PH4 Statistical Mchaics Probl Sht Aswrs Dostrat that tropy, as giv by th Boltza xprssio S = l Ω, is a xtsiv proprty Th bst way to do this is to argu clarly that Ω is ultiplicativ W ust prov that if o syst has tropy S ad aothr has S th wh cosidrd togthr thy hav tropy S = S + S ot that w ar ot cosidrig brigig th two systs ito cotact, so th rgy lvls ad thir populatios ar ot chagd W ar cosidrig a coposit syst of two sparat parts A icrostat of th coposit syst is spcifid wh th icrostat of ach of th systs is spcifid If th acrostat of syst has Ω icrostats ad th acrostat of syst has Ω icrostats, th wh syst is i o of its icrostats th syst ca b i ay o of its Ω icrostats For vry icrostat of syst thr ar Ω icrostats But sic syst ca xist i ay of its Ω icrostats it follows that thr ust b Ω Ω icrostats i th acrostat of th coposit syst: Ω =ΩΩ so that th tropy of th coposit syst is S = Tl Ω = t l Ω Ω = T l Ω+ T l Ω = S + S W hav show that th tropy for two isolatd systs is additiv, which as that tropy is a xtsiv quatity Dostrat that gravitatioal rgy is ot xtsiv: show that th gravitatioal rgy of a sphr of radius r ad uifor dsity varis with volu as V ad fid th xpot W shall calculat th wor do agaist th forc of gravity i assblig a sphr of attr by brigig togthr th costitut parts fro ifiity Wh all th attr is off at ifiity w ta th gravitatioal rgy to b zro Th gravitatioal rgy of th body is th wor do i brigig th attr togthr W cosidr a itrdiat stat of th syst: w hav a sphrical ass M of radius R ad w valuat th wor do i brigig up a xtra ass d fro ifiity to th surfac Th forc btw asss M ad sparatd by a distac r is giv by wto s law of gravitatio M F = G r Aswrs
Th th wor do i ovig a ass dm fro ifiity to distac R is R de = Fdr, whr th ius sig as that w ar doig wor by applyig a forc agaist th forc of gravitatio So itgratig up th wto xprssio w obtai GM de = dm R But i addig this xtra ass th radius will hav icrasd slightly Sic 4 M = π R ρ, whr ρ is th dsity, it follows that dm 4π R ρdr = W substitut for M ad dm i th wor xprssio: G 4 de = π R ρ 4πR ρdr R 6 4 = G π ρ R d R Th total gravitatioal rgy is foud by assblig th coplt syst, by buildig th radius up fro zro to its fial valu Upo itgratio w fid 6 5 E = G π ρ R 5 Fially w ust xprss this rgy i trs of th volu of th sphr 4 V = π R, so that Th w obtai R= V 4π E G 6 5 = π ρ 5 4π V Thus th gravitatioal rgy varis as th 5/ powr of volu; th xpot = 5/ This shows that gravitatioal rgy is ot xtsiv; xtsivity rquirs rgy to b proportioal to volu; th xpot would b uity I ivstigatig th coditios for th stablisht of quilibriu through th trasfr of thral rgy th fudatal rquirt is that th tropy of th quilibriu stat should b a axiu Equality of tpratur was stablishd fro th vaishig of th first drivativ of S What follows fro a cosidratio of th scod drivativ? Th total tropy of th coposit syst ay b writt S( E) = S( E) + S( Et E) Sttig th first drivativ of this to zro lads to th quality of th tpraturs of th two systs 5 Aswrs
Th scod drivativ of th total tropy is asily xprssd i trs of th first drivativs of th tpraturs of th two systs d S d d de de T de T = F H G I + KJ F H G Th rquirt that i quilibriu th tropy b a axiu as that th scod drivativ ust b gativ If w valuat th drivativ of th rciprocals w fid dt dt < T de T de or + >, C C sic th tpraturs ar qual i quilibriu Hr th C ar th thral capacitis of th subsysts Th iquality ust b satisfid for all allowd valus of C ad C I particular it ust b satisfid wh ithr syst bcos vaishigly sall so that its hat capacity approachs zro Th th iquality rquirs that th thral capacitis ust b positiv 4 Do particls flow fro high µ to low µ or vic vrsa? Explai your rasoig Th chag i tropy, wh particls flow is giv by S S S =, which ust b gratr tha or qual to zro by th Scod Law But sic S µ = T it follows that S = ( ) T µ µ sic tpratur is th sa i both systs W coclud that wh µ > µ th will b positiv; will icras I othr words, particls will flow fro high µ to low µ 5 I th drivatio of th Boltza factor th tropy of th bath was xpadd i powrs of th rgy of th syst of itrst Th highr ordr trs of th xpasio ar glctd Discuss th validity of this W drivd that whr E E T ( ) P E ( ) S ET E so that w xpad S as a Taylor sris: I KJ S E S S( ET E) = S( ET) E + E E Aswrs
Th drivativ i th scod tr is idiatly idtifid as th ivrs tpratur S = E T Th drivativ i th scod tr is th S = E E T T = T E But hr T Eis th ivrs of th thral capacity C (of th rsrvoir) Th th tropy xpasio bcos E E S( ET E) = S( ET) T T C ad th highr ordr trs ivolv drivativs of th thral capacity Th y poit of th argut is that th thral capacity of th rsrvoir is vry larg Th assuptio is that whil th rsrvoir dtris th proprtis of our syst of itrst, th syst of itrst ca hav o ffct o th rsrvoir This is what w a by a rsrvoir, ad it ay b capsulatd by sayig that its thral capacity is sstially ifiit Aothr way of looig at this is to say that sic thral capacity is a xtsiv quatity th i th liit that th rsrvoir is larg, its thral capacity will b larg Th th ivrs of th thral capacity will b sall ad th th third tr i th tropy xpasio ca b glctd Th highr-ordr trs ivolv drivativs of th thral capacity so that ths ca also b igord 6 Th Boltza factor could hav b drivd by xpadig Ω rathr tha by xpadig S I that cas, howvr, th xpasio caot b triatd Why ot? If w xpad Ω w gt Ω E Ω Ω( ET E) =Ω( ET) E + E E Sic S = lω, ad S E = T, it follows that th or Ω Ω Ω Ω = =,, E T E T ( ) tc E E Ω( ET E) =Ω( ET) + T T E E P( E) + T T So i this cas you crtaily ca t triat th xpasio But you ca su th ifiit sris (it is th xpotial fuctio) ad this givs th Boltza factor Aswrs 4
7 Show that l! = l By approxiatig this su by a itgral obtai Stirlig s = approxiatio: l! l Th factorial is giv by! = ( ) so that th logarith is l! = l{ ( ) } = l+ l + l + + l ( ) + l = l, = as rquird ow w approxiat th su by a itgral: l l d; = this corrspods to th lowr dottd li of th figur blow lx Evaluatio of th itgral givs 4 5 6 7 8 x l d= l so that w obtai Stirlig s approxiatio as l! l 8 Show that th Gibbs xprssio for tropy: S = P l P, rducs to th Boltza xprssio S = lω i th cas of a isolatd syst Aswrs 5
Th fudatal postulat of statistical chaics stats that for a isolatd syst all icrostats ar qually lily If this isolatd syst has Ω icrostats th th probability of ay o of ths icrostats is /Ω Th gralisd tropy xprssio is th S = P l P = l Ω Ω = l Ω Ω ow ach tr i th su is a costat, ad thr will b Ω such trs Thus th xprssio for tropy bcos S = l Ω as rquird 9 What is th coditio that th gotric progrssio i drivig th Bos-Eisti distributio is covrgt? Th gotric progrssio is This will b covrgt if I othr words o rquirs ( ) = ( ) T { } pv = T l ε µ ( ) ε µ T < ε > µ for all sigl-particl rgis ε ow th groud stat rgy will b zro (or vry clos to it), so that th coditio is o th chical pottial: µ < th chical pottial ust always b gativ ot that this rquirt applis to Bos particls; it dos ot apply to Frios Show that th tractory of a d haroic oscillator is a llips i phas spac What would th tractory b if th oscillator wr waly dapd Th displact of a sipl haroic oscillator volvs i ti as Th otu is giv by Th pair ( ) si ( ω ϕ ) x t = A t+ ( ) p = x = Aω cos ωt+ ϕ ( ) = si ( ω + ϕ ) () = cos( ω + ϕ ) x t A t p t B t Aswrs 6
ar s to spcify a llips i x-p spac (phas spac), sic ( xa) ( pb) + = I th cas of dapig thr will b a xpotial dcay supriposd o th si ad cosi So th phas poit will approach th poit x =, p = I th cas of wa dapig th phas poit will trac out ay cycls of th llips bfor th radius chags apprciably Th ffct of wa dapig is thus to caus th phas poit to gradually spiral ito th origi Th poit about wa dapig is that ay sigl cycl is still obsrvd to b sstially lliptical Why ca t th volutioary curv i phas spac itrsct? You d to dostrat that th volutio fro a poit is uiqu wto s quatios of otio ar scod ordr diffrtial quatios thr ar up to scod drivativs of th spatial coordiats A coplt solutio to th quatios thus ivolvs two costats of itgratio for ach dgr of frdo Ad ths costats could b th positio coordiat ad th otu copot at a giv istat i ti (Hailto s forulatio of dyaics th xtds th ida to a gralisd viw of coordiats ad ota) Such a coplt solutio of th quatios of otio givs a uiqu solutio; th volutio i ti is copltly dtrid ow a poit i phas spac rprsts a giv valu of positio ad otu So th futur (ad past) volutio of th syst, ad thus th phas spac tractory, is copltly dtrid fro this poit It th follows that th volutioary path through a poit i phas spac is uiqu Th two paths caot pass through th sa phas poit ad so th volutioary curv i phas spac caot itrsct itslf Startig fro th xprssio for th Gibbs factor for a ay-particl syst, writ dow th grad partitio fuctio Ξ ad show how it ay b xprssd as th product of Ξ, th grad partitio fuctio for th subsyst coprisig particls i th th sigl-particl stat O starts fro th Gibbs factor { E, (, V) µ } T P, ( V, T, µ ) = Ξ ( VT,, µ ) Th oralizatio costat Ξ is th grad partitio fuctio: {, (, ) µ } VT,, µ Ξ ( ) =, E V T whr, spcify th th quatu stat of th syst wh it cotais particls ow for a syst of idtical particls = E = ad a giv stat of th syst is spcifid by th occupatio of th sigl-particl stats : ε Aswrs 7
(,,, ) { } So i this cas th grad partitio fuctio is xprssd as ( ε µ ) Ξ VT,, µ = ( ) = { } { } ( ε µ ) T T ow th ar all idpdt, so that ( ) ( VT,, ) ε Ξ µ = µ T ow w hav th dfiitio ( µ ) ( ) T Ξ VT,, = ε µ, so that th grad partitio fuctio ay b xprssd as th product as rquird ( VT,, µ ) ( VT,, µ ) Ξ = Ξ It is iportat to ot th logic of this aswr It is ot good ough to start fro Ξ ad th to argu that it ust b ultiplicativ sic th (pv) cotributios ust b additiv; this is ot what is asd for That was th way thigs wr argud i th txt This probl cosidrs th probability distributio for th rgy fluctuatios i th caoical sbl Th ots of th rgy fluctuatios ar dfid by β E ( E ε) σ = Z whr β = T ad ε is a arbitrary (at this stag) rgy Show that βε βε Zσ = { Z } β ad us this to prov that th fluctuatios i a idal gas oby a oral distributio aroud ε (You rally d to us a coputr algbra syst to do this probl) Th partitio fuctio is giv by so that Z = β E βε ( E ) Z β = ε If w diffrtiat this tis with rspct to β th this brigs dow factors (E ε) o th right had sid: Aswrs 8
Multiply this by βε, to obtai βε ( E ) { Z } ( E ε ) β = ε β βε β Z E β βε E { } = ( ε ) whrupo w obsrv th right had sid to b Zσ Ad thus w fid as rquird βε Zσ = Z β βε { } Th th ot of th rgy fluctuatio distributio is th giv by σ = β Z βε { Z } βε ad for th idal gas th partitio fuctio Z is giv by T V Z = π V = π β Th th th ot ay b writt as σ = β ad, i particular, w fid for th first ot βε {( / β ) } ( / β ) σ = ε + β = T ε If w ow choos th rgy ε, about which th ots ar valuatd, to b th a rgy T/ th th first ot will vaish Thus w adopt this valu for ε Th ots ay th b valuatd as σ = σ = σ = β σ = 4 ( ) βε σ = 7 + 6 4β tc β Aswrs 9
Th highr-ordr ots ay b calculatd i a straightforward, if tdious, ar Th oral distributio (with zro a) is charactrisd by th urical valu of th disiolss rducd ots = σ σ I gral ths ots will b fuctios of th ubr of particls W fid = = 4 = = 4 = + 4 5 5 = 4 + 6 6 6 = 5 + + tc Th i th throdyaic liit,, w s that th odd rducd ots vaish whil th v os td to th valus = 4 6 8 = = = 5 = 5 tc Th v rducd ots of th oral distributio ar giv by ( ) = 5 W s that ths valuat to th ubrs calculatd abov Ad thus w coclud that i th throdyaic liit th rgy fluctuatios of a idal gas oby a oral distributio 4 For a sigl-copot syst with a variabl ubr of particls, th Gibbs fr rgy is G = G T, p, Sic is th a fuctio of tpratur, prssur ad ubr of particls: ( ) oly xtsiv variabl upo which G dpds, show that th chical pottial for this syst is qual to th Gibbs fr rgy pr particl: G = µ If th syst is icrasd by a factor x th th xtsiv variabls G ad will b icrasd by this factor: Ad i particular if x = / th (,, ) GT (, px, ) xg T p = Aswrs
(,, ) GT (, p,) GT p = ; th Gibbs fr rgy for a syst of particls is tis th Gibbs fr rgy pr particl But th gibbs fr rgy pr particl ay b xprsss as GT (, p, ) = GT (, p, ) Howvr w idtify th drivativ of th Gibbs fr rgy as th chical pottial Thus w obtai th rquird rsult G = µ 5 Us th dfiitio of th Gibbs fr rgy togthr with th rsult of th prvious qustio to obtai th Eulr rlatio of Appdix G is dfid as G = E TS+ pv But th prvious Probl givs G = µ Thus w hav E TS+ pv = µ ad th Eulr rlatio th follows idiatly E = TS pv + µ 6 Th rgy of a haroic oscillator ay b writt as ω x + p so it is quadratic i both positio ad otu thus, classically, quipartitio should apply Th rgy ε = + ω Show that th partitio lvls of th quatu haroic oscillator ar giv by ( ) fuctio of this syst is giv by ω Z = cosch T ad that th itral rgy is giv by ω ω ω E = ω coth = + T T ω Show that at high tpraturs E ay b xpadd as ω E = T + + T Idtify th trs i this xpasio Th partitio fuctio is dfid by Upo substitutio for ε w th hav Z = ε T = Aswrs
Z = = = ( + ) ω T ω T ( ) W obsrv th scod su hr to b a gotric progrssio, which is asily sud: ω T ( ) = ω T Thus Z is giv by which w idtify to b as rquird = Z = = = ω T ω T ω T ω T ω T ω Z = cosch, T Th itral rgy is foud fro th partitio fuctio as l Z E = T T Upo diffrtiatio w fid l Z ω ω = coth T T T so that ω ω ω E = ω coth = + T T ω Th high tpratur xpasio of this xprssio ay b foud usig th xpasio of th hyprbolic cotagt: givig 5 x x x coth x = + + +, x 45 945 5 ω ω ω E T ω = + + + T 7 T 4 T Th first tr, T, is th high-tpratur quipartitio cotributio to th itral rgy Thr follows a sris i ivrs powrs of T Ths giv th corrctios as th tpratur gts lowr Aswrs