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1 oil XIPT Lecture 8 Boltzmnn Eqution continued 81 Lst time derived Boltzmnn eqution oottuie + E tooth " nd obtined generl expression for collion integrl We now wnt to get some physics out of th eqution But first need to dcuss conservtion lws Conservtion lws We sw before tht conserved quntity X( ii F stfies Jdfxir F (fetho o Note : before integrted lso over F but since ( cn locl th stfied for ech F Going through H orem exct sme Steps s in derivtion of ( Ii 2 ; F Pi I E sum nd verge gt Jdfxir tfdrde F (fetho dpidrffrlv fz][ Xlpjlttlpz [ fifz f 1 PIT X( O Since by definition X conserved if XCF +X( Fz XIPY + XIPI NOT : The Conservtion lws re of course more generl thn Boltzmnn eqution They pproximtions mde in re good deriving check tht Boltzmnn eqution do not brek required conservtions
2 Etvip If 8 2 Conserved quntities re X 1 Prticle number momentum EIFI energy Equilibrium on o ( h Boltzmnn ftplftpz ftpyftpzy lst hve tiny In where t h function from Here superscript zero refers to n hve lnf lf + lnf ( Fc hf lpi + ln FTPI tht lnf invrint under collions 1 I nd E re only invrints must hve ln f 131Mt Tip e with ptfz p nd J constnts writing m like th Is Convention Th mens f cexp [ p( in ] If E cn write th 2 in f cexpfztntcp mil ] Where hve bsorbed or nd vh into coefficient C which in turn determined by normliztion
3 Stokes i Locl 83 The coefficients in function cn depend on position f lfp ( exp [ PIE M F } with p PC F t p MIF t 1 nd 55 ( i t Th wht clled locl Becuse of loclity of collion term locl not function ffected by it ie looked to otlou Locl hover not nd refore not solution to Boltzmnn eqution th mens tht it ffected by Note tht if th gives re to dynmics on scle of collion time of time scles : Collions quickly follod by streming slor dynmics of hydrodynmic One cn in fct Streming tht re slow terms hve seprtion estblh locl terms vribles derive equtions of when combined with conservtion lws quntities t one Obtins giving hydrodynmics Depending diff on re to sion het of eq Wvier ete
4 Relxtion time pproximtion 84 with se seprtion of time scles in mind use following Simplified mom of collion cn integrl *1 t where f f C F Fit locl dtribution function In bsence of streming terms nd with Jf f fo hve # r st ot T He Stlrip Sfc F pit oe tht system towrds locl scle given by relxes exponentilly everywhere dtribution on relxtion time T ( which in time generl function of F but tke it here to be constnt Q : which locl dtribution function does one use in relxtion time pproximtion?
5 Consider As Vcosity 8 5 Let s now s wy of demonstrtion clculte vcosity of fluid Imgine following setup ^ Z mmmo F Ux ( z > X A with F estblhes plte drgged force flow pttern flcoso loose t Z in where fluid fuicl height flows in with 4 (+1 direction speed consequence moment in re net flux of director tht in resulting drg free counterblnces F re Pzx The momentum flux density drg force per given by p* y f where y vcosity Let s first understnd phenomenologiclly how th comes bout plne t fixed z nd sk wht momentum flow through th surfce \ TI " T z :
6 Inme Enme mug 86 let s ssume tht ll prticles with negtive velocity cme from ttlcoso nd refore hve verge x speed U ( Ztlcoso Similrity prticles moving up hve speed uxlz lcosol l#7tnfodiplrvzmufz+ecoot The totl flow of Momentum in 2 direction n Tp net Plilvzmuxlrt ] ( odi nim[ Sdipklrzuxhtjfdrpkluzotztlosotf ;] now since coso 1 nd V / verge of { vz o th gives where off # Sdi Ir Pk v ir verge prticle speed Sdi 9V 0 [ Mxll velocity 9 fuitidrv ( Itl "2exp( dtribution feet " " (Ftktlkexpl FED Autogethv th gives us jtpx Pxz tznmlvr 0 or y tznmlrr
7 Generlly ( Also 87 Vcosity s kly inhomogeneous gs Lets solve sme problem illustrting Some generl strtegies towrds solving on wy Boltzmnn eqution from First if locl f lr 5 only kly perturb wy cn write ff tjf ( Ft + Eno of t E ( fotsfl (f c since collion integrl zero for f right hnd side of order St For vcosity cse M Const T const nd J Uxlz I I o so left hnd side becomes Pz 0f Oz The term 00 higher simply order in St nd + constnts cn refore drop it t F nl on nd J cn ll depend µ T e see g more complicted expression giving Continuing ot Oux Px o dux of Eet Oz OZ Ot dux since Tip Th e f ~ Aror
8 The n n ni In relxtion time pproximtion n 88 P + gu f To Clculte momentum flux with th result Pyz n df Pxvzorf HEoff fdf Pxuz P xf n Et df mvimui? he ystzmvxz < Ennui Et of 4 CKBTT Kt oz nd refore y nktstt n # vin 1 Uv ginml Vv in our i e prefctor of insted of zt rough Ip Clcultion difference Immteril re eqully pproximte clcultions since but se th Second clcultion contined importnt elements of typicl clcultion
9 Einstein It Im[ et In Dirc E µ( Boltzmnn Quntum systems 89 The Boltzmnn eqution description for quntum systems hve been similr to clssicl description very dcussing Bose Insted of Mxll dtribution or Fermi function form : form insted f ( i E e I exp ( E ( k Ttn Fit ± The collion term lso hs to be modified to tke Cre of stttics ddition semiclssicl equtions of motion for n electron wuepckrt ( in metl re { ( E xsi nic eixb In first eqution second term mostly msing in textbooks ws introduced by Krplns until in but lmost completely ignored Luttingr in it ws reinterpreted when recently very of Berry curvture nd terms < D EKUE 1 15 nuk>i The second modifiction compred moment orbitl comes from mgnetic with textbooks contribution
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