プラズマ基礎(chǔ)數(shù)學(xué) 図子 秀樹

1、1基礎(chǔ)數(shù)學(xué)基礎(chǔ)數(shù)學(xué)図子秀樹図子秀樹2007 Collision Processes (4-17,18) 1-1) collision dynamics and “quasi-collision potential filed” 1-2) Derivation of Fokker Planck equationConservation laws in global physical quantities (4-24,25) 2-1) particle density, momentum,energy, charge density 2-2) derivation of conservation

2、laws3. Maxwell equations (5-1,2) 3-1) derivation of 1st set of Maxwell equations 3-2) derivation of 2nd set of Maxwell equationsAnalysis of fluctuating signals in plasma (5-8,9) 4-1) digital data acquisition of analog signals and FFT methods 4-2) Physical meaning of frequency domain講義內(nèi)容講義內(nèi)容2Plasma D

3、efinition of “plasma”= Nature of “plasma” = Description of “plasma” element ? interaction between elements ? transfer information among element ?31. Collision Processes 1-1) interaction between plasma particles Rutherford formula equation of motion effective cross section 1-2) Kinetic description fo

4、r plasma Fokker Planck eq.4“衝突”関連key word考 運(yùn)動(dòng)量保存、保存、向、方向、弾性衝突、非弾性衝突、反発（係數(shù)）、衝突平均自由行程、衝突斷面積、衝突周波數(shù)（時(shí)間）5In burning plasma, we have to considertwo types of collisions 1) Coulomb collision 2) Nuclear fusion Collision6Characteristics in Coulomb Collisions (Arzimovichs view)“Collisional Transport in Magneti

5、zed Plasma”Halendar, Sigmar 2002Which one is a trajectory of the test charged particlein a plasma?7Spizters viewtrajectory in velocity space(a) Initial(b) After Nth collisions(c) After 10Nth collisions( 1962 Phys. of Fully Ionized Gases )Do you find rules to describe and D ?8Wessons view1) Collision

6、s between test particle and field particles2) Test particle: ions or electrons3) Field particles: ions and electrons V|V|V|VVV9Neutral Beam Injection Ebeam0.1-0.3 MeVWWW MPIH+10Te increase leads to enhanced scattered spectrumWcritWcrit)(8 .14keVTeWcritKurimoto, Zushi 1997eesnT2/3 11中衝突記述 衝突過程運(yùn)動(dòng)方程式組入

7、？ 場(chǎng)合相互作用何？ 繰返衝突過程記述？ 粒子衝突表？ 相互作用記述？12Collision dynamics in Coulomb fieldScattering angleImpact parameterInitial velocity vCoulomb fieldLandaus text2f f13Energy & Angular momentumrZerUrUrrmEmrM022224)(),(2, 14,2,/)(2/4)(),(2,2202222 brMrUEmdrrMrZerUrUrrmEmrM15Exercise Idue date; 23 April till noo

8、n 1. Derive Rutherford scattering formula hints: 2. Conversion from two body collision equations into one particle motion in a central field hints: conversion from v1,v2 into relative velocity and velocity of center of mass bmvMmvE,21216Coulomb collision cross section22420 cotmvbZee202904mveZb =90 s

9、cattering, Scattering cross section b902220429064EeZ 17Collision in shielded Coulomb fieldbdbnvdtNetargt 21) Total target ions2) Limit in bDebyemaxb 18Dominant Collision in Momentum change322900902029022|1ln42902122222cos12 cos1lnvvvnmvnmvbbbdbnmvdtdpbdbnmvtpbdbtnvmvmvpbbDD 90blnlnD 902cot2bb Integr

10、ation from bmin to Debye length19Dominant Collision in Momentum change32042322220422908lnln21644ln4vvmneZvvmvmveZnvvnmvdtpd 90blnlnD 902cot2bb 220429064EeZ 20 個(gè)荷電粒子標(biāo)的粒子作靜電場(chǎng)“運(yùn)動(dòng)量”微少変化。 “衝突”定義。 単位時(shí)間內(nèi)運(yùn)動(dòng)量変化量“有効場(chǎng)”運(yùn)動(dòng)方程式表現(xiàn)値微小散亂集積効果表。 衝突間運(yùn)動(dòng)量変化量入射粒子速度乗反比例。 初期運(yùn)動(dòng)方向運(yùn)動(dòng)量変化率、速度乗反比例、初期運(yùn)動(dòng)方向向。21一般化 質(zhì)量mi、電荷ei、速度vi入射粒子質(zhì)

11、量mj、電荷ej、速度vj標(biāo)的粒子衝突。體問題取扱、個(gè)同時(shí)衝突仮定。 中心力場(chǎng)換算質(zhì)量m持粒子散亂表現(xiàn) 標(biāo)的粒子既知分布関數(shù)f(vj)従。 粒子分布関數(shù)発展記述！ 22Generalization of momentum eq.3ijijijijvvAdtpdijijijiijijijjiijmm/mmmmmlneZA,vvv1412042m mm mTaking into account v-4 dependence of 90 ,Characteristics of “Force”1)2)3)23Similarity to electro-static force3ijijijijvvAd

12、tpd)() (414) (403303011rdVrxdxrrxrrqEEqdtpd ) (41)(30rdrrr , , 1xatqxatqxxr24Generalization of momentum eq.3ijijijijvvAdtpdijijijiijijijjiijmm/mmmmmlneZA,vvv1412042m mm m HAmdtdpvijiTaking into account v-4 dependence of 90 ,Introduce a potential H in velocity space2202434ln ,)(1ijijjjjimZeAvdvvfmmH

13、i; test particles, j: field particles25Wessons viewParallel Motion in Velocity spaceCan be interpreted by a potential H. V|V|V|VVVMotion in the Perpendicular direction ? 26 ,)(13jijjjjivdvvfmmH HAdtvdHAmdtdpvjivijivivjjiijvvvTest particle; iField particles; j27Potential H for field particles with Ma

14、xwell distribution2232/32exp2)(thjjthjjjjvvvnvf ,)(1)(3jjijjjiivdvvvfmmvHAssume the isotropic distribution function fj(vj) in velocity spaceUse the spherical coordinatesVi is set to Z-axis.28 200023sinsindvvdddrdvdvvdjjijjvdvvvf3)(22cos2)(jjiijjvvvvvf 1) Volume element d3v in velocity spacejjjijjdvv

15、vvvfd2)(cos2 jijjvvvf)(021122)(cos2cos2jjjjjiidvvfvvvvvd jijijijjjjijjjvvvvvvdvvfvjjiivvdvvfvvvvv1)(2221)(2021102cos2 2) 3) jjijjvdvvvf3)(4)29iivjjjvjjjijijjijjjiidvvfvdvvfvvmmvdvvvfmmvH)()(141)(1)(023 function ;exp22/102errordyyvvvnmmHxithjijji 30Exercise IIdue date; 23 April till noon,vvvnmm)v(H,vdv)v(fmmHithjijjiijijjjji21 13 function ;)exp(202Errordyyxx 222202j2i4j2)( )()(2)2/(12lnZZen xxxxxvvvmmmHAdtvdthjthjijiivji 1) 2) 31Potential H for field particles with Maxwell distribution, E ,)(13jijjjjivdvvfmmH) (41)