Self-organization and self-consistent equilibrium kinetics in electrodynamic systems with intense charged beams
Equilibrium states rising as self-organization result in unstable plasma with fast charged beam system are considered. Self-consistent states described by non-linear Bolzmann-Vlasov equation are obtained for harmonic, solitary and shock waves. Kinetics of captured and slipping beam particles is disc...
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| Published in: | Вопросы атомной науки и техники |
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| Date: | 2001 |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2001
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| Cite this: | Self-organization and self-consistent equilibrium kinetics in electrodynamic systems with intense charged beams / V.K. Grishin // Вопросы атомной науки и техники. — 2001. — № 6. — С. 115-117. — Бібліогр.: 5 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860176606572576768 |
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| author | Grishin, V.K. |
| author_facet | Grishin, V.K. |
| citation_txt | Self-organization and self-consistent equilibrium kinetics in electrodynamic systems with intense charged beams / V.K. Grishin // Вопросы атомной науки и техники. — 2001. — № 6. — С. 115-117. — Бібліогр.: 5 назв. — англ. |
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| container_title | Вопросы атомной науки и техники |
| description | Equilibrium states rising as self-organization result in unstable plasma with fast charged beam system are considered. Self-consistent states described by non-linear Bolzmann-Vlasov equation are obtained for harmonic, solitary and shock waves. Kinetics of captured and slipping beam particles is discussed.
|
| first_indexed | 2025-12-07T18:00:30Z |
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SELF-ORGANIZATION AND SELF-CONSISTENT EQUILIBRIUM
KINETICS IN ELECTRODYNAMIC SYSTEMS WITH INTENSE
CHARGED BEAMS
V.K. Grishin
Skobeltsyn Institute of Nuclear Physics of Lomonosov State University
Moscow, Russia
Equilibrium states rising as self-organization result in unstable plasma with fast charged beam system are
considered. Self-consistent states described by non-linear Bolzmann-Vlasov equation are obtained for harmonic,
solitary and shock waves. Kinetics of captured and slipping beam particles is discussed.
PACS: 52.40M
1. INTRODUCTION
In present paper series of aspects of wide-spread
subjects bound with phenomenon of self-organizing in
open non-equilibrium systems are considered. Among
last in modern physics different electrodynamic
schemes, using of intensive beams of charged particles
for fundamental and technological applications, are
actively explored. Further we will be restricted to one of
major applications of charged beams, in particular
beams of relativistic electrons, for generation of intense
microwave radiation.
It is known open systems under influence of external
perturbations (for example, flows of energy and а
matter) can lose stability, that is usually accompanied
by the various phenomena of self-organizing, i.e.
modification of an initial condition by formation of new
structures [1,2]. Here process of new formation is
already stopped owing to various mechanisms of
nonlinear saturation and stabilization.
In the elementary view a scheme for generation of
microwave radiation consists of the open system (base
or "cold" system) in which a beam of fast charged
particles is injected. As result the initial system loses a
stability and in it the irreversible processes - generation
of electromagnetic waves and nonlinear saturation of a
wave amplitude - are developed.
The nonlinear saturation of a wave appears because
of particle capture (trapping) by the same wave. In the
total the new structure of self-consistent state a wave -
modulated beam will be formed. New wave - beam state
aquires the rather important property: capture of
particles and their phase mixing translates a state of
system in essentially nonlinear one. Therefore a process
of generation is sated, and the new state results
irreversible (even in absence of a true dissipation).
Thus, our case is one of vivid examples of non-
equilibrium systems above mentioned with nonlinear
saturation and irreversibility.
In general new structure arises as the compromise of
processes of generation, i.e. redistribution of initial
beam energy, and nonlinear stabilization and, as the
consequence, it is active dynamic system. Therefore for
research its dynamics it is extremely important to
establish equilibrium state, in which the average
parameters of system are stationary.
Three cases are considered below, the distinctions
between which are determined by the initial conditions:
periodic (harmonic and quasi- harmonic) wave, solitary
and shock waves. The description is limited to the most
evident one-dimensional motion of a beam in plasma. It
is supposed that the beam of particles is immersed in a
strong longitudinal М electromagnetic slow waves is
taken into account only. Density of plasma exceeds
essentially a beam one, and plasma electrons are non-
magnetized. Besides the plasma particles are not
captured and can be described in hydrodynamic
approximation.
Further, it is necessary to explain physical
conditions at which there is a formation of equilibrium
state. Physical reason for instability and self-
organization is self-focusing of particles in systems with
slow wave (plasma, travelling wave tubes and so on).
So, in plasma Coulomb's force between the same
charged moving particle is attractive if the effective
frequency of signal, excited by both particles, is less
than Langmuir's one and magnitude of the plasma
dielectric constant is negative:
01 2
2
<→
ω
ω−=ε L
p . (1)
2. SIMPLE WAVES
In ordinary conditions an injected beam has the
extended length up to ten meters and more. Such beam
excites "almost" harmonic electromagnetic wave
(further harmonic wave is termed as a simple one).
Usually this process has resonant character if wave
frequency is close to frequency of a slow beam wave
and of cold system intrinsic one (see details [2]). Here
equilibrium state corresponds (meets) to a stage with
complete phase mixing of particles and their
synchronization with an excited wave. Thus the beam
becomes energetically inhomogeneous.
In general an equilibrium state of magnetized beam -
wave is described in phase space by Bolzmann-Vlasov
equations:
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2001, № 6, p. 115-117. 115
0=
∂
∂+
∂
∂+
∂
∂
z
F
p
FeE
t
F
z v , (2)
bяp Ediv π ρε 4)( = , (3)
∫∫ ==ρ FdpejFdpe bb v; ,
where F is the beam phase density, p and v are linear
momentum and velocity of particles.
In equilibrium state tzz bv−→ . So, phase
density F is determined in beam frame by equation
0=
∂
∂+
∂
∂
z
F
p
FeE z v , (4)
and condition 0=zbEj is irreversibility one of
non-linear state.
A solution of (4) is
)( 0 HHFF −= , (5)
where ∫= dpW v is particle energy,
∫= dpEeU z is field potential.
Due to finiteness of field magnitude and energy of
particle relative motion,
>
≤→−
= ∑
≥
+
0
0
0
0
0
)( 0
HH
HHHHF
F S
SS
S
(6)
and
2
1
0
0
1 =→=ρ ∑
≥
+ SUC
S
S
Sb (7)
where →SC are constants.
For simple (harmonic) wave in beam frame
(because bp h vv ==ω
k )
kzEEZ sin0→ .
Linearity of Maxwell's equation provokes linear
connection between field and beam density
∫ →
∂
∂= ZZ ConstEFdpe
z
EL̂ (8)
Due to non-relativistic motion of particles in beam
frame
U
m
pH −=
0
2
2
, (9)
U
m
pFHHFF −→−=
0
2
000 2
at 0HH ≤ . A typical particle distribution in beam
coordinate system is presented in Fig. 1.
Note that the self-consistent state of beam contains
not only captured particles, but also some fraction of
slipping ones. And all the particles have a developed
kinetics. So, half of particles overtakes a wave, and
another lags behind. But in the whole beam and wave
are synchronized. And it can be represented as a
consequence of bunches harmonically modulated.
Physically this type of kinetics becomes possible if
centers of every beam bunches are located in points kzn
= (2n + 1) π where n is integer. Thus the foremost parts
bunches appear in a braking wave phase, and other
particles are in accelerating one. It is seen from the field
equation that in possible if 0<ε p that is confirmed by
physical consideration (see above).
Fig. 1. Phase density distribution of F (a)) and
phase trajectories (b)) of beam particles. Dotted curve
is separatrix
The consideration of equilibrium states allows
receiving a broad circle of estimates, useful for
operational uses. So, utilizing the laws of conservation
of energy and momentum in system, it is possible to
estimate parameters of a limiting mode of generation of
an electromagnetic radiation (generation efficiency,
state and temperature of a beam, frequency shift at ge-
neration and so on, look more in detail for example [4]).
3. SOLITARY WAVE – SOLITION
Physical reason for forming of an equilibrium state
"solitary wave - short particle bunch" is self-focusing of
particles in systems with slow wave (plasma, travelling
wave tubes and so on). But self-consistent state obtains
new character including more complex non-linear field -
beam density connection that provides, since [5],
nonlinear stability of a soliton.
For wave potential U in beam frame
)(4)(42
2
Uee
dz
UdEediv pb ρπ=ρ+ρπ==
(10)
where ρb (U) includes captured and slipping particles,
but ρp (U) contains only slipping ones. Since (6) - (7)
∑
≥
ρ=ρ
1n
n
nU , (11)
116
and we have the following result:
dU
d
dz
Ud Φ−=2
2
, (12)
where "potential"
∫ ρπ−=Φ
U
due4 . (13)
According to equation (12) periodic solutions arise if
potential contains negative terms [5], Fig. 2.
Fig. 2. Potential wells Φ(U) (solid lines) and
energy levels of oscillator (dotted lines). Upper level in
curve 1 corresponds to solitary wave with pedestal.
Level in curve 2 and lower one in curve 1 correspond to
periodical waves
In particular if non-linear part of sum density
contains only a term of second order, the equation (12)
transforms to Korteweg-De Vries kind one. In result
( )
Const
Lzch
U
U +=
/2
0 (14)
where solition length L is determined by beam
parameters [4], Fig. 3.
Fig. 3. Phase trajectories of particles and field force
in solitary wave
4. SHOCK EQULIBRIUM WAVE IN FIELD
ABSORBING MEDIUM
Consider a system consisting from “collision-less"
fast beam propagating in absorbing plasma. Beam has a
sharp front. The magnitude of plasma current is
determined by a equation containing the collision
frequency ν:
z
L
p
p Evj
t
j
π
ω=+
∂
∂
4
2
. (15)
Now we have for the equilibrium front propagation
dz
deUv
dz
dU
dz
Ud bLL ρπ=ω−ω+ 4
22
3
3
32 vv
. (16)
Dissipation changes cardinally a solution character of
(12): upper trajectory in Fig. 2 begins to oscillate but
“falls” finally on the potential bottom [5]. Here slipping
particles overtaking beam are “working” on wave for
dissipation compensation. Then reaching the wave front
are lagging, Fig. 4.
Fig. 4. Phase trajectories of beam particles in
laboratory frame. Hatched areas are capture zones
In conclusion it is necessary to say that the results
obtained do not hard restrictions on beam parameters.
But all the equilibrium states are characterized by very
developed kinetics. Therefore resulting energetic spread
in beam states is sufficient for stability (with respects to
initial kinds of perturbations). The latter permits to use
proposed approach for estimations in particular of
diverse beam schemes efficiency.
REFERENCES
1. P. Glansdoff, I. Prigogine. Thermodynamic
Theory of Structure. Stability and Fluctuations.
1971, New York: “J. Wiley”.
2. H. Haken. Synergetics, an Introduction //
Rev. Mod. Phys. 1975, v. 47, p. 67.
3. V.E. Fortov. Entsiclopedia
nizkotemperaturnoi plasmy, kniga 4, razdel 10
(Plasmennaia electronika). Moscow: “Nauka”,
2000, p. 1.
4. V.K. Grishin. Ravnovesnye volny v electrod-
inamicheskikh structurakh s intensivnymi
pychkami zariazhennykh chastits, doctor thesis.
Moscow: MSU, 1987, 311 p.
5. B.B. Kadomtsev. Kollectivnye yavleniya v
plasme. Moscow: “Nauka”, 1976.
117
|
| id | nasplib_isofts_kiev_ua-123456789-79437 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T18:00:30Z |
| publishDate | 2001 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Grishin, V.K. 2015-04-01T19:54:34Z 2015-04-01T19:54:34Z 2001 Self-organization and self-consistent equilibrium kinetics in electrodynamic systems with intense charged beams / V.K. Grishin // Вопросы атомной науки и техники. — 2001. — № 6. — С. 115-117. — Бібліогр.: 5 назв. — англ. 1562-6016 PACS: 52.40M https://nasplib.isofts.kiev.ua/handle/123456789/79437 Equilibrium states rising as self-organization result in unstable plasma with fast charged beam system are considered. Self-consistent states described by non-linear Bolzmann-Vlasov equation are obtained for harmonic, solitary and shock waves. Kinetics of captured and slipping beam particles is discussed. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Electrodynamics of high energies in matter and strong fields Self-organization and self-consistent equilibrium kinetics in electrodynamic systems with intense charged beams Самоорганизация и самосогласованная равновесная кинетика в электродинамических системах с быстрыми заряженными пучками Article published earlier |
| spellingShingle | Self-organization and self-consistent equilibrium kinetics in electrodynamic systems with intense charged beams Grishin, V.K. Electrodynamics of high energies in matter and strong fields |
| title | Self-organization and self-consistent equilibrium kinetics in electrodynamic systems with intense charged beams |
| title_alt | Самоорганизация и самосогласованная равновесная кинетика в электродинамических системах с быстрыми заряженными пучками |
| title_full | Self-organization and self-consistent equilibrium kinetics in electrodynamic systems with intense charged beams |
| title_fullStr | Self-organization and self-consistent equilibrium kinetics in electrodynamic systems with intense charged beams |
| title_full_unstemmed | Self-organization and self-consistent equilibrium kinetics in electrodynamic systems with intense charged beams |
| title_short | Self-organization and self-consistent equilibrium kinetics in electrodynamic systems with intense charged beams |
| title_sort | self-organization and self-consistent equilibrium kinetics in electrodynamic systems with intense charged beams |
| topic | Electrodynamics of high energies in matter and strong fields |
| topic_facet | Electrodynamics of high energies in matter and strong fields |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/79437 |
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