appearance in its neighborhood
a stochastic layer. Width of this layer is proportional to
disturbance [2]. In the case h=0 at ν> 0 point ẋ=0 ,
x=0 is stable focus. If h=0 , ν < 0 the particle
motion should become infinite. The damping ( ν ≠0 ) at
h≠0 can lead to disappearance of splitting. Using
Melnikov’s method (see for example, [3]) it is possible to
show, that the splitting of separatrix is saved, if the value
of damping fulfills to inequality
νπ h /4 ch π / 2 . (2)
From (2) follows, that the stochastic layer generated by
an external disturbance, is collapsed much faster at
disturbance with large frequency ( π >>1), than
stochastic layer generated by low frequency disturbance.
This result is in full compliance with results of papers on
modulation diffusion [3-6], and also with results of paper
[7]. We shall compare the passing time for separatrix
region when there is an external disturbance and when it
isn’t. At external disturbance ( h =0) the change of
maximum velocity of particle on small times ( ν t << 1
) can be evaluated by formula
Δ ˙xmax=− ˙ymax⋅ν
2 t , (3)
where y t satisfies the equation of mathematical
pendulum ÿsin y=0 .
3. THE RESULTS OF NUMERICAL
ANALYSIS
The transition of particles through separatrix was
investigated by numerical solution of equation (1) for
different values of parameters h , ν , and various
initial conditions. The initial conditions for ν0 were
selected on invariant curves outside of separatrix, for
ν0 s - inside separatrix (see Fig. 1).
Fig.1. Initial conditions for particles ( V 0 min =–
134 Problems of Atomic Science and Technology. Series: Plasma Physics (11). 2005. № 2. P. 134-136
mailto:tolstoluzhsky@kipt.kharkov.ua
V 0 max ):
а) ν0 , V 0 max =2.3; b) ν0 , V 0 max =1.7
On the Figs. 2,3 the typical dependencies of particle
motion velocity on time, and the time of passing through
separatrix for different values of damping v without
disturbance h=0 are shown.
a
b
Fig. 2. a) dependence of particle velocity on time ;
b) distribution of trapping time for transient-time particles;
at values ν =0.0025, V0 max =2.3, h=0
a
b
Fig. 3. a)dependence of particle velocity on time ;
b) distribution of time for particles exit from trapping;
at values ν = -0.0025, V0 max =1.7
The change of maximum particle velocity, as it is visible
from these plots, is well described by the formula (3).
If there is an external periodic disturbance one can see the
splitting of separatrix and dynamics of particle motion changes
essentially. At first, the particle can move chaotically,
becoming whether trapped or transient-time (see Fig. 4).
a
b
Fig. 4. Dependence of particle velocity on time:
а) ν =0.0025, V0 max=2.3, h=0.23, Ω=1.5;
b) ν =-0.00075, V0 max=1.7, h=0.23, Ω=1.5
The character of its motion, and also the values of
velocities getting by a particle, do not depend on the
dissipation sign; and for all run time (t ~ 4000) the
maximum values of velocities, accessible by particle,
practically did not vary. Spectrum of its motion is wide,
and the correlation function droops rapidly (see Fig. 5).
Fig.5. Spectrum and correlation function of particle
velocity;
ν = -0.00075, V0 max =1.7, h =0.23, Ω =1.5
Secondly, except for such "long-lived" chaotic motion, one
can see regimes, when the motion of particle is regular,
though has multifrequency character (see Fig. 6).
a
b
Fig. 6. a) dependence of particle velocity on time ;
b) distribution of trapping time for transient-time particles;
at values ν=0.0025, V0 max=2.3,h=0.23, Ω=1.8
Most important peculiarity of such motion is the fact, that
despite of the availability of damping particle for all run
time remains transient-time and does not fall into
stochastic layer, while without an external disturbance the
particle becomes trapped during the time that is smaller
almost on the order. The motion of such particles has line
spectrum, and the correlation function oscillates with
slowly drooping amplitude.
3. BEAM-PLASMA INTERACTION
The dynamics of motion of thin electron beam with
radius b interacting in conditions of Cherenkov's resonance
with the magnetized plasma waveguide with radius a,
under acting of the external periodic electrical field with
given amplitude Ez=E0cos(ω0t), is investigated. For thin
beam it is possible to neglect the effect of beam
stratification in the field of wave. The system is placed into
a strong magnetic field, so that the motion of beam and
plasma particles is one-dimensional. Besides, let’s consider
that plasma is linear.
The universal non-linear set of equations, describing the
dynamics of excitation of plasma waves by electron
beam, is well-known [8]. It is easily possible to take into
account the presence of the external monochromatic field
of given amplitude, by introducing additional addend into
equation of motion of beam electrons [9].
The numerical solution of a non-linear set of equations
was carried out at fixed density of beam and plasma. Linear
increment was equal δ=0.05, initial velocity of beam
ν0=V0/Vph=1.0. As far as δ<< ΔΩ (ΔΩ=1.0 – distance
between harmonics), is possible to use only single-mode
approximation for a field. The initial values of field
amplitude are: Rе E = Im E = 2 10- 4.
135
The results of numerical solution of the set of equations
are shown on Figs. 7, 8 for various values of external field
amplitude 0 0 pheE / m Vε ω= . In these figures one can see
the dependencies of field amplitude on time, power
spectrum Sω, and correlation function Cf of oscillations at
ε0=0 - Fig. 7, at ε0=0.063, Ω=0.0015 - Fig. 8,
accordingly.
How one can see from the plots (see Fig. 7) the
exponential increasing of amplitude of the field with
linear increment δ, change into oscillations conditioned
by phase oscillations of bunch of beam particles, trapped
by wave. The frequency of oscillations is about δ,
modulation depth of amplitude is, approximately, half of
maximal [9].
s
Fig.7. Amplitude of external field ε0=0
Spectrum of oscillations has narrow peak on the base
frequency and two satellites with shifting of frequency
~δ=0.05.
The correlation function, oscillating on the base frequency,
droops slowly during the time. Physically such behavior of
correlation function can be explained by such a way.
At the beam motion in the field of wave with oscillating
amplitude, the stochastic instability of motion develops [10].
The stochastic instability acts first of all on particles located
in a stochastic layer in vicinity of separatrix. While the basic
group of particles which are generating a bunch, is far from
separatrix in islands of stability. Therefore, the influence of
neighboring non-linear resonances on motion of bunch is not
enough, and also the smoothing of amplitude oscillations
occurs with characteristic time, which is much more than the
time of splitting of particles’ motion correlations.
The presence of the external signal (Ω=0.0015, ε0=0.063)
significantly changes the dynamics of instability. At first (see
Fig. 8), as well as in the previous case, the exponential
increasing of field amplitude, limits by trapping of beam
particles into potential well of excited wave. The level of this
field exceeds the level, necessary for overlapping non-linear
resonances between field of wave and external field. Under
the influence of the external field there is a misalignment of
motion of beam bunch and of fundamental wave. It leads to
more fast chaotization of beam particles motion and, in turn,
to chaotic modulation of field amplitude (300<τ). Spectrum
of the field, though has a maximum on a base frequency, is
notably widened. The correlation function of this field
droops during the time fast enough.
4. CONCLUSIONS
Thus, the stochastic layer, which is generated in vicinity
of separatrix, can have the property of a potential barrier,
which resists the passing of particles through it.
Fig.8. Amplitude of external field ε0=0.063, Ω=0.0015
The properties of this stochastic layer essentially depend
on moving direction of particles (transient-time - trapped,
entrapped - transient-time). Moreover, the stochastic layer
is less transparent for the passing transient-time - trapped
and is more transparent for the passing trapped - transient-
time. The lifetime in the stochastic layer can be
anomalously big. This result is in the good agreement
with results of work [1].
REFFERENCES
1. G. M. Zaslavsky // Physics Today. 1999, N8, p.39.
2. G.M. Zaslavsky. Chaos in Dynamic Systems. New
York: Harwood. 1984.
3. A.J. Lichtenberg, M.A. Liberman. Regular and
Stochastic Motion. New York: Springer-Verlag. Berlin:
Heildelberg. 1983.
4. F.M. Izrailev, B.V. Chirikov, D.L. Shepelyanskij //
Preprint, Novosibirsk: IYaF SO AN SSSR, 80-209, 1980
(in Russian).
5. F.M. Izrailev, B.V. Chirikov, D.L. Shepelyanskij //
Preprint, Novosibirsk: IYaF SO AN SSSR, 80-211, 1980
(in Russian).
6. J.L. Теnnison // Physica. 1982, v.50, p. 123.
7. V.A. Balakirev, V.A. Buts, Yu.P. Machekhin,
A.P.Tolstoluzhsky // Pis’ma v ZhTF. 1983, v.9, № 23, p.
1427 (in Russian).
8. I.N. Onishchenko et al. // Pis’ma v ZhETF. 1970, issue
1(6), p. 407 (in Russian).
V.D. Shapiro, V.I. Shevchenko // Preprint, Kharkov:
KhFTI, 72-24, 1973 (in Russian).
9. V.A. Buts, O.V. Manuilenko, A.P. Tolstoluzhsky //
UFZh. 1994, v.39, №4, p.23-28 (in Russian).
10. V.A. Balakirev, V.A. Buts, A.P. Tolstoluzhsky, Yu.A.
Turkin // UFZh. 1983, v.32 №8, p. 1270 (in Russian).
ОСОБЕННОСТИ ПРОХОЖДЕНИЯ ЧАСТИЦ ЧЕРЕЗ РАСЩЕПЛЕННУЮ СЕПАРАТРИСУ
В.А. Буц, А.П. Толстолужский
На примере движения математического маятника с затуханием под действием внешнего периодического
возмущения исследована динамика прохождения заряженной частицы через стохастический слой,
образованный расщепленной сепаратрисой. Показано, что такой слой обладает свойствами эффективного
потенциального барьера, который может препятствовать прохождению заряженных частиц через него. Время
нахождения частицы в таком слое может быть аномально большим.
136
ОСОБЛИВОСТІ ПРОХОДЖЕННЯ ЧАСТИНОК ЧЕРЕЗ РОЗЩЕПЛЕНУ СЕПАРАТРИСУ
В.О. Буц, О.П. Толстолужський
На прикладі руху математичного маятника із загасанням під дією зовнішнього періодичного збурення
досліджена динаміка проходження зарядженої частинки через стохастичний шар, утворений розщепленою
сепаратрисою. Показано, що такий шар має властивості ефективного потенційного бар'єру, який може
перешкоджати проходженню заряджених частинок через нього. Час знаходження частинки в такому шарі може
бути аномально великим.
137
|
| id | nasplib_isofts_kiev_ua-123456789-79799 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T16:42:37Z |
| publishDate | 2005 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Buts, V.A. Tolstoluzhsky, A.P. 2015-04-04T19:48:05Z 2015-04-04T19:48:05Z 2005 The peculiarities of particles transition through split separatrix / V.A. Buts, A.P. Tolstoluzhsky // Вопросы атомной науки и техники. — 2005. — № 2. — С. 134-136. — Бібліогр.: 10 назв. — англ. 1562-6016 PACS: 05.45.–a, 45.50.–j https://nasplib.isofts.kiev.ua/handle/123456789/79799 The dynamics of charged particle transition through a stochastic layer generated by a split separatrix is investigated on an example of mathematical pendulum motion with damping under acting of external periodic disturbance. It is shown that such layer has properties of an effective potential barrier, which can prevent to transition of charged particles through it. The dwelling time of particle in such layer can be anomalous large. На прикладі руху математичного маятника із загасанням під дією зовнішнього періодичного збурення досліджена динаміка проходження зарядженої частинки через стохастичний шар, утворений розщепленою сепаратрисою. Показано, що такий шар має властивості ефективного потенційного бар'єру, який може перешкоджати проходженню заряджених частинок через нього. Час знаходження частинки в такому шарі може бути аномально великим. На примере движения математического маятника с затуханием под действием внешнего периодического возмущения исследована динамика прохождения заряженной частицы через стохастический слой, образованный расщепленной сепаратрисой. Показано, что такой слой обладает свойствами эффективного потенциального барьера, который может препятствовать прохождению заряженных частиц через него. Время нахождения частицы в таком слое может быть аномально большим. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Plasma electronics The peculiarities of particles transition through split separatrix Особливості проходження частинок через розщеплену сепаратрису Особенности прохождения частиц через расщепленную сепаратрису Article published earlier |
| spellingShingle | The peculiarities of particles transition through split separatrix Buts, V.A. Tolstoluzhsky, A.P. Plasma electronics |
| title | The peculiarities of particles transition through split separatrix |
| title_alt | Особливості проходження частинок через розщеплену сепаратрису Особенности прохождения частиц через расщепленную сепаратрису |
| title_full | The peculiarities of particles transition through split separatrix |
| title_fullStr | The peculiarities of particles transition through split separatrix |
| title_full_unstemmed | The peculiarities of particles transition through split separatrix |
| title_short | The peculiarities of particles transition through split separatrix |
| title_sort | peculiarities of particles transition through split separatrix |
| topic | Plasma electronics |
| topic_facet | Plasma electronics |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/79799 |
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