PHYC - 505: Statistical Mechanics Homework Assignment 4 Solutions

Transcription

1 PHYC - 55: Statistical Mechaics Homewor Assigmet 4 Solutios Due February 5, Cosider a ifiite classical chai of idetical masses coupled by earest eighbor sprigs with idetical sprig costats. a Write dow the Hamiltoia of this system ad the set of ordiary differetial euatios which are the Hamiltoia euatios of motio. Cosider the N masses coected to each other with idetical sprigs with sprig costat, i Problem 1 of Homewor, with periodic boudary coditios so that ν N + 1 st mass is euivalet to the 1 st mass ad so o. We will let N towards the ed of the calculatios. Let x ad p be the positio ad mometum of the th mass with respect to its positio whe oe of the sprigs are stretched or compressed. The the Hamiltoia of the system is give by H lim N [ N 1 The Hamiltoia euatios of motio are give by For our system these become ] x +1 x + p m ẋ j H p j ad ṗ j H x j ad ẋ j H p j p j p j p j m ṗ j H x j x j x j lim N [ N 1 ] x +1 x + p m [ x j+1 x j + x j x j 1 ] + p j m lim N [ N 1 ] x +1 x + p m [ x j+1 x j + x j x j 1 ] + p j m [x j x j+1 + x j x j 1 ] 1 of 8

2 Taig the time derivative of ẋ j we have that ẍ j ṗj m m [x j x j+1 x j 1 ] ω [x j x j+1 x j 1 ] This choice of geeralized coordiates is t very useful sice it leaves the euatios of motio coupled. As i Problem 1 of Homewor, defie the discrete Fourier trasform of the x j s to be x N 1 x j e ij, π N ν where ν taes o iteger values betwee ad N 1, iclusively, ad so taes o the values from to π, ad we see that the decoupled euatios of motio become ẍ ω x where ω ω si see Homewor. From classical mechaics we ow that the geeral solutio to such a euatio of motio is x t x cos ω t + p ω si ω t yieldig p t mẋ ω x si ω t + p cos ω t Note that these euatios of motio are writte i terms of ormal mode coordiates of the system. If oe would lie to explicitly obtai x j t s, which are the origial coordiates of the masses, oe eeds to ivert the Fourier trasform. I the limit N >> 1, the iverse discrete Fourier trasform becomes x j t 1 x j te ij N 1 x j te ij ν N 1 N x j te ij N π 1 π 1 π x j te ij x j te ij d Pluggig i our geeral solutio ito this Fourier trasform we have that x j t 1 e ij x cos ω t + p si ω t d π ω 1 e e x ij iω t + e iωt + p e iω t e iωt d π ω i 1 x e ij e iωt + e ij e iωt + p e ij e iωt e ij e iωt d 4π iω 1 x e ij e iωt + e ij e iωt t + p e ij e iωt + e ij e iωt dt d 4π of 8

3 Pluggig i our expressio for ω, we have that the itegrals tae the form of Bessel fuctios as 1 π x j t x e i j ωt si + e i j ωt si d π t + p dt 1 π e i j ωt si + e i j ωt si d π 1 π x e i jx ωt six + e i jx ωt six dx π t + p dt 1 π e i jx ωt six + e i jx ω t six dx π x J j ω t + J j ω t + p Now we ow that for iteger values of j, that so the above euatio ca be writte as t J j ω t 1 j J j ω t J j ω t x j t x J j ω t + J j ω t + p dt J j ω t + J j ω t t dt J j ω t + J j ω t Now we also ow that the series defiig the Bessel fuctio of the first id is J j αt m m 1 m m!γm + j + 1 m+j αt ad for iteger values of j, we also have that 1 m J j αt αt m+j m!γm + j + 1 m 1 m αt m!γm + j + 1 1m+j m 1 m αt m!γm + j + 1 1m+j m 1 m m+j αt m!γm + j + 1 m 1 m m+j αt m!γm + j + 1 So our x j t becomes J j αt x j t x J j ω t + p x J j ω t + p ω x J j ω t + p ω x J j ω t + p t dt J j ω t ωt ωt dτj j τ dτj j τ m+j m+j J j+m+1 ω t [Re j > 1] m 3 of 8

4 ad so we have that p j t is p j t 1 e ij p td π x 1 π x 1 π x x x 1 π 1 π 1 t π e ij ω si ω t d + p 1 e ij cos ω t d π e ij e iωt e iωt ω d + p 1 e ij eiωt + e iωt d i π e ij iω e iω t e iωt d + p 1 π e ij+iω t + e ij iωt d π e ij e iω t e iωt d + p J j ω t e ij e iωt e iωt d + p J j ω t x t J j ω t + p J j ω x ω J j 1 ω t j t J j ω t + p J j ω b Write dow explicitly the partial differetial euatio, which is the Liouville euatio for the system, obeyed by the Gibbs esemble desity R. Liouville s euatio for the phase desity is give by t H H x p p x ṗ + ẋ p x Usig our previous fidigs, we ca write dow the Liouville euatio for the chai of masses ad sprigs as t [x x +1 x 1 ] + p p m x It does ot loo lie the above euatio ca be solved trivially. Note that, as the solutio clearly shows, the Hamiltoia ca be re-expressed i the ormal coordiates as H Hˆx, ˆp [ ] m ω x + ˆp m which meas that each mode acts lie a idepedet mass ad sprig system with mass m ad sprig costat si. The the Liouville euatio becomes t ˆx, ˆp, t Let s defie the followig ew variables [ mωx ˆp ] p m x X 1 m mω ˆx + ˆp P 1 m mω ˆx ˆp 4 of 8

5 The the Hamiltoia becomes HX, P 1 X + P Now we ca thi of R as a fuctio of X s ad P s ad of t, ad write the Liouville euatio as t X, P, t H H X P P X which gives us the first order partial differetial euatio t X, P, t + If we defie polar coordiates r ad θ such that the partial differetial euatio becomes P X X P X r siθ P r cosθ t r, θ, t + Note that r X + P E, where the eergy of each mode, E is costat as all of them are closed systems, ad θ ta 1 X P. Therefore Rr, θ, t ca be thought as a fuctio of θ ad t oly, as r s are all costat parameters. We ca use the method of characteristics to solve the above euatio. See ay boo o partial differetial euatios about the details of the method. The characteristic euatios for the above euatio are Solutio of these give dt ds 1, θ dθ ds 1. θ s s + c, ts s + c, ad the iitial coditios are such that t ad θ τ. Therefore we have θ s s + τ, t s. The euatio we ow eed to solve is the ordiary differetial euatio drs, τ ds, whose solutio is a costat with respect to s, ad a fuctio of all the τ s Rτ 1, τ,, τ 3N. Trasformig bac to the origial coordiates through τ θ t we get or Rθ, t; E R θ 1 t, θ t,, θ 3N t; E Rθ, t; E R ta 1 X1 t, ta 1 X t,, ta 1 X3N t; E P 1 P P 3N ad remember that X 1 m mω ˆx + ˆp P 1 m mω ˆx ˆp 5 of 8

6 c Commet o the problem of solutio of these euatios. Solve them if you ca; mae isightful commets :- if you caot. Careful ow, do ot haste to give up o tryig to solve them. Let Gx, p, t be a fuctio that satisfies the Liouville euatio. Cosider H G H G + H G x p x p x p H G H G + H G p x p x p x Subtract the secod euatio from the first ad sum over to get { H G H G } { H G H } G x p p x x p p x Note that the above euatio is just [H, G] P B. We are iterested i fidig dσt dτ R l R. dt t G t where dτ d 3N d 3N p. Note that R l R obeys the Liouville euatio. Tae G R l R so that we have dσt dt t R l R [ dτ H R l R H ] R l R. x p p x Now we ca itegrate out x s i the first itegral ad ps i the secod to get dσt { d 3N p H R l R + d 3N x H R l R p x but H p R l R x ad x H x R l R x must vaish for ifiitely large values of x s ad p s. Therefore dσt dt. x are zero as ay physical R, ad thus R l R, Quatum mechaical aalogue: I a uatum mechaical calculatio we have dσt dt t R l R Tr t l R, as Tr t. The evolutio i this case is give by the Vo Neuma euatio Therefore i dσ dt t i [H, R]. Tr [H, R] Tr HR l R Tr RH l R Tr HR l R Tr H l RR Tr H [R, l R], where we used the cyclic property of trace ad [R, l R]. }, 6 of 8

7 d Cosider the itegral over all phase space of the uatity R l R. Do you thi that uatity icreases, decreases, or remais the same as time progresses? The uatity σt d 3N xd 3N pr l R, called esemble averaged idex of probability by Gibbs, has sometimes bee iterpreted as the idicator of a system s relaxatio to euilibrium. It is a costat i time for Gibbs esembles. Let s see how that comes about. σt d 3N xd 3N pr l R l R t, where represet a esemble average over all possible states of the system. Note that d l R dt 1 dr R dt, where dr dt follows from Liouvilles theorem. This meas that for each copy of the system, that is whe you follow each sigle trajectory i the phase space, l R is a costat i time. If you perform a esemble average a uatity that is costat i each member of the esemble, you will surely get a costat uatity. Therefore d dt l R This meas that σt caot describe a approach to euilibrium, see Boltzmas H-theorem for a discussio. e Commet o the uatum mechaical couterpart of this system regardig each of a, b, c, ad d above. I order to solve the uatum mechaical couterpart of this problem, oe may start with the Schrödiger euatio for N coupled uatum mechaical harmoic oscillators. Agai it is possible to decouple the euatios by a appropriate trasform. The Hamiltioia ca be writte i terms of ladder operators which would mae the calculatios very coveiet to do. I the uatum mechaical versio, istead of the Liouville euatio we would use the Vo Neuma euatio ad solve for the desity matrix. Whe dealig with the phase space desity i uatum mechaics, oe eeds to remember that because of the ucertaity priciple, the smallest volume elemet i the phase space will be h 3 where h is Plac s costat. So coversios from the summatios to itegrals will differ by a factor i uatum mechaics. Also, oce we obtai the desity matrix, the average value of a observable will be give by O Tr RO 7 of 8

8 . Specify the physical coditios uder which the members of a Gibbs esemble might iteract with oe aother ad write dow a example of the Liouville euatio i such a case. The metal copies of the Gibbs esemble would ot iteract with each other as they are merely abstractios employed to represet differet states of a system. 3. Read ay of the may statistical mechaics boos available to you about Gibbs esemble theory ad describe briefly what a grad caoical esemble is, what ids of systems it is relevat to, ad what the meaig of the uatity chemical potetial that appears i that cotext is. Gibbs itroduced the grad caoical esemble to geeralize the caoical esemble to iclude particle trasfer. I a microcaoical esemble copies of the system are closed, ad thus their eergy is precisely defied. I a caoical esemble, system ca iteract with a reservoir a exchage eergy, thus eergy of a system ca fluctuate. I a grad caoical esemble, the system iteracts with the reservoir to exchage eergy ad particles. Therefore i a grad caoical esemble, eergy ad umber of particles i each metal copy of the system ca both fluctuate. A grad caoical esemble would be useful i describig a system that ivolves chemical reactios as the umber of particles i the system is liely to chage durig such pheomea. Aother cool example system would be a cup filled with hot liuid that cools dow by evaporatio. The chemical potetial ca be defied as the chage i the eergy of the system as a result of the chage i its umber of particles, whe every other thermodyamical variable that describes the state of the system, such as etropy, volume, etc., is ept costat. It has the uits of eergy/molecule, or otherwise stated, for a sigle species of particles i a grad caoical esemble, the chemical potetial ca be defied as the chage i iteral eergy gaied per particles added to the system whe all other thermodyamic variables are held costat. 8 of 8