Weights Interpreting W and lnw What is β? Some Endnotes = n!ω if we neglect the zero point energy then ( )

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1 Sprg Ch 35: Statstcal chacs ad Chcal Ktcs Wghts... 9 Itrprtg W ad lw... 3 What s? Lt s loo at So Edots Chaptr 3: Fudatal Prcpls of Stat ch fro a spl odl (drvato of oltza dstrbuto, also s addtoal lctur ots o wbst) Suppos w hav a st of dpdt oscllators ( partcls ) ε!ω f w glct th zro pot rgy th Th rgy at lvl s ε ε whr,,,3,... ε!ω A stat of th coplt syst (gstat of Haltoa),, 3... Th total rgy s E ε+ ε + ε so E ε thr ar ay ways to gt ths rgy hgh lvl of dgracy For xapl: 3 oscllators that total to rgy of 5ε 3 E # of stat prutatos ! total # of stats (g. Last row, oscllator s at ε, s at ε, 3 s at ε, totalg to 5 ε ) Aothr xapl: 4 oscllators, E 6ε 3 4 Chaptr 3: Fudatal Prcpls of Stat ch fro a spl odl 8 E # of stat prutatos x C(4,)/ ay ay or cobatos Ths way spcfs th rgy lvl of ach oscllator, vry tdous

2 Sprg Ch 35: Statstcal chacs ad Chcal Ktcs Wghts W ca wrt t aothr way (Cofgurato Tabl), whch wll b or covt. # oscllators at rgy lvl ε ε ε 3 ε 4ε 5ε 6ε st row of prvous tabl 3 d row 3 rd 4 th.... (ach o of ths rows s calld a dffrt cofgurato) Each of th trs ths tabl ca b rprstd by (g. At st row, ε, 3 ) (total # of oscllators) ε E (total rgy) th ubr of oscllators at rgy How ay stats (dffrt ways to a t) ar thr a crta cofgurato? Fdg th prutato for ach box # of ways to put to box ox ox!!! ( ) ( )! ( ) ( )!!( )!!! (choos fro total partcls) (fro th rag choos partcls) (fro rag choos ) Puttg t togthr (ultply # of possblts for ach rgy lvl):! ( )! ( )! ( )!!! W!!!... ( )! ( )!!...! Chaptr 3: Fudatal Prcpls of Stat ch fro a spl odl 9

3 Sprg Ch 35: Statstcal chacs ad Chcal Ktcs! gral: W (!), (,,,...) W : th Wght of a cofgurato, # of possblts (crostats) for a crta cofgurato Eg. What s th wght of a cofgurato wth 4 hads ad 6 tals for a co flp? W!!!.37!!! 4!6! ( ) h t Suppos ow w hav oscllators ad E 5ε ε ε ε 3 ε 4ε 5ε W (ubr of ways to dstrbut) 999!!999! !!!998!!!!998!!!!997! ! 4!3!996! 6! 5!995! 5 (You ca th of ths as a draw ad popl gttg x przs, th last row, 5 popl w prz ach. I th frst row prso ws all 5 przs, whch s lss lly tha row 5) I ths aalogy th agtud of vry prc s qual, ad f sobody wo a prc th lottry tct s put bac th x. So a prso ca draw ultpl przs. Th olcular aalogy s that a olcul ca gt furthr xctd, o attr ts currt rgy lvl. Notc that row 4 s wght s llo ts bggr (hc or lly) tha row. Not all cofguratos ar qually lly. For xapl t s (of cours) uch or lly that 5 dffrt popl w a prz, tha that th prz gos 5 ts to th sa wr. Possbl but ully. O would xpct foul play! If s vry larg, a sgl cofgurato wll hav th largst W ad s hc th ost lly, ths wll bco th doat cofgurato. You ca s th ffct th abov tabl. Chaptr 3: Fudatal Prcpls of Stat ch fro a spl odl 3

4 Sprg Ch 35: Statstcal chacs ad Chcal Ktcs W s a hghly pad dstrbuto, ad * wll b th axu th dstrbuto, or th ost lly dstrbuto Fdg th dstrbuto wth axu wght: Costrats: W ε E! ( )!!!!!.. lw l! l!!!... lw l! l! + l! + l!... lw l! l! lw l l (apply Strlg approxato) Wat to axz F lw ε E λ Whr ε E ad λ ar costrats for total rgy ad total # of partcls, λ, ar calld dtrd Lagraga ultplrs (to b dtrd). W wat F to b axu udr varato of all paratrs, hc F ε E (rgy costrat) F (partcl # costrat) λ l F (Nw codtos to dtr ) Ths s gral fatur of th Lagrag ultplr procdur: Th costrats ar obtad by rqurg statoarty w.r.t. th Lagrag ultplrs. Fd th drvatv of lw lw l l + Chaptr 3: Fudatal Prcpls of Stat ch fro a spl odl 3

5 Sprg Ch 35: Statstcal chacs ad Chcal Ktcs ε ~ (oly drvatv for survvs fro th su) l lw l l + So drvatv of F F l ε λ ε λ l ε λ sg th costrat λ ε λ ε ε ε ε P (th rlatv populato of rgy lvl ) ε Total Ergy ε E l ε ε εl E whr ε ad s th xpctd ( ~ avrag) ubr of partcls at lvl Ths dstrbuto s so pad that th oly rlvat cotrbuto s that of *. It s th ost lly dstrbuto ad th fluctuato aroud ths pa s vry sall. Th oly rlvat dstrbuto s * (,,, 3,...). Itrprtg W * ad lw * l W * l!! l l Chaptr 3: Fudatal Prcpls of Stat ch fro a spl odl 3

6 Sprg Ch 35: Statstcal chacs ad Chcal Ktcs l l l l l l l P l P + l l Pl P l P sc P lw * P l P lw lw * P l P Ths loos vry slar to tropy as dscussd bfor. It s ssg th oltza factor, so a ss axzg W (pr partcl) s l axzg tropy. Itrprtato: Etropy ca oly cras bcaus t gos to th ost lly dstrbuto. I th coputr lab w wll llustrat ths bhavor: dffuso towards th ost lly dstrbuto. What s? You ca guss t s sohow rlatd to tpratur. Lt us vstgat. Gv that w ow ε (ot asy, you d a coputr) ε ε E w ca solv for f w assu w ow E,, Th dfto of tpratur (drvato) d TdS at costat V, T S V, tpratur s dfd as th chag ovr chag S oth ad S ar a fucto of howvr, d εp P ε V, N S d ad ds Δ S T l " " Δ ΔS V, N V, N V, N d Chaptr 3: Fudatal Prcpls of Stat ch fro a spl odl 33

7 Sprg Ch 35: Statstcal chacs ad Chcal Ktcs S P l P S P l P + P P P P l P S ε l P l ε l P ε l P P ε + l P Sc P, P, also Ad w fd: T d ds d S S d ε P P P has uts of T rgy acts as a covrso factor btw T ad rgy Lt s loo at T T T T T T T C T V T T T C T V εp Chaptr 3: Fudatal Prcpls of Stat ch fro a spl odl 34

8 Sprg Ch 35: Statstcal chacs ad Chcal Ktcs ε ε + + ε ε ε ε ε ε P + ε j ε ε + ε ε ε ε ε ε j j jpj j ε ε (th varac rgy) ( ε ε ) P So C V ( ε ε ) (f w chc th uts w gt J/K) T j So Edots lw Pl P lw Proportoal to tropy S rachg qulbru, axz tropy or lw So furthr justfcato that rlats to /T (dscusso s a bt brf). W dpds o W ad W( ) allow for tpratur coducto th total # of cofgurato: W W F lww + lw s S Chaptr 3: Fudatal Prcpls of Stat ch fro a spl odl 35

9 Sprg Ch 35: Statstcal chacs ad Chcal Ktcs Chaptr 3: Fudatal Prcpls of Stat ch fro a spl odl 36 l W F l W F otto l: T qulbrats for th axally lly dstrbuto.