The peculiarities of particle dynamics in the Fermi acceleration scheme
With examples of discrete and distributed mathematical models of the Fermi acceleration mechanism, a usefulness, or even necessity, of taking into account of singular solutions is demonstrated. Also the role is shown of those parts of phase space where the uniqueness theorem conditions to form the d...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
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irk-123456789-1123542017-01-21T03:02:47Z The peculiarities of particle dynamics in the Fermi acceleration scheme Buts, V.A. Динамика пучков With examples of discrete and distributed mathematical models of the Fermi acceleration mechanism, a usefulness, or even necessity, of taking into account of singular solutions is demonstrated. Also the role is shown of those parts of phase space where the uniqueness theorem conditions to form the dynamics of physical systems are broken. It was found that the dynamics of particles in discrete and distributed mathematical schemes of Fermi acceleration can be significantly different. The difference is due to the fact that the distributed model takes into account the effects of phase space where conditions do not correspond to those necessary for application of the uniqueness theorem. The role of singular solutions is under discussion as well. На прикладах дискретної та розподіленої математичних моделей механізму прискорення Фермі показана корисність, а в деяких випадках, і необхідність врахування особливих рішень. Також показана роль впливу областей фазового простору, в яких порушуються умови виконання теореми єдності на динаміку досліджуваних фізичних систем. Показано, що динаміка частинок у розподіленій та дискретній схемах прискорення Фермі може істотно відрізнятися. Ця відмінність пов'язана з урахуванням у розподіленій моделі впливу областей фазового простору, у яких порушена теорема єдності. Обговорюється роль особливих рішень. На примерах дискретной и распределенной математических моделей механизма ускорения Ферми показана полезность, а в некоторых случаях, и необходимость учета особых решений. Также показана роль влияния областей фазового пространства, в которых нарушаются условия выполнения теоремы единственности на динамику изучаемых физических систем. Показано, что динамика частиц в распределенной и дискретной схемах ускорения Ферми может существенно отличаться. Это различие связано с учетом в распределенной модели влияния областей фазового пространства, в которых нарушена теорема единственности. Обсуждается роль особых решений. 2015 Article The peculiarities of particle dynamics in the fermi acceleration scheme / V.A. Buts // Вопросы атомной науки и техники. — 2015. — № 6. — С. 73-76. — Бібліогр.: 9 назв. — англ. 1562-6016 PACS: 05.45.-a http://dspace.nbuv.gov.ua/handle/123456789/112354 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Динамика пучков Динамика пучков |
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Динамика пучков Динамика пучков Buts, V.A. The peculiarities of particle dynamics in the Fermi acceleration scheme Вопросы атомной науки и техники |
| description |
With examples of discrete and distributed mathematical models of the Fermi acceleration mechanism, a usefulness, or even necessity, of taking into account of singular solutions is demonstrated. Also the role is shown of those parts of phase space where the uniqueness theorem conditions to form the dynamics of physical systems are broken. It was found that the dynamics of particles in discrete and distributed mathematical schemes of Fermi acceleration can be significantly different. The difference is due to the fact that the distributed model takes into account the effects of phase space where conditions do not correspond to those necessary for application of the uniqueness theorem. The role of singular solutions is under discussion as well. |
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Article |
| author |
Buts, V.A. |
| author_facet |
Buts, V.A. |
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Buts, V.A. |
| title |
The peculiarities of particle dynamics in the Fermi acceleration scheme |
| title_short |
The peculiarities of particle dynamics in the Fermi acceleration scheme |
| title_full |
The peculiarities of particle dynamics in the Fermi acceleration scheme |
| title_fullStr |
The peculiarities of particle dynamics in the Fermi acceleration scheme |
| title_full_unstemmed |
The peculiarities of particle dynamics in the Fermi acceleration scheme |
| title_sort |
peculiarities of particle dynamics in the fermi acceleration scheme |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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2015 |
| topic_facet |
Динамика пучков |
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http://dspace.nbuv.gov.ua/handle/123456789/112354 |
| citation_txt |
The peculiarities of particle dynamics in the fermi acceleration scheme / V.A. Buts // Вопросы атомной науки и техники. — 2015. — № 6. — С. 73-76. — Бібліогр.: 9 назв. — англ. |
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ISSN 1562-6016. ВАНТ. 2015. №6(100) 73
ДИНАМИКА ПУЧКОВ
THE PECULIARITIES OF PARTICLE DYNAMICS
IN THE FERMI ACCELERATION SCHEME
V.A. Buts
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
E-mail: vbuts@kipt.kharkov.ua
With examples of discrete and distributed mathematical models of the Fermi acceleration mechanism, a useful-
ness, or even necessity, of taking into account of singular solutions is demonstrated. Also the role is shown of those
parts of phase space where the uniqueness theorem conditions to form the dynamics of physical systems are broken.
It was found that the dynamics of particles in discrete and distributed mathematical schemes of Fermi acceleration
can be significantly different. The difference is due to the fact that the distributed model takes into account the ef-
fects of phase space where conditions do not correspond to those necessary for application of the uniqueness theo-
rem. The role of singular solutions is under discussion as well.
PACS: 05.45.-a
INTRODUCTION
It was shown in [1 - 4] that, when analyzing the dy-
namics of physical systems, it is necessary to take into
account the influence of the phase space where the
uniqueness theorem conditions are not fulfilled. In par-
ticular, consideration must be given to the singular solu-
tions of ordinary differential equations (ODE). Note,
that singular solutions are those where the conditions of
uniqueness theorem are not met. Consideration of such
solutions, as was found out in the mentioned works,
allows one to expand significantly the range of parame-
ters of physical systems under study, in which the re-
gimes with chaotic behavior can be realized. In particu-
lar, with taking into account singular solutions, the re-
gimes with chaotic behavior can be realized in systems
with one degree of freedom, and even in systems that
are fully integrable. The singular solutions have to be
taken into consideration not only in the existing models
of the physical processes, but also at mathematical
modeling of those physical processes. This does mostly
occur in cases when the discrete mathematical models
are used as models of real physical processes. If in the
course of modeling the information about the areas of
phase space, where the uniqueness theorem gets broken
is lost then the results of the analysis of such models
can not reflect the real dynamics of the studied systems.
In the present paper, these peculiarities of influence of
the singular solutions on the dynamics of the studied
systems will be illustrated.
The scheme of acceleration of charged particles by
colliding with magnetic clouds in space, proposed by
Fermi [5, 6], is usually being modeled by particle oscil-
lations between two reflecting walls, with position of
one of them (or both) are oscillating. The reflection off
the walls is an instantaneous process of elastic reflec-
tion. In some cases, an energy loss during reflection
process is taken into account. Such a model of Fermi
acceleration will be called below in the text as the dis-
crete model. To date, discrete schemes with regularly
moving walls have been studied in detail, and various
regimes of particle dynamics in similar schemes were
found. The review of the results of analysis of such
schemes can be found, for example, in [7 - 9].
In this paper the scheme of interaction of particles
with reflecting walls is modified to be closer to the
natural scheme. Namely, it is assumed that the particles
move in a certain potential, which has several maxima
and each of them can reflect the particle. The position
of maxima may vary on regular periodic law. In the
simplest case, it looks like the movement of particles
between two reflective peaks of potential whose posi-
tions are changing periodically. Such a scheme of ac-
celeration can be called a distributed scheme. The dy-
namics of particles in such a scheme differs from that
of the particles in the discrete scheme with most signif-
icant difference consisting in the fact that chaotic re-
gimes have a much greater volume in parameter space
than the volume in the case of a discrete pattern. In
most cases, as in the discrete model, there is a maxi-
mum energy gained by particles. The main reason is an
appearance in a distributed model of the areas where
the conditions of uniqueness theorem are not hold. The
discussion about possible reasons of difference in the
particle dynamics in the discrete and the distributed
model ends the paper.
Fig. 1. Phase portrait of the system (1) ( 1x x= )
1. SINGULAR SOLUTIONS OF ODE
The author does not know the examples with singu-
lar solution in use when analyzing the dynamics of
physical systems. Let us briefly describe an example
when such solutions are necessary to be taken into ac-
count. This model is a model of а nonlinear oscillator
of the type:
0.5 0
2
xx x x
x
− + ⋅ =
. (1)
Equation (1) has the following integral:
( )2 2 2 0x R x Rϕ = − + − = . Moreover, this equation
mailto:vbuts@kipt.kharkov.ua
ISSN 1562-6016. ВАНТ. 2015. №6(100) 74
has a general solution, expressed by elementary func-
tions: ( )1 sinx R t C= ⋅ ± + . Looking at this general
solution and at the integral, it is difficult to imagine that
the dynamics of a nonlinear oscillator (1) can be com-
plex, chaotic. Indeed, the considered oscillator has only
one degree of freedom, has an integral and even has a
general solution. Such systems should not have complex
irregular dynamics. In reality, however, this dynamic is
chaotic. The reason for this is the presence of singular
solutions. The phase portrait of the equation (1) is
shown in Fig. 1. The phase trajectories are circles
whose centers are located on the axis 1 0x x≡ = . The
radius of these circles is the distance from the center
point to the point of singular solution. Moving along
any of these circles (depending on the choice of initial
conditions), the phase trajectory gets to the point of
singular solution. At this point, the trajectory can ran-
domly pass to any circle which passes through this
point. More details about analysis of this dynamics one
can found in [1 - 4]. Note that the transitions from one
circle to another circle are completely randomly. This
example shows that the presences of singular solutions
can quality change of the known dynamics, even in such
simple systems. Thus, the presence of singular solutions
creates an additional mechanism for the appearance of
unpredictability and irreversibility.
2. DISCRETE AND CONTINUOUS
SCHEME FERMI ACCELERATION
The simplest discrete scheme Fermi acceleration is
shown in Fig. 2. In this scheme, the particle moves be-
tween the two elastically reflecting walls, one of which
(lower) varies periodically. This and similar schemes
are studied in detail (see, for example, [6 - 9]).
Fig. 2. Discrete scheme of Fermi acceleration
The most important fact is availability of the analyti-
cal criterion of the appearance of stochastic acceleration.
Let the parameters acceleration schemes satisfy the fol-
lowing inequalities: / 1a l << ; 2a ε= ; 0/ 1u v V= >> ;
u u∆ << . Here b − distance between fixed walls; 0V − ve-
locity of the moveable wall; v − velocity of the particle.
Then the criterion of occurrence of chaotic dynamics
is the inequality:
22 / 1K l au= > . (2)
Instead of the discrete model, we introduce continu-
ous scheme Fermi acceleration. To do this, we assume
that the particle moves in the potential well, one of the
walls which periodically vibrate (Fig. 3).
The equations of motion of a particle in this will
have the form:
0 1x x= , (3)
1 0( )x F x= ,
where
( )( ) ( )0 2 2
00
1 1(x )
xx cos
F
bb tε ω
= − −
+− + ⋅
.
Fig. 3. The force, which are acting on the particle
in a distributed scheme of Fermi acceleration
Here, as in the discrete model, 2b − the distance be-
tween the peaks of the stationary potential. The analogy
between discrete and continuous models is visible.
Therefore, one would expect that if the condition
22 / 1K l au= > is not fulfilled, the dynamics of a dis-
tributed scheme, as well as the dynamics of the discrete
scheme, will be regular. However, the dynamics of a
distributed system does not obey this criterion. Indeed,
let choose the following parameters of the distribution
scheme: 21; 1; 10; 100b vε ω= = = = ; 0 (0) 1x = ;
1(0) 100x = . Then the parameter K is small:
34 10 1K −= ⋅ << . A typical dependence of the particle
position on the time, its speed, spectrum of motion of
the particle and its correlation function are shown in
Figs. 4-7. At the numerical calculations we use:
810TOL CTOL −= = ; 518 2.622 10m N= ⇒ = ⋅ ; length
of realization was 100.
It is clear from these figures that the dynamics of
the particles is irregular. Such a difference in the dy-
namics of seemingly similar schemes, is due to the fact
that in a distributed system on the dynamics of the par-
ticles affects the region of the phase space
( 0 1; 0x b x= ± = ), in the vicinity of which violated the
conditions of uniqueness theorem.
Thus at modeling of physical systems it is neces-
sary to take into account the presence of the phase are-
as, in which the uniqueness theorem is violated. The
ignoring these areas can lead to incorrect results. Most
often this happens when execute the transition from
distributed models to discrete models.
To illustrate this feature, consider as the simplest
example of such task. Suppose we want to determine
the dynamics of a particle in a potential well whose
walls are fixed ( 0ε = ). Such a problem can be solved
using the system of equations (2), i.e., analyzing dis-
tributed scheme. The result of this analysis is the same
as in the analysis of similar discrete scheme. Now sup-
pose that the position of one of the walls of the poten-
tial vibrates slightly. The amplitude of this vibration is
low, and the frequency is sufficiently large. Thus, it is
assumed that on the system acts small high-frequency
disturbances. The question is: "What effects will this
perturbation in the discrete and distributed schemes?".
l
a
b
2ε
ISSN 1562-6016. ВАНТ. 2015. №6(100) 75
Fig. 4. The dependence of the position of a particle
in a distributed scheme Fermi acceleration
Fig. 5. The dependence of the velocity of a particle
in a distributed scheme Fermi acceleration
Fig. 6. The spectrum of motion of a particle
Fig. 7. The autocorrelation function of the particle
motion
As for the discrete model, we can expect that the
dynamics of particles will remain regular and little
changed under the influence of a small perturbation, if
the parameter K is small. The distributed scheme, as
will be seen below, is very sensitive to such perturba-
tions. As an example, consider the following parameters
acceleration scheme: 10b = ; 0 (0) 1x = ; 1(0) 200x = ;
0.005; 1000ε ω= = ; 0.1K = ; 32.770N = ; length of
"realization": 100; TOL=CTOL=10-10. For these values
of the parameters of particle the dynamics in a discrete
pattern is a regular. In the distributed scheme such per-
turbation qualitative changes the dynamics of particles.
This dynamics becomes chaotic. As an example, in
Figs. 8 and 9 shows the dependence of position and
velocity of the particles in the distributed system. We
see a noticeable influence of a small perturbation.
Moreover, in the Figs. 10, 11 are presented statistical
processing of these dynamics. It can be seen that the
spectrum is broadened, and the correlation function
decreases sufficiently rapidly.
This result indicates that, simulating the dynamics
of particles in the potentials the discrete models must
be used with caution. In such models, there is no region
of phase space in which the uniqueness theorem is vio-
lated. Therefore, they are little sensitive to small per-
turbations. The real physical systems can be abnormal-
ly sensitive to small perturbations.
Fig. 8. The dependence of the position of a particle
in a distributed scheme Fermi acceleration
Fig. 9. The dependence of the velocity of a particle
in a distributed scheme Fermi acceleration
Fig. 10. The spectrum of motion of a particle
Fig. 11. The autocorrelation function of the particle
motion
CONCLUSIONS
Let us formulate the most important results:
1. One of the most important results is the proof of
the fact that singular solutions must be considered
when analyzing the dynamics of various physical sys-
tems. Accounting of these solutions can significantly
extend the range of parameters studied systems in
which can be realized chaotic regimes. Such regimes
may appear in systems with one degree of freedom, and
even in systems completely integrable. This fact vio-
lates one of the basic paradigms of dynamic chaos that
chaotic regimes can occur only in systems with 1.5 or
more degrees of freedom. In addition, the presence of
singular solutions provides an additional mechanism
for the appearance of unpredictability and irreversibil-
ISSN 1562-6016. ВАНТ. 2015. №6(100) 76
ity of the dynamics of physical systems. Let us remind
that there are two common mechanisms of unpredicta-
bility and irreversibility. The first is the mechanism of
local instability and uncertainty (even though very
small) in the initial conditions. The second mechanism
is associated with the presence of noise.
2. The physical nature of chaotic dynamics, which
was appear as the result taking into account singular
solutions is different from the nature of the appearance
of the conventional dynamic chaos. The unpredictability
of the phase trajectories in this mechanism occurs only
when these trajectories passing points of singular solu-
tions. Such mechanism of occurrence unpredictability is
more like to the mechanism unpredictability which ap-
pear when one throw the dice with an unlimited number
of faces.
3. Note also that even in those cases when the phase
trajectories do not fall strictly in the points of singular
solutions, but are close enough to them, singular solu-
tions must also be taken into account. In this case, the
structure of the phase space in the vicinity of singular
solutions is such that even small perturbations can sig-
nificantly change the dynamics of the physical system.
This feature of the particle dynamics was visible in the
distributed scheme Fermi acceleration. These features
one must to have in the mind at modeling physical pro-
cesses. Especially during the transition from continuous
mathematical models to discrete models.
4. In [3, 4] it was shown that the number of singu-
lar solutions is growing rapidly with growing of the
number of the freedom in physical systems, which stud-
ied. Therefore, the results formulated above are particu-
larly important for more complex systems, for systems
with many degrees of freedom.
The above, as well as in [1 - 4], the main attention
was paid to the fact that the inclusion of singular solu-
tions significantly expands the range of parameters of
physical systems with chaotic behavior. However, it
can be expected that the taking into account of such
solutions can be useful for opening some new features
of the dynamics of physical systems which one studied.
REFERENCES
1. V.A. Buts. Anomalous influence of small perturba-
tions // Physical bases of instrumentation. Novem-
ber. 2011, p. 76-87.
2. V.A. Buts. Chaotic motion of dynamic systems
with “one” degree of freedom // Problems of Atom-
ic Science and Technology. Series “Nuclear Phys-
ics Investigations”. 2012, № 1, p. 328-332.
3. V.А. Buts. Singular solutions and dynamic chaos //
Problems of Atomic Science and Technology. 2015,
№ 4, p. 232-236.
4. VA Buts. The role of singular solutions in analyz-
ing the dynamics of physical systems // Physical
bases of instrumentation. December, 2015.
5. E. Fermi. On the Origin of the Cosmic Radiation //
Phys. Rev. 1949, v. 75, p. 1169.
6. S. Ulam // Proc. 4th Berkeley Symp. on Math, and
Probability. 1961, v. 3, p. 315.
7. A.J. Lichtenberg, M.A. Lieberman. Regular and
Stochastic Motion. Springer-Verlag, 1983, p. 499.
8. G.M. Zaslavsky, B.V. Chirikov About mechanism
Fermi acceleration in one dimension case // Dokl.
USSR Academy of Sciences. 1964, v. 159, p. 306.
9. G.M. Zaslavsky. Stochasticity of dynamic systems.
Moscow: “Science”, 1984.
Article received 20.10.2015
ОСОБЕННОСТИ ДИНАМИКИ ЧАСТИЦ В СХЕМЕ УСКОРЕНИЯ ФЕРМИ
В.А. Буц
На примерах дискретной и распределенной математических моделей механизма ускорения Ферми пока-
зана полезность, а в некоторых случаях, и необходимость учета особых решений. Также показана роль вли-
яния областей фазового пространства, в которых нарушаются условия выполнения теоремы единственности
на динамику изучаемых физических систем. Показано, что динамика частиц в распределенной и дискретной
схемах ускорения Ферми может существенно отличаться. Это различие связано с учетом в распределенной
модели влияния областей фазового пространства, в которых нарушена теорема единственности. Обсуждает-
ся роль особых решений.
ОСОБЛИВОСТІ ДИНАМІКИ ЧАСТИНОК У СХЕМІ ПРИСКОРЕННЯ ФЕРМІ
В.А. Буц
На прикладах дискретної та розподіленої математичних моделей механізму прискорення Фермі показана
корисність, а в деяких випадках, і необхідність врахування особливих рішень. Також показана роль впливу
областей фазового простору, в яких порушуються умови виконання теореми єдності на динаміку досліджу-
ваних фізичних систем. Показано, що динаміка частинок у розподіленій та дискретній схемах прискорення
Фермі може істотно відрізнятися. Ця відмінність пов'язана з урахуванням у розподіленій моделі впливу об-
ластей фазового простору, у яких порушена теорема єдності. Обговорюється роль особливих рішень.
|