Wavelength regimes, where the deviations from the multi-

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Abstract

The self-consistent kinetic description of dusty plasmas, which takes into account the absorption of the plasma species on the dust particles leading to fluctuations of the collected fluxes and dust charge, has been proposed more than a decade ago and developed in later works. The predictions of these models have hardly been tested experimentally. The importance of such experimental tests is twofold. First of all, they provide a crucial benchmark of fundamental theoretical concepts of plasmas. Furthermore, they can be developed into new diagnostics of dust particles as the pre- dicted effects depend on dust size, density, and charge. In this work we evaluate low-frequency responses of the kinetic model in order to investigate regimes where observa- tions of kinetic effects are feasible. We study the dispersion relation and the damping rate of the dust acoustic waves under the weak growth rate approximation. Analytical for- mulas are derived for the imaginary parts of the low frequency responses and a methodology for the calculation of the real and imaginary parts of the frequency is outlined.

Wavelength regimes, where the deviations from the multi-

component approach with ﬁxed particle charge are most

pronounced are discussed and optimal sets of dusty plasma

parameters for realistic experimental set-ups are proposed.

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than we are, wiser even than their discoverers; that we get more out of them than was originally put into them”

Heinrich Hertz

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To my scientiﬁc ”too young to be” mama, mS Svetlana Ratynskaia, for initiating me

and guiding me in the exciting ﬁeld of Complex Plasmas and for fruitful conversations

on Kinetic Theory, Discharges and Viscoelastic Materials (only theoretical so far) .

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Kinetic Theory of Plasmas

1.1 Concepts of statistical physics

A macroscopic system is a system formed of a very large number of particles, such as molecules or atoms, and therefore has a very large number of degrees of freedom. In classical mechanics a complete specification of the generalized coordinates q and the generalized momenta p of a macroscopic system at any time will allow calculation of all physical quantities at all other times through Hamilton’s equations. Similarly, in quantum mechanics a complete specification of the quantum state ψ of a system through the quantum numbers at any time will allow calculations for all observable quantities at all other times through Schrodinger’s equation. Let a mechanical macroscopic system have n degrees of freedom, this implies that a full definition of its state will require n variables for the coordinates and n variables for the momenta.

For the exact solution of the system’s behavior n sets of Hamilton’s equations should be constructed and solved simultaneously, in total we would have 2n+1 coupled first order differential equations. It is obvious that this is an impracticable task even if all the initial conditions are known. Moreover, the initial conditions are usually not available or we are not interested into a complete specification of such a complex system. Therefore, statistical methods must be employed for the description of the system in terms of probability concepts. Central to this description are the concepts of statistical ensemble, phase space and statistical distribution function.

Instead of focusing on a unique system, we focus on an ensemble consisting of a very large number of identical realizations of our original system, all prepared subject to whatever conditions are specified [1, 2, 3]. These macroscopic conditions, for example fixed energy or temperature, make only some states accessible and it is self-evident that the number of systems in the ensemble should be equal to the number of accessible states, mathematically this is expressed through a set of restrictions in the averaging process. In general, the systems in the ensemble will be in different states and characterized by different macroscopic parameters.

It should be emphasized that the various systems that compose an ensemble are

not interacting with each other, each is carrying out its own independent behavior.

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By deﬁnition, the concept of ensemble is directly linked to the large number of experiments that can be conducted on a macroscopic system and the concept of probability space in mathematics.

Another central concept, first introduced in analytical mechanics, is the phase space. Let a system be described by n generalized coordinates q _i , i ∈ {1, n} and n generalized momenta p j , j ∈ {1, n}. Then the state of the system at constant time will be described by the pair {q i , p _j }, which is represented by a point in a 2n- dimensional phase space. It is obvious that the phase space is actually constructed out of the configuration space and the momentum space. (Use of momentum space, instead of velocity space, is more convenient being closer to the Hamiltonian for- mulation and easier to modify under relativistic considerations.) As the state of the system changes with time, the point in phase space moves along a curve called phase trajectory. The flow in phase space describes the future evolution of the system, making paths in phase space unable to cross, nevertheless they can intersect creating closed curves that correspond to oscillatory motion.

The key feature in the path between microscopic and macroscopic behavior is the statistical distribution function. We deﬁne that; the probability that the coordinates q i and momenta p i have values in the inﬁnitesimal intervals q i , p i and q i + dq i , p i + dp i , which is the probability of states in a volume element of the phase-space, is given by dw = ρ(p ₁ , p ₂ , ..., p _n , q ₁ , q ₂ , ..., q _n )dpdq with ρ a function of all coordinates-momenta representing the density of the probability distribution in phase space. This function is called statistical distribution function and is subject to the normalization condition ^∫ pdqdp = 1, expressing that the sum of probabilities all over the probability space must be unity.

For any quantity f (q, p) the mean value can be calculated by multiplying each of its possible values on the ensemble set by the corresponding probability and integrating all over the states, ⟨f⟩ = ^∫ ρ(p, q)f (p, q)dqdp. We can see that the actual time-dependence of the phase space trajectory is of no importance for the ensemble average, instead we have connected a probability to each phase space point that this microstate can be reached. In the traditional computation of the mean value, the set of Hamilton’s equations should have been solved and the formula f = e lim

T →+∞

1 T

∫ _T

0 f (q(t)p(t))dt used, with the time dependence q(t),p(t) dictated by the Hamiltonian solutions. It is imperative for the results of statistical mechanics to be accurate that ⟨f⟩ = f , this is the statement of the quasi ergodic hypothesis: ^e the time averaged mean value and the ensembled averaged mean value are identical.

Hence, ensemble averaging enables us to determine the mean value of a physical quantity without following the variation of its actual value with time.

1.2 The Liouville Equation

A. Derivation from the phase-space points conservation [4, 5]: The Liouville

equation is a cardinal equation of statistical mechanics, it depicts the evolution of the

phase-space distribution function for conservative Hamiltonian systems. Moreover,

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1.2. THE LIOUVILLE EQUATION

it supplies a complete description of the system both at equilibrium and away from it. Let us consider an arbitrary volume ω in any region of the phase space and let the surface enclosing it (∂ω) be denoted by σ and dω = d ^3N p · d ^3N q. The rate at which the number of phase-space points in ω changes with time will be given by

∂

∂t

∫

ω

ρdω, (1.1)

while the net rate of phase-space points ﬂowing out of the boundary surface will be

given by _∫

S

ρ(u. n)dσ. b (1.2)

Due to the fact that there are no sources or sinks in the phase-space (in the ensemble of systems under consideration there will not be new members admitted or old ones expelled), the number of phase space points must be conserved. Hence, the rate of change of phase-space points in a volume must be due to the ﬂow of phase-space points from the boundary surface. We combine Eqs.(1.1,1.2), use the divergence theorem for the surface integral and the fact that the volume ω is arbitrary, to ﬁnd

∂

∂t

∫

ω

ρdω +

∫

S

ρ(u. n)dσ = 0 b ⇒

∫

ω

∂ρ

∂t dω +

∫

ω ∇(ρu)dω = 0 ⇒

∫

ω

[ ∂ρ

∂t + ∇(ρu) ]

dω = 0 ⇒ ∂ρ

∂t + ∇(ρu) = 0. (1.3) In the 6N -dimensional phase space, divergence implies:

∑ 3N i=1

( ∂

∂q _i q ^b i + ∂

∂p _i p ^b i

) , while

6N-dimensional phase-space ﬂow implies:

∑ 3N i=1

( ˙ q i q b i + ˙ p i p b i ). Hence, if we substitute, we result in the relation

∂ρ

∂t +

∑ 3N i=1

( ∂(ρ ˙ q i )

∂q _i + ∂(ρ ˙ p i )

∂p _i )

= 0 (1.4)

This is the most general form of the standard Liouville equation, valid for non- Hamiltonian systems too, with only requirement the absence of sinks/sources. For a Hamiltonian system, the generalized coordinates and momenta are subject to Hamilton’s equations, ˙ p i = − ^∂H _∂q

_i

, ˙ q i = ^∂H _∂p

i

. Taking the proper partial derivatives of these equations, we end up with ^{∂ ˙} _∂p ^p

ⁱ

i

= − _∂p ^∂

²_i

_∂q ^H

_i

, ^{∂ ˙} _∂q ^q

ⁱ

i

= − _∂p ^∂

²_i

_∂q ^H

_i

, and if adding by parts ^{∂ ˙} _∂p ^p

ⁱ

i

+ ^{∂ ˙} _∂q ^q

ⁱ

i

= 0. Hence, for Eq.(1.4)

∂ρ

∂t +

∑ 3N i=1

( ∂ρ

∂q _i q ˙ i + ∂ρ

∂p _i p ˙ i

) + ρ ·

∑ 3N i=1

( ∂ ˙ q i

∂q _i + ∂ ˙ p i

∂p _i )

= 0 ⇒

∂ρ

∂t +

∑ 3N i=1

( ∂ρ

∂q i

˙ q _i + ∂ρ

∂p i

˙ p _i

)

= 0 ⇒ ∂ρ

∂t + ˙ q · ∇ q ρ + ˙ p · ∇ p ρ = 0 ⇒ dρ

dt = 0 (1.5)

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The last equation is the Liouville equation for Hamiltonian systems in three equivalent forms. The Liouville equation is a conservation theorem that expresses the fact that phase-space points of the ensemble can neither be created or destroyed.

Another way to think of it is as the 6N-dimensional analog of the equation of continuity of an incompressible ﬂuid. It also implies that the local density of phase- space points, as viewed by an observer moving with the points is constant.

B. Derivation from the Dirac-form distribution functions [6] : The previous derivation is common in statistical mechanics literature and it is based on the conservation of phase-space points with the phase-space distribution function treated as an unknown. In plasma physics literature, there is another popular derivation, it does not use any conjecture only an explicit functional form for the distribution function. Since ordinary plasmas consist of electrons and ions that can be treated as point particles with given mass and charge, knowledge of the particle orbits in conﬁguration and momentum space Q _i (t), P _i (t) means that the density of a system of N particles at a speciﬁc time will be a point in the 6N-dimensional phase space given by the product of delta functions, due to the fact that 6 coordinates are enough to fully describe a point particle in its own phase space. Hence for the distribution function we will have that

ρ =

3N ∏

i=1

δ(q _i − Q i (t))δ(p _i − P i (t)) (1.6) We take the partial derivative of the expression with respect to time and use

∂

∂t [δ(x j − X j (t))] = − ˙x i

∂

∂x j

[δ(x j − X j (t))] ,

which can be easily derived using the chain diﬀerentiation rule and some properties of the Dirac function.

∂ρ

∂t = ∂

∂t {

_3N

∏

i=1

δ(q

i

− Q

i

(t))δ(p

i

− P

i

(t)) }

⇒

∂ρ

∂t =

∑

3N j=1

∂

∂t {δ(q

j

− Q

j

(t))δ(p

j

− P

j

(t)) }

∏

3N i=1 i̸=j

δ(q

i

− Q

i

(t))δ(p

i

− P

i

(t)) ⇒

∂ρ

∂t = −

∑

3N j=1

˙ q

_j

∂

∂q

j

∏

3N i=1

δ(q

_i

− Q

i

(t))δ(p

_i

− P

i

(t)) −

∑

3N j=1

˙ p

_j

∂

∂p

j

∏

3N i=1

δ(q

_i

− Q

i

(t))δ(p

_i

− P

i

(t)) ⇒

∂ρ

∂t +

∑

3N j=1

˙ q

j

∂ρ

∂q

_j

+

∑

3N j=1

˙ p

j

∂ρ

∂p

_j

= 0 ⇒ ∂ρ

∂t +

∑

3N i=1

( ∂ρ

∂q

_i

q ˙

i

+ ∂ρ

∂p

_i

p ˙

i

)

= 0 ⇒ dρ

dt = 0 (1.7)

The ﬁnal equations are exactly the same with the Liouville equation, at least its

convective derivative form. The conservation form, which is the one described by

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1.3. THE KLIMONTOVICH EQUATION

Eq.(1.4), can also be reached by this description, with the use Newton’s second law of motion and the Lorentz force.

C. Discussion: As far as dusty plasmas are concerned, in their analysis we must keep in mind that they are non-Hamiltonian systems. More precisely, the fundamental equation in the kinetic model of the dust particles is of the form of the Eq.(1.4), which was derived from the conservation assumption without use of Hamilton’s equations. On the other hand, the fundamental equation in the kinetic model of the plasma species will not be of the form of the standard Liouville equation presented here, but of the form of the generalized Liouville equation due to the fact that dust acts as a sink for plasma particles [7, 8, 9].

1.3 The Klimontovich equation

The Liouville equation describes the behavior of systems in a 6N -dimensional phase- space, where N is the number of particles. On the other hand, the Klimontovich equation describes the behavior of individual particles in a 6-dimensional phase- space. While in the Liouville description a system is represented by a point in phase- space at each time, in the Klimontovich description a system will be presented by N points in phase space [6, 10]. Hence, for point particles of given mass and charge the exact distribution function will be a sum of delta functions instead of a product of delta functions. Since this description is used for plasma systems, we can drop the Hamiltonian notation and use {r, v}, for a species α the exact distribution function will be

f _α (r, v, t) =

∑ N i=1

δ(r − R i (t))δ(v − V i (t)). (1.8)

The behavior of the particles is governed by Newton’s second law for the Lorentz force. The positions and velocities of the particles will therefore be determined by

R ˙ i (t) = V i (t), m α V ˙ i (t) = e α E ^m [R i (t), t] + e α

c V i (t) × B ^m [R i (t), t] (1.9)

where the ﬁelds E ^m and B ^m are the microscopic ﬁelds produced self-consistently by

the point particles themselves together with any externally applied ﬁelds. Whereas,

in the equation of motion of each particle the portion of ﬁelds produced by the

particle itself should not be taken into account. The microscopic ﬁelds satisfy the set

of Maxwell equations with the sources (current and charge density) being produced

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by the particles themselves,

∇ · E ^m (r, t) = 4πp ^m (r, t)

∇ · B ^m (r, t) = 0

∇ × E ^m (r, t) = − 1 c

∂B ^m (r, t)

∂t

∇ × B ^m (r, t) = − 4π

c J ^m (r, t) + 1 c

∂E ^m (r, t)

∂t p ^m (r, t) = ^∑

α

e _α

∫

f _α (r, v, t)dv J ^m (r, t) = ^∑

α

e _α

∫

f _α (r, v, t)vdv. (1.10) It is obvious that the set of equations (1.10) can be solved to determine the exact fields in terms of particle orbits, while the set of equations (1.9) can be solved to determine the exact particle orbits in phase space in terms of the fields. Hence, if the fields and phase-space co-ordinates for all particles where known at one time, then we could compute them for all following times.

From the partial derivative of Eq.(1.8), with use of the Dirac function properties as in the Liouville equation, we will end up in

∂f _α (r, v, t)

∂t + v ∇ r f _α (r, v, t) + q _α m α

(

E ^m (r, t) + v

c × B ^m (r, t) )

· ∇ v f _α (r, v, t) = 0 (1.11) This is the convective derivative form that expresses that the density of particles of species α is constant as measured along the orbit of a hypothetical particle. Using a basic property of the Lorentz force, ∇ v (F L ) = 0 , and the fact that velocity and space are independent variables we can also obtain the continuity form

∂f

α

(r, v, t)

∂t + ∇

r

· (vf

α

(r, v, t)) + ∇

v

· { q

α

m

α

(

E

^m

(r, t) + v

c × B

^m

(r, t) )

f

α

(r, v, t) }

= 0 (1.12) Finally, we should note that from the deﬁnition of the system’s distribution function ρ and the particle’s distribution function f α , it is obvious that integration of ρ over the phase space coordinates of all particles but one will lead to f α . Similarly, integration of the Liouville equation of any form (even Generalized Liouville Equation forms), will lead to the corresponding Klimontovich equation. Both descriptions are exact and contain all of the particle orbits.

1.4 The plasma kinetic equation

For the transition from the exact particle distribution function, that contains unnec-

essary microscopic information, to the smooth distribution function, that contains

macroscopic information, we split the density and electromagnetic ﬁelds into ensem-

ble averaged and ﬂuctuating parts. The ensemble averaged part can be considered as

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1.4. THE PLASMA KINETIC EQUATION

the space-time average of a physical quantity over scales substantially greater than those associated with the ﬂuctuations, according to the quasi-ergodic hypothesis.

f _α (r, v, t) = Φ _α (r, v, t) + δf _α (r, v, t) E ^m (r, t) = ⟨E(r, t)⟩ + δE(r, t)

B ^m (r, t) = ⟨B(r, t)⟩ + δB(r, t) (1.13) Let us consider a hierarchical structure to the characteristic scales in space-time variations [11, 12]. We begin with the relaxation time τ ₀ of a hydrodynamic quantity, it can be approximated by the ratio of the system’s scale L to the sound velocity c s , τ 0 = L/c s . The characteristic time τ 1 for a single particle distribution function to relax to its local equilibrium values is approximated by the ratio of the mean free path in Coulomb collisions to the thermal velocity of the particles, which is the inverse of the collision frequency, τ 1 = l _{mf p} /u _th = 1/ν c . Finally, the characteristic time τ ₂ for the relaxation of a pair correlation function is estimated by the average time for a particle to travel over a correlation distance, hence it is approximately the ratio between the Debye length and the thermal velocity,which is the inverse of plasma frequency τ ₂ = λ _D /u _th = 1/ω _p . From the definition of plasma, for the collective effects to dominate over collisional effects we have ω p ≫ ν c , this leads to the inequalities L ≫ l mf p ≫ λ D , τ 0 ≫ τ 1 ≫ τ 2 . This space-time scale separation is useful in the mathematic treatment of the equations, enabling us to treat ensemble averaged quantities as constant in the fluctuations space-time scales.

We substitute (1.13) in the Klimontovich equation and ensemble average, using

⟨δA⟩ = 0, ⟨δA·δB⟩ ̸= 0 and interchanging the ensemble operator with space-time op- erators we reach an equation for the average component of the distribution function Φ α , the plasma kinetic equation [12],

[ ∂

∂t + v · ∇ r + q α

m _α ( ⟨E⟩ + v

c × ⟨B⟩) · ∇ v

]

Φ α = − q α

m _α ⟨(δE + v

c × δB) · ∇ v δf α ⟩ (1.14) where the left side is dependent on smooth ensemble averaged quantities representing collective eﬀects, while the right side is dependent on the ensemble average of the product of very spikey quantities representing discrete-nature collisional eﬀects.

With a dimensional analysis it can be shown that the ratio will be equal to ^ω _ν

^p

c

≃ Λ, where Λ ≫ 1 is the number of particles in a Debye sphere [6].

We subtract the plasma kinetic equation from the decomposed Klimontovich equation, neglecting non-linear second order ﬂuctuation terms and their ensemble averages we obtain an equation for the ﬂuctuation part,

{ ∂

∂t + v · ∇ r + q _α

m _α ( ⟨E⟩ + v

c × ⟨B⟩) · ∇ v

}

δf _α = − q _α

m _α (δE + v

c × δB) · ∇ v Φ _α ,

(1.15)

a similar decomposition of the Maxwell equations will complete the description.

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Considering unmagnetized plasmas and assuming that the ensemble averaged electric field can be neglected the above equations are significantly simplified. More- over, the set of Maxwell equations is replaced by Poisson’s equation when considering longitudinal fields, this equation will also be decomposed in normal and random components with the quasineutrality condition making the normal part zero. Hence, the set of equations will become:

{ ∂

∂t + v · ∂

∂r }

Φ ^α _p = −e α ⟨δE · ∂

∂p δf _p ^α ⟩ { ∂

∂t + v · ∂

∂r }

δf _p ^α = −e α δE · ∂

∂p Φ ^α _p (1.16)

∇ r · δE = 4π ^∑

α

e _α

∫

δf _p ^α d ³ p (2π) ³ .

Since the dependence of the plasma kinetic equations on the electric field fluctuations vanishes if we substitute from the Poisson equation, this set of equations can be viewed as a nonlinear system of two coupled equations with unknowns δf _p ^α , Φ ^α _p . The equation of δf _p ^α can be viewed as an inhomogeneous differential equation. The general solution will be the sum of the homogeneous and the particular solution.

The homogeneous solution corresponds to the spontaneous fluctuations in a gas of non-interacting particles due to random thermal motion of discrete particles, this part would exist even without electromagnetic interactions between the particles and represents the system’s discreteness (discreetness is mathematically expressed through delta functions), it is usually called free particle fluctuations and denoted by δf p ^α,(0) . The inhomogeneous solution corresponds to the induced fluctuations due to the microscopic electromagnetic interactions between charged discrete particles such fluctuations are ignored in the averaged fields, due to the average taken over large spatial/temporal scales. Overall, we have δf _p ^α = δf p ^α,(0) + δf p ^α,(ind) . Mak- ing this decomposition in the fluctuation part, we Fourier transform in space and time. In this part we shall use the fact that the temporal and spatial scales of the fluctuations are much faster and shorter than changes of the averaged distribution part in time and space. Hence, when treating the fluctuation part and considering its variations, the averaged part can be treated as a constant parameter. Thus, when we Fourier transform the fluctuation part, the normal part of the distribution function will be constant and the right side of the fluctuation equation will not be a product of functions (that would imply a convolution in Fourier space) [12, 13].

The free particle ﬂuctuations are described by the property,

⟨δf _p,k,ω ^α,(0) δf _p ^β,(0)

_′

_,k

_′

_,ω

_′

⟩ = Φ ^α p δ _α,β δ(p − p ^′ )δ(ω − k · v)δ(ω + ω ^′ )δ(k + k ^′ ), (1.17) and the induced particle ﬂuctuations are given by,

δf _p,k,ω ^α(ind) = −ı e α

ω − k · v k k · ∂Φ ^α _p

∂p δE _k,ω . (1.18)

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1.4. THE PLASMA KINETIC EQUATION

Substitution of the Poisson equation in the Eq.(1.18) will lead to a relation between the induced part and the free part of the ﬂuctuations. Finally, substitution of the Poisson equation and the relations (1.17,1.18) in the equation for the averaged component, will lead to the Lenard-Balescu integral [14, 15]:

{ ∂

∂t + v · ∂

∂r }

Φ

^α_p

= 2 ∑

β

e

²_α

e

²_β

∂

∂p

_i

∫ k

i

k

j

d

³

k

⁴

|ϵ

k,k·v

|

²

δ(k ·v−k·v

^′

) d

³

p

^′

(2π)

³

[

Φ

^β_p_′

∂Φ

^α_p

∂p

_j

− Φ

^αp

∂Φ

^β_p_′

∂p

^′_j

]

(1.19) with the collision term having the following properties:

1. for Φ p ≥ 0 at t = 0, Φ p ≥ 0 for all successive times.

2. the particle density is independent of time.

3. the mean velocity is independent of time.

4. the mean kinetic energy is independent of time.

5. any Maxwellian distribution is a stationary solution.

6. as time approaches inﬁnity, any solution approaches a Maxwellian distribu-

tion. Hence, these are the only stationary solutions.

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Bibliography

[1] L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Volume 5:

Statistical Physics, Part 1 (Pergamon Press, Oxford, 1959).

[2] F. Rief, McGraw-Hill Series in Fundamentals of Physics:Fundamentals of Sta- tistical and Thermal Physics (McGraw Hill, New York, 1965).

[3] P. Attard, Thermodynamics and Statistical Mechanics: Equilibrium by Entropy Maximisation (Academic Press, Sydney, 2002).

[4] D. Momeni and A. Afrazeh, The Liouville Equation : A rapid review, arXiv:0904.1981v4 [math-ph] (2009).

[5] R. C. Tolman, The Principles of Statistical Mechanics (Clarendon Press, Ox- ford, 1938).

[6] D. R. Nicholson, Introduction To Plasma Theory (Krieger Publishing Company, Florida, 1992).

[7] W. H. Steeb, Physica A 95, 181 (1979).

[8] J. D. Ramshaw, Europhys. Lett. 59, 319 (2002).

[9] V. N. Tsytovich and U. de Angelis, Phys. Plasmas 6, 1093 (1999).

[10] Yu. L. Klimontovich, Statistical Physics (CRC Press, 1986).

[11] R. L. Liboﬀ, Kinetic Theory: Classical, Quantum and Relativistic Descrip- tions,(Prentice Hall Advanced Reference Series) (Prentice Hall,1990).

[12] S. Ichimaru, Statistical Plasma Physics, Volume I: Basic Principles (Frontiers in Physics) (Addison-Wesley Publishing Company, Tokyo, 1992).

[13] V. N. Tsytovich, Lectures on non-linear plasma kinetics (Springer, Berlin, 1995).

[14] A. Lenard, On Bogoliubov’s Kinetic Equation for a spatially homogeneous plasma, Annals of Physics 10, 390 (1960).

[15] R. Balescu, Equilibrium and Nonequilibrium Statistical Mechanics (John Wiley

and Sons, New York, 1975).

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Chapter 2

Kinetic Theory of Dusty Plasmas

2.1 Dust Charging and growth

In multi-component plasmas, the charge of all species is fixed. On the contrary, in dusty plasmas the grain charge varies and its value is determined by local plasma parameters. A simple way to consider the charging process is: when embedding uncharged grains in a plasma, due to the thermal ion mobility being less than the thermal electron mobility, the grain becomes negatively charged. Hence, the dust grain starts repelling the electrons while the ions are attracted until the fluxes become equal. The dust grain will acquire the floating potential ϕ _s and its equilibrium charge value q _eq . The dust grains act as sinks for the plasma species and the fluctuations on the local plasma parameters will lead to charge fluctuations. In general, the charging equation will have the form

dq

dt = ^∑ I in − ^∑ I out , (2.1)

where q is the dust charge, I _in are the particle currents absorbed by the dust particles and I out are the currents emitted by the dust particles. In the quasi-stationary equilibrium we have dq/dt = 0 and the equilibrium charge is found by

∑ I _in = ^∑ I _out . (2.2)

The absorbed currents will be due to the ion/electron ﬂuxes to the grain, hence I _in = I _i − I e . The emitted currents will be due to secondary electron emission, thermionic emission, ﬁeld emission and photoelectric emission, thus I out = −I ph − I _sec − I therm − I f ld . In the regime, where the emitted currents are negligible the dust charge will be negative, otherwise, it can also obtain positive values. Overall, we have

dq

dt = I i − I e + I ph + I sec + I therm + I f ld (2.3)

Photo-electric emission is caused by extensive UV-ﬂuxes. When dust particles

are subject to intense photon radiation with the energy hf exceeding the photoe-

mission work function of the dust material, electrons will be emitted by the surface

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of the grain. An expression for the photoemission current is [1, 2]

I _ph =

{ 4πa ² eΓ _ν µ, ϕ _s ≤ 0

4πa ² eΓ _ν µ exp ( − ^eϕ _T

_ph^s

), ϕ _s > 0 (2.4) where Γ ν is the UV ﬂux, µ is the photoemission eﬃciency of the material (µ ≃ 1 for metals,µ ≃ 0.1 for dielectrics), while for the UV-radiation isotropic source is assumed and that the photoelectrons have a Maxwellian energy spectrum with a temperature T _ph . The dependence on the surface potential is straightforward, since a positively charged grain can recapture a fraction of its own photoelectrons.

Thermionic emission appears when the grain acquires suﬃciently high tem- peratures to operate as a thermal source. The thermal energy given to the grain overcomes the forces restraining it and electrons are emitted from its surface. The current will be dependent on the work function of the grain material and for equilibrium plasma characterized by a temperature T we have the expression [1],

I _therm =

{ _{(4πaT )}

2

em

e

h

³

exp ( − ^W _T ), ϕ s ≤ 0

(4πaT )

²

em

e

h

³

exp ( − ^W _T )(1 + ^eϕ _T

^s

) exp ( − ^eϕ _T

^s

), ϕ s > 0 (2.5) Secondary electron emission is due to particles (mainly electrons) of high energy impinging on the grain. The current is connected to that of the primary electrons through I sec = δI e , where δ is the secondary emission yield, depending on both the impact energy and grain material. Relations of δ(E) can be found in Ref. [2].

Finally, ﬁeld emission is induced due to strong electrostatic ﬁelds in the vicinity of small dust grains. Then electrons are emitted from the dust surface due to quantum tunneling.

For completeness we should also mention the relations for the particle currents for thermal distributions and use of the Orbit Motion Limited approach, which will be described in details in the next section. Using the capture cross-sections derived from OML it is quite straightforward to show that [1],

I e = √

8πa ² en e u T e exp ( eϕ s

T _e ) I i = √

8πa ² en i u _{T i} (1 − eϕ s

T _i ) (2.6)

We also notice that dust particles grow in size. The process of particle growth

depends on the form of the material and the plasma surroundings conditions, but it

is, in general, thought to proceed via these steps [3]: nucleation, that is a series of

chemical reactions in gas phase during which gas monomers form macromolecules,

coagulation, that is the process in which two particles collide to form a larger parti-

cle, during this phase negative charge is accumulated and strong Coulomb repulsion

impedes further growth of this form, surface growth, during which plasma particles

stick to the existing dust grain due to absorption.

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2.2. THE ORBIT MOTION LIMITED APPROACH

2.2 The Orbit Motion Limited Approach

The two basic assumptions of the Orbit Motion Limited (OML) approach are [4, 5]:

the dust grain is isolated in the sense that other dust grains do not aﬀect the motion of electrons and ions in its vicinity, electrons and ions do not experience collisions during the approach to the grain. These assumptions are satisﬁed when a ≪ λ Dd < l _{mf p} .

The heavy grain is considered to be an inﬁnitely massive scattering center for the electrons and ions colliding with it. Due to the conservation of angular momentum of the plasma particle we have m α ub = m α r ² ϕ, where u is the velocity of the ˙ particle when it enters the mean free path sphere and b is the impact parameter.

Conservation of the total energy of the particle also imposes the requirement that 1

2 m _α u ² + e _α ϕ _∞ = 1

2 m _α ˙r ² + 1

2 m _α r ² ϕ ˙ ² + e _α ϕ _s , (2.7) where ϕ _∞ ≃ 0 is the potential on the mean free path surface, ϕ s is the potential at the surface of the grain. At the closest approach r = r min = a and ˙r = 0, eliminating the angular velocity and solving for the impact parameter will result in b ² = b ² _max = (1 − ^2e _m

^α_α

^ϕ _u

2^s

)a ² . The absorption cross-sections will be πb ² _max .

In the case that the currents emitted from the grain are negligible, the grain will be negatively charged and we will have ϕ _s < 0. Hence, there will be an attractive potential for the ions, which implies that an ion with any velocity can be absorbed by the grain. On the other hand, there will be a repulsive potential for the electrons, which implies that electrons should have velocity larger than a minimum value in order to reach the grain. This velocity can be found by equating the initial kinetic energy of the electron with the potential energy of the electron on the grain, in that case the electron barely reaches the grain surface with zero kinetic energy, the result will be u _min =

√ 2e |ϕ

s

|

m

e

. Combining our results we end up with, σ e (u) =

 

 πa ²

(

1 − ^2e _m ^|ϕ

_e

_u

^s2

^|

) , u >

√ 2e |ϕ

s

| m

e

0, u <

√ 2e |ϕ

s

| m

e

(2.8)

σ i (u) = πa ² (

1 + 2e |ϕ s | m _e u ²

)

(2.9) The charge of the particle is related to the surface potential through the relation q = Cϕ s , where C is the capacitance of the dust particle in the plasma. For spherical dust grains with a ≪ λ D we have C = 4πϵ ₀ a, and we actually have the potential of a point particle ϕ s = _4πϵ ^q

0

a or in cgs units ϕ s = q/a.

2.3 Basic Dusty plasma parameters

In this section, we introduce the dusty plasma parameter notations and some con-

ventions used in the forthcoming presentation of the kinetic model.

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The notations we will follow for the charges are e _e = −e, e i = +e, while for the equilibrium dust charge we will have q _eq = −eZ d , where Z _d is the characteristic charge number (with values near 1000 for sub-micron dust grains).The dust radius will be denoted by a and its values are within the range (100nm,10µm), approximately. We also introduce average kinetic particle energies and denote them by T e , T i , T d for electrons,ions and dust particles respectively, these quantities are useful in order of magnitude estimations regarding the range of validity of our main assumptions. Common values are T _e ≃ (1 − 3)eV, T i ≃ 0.03eV,T d ≃ T i .

The main parameters describing the system are the following dimensionless quantities:

z = Z _d e ² aT e

, P = n _d Z _d n e

, τ = T _i T e

, τ _d = T _d (1 + P )

T i Z d P (2.10) The dimensionless charge parameter, z, has values near unity, this implies that the equilibrium dust charge is proportional to the dust size. The ion to electron temperature ratio, τ , has the values τ ≃ 1 in Q-machines and tokamaks and τ ≃ 0.01 in most discharge plasmas. The dimensionless density parameter, P , not to be confused with the Havnes parameter deﬁned by a normalization on ion density P _hav = n _d Z _d /n i , gives the quasineutrality condition n i = n e (1 + P ).

We also introduce the thermal velocities for each species u _{T α} =

√ T

α

m

α

, the Debye lengths λ ² _Dα = _4πe ^T

2^α

α

n

α

and the plasma frequencies ω _pα ² = ^4πn _m

^α

^e

²^α

α

. The total Debye length will be given by λ ⁻² _D = ^∑

α

λ ⁻² _Dα . We should also note that only in this case the subscript α implies all the plasma components, from now on its use will imply only the plasma species α = {i, e}. All ﬂuctuations will be denoted by δ in front of the physical quantity, the normal components of the distributions will be denoted by Φ and the dust velocities/momenta by v ^′ , p ^′ respectively. Finally, we work on c.g.s units.

2.4 Basic assumptions of the kinetic model

In order to formulate a kinetic model of dusty plasmas, some basic postulates are required, not only to make a statistical description possible but also to make it mathematically tangible. Order of magnitude estimations can be made for every assumption, in order to access regimes where the model is valid.

1. All plasma components are in the gaseous state, where the dust kinetic energy substantially exceeds the interaction energy.

This assumption is connected with the dimensionless coupling parameter Γ

[6, 7, 8] that is simply the ratio of the average Coulomb energy between

neighboring particles (unscreened) to the average kinetic energy of a particle,

Γ = ^Q _T

²

n ^1/3 . In the case that Γ > 1 the motions are strongly controlled by the

electric ﬁelds in the near vicinity of the individual grains, the electrons and ions

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2.4. BASIC ASSUMPTIONS OF THE KINETIC MODEL

become trapped in the potential wells of the dust grains and the dust grains organize their average position in order to minimize the average electrostatic energy, forming crystallized structures. Hence, for the components to be in the gaseous state, Γ ≪ 1, for electrons and ions this is easily satisﬁed. On the other hand, the dust grains have charge Q = −eZ d with Z _d larger than 1000 in most cases, Γ d = ^Z

2 d

e

²

T

d

n ^1/3 _d . The relation is quadratic to the charge, thus even though the dust temperature and density are small related to the plasma species values, Γ _d ≪ 1 will impose a limitation on the dust density, with the introduction of the dust Debye length this can be conveniently rewritten as n _d λ ³ _Dd ≫ 1.

2. The dust density parameter P must be large enough for the electron/ion binary collisions to be neglected compared to the dust-electron/ion collisions.

The Coulomb collisions in multi-component plasmas can be of three types:

ion-ion collisions, ion-electron collisions and electron-electron collisions, each with its own mean time between collisions and mean collision frequency. The largest collision frequency is the ion-ion collision frequency (ν i,i > ν i,e , ν e,e ) with the mean time between collisions given by the relation [9]

τ _i,i ∼ = 3 ^−3/2 4π √

m _i (T _i ) ^3/2

n i e ⁴ ln Λ ⇒ τ i,i ∝

√ m _i (T _i ) ^3/2

n i e ⁴ ⇒ ν i,i ∝ n _i e ⁴

√ m _i (T _i ) ^3/2 , where ln Λ is the Coulomb logarithm with a typical value near 10 in plasmas, which makes the neglected factor of the order of unity.

Another fundamental time-scale in dusty plasmas is characterized by the charging frequency ν _ch , which is the inverse of time for the dust grains to reach the equilibrium charge, using the OML cross-sections and thermal distributions we can obtain an order of magnitude

ν ch = aω _pi λ _Di √

2π (1 + τ + z) ⇒ ν ch ∝ aω _pi

λ Di ⇒ ν ch ∝ an _i e ²

√ m _i T _i ,

where we used the fact that z is of the order of unity and the deﬁnitions of the Debye length and plasma frequency.

Moreover, only in dusty plasmas, there will be inelastic charging collisions of the plasma species with dust grains apart from the usual elastic Coulomb collisions. From the OML model we can estimate that ν _d,i ^inel ∝ πa ² n _d _τ ^z

√ T

i

m

i

and ν _d,e ^inel ∝ πa ² n _d

√ T

e

m

e

, it is obvious that ν _d,i înel ≫ ν _d,e înel . Meanwhile, for the Coulomb collisions of ions with dust we acquire ν _d,i ^Coul ≃ _τ ¹ ν _d,i înel .

With use of the dimensionless charge and the dust density parameter it is

straightforward to show that ν _ch ∝ τν _d,i ^Coul /P and ν _ch ∝ Z d τ ν i,i , thus by

combining we end up with ν _d,i ^Coul ∝ P Z d ν ii . Hence, for P Z d ≫ 1 the smallest

frequency related to dust-plasma Coulomb interactions is much larger than the

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largest frequency related to plasma-plasma Coulomb interactions, this means that a necessary condition for the electron/ion binary collisions to be negligible is P ≫ _Z ¹

_d

. This condition is easily fulﬁlled in dusty plasma experiments.

3. The size of the dust grains is small compared to the Debye radius, a ≪ λ Dd . 4. The dust grains are characterized by positions, momenta and dust charge. The dust charge is an independent variable and is not a function of the position of the grain.

5. The charge on the grain is suﬃciently large (Z d ≫ 1) and therefore the dust charge ﬂuctuations are small.

Mathematically, this is expressed through ⟨(δq) ² ⟩ ≪ (⟨q⟩) ² . This assumption is used to simplify the expression for the natural dust ﬂuctuations and for the computation of integrals of the distribution function with respect to the charges, that lead to the key expressions for the total dielectric function and the eﬀective charge.

6. Only the discreetness of the dust component is taken into account while the electrons and ions are treated as continuous ﬂuids in the phase space.

We recall from the kinetic theory of plasmas [10, 11, 12, 13], that fluctuations are separated into field induced fluctuations (that depend on the fluctuations of the electric or magnetic fields δE, δB connected to the microscopic interactions between particles) and into natural fluctuations (that correspond to the spontaneous fluctuations in a non-interacting gas system due to the thermal motion of discrete particles). Since we treat electrons/ions as continuous flu- ids, there should be no sign of particle discreteness, and the only fluctuations shall be field induced. Thus, we have δf _p ^α = δf p ^α,(0) + δf p ^α,(ind) = δf p ^α,(ind) . This assumption is crucial in the simplification of the kinetic model, it makes it possible to express all fluctuations(δE, δf p ^α,(ind) , δf p ^d,(ind) ) through the free dust particle fluctuations related to dust discreetness. Physically, it is connected to the fact that the frequencies related to dust discreetness are much smaller than those related to ion/electron discreetness due to smaller dust velocities and number densities. Hence, this kinetic model is more adequate for the description of low frequency responses.

7. The discreteness of the dust component creates fluctuations in all the components of the dusty plasma including electrons and ions which have only the fluctuations induced by dust fluctuations and by the electric field.

This is the result of treating dust particles as sinks for the electrons and ions and of decomposing the Klimontovich equation in the normal and ﬂuctuation component.

8. The dust velocity is much less than the electron/ion velocities.

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2.5. GENERALIZATION OF THE BOGOLIUBOV-KLIMONTOVICH SCHEME

In any exact computation of the charging cross-sections of electrons and ions we should have a result of the form σ _α (q, u _r ), u _r = | u α −u d | [14]. The presence of the relative velocity would complicate all the mathematical derivations signiﬁcantly and can be simpliﬁed for u _d ≪ u α , due to the massive dust grains this is generally valid.

Apart from these assumptions, mentioned explicitly in Ref. [15], there is a number of additional assumptions taken into account in the developed mathematical framework.

1. The external magnetic and electric ﬁelds are considered zero.

2. The source of plasma particles does not have a ﬂuctuating part.

3. The currents on the dust particles are split in external currents and particle currents. The external currents do not have a ﬂuctuating part.

4. The size of the dust particles is considered constant and same for all the grains.

The grains are considered spherical in the computation of the charging collision cross-sections.

It is worth mentioning, that the kinetic model has been extended to include collisions with neutrals [16].

2.5 Generalization of the Bogoliubov-Klimontovich Scheme

The charging processes present only in dusty plasmas, change the collisional and collective processes involved in the system and therefore change the form of the kinetic equations for both the plasma species and the dust particles.

In case of the dust particles, the charge is considered as a new independent phase variable. Hence, the phase-space for a system consisting of N _d dust particles will be 7N _d -dimensional and the density will have the form

N (r ₁ , p ₁ , q ₁ , ..., r _N

_d

, p _N

_d

, q _N

_d

, t) =

N

d

∏

i=1

δ(r _i − X i (t))δ(p _i − p i (t))δ(q _i − Q i (t)).

Liouville’s equation, the expression of conservation of probability in the phase space, will now obtain the form

∂N

∂t +

N

d

∑

i=1

∇ r

i

( ˙ r i N ) +

N

d

∑

i=1

∇ p

_i

( ˙ p _i N ) +

N

d

∑

i=1

∂

∂q _i ( ˙ q i N ) = 0 (2.11)

The density N represents the joint probability that the dust particle 1 has coordi-

nates between (r 1 , p ₁ , q 1 ) and (r 1 +dr 1 , p ₁ +dp ₁ , q 1 +dq 1 ), the dust particle 2 has co-

ordinates between (r ₂ , p ₂ , q ₂ ) and (r ₂ +dr ₂ , p ₂ +dp ₂ , q ₂ +dq ₂ )... and the dust particle

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N _d has coordinates between (r _N

_d

, p _N

_d

, q _N

_d

) and (r _N

_d

+dr _N

_d

, p _N

_d

+dp _N

_d

, q _N

_d

+dq _N

_d

).

Hence, if we integrate over the phase-space coordinates of all particles except one, we will end up with the reduced probability distribution of one particle f _d (r, p, q, t) that satisﬁes the equation

∂f _p ^d

∂t + ∇ r ( ˙rf _p ^d ) + ∇ p ( ˙ pf _p ^d ) + ∂

∂q ( ˙ qf _p ^d ) = 0 (2.12) So far, we only did an abstract extension to the phase-space. The physical behavior of the dust particles is governed by a set of three dynamic equations, that comple- ment the statistical equation. These are the deﬁnition of conjugate momentum , Newton’s second Law of motion for the Lorentz force and the charging equation.

˙

r = p/m d ; p = qE + (q/m ˙ d )p × B; I = ˙ q = I ext + ^∑

α

I α (2.13) If we use the fact that momentum/space are independent variables, neglect the magnetic ﬁeld and substitute for all terms, we end up with the generalized form of the Klimontovich equation for dust particles,

( ∂

∂t + v · ∂

∂r + qE · ∂

∂p )

f _p ^d (r, q, t) + ∂

∂q (

I ext + ^∑

α

I α

)

f _p ^d (r, q, t) = 0. (2.14)

The current I _α (q, r, t) is the current of plasma particles collected by the grain, it will be the sum of current ﬂuxes on the grain, thus we just integrate the current ﬂux e _α σ _α (q, u)u all over the distribution function of the plasma species,

I α (q, r, t) =

∫

e α σ α (q, u)uf _p ^α (r, t) d ³ p

(2π) ³ (2.15)

The dependence of the particle current on the plasma species distributions, introduces coupling between the Klimontovich equations of the three species, making the mathematical analysis complicated.

In case of the plasma species, the phase space will remain 6N α -dimensional, the diﬀerence with the multicomponent kinetic theory is that probability is not conserved in the usual way, ions/electrons are lost due to the charging process and the dust particles act as sinks for the plasma species distribution and hence, an external source of plasma particles will be added, which we assume that only has a normal component. The generalized Klimontovich equation for the plasma species will be given by

( ∂

∂t + v · ∂

∂r + e _α E · ∂

∂p )

f _p ^α (r, t) = S _α − ^(∫ σ _α (q, u)uf _p ^d

′

(q)dq d ³ p ^′ (2π) ³

) f _p ^α

(2.16)

We can see that the role of dust particles as sinks introduces an extra dependence

between the Klimontovich equations.

Wavelength regimes, where the deviations from the multi-

Abstract