Study of Discrete-Particle Effects in a One-Dimensional Plasma Simulation with the Krook Type Collision Model

Transcription

1 Study of Disrete-Partile Effets in a One-Dimensional Plasma Simulation with the Kroo Type Collision Model Po-Yen Lai 1, Liu Chen 2, 3, Y. R. Lin-Liu 1, 4 and Shih-Hung Chen 1, 4, * 1 Department of Physis, National Central University, Jhongli 32001, Taiwan 2 Department of Physis and Astronomy, University of California, Irvine, CA 92697, USA 3 Institute for Fusion Theory and Simulation, Zhejiang University, Hangzhou, China 4 Center for Mathematis and Theoretial Physis, National Central University, Jhongli 32001, Taiwan ABSTRACT The thermal relaxation time of a one-dimensional plasma has been demonstrated to sale with N 2 D due to disrete partile effets by ollisionless partile-in-ell simulations, where N D is the partile number in a Debye length. The N 2 D saling is onsistent with the theoretial analysis based on the Balesu-Lenard-Landau ineti equation. However, it was found that the thermal relaxation time is anomalously shortened to sale with N D while externally introduing the Kroo type ollision model in the one-dimensional eletrostati PIC simulation. In order to understand the disrete partile effets enhaned by the Kroo type ollision model, the superposition priniple of dressed test partiles was applied to derive the modified Balesu-Lenard-Landau ineti equation. The theoretial results are shown to be in good agreement with the simulation results when the ollisional effets dominate the plasma system. PACS numbers: Ln, Fs, Kn, y * Eletroni address: hensh@nu.edu.tw 1

2 I. INTRODUCTION The partile simulation [1] - [2] has been widely used to study the ompliated nature of plasmas. This numerial sheme was suessfully applied to effiiently simulate the olletive behaviors of ollisionless plasmas using only few maro partiles instead of a large number of real partiles. [3] - [5] However, the dereasing number of maro partiles in a Debye ubi (N D ) enhanes the disreteness and graininess of the partile distribution, whih indue flutuating eletri fields in the plasma system. The disrete-partile effets also fore the plasma system to evolve from an arbitrary initial distribution to a Maxwellian within a finite time period in a typial partile simulation. The time period was thus defined as the relaxation time or the numerial thermalization time ( R ) in previous studies. [1], [6] - [7] The numerial thermalization due to disrete-partile effets essentially affets the auray and onvergene of a partile simulation. Dawson [1], [8] was the first researher to show the saling of R with N D, i.e., R N 2 D, using a single-speies one-dimensional (1D) eletrostati (ES) sheet model, where N D n 0 λ D is the number of partiles per Debye length (denoted as λ D ) and n 0 is the orresponding number density of partiles. The numerial thermalization time R is independent of the first order of the plasma parameter N D beause that the evolution of the partile distribution obeys the Lenard-Balesu equation in a 1D model. [9], [10] The saling law R N D with N D n 0 λ 2 D was demonstrated by a two-dimensional (2D) ES partile-in-ell (PIC) simulation [11] and the theoretial analysis using Landau ineti equation with the test-partile approximation. [12] For a three-dimensional (3D) ES PIC model, the saling of the thermalization time with N D, i.e., R N D /lnn D is expeted with N D n 0 λ 3 D. [11] The PIC simulation has been widely used to simulate more pratial plasma systems with wea partile ollisions. [6], [7], [13] - [21] The Monte-Carlo method is applied in the PIC simulation for additional onsiderations of partile ollisions in plasmas. It was ommonly supposed that the external addition of wea ollisions into the PIC simulation do not onsiderably hange the auray and the onvergene of the simulation results, so the numerial parameters an be safely hosen within the limits of typial numerial riteria. [6] However, Tuner observed that R an be quantitatively redued when the Monte-Carlo method are externally added into 1D ES PIC simulations [6]. The breadown of N 2 D saling of R was demonstrated in his simulation wor using a simplified head-on ollision model while varying some ey numerial parameters. Aording to Turner s simulation results, R is proportional to 2

3 N 2 D only when N D 0 / p <<1, otherwise, R is saled with N D. Here 0 is the effetive ollision rate and p is the plasma frequeny. A following wor proposed a more general saling law of R with a new sale parameter N 2 D 0 p on the basis of 1D ES PIC simulations with pratial ollision models. [7] Although the thermalization proesses of plasmas are more ompliated than Turner s observation due to the onversion between the longitudinal and transverse momentum, the simulation results still onform to Turner s N D saling law when N 2 D 0 p 1. The literatures about the theoretial analysis of disrete-partile effets in the PIC simulation are very rare. Yoo and Abraham-Shrauner analyzed the numerial thermalization of a 2D ollisionless ES plasma due to disrete-partile effets using the Landau ineti equation with the test-partile approximation, and their theoretial results are onsistent with the PIC simulation results. [12] The wor was then extended to study 2D ollisionless magnetized plasmas by Hsu et al. [22] However, the issue remains of great interests to theoretially explore how the enhaned disrete-partile effets hange the saling of R in a ollisional 1D PIC simulation. In this study, 1D ES PIC simulations were performed to examine the saling of the numerial thermalization time with N D for the thermalization purely indued by the disrete partile effets and that enhaned by the Kroo type ollisions [23], respetively. The disrete partile effets in the plasma system was then analyzed by the theory derived from the Klimontovih equation with the onsideration of the Kroo type ollision operator. [24] The superposition priniple of dressed test partiles presented by Rostoer [25] and Ihimaru [26] - [28] was applied to solve the ollision term, whih is ontributed by the flutuations in the plasma system, in the plasma ineti equation for obtaining the modified Balesu-Lenard-Landau ineti equation. [29] The theoretial results an provide the physial explanation and the analytial saling law for the numerial thermalization in both ollisionless and ollisional plasmas due to disrete partiles effets. The ollisional 1D ES PIC model and the simulation results for examining the saling of the numerial thermalization time R with N D are shown in Se. II. The derivation of the modified Balesu-Lenard-Landau ineti equation and the analytial results are shown in Se. III, and their omparisons with the simulation results are disussed in Se. IV. Finally, the study is onluded in Se. V. 3

4 II. COLLISIONAL 1D ES PIC SIMULATION AND THERMALIZATION TIME A typial PIC simulation ode is generally used to simulate the olletive behaviors of ollisionless plasmas from the mirosopi point of view, but numerial flutuations, whih are indued by the partile disreteness, thermalize the plasmas. The additional ollisions in the PIC simulation an enhane the disrete partile effets to aelerate the plasma thermalization. The phenomena an be explained using the ineti equation f f qe f f v t x m v t, (1) f f where f + t ν t dis, f is the normalized distribution funtion of partiles, and m and q are the partile rest mass and harge, respetively. The term f t on the right-hand side of Eq. (1) desribes the rate of hange of the distribution funtion f due to additional ollisions ( f ) and disrete-partile effets f/ t] dis, respetively. If the orresponding ollision term f t is equal to zero in the PIC simulation, then the PIC simulation lie the homogeneous solution of Eq. (1), whih an ideally desribe the behaviors of ollisionless plasmas. The atual ollision effets in the PIC simulation as desribed by Eq. (1) are disussed in the next setion. The plasma system is assumed to be homogeneous in both y- and z- diretions in the simulation for simpliity. Only partile dynamis in x and v x spae is onsidered here with an immobile ion baground for the quasi-neutrality approximation. Partiles interat with eah other and with the neutralized ion baground through the eletri field E, whih is determined by the partile distribution and the stati ion baground, an be alulated using the Gauss s law E 3 4 qn0 fd v 1 x, (2) where n 0 is the mean plasma number density. The numerial solvers of fields and partile motions in the simulation follow the standard 1D ES PIC algorithm. [3] In this study, the Kroo type ollision model is used in the PIC simulation and the 4

5 theoretial analysis. The ollision operator ( f ) in Eq. (1) an be represented as f f f ν, (3) where f M is a Maxwellian distribution. The distribution funtion of plasmas under the effets of Kroo type ollisions evolves toward a Maxwellian on a time sale of the order 0 due to ollisional flutuations. In PIC simulations, the Kroo type ollision model is implemented using the Monte-Carlo method, and 0 is an adjustable parameter to determine the probability of partile ollisions. The null ollision method [2] is used to determine whether the ollision our in simulations or not for every partile at eah time step. If the ollision ours, the partile is randomly sattered from its original momentum to the value determined by the Maxwellian distribution funtion, as desribed by Eq. (3). 0 M In order to determine the thermalization time in PIC simulations, the hi-square goodness-of-fit test is used to examine the time required for the plasma distribution evolves toward the Maxwellian distribution to a ertain degree. [7] At eah time step, all partiles are sorted by dividing them into M bins in the veloity spae to perform hi-squared statisti testing, i.e. 2 M 2 ( N n), (4) n 1 where N is the number of partiles in the -th bin and n is the number of partiles estimated by the Maxwellian distribution in the same bin. The hi-square values deay exponentially with time when the plasma distribution gradually approahes to the Maxwellian distribution. The fitted equation an be obtained by these hi-square 2 2t 3 values as e, where 1 and 2 are the orresponding oeffiients 1 2 depending on the number of bins (M) and the initial distribution of partiles, respetively. The reiproal of the oeffiient 3 an be defined as the relaxation time. Sine the Kroo type ollision indues the intrinsi thermal relaxation, 3 an be represented as the sum 0 R ; where, R is thermalization time due to disrete-partile effets. In order to examine the thermal relaxation in the 1D ES PIC simulation, partiles are initially separated with an even spae in the simulation box and the initial veloity distribution is set as the Fermi-lie distribution, i.e., 3 f z u t u u u u, (5) (, z, 0) ( ) 3 z z 4u

6 where u z v z v th, u 5, v th is the thermal veloity, and xis the Heaviside step funtion suh that xfor xand xfor x. The plasma thermalization from the Fermi-lie distribution to the Maxwallian distribution is then examined by the 1D ES PIC simulation as shown in Fig. 1. The simulation results learly show that the thermalization time τ R sales with N 2 D for ollisionless plasmas (shown as Fig. 1 (a)), and τ R linearly sales with N D while the ollision rate is 0 pe = 110 (shown as Fig. 1 (b)). The Kroo type ollision model, whih is lie other ollision models used in many different simulation wors [30]-[32], enhane the disrete partile effets, suh that R is dramatially hanged to sale with N D. The enhanement of the disrete partile effets due to the additional ollisions is theoretially analyzed in the following setion. III. BALESCU-LENARD-LANDAU KINETIC EQUATION BASED ON THE KROOK TYPE COLLISION OPERATOR In order to analyze the disrete-partile effets on the saling of the plasma thermalization time with respet to the plasma parameter N D, the equation for temporal evolution of the distribution funtion, i.e., f/ t] dis, needs to be derived. Consider a 1D uniform single-speies plasma system, in whih ions form a stati baground for maintaining quasi-neutrality. The disrete partile distribution is expressed as the Klimontovih distribution funtion, i.e., N tot, ; N x v t x x t v v t, (6) i i i where x i (t) and v i (t) represent the trajetory of the ith partile in the phase spae, and N tot is the total number of partiles. The distribution funtion N satisfies the ontinuity equation as N N qe N v 0, (7) t x m v where m and q are the rest mass and harge of partiles, respetively. If the partile ollisions beome effetive in the plasma system, the Monte-Carlo method is added into the PIC simulation to model the ollisional effets. The orresponding equation needs to inlude a ollision operator, Eq. (7) then beomes N N qe N v ν t x m v N. (8) The partile distribution as defined in Eq. (6) an be divided into the averaged 6

7 and flutuation parts, i.e., N x, v; t F v; t N x, v ; t, (9) where F is the ensemble average of the Klimontovih distribution funtion N over sales greater than those assoiated with the flutuations N in the partile distribution. The eletri field an also be expressed as the ombination of the averaged quantity and flutuations, i.e., E = E + E, (10) where denotes the ensemble average and E is zero beause of the quasi-neutral approximation. The magneti field is ignored under the eletrostati approximation. Substitute Eqs. (9) and (10) into Eq. (8) and tae ensemble averaging, we obtain the exat form of the plasma ineti equation F v F F q ν N E t x m v. (11) The right-hand side of Eq. (11) represents the ollisional effet due to the disrete-partile nature of the plasma and is the term we would lie to derive, i.e., F v; t q N E t m v dis. (12) The flutuation part of the plasma density N in Eq. (8) an be expressed as N v N ν N q E F. (13) t x m v N Noting ~ O( 1 ) 1, high order terms in Eq. (13) are negleted. F N tot Assume these flutuations our in an infinite and uniform plasma system, suh that the observation of the system an be extended over whole spae and time by adopting the periodi boundary onditions. The Fourier transform an be applied on Eq. (12) to obtain F v; t q 1 ˆ ˆ ˆ ˆ. t m v 2 dis N E E N d (14) The orresponding Fourier omponents of density flutuations Nˆ an be obtained by solving Eq. (13), and solutions are given by the sum of homogeneous and speial solutions, i.e., H S. N N N (15) 7

8 The homogeneous solution, whih represents the spontaneous flutuation indued by the random thermal motion of disrete partiles without eletromagneti fields, an be written as Fˆ,0 N, (16) ˆ H where = vi, and F,0 is a onstant that an be determined by the initial ondition. The speial solution, whih is orresponding to the indued flutuation by eletromagneti interations, an be expressed as S q 1 ˆ F N i E m v. (17) The eletri flutuation Eˆ, whih is self-onsistently indued by the density flutuation, is governed by the Gauss's law i Eˆ 4 q Nˆ Nˆ dv H S. (18) Substitute Eq. (17) into Eq. (18), the eletri flutuation is then given by Here is the dieletri response funtion, i.e., ˆ 4 q 1 ˆ H E i N dv,. (19) ν 2 p dv ' f, 1. (20) v ' iν v ' In Eq. (20), f denotes the normalized distribution funtion, i.e., F(v)=n 0 f (v) and f ( v) dv 1 with n 0 being the plasma density. The density flutuation ˆ H N represents the stohasti quantity arising from the thermal motion of disrete partiles, and the statistial nature arries over in the eletri flutuation as shown in Eq. (19). The disrete partiles an polarize the medium and indue a shielding loud around themselves to ause the density flutuation through the eletromagneti interations. The eletri flutuations indued by the shielding loud and additional ollision effets are impliitly represented by the dieletri response funtion, as shown in Eq. (19). So the flutuation an be oneptually expressed by superposing the fields of dressed test partiles without onsidering the orrelation between them, sine the mean ollision 8

9 time is muh greater than the pair orrelation time between two partiles. The total dressed partile fields Nˆ thus onsists of the bare-partile field ˆ H N and the ˆ S shield-partile field N, as shown in Eq. (15). [24] The spetral funtion ontributed by the bare-partile fields, as shown in Eq. (14), an be expressed as H 4 q 1 H H N = E i N N dv ', (21), The spetral funtion at the right hand-side of Eq. (21) an be expressed as H ν H N N v v' n f v, (22) sine partiles move along the unperturbed trajetories without orrelations. Hene, Eq. (21) an be re-written as H 4 q N E = i n0 f v, ν 0. (23) Here, the dieletri funtion (, ) is defined by Eq. (20), and (, ) v satisfies the relation (, + )= *(, ). The ollision term in the ineti equation due to the bare-partile field an be obtained by substituting Eq. (23) into Eq. (14), i.e., ( H ) 4 p 1 ν ν dv ' 2 2 dis 0, v ' iν f v; t v ' v 2 i f v' t n v v ' where, the expression of the modified delta funtion an be written as 1 2ν v ' v 4ν v ' v 2 iν = 2 2 ν f v v ; (24). (25) In a similar way, the spetral funtion ontributed by the shield-partile fields in Eq. (14) an be expressed as S q F v 1 N E i E E m v. (26) Aording to Eq. (19) and Eq. (22), the spetral funtion of eletri flutuation Eˆ at the right-hand side of Eq. (26) an be expressed as 9

10 2 ˆ ˆ 4 q 1 E E n 2 0 f v dv. (27), Therefore, the ollision term of the ineti equation due to the shield-partile field an be obtained by substituting Eqs. (26) and (27) into Eq. (14), i.e., ( S ) 4 ; p 1 ν ' 2 ν dv ' 2 2 dis 0, v ' iν f v t v v i f v f v' t n v v ν ν. (28) Combine Eqs. (24) and (28), we an obtain the Balesu-Lenard ollision term with the onsideration of artifiial ollisions as 4 ; p 1 ν ' 2 ν ' 2 2 dis 0, v ' iν f v t v v i dv f v f v t n v v ' v ν ' (29) In statistial equilibrium, the Balesu-Lenard ollision term is typially dominated by those Fourier omponents > 1 D [23] and orrespondingly (,v i ) 1. Equation (29) then beomes the following Landau equation [27], i.e., 4 f v; t p 1 t n0 v v ' v dis 2 dv ' ν v ' v 2 iν f v' f v (30) It is worth to note that is an arbitrary ollision operator in Eq. (30), suh that it an be applied to analyze the thermal relaxation of the plasma system simulated by the 1D ES PIC ode with any pratial ollision model. In the study, the Kroo type ollision model is used in the PIC simulation. Substitute Eq. (30) and Eq. (3) into Eq. (1) and normalize all of the variables, we an obtain the ollision term, whih inludes the disrete partile effet and Kroo type ollisions: f 0 p f f M 1 1 du ' u ' u 2 i f u ' f u N V ν u u 0 2 z z z z z D p z ' z, (33) where, the dimensionless parameters are defined as t p, D, u z v z p D ). Equation (33) an be numerially solved using the finite differene sheme to obtain the time-evolution of the distribution funtion, and the analytial results an provide physial explanations for the numerial thermalization observed in the 1D PIC 10

11 simulations. IV. RESULTS AND DISCUSSIONS The numerial results, as shown in Fig. 1, already demonstrate that the addition of the Kroo type ollision model in the 1D ES PIC simulation quantitatively redues the plasma thermalization time and hange its saling with N D from a quadrati form into a linear form. So the Kroo type ollision model atually enhanes the disrete partile effets in the PIC simulations, and the results an be theoretially examined by solving the modified Balesu-Lenard-Landau equation, i.e., Eq. (33) derived in the previous setion. Fig. 2 shows the plasma thermalization time, whih is normalized by the ollision frequeny, versus the plasma parameter N D at 0 pe and , respetively. The solid line (for 0 pe ) and the dotted line (for 0 pe ) in Fig. 2 are the plasma thermalization time alulated by solving Eq. (33), and these two theoretial urves onsistently demonstrate the linear saling with N D for the ollisional plasma system. The simulation results (hollow irle for 0 pe and rosses for 0 pe in Fig. 2) also show the linear saling of N D, but the simulated thermalization time onsiderably deviates from the theoretial results at 0 pe The disrepany between the simulation results and theoretial analysis at 0 pe is due to that high-order disrete partile effets are negleted in the theory. The lowest order disrete partile effets dominate the modified Balesu-Lenard-Landau equation while the ollision frequeny is inreased, suh that the theoretial results agree well with the simulation results at 0 pe The disrete partile effets an be further examined by observing the time-evolution of the partile distribution in the veloity spae, espeially the partile number near the edge of the Fermi-lie distribution at u z ~ u, as shown in Fig 3. At 0 pe , the partile number simulated by the PIC simulation (denoted as 11

12 hollow squares in Fig. 3(a)) at u z ~ u inreases muh faster than the values alulated by the analytial results (the solid line in Fig. 3(a)) within the early plasma osillation periods, then both values onsistently inrease with a gentle slope. It should be noted that the partile number simulated by the PIC simulation hanges very fast within the first two plasma osillation periods due to the initial transient behavior, whih has been demonstrated by Dawson. [8] The Kroo type ollision with 0 pe hardly affets the initial transient behavior beause, as shown in Figs. 3(a) and (b), the simulated partile numbers for different ollision frequenies are almost the same at time 1 ~2 pe. After the initial transient period, the partile orrelations are adjusted to the forms appropriate for the given veloity distribution, and partile ollisions and disrete partile effets dominate the plasma thermalization. Therefore, the disrepany between simulation results and analytial results in the beginning stage of the plasma thermalization (after the initial transient period) indiates that the high-order disrete partile effets, whih are negleted in the theoretial analysis, are short-term effets. The lowest order disrete partile effets finally dominate the thermalization proesses as shown in Fig. 3(a). Moreover, it is worth noting that the plasma thermalization in the PIC simulation is mainly indued by the disrete partile effets, whih are then enhaned by the additional Kroo type ollision model. If the disrete partile effets are negleted in the theoretial analysis, the plasma thermalization indued only by additional ollisions beomes ineffiient, as shown by the dashed line in Fig. 3(a). When 0 pe is raised to 210 2, the theoretial results show good agreement with the simulation results as shown in Fig. 3(b), sine the lowest order disrete partile effets dominate the plasma thermalization. Therefore, at higher 0 pe, the modified Balesu-Lenard-Landau equation derived in the study an orretly desribe the plasma thermalization observed in the 1D ES PIC simulation and provide the saling law whih is onsistent with that was derived from previous wors. [6], [7] V. CONCLUSION The modified Balesu-Lenard-Landau equation has been derived using the superposition method of dressed test partiles in this study. The theoretial analysis learly shows that the plasma thermalization is mainly indued by the disrete partile effets. The artifiial ollision model, e.g. the Kroo type ollision model, added in 12

13 the theoretial analysis and the PIC simulation enhanes the lowest order disrete partile effets, suh that the plasma thermalization time beomes linearly saled with N D, as observed by Turner. [6] For the theoretial study of wealy ollisional plasma systems, the high order disrete partile effets are essential for the orret desription of plasma thermalization proesses, as demonstrated by the PIC simulation in the present study. The improvement of the theoretial wor an be arried on using the BBGKY hierarhy with the onsideration of high order terms of N D. ACKNOWLEDGMENTS This is a report of wor sponsored by Grant No. MOST M from the Ministry of Siene and Tehnology, Taiwan. The wor is also supported by US DoE and ITER-CN Program. The authors would lie to anowledge the help of the National Center for High-Performane Computing in providing resoures under the national projet, Taiwan Knowledge Innovation National Grid. REFERENCES 1. J. M. Dawson, Phys. Fluids 5, 445 (1962). 2. C. K. Birdsall, IEEE Trans. Plasma Si. 19, 65 (1991). 3. C. K. Birdsall and A. B. Langdon, Plasma Physis via Computer Simulation (MGraw-Hill, New Yor, 1985). 4. J. P. Verbonoeur, Plasma Phys. Control. Fusion 47, A231 (2005). 5. R. W. Honey and J. W. Eastwood, Computer Simulation using Partiles (Hilger, Bristol, U.K., 1988), Chap M. M. Turner, Phys. of Plasmas 13, (2006). 7. P. Y. Lai, T. Y. Lin, Y. R. Lin-Liu, and S. H. Chen, Phys. Plasmas 21, (2014). 8. J. M. Dawson, Phys. Fluids 7, 419 (1964). 9. A. Lenard, Ann. Phys. (New Yor), 10, 390 (1960). 10. R. Balesu, Phys. Fluids, 3, 52 (1960). 11. D. Montgomery and C. W. Nielson, Phys. Fluids 13, 1405 (1970). 12. M. Y. Yoo and B. Abraham-Shrauner, Physia 68, 133 (1973). 13. M. Surendra and D. B. Graves, Phys. Rev. Lett. 66, 1469 (1991). 14. H. C. Kim and J. K. Lee, Phys. Rev. Lett. 93, (2004). 15. M. M. Turner, A. Derzsi, Z. Donó, D. Eremin, S. J. Kelly, T. Lafleur, and T. Mussenbro, Phys. Plasmas 20, (2013). 16. A. J. Kemp, R. E. W. Pfund, and J. Meyer-ter-Vehn, Phys. Plasmas 11, 5648 (2004). 17. F. Pérez, L. Gremillet, A. Deoster, M. Drouin, and E. Lefebvre, Phys. Plasmas 19, 13

14 (2012). 18. R. Mishra, P. Leblan, Y. Sentou, M. S. Wei, and F. N. Beg, Phys. Plasmas 20, (2013). 19. M. M. Oppenheim, Y. Dimant, and L. P. Dyrud, Ann. Geophys., 26, 543 (2008). 20. A. J. Kemp, B. I. Cohen, and L. Divol, Phys. Plasmas, 17, (2010). 21. T. Ma, H. Sawada, P. K. Patel, C. D. Chen, L. Divol, D. P. Higginson, A. J. Kemp, M. H. Key, D. J. Larson, S. Le Pape, A. Lin, A. G. MaPhee, H. S. MLean, Y. Ping, R. B. Stephens, S. C. Wils, and F. N. Beg, Phys. Rev. Lett. 108, (2012). 22. J. Y. Hsu, D. Montgomery, and G. Joye, J. Plasmas Phys. 12, 21 (1974). 23. P. L. Bhatnagar, E. P. Gross, and M. Kroo, Phys. Rev. 94, 511 (1954). 24. Y. L. Klimontovih, The Statistial Theory of Non-Equilibrium Proesses in a Plasma, Wiley, (1967). 25. N. Rostoer, Phys. Fluids 7, 479 (1964). 26. S. Ihimaru, J. Phys. So. Japan 19, 1207 (1964). 27. S. Ihimaru, Phys. Rev. 140 (1965). 28. S. Ihimaru, Statistial Plasma Physis Vol I: Basi Priniples, Addison-Wesley, (1992). 29. L. D. Landau, Phys. Z. Sowjetunion 10, 154 (1936). 30. K. Nanbu, J. Phys. So. Jpn, 50, 3154 (1981). 31. G. Yan and L. Yuan, Physia D, 154, 43 (2001). 32. C. D. Landon and N. G. Hadjionstantinou, Proeedings of the 27th International Symposium on Rarefied Gas Dynamis, (2010). 14

15 FIGURE CAPTIONS: FIG. 1. The plasma thermalization time alulated by 1D ES PIC simulations (a) without the onsideration of partile ollisions and (b) with Kroo type ollisions at 0 pe = 110. The dashed lines show the saling of the thermalization time with N D. FIG. 2. Figure shows the normalized thermalization time alulated by PIC simulations at 0 pe (rosses) and 0 pe 2 10 (hollow irles). The orresponding 2 analytial values are alulated by solving Eq. (33) at 0 pe 1 10 (the solid line) and 0 pe (the dashed line). All these results show the linear saling with N D. FIG. 3. The time evolution of partile numbers at u z ~ u in the veloity spae alulated by 1D ES PIC simulation (hollow squares) and by solving Eq. (33) (solid lines) at (a) 0 pe and (b) 0 pe 2 10, respetively. The partile number in a Debye length N D = 20. The dashed line in (a) shows the theoretial results alulated by solving Eq. (33) with the neglet of disrete partile effets. 15

16 FIG. 1 16

17 FIG. 2 FIG. 3 17

18

19

20