Zeroth moment of the Boltzmann Equation The Equation of Continuity (Particle conservation)

Transcription

1 Plasas as Fluids At this poit w d to us a ub of basi quatios that dsib plasas as fluids. Whil it is possibl to alulat ths quatios fo fist piipls, usig Maxwll s ltoagti fild quatios ad Maxwll s vloity distibutio (Ma that guy did vythig!) th poss is tdious, faily diffiult ad vy ti osuig. As th ai of this lass is to dvlop a basi fl fo plasas (wisdo ath tha book sats) i this lass w will siply assu that th quatios a ot. Fo thos of you who fl o advtuso, ots goig though thos divatios a foud at // Plas aliz that thos divatios took ~ 4 lass piods to oplt. Th basi quatios a follows: Boltza s Equatio df f ( f) + ( f) + v v a dt This is lativly asy to pov df fd fd f + v + dt dt v dt f ( f) + ( f) + v v a F f v ( f )+ ( f )+ v Zoth ot of th Boltza Equatio Th Equatio of Cotiuity (Patil osvatio) f ( )+ v Mots a divd by ultiplyig by v ot f(v) ad itgatig ov all vloity. Thus what w a sig is a asu of th avag of this patiula paat. Fist ot of th Boltza quatio Motu Cosvatio This is also kow as th fluid quatio of otio + v v M v f + q + P E v B otu lost via ollisios otu hag via patil gai/ loss Poisso s Equatio (whih os staight fo Maxwll s E-M quatios) ρ Φ ε ( i) ε

2 Boltza Dsity Rlatio xp Φ kt Now w hav th basi quatios (Th a a fw high ots that o ito play lswh that w will ot hav to dal with i this lass.) w a bgi to s how plasas at as a whol. Rb plasas hav ths olltiv bhavios whih w a goig to xplo ow. (W d to udstad ths bhavios so that w a udstad ay of th diagostis usd i plasas.) Th fist bhavio that w will xplo is th ability of th plasa to shild out stati lti filds. This bhavio should ak ss as ay stog lti fild i a plasa will spaat th gativ ad positiv hag ais. Th dista ov whih a fild a ptat is kow as th Dby lgth. (Not that ltoagti filds a also b shildd out BUT th bhavio is vy difft. W will gt to this soo.) Dby Lgth W a ow alulat th Dby lgth a fftiv lgth ov whih a plasa will shild a lti fild. (Th lgth is th / dista fo duig a pottial.) Fist, w hav Poisso s quatio ρ Φ ε ( i) ε W ak th futh assuptio that th dsity of th ltos i abs of th pottial is th sa as th io dsity. (W ak this assuptio baus if ith th ltos o th ios w to lav a aa, a sigifiat lti fild would b stup to ty to pull th bak togth.) Thus, i Pluggig this ad th Boltza latio ito Poisso s quatio givs Φ Φ xp ε kt Now, usig th Taylo sis xpasio of th xpotial, whih w assu is appoxiatly, givs + + Φ Φ Φ O ε kt kt Φ εkt Solvig th difftial quatio lavs

3 λ Φ Φ Dby εkt xp λ Dby wh Bulk Motios At this poit, w d to dal with so of th bulk otios that ou i plasas. Ths a ot sigl patil otios but ath olltiv otio of all/ost of th hag spis i th plasa. Th fist, ad ost ipotat is th ltostati plasa osillatio, givig is to th plasa fquy. [This but just o of a vy wid vaity of wavs i plasas.] Ths osillatios ou baus o of th spis bos displad fo th oth. Wh it alats bak towad th oth spis, i gais too uh gy ad ov shoots. To div th plasa fquy, w will assu th siplst of gotis ad plasas. ) No xtal filds. (This a b laxd ad th sa sult a b obtaid.) ) No ado otio of th patils. (H, all patils of a spis ov at th sa vloity at th sa poit i spa. This a b laxd ad o a gt th sa sult it is just had to do.) 3) Oly th ltos ov. (This is ot a bad assuptio fo ay aspts of plasas.) 4) Th plasa is o-disioal ad of suh a lgth that th walls do ot iflu th sult. (This iplis that w a osidig just gios that a at last sval λ dby fo th walls.) Fo Maxwll s quatios w hav, E ρε E (W will igo th idud agti fild.) Th ou quatio of otio (otu osvatio) ad otiuity (patil osvatio) bo Cotiuity Equatio f ( )+ v ( )+ v Egy Equatio

4 + v v M v f P+ q E+ v B otu lost via ollisios + v v q( E) otu hag via patil gai/ loss Fo this patiula wav, w a osidig dviatios fo hag utality. Thus w will hav a idud lti fild giv by ε E ρ ( i ) W hav th its that a hagig with ti, E, ad <v >. W will xpad ah of ths its to podu a ti avag t, dotd with a ad a osillatig t dotd with a. Thus E E + E - but E! + v v + v - but v! Th ou osvatio of otu (gy) quatio bos ( v + v) + ( + ) ( + ) q v ( + ) v v v E E t ( v) q + ( v) ( v) ( E) Now th sod t o th lft-had sid is sall opad to th oth two. (Two osillatig ts as opposd to o.) Thus w a lft with ( v) q i( ωt βx) iplyig w hav ( E) - lt v - a tavllig wav iωv E Now w a do th sa thig to th otiuity quatio + v t (( + ))+ ( + ) v ( ( )+ ) v Wh agai w hav doppd th high od ts. Thus

5 ( v ) ( ) - lttig t i βv iω Fially, w solv Poisso s Equatio ε E ε E + iβεe ( i ) i ωt βx This givs us th quatios ad th ukows iβv iω, iβεe, iωv E. Cobiig th last two to liiat E givs ωβε. Plaig this ito th fist to liiat givs ω p ; ω p fp / π, ε th agula lto fquy of th plasa. This is also kow as th dispsio latio. v Typially fo poss plasas th dsity is ~ Thus, f p ~ to GHz. This is, i so ss, th siplst osillatio that a xist i a plasa. Not that this is ot a wav i th typial ss! (Egy dos ot ov i this osillatio, th goup vloity, d ω, is zo. H β is th wav vto.) dβ Th a uous oth osillatios that a wavs that a tasf gy. Thy a b dividd ito ltostati wavs ad ltoagti wavs. W will dal fist with th ltostati wavs. Eltostati wavs i plasas Lt us go bak to ou fudatal fluid quatios, th otiuity quatio, th gy quatio, ad Poisso s quatio. As bfo w will igo th ollisio ts ad assu that th agti fild is zo. H howv, w will ilud th pssu vaiatios of th spis. Poisso s quatio ε E ρ ( i ) Cotiuity Equatio

6 f ( )+ v ( )+ v Egy Equatio + v v M v f + q + P E v B otu lost via ollisios otu hag via patil gai/ loss + v v q( E) P This would b idtial to th plasa osillatios xpt fo th pssu t. W will dal with that t fist. W kow fo th idal gas law that p kt. Th, assuig a isotopi, o o disioal, plasa P p kt This is tu povidd that th opssio is isothal. I oth wods th tpatu stays th sa duig th opssio. Oft, this is ot tu. Rath, w hav adiabati opssio, wh th tpatu hags. I this as, it a b show usig thodyais that p C γ. H C is a ostat ad γ Cp CV is th atio of th spifi hats. W a s fo th abov quatio that γ p ( C ) γ C ( ) γ C γ γ γc γp N + Futh, it a b show that γ wh N is th ub of dgs of fdo. N Thus fo N, γ 3. (This is a ud appoxiatio but it woks fo ou ds.) Thus, ou quatio of otio bos + v v q( E) γ kt.

7 As bfo, w will assu that th dsity, vloity ad lti fild osists of a tiavagd t ad a osillatig t. Thus, E E + E - but E! + v v + v - but v! lttig v, E, i ( ω t β x ) W a ow follow th divatio that w ad bfo but this ti w will add ou additioal t. Assuig that w a xaiig ltos, ou osvatio of otu (gy) quatio bos } v + v } ( + ) + + kt } + } v v v v ( + ) E + E + γ ( v ) E γkt i ωv E + iγkt β Now w a do th sa thig to th otiuity quatio + v t (( + ))+ ( + ) v ( ( )+ ) v v ( ) t i βv iω Fially, w solv Poisso s Equatio ε E ε E + iβεe ( i ) This givs us th quatios ad th ukows i βv iω, iβεe, i ωv E + iγkt β.

8 Cobiig th last two to liiat E givs γkt β v + βεω ω whih is vy siila to what w got bfo xpt w ow hav a sod t. Plaig this ito th fist to liiat givs γkt ω + β ε γkt ωp + γβ - wh γ is th lto soud spd Not that h th goup vloity is o-zo, aig that gy a b aid by th wav. Now, lt us assu that w a dalig with (positiv) ios. H, howv, th lti fild is dtid by th ltos, ot th ios. Thus, w d to pla th lti fild with th gadit of th pottial ad us Boltza s latio o th lto dsity. Thus, ou quatios bo + v t βv ω (as bfo) i + v v q( E) γkt ( φ) γkti } v + v } i( + ) + + } + } v v v v ( + ) φ + φ kti γ + i φ γkt i ωv βφ + γktβ i ( v ) i Now, w a t us Poisso s quatio but ath w assu that th hag i loal dsity a b odld with Boltza s quatio. E.g. that th loal io dsity is th sa as th loal lto dsity ad that th lto dsity is giv by

9 i φ / kt φ + kt } φ + φ / kt φ / kt (This appoxiatio auss so sall o) φ kt This givs us ou th quatios ad th ukows φ, βv ω, iωv βφ + γktiβ kt Puttig togth th fist two givs v ω φ. β kt Pluggig this ad th fist ito th thid givs ω γ β kt + kti i This is th io-aousti o io-soud wavs. All of th abov a just a fw xapls of ltostati wavs. Th a ay o ltostati wavs. W will ow dal bifly with ltoagti wavs. It is ipotat to ot that w will oly look at th vy siplst ass. Th a a wid vaity of ltoagti wavs that a sustaid i plasas. Fist, stadad light wavs xist. This os ditly fo Maxwll s quatio. E E E σµ + εµ H H H H E ± η η iµω iµω µ ; if σ γ σ + iεω ε sig dt i d by gowth / day ( gowth >, day > + ) π β ; λ ω εµ I plasas, this is ot quit ot. What happs if th wavs a itatig with th plasa? Fo Maxwll s quatios w hav

10 E B H Jf + D D ρ f ( i ) B ds Howv th ut is ot zo. This hags th filds. To solv th pobl, w will osid oly th ti vayig opots. Takig th ul of th fist quatio ad th ti divativ of th sod quatio givs ( E) ( E) E B µ B Jf + εe W a obi ths to giv ( E) E µjf εµ E ρ } + E E J + β β β iωµ E f εµω - - εµ ( β ω ) E iωµ Jf If th light is at a high fquy, th th ios a fftivly fixd. Thus th ut is alost tily du to th otio of th ltos. Th th ut a b giv as Jf v fo th quatio of otio v F d E dt so that i E J f ω pluggig this i to Maxwll s quatio givs µ E ( β ω ) E µ ω β E ω β ε E ω ω β E ( p ) ω ωp + β This is th dispsio latio of ltoagti wavs i a plasa.

11 Th is a vy usful appliatio of this dispsio latio. Th is a vy itstig phoo that ous, kow as utoff.