Solution to problem set 2, Phys 210A

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1 Solutio to problem set 2, Phys 20A Zhiyua Su Dated: April 27, 206 I. INFORMATION ENTROPY AND BOLTZMANN DISTRIBUTION a Usig Lagrage multipliers, we wat to maximize f p lp λe α From p f 0, we get p p lp λ E p α p lp λe α 0, p e λe α, 2 givig us the Boltzma distributio. There are two further coditios to satisfy, the eergy E p E ad the ormalizatio coditio p, which determie the multipliers λ ad α. b Thus S/k B is the same as the Shao etropy. S βe + lz k B e βe β E Z + e βe Z lz + βe e βe Z lz e βe e βe l Z Z q lq. 3 II. CLASSICAL IDEAL GAS a where Z T re βh N N p2 β N! 2π 3N dp i dq i e 2m V N i 2mπ N! 2π 3N β V N 3N/2 i N! λ 3N 4 2πβ 2π λβ m mk B T 5

2 2 Figure. Itegral cotour of Eq. 6. Figure from LECTURES/BOOK_STATMECH.pdf. is the thermal wavelegth. b ΩE ɛ+i ɛ i dβ 2πi ZβeβE. 6 I priciple, the partitio fuctio as a fuctio of β is oly defied o the right half complex plae. However, i the case of ideal gas, we ca exted it to the left half plae by aalytical cotiuatio, which allows us to perform cotour itegral. The itegral cotour is show i Fig.. We just cosider the case whe 3N is a eve umber, so that the itegrad does t have a brach cut o the egative real axis. There is a pole at β 0. By expadig e βe, we get the residue. c ΩE 2πi Res[Zβ] β0 3 2 N! E 2 3 N V N N! 3N 2π m 3 2 N 3N m V N E 2 3 N 2π 2 N! Γ 3N/2. 7 E is the average eergy S βe + lz k B βe + l N! V N λ 3N βe + N lv 3N lλ N ln + N + OlN βe + N l V N 3N 2 lβ 3N l 2π 2 m + N. 8 E H β lz 3 2 Nβ. 9 Plug the above expressio for β ito Eq. 8, we get the expressio for etropy of caoical esemble of the ideal gas as a fuctio of the average eergy: S 3 N k B 2 l m 2E 2π 2 3N + l V N

3 Compared to micro caoical esemble with the same eergy, it is differet by N 3 2 l 2 3. d F k B T lz k B T N lv + 3N 2 lk BT 3N 2 l 2π 2 m ln! 3. From df SdT pdv, the etropy is the same as the oe got from Legedre trasform S T F k B N lv + 3N 2 lk BT 3N 2π 2 l 2 m ln! N k B N l V N lmk BT 2π The pressure is The eergy is The heat capacity is The variace of eergy is p V F Nk BT V. 3 E H β lz 3 2 Nk BT. 4 C V 3 2 Nk B. 5 H E 2 2 β lz k BT 3 2 T F k BT 2 C V 3 2 Nk2 B T 2. 6 Which meas the stadard deviatio of eergy scales as N, ad is egligible compared to E i the thermal dyamic limit. III. ANHARMONIC OSCILLATOR The average eergy ca be directly calculated from the partitio fuctio Zβ. We will fid ad therefore N lz 2 lβ lβ + terms idepedet of β. 7 I detail, the partitio fuctio of each oscillator is p2 β Z i dpe 2m dxe βε 0 x a 2mπ dxe βε 0 x a β 2mπ β 2 dxe βε 0 x a 0 2mπ β 2 β / dte ε 0 a t Therefore, E/N β lz/n 2 + k BT. 8 0 N lz lz i 2 lβ lβ + terms idepedet of β. 20 9

4 4 IV. POLYMER ELASTICITY a N+ H F r N+ F r i r i F i N a i. 2 b The kietic eergy part of the distributio fuctio ivolves the degree of freedom of the caoical mometums, which do t affect the distributio i caoical coordiates, thus we do t cosider the mometums. Z Z i. 22 i i Ad Z i d 3 a i δ a i ae βh i d 3 a i δ a i ae βf a i dωdr r 2 δr ae βf a i a 2 dωe βfacosθ a 2 2π 2 sihβaf. 23 βfa The free eergy of each lik is F i β lz i β l a 2 2π 2 βfa sihβaf. 24 c Assume F is i z directio. a iz F H i F β lz i F β lβ + lsihafβ β F + af cothaβf. 25 For small F, expad to liear order i F, we have a iz 3 a2 βf 3 a2 k B T F. 26 ad R z N a iz N 3 a2 k B T F. 27 V. ISOBARIC ENSEMBLE a The Isobaric distributio fuctio is ρv,γ Z exp βh P ev. 28

5 5 where P e is the pressure of the eviromet. The Hamiltoia of the fluid-pisto system is Therefore, the distributio fuctio should be H H N + H pisto H N Γ + Mg V A + 2M p2. 29 ρp,v,γ Z exp βh N Γ + Mg V A + Z exp βh N + 2M p2 β Z exp βh N + 2M p2 βp V 2M p2 βp e V P e + Mg V V A. 30 where the ew pressure P is defie as P P e + Mg V A, ad Z is a ormalizatio factor. The mometum of the pisto p is decoupled from the rest of the Hamiltoia, ad does t affect the distributio of the other variables. The expectatio value of the operator OΓ is Ô dpdv dγ Z ÔΓ exp βh N + 2M p2 βp V dv dγ Q ÔΓ exp βh N βp V dv exp βp V dγ Q ÔΓ exp βh N. 3 where b Qβ,P,N dv dγ exp βh N βp V Qβ,P,N dv explz N β,v,n βp V dv Z N β,v,nexp βp V. 32 dv exp βf N βp V 33 At the saddle poit, V F N P. Oly keepig the saddle poit cotributio to the itegral, we have Therefore, c Thus d Qβ,P,N exp βf N + P V. 34 Gβ,P,N β lq F N + P V. 35 dg SdT + V dp + µdn. 36 S T G P,N, 37 V P G T,N. 38 V V 2 V 2 V 2 β 2 2 P lq β 2 P G β P V k B T V κ T. 39 Therefore, the stadard deviatio of V scales with V, ad is small compared to V.

6 6 e The distributio i V is P V Q exp βf N βp V 40 Aroud the saddle poit V V, we have βf N + P V βg V F N V V βg + V V V κ T After all, the distributio is goig to be a guassia distributio with the variace V V 2. So P V D exp V V 2 2 k B T V κ T 42 where D is a ormalizatio factor.