Phys 344 Lect 17 March 16 th,

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1 Phs 344 Lct 7 March 6 th, 7 Fri. 3/ 6 C , S. B. Gaussian, quipartion, Mawll HW7 S. 3,36, 4 B.,,3 Mon. 3/9 Wd. 3/ Fri. 3/3 S Partition Function S A.5 Q.M. Background: Bos and Frmi S 7. - Q. Stats, Bos- and Frmions HW 8 S. 44, 48, 49, 5 HW9 S., HW S. 3, 8, 9,,, 3ac, 8 HW5,6, 7 It s a Gas Last tim w considrd a fw spcific sstms dipol, instin solid, and rotating molcul. Of cours, our othr faorit sstm is an idal gas. So now w ll appl our nw found tool to that. In point of fact, to compltl spcif a gas molcul s micro-stat, ou d nd to spcif r frdom it has: what kind of molcul it is, th nuclar stat, th lctronic stat, th rotational stat, th ibrational stat, th lational stat, and, oh ah, its position, and othr stuff too. ach of ths could ha nrg ramifications, and thus ffct th probabilit. P( s) ( s) ( ib + + rot + position + lctroni c+ ) nuclar But, rmmbr th old mantra that probabilitis of inddpndnt frdoms multipl. So, w can sparat out th diffrnt frdoms and spak just of th probabilit of bing in a particular rotational stat, lational stat, or whatr. + ib+ rot+ position+ lctroni c+ nuclar tran ib pos l all. stats ( ) ( ) ( ) ( ) ( ) ( nuc) ib pos l nuc ( s) ( r ) ( ir) ( position ) ( r ) ( ir) ( ) position... P ( s)... P( s) P( srot) P( spo... ib pos ib So, whn w wr considring th statistics of rotational stats last tim, and compltl ignoring an othr frdoms, that was prfctl alid. Similarl, w can now considr just th frdoms of lation and position indpndntl. In fact, w ll start with position sinc it s darn simpl and w gt a familiar rsult. Th ponntial Atmosphr, risitd ( ) P( s ) mg mg/ kt In th arth s graitational, () mg, so P( s ). Now, th rlati dnsitis of particls at two lations should scal with thir mg kt D high/ ( high ) mg( / kt probabilitis, so. mglow/ kt D( low) Dos that look familiar? So w quit simpl find that th dnsit of air dcrass ponntiall with lation. ib pos pos l nuc

2 Phs 344 Lct 7 March 6 th, 7 Now w ll mo on to th locit dpndnc. Oka, mab not quit t. First w ll look at a fairl gnral rsult which happns to appl to th locit dpndnc, as wll as man othrs. In fact, w inokd this a fw tims in th past. 6. Th quipartition Thorm What s so spcial? If ou rcall, for gasss in particular, cpt for spcific rangs of tmpratur, th quipartition thorm holds. It s a powrful rul of thumb. W run into it or and or again for diffrnt spcific sstms. Using th Boltmann statistics w can show wr it s gnralit coms form, and w can plain wh it dosn t appl for all tmpraturs. Now that w ha th Boltmann factor, it s quit as to pro this is tru for an quadratic nrg trm, that is, an trm of th form ( q) cq whr c is a constant and q is a frdom such as, p, L, o Not, cq hr is a stand-in for ½ m or ½ L /I or th all ha th sam basic form. o Studnt Qustion: Th author sas w ll trat on dgr of frdom as our sstm, what dos h man? Sa ou got a particl that s fr to mo in,, and. Th frdoms to mo in ach dirction ar indpndnt of th othr dirctions. So ou can dlop th statistics associatd with ach dgr indpndntl. For ampl, ou can sa what th probabilit is that th particl s moing with a crtain rgardlss of what it s doing in and. So, th math looks just lik it would if ou had 3 -D particls instad of 3-D particl. In this wa h suggsts that ach dgr of frdom b tratd as a sstm for us to anal. W ll start b finding th partition function, and from thr w ll find th arag nrg. o ( ) cq cq c q q stat stat stat o I rwrittn this in trms of o c q q c q c q bcaus, as long as th stp si in q is quit small in comparison to kt/c, w ha c q <<, thn it s not much of a strtch of th imagination to lt it bcom diffrntiall small, i.., turn th discrt sum into a continuous on. d c q o How to do this intgral: Appndi B. This is awfull cut

3 Phs 344 Lct 7 March 6 th, 7 3 d d d d ( + ) dd d r rdφdr r rdr r o So, c q. c o Now q ( / ) kt 3 / c q o Tada! o Don t forgt th approimation. This rsult followd from th condition that c q <<. If this is not tru, th discrt sum looks nothing lik th continuous intgral. Rwriting this condition gis c( q) << kt << q kt. Onl if th nrg stps ar quit small compard to kt is th partition thorm alid. This maks sns bcaus, n if th hat-bath would happil gi a particl kt of nrg, th particl can onl accpt discrt multipls of, no lss and nothing in btwn. Prp. For 6.3 You will do somthing similar, but for a linar dgr of frdom. 6.4 Th Mawll Spd Distribution Prp for 6.4. You will rpat this following work for -D instad of a 3-D gas. Translational kintic nrg is m + m + m, ach of ths thr rprsnts a dgr of frdom, and ach dpnds quadraticall, so lads to a ½ kt. For ampl, if w mak th substitutions m c and q, thn w gt actl what w just prod ½ kt. 3 K.. m m + m + m kt + kt + kt kt 3kT m But what dos th whol distribution look lik? For that, w rturn to th s prssion X X s, whr X, th spd. W ll build up th quation s and thus s how a gin spd wighs into th arag. s (absolut alu sign sinc w r intrstd in spd which is just a magnitud). Now w must sum or th stats. Considring ach componnt of locit as a sparat dgr of frdom, w can indpndntl sum or stats of - motion, - motion, and -motion.

4 Phs 344 Lct 7 March 6 th, 7 4 stat stat stat -stat# So w can switch from summing or stats to summing or thir corrsponding locit componnts. W d sa that thr s stat of -motion pr, on stat of -motion pr, so w ha sum or ths thr indpndnt stats of motion indiiduall: m( + + ) stat stat + Allowd alus, i.., thr is on -motion stat pr ach alu stat. Just as in our quipartition argumnt, if th tmpratur is high nough, w can approimat ths thr sums with thr intgrals: stat m( + + ) dd d stat 8 m( + + ) d d o Rang: Th intgrals would b or all alus of th locit componnts, but sinc I m just intrstd in th spd, I can multipl b 8 and confin mslf to considring just th positi octant. Now, w ha thr, orthogonal ariabls, not unlik,, and. For that mattr + +, quit similar to r + d +. Borrowing a procdur from ral spac, w can form from Cartsian to polar coordinats. 8 4 stat stat 3 d sin θdφ dθ θ φ Dtrmining Pr-factor o Bfor going an furthr, w can paus and figur out actl what is so w can gt rid of ths unsightl ariabls. Looking back whr w startd from,, if w just didn t ha that s in thr, w d simpl ha / which of cours is. s. So tracing back through our work, that mans if w rmo on factor of from our intgrand, w ll ha /, i..,. 4 stat d aluating th intgral: back to Appndi B. d

5 Phs 344 Lct 7 March 6 th, 7 5 Not that So, d d m d d d d d d but ( ) m m ( m ) m ( ) 3 ( m) 3 d 4 m o / 4 m 4 stat stat stat d 3 ( ) m m 3/ 3/ m So, 3 4 d. Rsults. o Arag Spd. Sur, w can aluat this intgral, with th hlp of a 8T chang of ariabls to q m and appndi B, to find. m o Probabilitis. W can also find th probabilitis of ach spd. Sinc P() w can rwrit th intgral in this form and rad off 3/ m what s plaing th roll of P(). 4 d, I rwrittn th intgral smbol as th sum smbol to mak th two quations m mor paralll. Clarl P( ) 4 d. It is worth noting that d is an infinitsimall small constant, thus so is th probabilit of haing an spcific locit (sinc, classicall spaking, thr ar infinit diffrnt locitis to choos from, th probabilit of an spcific on is ). But it s rall th rlati probabilitis of diffrnt spds that w r intrstd in, so w ll just considr d as a constant. o Most Probabl Spd. Thn th most probabl spd can b found b maimiing th probabilit function (taking its driati and stting qual to.) this ilds ma/ min (, kt / m, ). Plugging ths thr alus back in rals that and infinit ar minima, whr th probabilit gos to. 3/

6 Phs 344 Lct 7 March 6 th, 7 6 o Distribution. P()/d ma a rms Prp for You will find fill in som of th math to actuall aluat th arag spd. I rcommnd doing a chang of ariabl or two to mak th intgral look lik stuff* d which can b intgratd b parts or lik othr.stuff* a d, in which cas ou can us th incradibl hand trick mplod in appndi B. 6.5 Partition Functions and Fr nrg So far, w applid th Boltmann formalism dirctl to singl-mmbr, or micro-sstms. But n for ths simpl sstms, w somtims found that mor than on micro-stat had th sam nrg, so w got usd to daling with a stat S( ) / k Ω. Of cours, Ω ( ) so dgnrac: ( ) stat stat Ω ( ST) F F stat ( ) ( ) Now hr s th shortcut to th book s rsult. Lt s appl th Boltmann formalism not to a singl-mmbr, micro-sstm, but to a whol macro-sstm at nrg U and tmpratur T. Wh not? o Thn th partition function for th whol sstm must b stat ( U, T ) Ω, but of cours r micro-stat of stat ( ) th macro-sstm has th sam nrg, U, sinc w dcidd to onl look at such micro-stats. So thr s just on trm in th sum or nrg, U S / k U ( U TS ) F ( U, T ) Ω( U ) F ktln ( U, T). Rcall, for a sstm at constant T, a procss that rducs its F incrass th unirs s ntrop, thus is a faorabl procss. Anothr wa to put it, th macrostat with lowst F is th most probabl. is somthing lik a (wightd) count of stats. W can s b this nw rlationship that rducing F incrass, as would b pctd. This F rlationship is usful bcaus w can oftn figur out, and bcaus w ha partial driatis of F for finding S, P, and µ. F F F o S, P µ T V N V, N T, N V, T F

Differential Equations

Differential Equations UNIT I Diffrntial Equations.0 INTRODUCTION W li in a world of intrrlatd changing ntitis. Th locit of a falling bod changs with distanc, th position of th arth changs with tim, th ara of a circl changs