NATONAL UNVERSTY OF SNGAPORE PC5 ADVANCED STATSTCAL MECHANCS (Semeser : AY 1-13) Tme Allowed: Hours NSTRUCTONS TO CANDDATES 1. Ths examnaon paper conans 5 quesons and comprses 4 prned pages.. Answer all he quesons. 3. Answers o he quesons are o be wren n he answer books. 4. Ths s a CLOSED OOK examnaon. 5. Each queson carres marks. 1
1. Answer brefly: a. Wha s he Gbbs phase rule and how s derved? b. Sae he equparon heorem for classcal sysems and from hs, derve he Dulong-Pe law for solds. c. Gve he mahemacal equaon ha saes he global concavy propery of he enropy S as a funcon of nernal energy U. d. Sae he connuy equaon (for probably, parcle number, or elecrc charge). Sae he Fck s law. Combne he wo o derve he dffuson equaon. a) f = r M +, where r s he number of componens (.e., dfferen ypes of molecules), M s he number of phases, and f s he number of nensve varables ha can be vared and sll s n ha phase. s derved by consderng he Gbbs-Duhem relaon n each phase. b) n classcal sysems, each quadrac form of he erm n he Hamlonan ges a canoncal average of (1/)k T. n a sold lace of N aoms, each aom has hree knec energy erms and hree poenal energy erms and oal average nernal energy s 3Nk T. Thus he hea capacy s 3Nk : hs s Dulong-Pe law. c) S(λU 1 + (1-λ)U ) λs(u 1 ) + (1-λ)S(U ), λ 1. n n d) + j=, j = D n, = D n.. Consder one sngle parcle of mass m movng n he one-dmensonal doman x L. Excep when he parcle colldes elascally wh he boundares, he parcle moves freely wh knec energy p /(m) and no poenal energy. a. Compue he paron funcon Z a emperaure T n a canoncal ensemble. From hs, deermne he average knec energy p /( m ), hea capacy C, and he force f he parcle exers on one of he confnng boundares (walls). b. Calculae he phase-space volume Γ(U) correspondng o he energes of he parcle less han a gven U. Assumng he olzmann enropy formula Γ( U ) SU ( ) = k ln, where k s he olzmann consan and h s he Planck h consan, deermne he sysem emperaure T, hea capacy C, as well as force f exered on he boundary by he parcle. c. Dscuss f he resuls n par a (canoncal ensemble) and par b (mcro-canoncal ensemble) above are equvalen for he one parcle problem. Explan why.
a) The paron funcon s b) L + β p /( m) L π. β=1/(k T). From Z h 1 Z = dx dpe = mk T h p ln Z 1 1 F kt we obanu = = = = kt. C=dU/dT = k m β β /. f = = L L = -k T ln Z). L Γ ( U ) = dx dp = L mu. From hs, S=k ln(γ(u)/h), 1/T= S/ U=k (U), or p < U m U=(1/)k T, and C = du/dt = (1/) k. f = - F/ L=k T/L (from F=U-TS). c) nernal energy U, hea capacy C, and force f are he same, bu he enropy S, free energy F are no. The wo ensembles are no compleely equvalen (snce we are no n he hermodynamc lm). hnk he fac U, C, f, are he same s accdenal. (F 3. Consder an sng model defned on he graphs shown below n he nex page, known as Cayley rees. The frs hree generaons of he rees are shown. We assume ha each se denoed by an open crcle has an sng spn σ = ±1 and each lnk has a neares neghbor neracon, J σ σ j. For example, he frs generaon of he graph s assocaed wh he energy E = J σ σ 1 J σ σ J σ σ 3. a. Deermne he canoncal paron funcon Z 1 and Z of he sng model on he frs and second generaon Cayley rees. b. Derve a general formula for Z N for he N-h generaon Cayley ree. 3
c. Dscuss f he sysem has a phase ranson a a fne emperaure T > when N approaches nfny. a) Sum over he ou spns 1,,3 frs, we fnd ( K=βJ=J/(k T) ) Z1 = e e e = e + e = ( e + e ) σ σ1 σ σ3 σ 3 Kσσ 1 Kσσ Kσσ 3 Kσ Kσ K K 3 ( ). For Z we also sum over he ouer spns frs, we fnd ( ) 9 Z = e + e. K K b) For he general case, we use hgh-emperaure expanson. Snce he ree graphs canno have loops, all he anh(x) pars are, and we only have he frs erm. Z N = S cosh L (K) where S = L +1 s he number of ses, and L s number of lnks, L=3( N -1). c) No phase ranson as he paron funcon s he same, upo a consan facor, as he one-dmensonal sng model. 4. Arsoelan physcs says ha he velocy of a parcle s proporonal o he force appled o. We consder such a parcle conneced o a sprng o form an oscllaor experencng a random force (whe nose) wh he equaon dx mγ = kx+ R( ), d R ( ) =, RR ( ) ( ') = mγktδ( '), where γ s he dampng parameer, m s mass, k s force consan, x s he poson of he parcle whch s a funcon of me. The random force R() s he sandard whe nose. a. Derve a formal soluon x() expressed n erms of he random force R(). b. Derve he assocaed Fokker-Planck equaon for he average probably dsrbuon Px (,) of he poson varable x. c. Show ha n he long-me lm when equlbrum s reached, he dsrbuon s gven by he Gbbs dsrbuon proporonal o exp[ (1/ ) kx /( k T)]. a) The soluon x() s obaned by he mehod of varaon of a consan, where we frs le R()=, hen x()=ae -c where c = k/(mγ). Then we le A -> A() and subsue back no he equaon o oban equaon for A(). Afer negraon we ge c Rs () c( s) x() = Ae + e ds. mγ 4
b) We can follow he sandard dervaon of Zwanzg, bu he equaon s dencal o he sandard one reaed n class f we denfy x as velocy v, and some change of varables. So he Fokker-Planck equaon s he same (skp he dervaon) P k ( x P ) kt P = +. mγ x mγ x c) We can do n wo ways, eher o verfy ha exp(-(1/)kx /(k T)) sasfes he Fokker-Planck equaon wh <P>/ =, or solve he equaon kx<p> + k T <P>/ x = cons = (he consan has o be n order for x P( x) dx fne.) + 5. A quanum harmonc oscllaor n hermal equlbrum wh he Hamlonan, p 1 H = + k x, s drven by an exernal me-dependen force f() when >, so ha m he oal Hamlonan s explcly me-dependen, H() = H f() x. Noe ha p and x are operaors sasfyng he canoncal commuaon relaon, [x, p] = ħ, and he mass m, he force consan k, and he exernal force f() are c-numbers. a. Gve he defnons of he p () and x (), he neracon pcure momenum and poson operaor wh respec o H, and fnd explcly he me-dependence n erms of he orgnal Schrödnger pcure operaor p and x. b. Sae he equaon ha he neracon pcure densy marx ρ () mus sasfy. Solve hs equaon perurbavely o he lowes order (.e. frs order) n f(). c. ased on he resul of par b, derve he quanum expecaon value of he poson x( ) as x() = Tr ρ ()x() = G (, ') f('). Gve he explc form of he Green s funcon a) The neracon pcure operaors are H p () e pe G (, '). H =, and H x () e xe H =. dx () 1 p () The assocaed Hesenberg equaons are = [ x ( ), H] =, and d m dp () 1 [ (), ] = p H = kx (). We have used he fac H =H (). Snce he d equaons are dencal o he classcal verson of a harmonc oscllaor, we have he p soluon (explc me dependences) as x ( ) = x cos( ω) + sn( ω), and ωm 5
p ( ) = xmωcos( ω) + p cos( ω), where x and p are Schrödnger operaors sasfyng [x,p]=ħ. dρ () b) The densy marx n neracon pcure sasfes = [ V ( ), ρ ( )]. The d lowes order soluon s ρ ( ) = ρ () [ V ( '), ρ ()] d ' + O( V ) where V() = fx () (). c) Compue he average of x() usng he resul of par c and par a n he neracon pcure, usng he cyclc propery of race, we ge sn [ ω( ') ] G(, ') = θ ( ') [ x ( ), x ( ')] =. ωm -- End of Paper -- [WJS] 6