10.40 Appendix Connection to Thermodynamics and Derivation of Boltzmann Distribution

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1 10.40 Appendx Connecton to Thermodynamcs Dervaton of Boltzmann Dstrbuton Bernhardt L. Trout Outlne Cannoncal ensemble Maxmumtermmethod Most probable dstrbuton Ensembles contnued: Canoncal, Mcrocanoncal, Gr Canoncal, etc. Connecton to thermodynamcs Relaton of thermodynamc quanttes to Q 0.1 Canoncal ensemble In the Canoncal ensemble, each system has constant,v, T. After equlbraton, remove all of the systems from the bath, put them all together: Apply postulate 2 to the ensemble of systems, also called a supersystem. Let n = number of systems wth energy E. Also, = n E tot = n E. If we know all E s, then the state of the entre ensemble would be well-defned. For example, let s analyze an ensemble wth 4 systems, labeled A, B, C, D, where A B C D E 2 E 3 E 2 E 1 Then, E tot =E 1 +2E 2 +E 3 Also, the dstrbuton of the systems, n =(n 1,n 2, n 3,...)=(1, 2, 1). But there are many dfferent supersystems consstent wth ths dstrbuton. In fact, the number of supersystems consstent wth ths dstrbuton s Ω tot ( n )= Q! n! = 4! 1!2!1! =12 What s the probablty of observng a gven quantum state, e.g. E? In other words,whatsthefractonofsystemsntheensemblenthestatee? 1

2 10.40: Fall 2003 Appendx 2 (Ensemble of systems),v,e A,V,E B,V,E C......,V,E (Bath at temperature T) Fgure 1: Canoncal ensemble The answer s n. However, t may be the case that many dstrbutons fulfll the condtons of the ensemble, (,V,E tot ). For example, assume that there are two: n 1 =1,n 2 =2,n 3 =1;Ω tot =12 n 1 =2,n 2 =0,n 3 =2;Ω tot =6 Then the probablty of observng, for example, E 3 s 1 4 n the frst dstrbuton 1 2 n the second. The probablty n the case where both dstrbutons make up the ensemble s: µ p 3 = = In general, µ 1 p = n Ω tot ( n )n ( n ) n Ω tot ( n ) where the sum s over all dstrbutons satsfyng the condtons of (,V,E tot ). Then, for example, we could compute ensemble averages of mechancal quanttes: E = he = X E 2

3 10.40: Fall 2003 Appendx 3,V,E A,V,E B,V,E,V,E C Fgure 2: Canoncal ensemble formng ts own bath = h = X p, where p s the pressure. 0.2 Maxmum term method Recall: where p = µ 1 n Ω tot ( n )n ( n ) n Ω tot ( n ) Ω tot ( n )=! Q n!. We can call ths ds- As, n,foreach. Thus, the most probable dstrbuton becomes domnant. trbuton, n. Let n = n n the n dstrbuton. Then p = 1 Ω tot ( n )n Ω tot ( n ) = n 0.3 Most probable dstrbuton Whch dstrbuton gves the largest Ω tot? Solve va method of undetermned multplers: Take natural log of Ω tot. ln (Ω tot ( µ! n )) = ln Q n =! Ã! Ã! X X n ln n X n ln n, where we have swtched the ndex from to used Strlng s approxmaton, whch becomes exact as n : ln y! y ln y y. We wsh to fnd the set of n s, whch maxmze Ω tot ( n ) hence ln(ω tot ( n )) : " ln (Ω tot ( n )) α X n β X # n E n =0, =1, 2, 3,... 3

4 10.40: Fall 2003 Appendx 4 where α β are the undetermned multplers. Carryng out the dfferentaton yelds Ã! X ln n ln n α βe =0, =1, 2, 3,... or n = e α e βe, =1, 2, 3,... Recallng that = X n or yelds X e α e βe =1 e α = X e βe. where Also, he = n E = e α e βe E = p = n = e α e βe = e βe e βe, Q = X e βe e βe E e βe, as we dscussed n the last lecture, s the partton functon, the normalzaton factor. 0.4 Canoncal ensemble contnued connecton to thermodynamcs Recall from last tme, va the maxmum-term method n the canoncal ensemble: he = n E = e α e βe E = e βe E e βe where, p = n = e α e βe e βe =, e βe Q = X e βe, as we dscussed n the last lecture, s the partton functon, the normalzaton factor. In addton, as we have shown: E = he = X p E 4

5 10.40: Fall 2003 Appendx 5 where s the pressure. = h = X p, If we dfferentate the equaton for he, dhe = X E dp + X p de = 1 X (ln p +lnq) dp + X µ E p dv. (1) β Recall that the pressure, µ E = or µ E =. Ths yelds for equaton 1 dhe = 1 X ln p dp 1 X ln Qdp + X p dv. β β [ote that d X p ln p = X ln p dp + X p d (ln p ) = X ln p dp + X p dp p. (2) Snce, X p =1, X dp =0.] Thus, the rght term n equaton 2 s equal to 0. Ths yelds dhe = 1 β d X p ln p h dv. Recallng from the combned frst second laws (n ntensve form, notng that snce s a constant, ntensve extensve forms are equvalent): du = TdS dv Snce U he p hp, TdS 1 β d X p ln p. 5

6 10.40: Fall 2003 Appendx 6 Let X = X p ln p. Then, ds = 1 dx. (3) βt We know that the left sde of the equaton s an exact dfferental, so the rght sde 1 must be too, thus, βt must be a functon of X. Ths means that ds = φ(x)dx = df (X). Integratng, S = f(x)+const, (4) where we can set the arbtrary constant, const, equal to 0 for convenence. ow we can make use of the addtve property of S, we can dvde a system nto two parts, A B. Ths yelds: ote that S = S A + S B = f(x A )+f(x B ). (5) X A+B = X, p, ln p,, where s the ndex for the possble states of A s the ndex for the possble states of B. Then X A+B = X p A pb (ln pa +lnp B ), = X p A ln p A X p B ln p B Thus, from equaton 5, = X A + X B. S = f(x A )+f(x B )=f(x A + X B ). For ths to be so, where k s a constant. Thus, f(x) =kx, S = k X p ln p. (6) From equatons 3 4, 1 βt = k, thus, β = 1 kt. We desgnate k as Boltzmann s constant, a unversal constant. 6

7 10.40: Fall 2003 Appendx Mcrocanoncal, Gr Canoncal, other ensembles Recallng the formulaton for S from equaton 6, notng that n the mcrocanoncal ensemble, p = 1 Ω, where we recall that Ω s the total number of states wth the same energy, then S = k X p ln p = k X 1 Ω ln 1 Ω = k ln Ω(,V,E). Ths s Boltzmann s famous formula for the entropy. In the Gr Canoncal ensemble, the number of partcles n each system s allowed to fluctuate, but µ s kept constant. Ths s called the (V,T,µ) ensemble. Also, there are other ensembles, such as (,,T), etc. otethatfromananalyss of fluctuatons (Lecture 27), we shall see that n the macroscopc lmt of a large number of systems, all of these ensembles are equvalent. 0.6 Relaton of thermodynamc quanttes to Q Recall that S = k X p ln p p = e βe Q Q = X e βe luggng n the formula for p nto that for S yelds S = k X = k X e βe Q e βe Q e βe ln Q µ E kt ln Q = he T + k ln Q Recallng our defntons from macroscopc thermodynamcs the fact that U he yelds A= kt ln Q Smlarly, µ µ A ln Q S = = kt + k ln Q T V, T V, µ µ A ln Q = = kt T, T, µ ln Q U = A + TS = kt 2 T 7 V,.

8 10.40: Fall 2003 Appendx 8 Thus, all thermodynamc propertes can be wrtten n terms of the partton functon, Q(,V,T)! In order to compute Q, all we need are the possble energy levels of the system. We can obtan these from solvng the equatons of quantum mechancs. 8