Brownian particle in non-equilibrium plasma
The stationary distribution function of Brownian particles in a nonequilibrium dusty plasma is calculated with regard to electron and ion absorption by grains. The distribution is shown to be considerably different from the distribution function of ordinary Brownian particles in thermal equilibriu...
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Lev, B.I. Tymchyshyn, V.B. Zagorodny, A.G. 2017-06-11T16:51:04Z 2017-06-11T16:51:04Z 2009 Brownian particle in non-equilibrium plasma / B.I. Lev, V.B. Tymchyshyn, A.G. Zagorodny // Condensed Matter Physics. — 2009. — Т. 12, № 4. — С. 593-602. — Бібліогр.: 19 назв. — англ. 1607-324X PACS: 05.70.Rr, 05.20.Dd, 05.40.-a DOI:10.5488/CMP.12.4.593 https://nasplib.isofts.kiev.ua/handle/123456789/120332 The stationary distribution function of Brownian particles in a nonequilibrium dusty plasma is calculated with regard to electron and ion absorption by grains. The distribution is shown to be considerably different from the distribution function of ordinary Brownian particles in thermal equilibrium. A criterion for the grain-structure formation in a nonequilibrium dusty plasma is derived. Знайдена стацiонарна функцiя розподiлу для Броунiвських частинок, що знаходяться в в нерiвноважнiй запорошенiй плазмi. Для цього використано нелiнiйне рiвняння Фоккера-Планка. Сила, що дiє на частинку, враховує безпосереднiй вплив плазми за рахунок змiни поля рухомої частинки при абсорбцiї зарядiв на її поверхнi та звичайне тертя порошинки в плазмi. Показано, що стацiонарна функцiя розподiлу по швидкостях для частинок не вiдповiдає рiвноважному розподiлу для звичайних Броунiвських частинок. Знайденi критерiї формування структур в системi таких частинок в нерiвноважнiй плазмi. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Brownian particle in non-equilibrium plasma Броунiвська частинка у нерiвноважнiй плазмi Article published earlier |
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Brownian particle in non-equilibrium plasma |
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Brownian particle in non-equilibrium plasma Lev, B.I. Tymchyshyn, V.B. Zagorodny, A.G. |
| title_short |
Brownian particle in non-equilibrium plasma |
| title_full |
Brownian particle in non-equilibrium plasma |
| title_fullStr |
Brownian particle in non-equilibrium plasma |
| title_full_unstemmed |
Brownian particle in non-equilibrium plasma |
| title_sort |
brownian particle in non-equilibrium plasma |
| author |
Lev, B.I. Tymchyshyn, V.B. Zagorodny, A.G. |
| author_facet |
Lev, B.I. Tymchyshyn, V.B. Zagorodny, A.G. |
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2009 |
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Condensed Matter Physics |
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Інститут фізики конденсованих систем НАН України |
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Article |
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Броунiвська частинка у нерiвноважнiй плазмi |
| description |
The stationary distribution function of Brownian particles in a nonequilibrium dusty plasma is calculated with
regard to electron and ion absorption by grains. The distribution is shown to be considerably different from the
distribution function of ordinary Brownian particles in thermal equilibrium. A criterion for the grain-structure
formation in a nonequilibrium dusty plasma is derived.
Знайдена стацiонарна функцiя розподiлу для Броунiвських частинок, що знаходяться в в нерiвноважнiй запорошенiй плазмi. Для цього використано нелiнiйне рiвняння Фоккера-Планка. Сила, що дiє на частинку, враховує безпосереднiй вплив плазми за рахунок змiни поля рухомої частинки при абсорбцiї зарядiв на її поверхнi та звичайне тертя порошинки в плазмi. Показано, що стацiонарна функцiя розподiлу по швидкостях для частинок не вiдповiдає рiвноважному розподiлу для звичайних Броунiвських частинок. Знайденi критерiї формування структур в системi таких частинок в нерiвноважнiй плазмi.
|
| issn |
1607-324X |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/120332 |
| citation_txt |
Brownian particle in non-equilibrium plasma / B.I. Lev, V.B. Tymchyshyn, A.G. Zagorodny // Condensed Matter Physics. — 2009. — Т. 12, № 4. — С. 593-602. — Бібліогр.: 19 назв. — англ. |
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| first_indexed |
2025-11-25T23:28:33Z |
| last_indexed |
2025-11-25T23:28:33Z |
| _version_ |
1850581071331065856 |
| fulltext |
Condensed Matter Physics 2009, Vol. 12, No 4, pp. 593–602
Brownian particle in non-equilibrium plasma
B.I. Lev, V.B. Tymchyshyn, A.G. Zagorodny
Bogolyubov Institute for Theoretical Physics, NAS Ukraine, Metrologichna 14-b,03680, Kyiv, Ukraine
Received June 1, 2009
The stationary distribution function of Brownian particles in a nonequilibrium dusty plasma is calculated with
regard to electron and ion absorption by grains. The distribution is shown to be considerably different from the
distribution function of ordinary Brownian particles in thermal equilibrium. A criterion for the grain-structure
formation in a nonequilibrium dusty plasma is derived.
Key words: Brownian motion, Langevine equation, Fokker-Planck equation, nonequilibrium
PACS: 05.70.Rr, 05.20.Dd, 05.40.-a
1. Introduction
The physics of dusty plasma attracts growing attention both as an academic problem and in
view of technological applications; very recently it has become a subject of interest in the field of
condense matter. A dust particle introduced in a plasma changes the properties of the latter. In as-
trophysics, dusty plasma has been found in planetary rings, in interplanetary and interstellar clouds
as well as comets. Particles immersed in a laboratory plasma usually acquire negative charges since
the diffusion flux of the electron component is higher than the ion flux. The parameters of a sys-
tem of dusty particles can satisfy the condition for the formation of a Wigner or Coulomb crystal
[1]. The dynamics of the formation and melting of such crystals was studied in [1] The particle
dynamics during the structure formation process and the process of melting of a crystal structure
should be described in terms of relevant physical conditions; it directly depends on the temperature
and concentration of the relevant plasma component. A variety of physical processes can occur in
a plasma, only sequential analysis can provide answers to many questions and thus enable us to
understand the behavior of a system of particles immersed in a plasma. A very important issue is
the accuracy of description of the dusty-plasma effective potential. The basic idea is to describe the
absorption of electrons and ions by macroparticles in terms of effective point sinks appearing in the
equation of plasma dynamics [1,3]. The approach makes it possible to find an explicit relation for
the potential and the particle density distribution near the grains. The energy of grain interaction
with the charge induced by the grain and the force acting on a moving dust particle depend on
the particle velocity. In this case negative friction of the moving grains is possible. The standard
approach to the description of the particle-velocity distribution function cannot be applied now
and there arises the problem of how to find the average velocity of dust particles and to obtain
a condition of crystal-structure formation in the grain subsystem. of macroparticles introduced in
the plasma. In this paper we propose a self-consistent approach to the description of friction and
diffusion of dust particles (grains) treated as Brownian particles[3]. The approach employs the non-
linear Fokker-Planck equation with regard to the velocity dependence of both friction and diffusion
coefficients. We find the distribution function for dust particles in a plasma and derive conditions
of a crystal structure formation in a system of dust macroparticles immersed in a weakly ionized
plasma. We formulate a basic model for the general case of macro-particle Brownian motion in non-
linear media. Then we consider the macro-particle screening in a weakly ionized plasma and find
the force that acts on each moving particle in such a system. With the force being known, we find
the dust-particle distribution function and derive the structure-formation condition for this system.
We conclude with the discussion of a possibility to observe the formation pattern experimentally.
c© B.I. Lev, V.B. Tymchyshyn, A.G. Zagorodny 593
B.I. Lev, V.B. Tymchyshyn, A.G. Zagorodny
2. Brownian motion of dusty particle in nonequilibrium plasma
It is a matter of general knowledge that any statistical system interacting with the environ-
ment reaches its equilibrium state during the relaxation time. The properties of the combined
system are determined by the physical peculiarities of the statistical system and the characteris-
tics of the environment. Equilibrium state of a macroscopic system can be determined for some
ideal conditions [4–7]. By virtue of the effect of the environment, the thermodynamic parameters
of this system become equal to those for the thermal bath. In the case of an open system the
combined parameters cannot be found. Such systems can, however, manifest equilibrium-like be-
havior. For example, in the case of a dusty plasma consisting of electrons, ions, and grains which
absorb all the encountered electrons and ions, we can define the equilibrium state as a stationary
state with equalized energy interchange between the grains and the environment. This state ari-
ses due to the balance between the direct energy transfer from the environment to the grain and
the energy losses due to the processes that accompany the grain-motion relaxation. Experimental
measurements of the grain temperature [8] show that the kinetic grain temperature can exceed
the ion temperature by about two orders. This confirms that the behavior of the system can be
indeed equilibrium-like. There are many other examples of such systems both in nature and in
laboratories. These are hot electrons in semiconductors [6], photons scattered by an inhomoge-
neous medium with frequency-dependent diffraction coefficient [9], high-energy particles generated
in the course of nuclear collisions in accelerators, ordinary Brownian particles in a medium with
particle-velocity-dependent friction coefficient, etc. For such systems, characteristics of the media
can be taken into account through the friction coefficient that depends on the phase-space point
as a random quantity. Colloid particles in suspensions and grains in dusty plasmas provide good
examples of systems with nonlinear particle-velocity dependencies of the kinetic coefficients. To
describe the Brownian motion in a system with foreign particles is a key problem of the statis-
tical physics of open systems. Stationary distributions of Brownian particles in a dusty plasma
with a specified velocity dependence of the friction and diffusion coefficients were found in [10]
for the case of the collisionless plasma background. The stationary distribution function has a
maximum for some value of the velocity that determines the mean kinetic energy of the grains;
the stationary one-particle distribution function is non-Maxwellian. The traditional description of
the non-equilibrium Brownian motion employs phenomenological expressions for the friction and
diffusion coefficients. Usually, the velocity dependencies of these coefficients are postulated, with a
high level of uncertainty being thus introduced in the application of such models to real systems.
Moreover, in the case of an open system there exist several types of Fokker-Planck equations which
can be related to the nonlinear Langevine equation [11,12]. The dynamics of an ordinary Brownian
particle is described in terms of velocity by the Langevin equation
v̇ = −γv + F (t), (1)
where γ is the coefficient of friction, and F (t) is a random force accounting for the irregular action
of the environment on a separate particle. The average over the statistical ensemble 〈F (t)〉 = 0,
but 〈F (t)F (t′)〉 = φ2δ(t − t′), which corresponds to the white-noise condition and thus accounts
for the correlations in the particle motion under the action of a random force. The random effect
of the environment can be taken into account only through the correlations between fluctuations
at various time instants 〈L(t)L(t′)〉 = φ(t − t′). The mean value of the correlations φδ(t − t′) is
nonzero only during the correlation time. Therefore, the function must have a sharp peak as the
time interval tends to zero, which corresponds to the condition that characterizes the white noise
[12,13]. In the most general case we can use the Newton dynamic equation given by
v̇ = f(v) + F (t), (2)
where f(v) ≡ F (v)
M
, M is mass of Brownian particle and F (v) is the resulting force that acts on an
individual particle including the contribution of the friction force. In the general case the resulting
force in a nonlinear medium can depend on the particle velocity. This form of the dynamic equation
for a Brownian particle can be regarded as nonlinear Langevine equation. The true nonlinear
594
Brownian particle in non-equilibrium plasma
Langevine equation should be equivalent to the equation for the probability distribution function
describing the physical process under consideration. Currently, there exist two different approaches
to the solution of this problem. If one considers a Brownian particle for which the coefficient depends
on the velocity at the starting point, then the equation for the non-equilibrium distribution function
can be obtained in the Ito form. On the other hand, if this coefficient depends on the velocity before
and after the transition, then the diffusion equation can be written in the Stratonovich form:
∂ρ
∂t
= −
∂
∂v
f(v)ρ +
σ2
2
∂
∂v
g(v)
∂
∂v
g(v)ρ. (3)
In what follows only the Stratonovich presentation is used because both presentations are
related [12,13]. All the solutions of these equations for the case of long evolution time possess a
fundamental property, namely, they reduce to the stationary solution which can be regarded as an
“equilibrium” solution for this system. Such stationary solution complies with the H-theorem in
its general meaning [12]. The processes under consideration require physical interpretation. The
equation for the non-equilibrium distribution function can thus be rewritten in a more usual form
of the local probability conservation law, i. e.,
∂ρ(v, t)
∂t
=
∂J(ρ(v, t))
∂v
. (4)
In turn, the probability flow can be written as:
J = −
(
f(v) −
σ2
2
g(v)
∂
∂v
g(v)
)
ρ +
σ2
2
g2(v)
∂
∂v
ρ. (5)
The stationary solution of the Fokker-Planck equation for J(ρ(v, t)) = 0 is given by
ρs(v) = A exp
{
∫
2f(v′)dv′
σ2g2(v′)
− ln g(v)
}
. (6)
The equilibrium distribution function as a stationary solution in the general non-equilibrium
case is given by
ρs(v) = A exp {−U(v)} , (7)
where
U(v) = ln g(v) −
∫
2f(v′)
g2(v)
dv′. (8)
This distribution function has an extreme value, which can be found as a solution of the equation
U ′(ṽ) =
1
D(v)
(D′(v) − f(v)) , (9)
where symbol ′ denotes the energy derivative. This equation is equivalent to the one
D′(ṽ) = f(ṽ), (10)
which gives a relation between the dissipation and diffusion in the stationary case and completely
determines the new “equilibrium” state of the system. The stationary non-equilibrium distribution
function is given by
ρs(v) = exp {−U(ṽ)} exp
(
−U ′′(ṽ)v2
)
, (11)
where
U ′′(ṽ) = −
1
D(ṽ)
(D′′(ṽ) + f ′(ṽ)) . (12)
This “equilibrium” distribution function is Gaussian in all the cases. When the system dissipa-
tion f(v) is described by the nonlinear function of state and the diffusion coefficient depends on
595
B.I. Lev, V.B. Tymchyshyn, A.G. Zagorodny
the velocity, many interesting cases, including the noise-induced transition in a more stable new
“equilibrium” state, can be considered. The velocity dependence of the diffusion coefficient can be
also derived from the linear Langevin equation in terms of the theory of Markovian processes with
regard to the non-equilibrium fluctuations of various coefficients which determine the function.
This approach can also be used to consider a very simple picture of the Brownian motion in het-
erogeneous media. For this case the characteristics of the medium can be taken into consideration
by means of a varying friction coefficient which depends on the space point as a random quantity.
This means that we have to put in the Langevine equation, i. e., the friction coefficient consists
of the constant part which determines the average friction coefficient, and the random part which
describes the effect of random variations of the friction properties of the matter. In the case of
white-noise spectrum of density fluctuations or other parameters which characterize the matter,
the Fokker-Planck equation can be written as [13]:
∂ρ(v, t)
∂t
=
∂
∂v
(γvρ(v, t)) +
σ2
2
∂2
∂v2
v2ρ(v, t). (13)
The stationary solution of this equation is given by [13]:
ρs(ε, t) = Nv−( 2γ
σ2
+1), (14)
which can be verified by direct substitution into equation (13). This stationary solution considerably
differs from that for the case of velocity-independent kinetic coefficients. Let us assume that there
exists a mechanism responsible for the velocity limitation. For example, for the dissipation function
f(ε) = γε − ε2, such limitation follows from the second part. The absorption parameter can be
presented as γt = γ + ξt where the second part describes the random change of the environmental
effect. In this case the Fokker-Planck equation has the form [13]:
∂ρ(v, t)
∂t
=
∂
∂v
(
(γv2)ρ(v, t)
)
+
σ2
2
∂2
∂v2
v2ρ(v, t). (15)
The stationary solution of this equation is given by
ρs(v, t) = Nv−( 2γ
σ2
+1) exp
{
−
2
σ2
v
}
. (16)
The latter statement can be verified by a direct substitution into equation (13). This stationary
solution considerably differs from that for the case of velocity-independent kinetic coefficients. For
Rayleigh’s phenomenological model of active friction γt = γ − αv2 ≡ α(v2 − v2
0), where v2
0 = g
α
with the constant diffusion coefficient D(v) ≈ D0, the stationary velocity distribution function is
given by
ρs(v, t) = N exp
{
−
α
4D0
(v2 − v2
0)
2
}
, (17)
where N is a normalization constant. In the weak noise limiting case α
4D0
→ 0, this distribution
is transformed into δ -the function and thus the dispersion of the kinetic energy is completely
neglected. The Brownian particle then moves with constant kinetic energy
mgv2
0
2 while its velocity
is randomly changed with time. For this simple model, we can find the condition for the formation
of a spatial periodic structure with regard to the potential interaction energy. In the next section
we consider a more realistic model of grain behavior in a dusty plasma.
3. Macroparticle screening in weakly ionized plasma
The purpose of this section is to find the force acting on an individual particle in a weakly
ionized plasma. The appropriate calculations for the case of collisionless plasma were obtained in
[14,15] on the basis of the microscopic treatment. In the case of the grain moving in weakly-ionized
596
Brownian particle in non-equilibrium plasma
plasma the basic set of equations for individual component concentration variations can be written
in the form [1,2]
∂nα(r, t)
∂t
− divΓα = I0 − βner, t)nir, t) − Sαδ(r − vt) (18)
and the Poisson equation for the potential is given by
∆Φ(r, t) = −4π
∑
α
eαnα(r, t) − 4πqδ(r − vt), (19)
where v is velocity of grain, q and S can be treated as constant and determined concentrated
charge and intensity of the point sink. As far as its form is rather involved, we shall not reproduce
it here. Instead we give an expression for the force that should be considered in order to describe
the dynamics of dust particles in the plasma. The force acting on the grain due to the existence of
an induced potential is described by the relation
Φ(r, t) = (q + S)
exp(−kD)r
r
−
S
r
, (20)
where kD is inverse screening length. For the moving grains, electrostatic potential has been ob-
tained in article [20] and it has got a very cumbersome form, which we are not going to cite here,
but we do cite the force which must be given consideration to describe the dynamic dusty particles
in plasma. In fact, the force acting on the grain due to the existence of the induced potential is
described by the relation F (v, α) = −q limr→vt
∂Φ(r,t)
∂r
. The force produced by the particle-cloud
interaction is given by [20]:
F (ṽ, α) =
iq2k2
D
π
∞
∫
0
dx
ṽ2(µ1 − µ2)d
{
2iṽ
t + d
t + 1
(µ1µ2) + µ1
[
iµ1ṽ
t + d
t + 1
− x(1 − α)
]
ln
µ1 − 1
µ1 + 1
−µ2
[
iµ2ṽ
t + d
t + 1
− x(1 − α)
]
ln
µ2 − 1
µ2 + 1
}
. (21)
We introduce a coefficient
µ1,2 =
i
[
x2(1 + d) + t+d
t+1
]
−2xṽd
±
√
4x2d(1 + x2) −
[
x2(1 + d) + t+d
t+1
]2
−2xṽd
, (22)
where ṽ = v
DikD
, d = Di
De
, t = Ti
Te
, α = − S̃
q
, here Di and De are ion and electron diffusion coefficient,
Ti and Te - temperature of ion and electron plasma components respectively. q – charge of particle
that can be calculated as:
q = −
Ter0
e
ln
(
µe
µi
)
, (23)
µi and µe are ion and electron mobilities respectively, e is electron charge. The model of a point
sink makes it possible to obtain an analytic solution of the problem of grain screening and to find
explicitly the asymptotical behavior of the potential and the force which acts on several grains in
a weakly ionized plasma. After this we can explain the dynamics of the grain system and estimate
the probable conditions for crystal structure formation in such systems. To this end, we can obtain
the distribution function for the grains regarded as Brownian particles in non-equilibrium media.
4. Stationary distribution function of grains in nonequilibrium plasma
As follows from the previous section, the description of an aggregate dust particle can use the
general form of the kinetic equation for particle distributions in various media. First of all we use
the standard Fokker-Planck equation with velocity-independent diffusion coefficient. This equation
can be written in the form:
∂ρ
∂t
= −
∂
∂v
(γρ) +
∂2
∂v2
Dρ. (24)
597
B.I. Lev, V.B. Tymchyshyn, A.G. Zagorodny
The solution of the latter is
ρ(v, α) =
C
D(v, α)
exp
{
−
∫
f(v, α)v
D(v, α)
dv
}
(25)
its value is maximum for the velocity that depends on the coupling parameter. This velocity can be
treated as the equilibrium velocity of a Brownian grain in the plasma. Obviously, this velocity dif-
fers from the conventional thermal velocity. Its value is determined by the energy balance between
the energy obtained by the grain due to ion absorption from the plasma, and the energy dissipation
associated with the friction. Notice that large velocity cannot be attained since the distribution
function has a maximum at some velocity. In order to get some analytical estimates let us consider
[16,17] of the order of one. In such a case the stationary solution for the distribution function
possesses a maximum for non-zero velocity. Figure 1 shows the result of computer calculations of
a stationary distribution function for the case of constant diffusion coefficients Di = 0.0542 cm2
sec ,
De = 2580 cm2
sec , D = 0.016 cm2
sec and η = 2, 27 × 10−4 cm
cm sec As follows from the picture, the stati-
onary distribution function has a sharp peak for nonzero grain velocity that depends on plasma
temperature. Another important result is the asymmetric smearing of the distribution function
that depends on possible fluctuations of the coupling parameter. The dependence of the probabi-
lity distribution function on the grain velocity and temperature is given in figure 2. Here and in
what follows for the sake of simplicity we consider the case of isothermal plasma T = Ti = Te.
Figure 1. Probability distribution function for constant diffusion coefficients for different tem-
peratures (T = Te = Ti) when α is stable. The plot in the upper right corner shows the difference
between the probability distribution function for stable α and for the case when α fluctuates
with dispersed σ density.
The dependence of the distribution function on the grain velocity and radius for fixed temperature
is shown in figure 3. The dependence of the velocity corresponding to the local maximum of the
distribution function on the grain radius and temperature is given in figures 4 and 5.
Figures 6 and 7 present the dependence of the grain distribution function on the grain velocity
and plasma temperature. The numerical calculations show that the qualitative changes of the
grain distribution function can occur with the variation of both plasma and grain subsystem
parameters If the distribution function has a local maximum for zero velocity, then the condition
of Wigner crystal structure formation is completely determined by concentration of grains. In this
case there exists only repulsive electrostatic interaction and mean concentration can determine
598
Brownian particle in non-equilibrium plasma
Figure 2. The dependence of the probability
distribution function on the grain velocity and
temperature (T = Te = Ti). Grain radius is fi-
xed (40 mkm).
Figure 3. The dependence of the probabi-
lity distribution function on the grain velocity
and radius. Temperature (T = Te = Ti) is fi-
xed(10 eV).
Figure 4. Grain-radius dependence of the
grain velocity in the local maximum of the
probability distribution function. The temper-
ature is fixed (T = Te = Ti).
Figure 5. Temperature dependence of the
grain velocity of in the local maximum of the
probability distribution function.
spatially homogeneous distribution of grains. If the mean kinetic energy is nonzero, as in our case,
and is equal to the potential interaction energy, one may expect a spatially ordered grain structure
to be formed in the dusty plasma (Wigner-crystal-type structure). An essential element in any
self-organization of condensation in a many-particle system is the condition that the potential due
to the nearest- neighbor force must be greater than the thermal energy. In our case the criterion
is satisfied if the average value of the velocity is zero. In more realistic cases, however, the average
velocity is determined by the maximum distribution function which depends on the grain radii and
plasma temperature;. In the present case we can find the critical parameter for the formation of
such structures in a weakly ionized dusty plasma. We find the kinetic energy of the dusty plasma
and compare it to the interaction energy for an average distance. Thus we obtain a relation given
by
(
mg
mi
)
1
2
'
ZgR
ZiλD
. (26)
The latter may be used to find the parameter of the periodic distribution of grains in the plasma.
This condition determines the parameters of the plasma and grains for which a periodic grain
599
B.I. Lev, V.B. Tymchyshyn, A.G. Zagorodny
structure can be formed in the plasma. For a real parameter of grains R ∼ 10µm − 200µm, the
Debye lengths λD ∼ 1000µm, and the relation between the masses being given by M ∼ 104mi, the
periodic structure can be realized for Zg ∼ 2 · 104Zi which is a realistic parameter for the grains
in a dusty plasma.
Figure 6. Grain-velocity dependence of the probability distribution function for different plasma
temperatures (T = Te = Ti). The radius is fixed (40 mkm). As temperature increases, the prob-
ability distribution function acquires two local maxima. For some values of the parameters, one
may expect a bifurcation to appear.
Figure 7. Grain-velocity dependence of the probability distribution for different grain radii. The
temperature (T = Te = Ti) is fixed(10 eV). As the radius decreases, the probability distribution
function can acquire two local maxima. For some values of the parameters, one may expect a
bifurcation to appear.
It should be noted that the latter estimates can also be used in the case of a fully-ionized
plasma since the kinetic coefficients were derived disregarding electron and ion collisions with
neutrals. Obviously, such collisions are extremely important in the case of a weakly ionized plasma.
Nevertheless, the idea to calculate the average kinetic energy from the Fokker-Planck equation with
velocity-dependent kinetic coefficients and then to compare this energy to the interaction energy
still remains valid in the case of weakly ionized plasmas. In this case the main contribution to the
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Brownian particle in non-equilibrium plasma
diffusion is associated with the grain interaction with neutrals and thus the diffusion coefficient
can be approximated by the appropriate expression for neutral grains in a neutral gas. As for
the friction coefficient, it considerably depends on the velocity [16–19]. Moreover, under certain
conditions [16–19] it can take negative values, as in the case of a fully-ionized plasma. Thus, the
conditions for the formation of a dusty crystal in a weakly ionized plasma can be also estimated.
5. Conclusion
To conclude we notice that a new approach to the problem of grain structure formation in
dusty plasmas is proposed. The stationary non-equilibrium solution of the Fokker-Planck equation
for the Brownian motion of grains in a plasma is obtained. This solution can be interpreted as
an “equilibrium” distribution function for a new steady state in the open system. This state is
characterized by the “equilibrium” velocity of foreign particles. The approach takes into account
probable transitions between different states of the system induced by the energy dissipation and
the interaction with the environment dependent on the velocity. For a dusty plasma, as for an
open system, the fluctuation-dissipation theorem in the Einstein-relation form is inapplicable.
Therefore, microscopic approaches to this problem in terms of specific models are important and
useful. Nonlinear models describing stationary states of the system are proposed and conditions of
periodic structure formation in the grain subsystem are obtained. The presented numerical results
make it possible to find the condition for the formation of a spatial periodic structure in the grain
system. This research is partially supported by the joint NASU–RFFR grant.
References
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3. Lev B.I., Zagorodny A.G., Phys. Lett. A, 2009, 373, 158.
4. Balesku R. Equilibrium and Nonequilibrium Statistical Mechanics. J. Wiley and Sons, New York, 1978.
5. Landau L.D. Statistical Mechanics. Nauka, Moskow, 1973.
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9. Gardiner C.W., Zoller P. Quantum Noise. Springer, 2000; Wells D.F., Milburn G.J. Quantum Optics.
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11. Klimontovich Yu.L. Theory of Open System. Kluwer A. Publication, Dordrecht, 1995.
12. van Kampen N.G. Stochastic Process in Physics and Chemistry. North-Holland, Amsterdam, 1990.
13. Horsthemke W., Lefever R. Noise-induced transition (Theory, applications in physics, chemistry and
biology). Springer-Verlag, New-York, 1984.
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15. Schram P.P.J.M., Sitenko A.G., Trigger S.A., Zagorodny A.G., Phys. Rev. E, 2000, 63, 016403.
16. Zagorodny A.G., Filippov A.V., Pal’ A.F., Starostin A.N., Momot A.I. J. Phys. Stud., 2007, 11, 158.
17. Khrapak S.V., Zhdanov S.A., Ivlev A.V., Morfill G.E., J. Appl. Phys., 2007, 101, 033307.
18. Vladimirov S.V., Khrapak S.A., Chaudhuri M., Morfill G.E., Phys. Rev. Lett., 2008, 100, 055002.
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601
B.I. Lev, V.B. Tymchyshyn, A.G. Zagorodny
Броунiвська частинка у нерiвноважнiй плазмi
Б.I. Лев, В.Б. Тимчишин, А.Г. Загороднiй
Iнститут теоретичної фiзики iм. Боголюбова НАН України, Метрологiчна 14-б,03680, Київ, Україна
Отримано 1 червня 2009 р.
Знайдена стацiонарна функцiя розподiлу для Броунiвських частинок, що знаходяться в в нерiвно-
важнiй запорошенiй плазмi. Для цього використано нелiнiйне рiвняння Фоккера-Планка. Сила, що
дiє на частинку, враховує безпосереднiй вплив плазми за рахунок змiни поля рухомої частинки при
абсорбцiї зарядiв на її поверхнi та звичайне тертя порошинки в плазмi. Показано, що стацiонарна
функцiя розподiлу по швидкостях для частинок не вiдповiдає рiвноважному розподiлу для звичайних
Броунiвських частинок. Знайденi критерiї формування структур в системi таких частинок в нерiвно-
важнiй плазмi.
Ключовi слова: Броунiвський рух, рiвняння Ланжевена, рiвняння Фоккера-Планка
PACS: 05.70.Rr, 05.20.Dd, 05.40.-a
602
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