Introduction to Kinetic Simulation of Magnetized Plasma

Transcription

1 Introduction to Kinetic Simulation of Magnetized Plasma Jae-Min Kwon National Fusion Research Institute, Korea 018 EASW8 July 30 Aug 3, 018 1

2 Outline Introduction to kinetic plasma model Very brief on essential things to understand kinetic simulation Reduced kinetic models for magnetized plasma 5D gyrokinetic, 4D bounce-averaged kinetic, 3D fluid with kinetic closure Numerical methods for kinetic simulation of magnetized plasma Particle-in-Cell methods, and related numerical issues Help students understand the basic idea behind the models and read related literatures for further studies

3 Introduction to Kinetic Plasma Model 3

4 Kinetic Plasma Model The most general description of physical system with many particles Each particle satisfies the following equations of motion dd xx dddd = vv, dd vv mm dddd = FF xx, tt = qqee + vv cc BB Number of particles in dd 3 xxdd 3 vv ff xx, vv, tt dd 3 xxxx 3 vv 4

5 Klimontovich Equation Exact description of classical particles interacting with self-consistent electromagnetic forces NN FF xx, vv, tt = δδ xx XX pp tt pp=1 δδ vv VV pp tt dd dddd XX pp tt = VV pp tt, mm ss dd dddd VV pp tt = qqee mm XX pp tt, tt + qq cc VV pp tt BB mm XX pp tt, tt Then, FF satisfies FF + vv FF + qq mm EEmm + vv cc BBmm vv FF = 0 Note that this equation contains whole spatio-temporal scales all the way down to particle distances. 5

6 Klimontovich Equation Since we are not interested in physical phenomena occurring in super micro-scales (actually, the equation itself is not valid in such scales), We separate quantities into two scales i.e. smooth part in large scale and non-smooth part in small scale, and We keep only the smooth part in left hand side and through out all the remaining into the right hand side and call them collision FF = ff + δδδδ, EE mm = EE + δδδδ, BB mm = BB + δδδδ ff + vv ff + qq mm EE + vv cc BB vvff = qq mm δδδδ + vv cc δδδδ vvδδδδ CC ρρ = qq ss ff ss dd vv ss ȷȷ = qq ss ff ss vvdd vv ss BB = 0 BB = EE EE = ρρ εε 0 EE = BB μμ 0 ȷȷ 6

7 Collision Operator Collision term can be derived by following BBGKY hierarchy and truncating higher order interaction terms (Introduction to Plasma Theory, Nicholson) CC ff = vv AAff vv + 1 vv vv : BBff vv AA vv, tt 8ππnn 0ee 4 llllλ mm ee BB vv, tt 4ππnn 0ee 4 llllλ mm ee vv dd vv ff vv, tt vv vv vv vv dd vv vv vv ff vv, tt For high temperature plasmas, the collision frequency becomes very small νν cccccccc nn/tt 3/ In tokamak plasma νν 1 cccccccc = 10 ~ 100 mmmm collisionless 7

8 Kinetic Phase Mixing vv + vv ff = 00 vv xx xx ff ff vv vv 8

9 Collisionless No Collision ff = CC(ff, ff) ff ff ee vv vv vv As Δvv 0, CC ff, ff ~ ff vv increases faster than streaming term Collision becomes important as fine scale structures are developed in velocity space 9

10 Kinetic Phase Mixing vv xx 10

11 Kinetic Plasma Model + vv + FF mm vvff = CC ff, ff Streaming motions of particles + Mutual interactions in large scales Tend to drive complicated phase space structures Mutual interactions in very short scales ~ Random collision Tend to erase fine scale phase space structures Weaker for small density and high temperature system. Stronger for smaller velocity space scales ff ee vv vv vv 11

12 Kinetic vs Fluid Shifted Gaussian distribution function ff xx, vv, tt = ππ 3/ dd vvff nn( xx) TT xx mm = nn 3/ mm vv uu(xx) exp[ ] TT(xx) Navier-Stokes (Fluid) Equation nn uu + uu uu = pp + μμ uu pp = nnnn uu vv tt = TT mm vv 1

13 Spatio-Temporal Scales of Fusion Plasma S.Either et al, IBM J. RES. & DEV. Vol

14 Turbulence in Fusion Device Fusion plasma confined by external magnetic field strongly magnetized plasma ρρ RR 14

15 Turbulence in Fusion Device In Tokamak, anomalous heat and particle transport driven by micro-scale fluctuations with relatively low frequency ff 300 kkkkkk and kk ρρ ii 1 Collisionless plasma νν cc /ff 1 MIR measured nn ee fluctuation on KSTAR L-mode (J.A. Lee et al, PoP 5, 0513(018)) ECEI measured TT ee fluctuation with ff 100 kkkkkk (M.J. Choi, 018) 15

16 Kinetic Plasma Model: Problem Size Problem size for KSTAR plasma Number of grids: NN xx NN yy NN zz NN vvxx NN vvyy NN vvzz ~ Electron-Ion mass ratio ~ 1:3600 time scale disparity ~ 100 Even for limited spatio-temporal scales, simulations based on brute-force approaches are practically impossible Reduced models are essential 16

17 Model Hierarchy for Magnetized Plasma 6D Boltzmann 5D Gyrokinetic (4D Kinetic) ff ss + vv ff ss + qq ss mm ss ρρ = qq ss ff ss ddvv ss EE + vv BB vv ff ss = CC ff ss jj = qq ss ff ss vvddvv ss 3D Fluid (Gyro-Landau-Fluid, MHD) BB = 00 BB = EE EE = ρρ εε 00 EE = BB μμ 00 jj 1D Diffusion 17

18 18 5D Gyrokinetic Model

19 Gyro Motion in Magnetized Plasma Electron Magnetic field μμ vv ΩΩ ii Adiabatic invariant of motion for time scales slower than ΩΩ ii 11 19

20 Basic Idea of Gyrokinetic Model Gyrokinetic orderings Small fluctuation: δδδδ ff 0 ~ eeeeee TT ~ δδδδ BB 0 1 Frieman, Chen, Phys. Fluids 5, 50(198) Hahm, Lee et al, Phys. Fluids 31, 1940(1988) Hahm, Phys. Fluids 31, 670(1988) Brizard, Hahm, Rev. Mod. Phys. 79, 41(007) Low frequency: ωω Ω ii 1 Fast MHD waves and cyclotron waves are ruled out (high freq. GK; Kolesnikov et al, Phys. Plasmas 14, 07506(007)) Anisotropic fluctuation: kk kk 1, kk ρρ ii ~ 1 Mild non-uniformity in plasma profiles, background magnetic field: ρρ ii LL TT,nn 1 Low beta: ββ 1 Free energy to drive turbulence (GK with strong gradient; Shear Alfven wave only (GK with Compressional Alfven; Brizard, Hahm 07 ) Hahm et al, Phys. Plasmas 16, 0305(009)) 0

21 Basic Idea of Gyrokinetic Model Guiding center transformation particle space xx, vv guiding center space XX, vv, μμ, θθ θθ: gyro-angle average out XX = xx ρρ ρρ = bb vv Ω Ω = eebb 0 mmmm ρρ θθ + vv = bb vv μμ = vv BB XX xx vv = vv bb + vv ee 1

22 Basic Idea of Gyrokinetic Model Schematics of guiding center transformation in Gyrokinetic model Particle Space ( xx, vv) Vlasov Equation (6D) Reduced Maxwell Equation (3D) Guiding Center Space ( XX, vv, μμ) GK Equation (5D) Recued Maxwell Equation (3D) Solve Vlasov equation in guiding center space and evaluate sources (nn ss, jj ss ) Transform sources (nn ss, jj ss ) to particle space Solve recued Maxwell equations to obtain EM fields Transform EM fields to guiding center space

23 Gyrokinetic Vlasov Equation for Low-ββ Plasma Transform original 6D Vlasov equation in particle space into guiding center space Take gyro-angle average to remove θθ reduction to 5D ff(xx, vv, μμ, tt), retaining only slow time scales Δtt 1/Ω ii ff + vv bb + μμ bb BB BB + cc bb δδδδ ff BB 0 XX + qq mm bb μμ BB bb δδδδ 1 cc δδaa ff vv = 0 δδδδ = δδδδ vv cc δδaa bb = bb + vv BB bb bb bb = gyro-phase averaged fluctuations 3

24 Gyrokinetic Model Simple View Gyrokinetic description of magnetized plasmas Motion of charged particle d x = v dt d q q v = δφ + v ( B0 + δb) dt m mc Motion of charged ring centered at XX ddxx dddd = vv bb + μμ bb BB BB + cc bb δδδδ BB 0 ddvv dddd = qq bb mm μμ BB + bb δδδδ + 1 cc δδaa 4

25 Gyrokinetic Model Simple View FF BB drift motion of charged particle drift motion of gyro-center in FF BB direction FF BB 5

26 Gyrokinetic Model Simple View Grad-B + Curvature drift dd XX dddd vv bb + δδaa bb cccc + mmvv + μμμμ bb BB qqbb + cc BB 0 bb δδδδ ExB drift Parallel motion along perturbed magnetic field ddvv dddd qq mm μμ bb BB + bb δδδδ + 1 cc mirror force δδaa Parallel E-field GK equations of motion are nothing but a combination of familiar drift motions ensuring phase space volume conservation and making them Hamiltonian flows BB ff + XX dd XX dddd BB ff + vv ddvv dddd BB ff = 0 6

27 Gyrokinetic Model Simple View Poisson equation with enhanced polarization shielding (1 + ρρ ii λλ ) δδδδ xx, tt = 4ππ DDDD ss qq ss NN ss Density from charged rings Additional shielding by polarization charges carried by charged rings Significantly enhanced compared to Debye shielding Ampere equation without displacement current, also for δδ AA = bbδδaa for low-ββ δδbb bbδδaa 4ππ cc SS JJ ss xx ii=1 NNgg δδaa = 4ππ cc ss JJ ss Parallel current carried by charged rings XX BB 0 7

28 8 Gyrokinetic Model Simple View Gyro-averaged potentials δδδδ, δδaa felt by charged ring = + + = + N l l i i i X N d X x x X dx d )) ( ( 1 )) ( ( 1 ) ( ) ) ( ( 1 θ ρ δφ θ θ ρ δφ π δφ θ ρ δ θ π δφ π π X ) ( l ρ i θ or in Fourier space (as is often done in continuum codes) ( ) ( ) ( ) ( ) + Ω = = = + = + k d e v k J k dk e d e k d dk e k d X x x X dx d X ik i X ik ik X ik i i i i cos 3 0 ) ( ) ˆ( 1 ) ˆ( 1 1 ) ˆ( 1 1 )) ( ( 1 ) ( ) ) ( ( 1 δφ π θ δφ π π θ δφ π π θ θ ρ δφ π δφ θ ρ δ θ π δφ π θ ρ π θ ρ π π Integration can be approximated by a few points sum

29 Reduced Problem Size Problem size for KSTAR plasma Number of grids: NN xx NN yy NN zz NN vvxx NN vvyy NN vvzz ~ Number of grids: NN xx NN yy NN zz NN vv NN μμ ~ Electron-Ion mass ratio ~ 1:3600 Time scale disparity ~ 100 Fluid Model? 9

30 4D Bounce-Averaged Kinetic Model 30

31 Bounce Motion in Magnetized Plasma vv FF = μμ BB Trapped particles vv vv = 1 BB BB mmmmmm Passing particles Passing particles vv Passing and trapped particles behave very differently Passing particles move freely along magnetic field line For slow perturbation, fluid model works well Trapped particles show non-trivial responses to slow perturbations Hard to capture in fluid model 31

32 Bounce Motion in Magnetized Plasma JJ 11 = PP dd ll = mmvv 11 qqqqqq RRRRRR = ππππvv ΩΩ μμ JJ = PP dddd mmvv zz dddd = mmvv zz dddd = ππππvv zz, vv zz = vv zz cccccc ωω bb tt ωω bb 3 Figure from Geomagnetism, Nathani Basavaiah

33 Bounce Motion in Magnetized Plasma Trapped electrons in fusion device Motions along banana are very fast detailed position is negligible Toroidal precession motions are slow comparable with ion transit motions Trapped electron bounce-centers behave like ions resonate with ion scale turbulence 33

34 Bounce-Averaged Kinetic Model Bounce-center coordinate Radial position of bounce center: ββ = rr Toroidal angle of bounce center at outer mid-plane: αα = ζζ 1 sin θθ/ Bounce phase: Ψ = ππ + ssssss vv sin κκ pp Ignorable variable, averaged out Bounce invariant: II qqrr 0 mmεεεεbb 0 κκ pp Further reduction of 5D gyrokinetic equation to 4D bounce-averaged kinetic model Fong, Hahm, Phys. Plasmas 6, 188 (1999) Qi, Kwon, Hahm, Jo, Phys. Plasmas 3, 06513(016) Kwon, Qi, Yi, Hahm, Comput. Phys. Commun. 177, 775(017) + ππ/ αα ββ Ψ 34

35 Bounce-Averaged Kinetic Model Hamiltonian and equations of motion: HH ββ, αα, ΙΙ, μμ = μμbb ΔΔ εε + κκ qq εε + εεεεbb 0 qqrr 0 mm II + qq ss φφ bb ddββ = cc φφ bb dddd αα ddαα dddd = cc ee HH 0 cc φφ bb ββ ββ dddd dddd = 0 dddd dddd = 0 Bounce averaged kinetic equation: dddd dddd = + ddββ dddd ββ + ddαα dddd αα = 0 35

36 Bounce-Averaged Kinetic Model Numerical calculation of bounce average φφ bb φφ dddd ll dddd ll = 1 TT φφ dddd ll Approximation of bounce orbit by unperturbed guiding center motion 36

37 Reduced Problem Size Problem size for KSTAR plasma Number of grids: NN xx NN yy NN zz NN vvxx NN vvyy NN vvzz ~ Number of grids: NN xx NN yy NN zz NN vv NN μμ ~ 10 9 Electron-Ion mass ratio ~ 1:3600 time scale disparity ~ 100 Δtt CCCCCC Δtt for ions only 37

38 Fluid Model with Kinetic Closure 38

39 Kinetic Phase Mixing vv + vv ff = 0 vv xx xx ff δδδδ(0) δδδδ tt < δδδδ(0) ff vv Decay rate? vv 39

40 Kinetic Closure: How to mimic kinetic process? Fluid Model For neutral gas + (Fick s law) VVVV = 0 with particle flux Γ = nnnn = DD Then, we have = DD xx nn Let s assume a stationary solution nn xx, tt = nn 0 and put a perturbation at tt = 0 as δδδδ xx, 0 = nn 1 ee iiiiii. Then, the solution of the equation becomes δδδδ xx, tt = δδδδ xx, 0 ee DDkk tt 40

41 Kinetic Closure: How to mimic kinetic process? Kinetic Model For neutral gas + vv = 0 With a stationary solution, ff 0 vv = nn 0 ππvv tt ee vv vv tt + (note that ddddff0 = nn 0 ) Let s put a perturbation at tt = 0, ff 1 xx, vv = nn 1ee iiiiii ππvv tt Then the solution of the kinetic equation becomes ee vv vv tt + (note that ddddff1 = nn 1 ee iiiiii ) ff xx, vv, tt = nn 0 + nn 1 ee iiii xx vvvv 1 ππvv tt ee vv /vv tt 41

42 Kinetic Closure: How to mimic kinetic process? Kinetic Model ff xx, vv, tt = nn 0 + nn 1 ee iiii xx vvvv 1 ππvv tt ee vv /vv tt The density evolution becomes nn xx, tt = dddddd = nn 0 + nn 1 ee iiiiii ππvv ddddee iiiiiiii ee vv vv tt = nn 0 + nn 1 ee iiiiii ee kk vv tt tt / tt i.e. δδδδ xx, tt = δδδδ xx, 0 ee kk vv tt tt / Kinetic vs Fluid kinetic ee kk vv tt tt / fluid ee DDkk tt 4

43 Kinetic Closure: How to mimic kinetic process? kinetic ee kk vv tt tt / dddd ee kk vv tt tt / = kk vv tt ππ fluid ee DDkk tt dddd ee DDkktt = DDkk tt Match two time responses ~ match the two areas DD = ππ vv tt kk Hammett et al, Phy Rev Lett 64, 3019(1990) Hammett et al, Phys Fluids B 4, 05(199) 43

44 Kinetic Closure: How to mimic kinetic process? With DD kk = ππ vv tt kk, the particle flux can be written as ΓΓ kk = DD kk iiiinn kk In real space, Γ = 1 ππ + ddddee iiiiii Γ kk = 1 ππ + ddddee iiiixx iiii nn kk kk Using delta function identities 1 = kk + δδ kkkkk ddddd, δδ kkxx = lim εε 0 1 ππ εε εε +kk xx Γ = vv tt ππ 3/ 0 ddxx nn xx+xx nn xx xx xx ( c.f. conventional closure Γ = DD ) i.e. DD kk is a non-local integral operator 44

45 45 Hammett et al, Phys Fluids B 4, 05(199) Beer, Hammett, Phys. Plasmas 3, 4046(1996) P. Snyder, Ph.D. Thesis (1999)

46 Numerical Methods for Kinetic Plasma Simulation Continuum (Eulerian) Method Discretize 5D/4D phase space, and apply FDM, FVM, FEM Computationally expensive, but enable high quality simulation Particle-in-Cell (Lagrangian) Method Computationally cheap (relative to continuum method) Noise issues Semi-Lagrangian Method 46

47 Particle-in-Cell Method for Kinetic Simulation C.K. Birdsall and A.B. Langdon, Plasma Physics via Computer Simulation, McGraw-Hill, 1985 R.W. Hockney and J.W. Eastwood, Computer Simulation using Particles, IPP, 1988 W.W. Lee, J. Comput. Phys. 7, 43 (1987) 47

48 From Klimontovich Equation to PIC FF + vv FF + qq mm EEmm + vv cc BBmm vv FF = 0 FF xx, vv, tt = NN pp=1 δδ xx XX pp tt δδ vv VV pp tt dd dddd XX pp tt = VV pp tt mm ss dd dddd VV pp tt = qqee mm XX pp tt, tt + qq cc VV pp tt BB mm XX pp tt, tt NN We want simulate NN-particle system with NN ss NN, GG = ss pp=1 δδ xx XXpp tt δδ vv VV pp tt NN ss FF WW xx, vv GG = WW pp δδ xx XX pp tt δδ vv VV pp tt pp=1 {WW pp } depends on marker particle loading scheme i.e. how to set {XX pp 0, VV pp (0)} For example, if GG xx, vv, tt = 0 FF(xx, vv, tt = 0) WW pp = NN NN ss More sophisticated schemes to minimize loading noise: quite starting scheme, optimal loading scheme etc. (J. Denavit and J.M. Walsh, Plasma Phys. Control. Fusion 6, 09 (1981))

49 Poisson Equation: φφ = dddddd φφ xx, tt = jj φφ jj tt SS jj xx = jj φφ jj tt SS xx xx jj xx jj xx xx If we write electrostatic potential as φφ xx, tt = jj φφ jj tt SS jj xx using a set of basis function SS jj xx, the Poisson equation becomes jj φφ jj tt SS jj xx = 4ππππ dddd FF xx, vv, tt = 4ππππ dddd WW pp δδ xx XX pp tt NN pp=0 δδ vv VV pp tt P.M. Prenter, Splines and Variational Methods, John Wiley & Sons

50 Poisson Equation: φφ = dddddd jj φφ jj tt SS jj xx = 4ππππ dddd FF xx, vv, tt = 4ππππ dddd WW pp δδ xx XX pp tt NN pp=0 δδ vv VV pp tt Appling dddd SS ii (xx) on both sides, jj dddd SS ii xx SS jj xx φφ jj (tt) = 4ππππ WW pp SS ii pp XX pp tt XX pp tt Note that if S is spline function, it satisfies SS xx yy = SS yy xx xx jj 1 xx jj xx jj+1 WW pp SS ii XX pp tt = WW pp SS XXpp tt (xx ii ) = dddd WW pp SS XXpp tt (xx ii )δδ xx XX pp tt pp pp pp δδ vv VV pp tt i.e. the distribution function can be interpreted as a set of particles with spatial shape SS xx P.M. Prenter, Splines and Variational Methods, John Wiley & Sons

51 Collision in PIC Simulation Fokker-Planck equation ff vv, tt = αα vv ff vv, tt + [ββ vv ff vv, tt ] vv It can be interpreted as a distribution function for particles with characteristic equation ddvv dddd = αα vv, tt + ββ vv, tt ξξ tt, ξξ tt ξξ tt = δδ tt tt Due to the drag and diffusion on the right hand side, particle trajectories are scattered as Δvv = ααδtt + 3 RR 0.5 ββδtt RR is a random number in [0, 1] J. Grasman and O.A. van Herwaarden, Asymptotic Methods for the Fokker-Planck Equation and the Exit Problem in Applications, Springer

52 Example: nd order Runge-Kutta dddd dddd = ZZ zz, Φ Start Calculate charge density using zz pp nn Solve Poisson equation for Φ nn Initialize Poisson solver Initialize zz pp 0 xx pp 0, vv pp 0 Push Particles zz pp nn+ 1 = zz pp nn + Δtt ZZ( zz pp nn, Φ nn ) Calculate charge density using zz pp nn+ 1 Diagnosis End Solve Poisson equation for Φ nn+1 Push Particles zz pp nn+1 = zz pp nn + Δtt ZZ( zz pp nn+ 1, Φ nn+1 )

53 Numerical Issues in PIC Simulation 53

54 NN-Particle System Temperature

55 NN-Particle System Fourier representation of potential and distribution function ΦΦ xx = ee iiiiii ΦΦ kk, kk ΦΦ kk = 1 LL ddddee iiiiii ΦΦ(xx) ff xx, vv, tt = ww pp δδ xx xx pp δδ(vv vv pp ), pp ff kk tt = 1 LL ddddee iiiiii ff xx, vv, tt = 1 LL pp ww pp ee iiiixx ppδδ vv vv pp Poisson equation in Fourier space ΦΦ = 4ππππ = 4ππππ dddddd kk ΦΦ kk = 4ππππ LL pp ww pp ee iiiixx pp Apply a filter SS kk on RHS ΦΦ kk = kk ΦΦ kk = 4ππee LL SS kk ww pp ee iiiixx pp pp 4ππππ LLkk SS kk ww pp ww ppp ee iiii(xx pp xx pp ) pp ppp

56 NN-Particle System Φ kk = 4ππππ LLkk SS kk ww pp ww ppp ee iiii(xx pp xx pp ) pp ppp For randomly scattered particles and NN 1, we can simplify RHS Φ kk 4ππee LLkk SS kk ww pp = pp 4ππee LLkk SS kk ww NN ee Φ kk TT 4ππee LLTTkk SS kk ww NN = 4ππππee TT 1 nn LL kk 4 SS kk ww NN = SS kk ww λλ 4 DD kk 4 NN ffffffffffffffffffffff eeeeeeeeeeee ttttttttttttt eeeeeeeeeeee ~ 1 NN

57 Dielectric Response Function and Fluctuation Dissipation Theorem Suppose we put an external test charge ρρ eeeeee xx, tt = ρρ eeeeee kk, ωω exp[ii kkkk ωωωω ] kk,ωω Then, plasma will respond to this external perturbation and generate density fluctuation ρρ(kk, ωω) If we write the total charge density as ρρ tttttt (kk, ww), dielectric response function is defined as ρρ tttttt kk, ωω = ρρ eeeeee kk, ωω + ρρ kk, ωω ρρ eeeeee kk, ωω εε kk, ωω, εε kk, ωω ρρ tttttt kk, ωω = ρρ eeeeee kk, ωω Fluctuation Dissipation Theorem tells us how system responds to small fluctuations and provides useful information for fluctuation spectrum LL ππ δδδδ ωω, kk = TT ωω IIII 1 εε(kk, ωω) Lower bound of fluctuation (or noise) for N-body system

58 Fluctuation and Dissipation Theorem: Implication for PIC simulation PIC simulation employs finite number of particles, actually far fewer than the real number of particles (e.g. in medium fusion device NN rrrrrrrr ~ 10 1 NN ssssss = 10 9 ~ 10 1 ) Fluctuation (or noise) from simulation should be much bigger than physical ones However, we have numerical tools to control too big fluctuation (or noise): Number of particles Particle shape function SS kk Spatio-temporal discretization Δxx, Δtt ee Φ kk TT SS kk ww λλ 4 DD kk 4 NN The strategy is to employ these to suppress noise to a level not to affect key physics

59 Noises NN rrrrrrrr NN ssssssssssssssssssss Noises particle shape, filtering, smoothing, errors etc. Noises

60 Summary Reduced kinetic models for magnetized plasma 5D gyrokinetic, 4D bounce-averaged kinetic, 3D fluid with kinetic closure These enable us to perform kinetic simulation with reasonable computing cost Numerical methods for kinetic simulation of magnetized plasma Particle-in-Cell method Fluctuation dissipation theorem to understand discrete particle noise Verification and benchmark test are necessary, and experimental validation should be the next step very active area of on-going research in fusion 60