On anomalous diffusion in a plasma in velocity space

(1)

On anomalous diffusion in a plasma in velocity space

Citation for published version (APA):

Trigger, S. A., Ebeling, W., Heijst, van, G. J. F., Schram, P. P. J. M., & Sokolov, M. (2010). On anomalous diffusion in a plasma in velocity space. Physics of Plasmas, 17(4), 042102-1/8.

https://doi.org/10.1063/1.3377779

DOI:

10.1063/1.3377779

Document status and date: Published: 01/01/2010

Document Version:

Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)

Please check the document version of this publication:

• A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website.

• The final author version and the galley proof are versions of the publication after peer review.

• The final published version features the final layout of the paper including the volume, issue and page numbers.

Link to publication

General rights

Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain

• You may freely distribute the URL identifying the publication in the public portal.

If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, please follow below link for the End User Agreement:

www.tue.nl/taverne

Take down policy

If you believe that this document breaches copyright please contact us at: openaccess@tue.nl

providing details and we will investigate your claim.

(2)

On anomalous diffusion in a plasma in velocity space

S. A. Trigger,1,a兲 W. Ebeling,2G. J. F. van Heijst,3P. P. J. M. Schram,3and I. M. Sokolov2 1

Joint Institute for High Temperatures, Russian Academy of Sciences, Izhorskaya 13/19, 125412 Moscow, Russia

2

Institut für Physik, Humboldt-Universität zu Berlin, Newtonstraße 15, D-12489 Berlin, Germany 3

Eindhoven University of Technology, P.O. Box 513, MB 5600 Eindhoven, The Netherlands 共Received 15 February 2010; accepted 10 March 2010; published online 12 April 2010兲

The problem of anomalous diffusion in momentum space is considered for plasmalike systems on the basis of a new collision integral, which is appropriate for consideration of the probability transition function 共PTF兲 with long tails in momentum space. The generalized Fokker–Planck equation for description of diffusion 共in momentum space兲 of particles 共ions, grains, etc.兲 in a stochastic system of light particles共electrons or electrons and ions, respectively兲 is applied to the evolution of the momentum particle distribution in a plasma. In a plasma the developed approach is also applicable to the diffusion of particles with an arbitrary mass relation due to the small characteristic momentum transfer. The cases of an exponentially decreasing 共including a Boltzmann-like兲 kernel in the PTF in momentum space, as well as more general kernels, which create anomalous diffusion in velocity space due to the long tail in the PTF, are considered. Effective friction and diffusion coefficients for plasmalike systems are found. © 2010 American Institute of

Physics. 关doi:10.1063/1.3377779兴

I. INTRODUCTION

Diffusion in coordinate and in momentum 共velocity兲 space is of fundamental importance and has attracted a grow-ing interest durgrow-ing many years since the description of this process provides a simplified and effective key for address-ing of many problems of the kinetic theory.

Deviations from a linear time dependence, i.e., 具r2_{共t兲典⬃t, of the mean square displacement in coordinate}

space have been experimentally observed, in particular,

un-der essentially nonequilibrium conditions or for some disor-dered systems. The average square separation of a pair of particles passively moving in a turbulent flow grows, accord-ing to Richardson’s law, with the third power of time.1 For diffusion typical for glasses and related complex systems,2 the observed time dependence is slower than linear. These two types of anomalous diffusion are obviously character-ized as superdiffusion 具r2共t兲典⬃t␣ 共␣⬎1兲 and subdiffusion 共␣⬍1兲.3

For a description of these two diffusion regimes a number of effective models and methods have been sug-gested. The continuous time random walk model of Scher and Montroll,4 leading to subdiffusion behavior, provides a basis for understanding photoconductivity in strongly disor-dered and glassy semiconductors. The Levy-flight model,5 leading to superdiffusion, describes various phenomena as self-diffusion in micelle systems,6reaction, and transport in polymer systems,7 and is applicable even to the stochastic description of financial market indices.8 Both cases can be effectively described by generalized diffusion equations with fractional derivatives in time or in coordinates, respectively.9 For example, different aspects of the anomalous diffusion in coordinate space were considered within this scheme in Refs.

10and11.

However, recently a more general approach has been suggested in Refs. 12 and 13. This approach allows us to reproduce the results of the standard fractional differentiation method in coordinate space, when the latter is applicable, and enable to describe more complicated cases of anomalous diffusion processes. In Ref.14 this approach has also been applied to the case of diffusion in a time-dependent external field in coordinate space. In what follows, we concentrate on the problem of diffusion in momentum space in application to plasma systems.

Problems of diffusion in momentum space have been considered for plasmas in the fundamental study by Landau15 and later by Rostoker and Rosenbluth, Lenard, Balescu, Kli-montovich, and many others. Various theoretical and experi-mental aspects of these investigations can be found in Refs.

16–18.

Here our main interest is focused on anomalous diffu-sion in momentum space by using the methods developed in Refs.12and13for coordinate space. Recently, see Ref.19, a new kinetic equation for anomalous diffusion in velocity space has been derived on the basis of an appropriate expan-sion of probability transition function 共PTF兲 and some par-ticular problems were investigated on this basis. In the present paper the problem of anomalous diffusion in momen-tum 共velocity兲 space will be considered for plasmalike systems.

Some aspects of anomalous diffusion in velocity space have been investigated during the past decade in a number of studies.20–23 In particular, in Ref. 22 the phenomenological equation for anomalous diffusion in velocity space for a magnetized plasma has been obtained on the basis of the Langevin model with the linear friction proportional to the particle velocity and with non-Gauss noise. The correspond-ing Fokker–Planck-type equation included a diffusion term with a fractional derivative and the usual drift term with the

a兲_{Electronic mail: satron@mail.ru.}

(3)

first momentum derivative. The applicability of this equation is limited due to phenomenological nature of the considered model. We should mention the analogy of this equation with the structure of the equation for anomalous diffusion in co-ordinate space under action of an external field.11,14An equa-tion similar to that in Ref.22 has recently been applied in Ref.23to describe the evolution of the velocity distribution function of strongly nonequilibrium hot and rarified plasmas. Such types of plasma exist, e.g., in tokomaks. On the whole, in comparison with anomalous diffusion in coordinate space, anomalous diffusion in velocity space has been studied to a modest extent.

In this paper the problem of anomalous diffusion in mo-mentum 共velocity兲 space will be considered for plasmalike systems. In spite of formal similarity, the physical共and math-ematical兲 nature of diffusion in momentum space is very different from that in coordinate space. This is clear already from the fact that momentum conservation, which takes place in momentum space, has no analogy in coordinate space.

Diffusion in velocity space for the cases of normal and anomalous behavior of the PTF is presented in Sec. II. Start-ing from the argumentation based on the Boltzmann-type of the PTF, we describe the new approach to the kinetic equa-tion, which in fact can be applied to the wide class of PTF functions based on the prescribed distribution function for one共light兲 sort of particles. The anomalous diffusion in mo-mentum space for plasma is analyzed in Sec. III on the basis of the Boltzmann-type kernel for PTF. Models of anomalous diffusion for plasmalike systems are considered in Sec. IV. In Sec. V the generalized Fokker–Planck equation for diffusion, written for the formal Fourier component f共s,t兲 of the distri-bution function f共p,t兲, is represented in partial derivatives in velocity space. This representation is possible only in par-ticular cases of the power dependence of the coefficients in the generalized diffusion equation.

II. CALCULATION OF THE DIFFUSION IN THE VELOCITY SPACE ON THE BASIS OF A MASTER-TYPE EQUATION

Let us consider now the main problem formulated in the introduction, namely, diffusion in momentum space 共V-space兲 on the basis of the master equation, which de-scribes the balance of grains entering and leaving point p at time t共see, e.g., Refs.24and25兲,

df共p,t兲

dt =

冕

dp兵w共p,p

⬘

兲f共p

⬘

,t兲 − w共p

兲 w共p

⬘

,p兲=

fst_共p兲

fst共p

⬘

兲. 共2兲

In the following analysis we use a form of the master equation26 equivalent to Eq.共1兲,

df共p,t兲

dt =

冕

dq兵W共q,p + q兲f共p + q,t兲 − W共q,p兲f共p,t兲其. 共3兲

再

d

r

qq2W共q兲. 共6兲

If we neglect the p-dependence of the PTF at all, we arrive at

A_␣= 0 共while the diffusion coefficient is constant兲. In this approach, which is known to be incorrect, the coefficient A_␣ for the Fokker–Planck equation can be determined on the basis of the argument that the stationary distribution function is Maxwellian. In this way we arrive at the standard form of the coefficient MTA_␣共p兲=p_␣B, which is one of the forms of

Einstein’s relation. For systems far from equilibrium this ar-gument is not acceptable.

Following Ref.1we now generalize the Fokker–Planck approach to find the coefficients of the kinetic equation, which are applicable also to slowly decreasing PTFs. We apply a more general approach, based on the difference in the velocities for light and heavy particles. For calculation of the function A_␣we have to take into account that the function

W共q,p兲 is scalar and depends on the variables q,q·p,p.

Ex-panding W共q,p兲 on q·p one arrives at the following ap-proximate representation of the function W共q,p兲:

(4)

W共q,p兲 ⯝ W共q兲 + W˜

⬘

共q兲共q · p兲 +1₂W˜

⬙

共q兲共q · p兲2, 共7兲 where W˜

⬘

共q兲⬅⳵W共q,q·p兲/⳵共qp兲兩_q·p=0 and W˜

⬙

共q兲

⬅⳵2_W_{共q,q·p兲/}_⳵_共qp兲2_兩

q·p=0.

Then, with the necessary accuracy, A_␣ equals

A_␣共p兲 =

冕

drqq_␣q_␤p_␤W˜

⬘

共q兲 = p_␣

冕

drqq_␣q_␣W˜

⬘

共q兲 = p␣ r

冕

d r qq2W˜

⬘

共q兲. 共8兲

If the function W共q,p兲 satisfies the equality W˜

_⬘

_共q兲

= W共q兲/2MT, we obtain the known Einstein relation

MTA_␣共p兲 = p_␣B. 共9兲

Let us check this relation for Boltzmann-type collisions, which are described by the PTF W共q,p兲=wB共q,p兲,13

wB共q,p兲 = 2␲ ␮2_q

冕

q/2␮ ⬁ duud␴ do

冋

arccos

冉

1 − q2 2␮2_u2

冊

,u

册

⫻fb共u2+v2− q · v/␮兲, 共10兲

where共p=Mv兲 and d␴共␹, u兲/do,␮, and fbare the

_{共q兲⬃1/q}␤ with␤ⱕr+2.

Inserting expansion 共7兲 for W共q,p兲 in Eq. 共3兲共as an example, we choose r = 3; the analysis for arbitrary r runs in a similar way兲 we arrive at a new collision term in the kinetic equation, which can be considered as a generalization of the Fokker–Planck equation for anomalous diffusion in velocity space,19

df共s,t兲

dt = A共s兲f共s兲 + B␣共s兲

⳵f共s,t兲

⳵s_␣ . 共11兲

In fact, as shown in Ref.19, in expansion共7兲for W共q,p兲 we have to keep共with the necessary accuracy兲 only the terms linear in qp and p. The function f共s兲 in Eq. 共11兲 is the

Fourier-component f共s兲=兰关dp/共2␲兲3_{兴exp共ips兲f}

cos共qs兲 −sin共qs兲 qs

册

W ˜

_⬘

_共q兲.

Here we took into account the existence of the small param-eter␮/M and we omitted the small on this parameter terms of order p2 _{and W}

_⬙

_{in Eq.}_共7兲_.

For the isotropic function f共s兲= f共s兲 one can rewrite Eq.

共11兲in the form

df共s,t兲

dt = A共s兲fg共s兲 + B共s兲s

⳵f共s,t兲

⳵s . 共14兲

For the case of strongly decreasing PTF the exponent under the integrals for the functions A共s兲 and B共s兲 can be expanded as A共s兲 ⯝ = −s 2 6

冕

dqq 2_W_{共q兲, B共s兲 ⯝ −}1 3

冕

dqq 2_W˜

_⬘

_共q兲. 共15兲

= C exp

冋

− A0s2 2B0

册

. 共17兲

The corresponding normalized stationary momentum distri-bution is given by f共p兲 = NB0 3/2 共2␲A0兲3/2exp

冋

− B0p2 2A0

册

. 共18兲

Therefore, in Eq.共17兲the constant C = N, where N is the total number of particles in the system, which undergo diffusive motion. Equation共16兲and this distribution are the generali-zation of the Fokker–Planck case for normal diffusion in a nonequilibrium situation with strongly decreasing kernels

W共q兲, W

⬘

共q兲, when the prescribed PTF function W共q,p兲 is

determined, e.g., by some non-Maxwellian distribution of the small particles fb. To show this in an alternative way, let us

(5)

take the Fourier transformation of Eq. 共11兲 and the corre-sponding coefficients A and B_␣,

df共p,t兲 dt = − A0 ⳵2_f_共p,t兲 ⳵p2 − B0 ⳵关p␣f共p,t兲兴 ⳵p_␣ . 共19兲

We then arrive at a Fokker–Planck-type equation with friction coefficient ␤= −B0 and diffusion coefficient

D = −A0/M2. In general these coefficients 关Eq. 共15兲兴 do not satisfy the Einstein relation.

In the case of equilibrium W-function 共e.g., fb= fb

0_{, see} above兲 the equality W˜

_⬘

_{共q兲=W共q兲/2MT}

bis satisfied. Then we

find A共s兲/sB共s兲⬅A0/B0= MTb. In this case the Einstein

rela-tion between the diffusion and fricrela-tion coefficients D =␤T/M is satisfied and the standard Fokker–Planck equation

is valid. In the general case, however, the general equations

共11兲–共13兲have to be used.

III. DIFFUSION IN PLASMA SYSTEMS ON THE BASIS OF BOLTZMANN-TYPE COLLISIONS

Let us calculate the PTF for the case of Coulomb colli-sions. The differential cross section for the Coulomb scatter-ing d␴/do equals

d␴共␹,u兲 do =共Ze 2_/2_␮_u2_兲2 1 sin4␹ 2 =共Ze2/2␮u2兲216␮ 4_u4 q4 , 共20兲

where␹= arccos共1−q2/2␮2u2兲. Then wB Coul_{共q,p兲 =} 2␲ ␮2_q

冕

q/2␮ ⬁ duu共Ze2/2␮u2兲2 ⫻16␮4u4 q4 fb共u2+v2− q · v/␮兲 =8Z 2_e4_␲ q5

冕

q/2␮ ⬁ duufb共u2+v2− q · v/␮兲. 共21兲

It is necessary to stress that in the case of the Coulomb interaction the general equations共11兲and共14兲are applicable not only for diffusion of heavy particles in a light particle medium but also for arbitrary mass relations. The reason for this statement is the typical small transfer of momenta in the Coulomb systems.

Let us calculate now the coefficients A_␣共p兲 and B_␣␤共p兲 to compare the results with the linearized Landau kinetic equation, in which these coefficients depend on p. This im-plies that for the Coulomb interaction the expansion by 共q·p兲 has to be performed at finite p.

⬘

共y兲⬅⳵fb共y兲/⳵y. The procedure that we used here

im-plies that the long tails of the functions wBCoul共q兲 and w

˜_B

⬘

冑

␮ 共2␲兲3/2

冑

_T. 共28兲

Therefore, one can rewrite A_␣⬅−p_␣␯ie, where␯ieis the

char-acteristic frequency friction ions on electrons,

␯ie= 4

冑

2␲Z2e4n␮1/2 3MT3/2 ln

冉

qmax qmin

冊

. 共29兲

The corresponding friction force per unit volume Fieis equal

Fie= niMU␯ie, where U is the relative velocity of the

elec-trons and ions.27In fact, the divergence at large q handled by a cutoff is an artifact. This becomes clear when calculating the equilibrium function wB

Coul,0_{共q,p兲 more accurately,} with-out expansion on small values of q.

For the equilibrium distribution function fb

0_共u兲 = ne共␮/2␲T兲3/2exp共−␮u2/2T兲 the PTF function reads

(6)

wBCoul,0共q,p兲 = 4neZ2e4␲ q5 ⫻exp关−␮共v2_{− q · v/}_␮_兲/2T兴 ⫻

冕

q/2␮ ⬁ du2共␮/2␲T兲3/2exp关−␮u2/2T兴 =4neZ 2_e4_␮1/2

冑

2␲Tq5 ⫻exp关−␮共v2_{− q · v/}_␮_{+ q}2_/4_␮2_兲/2T兴, 共30兲 wB

⬘

Coul,0共q,p兲 = 2neZ2e4␮1/2

冑

2␲MT3q5 ⫻exp关−␮共v2_{− q · v/}_␮_{+ q}2_/4_␮2_兲/2T兴. 共31兲

Using the Fokker–Planck approximation for the coefficients

A_␣共p兲 and B_␣␤共p兲, and Eqs.共35兲and共31兲we find

rmin/rmax= Ze2/TrDⰆ1 for Ze2/បvTⰇ1, or rmin/rmax

=ប2_/2_␮_Tr

DⰆ1 for the opposite inequality Ze2/បvTⰆ1.

Here rDis the Debye radius andvT=

冑

T/␮is of the order of

the thermal velocity. In our approach the cutoff for the small momenta is satisfied automatically and corresponds to the second inequality共the “quantum” case兲.

For a weakly nonideal plasma this means a cutoff at the minimal momentum qmin=ប/rD. Then

J1= − 2␲C + 4␲ln

冉

rD rmin

冊

⯝ 4␲ln

冉

rD rmin

冊

, 共35兲

where r_min2 ⬅ប2_/2_␮_{T and C}_{⯝0.577 is the Euler constant.} It is easy to verify that W˜

⬘

共q兲=W共q兲/2MT and W˜

⬙

共q兲 = W共q兲/4M2_T2 _{关in the case under consideration W共q,p兲} ⬅wB

Coul,0_{共q,p兲兴. Therefore, for the equilibrium case the usual} Fokker–Planck equation for heavy particles 共ions or dusty particles in dusty plasmas兲 is, naturally, valid with a good accuracy, owing to the exponential convergence of the inte-grals in the coefficients A共s兲 and B␣共s兲 at high values of q. The term with W

⬙

in Eq.共7兲 is negligible, according to the above general statement. However, for small q the coeffi-cients A and B_␣have the logarithmical divergence typical for the Coulomb interaction because W⬃1/q5_{, just like W}

_⬘

_共q兲. As follows from Eq.共21兲, this divergence not only exists for equilibrium but for an arbitrary distribution function fb. The

simplest physical way to avoid this divergence is to cut the integrals for A and B_␣ in Eqs.共12兲 and共13兲for small q by the Debye radius 1/rD, following the well known Landau

procedure. We are more interested to find examples for non-exponential behavior of W, which may occur, e.g., for some specific non-Maxwellian distributions fb.

IV. MODELS OF ANOMALOUS DIFFUSION FOR COULOMB INTERACTION

Now we can calculate the coefficients for models of anomalous diffusion in plasmalike systems. At first we cal-culate the model of a Coulomb system with two species of particles with masses ␮ and MⰇ␮. Let us suppose that in the model under consideration the small particles are de-scribed by a prede-scribed stationary distribution fb= nb␾b/u0 3 共where␾bis the dimensionless distribution function and u0is the characteristic velocity for the distribution of the small particles兲 and␰⬅共u2₊_v2_{− q · v}_/_␮_兲/u

0 2 , Wa共q,p兲 =8␲Z 2_e4_n b u₀q5

冕

共q2_/4␮2₊_v2_−q·v_/␮兲/u 0 2 ⬁ d␰·␾b共␰兲. 共36兲

First, let us consider a power-law distribution ␾b共␰兲

= C/␰␥_共_␥_⬎1兲, Wa共q,p兲 =

冏

8␲Z 2_e4_nbC u0q5 ␰1−␥ 共1 −␥兲

冏

_␰0 ⬁ =8␲Z 2_e4_nbC u0q5 ␰01−␥ ␥− 1, 共37兲 where␰₀⬅共q2_/4_␮2₊_v2_{− q · v}_/_␮_兲/u 0 2_.

For the case p = 0 the value ␰0→˜␰0⬅q2/4␮2u0 2

册

W共q兲 = 4␲Ca

冕

0 ⬁ dq 1 q2␥+1

冋

册

. 共41兲

Comparing the reduced equation 共see below兲 in velocity space with the diffusion in coordinate space 关2␥+ 3↔␣ and W共q兲=Ca/q2␥+3兴 we can conclude that the integral

in the right-hand side of Eq. 共41兲 共3d case兲 converges if 3⬍2␥+ 3⬍5 or 0⬍␥⬍1. The inequality␥⬍1 implies con-vergence for small q共q→0兲 and the inequality␥⬎0 implies convergence for q→⬁. Likewise for the integral in B共s兲,

B共s兲 =4␲ s2

冕

0 ⬁ dqq2

冋

cos共qs兲 −sin共qs兲 qs

册

W ˜

_⬘

_共q兲, _共42兲

convergence of B共s兲 exists for small q if␥⬍0 and for large

q→⬁ if ␥⬎−3/2. Again, it is easy to show that the term

with W

⬙

can be omitted.

Therefore, for convergence of A and B for a large q we require convergence for A, which implies␥⬎0. For conver-gence for small q it is sufficient to have converconver-gence for B, implying ␥⬍0. Therefore, for the case of pure power-law behavior of the function fb共␰兲 convergence is absent. It is

also clear that the function fb共␰兲=C/␰␥ 共␥⬎1兲 cannot be

normalized. However, for anomalous diffusion in momentum space in reality the convergence for small q is always ob-tained, e.g., by a finite value ofv or by a change in the small q-behavior of W共q兲 by screening 共compare with the

ex-amples of anomalous diffusion in coordinate space13兲. There-fore, in the model under consideration, the “anomalous dif-fusion in velocity space” for a power-law behavior fb共u兲关and

as a consequence with a power-law dependence of W共q兲 and

W˜

⬘

共q兲兴 on large q exists if for large q the asymptotic

W˜

⬘

共q兲, and W˜

⬙

共q兲 possess a power-type q-dependence for a

large q.

As an example, this dependence can be taken as the power type for two functions W共q兲⬅a/qa _{and W}_˜

_⬘

_共q兲

⬅b/q␤_{, where}_␣_{⬎0 and}_␤_{are independent. Then, as follows} from the consideration above, convergence of the function W exists if 5⬎␣⬎3 共for asymptotically small and large q, re-spectively兲. For the function W˜

_⬘

_{共q兲 the convergence}

condi-tion is 5⬎␤⬎2 for asymptotically small and large q, respec-tively.

Finally for the function W˜

⬙

共q兲 the convergence condition is 7⬎␩⬎5 共for asymptotically small and large q, respec-tively兲. In this case the terms with W

⬙

can be omitted共for the same reasons as above兲.

For this example the kinetic equation共11兲reads

df共s,t兲 dt = P0s ␣−3_f_{共s,t兲 + s}␤−5_P 1si ⳵ ⳵sif共s,t兲, 共46兲 where P₀= 4␲a

冕

0 ⬁ d␨␨2−␣

冋

sin␨ ␨ − 1

册

, 共47兲

(8)

P1= 4␲b

冕

0

⬁

d␨␨2−␤

冋

cos␨−sin␨

␨

册

. 共48兲

Taking into account the isotropy in s-space we can rewrite Eq.共46兲in the form

df共s,t兲 dt = P0s ␣−3_f_{共s,t兲 + s}␤−4_P 1 ⳵ ⳵sf共s,t兲, 共49兲

Naturally, Eqs. 共46兲 and 共49兲 can be formally rewritten in momentum 共or in velocity兲 space via the fractional deriva-tives of various orders共see below兲. Therefore, as is easy to see, for the purely power-law behavior of the functions W共q兲 and W˜

⬘

共q兲 the solution with the convergent coefficients ex-ists for powers in the intervals mentioned above. We estab-lish that the universal type of anomalous diffusion in velocity space in the case under consideration exists if 5⬎␣⬎3 or 5⬎␤⬎2. This universality seems similar to the universality of the Levy distribution in coordinate space, where the power␣ of the dependence of the PTF in coordinate space,

W共q˜兲⬃C/q˜␣ on the displacement q˜ lies in the interval 0

⬍␮=␣− r⬍2 共r is the dimension of the coordinate space兲. As is easy to see the stationary solution of Eq.共49兲reads

f共s兲 = C

⬘

exp

冋

− P0s ␣−␤+2

共␣−␤+ 2兲P1

册

. 共50兲

Of course, the general description above is also valid for the more complicated functions W and W˜

⬘

, possessing a non-power dependence on q at small q and an asymptotical power dependence on q at large q. In this case the limitations for convergence are connected only with large values of q, namely, it is enough to provide the inequalities ␣⬎3 and

␤⬎2. Simple examples of such type of PTFs are 共in analogy with anomalous diffusion in coordinate space13兲

Wa共q兲 =1 − exp共−␥q n_兲 q␣ , W ˜ a

⬘

共q兲 =1 − exp共−␦q m_兲 q␤ . 共51兲

The corresponding kinetic equations in these cases cannot be written in partial derivatives and evolution of the system has to be described by Eq.共11兲, or for the isotropic case by Eq.

共14兲. If external forces are present, they have to be included in the usual way in the left side of Eq.共11兲. Physically, this type of PTF behavior can appear, in particular, for the case of a turbulent plasma, when the development of some instabil-ity can create a strong chaotic electrical field or irregular chaotic motion of one sort of particles with a prescribed non-Maxvellian distribution function. In such a turbulent plasma scattering with large transferring momenta can play a crucial role.

V. REPRESENTATION IN MOMENTUM SPACE AND CONNECTION WITH THE FRACTIONAL DIFFERENTIATION APPROACH

As mentioned, in general Eq. 共11兲 cannot be written in terms of fractional differentiation. It confirms that the ap-proach to anomalous diffusion based on the Fourier

transfor-mation of the PTFs, in the form applied in this paper 共see also Refs.13and19兲 is a more general way for the problems

under consideration.

However, for the purely power-type dependence of the functions W共q兲⬅a/q␣_{and W}˜

_⬘

_{共q兲⬅b/q}␤_{, where}_␣_and_␤_are independent and satisfy the inequalities 5⬎␣⬎3 and 5⬎␤ ⬎2, Eq. 共46兲 is appropriate and can be represented after inverse Fourier transformation in the following form 共with the fractional derivatives兲:

df共p,t兲 dt = P0⌬

␯_f_{共p,t兲 + P1⌬}␭

冉

_{␭ +} ⳵

⳵pp

冊

f共p,t兲, 共52兲 where␯⬅共␣− 3兲共2⬎␯⬎0兲; ␭=␤− 5 共2⬎␭⬎0兲. Here we introduced the fractional differentiation operator in the mo-mentum space⌬␭f共p,t兲⬅兰dss2␭exp共−ips兲f共s,t兲.

Let us consider now formally a specific particular model of anomalous diffusion, for which we assume a structure of the PTF W共q,p兲 with a rapid 共say, exponential兲 decrease in the function W˜

⬘

共q兲. Therefore, the exponential function un-der the integrals in the coefficients B共s兲 can be expanded, implying B共s兲=B0 关or ␤= 5 and B0⬅ P1 in the notations of Eq.共46兲兴. At the same time the function W共q兲⬅a/q␣ has a purely power-law dependence on q. The kinetic equation

共49兲then reads

df共s,t兲 dt = P0s

␣−3_f_{共s,t兲 + B0}_si ⳵

⳵sif共s,t兲 共53兲

or in momentum space, according to Eq.共52兲,

df共p,t兲 dt = P0⌬

␯_f_{共p,t兲 − B0} ⳵

⳵pi

关pif共p,t兲兴. 共54兲

Equation 共54兲 is similar to the corresponding equation in Ref.22where a model of the Langevin equation with a con-stant friction frequency ␯₀⬅−B₀ has been considered. In Ref. 23 a similar model with ␯= 3/2 has been applied to estimate the fusion rate in a hot rarified plasma.

The stationary solution of Eq.共53兲is

f共s兲 = C

⬘

exp

冋

− P0s ␯

␯B0

册

. 共55兲

The corresponding distribution in p-space is propor-tional to the Levy-type distribution W共y,␣

⬘

兲

⬅共2y/␲兲兰0⬁dtt sin共yt兲exp共−t␣⬘兲,

f共p兲 = C

⬘

冕

d3s exp共− ips兲exp

冋

−P0s ␯ ␯B0

册

⬅4␲C

⬘

p

冕

0 ⬁ dss sin共ps兲exp

冋

− P0s ␯ ␯B0

册

, 共56兲

with y⬅p共␯B0/ P0兲1/␯and␣

⬘

⬅␯. As an example for the case

␯= 1 we find f共p兲,

f共p兲 = 8␲C

⬘

P0

B0关共p2+ 4P₀2/B₀2兲兴2. 共57兲

In the case␯= 1 the long tail of the distribution is propor-tional to p−4 and the distribution f共p兲 corresponds with the Cauchy–Lorentz distribution. Normalization of the

(9)

distribu-tion f共p兲 leads to the value C

⬘

= n/共2␲兲3_{, where n = N}_{/V is} the average density of particles undergoing diffusion in ve-locity space. A similar approach can be taken for other types of anomalous diffusion in velocity space.

VI. CONCLUSIONS

In the present paper the problem of anomalous diffusion for plasmalike systems in momentum共velocity兲 space is rig-orously analyzed. A new kinetic equation for anomalous dif-fusion in velocity space has been derived recently in Ref.19, without suggesting any stationary equilibrium distribution function. We applied this equation to a system of charged particles with different masses to describe diffusion of heavy particles共ions and charged grains兲 in the surrounding light particles 共electrons for the electron-ion plasma, electrons, and ions for dusty plasmas28兲. The distribution of the light particles can be non-Maxwellian, which is the cause of the appearance of long tails in the PTF. Conditions of conver-gence for the coefficients of the kinetic equation have been derived for a number of particular cases. It is found that a wide variety of anomalous processes in velocity space exists. In general the Einstein relation for such a situation is not applicable because the stationary state may be far from equi-librium. For the case of normal diffusion the friction and diffusion coefficients have been found explicitly for the non-equilibrium case. For the non-equilibrium case the known Fokker–Planck equation in plasma is reproduced as a par-ticular case.

ACKNOWLEDGMENTS

The authors are thankful to A. M. Ignatov and A. G. Zagorodny for the valuable discussions of some problems, reflected in this paper. S.A.T. would like to thank the Neth-erlands Organization for Scientific Research共NWO兲 for sup-port of his investigations on the problems of stochastic trans-port in gases, liquids, and plasmas.

1_{A. S. Monin and A. M. Yaglom, Statistical Fluid Mechanics: Mechanics}

of Turbulence共MIT Press, Cambridge, 1975兲, Vol. II, p. 77.

2_{B. Rinn, P. Maass, and J.-P. Bouchaud,}_{Phys. Rev. Lett.} _{84, 5403}_共2000兲. 3_{R. Metzler and J. Klafter,}_{Phys. Rep.} _{339, 1}_共2000兲.

4_{H. Scher and E. W. Montroll,}_{Phys. Rev. B} _{12, 2455}_共1975兲.

5_{J. Klafter, M. F. Schlezinger, and G. Zumofen,}_{Phys. Today} ₄₉_{共2兲, 33}

共1996兲.

6_{A. Ott, J.-P. Bouchaud, D. Langevin, and W. Urbach,}_{Phys. Rev. Lett.} _65,

2201共1990兲.

7_{I. M. Sokolov, J. May, and A. Blumen,}_{Phys. Rev. Lett.} _{79, 857}_共1997兲. 8_{P. Gopikrishnan, V. Plerou, L. A. N. Amaral, M. Meyer, and H. E. Stanley,}

Phys. Rev. E 60, 5305共1999兲.

9_{B. J. West, M. Bologna, and P. Grigolini, Physics of Fractal Operators}

共Springer-Verlag, New York, 2003兲, p. 194.

10_{H. C. Fogedby,}_{Phys. Rev. Lett.} _{73, 2517}_共1994兲.

11_{S. Jespersen, R. Metzler, and H. C. Fogedby,} _{Phys. Rev. E} _{59, 2736}

共1999兲.

12_{S. A. Trigger, G. J. F. van Heijst, and P. P. J. M. Schram,}_{Physica A} _347,

77共2005兲.

13_{S. A. Trigger, G. J. F. van Heijst, and P. P. J. M. Schram,}_{J. Phys.: Conf.}

Ser. 11, 37共2005兲.

14_{S. A. Trigger,}_{Phys. Lett. A} _{372, 1288}_共2008兲. 15_{L. D. Landau, Zh. Eksp. Teor. Fiz. 7, 203}_共1937兲. 16_{O. Klein, Ark. Mat., Astron, Fys, 16, 1}_共1921兲. 17_{H. A. Kramers,}_Physica_{共Amsterdam兲} _{7, 284}_共1940兲.

18_{B. A. Carreras, D. Newman, V. E. Lynch, and P. H. Diamond, Plasma}

Phys. Rep. 22, 740共1996兲;Phys. Plasmas 3, 2903共1996兲; L. W. Owen,

B. A. Carreras, R. Maingi, P. K. Mioduszewski, T. N. Carlstrom, and R. J. Groebner,Plasma Phys. Controlled Fusion 40, 717共1998兲.

19_{S. A. Trigger,}_{Phys. Lett. A} _{374, 134}_共2009兲. 20_{E. Barkai,}_{Phys. Rev. E} _{68, 055104}_{共R兲共2003兲.} 21_{E. Barkai,}_{J. Stat. Phys.} _{115, 1537}_共2004兲.

22_{A. V. Chechkin, V. Yu. Gonchar, and M. Szydlowski,}_{Phys. Plasmas} _{9, 78}

共2002兲.

23_{W. Ebeling and M. Yu. Romanovsky, Contrib. Plasma Phys. 49, 479}

共2009兲.

24_{J. L. Lebowitz and P. G. Bergmann,}_{Ann. Phys.}_共N.Y.兲 _{1, 1}_共1957兲. 25_{N. G. van Kampen, Stochastic Processes in Physics and Chemistry}

共North-Holland, Amsterdam, 1992兲, p. 83.

26_{E. M. Lifshitz and L. P. Pitaevsky, Physical Kinetics}_{共Nauka, Moscow,}

1979兲, p. 116 共in Russian兲.

27_{S. I. Braginskii, The Problems of Plasma Theory}_{共Gosatomizdat, Moscow,}

1963兲, Vol. 1, p. 197 共 in Russian兲.

28_{V. E. Fortov, A. V. Ivlev, S. A. Khrapak, A. G. Khrapak, and G. E. Morfill,}

Phys. Rep. 421, 1共2005兲.