Dynamical behaviour of stochastic dust charge fluctuations

Physics Letters A 376 (2012) 2552–2554

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

Dynamical behaviour of stochastic dust charge fluctuations Samiran Ghosh a,∗ , P.K. Shukla b,c a b c

Department of Applied Mathematics, University of Calcutta, 92, Acharya Prafulla Chandra Road, Kolkata 700 009, India RUB International Chair, International Center for Advanced Studies in Physical Sciences, Faculty of Physics and Astronomy, Ruhr University Bochum, D-44780 Bochum, Germany Department of Mechanical and Aeropsace Engineering, University of California San Diego, La Jolla, CA 92093, USA

a r t i c l e

i n f o

Article history: Received 6 June 2012 Accepted 30 June 2012 Available online 4 July 2012 Communicated by V.M. Agranovich

a b s t r a c t The dynamics of stochastic dust charge fluctuations is investigated. An inhomogeneous Burgers equation with a linear forcing term is derived in which the Burgers term is proportional to the diffusivity of dust charge fluctuations. This novel equation is solved analytically. Possibility of turbulence due to stochastic charge fluctuations is highlighted. © 2012 Published by Elsevier B.V.

Keywords: Charge fluctuations Diffusion process Burgers’ equation

A dusty plasma is a highly dissipative plasma medium composed of several charged particles (e.g. electrons, ions and micronsized dust particles) [1–3]. The distinctive property of a dusty plasma is that the charge Q on a dust particle fluctuates with time and thus becomes a new dynamical time dependent variable Q (t ) [4–6]. The dust charge fluctuations are of two types in nature: one is a collective [4–6] and the other is non-collective [2, 3,7]. The collective dust charge fluctuations occur due to the turbulence or other spatial and temporal variations of the surrounding plasma properties. This charge fluctuation causes an anomalous dissipation that leads to a non-Landau damping [4–6], instability [8] of linear collective modes, and also generates shock waves [9–13] associated with nonlinear acoustic modes. Non-collective dust charge fluctuations are treated as a result of the discreteness of the charging process, i.e. due to the discrete nature of the charge carriers. In this case, the charge fluctuations are random in nature unlike the case of collective fluctuations. Actually, the distribution of dust grain charges has a finite dispersion [14–16] and the charge of dust particles fluctuates continuously about its equilibrium mean value  Q . These random stochastic dust charge fluctuations can cause the dust particle heating [17–19], and also an instability for which the lattice wave becomes energywise unstable in a low pressure gas discharge plasma [20]. However, the dynamics of non-collective dust charge fluctuations is not well investigated. Therefore, it is instructive to study the dynamics of non-collective dust charge fluctuations.

*

Corresponding author. Tel.: +91 33 2350 8386/1397; fax: +91 33 2351 9755. E-mail addresses: [email protected] (S. Ghosh), [email protected] (P.K. Shukla). 0375-9601/$ – see front matter © 2012 Published by Elsevier B.V. http://dx.doi.org/10.1016/j.physleta.2012.06.034

Thus, in this Letter, we make an attempt to investigate the dynamics of the random stochastic dust charge fluctuations in the frame work of Uhlenbeck and Ornstein diffusion process [21,22]. Let us assume that dust particles acquire electrical charges by absorbing electrons and ions from the background plasma, and become negatively charged because of the higher thermal speed of electrons than that of ions. The dust charge Q (t ) is then determined from the charging equation [1,2]

dQ dt

= I = Ie + Ii ,

(1)

where I e and I i are the electron and ion currents flowing onto the surface of a spherical dust particle of radius r0 , and are given by



Ie = π

r02 ene

 I i = π r02 eni



8T e

π me 8T i

π mi

exp

 1−

Qe 4π0 r0 T e Qe

4π0 r0 T i

 ,

 .

(2)

Here ne(i ) , me(i ) , and T e(i ) are the electron (ion) number density, mass and temperature, respectively. The steady-state (equilibrium) charge  Q  is determined by the steady-state charging current  I e  +  I i  = 0. Due to normal statistical fluctuations associated with the discrete impacts of individual electrons and ions, the dust charge will fluctuate. Considering small charge fluctuations about the equilibrium charge, i.e. Q (t ) =  Q  +  Q (t ) (| Q (t )|  | Q |), Eq. (1) in the first order term becomes

d Q (t ) dt

= −γch  Q (t ),

(3)

S. Ghosh, P.K. Shukla / Physics Letters A 376 (2012) 2552–2554

γch−1  Ω −1

where

γch = −

∂ I  r0 ω =√ (1 + z + σ )  ∂ Q Q =| Q | 2π V thi 2 pi

(4)

is the steady-state √charging frequency, z = | Q |e /4π0 r0 T e , σ = T i /T e , V thi (= T i /mi ) the ion thermal speed, and ω pi (= ni0 e 2 /0mi ) the ion plasma frequency. The term γch  Q (t ) in Eq. (3) acts as a restoring force that tends to bring the charge equilibrium value. In order to study the dynamics of stochastic dust charge fluctuations, it is conceived that a Brownian force Γ (t ), which arises from the molecular bombardment, is responsible for dust charge fluctuations around the equilibrium value of a dust charge. Then, the stochastic variable x(t ) ≡  Q (t ) describes the Uhlenbeck and Ornstein [21,22] process, namely,

dx(t ) dt

=−

df (x) dx

+



2D Q Γ (t ),

(5)

df (x) dx

= γch x

⇒

f (x) =

1 2

γch x2

(6)

is the potential, D Q the constant diffusivity of the dust charge fluctuation, given by

D Q = γch σ Q2 ,

(7)

and σ Q2 is the variance of dust charge distributions, which under the assumption that the dust charge fluctuations are determined by the fluctuations of electron and ion fluxes in macroscopically equilibrium plasma, becomes

σ Q2 =

e





I

2 −∂ I /∂ Q



=π

Q =| Q |

z+σ z(1 + z + σ )

    Q e .

(8)

Characteristically of Eq. (5), the charging process is split into two parts:

A rapidly fluctuating environment can be characterized if the correlation time, viz. the memory time of the stochastic process t corr  t macro , the macroscopic time, viz. the relaxation time of the system, is towards a reference steady state under average environmental conditions. Concerning this, it is assumed that Γ (t ) represents the Gaussian white noise arising due to dust charge fluctuations and satisfies the following conditions:





Γ (t )Γ t







=δ t −t .

(9)

Note that initially there is no dust charge variation, i.e. x(0)(=  Q (0)) = 0. Thus, due this initial condition, the solution of Eq. (5) can be written in the form

x(t ) =

t



2D Q exp(−γch t )







Γ t exp γch t dt .

(10)

0

The temporal autocorrelation function of dust charge fluctuations is











x t + t x(t ) = σ Q2 exp −γch t .

(12)

  ∂ ∂ P (x, t ) ∂ df (x) = DQ + P (x, t ). ∂t ∂x ∂x dx

(13)

The stationary state (time independent) solution [22,25] of this equation is

N0 DQ



f (x)

exp −

DQ



(14)

,

where N 0 is a normalization constant. More precisely, following the procedure of Ref. [22], the non-stationary solution of (13) with df (x)/dx = γch x becomes

P (x, t ) =



1



σ Q 2π (1 − e−2γcht )

 (x − x e −γch t )2 . 2σ Q2 (1 − e −2γch t )

exp −

(15)

In the asymptotic limit t → ∞, it becomes the Gaussian distribution

lim P (x, t ) =

t →∞



1

exp −

√

σ Q 2π

x2 2σ Q2



(16)

.

However, to study the non-stationary process, we introduce

  f (x) P (x, t ) = Ψ (x, t ) exp −

(17)

2D Q

in Eq. (13). This substitution leads to the following Schrödinger equation in imaginary time [25]

(i) a slowly varying term df (x)/dx = γch x, (ii) a rapidly varying term Γ (t ).



Γ (t ) = 0,

Ω  γch ,

where t macro = Ω −1 , Ω is the characteristic oscillation frequency of the dust particles in the steady-state dusty plasma. In a gas discharge Argon plasma [23,24], for r0 = 1 μm, the steady-state charging frequency γch ∼ O (104 ) s−1 , whereas the macroscopic time i.e. relaxation time Ω ∼ O (10−3 ) s−1 . Thus, the assumption of the Gaussian white noise is quite justified for a gas discharge plasma. The Fokker–Planck equation of (5) for the probability density  function P (x, t ) (with the normalization condition P (x, t ) dx = 1) reads

P (x) =

where

⇒

2553

(11)

Consequently, the autocorrelation function vanishes only for t

γch−1 , so that t corr = γch−1 , the steady-state charging time. Thus, the assumption of the Gaussian white noise (rapidly varying stochastic term) due to dust charge fluctuations is justified only if

∂Ψ (x, t ) ∂ 2 Ψ (x, t ) = DQ − V (x)Ψ (x, t ), ∂t ∂ x2

(18)

where



V (x) =

1 (

df (x) 2 ) dx

2

2D Q

−

d2 f (x)



dx2

 =

γch2 x2 4D Q

−

γch 2

 .

(19)

Finally, we introduce the Hopf–Cole transformation [26]

u (x, t ) = −2D Q

 ∂  ln Ψ (x, t ) ∂x

(20)

to Eq. (18) and this leads to the following transformed equation

∂u ∂u ∂ 2u 2 +u = D Q 2 + γch x. ∂t ∂x ∂x

(21)

This is the well-known nonlinear diffusion equation or the Burgers equation [27] with a linear forcing term in the dust charge space (x, u ), where x(=  Q ) acts as a co-ordinate and u = dx/dt acts as velocity. The coefficient of the Burgers term is D Q , the diffusivity constant that arises due to stochastic dust charge fluctua2 tions (the rapidly varying term) and linear forcing term γch x arises due to the slowly varying term. To solve the inhomogeneous equation (21), following the procedure [28], let us substitute

u (x, t ) = γch x + Λ(t ) v





η ≡ Λ(t )x, ζ (t )

(22)

2554

S. Ghosh, P.K. Shukla / Physics Letters A 376 (2012) 2552–2554

into Eq. (21). This substitution leads to the following homogeneous Burgers equation

∂v ∂v ∂2v = DQ 2 , +v ∂ζ ∂η ∂η

(23)

subject to the following solvability condition

dΛ dt dζ dt

+ γch Λ = 0 = Λ2

⇒

Λ = e −γch t ,

⇒ ζ=

1  2γch

1 − e −2γch t



(24)

with the initial conditions: Λ(0) = 1 and ζ (0) = 1. The solution of the homogeneous Burgers equation (23) is





v (η, ζ ) = v 0 1 − tanh

v0



2D Q

 (η − v 0 ζ ) ,

References

(25)

where v 0 is the initial intensity. Finally, substituting this solution into Eq. (22) together with Eq. (24), we have the following solution of the inhomogeneous Burgers equation (21)



u (x, t ) = γch x + v 0 e −γch t 1 − tanh

 × xe

−γch t

v0 

−

2γch

1−e





v0 2D Q

−2γch t

 .



(26)

In the non-dimensional form this solution reads

U (ξ, τ ) = ξ + U 0 e



−τ





1 − tanh

× ξ e −τ −



U0 2σ˜ Q2

 ,

U0  1 − e −2 τ 2

(27)

where we have introduced the normalization: ξ = x/| Q |, τ = γch t, U = u /γch | Q |, U 0 = v 0 /γch | Q |, and σ˜ Q2 = σ Q2 /| Q |2 . From the relation (26) or (27), we note that in the asymptotic limit t → ∞ (τ → ∞) the velocity in the charge space (x, u = dx/dt ) or (ξ, U = dξ/dτ ), becomes

lim u (x, t ) = γch x

t →∞

⇒

lim U (ξ, τ ) = ξ.

τ →∞

The forcing term is proportional to a slowly varying term in the Uhlenbeck–Ornstein dust charge diffusion process [Eq. (5)]. Also, it appears from Eq. (28) that in the charge space (x, u ) or (ξ, U ), the linear force due to a slowly varying term in Uhlenbeck– Ornstein process [Eq. (5)] plays a stabilizing role in the asymptotic limit with velocity proportional to the force applied initially, independent of the initial conditions of the velocity field. This is a characteristic property of turbulence: the unpredictability of the velocity field of the Burgers equation [27,29] in the dust charge space. Thus, the result of the present investigation indicates the possibility of turbulence due to stochastic dust charge fluctuations.

(28)

Thus, the velocity associated with the Uhlenbeck and Ornstein dust charging process behaves asymptotically linear with x(ξ ). In conclusion, we have investigated the dynamical behaviour of stochastic dust charge fluctuations, which is shown to be governed by a Burgers equation with a linear forcing term in the dust charge space (x ≡  Q , u ≡ dx/dt ) or (ξ ≡ x/| Q |, U ≡ dξ/dτ ).

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