On the Anomalous Diffusion in Nonisothermal plasma A. A. Rukhadze1,a) and S. P. Sadykova2,b) 1)Prokhorov General Physics Institute, Russian Academy of Sciences, Vavilov Str., 38., Moscow, 119991, Russia 2)Humboldt-Universita¨t zu Berlin, Berlin, 12489, Germany, Newtonstr. 15 (Dated: 6 January 2012) In nonisothermal plasma at temperature T >> T diffusion plays decisive role at conditions of smooth e i inhomogeneity when the inhomogeneity size is larger than the Debye radius by more than T /T times. e i When the inhomogeneity is rather abrupt and the condition is violated, then during the sprpeading process 2 theMaxwellianrelaxationofionchargesbecomessignificant. Here,weconsiderthesetwophenomenatogether 1 and refer to the anomalous character of diffusion, i.e. anomalous diffusion. 0 2 PACS numbers: 52.25.Fi, 52.25.-b n Keywords: nonisothermal plasma, Maxwellian relaxation, ambipolar diffusion, anomalous diffusion a J 5 I. INTRODUCTION II. ANOMALOUS DIFFUSION MODEL ] h One can talk of plasma as a continous medium when First of all, we will consider the main equations of the p its size (and the inhomogenuity size) exceeds signifi- studied phenomenon and its constraints. - m cantly the Debye radius. In a case of nonisothermal The main constraints for the considered below equa- plasma T >> T Debye radius is determined through tions are: s e i a the ion temperature and is equal to vTi/ωLi, where ων >>ω2;k2v2 >>ων ;ν >>kv ,ω;ν >>kv ,ω l v = T /M is the ion thermal velocity and ω = i Te e e Te i Ti p Ti i Li (2) cs. Zpe4πbqeii2npngi0t/hMe ioisntchheariogne,Lnain0g-meuqruiflrleibqruieunmcyiownitdhenqsiit=y fHreeqreueνnec,iesν,i vare =electrTon/-mato-meleacntrdoniotnh-aertmomal cvoellloisciiotny i (en =q n ), here e - electron charge. Here and below Te e s e0 i i0 with m being the elepctron mass. Nonisothermal plasma y we use the CGS unit system. We write for general case can be only the weakly ionized plasma, in which the h but further on we present the results of our calculations dominant collisions are chargedparticle-atom, -molecule p only for the Hydrogen plasma Z=1. collisions, ω ∝ 1/τ, where τ is the characteristic time [ The diffusion process is characterized through the of spreading and k ∝ 1/L, where L is the characteristic 1 mean square displacement dependence on time: size of spreading. v 4 r2n(~r,t)d~r = ∼tγ, (1) 7 R n(~r,t)d~r 1 Equation of motion of electrons and ions has the 2 R following view: 1 If γ = 1 then diffusion is termed as normal, in all other 1. cases one refers to the anomalous character of diffusion: q E~ T i −ν ~v =0, q E~ − e∆n =0, (3) 0 γ >1 - superdiffusion, γ <1- subdiffusion. M i i e n e e 2 When the size of innhomogeneity of ionic component 1 perturbation is larger than the Debye radius by more Here qe = e, qi = Ze are the electron and ion v: than Te/Ti times, then spreading of the ionic compo- charges, Z ≤ |qi/q|, E~ are the self-consistent field, Xi nent pperturbation has a diffusive character and is de- ~vi, ~ve are ion and electron velocities, Ti = 0, fined by the time of the normal ambipolar diffusion1. Te = const, ni, ne are the ion and electron den- ar If it is less then the Maxwellian relaxation of ionic sities. charges becomes dominant during the spreading of 2 Field equation (Poisson equation) inhomogenuity2,3. We will introduce a reader below into the theory of divE~ =4πe(n −Zn ) (4) e i spreading of perturbation in nonisothermal plasma tak- ing into account both mentioned phenomena and show 3 Continuity equation of the ion component the anomalous character of diffusion. ∂n i +divn~v =0 (5) i i ∂t a)Electronic mail: rukh@fpl.gpi.ru; A. A. Rukhadze and S. P. 4 Equilibrium state satisfying the stationary equa- Sadykova contributedequallytothiswork. tions (3)-(5) at E~ =0 takes the following view: 0 b)Electronic mail: Corresponding author - saltanat@physik.hu- berlin.de. q n =q n =Zen =Const (6) e e0 i i0 i0 2 Let’s consider the small density perturbations : This solution corresponds to the Maxwellian tem- porary relaxation of the charge density perturba- n =n +δn , n =n +δn , (7) e e0 e i i0 i tion. where nα0 - denity of the homogenious back- B . In a case of dense plasma, when ω 2 >>k2v 2: Li S ground. These small density perturbations lead totheemergenceoftheweakself-consistentfieldE~. k2v 2 δn (~r,t)= d~kf(~k)exp i~k·~r−t S . (15) i Z (cid:18) ν (cid:19) i System of equations (3)-(5) at (6) for small perturba- The integral can be easily solved at Eq. (9), when tions(7)canbededucedtooneequationforperturbation f(~r) = N δ(~r) with the corresponding Fourier i0 δn . The rest magnitudes can be expressed through δn . i i transformf(~k)= Ni0 ,whereN isthetotalnum- Equation for δn has the following view: (2π)3 i0 i ber of perturbed particles at r = 0. The result is ∂δn v 2 well known4: (ω 2−v 2∆) i − S ω 2∆δn =0. (8) Li S Li i ∂t ν i N δn (~r,t)= i0 exp −r2/4Dt , (16) Hereω istheequlibriumLangmuirionfrequency,v 2 = i 8(πDt)3/2 Li S (cid:0) (cid:1) ZT /M isthesquaredionsoundvelocityandν =Const. e i whereD =v 2/ν isthecoefficientofambipolardi- Eq. (8) representsthe partialdifferentialequationofthe S i fusion and Eq. (16) describes the diffusive spread- first order with the constant coefficients. ing of the localized ion density perturbation (δ - Let us formulate the Koshi problem for Eq. (8), i.e. functional). we consider infinite plasma and set the following initial Let us determine the < r2(t) > Eq. (1) for the condition: ambipolar diffusion (16): δn (~r,0)=f(~r), (9) i =6Dt, (17) and take into account that limr→∞δni = 0. Using the Fourier transform here in Eq. (1) we used δni Eq. (18). As one can see this corresponds to the normal diffusion. δn (~r,t)= d~kδn (~k,t)exp(i~k·~r), i Z i Letuschoosethefollowingtwoinitialperturbationsf(~r), (10) 1 the first as representing the infinite wave: f(~k)= d~rf(~r)exp(−i~k·~r), (2π)3 Z f(~r)=n cos(~r·~k ) (18) i0 0 Eq. (8) will be reduced to the ordinary differential equation of the first time order. with the Fourier transform f(~k) = δ(~k−~k0)+δ(~k+~k0), here 2 ~k is the wave vector; and the second as repesenting the ∂δn (~k,t) k2ω 2v 2/ν 0 i =− Li S iδn (~k,t). (11) localized ion density perurbation mentioned above: ∂t ω 2+k2v 2 i Li S f(~r)=N δ(~r) (19) We obtain the solution of Eq. (11) with the initial con- i0 dition with the Fourier transform f(~k) = Ni0 , what will be (2π)3 δn (~k,0)=f(~k), discussedlater. We wouldlik to startwith the condition i (18). as following: Then, the equation (14) corresponding to the Maxwell k2ω 2v 2/ν relaxation will turn into δn (~k,t)=f(~k)exp −t Li S i . (12) i (cid:18) ω 2+k2v 2(cid:19) ω2 t Li S δn (~r,t)=n exp − Li cos(~r·~k ) i i0 0 (cid:18) ν (cid:19) Making the inverse Fourier transform (10), we can find i k2ω 2v 2/ν or in dimensionless form δn (~r,t)= d~kf(~k)exp i~k·~r−t Li S i . i Z (cid:18) ωLi2+k2vS2(cid:19) δni(R~,T) =exp −ωLiT cos(R~ ·~e ), (20) (13) 0 n (cid:18) ν (cid:19) i0 i Let us analyse the limitting cases of Eq. (13). here , ~k r = ~e , R = r/r , T = tω - dimensionless A . In a case of rare plasma, when ω →0: 0 D 0 D Li Li time. ω 2 The equation(15)correspondingto the ambipolardif- δn (~r,t)= d~kf(~k)exp i~k·~r−t Li i Z (cid:18) νi (cid:19) fusion will turn into ω 2 k2v2t =f(~r)exp −t Li . (14) δn (~r,t)=n exp − 0 S cos(~r·~k ) i i0 0 (cid:18) ν (cid:19) (cid:18) ν (cid:19) i i 3 or in dimensionless form As one can easily see the behaviour of the Maxwellian relaxation Eq. (20), ambipolar diffusive relaxation Eq. δn (R~,T) k2DT i =exp − 0 cos(R~ ·~e ), (21) (21), and the anomalous diffusion Eq. (16) is oscillatory 0 ni0 (cid:18) ωLi (cid:19) and fades down with the time as the exp(−αt)cos(βR) function. In a case for the localized ion density pertur- here~k0 is the same wave vector. bation relaxationEq. (16) the perturbations fades down Finally, the equation for anomalous diffusion (13) will as the exp(−γR2/t)/t3/2. become At the given oscillatory initial perturbation (18) no normal diffusion occurs: the initial perturbation gets k2Dω 2v 2 δn (~r,t)=n exp −t 0 Li S cos(~r·~k ). faded down with the time evolution keeping its initial i i0 (cid:18) ω 2+k2v 2(cid:19) 0 Li 0 S spatial functional form - cos(βR), i.e. only the amplitude decreases, and in this way revealing the anomalous or in dimensionless form nature of the diffusion in nonisothermal plasma. This δn (R~,T) k2Dω v 2 behaviour is peculiar for the Maxwellian relaxation (14) i =exp −t 0 Li S cos(R~ ·~e ). (22) n (cid:18) ω 2+k2v 2(cid:19) 0 independent on the type of initial condition. However, i0 Li 0 S whenwetakeanotherinitialcondition(δ(~r)-functional) As one can easily see in all cases, Eqs. (20)-(22), the forthecaseofdiffusiverelaxation(15)thenweobtainthe functionsgetseparatedbyvariablesT andR. Asaresult normal ambipolar diffusion (16). we can not define the mean square displacement. This In the Figures 2, 3, 4, and 5 comparison between means that at the given oscillatory initial perturbaion the obtained relative density perturbation relaxation for (18) no normal diffusion occurs: the initial perturbation anomalous diffusion Eq. (23) with δ(~r) as initial per- gets faded down with the time evolution keeping its ini- turbation (19) and normal ambipolar diffusion Eq. (16) tial functional form in space - cos(βR), i.e. only the and corresponding variances at the different ion Lang- amplitude decreases. This behaviour is peculiar for the muir frequencies and different time moments are shown. Maxwellian relaxation (14) independent on the type of In the Figs. 2, 3 the difference between the obtained initial condition. However, when we take another ini- curves for anomalous diffusion and the normal ambiplar tial condition (δ(~r) - functional) for the case of diffusive diffusion is quite significant, whereas with an increase of relaxation(15)thenweobtainthenormalambipolardif- eitherionLangmuirfrequencyortimetheanomalousdif- fusion (16). fusion gets to turn into the normal ambipolar diffusion. Finally, let us find the solution for the anomalous dif- This is also demonstrated in the correspoding Figs. d) fusion Eq. (13) with the second initial condition Eq. wheredependenceofthevarianceontimeandthefitting (19) as representing the localized ion charge perturba- curvesfor the correspondingdifferentLangmuirfrequen- tion. Havingsolvedthisequationandhavingmadesome ciesisshown. ThevariancewasestimatedusingEq. (1). simplifications, we have got the following: As we can see at the lowest frequency there is almost no diffusion Fig. 2 d), the variance does not depend on time. However, with further increase of the ion fre- δni(r,t) = rD3 d~kexp i~k·~r−tk2ωLi2vS2/νi quency the diffusion becomes anomalous < r2(t) >=6 αt Ni0/rD3 (2π)3 Z (cid:18) ωLi2+k2vS2(cid:19) (α=Const), whereas when the frequency and time mo- r3 10ωLi/vs sin(kr) ments increase the diffusion starts to turn into the nor- = D dkf(k)k mal: =αt (17). 2π2 Z r 0 k2v2ω2 /ν ω2 ×exp −t s Li i +δ(~r)r3 exp −t Li (cid:18) ω2 +k2v2(cid:19) D (cid:18) ν (cid:19) Li s i ACKNOWLEDGMENTS r3 ω2 10ωLi/vs sin(kr) − D exp −t Li dkk (23) 2π2 (cid:18) νi (cid:19)Z0 r One of the authors S.P. Sadykova would like to thank hercoauthorA.A.Rukhadzeforhisfruitfulcontribution to the work and discussions on the presented topic. S.P. III. RESULTS AND DISCUSSIONS Sadykovawouldliketoexpresshergratitudetoherfather P. S. Sadykov for his financial support of the work and In the Figure 1 a) the 3 dimensional view of the rela- for being all the way the great moral support for her. tive value ofdensity perturbationin dependence ontime T and absolute value of the radius vector R is shown. 1A.F. Alexandrov, L.S. Bogdankevich, A.A.Rukhadze, in Prici- Namely, Fig. 1 a) represents the anomalous relaxation ples of Plasma Elektrodynamics, Springer Series in Electro- as the combined effect Eq. (22) of the Maxwellian re- physics,Volume9,editedbyG.Ecker(Ruhr-UniversittBochum) laxation Eq. (20) and ambipolar diffusive relaxationEq. (Springer-Verlag,Berlin,Heidelberg,NewYork,Tokio,1984). 2M.V.Kuzelev A.A,Rukhadze, inBasic of Plasma Free Elektron (21). In Fig. 1 b) the comparison among the mentioned Lasers (Editionfrontier,Paris,1995). relaxationsandthelocalizediondensityperturbationre- 3M.V.KuzelevA.A,Rukhadze, inMethods of wave theory in dis- laxation Eq. (16) at t=1/ωLi is shown. persive media (WorldScienticPublishing,Singapore,2010) 4 1,0 T=1 0,8 0,6 ni0 T)/ 0,4 R, n(i 0,2 0,0 -0,2 -0,4 Initial perturb. Eq. (16) -0,6 Anomalous diff. Maxwell relax. -0,8 Diffusive relax. -1,0 0 2 4 6 8 R=r/rD a) b) FIG. 1. a) The anomalous diffusion Eq. (22) and b) comparison among the anomalous, Maxwellian relaxation Eq. (20), ambipolar diffusive relaxation Eq. (21) and the localized ion density perturbation relaxation Eq. (16) at t=1/ωLi is shown. Here,~e0||R~,Te=10eV,ni0 =1012cm−3,ωLi=1.3·109s−1,νi=1.2·109s−1,n0 =3·1016cm−3(atomdensity),rD =2.3·10−5m. 4L.D.Landau,E.M.Lufshitz,inHydrodynamics (Nauka,Moscow, 1982) 5 0 1 2 3 4 0 1 2 3 4 -4 1,6x10 -5 8,0x10 3n(r,t)/(N/r), anom. diff.ii0D468000000000 AAnmobmipaololaurs d dififf.f .Eq. (16) 34567,,,,,00000xxxxx1111100000-----55555 ii0Dn(r,t)/(N/r), anom. d 3 3n(r,t)/(N/r), anom. diff.ii0D23000000 68111,,,,,00024xxxxx1111100000-----55444 ii0Dn(r,t)/(N/r), anom. d 3 2,0x10-5 iff. 1000 4,0x10-5 iff. 2000 t=1/i 1,0x10-5 t=2/i 2,0x10-5 0 0,0 0 0,0 0,00 0,02 0,04 0,06 0,08 0,10 0,12 0,14 0,00 0,02 0,04 0,06 0,08 0,10 0,12 0,14 R=r/rD R=r/rD a) b) 0,50 0 1 2 3 4 2000 2,4x10-4 0,45 Anomalous diff. 0,40 3n(r,t)/(N/r), anom. diff.ii0D11268000000 Ambipolar diff. Eq. (16) 8112,,,,0260xxxx11110000----5444 ii0Dn(r,t)/(N/r), anom. d 3 2 00000,,,,,1223350505 iff. 0,10 400 4,0x10-5 Fitting, =0.4141+2.43(cid:215)10-5T 0,05 t=3/i Anomalous diff. 0 0,0 0,00 0,00 0,02 0,04 0,06 0,08 0,10 0,12 0,14 0 2 4 6 8 10 12 R=r/rD T= ti c) d) FIG. 2. Comparison between the obtained the relative density perturbation relaxation for anomalous diffusion Eq. (23) and normal ambipolar diffusion Eq. (16) at the different time moments T =tνi but fixed Langmuir frequency ωLi =1.6·107 s−1, ni0 =1.5·108 cm−3 and its variance Eq. (1) together with thevariance fit d) are shown. Here Te =10 eV, νi =1.2·109 s−1, n =3·1016 cm−3 (atom density). 0 6 -2 -1 -3 1,75x10 1,8x10 7,0x10 3), anom. diff.n(r,t)/(N/rii0D571112,,,,,,050255000500xxxxxx111111000000------333222 AAnmobmipaololaurs d dififf.tf =.E1q/. i(16) 4681111,,,,,,,0000246xxxxxxx11111110000000--------22221111iiDn(r,t)/(N/r), ambipol. diff.3 3n(r,t)/(N/r), ambipol. diff.ii0D123456,,,,,,000000xxxxxx111111000000------333333 AAnmobmipaololaurs d dififf.f .tE=q2./ (i16) 24681,,,,,00000xxxxx1111100000-----22221ii0Dn(r,t)/(N/r), anom. diff.3 2,0x10 0,00 0,0 0,0 0,0 0 1 2 3 4 5 6 0 1 2 3 4 5 6 R=r/rD R=r/rD a) b) -3 3,5x10 Anomalous diff. 5,0x10-2 7 Fitting, =0,64116-0,24226 T+0,25896 T2 diff. 3,0x10-3 Ambipolar diff. Eq. (16) 6 Anomalous diff. 3n(r,t)/(N/r), ambipol. ii0D1122,,,,0505xxxx11110000----3333 t=3/i 234,,,000xxx111000----2222ii0Dn(r,t)/(N/r), anom. diff.3 2 2345 -4 1,0x10 5,0x10 1 0,0 0,0 0 0 1 2 3 4 5 6 0 1 2 3 4 5 6 R=r/rD T= ti c) d) FIG. 3. Comparison between the obtained the relative density perturbation relaxation for anomalous diffusion Eq. (23) and normal ambipolar diffusion Eq. (16) at the different time moments T =tνi but fixed Langmuir frequency ωLi =1.3·109s−1, ni0 = 1012 cm−3 and its variance Eq. (1) together with the variance fit d) are shown. The Hydrogen plasma parameters are thesame as in theFig. 2. 7 -3 -3 1,2x10 5,0x10 Anomalous diff. 5,0x10-4 Anomalous diff. 3n(r,t)/(N/r), ambipol. diff.ii0D4681,,,,0000xxxx11110000----4443 Ambipolar diff. Eq. (16) 234,,,000xxx111000---333ii0D), anom. dn(r,t)/(N/r3 3 n(r,t)/(N/r)ii0D234,,,000xxx111000---444 Ambipolar diff. Eq. (16) 2,0x10-4 t=1/i 1,0x10-3iff. 1,0x10-4 t=2/i 0,0 0,0 0,0 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 R=r/rD R=r/rD a) b) 80 -4 2,5x10 Anomalous diff. 2 2 Fitting, =-5,65731+15,44451 T-0,05944 T Ambipolar diff. Eq. (16) 70 Anomalous diff. -4 3 n(r,t)/(N/r)ii0D12,,50xx1100-4 2 456000 -4 1,0x10 30 5,0x10-5 t=3/i 20 10 0,0 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 0 1 2 3 4 5 R=r/rD T =ti c) d) FIG. 4. Comparison between the obtained the relative density perturbation relaxation for anomalous diffusion Eq. (23) and normal ambipolar diffusion Eq. (16) at the different time moments T =tνi but fixed Langmuir frequency ωLi =3.3·109 s−1, ni0 =6.25·1012 cm−3 and itsvariance Eq. (1) together with thevariance fit d) are shown. The Hydrogen plasma parameters are thesame as in theFig. 2. 8 -5 1,6x10 4,0x10-5 Anomalous diff. -5 Anomalous diff. Ambipolar diff. Eq. (16) 1,4x10 Ambipolar diff. Eq. (16) -5 3 n(r,t)/(N/r)ii0D23,,00xx1100--55 3 n(r,t)/(N/r)ii0D811,,,002xxx111000--65 -6 6,0x10 1,0x10-5 t=1/i 4,0x10-6 t=2/i -6 2,0x10 0,0 0,0 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 0 4 8 12 16 20 24 28 32 36 R=r/rD R=r/rD a) b) 550 8,0x10-6 Anomalous diff. 500 Fitting, = -3,898+133,1T Ambipolar diff. Eq. (16) 450 Anomalous diff. 3 n(r,t)/(N/r)ii0D46,,00xx1100--66 2 233450500000 200 2,0x10-6 t=3/i 150 100 50 0,0 0 0 4 8 12 16 20 24 28 32 36 40 44 0 1 2 3 4 R=r/rD T=ti c) d) FIG. 5. Comparison between the obtained the relative density perturbation relaxation for anomalous diffusion Eq. (23) and normal ambipolar diffusion Eq. (16) at the different time moments T =tνi but fixed Langmuir frequency ωLi =9.7·109 s−1, ni0 =5.6·1013 cm−3 and its variance Eq. (1) together with the variance fit d) are shown. The Hydrogen plasma parameters are thesame as in theFig. 2

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