Brownian motion of grains and negative friction in dusty plasmas
Within the approximation of dominant charging collisions the explicit microscopic calculations of the Fokker-Planck kinetic coefficients for highlycharged grains moving in plasma are performed. It is shown that due to ion absorption by grain the friction coefficient can be negative and thus the...
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| Cite this: | Brownian motion of grains and negative friction in dusty plasmas / S.A. Trigger, A.G. Zagorodny // Condensed Matter Physics. — 2004. — Т. 7, № 3(39). — С. 629–638. — Бібліогр.: 21 назв. — англ. |
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| citation_txt | Brownian motion of grains and negative friction in dusty plasmas / S.A. Trigger, A.G. Zagorodny // Condensed Matter Physics. — 2004. — Т. 7, № 3(39). — С. 629–638. — Бібліогр.: 21 назв. — англ. |
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| description | Within the approximation of dominant charging collisions the explicit microscopic
calculations of the Fokker-Planck kinetic coefficients for highlycharged
grains moving in plasma are performed. It is shown that due to
ion absorption by grain the friction coefficient can be negative and thus the
appropriate threshold value of the grain charge is found. The stationary
solutions of the Fokker-Planck equation with the velocity dependent kinetic
coefficient are obtained and a considerable deviation of such solutions
from the Maxwellian distribution is established.
В наближенні домінуючих контактних зіткнень порошинок з частинками плазми виконано макроскопічні розрахунки кінетичних коефіцієнтів Фокера-Планка для заряджених порошинок, що рухаються у плазмі. Показано, що завдяки зіткненням з йонами коефіцієнт тертя може бути від’ємним, і знайдено порогове значення заряду порошинки. Знайдено розв’язок стаціонарного рівняння Фокера-Планка з кінетичними коефіцієнтами, що залежать від швидкості, і встановлено суттєву відмінність отриманих розв’язків від максвелових розподілів.
|
| first_indexed | 2025-12-01T21:45:17Z |
| format | Article |
| fulltext |
Condensed Matter Physics, 2004, Vol. 7, No. 3(39), pp. 629–638
Brownian motion of grains and
negative friction in dusty plasmas
S.A.Trigger 1,2 , A.G.Zagorodny∗3
1 Joint Institute for High Temperatures,
Russian Academy of Sciences,
13/19 Izhorskaya Str., Moscow 127412, Russia
2 Humboldt University,
110 Invalidenstr., D–10115 Berlin, Germany
3 Bogolyubov Institute for Theoretical Physics,
National Academy of Sciences of Ukraine,
14B Metrolohichna Str., Kiev 03143, Ukraine
Received February 2, 2004
Within the approximation of dominant charging collisions the explicit mi-
croscopic calculations of the Fokker-Planck kinetic coefficients for highly-
charged grains moving in plasma are performed. It is shown that due to
ion absorption by grain the friction coefficient can be negative and thus the
appropriate threshold value of the grain charge is found. The stationary
solutions of the Fokker-Planck equation with the velocity dependent kinet-
ic coefficient are obtained and a considerable deviation of such solutions
from the Maxwellian distribution is established.
Key words: dusty plasma, negative friction, effective temperature,
Fokker-Planck equation
PACS: 52.27.Lw, 52.20.Hv, 52.25.Fi
Description of the Brownian motion in the systems with particle or energy fluxes
still remains one of the key problems in the statistical physics of the open systems
[1]. Starting from the classical Lord Rayleigh work [2] many studies of the non-
equilibrium motion of Brownian particles with additional (inner or external) energy
supply have been performed. In particular, such studies are of great importance for
physical-chemical [3,4] and biological [5] systems in which non-equilibrium Brownian
particle motion is referred to as the motion of active Brownian particles. Recently
the dynamical and energetic aspects of motion for the active Brownian particles
have been described based on the Langevin equation and the appropriate Fokker-
Planck equation [6,7]. The possibility of negative friction (negative values of friction
coefficient) of the Brownian particles was regarded as a result of energy pumping.
∗E-mail: azagorodny@gluk.org
c© S.A.Trigger, A.G.Zagorodny 629
S.A.Trigger, A.G.Zagorodny
For some phenomenological dependences of the friction coefficient as a function of
the grain’s velocity, one-particle stationary non-Maxwellian distribution function
was found.
The traditional formulations of the non-equilibrium Brownian motion are based
on some phenomenological expressions for the friction and diffusion coefficients. In
particular, it means that deviations from the Einstein relation, as well as the velocity
dependence of these coefficients are postulated and high level of uncertainty for
the application of such models to the real systems takes place. Moreover, since in
the case of the open systems there exist few types of the Fokker-Planck equation
which can be related to the nonlinear Langevin equations (see, for example [1] and
references cited therein) the form of the Fokker-Planck equation itself could be a
matter of the choice. Here we will consider another situation, when the kinetic
coefficients can be calculated exactly on the basis of the microscopically derived
Fokker-Planck equation for dusty plasmas [8,9]. It will be shown that in the case
of strong interaction parameter Γ ≡ e2ZgZi/aTi � 1 (here Zg, Zi are the charge
numbers for the grains and ions, respectively, a is the grain radius, Ti is the ion
temperature) the negative friction coefficient appears for some velocity domain. If
the charging collisions are dominant, this domain is determined by the inequalities:
v2 < 10v2
Ti(Γ−1)/(3Γ−1) for Γ > 1, but (Γ−1) � 1,(v2
Ti ≡ Ti/mi); and v2 < 2v2
TiΓ
for Γ � 1. The physical reason for manifestation of negative friction is clear: the
cross-section for ion absorption by grain increases with the relative velocity between
the ion and grain decreases due to the charge-dependent part of the cross-section.
Therefore, for a moving highly-charged grain (Γ � 1) the total momentum transfer
from ions to the grain in the direction of grain velocity could be larger than in the
opposite direction.
Naturally, the Coulomb scattering and particle friction related to the ion-grain
and neutral-grain elastic collisions will increase the threshold for negative friction
and even can suppress it for some plasma parameters. However, the latter are rather
sophisticated processes, and their theoretical description within various approxima-
tions can shadow the physics of the phenomenon. Some estimate of the effect of such
processes will be also done below, but the detailed description of their action should
be a matter of further consideration. On the other hand, the problem of microscopic
investigation of negative friction in dusty plasmas is so fundamental that it deserves
a description of its simplest manifestation (which occurs in the case of dominant
charging collisions) to be considered in this paper.
We start from the Fokker-Planck equation for the spherical grains in dusty plas-
ma with typical narrow charge distribution around a negative value q = eeZg, which
permits to put all grain charges to be equal. We also ignore the increase of the grain
mass [10, 11] assuming that neutral atoms generated in the course of the surface
electron-ion recombination escape from the grain surface into a plasma. As it was
mentioned already in [9], where the subsequent kinetic theory with ion absorption
in dusty plasmas has been developed, the problem of correct momentum transfer
due to ion absorption and further surface recombination is important for description
of stationary state. Strictly speaking the stationary state can be reached only by
630
Brownian motion of grains
inclusion of these both processes. To avoid this complication, however, usually it is
suggested that the mass of grain should be constant, due to the big difference of ion
and grain masses. Recently in [12] it was shown that this assumption leads to the
violation of the Galiley invariance of the particle transition probability generated
by collisions. Nevertheless, the correct kinetic theory of ion absorption with further
surface ion recombination is still in the stage of development and will be published
elsewhere.
Then, for the conditions li � λD, a, where li is the ion mean free path length
and λD is the plasma screening (Debye) length, the friction and diffusion coefficients
in the Fokker-Planck kinetic equation β(q,v) and D(q,v) are given by [9]:
β(q,v) = −
∑
α=e,i
mα
mg
∫
dv′
v · v′
v2
σαg(q, |v − v
′|)|v − v
′|fα(r,v′, t),
D(q,v) =
∑
α=e,i
1
2
(
mα
mg
)2 ∫
dv′
(v · v′)2
v2
σαg(q, |v − v
′|)|v − v
′|fα(r,v′, t),
σgα(q, v) = πa2
(
1 − 2eαq
mαv2a
)
θ
(
1 − 2eαq
mαv2a
)
. (1)
Here σgα(q,v) is the cross-section for grain charging within the orbital motion limited
(OML) theory. In this approximation all electrons and ions approaching the grain
on the distance smaller than a are assumed to be absorbed. Subscript α = e, i labels
plasma particle species. The rest of notation is traditional. To include the processes
of the electron, ion and atom scattering we have to summarize the appropriate
coefficients on the different type of the processes. In the case of dominant charging
collisions integration in equation (1) can be performed explicitly. The ion part of β
(which exceeds the electron one at least in (miTe/meTi)
1/2 times) is as follows:
βi(q,v) = −
√
2π
mi
mg
a2nivTi [I1(η) + I2(η, Γ)] ,
I1(η) =
(
−1
2
+
1
4η
) √
π
η
Erf
√
η − 1
2
√
η
e−η,
I2(η, Γ) =
Γ
η
[
1
2
√
π
η
Erf
√
η − e−η
]
. (2)
Here na are the densities of the electrons and ions, Erf
√
η is the error function and
η ≡ η(v) = v2/2v2
Ti. It is easy to see that the terms I1(η) and I2(η, Γ) describe
the parts of βi(q,v) related to purely geometrical and charge-dependent collecting
cross-sections, respectively. The integration for Di(q,v) leads to the expression:
D(q,v) =
4
3
√
2π
(
mi
mg
)
a2nivTi
(
Ti
mg
) [
K1(η) + K2(η, Γ)
]
,
K1(η) =
3
16η
[
2(η − 1)e−η+ (2η2+ η + 1)
√
π
η
Erf
√
η
]
,
631
S.A.Trigger, A.G.Zagorodny
K2(η, Γ) =
3Γ
8η
(η + 1)
[
−2e−η +
√
π
η
Erf
√
η
]
. (3)
As follows from the physical reason and directly from equation (3), the coefficient
D(q,v) is always positive. At the same time for some values of η and Γ the friction
coefficient βi(q,v) can be negative. To find the root η(Γ) of the equation βi(η, Γ) = 0
let us consider two limiting cases η � 1 and η � 1. For η � 1 equation (2) gives
βi(η, Γ) = 2A
[
1 − Γ − η
5
(1 − 3Γ)
]
, (4)
where
A =
1
3
√
2π
(
mi
mg
)
a2nivTi.
Equation (4) has a root η(Γ), which exists and is small (according to the conditions
of applicability for this expansion) only for Γ > 1, but Γ − 1 � 1:
η(Γ) = 5
(Γ − 1)
3Γ − 1
' 5
2
(Γ − 1). (5)
Figure 1. Numerical solution of the
equation β(η,Γ) = 0 separating the pos-
itive and negative values of the friction
coefficient.
Therefore the function βi(η, Γ) is
negative at η < η(Γ), if Γ > 1. Equa-
tion (4) shows that for Γ < 1, when
βi is positive for all η, the derivative
(dβi(η, Γ)/dη)|η=0 changes its sign at
Γ = 1/3. Near η = 0 the friction βi
decreases as function η if Γ < 1/3 and
increases if Γ > 1/3.
For η � 1 and arbitrary values of Γ
equation (2) gives
βi(η, Γ) =
3
2
A
√
π
η
[
1 − 1 + 2Γ
2η
]
. (6)
It means that for Γ � 1 and large η
the equation βi(η1Γ) = 0 has a root
η1(Γ) ' (1 + 2Γ)/2 ∼ Γ. Naturally,
for all Γ > 1 there exists appropriate
η(Γ), which is the root of that equation.
This conclusion is confirmed by the exact numerical calculations of η(Γ) (figure 1).
Asymptotically (for η � max(1, Γ)) βi(η, Γ) tends to zero as
√
η−1 and is positive
for η > η(Γ). For the case of the negative friction (Γ > 1) the maximum of the
coefficient βi(η, Γ) is located at the point ηm(Γ) � 1:
ηm ' 3
2
(1 + 2Γ) , βi(ηm, Γ) ' A
√
2π
3(1 + 2Γ)
. (7)
632
Brownian motion of grains
For the diffusion coefficient Di(η, Γ) the expansions for η � 1 and η � 1 lead to
Di(η, Γ) = 4A
(
Ti
mg
) [
1 +
Γ
2
+
η
10
(1 + 2Γ)
]
, η � 1
Di(η, Γ) = 4A
(
Ti
mg
)
3
8
(πη)1/2
[
1 +
1
2η
(1 + 2Γ)
]
, η � 1. (8)
The typical behaviour of βi(η, Γ) and Di(η, Γ) calculated numerically based on the
equations (2), (3) is shown in figures 2, 3.
Figure 2. The velocity dependences of dimensionless friction coefficient β̃(η,Γ) =
I1(η) + I2(η,Γ) for different values of Γ.
Figure 3. The same dependences for the dimensionless diffusion coefficient
D̃(q, v) = K1(η) + K2(η,Γ).
The stationary solution of the Fokker-Planck equation (which is of the Itoh’s
form) with the kinetic coefficients (2), (3) for the grain distribution function fg(q, v)
is
fg(q, v) =
C
Di(q, v)
exp
−
v∫
0
dvv
βi(q, v)
Di(q, v)
, (9)
633
S.A.Trigger, A.G.Zagorodny
Figure 4. The same dependences for the distribution function fg(η,Γ).
where C is a constant, providing normalization
∫
dvfg(q, v) = 1. The velocity de-
pendence of this solution for different values of Γ is shown in figure 4.
In order to get some analytical estimates let us consider the vicinity of the point
Γ = 1. In such a case the integration in equation (9) leads to the non-Maxwellian
distribution function, which for Γ > 1 possesses a maximum at v 6= 0:
fg(q,v) =
(
mi
2πTi
)3/2 √
π
2Y3/2(1 + Γ/2)
[
1 +
miv
2
2T ∗
i
1
5
(1 + 2Γ)
]−1
× exp
{
−mgv
2
2T ∗
i
[
(1 − Γ) +
miv
2
2T ∗
i
1
10
(
5Γ2 + 4Γ − 3
)]}
. (10)
Here
Yν(Γ) =
∞∫
0
dηην−1 exp
[
−αη(1 − Γ) − γη2
]
=
= (2γ)−
ν
2
√
π
2
exp
[
α2(1 − Γ)2
8γ
]
D−ν
(
α(1 − Γ)√
2γ
)
,
T ∗
i = 2Ti
(
1 +
Γ
2
)
, α =
mg
2mi
(
1 +
Γ
2
)
,
γ =
α(5Γ2 + 4Γ − 3)
10(2 + Γ)
(11)
and Dν is the cylindrical parabolic function. Equation (10) is relevant, if the integral
over η converges, which is valid for positive γ (Γ > 0, 472). At the same time, as was
mentioned above, for the applicability of the expansions (4), (8) the inequality Γ <
1 + δ with δ � 1 is required. It is clear from equations (6), (8), that the asymptotic
behaviour of the distribution function is non-exponential fg(η)|η→∞ ∼ η−(mg/mi),
634
Brownian motion of grains
but for the values of Γ under consideration it is not essential for calculations of
the averages due to the rapid convergence of the integrals over η. In particular, the
average kinetic energy K of grains is
K(Γ) =
(
mg
mi
)
Ti
Y5/2(Γ)
Y3/2(Γ)
, (12)
that gives K = 2, 86(mg/mi)
1/2Ti for Γ = 1. Thus, the ion absorption by grains
can lead to grain heating and grain average kinetic energy (or their effective tem-
perature) can be much higher than the electron and ion temperatures. This effect
could be treated as the one giving qualitative explanation of the experimental da-
ta [13, 14], as it was already suggested in [8,9] based on the velocity independent
approximation for β and D. As it was shown in [7,8] for some plasma parameters
the effective temperature became negative, which has no physical sense. The same
statement was repeated in the recent paper [15], dealing with the hydrodynamic
consideration of anomalous heating of grains in dusty plasmas. As is clear from the
present paper and was shown already earlier (see e.g. [16]), the effective tempera-
ture, as well as the average kinetic energy of grain, calculated consequently based on
the theory developed in [8,9], with velocity dependent friction coefficient, is always
positive, as it follows from equations (10)–(12). At the same time the statement on
the possibility of anomalous temperature (or more exactly average kinetic energy)
of grains as the result of momentum transfer due to ion absorption is valid in the
model under consideration. The final conclusion about the anomalous heating due
to the ion absorption can be done based on the theory, taking into account surface
ion recombination and regeneration of neutral atoms.
Let us consider now the case Γ � 1. In this case the ratio Q = (Ti/mi)β(η, Γ)/D(η, Γ)
can be represented for all values of η with a good accuracy by the function Q =
(mg/mi)(Γ − η)/[(1 + Γ)(η + Γ)] that gives
fg =
C
D(η, Γ)
exp
{
mg
mi
[
(3Γ + 1)
(2Γ − 1)
ln
Γ(η + 1)
Γ + η
− 1
2
ln
η2 + (Γ + 1)η + Γ
Γ
]}
, (13)
where
Di(η, Γ)|Γ�1 '
3
2
A
Ti
mg
√
πη
[
1 +
Γ
(η + 3
4
√
πη)
]
. (14)
It is clear that fg(η, Γ) given by equation (13) is non-exponential.
Finally for the domain Γ < 0, 472 we can omit the term ∼ v4 in equation (10) and
the distribution fg(q, v) becomes Maxwellian with the effective grain temperature:
Teff =
2T ∗
i
1 − Γ
=
2Ti(1 + Γ
2
)
1 − Γ
. (15)
Equation (15) is similar to equation (69) for the effective temperature, which was
found in [8]. At the same time equation (15) is valid only for Γ < 0, 472, where the
635
S.A.Trigger, A.G.Zagorodny
effective temperature is positive. The above conclusions are in a good agreement with
the results of numerical calculations of fg(η, Γ) based on the equation (9) (figure 4).
If we take into account the processes of atom-grain and ion-grain elastic scat-
tering the coefficients β(η, Γ) and D(η, Γ) should include additional terms. To es-
timate, for example, contribution of atom-grain and ion-grain scattering we can
consider β(η) calculated based on the equation (2) with the appropriate transport
cross-sections. For the case η � 1 the friction coefficient is
βi(η, Γ) = 2A
[
1 − Γ + 4
na
ni
(
Tama
Timi
)1/2
− η
5
(1 − 3Γ) + Γ2 ln Λi
]
. (16)
The negative friction can exist for small η, if the Coulomb scattering is strongly
suppressed, when the Coulomb logarithm ln Λi is small [17–19], which is typical of
strong interaction. The root of the function β(η, Γ) is shifted to the region of large,
Γ, which is determined by the atom density na. The result of rough estimate for the
case λLi > λD > a (λLi is the Landau length) gives
η(Γ) =
5
3Γ − 1
[
Γ − 1 − 3
√
ZgΓ
πnia3
− 4na
ni
(
Tama
Timi
)1/2
]
. (17)
It is necessary to point out that completely positive Coulomb logarithm for dusty
plasma (for arbitrary relations between the values λD, λLi, a) has been introduced
in [20]. For a � λD the respective logarithm was found in [18].In general, a more
detailed consideration of scattering processes and the effects of strong interactions
between the grains [19], is needed for the exact description of the region of negative
friction. Of course, the realization of the negative value of the total friction coefficient
in the physical experiment requires special conditions, since other mechanisms of
positive friction exist. At the same time at the presence of ion flow there is an
effective mechanism of negative friction due to the ion scattering by grains [21],
which can be very important for dusty plasmas even when ion absorption is small.
In conclusion, ion absorption by grains can generate negative friction and pro-
vide a substantial increase of the average grain kinetic energy in comparison with the
temperatures of the other plasma components. Microscopical justification of negative
friction on the kinetic level is presented in the model without surface ion recombi-
nation and thus deviation of the grain distribution function from the Maxwellian
distribution is found. Further development of the theory with regard to surface re-
combination is needed in order to answer the question whether the conditions for
negative friction in real dusty plasmas do exist. For dusty plasma, as for an open
system, the fluctuation-dissipation theorem in the form of Einstein relation is not
applicable. Finally, we would like to point out once more that negative friction is
a widely spread phenomenon in physics, chemistry and biology, which is usually
described phenomenologically. Therefore microscopical approach to this problem for
specific models seems to be important and useful.
636
Brownian motion of grains
Acknowledgement
The authors are thankful to W.Ebeling, U.Erdmann, G.J.F. van Heijst, L.Schimansky-
Geier, and P.P.J.M.Schram for valuable discussions and to the Netherlands Organi-
zation for Scientific Research (NWO) for the support in this work. We also appreciate
V.Kubaichuk for the help in numerical calculations.
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S.A.Trigger, A.G.Zagorodny
Броунівський рух зернин і від’ємне тертя
у запорошеній плазмі
С.Трігер 1,2 , А.Загородній 3
1 Об’єднаний інститут низьких температур,
РАН, Москва, Росія
2 Університет Гумбольдта, Берлін, ФРН
3 Інститут теоретичної фізики ім. М.М. Боголюбова,
НАН України, Київ, Україна
Отримано 2 лютого 2004 р.
В наближенні домінуючих контактних зіткнень порошинок з частин-
ками плазми виконано макроскопічні розрахунки кінетичних коефі-
цієнтів Фокера-Планка для заряджених порошинок, що рухаються у
плазмі. Показано, що завдяки зіткненням з йонами коефіцієнт тертя
може бути від’ємним, і знайдено порогове значення заряду поро-
шинки. Знайдено розв’язок стаціонарного рівняння Фокера-Планка
з кінетичними коефіцієнтами, що залежать від швидкості, і встанов-
лено суттєву відмінність отриманих розв’язків від максвелових роз-
поділів.
Ключові слова: запорошена плазма, від’ємне тертя, ефективна
температура, рівняння Фокера-Планка
PACS: 52.27.Lw, 52.20.Hv, 52.25.Fi
638
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| id | nasplib_isofts_kiev_ua-123456789-119032 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1607-324X |
| language | English |
| last_indexed | 2025-12-01T21:45:17Z |
| publishDate | 2004 |
| publisher | Інститут фізики конденсованих систем НАН України |
| record_format | dspace |
| spelling | Trigger, S.A. Zagorodny, A.G. 2017-06-03T04:23:17Z 2017-06-03T04:23:17Z 2004 Brownian motion of grains and negative friction in dusty plasmas / S.A. Trigger, A.G. Zagorodny // Condensed Matter Physics. — 2004. — Т. 7, № 3(39). — С. 629–638. — Бібліогр.: 21 назв. — англ. 1607-324X PACS: 52.27.Lw, 52.20.Hv, 52.25.Fi DOI:10.5488/CMP.7.3.629 https://nasplib.isofts.kiev.ua/handle/123456789/119032 Within the approximation of dominant charging collisions the explicit microscopic calculations of the Fokker-Planck kinetic coefficients for highlycharged grains moving in plasma are performed. It is shown that due to ion absorption by grain the friction coefficient can be negative and thus the appropriate threshold value of the grain charge is found. The stationary solutions of the Fokker-Planck equation with the velocity dependent kinetic coefficient are obtained and a considerable deviation of such solutions from the Maxwellian distribution is established. В наближенні домінуючих контактних зіткнень порошинок з частинками плазми виконано макроскопічні розрахунки кінетичних коефіцієнтів Фокера-Планка для заряджених порошинок, що рухаються у плазмі. Показано, що завдяки зіткненням з йонами коефіцієнт тертя може бути від’ємним, і знайдено порогове значення заряду порошинки. Знайдено розв’язок стаціонарного рівняння Фокера-Планка з кінетичними коефіцієнтами, що залежать від швидкості, і встановлено суттєву відмінність отриманих розв’язків від максвелових розподілів. The authors are thankful to W.Ebeling, U.Erdmann, G.J.F. van Heijst, L.SchimanskyGeier, and P.P.J.M.Schram for valuable discussions and to the Netherlands Organization for Scientific Research (NWO) for the support in this work. We also appreciate V.Kubaichuk for the help in numerical calculations. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Brownian motion of grains and negative friction in dusty plasmas Броунівський рух зернин і від’ємне тертя у запорошеній плазмі Article published earlier |
| spellingShingle | Brownian motion of grains and negative friction in dusty plasmas Trigger, S.A. Zagorodny, A.G. |
| title | Brownian motion of grains and negative friction in dusty plasmas |
| title_alt | Броунівський рух зернин і від’ємне тертя у запорошеній плазмі |
| title_full | Brownian motion of grains and negative friction in dusty plasmas |
| title_fullStr | Brownian motion of grains and negative friction in dusty plasmas |
| title_full_unstemmed | Brownian motion of grains and negative friction in dusty plasmas |
| title_short | Brownian motion of grains and negative friction in dusty plasmas |
| title_sort | brownian motion of grains and negative friction in dusty plasmas |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/119032 |
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