From Maxwell s Eqs. and Newton s Laws to the Vlasov and Fokker-Planck Eqs.

Transcription

1 F Maxwell s Eqs. and Newtn s Laws t the Vlas and Fkke-Planck Eqs. Rbet A. Schill, J. Uniesity f Neada, Las Vegas Dept. f Electical and Cpute Engineeing 4505 Mayland Pakway; Bx Las Vegas, NV (70) /4183 Lab: (70) FAX: (70) e-ail: schill@ee.unl.edu URL:

2 Diffeent Kinetic They Appaches R.A.Schill, J. ABCs f Zs Wkshp [5/1-/98] 4

3 Elastic Binay Cllisins - Cente f Mass Syste C.M. c c Oigin c Cente f Mass (C.M.) Fae Cental Fce Tw bdy cllisins is equialent t a single bdy cllisin Mass Ments C + c 0 Cnseatin f Mass and Mentu & u cnst. c Relatie Mtin with Respect t C.M. Newtnian Fce c & & & M and M & ; M c c j F && ; educed ass Tque exeted by Cental Fce is ze & cnst. f tin Tque & Angula M. - educed ass + + j R.A.Schill, J. ABCs f Zs Wkshp [5/1-/98] 7

4 z~ u P Binay Cllisins in a Cental Fce - Single Bdy Pble (1) x' Apse Line y' P P-dP θ θ P ~ x, x O θ P -Ipact Paaete χ - Scatteing Angle P -Vetex Pnt. f Pat. Taj. (,θ ) OP - Apse Line (length ) O - Oigin (Paticle ) P - Paticle (Paticle ) u - Initial Velcity f Paticle - Reduced Mass f Paticle χ y~ χ+ dχ z Gening Equatins [Pla Cd. (,θ)] P Ext. Cental Fce (Inteactin Pt. Φ) F Φ Dynaics Kinetic Enegy Paticle T 1 & + θ & Angula Mentu Γ y $ θ& Tque I F dγ dt 0 Cental fce is adial N fce in θ diectin Cnstants f Mtin ()() tθ & t ( t ) θ & ( t ) Cnst. R.A.Schill, J. ABCs f Zs Wkshp [5/1-/98] 8 ext [( θ ) ( θ θ) θ] && & $ && && $ d Φ $ + + d Γ y Tt + Φ Tt + Φ Cnst.

5 Assuptins Initially plasa is in theal equilibiu Psitie test chage placed in plasa n - plasa density fa f test chage T e electn tepeatue; electn chage q -e Cllisinless plasa Since, i >> e the in dist. is assued nt affected by the test chage Vlas Equatin assuing an Maxwell-Bltzann Equilibiu f t Debye Shielding - Statistical Appach (1) f F f whee f (, ) e Geneally Spheically Syetic E-field f (, ) Culb Fce - F qe q $ dφ d f q d Φ f spheical cds d Methd f Sepaatin f Vaiables Bunday Cnditins (Maxwellian at infinity) n f d, ;. f, Ae e 3kT whee k Bltzann' s Cnst. e R.A.Schill, J. ABCs f Zs Wkshp [5/1-/98] 15 1 A qa Φ e

6 Debye Shielding - Statistical Appach () V, Vc V Vc R ( R) V( R) Φ Φ Φ R V R R λ D q Φ 4πε λ Φ c c D Electn Velcity Distibutin Functin 3 (, ) ( ) Φ π f n kt e e e Electn and In Densities Φ n e ne n i n q kt e Electns Shield Test Chage - Weak Field Appx. Tayl Expansin Applied: n e Pissn s Eq. Φ( ) Φ( ) λ D whee λd εkte nq Debye Ptential and Debye Length Nea test chage 0; Culb-like Ptential Fa f test chage; Pefect shielding kt q kt e e e λ D Φ q 4 πε e λ D - the Debye Length; Gd shielding 1 t 5 λ D Pt. inalid 0; ilates weak field appx. R.A.Schill, J. ABCs f Zs Wkshp [5/1-/98] 16

7 Ae. Fce Acting n a Test Paticle in a Plasa (1) u j u P j j+1 1 x 1 P 1 x n n z 1 x j P u'' j P z n x n j P j ~x j x n+1 n+1 x 3 z 3 P P χ j n+1 z n+1 Chaacteistics [MKS] Chaged test paticle ;, q, zj Dist. chaged field paticles (scattees); ~ z Fee space; ε,µ j Eq. f Mtin in Culb Field: CM s Lab. Fae CM: && 3 qq 4πε Mtin f C.M. is a cnstant LAB: F && && Cllisin with Field Paticles ( ) u sin ( χ ~$ j ) + j : F && z 3 P 3 u j + j+1 : F [ u ( χ j ) t] u Many paticles with elcity between and df qq 8πε u u ln f, q, u z usin χ x ~$ sin + d P + P P ax f ( ) d j Shielding effect: P ax λ D ; λ D >>P R.A.Schill, J. ABCs f Zs Wkshp [5/1-/98] 17

8 Ae. Fce Acting n a Test Paticle in a Plasa () Cnsequence f Integal Tuncatin Plasa is nealy in a state f thedynaic equilibiu Debye shielding akes sense Paticles d nt inteact and d nt undeg significant scatteing f ipact paaetes lage than the Debye length Justified cutff distance - P ax ~ λ D Nn-equilibiu plasas - cutff distance abiguus - λ D nt justified Exact alues f P ae n lnge apppiate Mean P --- u 3k( T + T ) qq Culb Lg - ln Λ ln( λ D P ) whee P 6πεk[ T + T] Aeage Fce Acting n Paticle due t all Scattes qq F ( ) d ln Λ 8πε Aeage Change in Paticle Velcity d t dt F Inadequate desciptin f tin IN the plasa R.A.Schill, J. ABCs f Zs Wkshp [5/1-/98] 18

9 tt 3 d dt i i z tt 4 t >t >t >t >0 x () i t t 0 () () d i j i t j t dt i Statistical Desciptin f Paticle t0 tt tt 1 Paticle Enseble Mdel N ( ) f abslutely identical test paticles with sae initial elcity Spatial Desciptin (tw diffeent ways t iew) Neglect - unif e plane in cnfiguatin space y Retain - unif e all cnfiguatin phase space Iagined as a spheical clud f phase pints cncentated at a pint in el. space, at tie t0; f (,, t 0) n δ( ) Ae. tin and speading f clud (ppeties sught) Cplete Desciptin f Clud Dynaics [i,jx,y,z] / supescipt -scatteing f ONE pat. in ed. f s Rate f change f fist ent - ate f change f the cente f gaity f the paticle clud in a ediu f field paticles Rate f change f ent - speading f clud sy. tens ank chaacteizes the 3-D ean squae deiatin Rate f change in highe de ents - negligible R.A.Schill, J. ABCs f Zs Wkshp [5/1-/98] 19

10 Rate f Change f Ments - Integal F i w i w f Rate f Change f Ments () wd i λd π i i 0 0 j λ D π uupdpdϕ f w d ij u u up dp dϕ ij i j 0 0 Cputed fs f i whee 1 L ui u f d 3 L 1+ L 1+ [ Φ( ) ] 1 u L δ iu 4π 3 u u i j ij L i qq ε j i i 1 4π j f d f d 1 L f ( ) d i j 4 π L [ Ψ ] ln Λ u uz$ R.A.Schill, J. ABCs f Zs Wkshp [5/1-/98] 1

11 Rate f Change f Ments-Obseatins Ψ ( ) ( ) Φ Ψ ( ) Φ ( ) f ( ) Highe de ents sall by fact f ln(λ) - - Culb Lgaith Special featue f Culb Ptential Fkke-Planck Appxiatin Spitze - cined the fllwing tes as diffusin tes and i i j + i k i k Rsenbluth Ptentials Φ andψ The elcity f the electn clud as a whle (elcity f the cente f gaity f the clud) in elcity space is elated t the ate f expansin f the clud in all diectins R.A.Schill, J. ABCs f Zs Wkshp [5/1-/98]

12 Cntinuity Equatin f Phase Pints in Phase Space z, z Ω (6) Integal F f Cntinuity Eq. f Phase Pints (Absence f Cllisins) t f t d J Ω t d,,,, Σ s 6 ( 5) Ω Σs x, x Σ y, y (5) s J (6) - Six di. flux f phase pints J 6 f ( t) a f ( t) J,, +,, + J J Kineatic Steaing Te Pint F f Cntinuity Eq. f Phase Pints (Absence f Cllisins) f t f t + J + J [ f] [ af] R.A.Schill, J. ABCs f Zs Wkshp [5/1-/98] 3

13 x t0- t0+ y (t0-)0 (t0+)>>0 ' z Ω (6) Σ (5) s Abupt Change in Velcity Cllisins - Geneal Discussin Cnsequence [Macscpic View] Abupt change in psitin and elcity Abupt changes in spatial psitin ae negligiblecntinuus f pint-t-pint In the cnfiguatin ptin f phase space, the phase pint will actually cut thugh the spatial suface f Σ (5) s bunding lue Ω (6) when enteing leaing Abupt changes in elcity ae dastic Befe cllisin afte cllisin At the instant f the cllisin, the phase pint in elcity space is annihilated at and ceated at a ete pint withut passing thugh inteediate pints in elcity space Des NOT cut suface Σ (5) s J can nt accunt f this Nea cllisins (sall ipact paaetes) Culb paticles ( lage ipact paaete) d nt exhibit dastic changes in elcity R.A.Schill, J. ABCs f Zs Wkshp [5/1-/98] 4

14 Cllisins - Culb Cllisins (ln Λ>>1) f t J cl (5) Σ s Neighbhd f Suface Abupt Change in Velcity 1 ext + ( f) + ( F f) J Fcl i f D f ik k Teat as Micscpic Jups - alst cntinuus Equatin f Cntinuity is alid f ete inteactins Tayl expansin f the kineatic steaing te abut the neighbhd f Σ (5) s J A f ( t) B f t i i,, + ij,, +... Salle the jup, the e cntinuus is the flw 1 A a [ ] F ext F cl B D i i i + i ; ij ij Plasa Kinetic Equatin - Culb Cllisins Kineatic steaing is due t the fce f cl Dynaic Fictin [Chandasekha] Diffusin Flux Many & Single Specie Plasa - Binay Cllisins J J ; F F ; D D cl cl cl cl ik ik j R.A.Schill, J. ABCs f Zs Wkshp [5/1-/98] 6

15 Rate f Change f the Ments f the Paticle Enseble Distibutin Functin f Paticle Enseble at t0 f (,, t 0) n δ( ) Fal Definitins i 1 1 if( t) d i j ( i i)( j j) f ( t) d n,, n,, Kinetic Equatin (Altenate F) f t t (,, ) ext cl Methd f Chaacteistics (Lagangian fae-fllw phase pint bits) df t dt (,, ) F cnsistency with u del, we assue thee ae n extenal fces. Only Culb cllisinal fce cntibutins ae assued t exist in deteining the fce f dynaical fictin and the diffusin tens. Rate f Change f Ments Ealuated at t0 R.A.Schill, J. ABCs f Zs Wkshp [5/1-/98] 7 F + f t + (,, ) f (,, t) J (,, t) Fcl J whee J f t D f t J cl cl,,,, di 1 df dt n dt d cl i D ik i i + F t k d df ( )( ) [ ] dt n dt d D i k 1 i i i j j ik j cl

16 Fkke-Planck Equatin Fkke-Planck Cllisin Te due t Culb Cllisins J cl i i f t 1 i k (,, ) f (,, t) Kinetic Equatin with Fkke-Planck Cllisin Te ext f(,, t) F + f (,, t) + f (,, t) t k F cl i i 1 i k i k and Dik 1 k R.A.Schill, J. ABCs f Zs Wkshp [5/1-/98] 8 i f t i t 1 i k t 1 f t f t,, :,, F f t D f t,,,, whee, i t L f t d + 1,, 1 i 4 π i j 1 L f( t) d i j π,, 4 i k (,, ) f (,, t)

17 Fkke-Planck Eqs. - Obseatins (1) The Diffusin Tens is elated t the initial ate f change f the tens that descibes the quadatic deiatins f the elcity f the ean in the test paticle enseble. D ( )( ) 1 d dt The Dynaic Fce f Fictin is pptinal t the Mean Fce. The Mean Fce is physical. The Dynaic Fce f Fictin is defined ut f cnenience. F F + Because f the sign diffeence assciated with the diffusin and the dynaic fce f fictin tes, they tend t be cpeting echaniss. Diffusin has the tendency f speading the distibutin in elcity space away f the aeage elcity wheeas the dynaical fictin has the tendency f slwing dwn speeding up paticles until they each an aeage elcity. Because f this cpeting effect, the Fkke-Planck cllisin tes yield a ze esult f Maxwellian distibutin functins. R.A.Schill, J. ABCs f Zs Wkshp [5/1-/98] 9

18 Fkke-Planck Eqs. - Obseatins () f in the Plasa Kinetic Equatin is due t all fces (extenal echanic and electagnetic, and Culbic). Petubatin schees ae used t sle these equatins. The they deelped pides a fundatin f studying tanspt. R.A.Schill, J. ABCs f Zs Wkshp [5/1-/98] 30