Kinetics of the electron distribution function formation and its self-similarity in the course of heating
In many important cases one should treat plasma transport kinetically. In more general sense the solutions of the Landau-Fokker-Planck (LFP) equation, which is one of the key ingredient of plasma kinetic equation, have much broader interest ranging from plasma physics to stellar dynamics. We examine...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2010
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| Цитувати: | Kinetics of the electron distribution function formation and its self-similarity in the course of heating / I.F. Potapenko // Вопросы атомной науки и техники. — 2010. — № 4. — С. 245-249. — Бібліогр.: 4 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860244247220846592 |
|---|---|
| author | Potapenko, I.F. |
| author_facet | Potapenko, I.F. |
| citation_txt | Kinetics of the electron distribution function formation and its self-similarity in the course of heating / I.F. Potapenko // Вопросы атомной науки и техники. — 2010. — № 4. — С. 245-249. — Бібліогр.: 4 назв. — англ. |
| collection | DSpace DC |
| description | In many important cases one should treat plasma transport kinetically. In more general sense the solutions of the Landau-Fokker-Planck (LFP) equation, which is one of the key ingredient of plasma kinetic equation, have much broader interest ranging from plasma physics to stellar dynamics. We examine the interplay of the non-linear colli-sional operator with the different heating terms: mono kinetic; hot ions, DC electrical field and a quasi-linear diffusion operator that models interaction of RF waves with a plasma. Throughout the paper obtained analytical asymptotic results are confirmed with high accuracy by the numerical computing of the non-linear kinetic equation and vice versa.
Во многих важных случаях перенос плазмы необходимо рассматривать кинетически. В более общем смысле решения уравнения Ландау-Фоккера-Планка, являющегося одним из ключевых компонентов кинетического уравнения плазмы, представляют более широкий интерес: от физики плазмы до динамики звезд. Мы исследуем взаимосвязь нелинейного столкновительного оператора с различными составляющими нагрева: однокомпонентная кинетика, горячие ионы, постоянное электрическое поле и квазилинейный диффузионный оператор, который моделирует взаимодействие ВЧ-волн с плазмой. Полученные в работе аналитические асимптотические результаты подтверждаются с высокой точностью численным моделированием нелинейного кинетического уравнения и наоборот.
У багатьох важливих випадках транспорт плазми треба розглядати кінетично. У більш загальному розумінні розв’язки рівняння Ландау-Фокера-Планка, яке є одним з ключових компонентів кінетичного рівняння плазми, становлять більш широку цікавість: від фізики плазми до динаміки зірок. Ми дослідимо взаємозв’язок нелінійного столкновительного оператора зіткнень з різними складовими нагріву: однокомпонентна кінетика, гарячі іони, постійне електричне поле та квазілінійний дифузійний оператор, який моделює взаємодію ВЧ-хвиль з плазмою. Отримані у роботі аналітичні асимптотичні результати підтверджуються з високою точністю числовим моделюванням нелінійного кінетичного рівняння та навпаки.
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| first_indexed | 2025-12-07T18:33:52Z |
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| fulltext |
KINETICS OF THE ELECTRON DISTRIBUTION FUNCTION FORMA-
TION AND ITS SELF-SIMILARITY IN THE COURSE OF HEATING
I.F. Potapenko
Keldysh Institute of Applied Mathematics, Moscow, Russia
E-mail: irina@keldysh.ru
In many important cases one should treat plasma transport kinetically. In more general sense the solutions of the
Landau-Fokker-Planck (LFP) equation, which is one of the key ingredient of plasma kinetic equation, have much
broader interest ranging from plasma physics to stellar dynamics. We examine the interplay of the non-linear colli-
sional operator with the different heating terms: mono kinetic; hot ions, DC electrical field and a quasi-linear diffu-
sion operator that models interaction of RF waves with a plasma. Throughout the paper obtained analytical asymp-
totic results are confirmed with high accuracy by the numerical computing of the non-linear kinetic equation and
vice versa.
PACS: 52. 65. -у
1. PRELIMINARIES
In many important cases one should treat plasma
transport kinetically. Examples are: the electron heat
transport in inertial confinement fusion; propagation of
the heat bursts, caused by edge-localized mode (ELM),
into scrape-off layer (SOL) of tokamaks. In more gen-
eral sense the solutions of the Landau-Fokker-Planck
(LFP) equation, which is one of the key ingredient of
plasma kinetic equation, have much broader interest
ranging from plasma physics to stellar dynamics (e.g.,
see [1-4] and the references therein). In certain cases, it
worthwhile to analyse simpler models of plasma trans-
port problems, solution of which, nevertheless, exhibit
some important features of the problem of interest and
also help to benchmark complex kinetic codes.
The non-linear LFP equation reads
21 1 ;
2i i j i j
f h g
f f
t v v v v v
α α
α α
⎧ ⎛ ⎞∂ ∂ ∂∂ ∂⎪= − + ⎜ ⎟⎨ ⎜ ⎟Γ ∂ ∂ ∂ ∂ ∂ ∂⎪ ⎝ ⎠⎩
α
⎫⎪
⎬
⎪⎭
( ) 1
4 2
1 / ( , ) ,
( , ) , 2 / ,
h m m dw f w t v w
g dw f w t v w e L m
α α β β
β
α β π
−= + −
= − Γ =
∑ ∫
∫
where L is the Coulomb logarithm, ie,=α .
A complexity of the LFP equation makes it almost
impossible to use analytical methods for applied prob-
lems. Therefore the role of numerical methods for this
equation is very important. The well-known specific
property of nonlinear kinetic equations is that a single
equation has several conservation laws (mass, energy,
momentum). In case of isotropic LFP equation this
means that equation can be written in two equivalent
divergent forms: conservation of mass and conservation
of energy. We have used implicit schemes with itera-
tions that guaranteed energy conservation with high
accuracy (till the order of machine errors, if necessary).
For linear equation when particle exchange energy with
the bulk plasma the only collision invariant is particle
mass (density).
We consider the isotropic electron distribution
and study the electron heating with a simple
model:
),( tvf
( ) ( )fH+ff,I=tf ∂∂ / , where ( ff,I ) is the
LFP collisional integral and ( )fH is the heating
source. We examine the interplay of the non-linear col-
lisional operator with the different heating terms: mono
kinetic distribution (the energy (particle) source and
sink can be provided by ion beams, neutral injection,
etc.); hot ions (two component plasma), and a quasi-
linear diffusion operator that models interaction of RF
waves with a plasma. We shortly review some selected
results of our works on the subject with a heating source
localized in the velocity space. Then a broader class of
the heating terms resulting in enhancement of the tail of
the distribution function is analytically analyzed. Ana-
lytical treatment of the nonlinear kinetic equation usu-
ally deals with rigorous simplifications (linearization of
the equation, taking into account small parameters, such
as mass ratio 1/ <<= ie mmρ , etc.). Throughout the
paper obtained analytical asymptotic results are con-
firmed with high accuracy by the numerical computing
of the non-linear kinetic equation and vice versa.
The investigation is mainly concentrated on the evo-
lution of the distribution function tails in high velocity
region ∞→∞→ tv , . To characterize the tail forma-
tion we use the following presentation of the distribu-
tion where )(),(),( ξξξ MftGtf ⋅= , where ,
is the thermal velocity, and Maxwell distribution
is . It is known that the Coulomb diffusion
influence is the utmost in the cold energetic region
22 / thvv=ξ
thv
ξ−≈ ef M
10 <≤ξ . In the high-velocity region 1>>ξ the LFP
non-linear parabolic equation degenerates because of
known Rutherford cross-section dependence on velocity
and acquires more pronounced hyperbolic type when
the transport term (the first derivative) becomes more
important . This circumstance
leads to inevitable retarding of the distribution tail for-
mation comparing to the relaxation in the thermal veloc-
ity region
ξξξ )(~)( 2/1 fffI +−
1~/~,1~ ettt =ξ , − the collision time. et
The particle density n and temperature T (average
energy ) expressed in energetic units are defined
through integrals
)(te
∫∫
∞∞
==
0
2/3
0
2/1 ).(),(,1),( tetfdtfd ξξξξξξ
________________________________________________________________
ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 2010. № 4.
Серия: Плазменная электроника и новые методы ускорения (7), с.245-249.
245
In the absence of energy (particle) sources the den-
sity and the thermal velocity (energy) are constant. The
unique equilibrium solution of the problem is the Max-
well distribution function. In this case the following
results were obtained from asymptotic analysis of the
distribution function behavior in the region 1>>ξ and
for time intervals larger than collision time . The
initial function is located in the thermal velocity region
and equals zero in high velocity region. The time period
when the relaxation in the bulk of the distribution is
finished is characterized by
1>>t
1~),( tG ξ (Fig.1). Asymp-
totic analysis shows that ),( tG ξ for 1, >>tξ can be
analytically expressed in terms of error function, has a
character of a propagating wave, which front moves
under the law with the constant front
width
3/2)3( tf =ξ
π=Δ )(tf and can be roughly estimated as
~ . )exp( 2/5ξ−
Fig.1. The graph of the function G(v,t) for different
time instants
2. HEATING LOCALIZED
IN THE VELOCITY SPACE
2.1. HEATING BY THE SOURCE
OF HOT ELECTRONS
Let us include into consideration the heating of the
electrons by the source of the hot particles ,
which is localized in a high energetic region
)( fH
1>>+ξ . In
a cold region the distribution is supposed Maxwellian.
The next simplified assumption is that the heating
source has small intensity σ<<1, so the density and the
energy of the system practically do not undergo notice-
able changes during the process under consideration.
Then asymptotic analysis shows that if the source is
localized at )/1log(~ σξ then for the time period
a non-equilibrium quasi steady-
state local distribution is formed. It is located inside the
momentum interval between the energy (particle) source
and the bulk of the particle distribution and reads
1))/1(ln(1 −<<<< σt
246
.)()()( 2/1∫
∞−
+≈
ξ
ξξ xHdxxff M
This non-equilibrium distribution has the form of
plateau (Fig.2). In general the functional dependence of
the quasi steady-state electron distribution is insensitive
to the extent to which the source and sink are located in
momentum space or the sink if any is localized at −ξ
and −+ < ξξ . If, in particular, the source and the sink
are mono kinetic distributions (like δ -type functions)
)],()([)( 2/1
−+
− −−−⋅⋅≅ ξξδξξδξσfH
then , where )()( ξσξ ξ Δ⋅+⋅= −eCf
)](exp[)][(][
)](exp[)][(][)(
−−−
+++
−−⋅−+−−
−−⋅−+−=Δ
ξξξξηξξη
ξξξξηξξηξ
and η[y] is a unit function. The non-equilibrium distri-
bution may differ from the equilibrium by tens of orders
in magnitude. Such extended knees of the distribution
functions can be observed in laboratory experiments
(additional heating in tokamaks, afterglow gas discharge
with metastable atoms), as well in magnetoshperic
plasmas.
However, the distribution tail is under heated in
comparison to the Maxwellian. Its formation is de-
scribed by aforesaid formulas in the region
1, >>>> + tξξ having wave-like character.
Fig.2. Logarithm of the steady-state non equilibrium
distribution function vs. squared velocity (arbitrary
units).The computations were carried out for the source
located at 36=ξ for theintensity 610−=σ
2.2. HEATING BY THE SOURCE OF HOT IONS
Now let pass to the classical problem of electron and
ion temperature relaxation considering a system of elec-
tron and ion LFP equations, with 0)( =fH ,
eqie TtTtT 2)()( =+ . We use two asymptotic parameters
1/ <<= ie mmρ and 1/)( <<= ei TTt ρε for analytical
analysis. We introduce the self-similar variables
then the equation for the
electron function reads:
),/(),( 223 tvvfvtf th αα ξ −≅
3/2
1/2
1/2 3/2
1 1[
].
e i
e e e
e
e
e e
df T f
T = I(f , f )+ + f +
dt ξ ρ T ξξ
dT
T ξ f
dt
⎛ ⎞∂∂
⎜ ⎟∂ ∂⎝ ⎠
e
e
.
For the temperature changing we have
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
− ∫
∞
0
2/3 ),(0,
3
2 tfdTt)(fT=
dt
dTT eeei
e
e ξξ .
The remarkable property of the above equation is
that the electron distribution is independent of the ion
distribution and depends only on the ion energy.
ei TT > The hot ions interact primarily with cold elec-
trons at 0≈ξ . From asymptotic analysis (singular per-
turbation theory) it is obtained that the perturbation of
the electron distribution function in a cold region has a
character of a boundary layer with a width of .
The electron function achieves its maximum absolute
deviation from its local (in time) Maxwell distribution at
3/2)(~ tε
0≅ξ :
]/)(9.21[),( 3/2
iie TTTetf −+⋅≅ − εξ ξ .
The applicability condition of the known formula for
temperatures is )(2/
eite TTCTT −≅ 1<<ε that is hundred
times less rigorous than the condition 1<<ρ . From nu-
merical simulation the condition is estimated as 1.0≤ε .
Note, from the formula the time dependence of the tem-
perature is . The relative deviation of the elec-
tron distribution
3/2~ tT
),(/),(),( tvftvftG Me=ξ from the
equilibrium is much larger in the tail region. It has a
wave character with a stable front, which propagates in
high velocity region being described by the above for-
mulas in the case without heating. For hot electrons
the electron tail is cooling while relaxation in
the high energetic region having the same character of a
propagating wave with (Fig.3).
ie TT >
tv f ~3
Fig.3. The graph of the deviation of the electron func-
tion ϕ for different time instants
Fig.4. The graph of the function G(v,t)
for different time instants
Now we consider the example when the initial elec-
tron and ion distribution functions are )1( −vδ and
)5.0( −vδ , correspondingly, and the ion function is con-
stant in time. For this case the plots demonstrate the
comparison of the numerical simulation results with
analytical results. First, the spreading of the function G
in high velocity region is presented (Fig.4). From the
time, when the parameter 05.0≤ε the curves do not
change their slope. Then the dependence of the energy,
the wave front velocity and the width front on time is
given (Figs.5-7). As can be seen the analytical and nu-
merical results are a good match.
Fig.5. The electron energy dependence on time
The deviation of the width front at the beginning is
understandable: the system “remembers the initial
state”. Another example considered here is interaction
of RF waves with plasma that is simulated by the quasi-
linear operator (usually 2D in velocity space) acting
within corresponding resonant region
.0,][)( 21
2/1 ξξξξ ξξ ≤≤≠⋅= − ifDfDfH qlql
Fig.6. The plot of the front width on time
Fig.7. The wave front velocity dependence on time
Fig.8. The function ),( tG ξ for the different time mo-
ments in a case of the quasi-linear operator action with-
in the energetic region 75.005.0 ≤≤ξ
247
For this case Fig.8 demonstrates the numerical result
of the temporal tail formation that has a character of a
propagating wave with the slope slightly changing in
time because of constant heating.
The same behavior of the distribution function tails
seems valid for any localized in the momentum space
heat source having time dependence as . 3/2~)( ttT
2.3. HEATING AND ACCELERATION BY DC
ELECTRICAL FIELD
The problem of runaway electrons is connected with
the solution of the 2D in the velocity space LFP equa-
tion with the DC electrical field action. The influence of
an electrical field which is small in comparison with
that of Dreiser 1/ <<= DEEγ is taken into account as
follows
( ) .// zvf+ff,I=tf ∂⋅∂∂ γ
In the direction of the electrical field the distribution
has the enhanced tail and it is depleted in the opposite
direction so the density of particles is preserved. During
the process of constant heating the thermal region of the
distribution function is close to the Maxwell distribution
because the parameter γ is small. Otherwise the maxi-
mum of the distribution corresponds to the velocity
),(
0
3
1
1
μμμ vfdvvdv ∫∫
∞
−
⋅= .
For the numerical simulation the boundary condition
for 0),( →tvf ∞→v is used, so that the numerical
distribution function is equal to machine zero. Fig.9
shows the electron distribution function that formed
under the DC field action when the initial temperature
rises two times, 01.0=γ . The distribution has an accel-
erated tail in the electrical field direction up to the criti-
cal velocity ( ). Further the
curve slightly changes the functional dependence. Thus
even with the additional transport term in the LFP equa-
tion the tail of the electron distribution is under heated.
2/1−≅γcrv 100/ 22 ≈thcr vv
Fig.9. The graph of the electron distribution function for
2D case for different )cos( Ev
rr
=μ
3. SELF-SIMILAR SOLUTIONS
WITH ACCELERATED TAILS
Unlike the cases considered above the situation
changes drastically when coefficient of quasi-linear dif-
fusion increases with velocity increasing. We consider a
special class of functions ,
where is the normalization constant and
is an adjustable parameter, for which it is possible to
construct a self-similar solution in high velocity region
2/12/3
0 )( −− ⋅⋅= tTDD p
ql ξ
0D 02/5 ≥> p
.
2/5
exp~)(
2/5
⎥
⎦
⎤
⎢
⎣
⎡
−
−∞→
−
p
f
pξξ
Here the variable is changing in time. 2
h
2 / vv=ξ
Two cases ought to be specified: for the so-
lution is the Maxwellian and for we have a
power-law tail
2/3=p
2/5=p
2/5−→ξf . The time dependence of the
temperature is as the above . 3/2~)( ttT
In numerical modeling the initial distribution is ap-
proximated on the mesh in the usual way, that is, it dif-
fers from zero at one point. Very rapidly, the solution
acquires a quasi equilibrium form in the thermal veloc-
ity region at the instant that corresponds to the
collision time. In this region the distribution functions
are close to each other throughout the entire relaxation
process for different coefficients of quasi-linear diffu-
sion. The main difference is observed in the region of
the distribution tails for
1~et
1>>ξ . For the case p=3/2 the
solution is Maxwellian. Numerical results show that at
the beginning the tail has Coulombian character and
then since the time it spreads into super ther-
mal region following the diffusion action (Fig.10).
0/1~ Dt
Fig.10. The graph of the function ),( tG ξ
for the quasi-linear coefficient
for different time moments
0015.00 =D
1000,...,100,50,10=t
p=2
),0(
),(
tf
tf ξ
ξ1/2
Fig.11. Temporal evolution of the electron distribution
function for p=3/2. In agreement with analytic results,
the distribution function approaches Maxwellian distri-
bution (dotted line)
248
REFERENCES Fig.11 shows the logarithm of the distribution func-
tion ),( tf ξ normalized on its value at zero velocity
for different time moments for ),0( tf 2=p . It dis-
plays the transition region between the Maxwellian part
and the enhanced tail. In the region the
distribution is visibly close to Maxwellian. The obtained
results can be used for the assessment of the impact of
ELMs on heat transport and sheath parameters.
thvv 4~0 ≤
1. H. Risken. The Fokker-Planck Equation-Methods of
Solution and Applications. Springer, Berlin, 1989,
p.437.
2. I.F. Potapenko, A.V. Bobylev, and E. Mossberg.
Deterministic and stochastic methods for nonlinear
Landau-Fokker-Planck kinetic equations with appli-
cations to plasma physics // Transp. Theory Stat.
Phys. 2008, v.37, p.113-170.
3. I.F. Potapenko, T.K. Soboleva, and S.I. Krashenin-
nikov. Electron heating and acceleration for the non-
linear kinetic equation // Il Nuovo Cimento. 2010,
v.33 (1), p.199-206.
ACKNOWLEDGEMENTS
This work is partially supported by the PFI DMS N3
of RAS and by the grant №09-02– (02-620) RFFI RAS
and NAS of Ukraine. 4. I.F. Potapenko, M. Bornatici, V.I. Karas`. Quasi
steady-state distributions for particles with power-
law interaction potentials // J. of Plasma Physics.
2005, v.71, p.859-875.
Статья поступила в редакцию 02.06.2010 г.
КИНЕТИКА ФОРМИРОВАНИЯ ФУНКЦИИ РАСПРЕДЕЛЕНИЯ ЭЛЕКТРОНОВ
И ЕЕ САМОПОДОБИЕ В ПРОЦЕССЕ НАГРЕВА
И.Ф. Потапенко
Во многих важных случаях перенос плазмы необходимо рассматривать кинетически. В более общем
смысле решения уравнения Ландау-Фоккера-Планка, являющегося одним из ключевых компонентов кине-
тического уравнения плазмы, представляют более широкий интерес: от физики плазмы до динамики звезд.
Мы исследуем взаимосвязь нелинейного столкновительного оператора с различными составляющими на-
грева: однокомпонентная кинетика, горячие ионы, постоянное электрическое поле и квазилинейный диффу-
зионный оператор, который моделирует взаимодействие ВЧ-волн с плазмой. Полученные в работе аналити-
ческие асимптотические результаты подтверждаются с высокой точностью численным моделированием не-
линейного кинетического уравнения и наоборот.
КІНЕТИКА ФОРМУВАННЯ ФУНКЦІЇ РОЗПОДІЛУ ЕЛЕКТРОНІВ
ТА ЇЇ САМОПОДІБНІСТЬ У ПРОЦЕСІ НАГРІВУ
І.Ф. Потапенко
У багатьох важливих випадках транспорт плазми треба розглядати кінетично. У більш загальному розу-
мінні розв’язки рівняння Ландау-Фокера-Планка, яке є одним з ключових компонентів кінетичного рівняння
плазми, становлять більш широку цікавість: від фізики плазми до динаміки зірок. Ми дослідимо взаємо-
зв’язок нелінійного столкновительного оператора зіткнень з різними складовими нагріву: однокомпонентна
кінетика, гарячі іони, постійне електричне поле та квазілінійний дифузійний оператор, який моделює взає-
модію ВЧ-хвиль з плазмою. Отримані у роботі аналітичні асимптотичні результати підтверджуються з висо-
кою точністю числовим моделюванням нелінійного кінетичного рівняння та навпаки.
249
Unlike the cases considered above the situation changes drastically when coefficient of quasi-linear diffusion increases with velocity increasing. We consider a special class of functions , where is the normalization constant and is an adjustable parameter, for which it is possible to construct a self-similar solution in high velocity region
ACKNOWLEDGEMENTS
|
| id | nasplib_isofts_kiev_ua-123456789-17340 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T18:33:52Z |
| publishDate | 2010 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Potapenko, I.F. 2011-02-25T14:05:51Z 2011-02-25T14:05:51Z 2010 Kinetics of the electron distribution function formation and its self-similarity in the course of heating / I.F. Potapenko // Вопросы атомной науки и техники. — 2010. — № 4. — С. 245-249. — Бібліогр.: 4 назв. — англ. 1562-6016 https://nasplib.isofts.kiev.ua/handle/123456789/17340 In many important cases one should treat plasma transport kinetically. In more general sense the solutions of the Landau-Fokker-Planck (LFP) equation, which is one of the key ingredient of plasma kinetic equation, have much broader interest ranging from plasma physics to stellar dynamics. We examine the interplay of the non-linear colli-sional operator with the different heating terms: mono kinetic; hot ions, DC electrical field and a quasi-linear diffusion operator that models interaction of RF waves with a plasma. Throughout the paper obtained analytical asymptotic results are confirmed with high accuracy by the numerical computing of the non-linear kinetic equation and vice versa. Во многих важных случаях перенос плазмы необходимо рассматривать кинетически. В более общем смысле решения уравнения Ландау-Фоккера-Планка, являющегося одним из ключевых компонентов кинетического уравнения плазмы, представляют более широкий интерес: от физики плазмы до динамики звезд. Мы исследуем взаимосвязь нелинейного столкновительного оператора с различными составляющими нагрева: однокомпонентная кинетика, горячие ионы, постоянное электрическое поле и квазилинейный диффузионный оператор, который моделирует взаимодействие ВЧ-волн с плазмой. Полученные в работе аналитические асимптотические результаты подтверждаются с высокой точностью численным моделированием нелинейного кинетического уравнения и наоборот. У багатьох важливих випадках транспорт плазми треба розглядати кінетично. У більш загальному розумінні розв’язки рівняння Ландау-Фокера-Планка, яке є одним з ключових компонентів кінетичного рівняння плазми, становлять більш широку цікавість: від фізики плазми до динаміки зірок. Ми дослідимо взаємозв’язок нелінійного столкновительного оператора зіткнень з різними складовими нагріву: однокомпонентна кінетика, гарячі іони, постійне електричне поле та квазілінійний дифузійний оператор, який моделює взаємодію ВЧ-хвиль з плазмою. Отримані у роботі аналітичні асимптотичні результати підтверджуються з високою точністю числовим моделюванням нелінійного кінетичного рівняння та навпаки. This work is partially supported by the PFI DMS N3 of RAS and by the grant №09-02– (02-620) RFFI RAS and NAS of Ukraine. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Нелинейные процессы в плазменных средах Kinetics of the electron distribution function formation and its self-similarity in the course of heating Кинетика формирования функции распределения электронов и ее самоподобие в процессе нагрева Кінетика формування функції розподілу електронів та її самоподібність у процесі нагріву Article published earlier |
| spellingShingle | Kinetics of the electron distribution function formation and its self-similarity in the course of heating Potapenko, I.F. Нелинейные процессы в плазменных средах |
| title | Kinetics of the electron distribution function formation and its self-similarity in the course of heating |
| title_alt | Кинетика формирования функции распределения электронов и ее самоподобие в процессе нагрева Кінетика формування функції розподілу електронів та її самоподібність у процесі нагріву |
| title_full | Kinetics of the electron distribution function formation and its self-similarity in the course of heating |
| title_fullStr | Kinetics of the electron distribution function formation and its self-similarity in the course of heating |
| title_full_unstemmed | Kinetics of the electron distribution function formation and its self-similarity in the course of heating |
| title_short | Kinetics of the electron distribution function formation and its self-similarity in the course of heating |
| title_sort | kinetics of the electron distribution function formation and its self-similarity in the course of heating |
| topic | Нелинейные процессы в плазменных средах |
| topic_facet | Нелинейные процессы в плазменных средах |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/17340 |
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