Electrostatic fluctuations in nonequilibrium plasmas with particle collisions

Physics Letters A 372 (2008) 6400–6403

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Physics Letters A www.elsevier.com/locate/pla

Electrostatic fluctuations in nonequilibrium plasmas with particle collisions Valery A. Puchkov IZMIRAN, Troitsk, Moscow Region, 142092, Russia

a r t i c l e

i n f o

Article history: Received 14 April 2008 Accepted 16 August 2008 Available online 2 September 2008 Communicated by F. Porcelli PACS: 52.25.Gj 05.40.+j

a b s t r a c t We present a method for calculation of plasma fluctuations in a nonequilibrium plasma with stable particle distribution functions. The method takes into account particle collisions and collective particle interactions which affect the collision process. In contrast to the known approaches, our method makes no reference to a specific form of the collision integrals and distribution functions. Using the developed method, we calculate the high-frequency spectrum of the electric field fluctuations in a collisional plasma with arbitrary particle distribution functions. © 2008 Published by Elsevier B.V.

Keywords: Electrostatic fluctuations Kinetic theory Collisions Nonequilibrium distribution functions

The theoretical study of plasma fluctuations is necessary for calculations of plasma kinetic coefficients and plasma diagnostics. By now, a general theory is developed for nonequilibrium plasma fluctuations in collisionless stable plasmas where the fluctuation spectra are determined by one-particle distribution functions (see [1–5] and references therein). But this theory becomes insufficient in many important cases. For example, it predicts a δ -function spectrum in the vicinity of the plasma frequency for the long-wave fluctuations with the wave length much greater than the Debye radius. Effect of particle collisions on the spectrum has been treated by the kinetic theory of plasma fluctuations [2,3,6,7]. This theory uses kinetic equations for the fluctuations with specified forms of the collision integral (such as Landau and Balescu–Lenard integrals). But employing such kinetic equations is a very complicated problem, and the fluctuation spectra have been found with the help of this method only for the so-called local equilibrium when the particle distribution functions are the Maxwellian ones with different temperatures and mean velocities. Using various kinds of model collision integrals [2,3,8] simplifies calculations of the spectra but restricts generality of the results obtained. In this Letter, we develop a new formalism for calculation of plasma fluctuations taking into account particle collisions. This method is based on direct solution of the Klimontovich equations and makes no reference to a specific form of collision integrals and distribution functions. Using our method, we calculate the spec-

E-mail address: [email protected]. 0375-9601/$ – see front matter © 2008 Published by Elsevier B.V. doi:10.1016/j.physleta.2008.08.058

trum of the electrostatic field fluctuations in a stable plasma where the fluctuations are determined by one-particle distribution functions. Our formula takes into account both two-particle collisions and collective particle interactions which make contributions to efficient collision frequencies. One of possible applications of our results is the ionospheric plasma with anisotropic electron temperatures (see, for example, [9]). In the ionospheric conditions such an anisotropy does not result in parametric instabilities, and the plasma remains stable and quasi-stationary at time scales greater than the reciprocal of the collision frequency. The electron temperature anisotropy affects the collision process resulting in modification of the intensity and the width of the incoherent scattering spectrum in the vicinity of the plasma frequency. For stable plasma when mode coupling is weak, longitudinal (electrostatic) and transverse fluctuations can be treated independently. Below, we concentrate on the electrostatic fluctuations which determine the incoherent scattering of electromagnetic waves. We start from the Klimontovich equations [3] for the microscopic (random) phase density N a of the particles of a species along with the Poisson equation for the electric field E

∂ Na ∂ Na ∂ Na +v + ea E = 0, ∂t ∂r ∂p

div E = 4πρ ,

(1)

where r and p are the coordinate and pulse, respectively, ea is the   charge, v = p/ma , ma is the mass, and ρ = a ea dp N a is the charge density. The ensemble average of N a gives the distribution function f a :  N a  = na f a (t , r, p), where na is the concentration. Take the Laplace–Fourier transform of (1) using the operator

V.A. Puchkov / Physics Letters A 372 (2008) 6400–6403

Qˆ t ,r (ω, k) = (2π )−4

∞





where δ E(t 1 , r1 )δ E(t 2 , r2 ) depends on τ = t 1 − t 2 , u = r1 − r2 due to the homogeneous and stationary case, and the Fourier (not Laplace) transform over τ is taken. Separating in Eq. (3) regular and fluctuation quantities: N a = na f a + δ N a ,  M = ε + δ  yields



dr exp −i (ω − i )t + ikr ,

dt 0

with  > 0 and  → 0 in final results. Then we get

=−



iea

N a (ω, k, p) −

ω − kv − i 

i

N a (0, k, p)

2π

ω − kv − i 



E(ω, k) =

Qˆ E(t , r)

∂ N a (t , r, p) ∂p





ε(ω, k)δ E (ω, k) +

Tˆ a =

4π ikea k2

a



where dp,

(2)

with N a (ω, k, p) = Qˆ N a (t , r, p), E(ω, k) = Qˆ E(t , r), and N a (0, k, p) ≡ N a (t = 0, k, p).  ˆ Let us apply the operator a Ta to the first Eq. (2). Integrating by parts the nonlinear term and employing the identity k2 = k j k j , we reduce Eq. (2) to the form





Qˆ t ,r (ω, k)

 M (t , r, ω, k) E k (t , r) = S nc (ω, k),  dp k j ∂ N a (t , r, p)/∂ p j 4π 2  M (t , r, ω, k) = 1 + 2 ea , ω − kv − i  k a S nc (ω, k) =

2 k



ea

a

dp N a (0, k, p)

ω − kv − i 

(3)

.

Here E k (t , r) = (kE(t , r))/k, and a new function  M is introduced which may be called the microscopic dielectric function. In the homogeneous and stationary case treated in this Letter,  M is related to the conventional dielectric function ε :  M (t , r , ω, k) = ε (ω, k), with

ε(ω, k) = 1 +

4π e 2na  dp k j ∂ f a (p)/∂ p j a . ω − kv − i  k2 a

(4)

Neglecting fluctuations of  M in the first Eq. (3) yields the well known formula for the collisionless theory [3]

ε(ω, k)δ E nc (ω, k) = S nc (ω, k),

(5)

where “nc” denotes the collisionless approximation, E (ω, k) = Enc (ω, k), δ E = E − E. In the homogeneous case E = 0, and k E = δ E. The right-hand side of (5) depends only on the phase density at t = 0 (see Eq. (3)) and is called the source term. Thus, collisionless approach corresponds to neglecting fluctuations of  M or (which is equivalent) to neglecting the fluctuations of phase density N a in the nonlinear term of Eq. (1). In order to take into account collisions, we should retain these fluctuations. According to the kinetic theory of plasma, “collisions” mean small scale, twoparticle correlations with a radius less than the Debye radius [3]. It is these correlations that are taken into account in the collisional terms of Eq. (10) (see below). These terms are of the order of νe f /|ω| where νe f is an efficient collision frequency (see below), and ω is the frequency of fluctuations. In the high-frequency limit this parameter is small. But for the long-wave fluctuations (which case is the most important in the incoherent scattering plasma line observations) collisional terms affect the spectrum, as collisionless terms become exponentially small. Our purpose is to obtain a collisional analog of Eq. (5). Then the spectrum of the electric field fluctuations (δ E δ E )ω,k can be calculated using the formula [3] nc



lim 2 δ E (ω, k)δ E ∗ (ω, k1 ) =

→0

(δ E δ E )ω,k =



dτ du

(2π )4

1 2π

˜  , k − k , ω , k) dω dk δ  (ω − ω

  × δ E k ω˜  + i , k = S nc (ω, k),

,

Tˆ a N a (ω, k, p),

6401

(δ E δ E )ω,k δ(k − k1 ),

exp(−i ωτ + iku)δ E δ E τ ,u ,

δ  (Ω, χ , ω, k) =

4π k2

 ea2

dp

k j ∂[δ N a (Ω, χ , p)]/∂ p j

ω − kv − i 

a

˜  = ω − i 1 . The form (7) results from the plasma theory and ω notation in which the Laplace transform of a function ϕ (t ) is denoted as ϕ (ω), rather than ϕ (i ω + ), as it should be according to the operational calculus notation [10]. The constant 1 satisfies the condition 0 < 1 < , which ensures the existence of all the Laplace transforms entering into (7). The limit in (6) is non-zero only when δ E and δ E ∗ depend on the same frequency ω . Therefore, we should select the terms proportional to δ E(ω) in the second term of the left-hand side of Eq. (7). As will be seen below, to make this selection we should reduce the second term to a form involving products of three fluctuation quantities. To do so, we find the expression for δ E (ω˜  + i , k ) from Eq. (7) changing variables: ω → ω , 1 → 2 , k → k , ω → (ω − i 1 + i ), k → k . Substituting the result into the second term of the same equation, we get

ε(ω, k)δ E (ω, k) + R 1 (ω, k) − G (ω, k) = S nc (ω, k),  ˜  + i , k ) kk k k δ E (ω R 1 = − dω dk dω dk    ˜  + i , k ) kk k k ε (ω   × δ  (ω − ω˜ , k − k , ω, k) × δ  (ω˜  − ω˜  + i , k − k , ω˜  + i , k ),  kk ˜  + i , k )δ  (ω − ω˜  , k − k , ω, k), G = − dω dk  δ E nc (ω kk

(8)

(9)

˜  = ω − i 2 , 0 < 2 < 1 < , δ E nc = S nc /ε (see (5)), where ω the products kk and k k in the nominators are scalar products, and the identity δ E k (k ) = (kk /kk )δ E (k ) is used. A similar procedure can be taken to transform δ  in G (ω, k). Then we get

ε(ω, k)δ E (ω, k) +

3

R j (ω, k)

j =1

= S (ω, k) ≡ S coll (ω, k) + S nc (ω, k),    k 4π ie 3na  j ∂ f a (p) a     kk k k R2 = dp d ω dk d ω dk  k k k ∂ p 2 kk m ω a j a

(10)

δ E nc (ω˜  + i , k )δ E (ω˜  + i , k ) ε(ω − ω˜  , k − k )[ω − ω˜  + k v − i ] × δ  (ω − ω˜  − ω˜  , k − k − k , ω − ω˜  , k − k ),  −4π ie 3  kk k j

×

R3 =

dp dω dk dω dk

a

a

ma ω2

kk k

δ E nc (ω˜  + i , k )δ E (ω˜  + i , k ) ω − ω˜  + k v − i  ∂ δ N a (ω − ω˜  − ω˜  , k − k − k , p), × ∂pj e2  kk a ˜  + i , k ) S coll = 4π dp dω dk  δ E nc (ω 2 ×

ω

(6)

(7)

ma a nc × δ Na ( −

ω

kk





ω˜ , k − k , p),

(11)

6402

V.A. Puchkov / Physics Letters A 372 (2008) 6400–6403

where the high-frequency approximation |ω| νe f , kv is employed, and k , k k. The terms R j in the left-hand side of (10) determine the collisional contribution to the dielectric function, while the term S coll in the right-hand side takes into account collisions in the fluctuation source. As we have mentioned, the hierarchy of , 1 , 2 is determined by the requirement of the existence of corresponding Laplace transforms. Physically, the condition  > 0 results in Landau damping (not growth) of plasma waves in the equilibrium. Similarly, correct ordering of 1 , 2 leads in the equilibrium state to the collisional broadening (not narrowing) of the plasma line in the fluctuation spectrum. Let us apply now the procedure (6) to both sides of Eq. (10). Then in the right-hand side we have lim→0 2 S (ω, k) S ∗ (ω, k1 ). This expression involves the second, third, and forth moments of the initial fluctuations. One can show that the multi-particle correlations make no contribution to the expression in the limit  → 0 [3]. Due to that, the third moments make no contribution to the right-hand side, and the contributions of the forth moment of fluctuating quantities ϕ j can be found making the replacement

(12) Before applying (6) to the left-hand side of Eq. (10) we can represent all the non-vanishing terms in R j as (13)

( S S )nc ω,k | D (ω, k)|2 2 2 2    16π ea e c + dω1 dk kα kβ /k2 4



ϕ j (Ω1 , k1 )ϕs (Ω2 , k2 ) =

δ(k1 + k2 ) (2π )2



dω1 (ϕ j ϕs )ω1 ,k1 , (ω1 − Ω1 + i )(ω1 + Ω2 − i )

where the frequencies Ω1 , Ω2 with | Im Ω1,2 | <  are combina˜  , ω˜  . tions of ω and the complex frequencies ω   The integration over ω and ω is taken along the real axis from −∞ to +∞. It can be made by  the residue theorem. The integrals  in each case are equal to ±2π i Res where is the sum over Res the residues in the upper (sign +) or lower (sign − ) half-plane of the complex plane. The position of each pole is determined by the hierarchy of the numbers , 1 , 2 (see above). After the integration, we obtain that the non-averaged factors ϕ3 , ϕ2 , and ϕ1 in the non-vanishing terms of the right-hand side of (13) are proportional to δ E (ω, k). Hence, all these factors give the spectrum (δ E δ E )ω,k when the procedure (6) is applied to the left-hand side of Eq. (10). Further calculations are similar to those for the conductivity (see [11]). Therefore, without going into detail we will give here the final result:

(15)

ma mc

nc + (kk /kk )2 (δna δnc )nc ω−ω1 ,k (δ E δ E )ω1 ,−k

 

(16)

,

(17)



where δna = dp δ N a are the particle density fluctuations, ( S S )nc is the spectrum of the collisionless source term, D = ε − 4π i σ /ω is the collisional dielectric function, and σ is the conductivity. All the spectral functions in the right-hand side are taken in the collisionless approximation due to the condition |ω| νe f . Taking into account that the electron to ion mass ratio me / mi 1 and using the approximation |ω| k v i , |ω| ω Li (where v i is a characteristic velocity of ions and ω Li = (4π e 2i ni /mi )1/2 is the ion Langmuir frequency), σ can be reduced to the form e e2

1



2π 2 ω3 me2

−



dk dp



|ε |2



kk

2  

kk

1 + εe (0)

ε (ω)

i

εi (ω) − εi∗

  2 ε i |2 2 e ne f e ( p ) + εe (ω) 1 + εe (0) 1 + εe (ω) e |

i



(18)

 −1   2 × 1 + εe (ω) − εe (0) 1 + εe (0) e i ni f i ( p ) , 

(19)

i



ε = 1 + a εa , εa = εa (k v , k ), εa (ω) = εa (ω + k v , k ), ε (ω) = 1 + εe (0) + i εi (ω + k v , k ), εe (0) = εe (0, k ). The efficient collision frequency mentioned above is proportional to σ : νe f = 4πω2 σ /ω2Le . The expressions (18), (19) are the nonequilib-

rium plasma conductivity which takes into account the dynamic polarization [11]. The second and the third terms in the nominator of Eqs. (15)–(17) are proportional to corresponding plasma emission coefficients calculated in [12]. Note that the nonequilibrium spectrum (15)–(17) cannot be expressed only in terms of the conductivity as in the equilibrium case. All collisionless spectral functions in (15)–(17) are determined by the one-particle distribution functions f a [2,3]. In the collisionless approximation, we should take D = ε and neglect all the terms in the nominator of (15)–(17), except ( S S )nc . Then, one has the known collisionless spectrum

 2 nc   (δ E δ E )nc ω,k = ( S S )ω,k / ε (ω, k) ,  2 2 ea na dp δ(ω − kv) f a ( p ). ( S S )nc ω,k = 2

π

(14)

a,c

nc × (δna δ E α )nc ω+ω1 ,−k (δnc δ E β )ω1 ,−k

where

which is consistent with (12). Thus the procedure (6) along with the decoupling procedure (12), (13) leads to an expression of the electric field fluctuation spectrum as a functional of the second moments of fluctuations. In the monograph [3] it is shown that considering the Laplace transform of the fluctuations as stationary is equivalent to the assumption μ = νe f /|ω| 1. The decoupling procedure (12), (13) means retaining the main collisional terms of the order of μ and neglecting the terms of the second order ˜μ2 . That is why within this accuracy we can replace all the spectral functions entering into the collisional terms of the explicit expression of the fluctuation spectrum (see below, Eq. (17)) by the collisionless approximations. Indeed, such a replacement leads to a relative error ˜μ in the terms being of the order ˜μ themselves, i.e. the resultant error will be ˜μ2 . The second moments entering into (12), (13) are related to corresponding spectral functions



1

ω

σ =i

ϕ1 ϕ2 ϕ3 ϕ4  → ϕ1 ϕ2 ϕ3 ϕ4  + ϕ1 ϕ3 ϕ2 ϕ4  + ϕ1 ϕ4 ϕ2 ϕ3 .

ϕ1 ϕ2 ϕ3 → ϕ1 ϕ2 ϕ3 + ϕ1 ϕ3 ϕ2 + ϕ2 ϕ3 ϕ1 ,

(δ E δ E )ω,k =

(20)

k

a

At the local equilibrium in the case of one species of ions the formula (15)–(17) takes the form

(δ E δ E )ω,k = −

1

T e D e (ω, k) + T i D i (ω, k)

2π 3 ω

| D (ω, k)|2

(21)

where T e,i are the electron and ion temperatures, respectively, D e,i are the imaginary parts of the partial contributions D e,i to the collisional dielectric function (D = 1 + D e + D i ). This expression agrees with the result [2]. Due to our choice of the operator Qˆ , sign D  = − sign ω , and the spectrum (21) is positive. For T e T i the term D i ≈ εi and



D e = εe − i



ω2Le (8π )1/2 |ee |3 e i ne −1 T e3 e 2i mi νei + ln 3 1/ 2 1/ 2 ω T i3 e e2me 3me T e T i

 ,

where νei is the electron–ion collision frequency [2]. The second term in the braces describes the increase of the efficient collision frequency due to the ion-sound oscillations (see, for example, [13]).

V.A. Puchkov / Physics Letters A 372 (2008) 6400–6403

It is evident that such an increase will result in a broadening of the plasma line spectrum. In the total equilibrium (T e = T i = T ) the nominator of (21) is equal to T D  , and D = ε − i (ω2Le /ω3 )νei . Such a form of the spectrum proportional to the imaginary part of the collisional dielectric function D is in agreement with the fluctuation–dissipation theorem [2]. References [1] W. Thompson, J. Hubbard, Rev. Mod. Phys. 32 (1960) 714; N. Rostoker, Nucl. Fusion 1 (1961) 101; M. Rosenbluth, N. Rostoker, Phys. Fluids 5 (1962) 776. [2] A.I. Akhiezer, I.A. Akhiezer, R. Polovin, A.G. Sitenko, K.N. Stepanov, Plasma Electrodynamics, Linear Theory, vol. 1, Pergamon, New York, 1975.

6403

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