The Boltzmann transport equation and the diffusion equation

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1 The Bonn npo eon n he on eon Sego Fnn gop Depen o Boec Engneeng T ne oeng gh popgon n ceng e wh npo heo The Bonn npo eon BTE bnce eonhp h ecbe he ow o pce n ceng n bobng e. The popgon o gh n opc b e cn be oee b he npo eon whee he phoon e ee he npoe pce. I we enoe he ng phoon en wh whch ene he nbe o phoon pe n oe pe n o nge eng n econ poon n e we cn we he BTE oow De n Hon 976: whee he pee o gh n he e he bopon coecen n o c - he ceng coecen n o c - he phe ncon o he pobb en o ceng phoon h e ong econ no econ n he oce e. h n o n epeen he nbe o phoon njece b he gh oce pe n oe pe n e pe n o nge poon e n econ. The e hn e o E. epeen he epo on o he ng phoon en. Ech one o he e on he gh hn e epeen pecc conbon o h on. The e he ne gn o phoon poon n econ e o he ow o phoon. The econ e he o o phoon n e o coon bopon n ceng. The h e he gn o phoon n e o ceng. Fn he oh e he gn o phoon e o he gh oce. e now ene oe o he ne e o ecbe phoon npo.

2 Ang phoon en: ene ch h epeen he nbe o phoon n h e n econ whn on. The n o e - -. Phoon nce:. epeen he nbe o phoon eng pe n e pe n e pepenc o n nge o econ whn on. The n o e Phoon en:. The phoon en he nbe o phoon pe n oe. The n e -. Phoon ence e: E E. The phoon ence e ene he nbe o phoon eng pe n e pe n e pepenc o he econ o popgon oe econ. The n e - -. Phoon cen en o phoon :. The phoon eco h epeen he ne ow o phoon. I econ pon n he econ o he ne whe pe ge he ne nbe o phoon ne pe n e pe n e n h econ. The n o e - -. The boe enon cn be eene o ecbe n eneg ne o phoon nbe b epcng he wo phoon wh eneg n b nocng co hν n enon hν he eneg pe phoon whee h Pnck conn n ν he gh eenc. A copee noence o ne e n ec opc cn be on n Hee e. 99. Epnon o he Bonn eon n phec honc To oe gh popgon n hgh ceng e e o boogc e e o epn he ng phoon en he oce e n he phe ncon no phec honc Kenbch n Kchke 99; Bo 99; Age 999. The o-ce P ppoon o he Bonn eon ee Secon. be on ch n epnon. A e o he copeene pope o he phec honc n ncon hϕ wh cen conn popee cn be epne n he pce ee W 99: h ϕ h

3 whee h e coecen nepenen o n ϕ n he eonhp ong ϕ n n co ϕ n n ϕ co. Accong we epn n no phec honc oow:. We e h he phe ncon on epen on.e. on he cone o he ceng nge. We cn h epn n egene pono b ecng h ncon H whch econ conno ogehe wh ee n he ne [-] h he gene egene ee epeenon W 99: H H P. 5 whee P he egene pono o oe n H H P. We hen we: P * whee he epeon oow o he on heoe o phec honc W 99 ne P * /. Hee co P co co. B bng hee epeon no E. we obn: The neg n honc: *. 7 cn be cce ng he ohogon pope o he phec * δ. The BTE h becoe: δ [ ]. 6 8 We hen p h eon b * n nege oe o obn he eonhp beween he pecc coecen n n he coecen o he phec honc epnon o :

4 *. 9 The neg oe cn be ee b wng he n coponen o he eco n e o phec honc. Th cn be one b ng he ecence eon o he oce egene ncon The e he oowng: P. co n / / / / ϕ n n / / / / ϕ. co / / ng hee epeon o he n coponen o pobe o cce he neg ng he ohogon eon o phec honc. We n h he eonhp beween he pecc coecen n oe no noe he coecen o he phec honc epnon o b on conn wh nce ngng o - o n ngng o - o :

5 . / / / / / / The P ppoon The epnon o he BTE no phec honc h e o n nne e o eon wh nce ngng o o n ngng o - o. Tncon o he pce ee e o he o-ce P ppoon. The eon o h ne h he e n he nce pce ee conn whch cn be wen n e o he oce egene ncon whch n n cn be wen n e o he egene pono P. The eonhp e he oowng: P co!! / ϕ e P. / P P 5 5

6 The P ppoon We now cone he P ppoon whch oen e o ecbe phoon gon n e. In he P ppoon e o o. In he P ppoon E. e o eon. The o : > 6 he econ o -: 7 he h o : 8 n he oh o :. 9 The coecen n e ee o he phoon en n o he phoon epece. In c: nce o n / n: 6

7 . co n n co n * * * * ϕ ϕ The e o o eon 6-9 o he P ppoon e h een o he oowng wo eon one c n one eco:. The eco eon obne b cobnng E ccong o he oowng o eonhp: 7.8. / 7.9] [7.7 / 7.9] [7.7 / Fo he gene enon o he coecen we n h n e gen b: co co co co co P > < co co co co co co co P 5 whee n E. we he e he c h he ceng pobb noe ccong o he conon whch een o. Theeoe whee he ege cone o he ceng nge co co > < co. The oce e n E. n e o onopoe e phec ec n poe e epece. We w nce he wh he bo n epece. The n epeon o he P eon e: S S S 6. co S > < 7 7

8 The ece ceng coecen Eon 6 n 7 how h n he P ppoon n co on ppe n he e -<co>. In h econ we ge phc enng o h e on he b o n n epoe b Zccn e. 99. Sppoe h phoon ee he pon P n econ ẑ. Th phoon w be cee pon P e hng ee nce. Then w be cee pon P e hng ee nce n o on. In gene we ee o he ceng pon P n he n-h oe ceng. We wn o ene he ece ceng coecen he nee o he ege nce pojece ong he h he phoon h o e o oe eo o he n econ o popgon. In ohe wo / epeen he ege nce beween wh e eece oopc ceng een. In he eon o we negec he bopon o he e nce we e on neee n ceng popee. The pobb en g o eng nce who eng ceng een ene g e. The oe ceng occ P whoe ege coone e Zccn e ;. 99: < < >< > > g. The econ oe ceng occ n co ϕ n n ϕ P co. Snce ϕ e no c coee he ege e o he coone o P e Zccn e ;. 99: < < < >< >< >< >< n >< n > < >< co ϕ >< n ϕ >< co > > > < co >. The h oe ceng occ P whee: n co ϕ co co ϕ n n ϕ n ϕ co n co ϕ n co ϕ co n ϕ n n ϕ co ϕ co n n ϕ n co ϕ n co co. The ege e o he coone o P e Zccn e ;. 99: < >< > < > < co > < co >. 8 9 In gene he n-h oe o ceng he ege e o he coone o he ceng pon e Zccn e ;. 99: P n < < n n >< > n- k < co > > < co > > n k n < co 8

9 n whee we he e he e o he geoec ee k wh <. In he o k hgh nbe o ceng een n < >< > n < > /[ < co > ] ge he coone o he cene o e o he c phoon bon. In pc he coone < > cn be nepee he ege nce beween conece eece oopc ceng een n nee ene he ece ceng coecen : n < co >. In he ce o oopc ceng < co > n. In he ce o ow ceng < co > n. The P eon n he n on eon SDE We now ece he P ppoon o nge eon o he phoon en E. 7 we obn :. Fo D D D S S whee we he ene he on coecen D /[ ]. B bng h epeon o n E. 6 we ge: D D D S S. Fo E. 6 S. B bng h epeon n b engng he e we n ge he P eon o he phoon en: D D D S S S. D B kng ew pon whch e oen e n he ce o gh popgon n boogc e E. 6 ece o he n on eon SDE. The pon e he oowng: Song ceng ege o <<. Th conon en h phoon on he ege w nego n eece oopc ceng een beoe beng bobe. In h ce D / / << n he econ e on he gh hn e o E. 6 ece o / D /

10 Te ce o he on o n S e ch gee hn he ege e beween coon /[ ]. Th conon cn be epee b he o ne: / << /D. Coneen: << D S << S. D In he eenc-on whee he honc e epenence gen b co ep-ω he e ee opeo becoe pcon b -ω. Hee ω he ng oon eenc o he nen oon whch ho no be cone wh he eenc o gh. Coneen h conon poe n ppe o he oon eenc gen b ω << /D. In he ce o boogc e he SDE bek own oon eence on he oe o GH Fhkn e The oce e oopc.e. S. 7 8 Wh hee pon he P eon E. 6 ece o he n on eon: D S n he phoon ee o he phoon en b Fck w: 9 D. In he eenc-on eon: whee k ω /D. / ω n he on eon ke he o o he Heho S k D Soon o he n on eon n he eenc-on The oon o he on eon o hoogeneo nne e connng honc oe pon oce o powe Pω gen b Bo e. 99: k P ω e ω. D The epc epeon o he ege phoon en DC n o he pe AC n phe Φ o he phoon-en we e Fhkn n Gon 99; Hke e. 99; Fnn e. 99: DC / / D PDC e D

11 AC / ω / / / D P ω e ω D / / / ω Φ ω D Φ 5 whee Φ he oce phe n n. Anc oon n he eenc-on he o been epoe o e-nne e Peon e. 99; Hke e. 99; Fnn e. 99 nne b Age e. 99 cnc n phec geoee Age e. 99. Eon 9 n ee o hoogeneo e. Fo ne e pecocop n oe one pc e h e e cocopc hoogeneo o h E. 9 n e ppcbe. B con opc gng o e eng he p bon o he e opc popee n E. 9 be genee o ccon o he p epenence o n D. Reeence: Age S. R. Opc oogph n ec gng Inee Pobe 5 R-R Age S. R.. Cope n D. T. Dep The Theoec B o he Deenon o Opc Phengh n Te: Tepo n Feenc An Ph. e. Bo Bo D. A. De Phoon Pobe o Sc n Dnc Popee o Tb e: Theo n Boec Appcon Ph.D. The Dep. o Phc ne o Pennn 996. Bo D. A.. A. O'e B. Chnce n A. G. oh Sceng o De Phoon Den We b Sphec Inhoogenee whn Tb e: Anc Soon n Appcon Poc.. Ac. Sc. SA Ce K.. n Zwee P. F. ne Tnpo heo Aon-Wee Pbhng Copn Reng che Po Ao onon Don Ono 967. De.. n.. Hon ce Reco An We ew ok 976 p.. Fnn S.. A. Fncechn n E. Gon Se-Inne-Geoe Bon Pobe o gh gon n Hgh Sceng e: Feenc-Don S n he Don Appoon. Op. Soc. A. B Fhkn. B. n E. Gon Popgon o Phoon-Den We n Song Sceng e Connng n Abobng Se-Inne Pne Bone b Sgh Ege. Op. Soc. A. A Fhkn. B. S. Fnn.. neven n E. Gon Gghe Phoon Den We n Tb e: Theo n Epeen Ph. Re. E Hnen. E. n T. D. gh ceng n pne ophee Spce Scence Reew Hke R. C.. O. Sn T. T. T T. C. Feng. S. ca n B.. Tobeg Bon Conon o he Don Eon n Re Tne. Op. Soc. A. A Hee F.. Peon. Pe n B. Won Recoene oence o Phc Qne n ec Appcon o gh AAP Repo o. 57 Aecn Ine o Phc Woob pp Ih A. We popgon n Sceng n Rno e Acec Pe ew ok Sn Fncco onon 978. /

12 Kenbch.-. n. Kchke Feenc- n Te-Don oeng o gh Tnpo n Rno e n ec Opc Toogph: Fncon Igng n onong Eo G.. e e. SPIE Bengh Whngon 99 pp Peon. S.. D. oon B. C. Won K. W. Ben n. R. kowc Feenc- Don Reecnce o he Deenon o he Sceng n Abopon Popee o Te App. Op W H. W. hec eho o Phc Aon-Wee Reng A 99 Chpe. Zccn G. E. Be P. Bcgon n Q. We Anc Reonhp o he Sc oen o Sceng Pon Coone o Phoon gon n Sceng e Pe App. Op