Lecture 21. Boltzmann Statistics (Ch. 6)

Transcription

1 Lctur. oltzmann tatstcs (Ch. 6) W hav followd th followng logc:. tatstcal tratmnt of solatd systms: multplcty ntropy th nd Law.. hrmodynamc tratmnt of systms n contact wth th hat rsrvor th mnmum fr nrgy prncpl. Howvr th ln btwn G and th procss of countng of accssbl mcrostats was not straghtforward. ow w want to larn how to statstcally trat a systm n contact wth a hat bath. h fundamntal assumpton stats that a closd (solatd) systm vsts vry on of ts mcrostats wth qual frquncy: all allowd stats of th systm ar qually probabl. hs statmnt appls to th combnd systm (th systm of ntrst th rsrvor). W wsh to translat ths statmnt nto a statmnt that appls to th systm of ntrst only. hus th quston: how oftn dos th systm vst ach of ts mcrostats bng n th thrmal qulbrum wth th rsrvor? h only nformaton w hav about th rsrvor s that t s at th tmpratur. Combnd systm const srvor - ystm a combnd (solatd) systm a hat rsrvor and a systm n thrmal contact

2 σ h undamntal Assumpton for an Isolatd ystm σ σ Isolatd th nrgy s consrvd. h nsmbl of all qu-nrgtc stats a mrocanoncal nsmbl h rgodc hypothss: an solatd systm n thrmal qulbrum volvng n tm wll pass through all th accssbl mcrostats stats at th sam rcurrnc rat.. all accssbl mcrostats ar qually probabl. h avrag ovr long tms wll qual th avrag ovr th nsmbl of all qunrgtc mcrostats: f w ta a snapshot of a systm wth mcrostats w wll fnd th systm n any of ths mcrostats wth th sam probablty. robablty of a partcular mcrostat of a mcrocanoncal nsmbl / (# of all accssbl mcrostats) h probablty of a crtan macrostat s dtrmnd by how many mcrostats corrspond to ths macrostat th multplcty of a gvn macrostat macrostat Ω robablty of a partcular macrostat (Ω of a partcular macrostat) / (# of all accssbl mcrostats) ot that th assumpton that a systm s solatd s mportant. If a systm s coupld to a hat rsrvor and s abl to xchang nrgy n ordr to rplac th systm s trajctory by an nsmbl w must dtrmn th rlatv occurrnc of stats wth dffrnt nrgs.

3 ystms n Contact wth th srvor srvor - ystm h systm any small macroscopc or mcroscopc objct. If th ntractons btwn th systm and th rsrvor ar wa w can assum that th spctrum of nrgy lvls of a waly-ntractng systm s th sam as that for an solatd systm. Exampl: a two-lvl systm n thrmal contact wth a hat bath. W as th followng quston: undr condtons of qulbrum btwn th systm and rsrvor what s th probablty ( ) of fndng th systm n a partcular quantum stat of nrgy? W assum wa ntracton btwn and so that thr nrgs ar addtv. h nrgy consrvaton n th solatd systm systmrsrvor : const Accordng to th fundamntal assumpton of thrmodynamcs all th stats of th combnd (solatd) systm ar qually probabl. y spcfyng th mcrostat of th systm w hav rducd Ω to and to. hus th probablty of occurrnc of a stuaton whr th systm s n stat s proportonal to th numbr of stats accssbl to th rsrvor. h total multplcty: ( ) Ω ( ) Ω ( ) Ω ( ) Ω ( ) Ω

4 ystms n Contact wth th srvor (cont.) ( ) ( ) ( ) Δ ( ) ( ) ( ) ( ) - - ( - ) ( - ) ( ) ( ) ( ) ( ) d d Lt s now us th fact that s much smallr than ( << ). h rato of th probablty that th systm s n quantum stat at nrgy to th probablty that th systm s n quantum stat at nrgy s: ( ) ( ) ( ) ( ) ( ) [ ] ( ) [ ] ( ) ( ) Δ Ω Ω xp xp / xp / xp ( ) d d d d μ (r. 6.9 addrsss th cas whn th nd s not nglgbl) Also w ll consdr th cas of fxd volum and numbr of partcls (th lattr lmtaton wll b rmovd latr whn w ll allow th systm to xchang partcls wth th hat bath

5 oltzmann actor ( ) ( ) Δ xp xp s th charactrstc of th hat rsrvor xp(- / ) s calld th oltzmann factor ( ) ( ) xp xp hs rsult shows that w do not hav to now anythng about th rsrvor xcpt that t mantans a constant tmpratur! h corrspondng probablty dstrbuton s nown as th canoncal dstrbuton. An nsmbl of dntcal systms all of whch ar n contact wth th sam hat rsrvor and dstrbutd ovr stats n accordanc wth th oltzmann dstrbuton s calld a canoncal nsmbl. rsrvor h fundamntal assumpton for an solatd systm has bn transformd nto th followng statmnt for th systm of ntrst whch s n thrmal qulbrum wth th thrmal rsrvor: th systm vsts ach mcrostat wth a frquncy proportonal to th oltzmann factor. Apparntly ths s what th systm actually dos but from th macroscopc pont of vw of thrmodynamcs t appars as f th systm s tryng to mnmz ts fr nrgy. Or convrsly ths s what th systm has to do n ordr to mnmz ts fr nrgy.

6 On of th most usful quatons n D... ( ) ( ) Δ xp xp rstly notc that only th nrgy dffrnc Δ - j coms nto th rsult so that provdd that both nrgs ar masurd from th sam orgn t dos not mattr what that orgn s. condly what mattrs n dtrmnng th rato of th occupaton numbrs s th rato of th nrgy dffrnc Δ to. uppos that and j. hn ( - j ) / -9 and ( ) ( ) j 9 8 h lowst nrgy lvl avalabl to a systm (.g. a molcul) s rfrrd to as th ground stat. If w masur all nrgs rlatv to and n s th numbr of molculs n ths stat than th numbr molculs wth nrgy > ( ) n n xp roblm 6.3. At vry hgh tmpratur (as n th vry arly unvrs) th proton and th nutron can b thought of as two dffrnt stats of th sam partcl calld th nuclon. nc th nutron s mass s hghr than th proton s by Δm.3-3 g ts nrgy s hghr by Δmc. What was th rato of th numbr of protons to th numbr of nutrons at K? ( n) ( p) m xp nc m c xp p Δmc xp 3.3 g xp 3.38 J/K 8 ( 3 m/s). 86 K

7 Mor problms roblm 6.. s oltzmann factors to drv th xponntal formula for th dnsty of an sothrmal atmosphr. h systm s a sngl ar molcul wth two stats: at th sa lvl (z ) at a hght z. h nrgs of ths two stats dffr only by th potntal nrgy mgz (th tmpratur dos not vary wth z): xp ( ) ( ) mgz xp xp At hom: A systm of partcls ar placd n a unform fld at 8K. h partcl concntratons masurd at two ponts along th fld ln 3 cm apart dffr by a factor of. nd th forc actng on th partcls. Answr:.9-9 mgz xp xp ( z) ρ xp A mxtur of two gass wth th molcular masss m and m (m >m ) s placd n a vry tall contanr. h contanr s n a unform gravtatonal fld th acclraton of fr fall g s gvn. ar th bottom of th contanr th concntratons of ths two typs of molculs ar n and n rspctvly (n >n ). nd th hght from th bottom whr ths two concntratons bcom qual. ln( n / n ) h Answr: ( m m )g ρ mgz

8 h artton uncton or th absolut valus of probablty (rathr than th rato of probablts) w nd an xplct formula for th normalzng factor /: ( ) xp xp ( ) β - w wll oftn us ths notaton h quantty th partton functon can b found from th normalzaton condton - th total probablty to fnd th systm n all allowd quantum stats s : ( ) xp( ) or ( ) xp( ) h ustandsumm n Grman h partton functon s calld functon bcaus t dpnds on th spctrum (thus ) tc. Exampl: a sngl partcl ( ) contnuous spctrum. xp d ( x) dx xp ( ) - h aras undr ths curvs must b xp th sam (). hus wth ncrasng ( ) / dcrass and ncrass. At ( ) - th systm s n ts ground stat () wth th probablty. <

9 x x Avrag alus n a Canoncal Ensmbl W hav dvlopd th tools that prmt us to calculat th avrag valu of dffrnt physcal quantts for a canoncal nsmbl of dntcal systms. In gnral f th systms n an nsmbl ar dstrbutd ovr thr accssbl stats n accordanc wth th dstrbuton ( ) th avrag valu of som quantty x ( ) can b found as: ( ) x ( ) ( ) In partcular for a canoncal nsmbl: Lt s apply ths rsult to th avrag (man) nrgy of th systms n a canoncal nsmbl (th avrag of th nrgs of th vstd mcrostats accordng to th frquncy of vsts): x ( ) x ( ) ( ) xp ( ) xp( ) xp( ) h avrag valus ar addtv. h avrag total nrgy tot of dntcal systms s: tot Anothr usful rprsntaton for th avrag nrgy (r. 6.6): xp β β xp ( β ) β ln ln β β ( ) thus f w now () w now th avrag nrgy!

10 Exampl: nrgy and hat capacty of a two-lvl systm th slop ~ E h partton functon: xp xp xp h avrag nrgy: -lnn ( ) xp xp xp / (chc that th sam rsult follows from ) β C h hat capacty at constant volum: C ( / ) xp( / ) [ xp( / )] xp( / )

11 artton uncton and Hlmholtz r Enrgy ow w can rlat to : Comparng ths wth th xprsson for th avrag nrgy: ( ) β xp xp hs quaton provds th conncton btwn th mcroscopc world whch w spcfd wth mcrostats and th macroscopc world whch w dscrbd wth. If w now () w now vrythng w want to now about th thrmal bhavor of a systm. W can comput all th thrmodynamc proprts: ln ln ln ln μ

12 n Ω Mcrocanoncal Canoncal Our dscrpton of th mcrocanoncal and canoncal nsmbls was basd on countng th numbr of accssbl mcrostats. Lt s compar ths two cass: mcrocanoncal nsmbl or an solatd systm th multplcty Ω provds th numbr of accssbl mcrostats. h constrant n calculatng th stats: const or a fxd th man tmpratur s spcfd but can fluctuat. - th probablty of fndng a systm n on of th accssbl stats canoncal nsmbl ( ) ln Ω ( ) ln or a systm n thrmal contact wth rsrvor th partton functon provds th # of accssbl mcrostats. h constrant: const or a fxd th man nrgy s spcfd but can fluctuat. n En - th probablty of fndng a systm n on of ths stats - n qulbrum rachs a maxmum - n qulbrum rachs a mnmum or th canoncal nsmbl th rol of s smlar to that of th multplcty Ω for th mcrocanoncal nsmbl. hs quaton gvs th fundamntal rlaton btwn statstcal mchancs and thrmodynamcs for gvn valus of and just as lnω gvs th fundamntal rlaton btwn statstcal mchancs and thrmodynamcs for gvn valus of and. μ μ

13 oltzmann tatstcs: classcal (low-dnsty) lmt W hav dvlopd th formalsm for calculatng th thrmodynamc proprts of th systms whos partcls can occupy partcular quantum stats rgardlss of th othr partcls (classcal systms). In othr words w gnord all nd of ntractons btwn th partcls. Howvr th occupaton numbrs ar not fr from th rul of quantum mchancs. In quantum mchancs vn f th partcls do not ntract through forcs thy stll mght partcpat n th so-calld xchang ntracton whch s dpndnt on th spn of ntractng partcls (ths s rlatd to th prncpl of ndstngushablty of lmntary partcls w ll consdr bosons and frmons n Lctur 3). hs typ of ntractons bcoms mportant f th partcls ar n th sam quantum stat (th sam st of quantum numbrs) and thr wav functons ovrlap n spac: strong xchang ntracton wa xchang ntracton ( n) / 3 th d rogl wavlngth λ th avrag dstanc btw partcls /3 h h λ ~ (for p m molcul λ ~ - h ( n) oltzmann m at ) << statstcs m appls olatons of th oltzmann statstcs ar obsrvd f th th dnsty of partcls s vry larg (nutron stars) or partcls ar vry lght (lctrons n mtals photons) or thy ar at vry low tmpraturs (lqud hlum) Howvr n th lmt of small dnsty of partcls th dstnctons btwn oltzmann rm and os-enstn statstcs vansh.

14 Dgnrat Enrgy Lvls If svral quantum stats of th systm (dffrnt sts of quantum numbrs) corrspond to th sam nrgy lvl ths lvl s calld dgnrat. h probablty to fnd th systm n on of ths dgnrat stats s th sam for all th dgnrat stats. hus th total probablty to fnd th systm n a stat wth nrgy s d xp whr d s th dgr of dgnracy. ( ) Exampl: h nrgy lvls of an lctron n th hydrogn atom: ang th dgnracy of nrgy lvls nto account th partton functon should b modfd: d 8 d r d n (for a contnuous spctrum w nd anothr approach) 3.6 ( n ) n whr n... s th prncpl quantum numbr (ths lvls ar obtand by solvng th chrödngr quaton for th Coulomb potntal). In addton to n th stats of th lctron n th H atom ar charactrzd wth thr othr quantum numbrs: th orbtal quantum numbr l max... n th projcton of th orbtal momntum m l - l - l... l -l and th projcton of spn s ±/. In th absnc of th xtrnal magntc fld all lctron stats wth th sam n ar dgnrat (th proprty of Coulomb potntal). h dgr of dgnracy n ths cas: d l max n l mn [ l ] n xp d ( n ) n

15 roblm (fnal 5 partton functon) Consdr a systm of dstngushabl partcls wth fv mcrostats wth nrgs and ( ) n qulbrum wth a rsrvor at tmpratur.5.. nd th partton functon of th systm.. nd th avrag nrgy of a partcl. 3. What s th avrag nrgy of such partcls? 3xp xp.6.8. th avrag nrgy of a sngl partcl: th sam rsult you d gt from ths: 3xp xp 3xp xp ( ). 3 3 xp β ( ) ( ) xp( ) ( ) 3xp( ) xp( ) th avrag nrgy of such partcls:.3 3.

16 roblm (partton functon avrag nrgy) Consdr a systm of partcls wth only 3 possbl nrgy lvls sparatd by (lt th ground stat nrgy b ). h systm occups a fxd volum and s n thrmal qulbrum wth a rsrvor at tmpratur. Ignor ntractons btwn partcls and assum that oltzmann statstcs appls. (a) () What s th partton functon for a sngl partcl n th systm? (b) (5) What s th avrag nrgy pr partcl? (c) (5) What s probablty that th lvl s occupd n th hgh tmpratur lmt >>? Explan your answr on physcal grounds. (d) (5) What s th avrag nrgy pr partcl n th hgh tmpratur lmt >>? () (3) At what tmpratur s th ground stat. tms as lly to b occupd as th lvl? (f) (5) nd th hat capacty of th systm C analyz th low- ( <<) and hgh- ( >> ) lmts and stch C as a functon of. Explan your answr on physcal grounds. d xp (a) ( ) (b) ( ) ( ) (c) β 3 all 3 lvls ar populatd wth th sam probablty (d)

17 roblm (partton functon avrag nrgy) () ( ) ln. ln.. xp (f) ( ) ( ) ( )( ) ( ) [ ] [ ] [ ][ ] ( )( ) [ ] [ ] [ ] β β d d d d d d d d C Low (β>>): [ ] C 3 [ ] 3 3 C hgh (β<<): C

18 H h H roblm (th avrag valus) A gas s placd n a vry tall contanr at th tmpratur. h contanr s n a unform gravtatonal fld th acclraton of fr fall g s gvn. nd th avrag potntal nrgy of th molculs. d h h dh mgh n # of molculs wthn dh: ( ) xp ( ara) dh mgh mgh n xp ( ara) dh H mgh n xp ( ara) dh mgh y mgh ( ) y xp( y) mgh xp ( y) dy mg dy mg mgh y xp mgh xp ( y) ( y) dy dy A y xp d dy ( y) dy [ y xp( y) ] xp( y) y xp( y) A A A [ y xp( y) ] xp( y) dy y xp( y) dy y xp( y) dy A xp( A) xp( A) A xp ( y) dy xp( A) A or a vry tall contanr (mgh/ ): A dy

19 artton uncton for a Hydrogn Atom (r. 6.9) Any rfrnc nrgy can b chosn. Lt s choos n th ground stat:. 3. tc. h partton functon: r xp 3.6 xp ( n ) / d n (a) Estmat th partton functon for a hydrogn atom at 58K (.5 ) by tang nto account only thr lowst nrgy stats w can forgt about th spn dgnracy t s th sam for all th lvls th only factor that mattrs s n xp xp 9 xp Howvr f w ta nto account all dscrt lvls th full partton functon dvrgs: 3.6 n 3. 6 n xp > xp n n n

20 artton uncton for a Hydrogn Atom (cont.) Intutvly only th lowst lvls should mattr at >>. o rsolv ths paradox lt s go bac to our assumptons: w nglctd th trm d n ( d d ) d If w p ths trm thn w oltzmann factor E xp or a H atom n ts ground stat ~(. nm) 3 and at th atmosphrc prssur ~ -6 (nglgbl corrcton). Howvr ths volum ncrass as n 3 (th ohr radus grows as n) and for n s alrady ~. h trms caus th oltzmann factors to dcras xponntally and ths rhabltats our physcal ntuton: th corrct partton functon wll b domnatd by just a fw lowst nrgy lvls.