Large Systems (Section 2.4)

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Frederica Kennedy
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1 Larg Sstms (Sction.4) Rad Schrodr sction.4 about small numbrs, larg numbrs (i ), and 3 10 VERY larg numbrs (i.. 10 ) (Mak sur ou can do Probs..1,.13) o Eonntial of a vr larg numbr is a larg numbr In a sstm with a larg numbr of articls, MULTIPLICITY as a function of aramtr that dfins MCROSTTES is VERY sharl akd at valu of aramtr corrsonding to most robabl MCROSTTE STIRLIG S PPROXIMTIO Evn for a fw hundrd articls, multilicitis for Einstin solid wr larg numbrs For larg, can aroimat! π and ln! ln o carful with choic of vrsion Eaml: Prob..16

2 MULTIPLICITY for LRGE EISTEI SOLID For Einstin solid with oscillators, numbr of MICROSTTES with nrg E hf is ( ) ( + 1 )! ( + )!,! ( 1 )!!! o Will assum >> (i.. high tmratur) Us Stirling s aroimation to rss natural logarithm of multilicit as ln ln +! ln! ln! + ln + ln ln Us ln(1 + ) (( )) ( ) ( ) ( ) ( ) lim to writ 0 ( ) ln + ln + and thus gt o ln ln + + o us assumtion of >> to discard last trm comard to Eonntiat to gt (, ) o This is numbr of microstats for a singl Einstin solid, OT a comosit sstm consisting of intracting sstms o (, ) For fid, (, ) is a vr larg numbr incrass VERY uickl with incrasing (numbr of nrg units in sstm) Homwork: Prob.18

3 MULTIPLICITY FOR LRGE COMPOSITE SYSTEM If two larg intracting Einstin solids ( and ) shar fid amount of nrg ( + units), multilicit is VERY sharl akd at distribution of nrg corrsonding to most robabl macrostat Multilicit of joint sstm in macrostat for which has units of nrg and has units of nrg is o Could find valu of that maimis, subjct of constraint +. most robabl distribution of nrg SHRPESS of MULTIPLICITY for LRGE COMPOSITE SYSTEM To s sha of multilicit, will look at scial cas of o Thn ( ) o Maimum multilicit is ma n at To look at sha of multilicit and nar ak, lt + o Givs o Us lim ln(1 + ) 0 to show ( ) o This givs ma ( )

4 ( ) What is th sha of? o Gaussian: P( ) ma 1 σ σ π Width of ak: P( ) o Sas that for macroscoic sstm (i.. So ( ) is a Gaussian of width Pma at ± σ cntrd on ), robabilit of finding 11 mor than about 1 art in 10 awa from its most robabl valu is vr small. MI POIT of EXMPLE: ll MICROSTTES of ISOLTED comosit sstm ar uall likl but most robabl MCROSTTE of macroscoic sstm in thrmal uilibrium is ovrwhlmingl robabl HOMEWORK: Prob. Prob.3

5 Indistinguishabilit: Th Idal Gas (sction.5) MULTIPLICITY: IDEL GS COSISTIG OF SIGLE TOM in volum V How do w count accssibl microstats of a singl atom of mass m with kintic nrg U containd in volum V? For now, follow tratmnt in sction.5 o d 6 numbrs to scif microstat of singl atom 3 coordinats to scif location and 3 comonnts of momntum o If ffctiv volum of atom is of ordr thn ositional dgrs of V frdom contribut factor of to multilicit for singl atom. o Comonnts of momntum ar constraind b + + mu ccssibl momntum stats li of surfac of shr of radius mu ccssibl volum, V, of momntum sac is surfac ara of shr of radius mu tims a shll thicknssδ. o If th uncrtaintis in comonnts of momntum ar numbr of accssibl momntum stats is of ordr thn V o Coordinats and comonnts of momntum ar constraind b Hisnbrg uncrtaint rincil ( )( ) ( )( ) ( )( ) h o So multilicit for singl atom in volum V is V V V V 1 3 h

6 TOMS of MOTOMIC IDEL GS in volum V How do w count accssibl microstats for atoms with kintic nrg U in volum V? o If ffctiv volum of atom is of ordr thn ositional dgrs of V frdom contribut factor of to multilicit for atoms. o Constraint on comonnts of momntum is L mu ccssibl volum of 3 dimnsional momntum sac is surfac ara of 3 dimnsional shr of radius r mu tims a shll thicknss δ IF atoms ar distinguishabl, thn multilicit for -atom gas is V (ara of 3 - dimnsion shr of radius r ( ) mu ) δ which bcoms V h 3 (ara of 3 - dimnsion shr of radius r mu ) δ using Hisnbrg Uncrtaint Princil

7 toms of Idal Gas ar IDISTIGUISHLE so numbr of microstats for distinguishabl cas ovrcounts multilicit for indistinguishabl cas b! o So multilicit for gas of indistinguishabl atoms is 1 V (ara of 3 - dimnsion shr of radius r 3! h mu ) δ o What is th ara of a 3 dimnsional shr of radius d / π d 1 ndi 4 shows d () r r d Γ whr ( n + 1) nγ( n) n! Γ and 3 Γ π r mu? So multilicit for atom IDEL GS of IDISTIGUISHLE atoms is 1 V! h π 3 / 3 / ( ) ( mu ) δ ( ) ( mu ) δ h 3 3 1!! 3 /! V π Will vntuall nd natural logarithm of multilicit so onl, V and U dndnc mattrs. U, V, f V U 3 / o Usful to writ ( ) ( )

Introduction to Condensed Matter Physics

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