Entropy: From the Boltzmann equation to the Maxwell Boltzmann distribution


 Kathryn Hunter
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1 Enropy: From he Bolzmann equaion o he Maxwell Bolzmann disribuion A formula o relae enropy o probabiliy Ofen i is a lo more useful o hink abou enropy in erms of he probabiliy wih which differen saes are occupied. Les see if we can describe enropy as a funcion of he probabiliy disribuion beween differen saes. W N paricles sirling N n 1 n 2...n (N e) N W N paricles (n 1 e) n 1 (n 2 e)n 2...(n e) n wih N N n n 1 n 1 n 2 n 2...n p i N n i W N paricles akeln 1 n p 1 n 1 p 2 n 2...p lnw N paricles "# n i divide N paricles lnw 1paricle "# p i imes k k lnw 1paricle "k# p i S 1paricle and S NA "N A k# p i ln p "R# p i Think abou how his equaion behaves for a momen. If any one of he saes has a probabiliy of occurring equal o 1, hen he of ha sae is 0 and he probabiliy of all he oher saes has o be 0 (he sum over all probabiliies has o be 1). So he enropy of such a sysem is 0. This is exacly wha we would have expeced. In our coin flipping case here was only one way o be all heads and he W of ha configuraion was 1. Also, and his is no so obvious, he maximal enropy will resul if all saes are
2 equally populaed (if you wan o see a mahemaical proof of his check ou Ken Dill s book on page 85). In oher words a compleely fla disribuion of probabiliies, has he highes enropy. Calculaing wih our formulas for enropy We now have spen a lo of ime alking abou coin flips and erahedrons and derived wo equaions for enropy. I is now ime o pu some flesh on he bones of his heory and use i. Firs, les look a a simple molecular example. Les say we have a lysine sidechain ha is being pulled sraigh by an exernal elecrosaic field. We now urn of he field insananeously. Then we wach he reacion in which he sreched ou lysine slowly sars o adop a random conformaion. Wha is he sandard free energy for his reacion? Our reacion is: Lysine ordered > Lysine disordered And ΔG 0 ΔH 0  T ΔS 0 For now we can assume ha ΔH 0 is zero for his reacion so ΔG 0  T ΔS 0 How can we calculae ΔS 0? Simple, we jus use he Bolzmann Equaion. S 0 RlnW Of course here is only one way o be fully exended so he W for he sreched ou form is 1 and S 0 0. How many ways are here for he Lysine o be disordered? One migh be emped o hink ha his number is infiniy, bu common chemical knowledge ells us oherwise.
3 Because of seric clashes, each of he carbon bonds will adop only one of hree differen roamers. So if we assume ha all combinaions of roamers would be possible we would ge a W for our compleely disordered form of: W so S 0 disordered Rln81 hen R " 4.4 ΔG 0 ordered >disordered T (ΔS 0 diordered  ΔS 0 ordered) T (ΔS 0 diordered  0) T R 4.4 and wih kcal/mol G 0 ordered >disordered 2.64 kcal/mol A common quesion / objecion. Jus when I urn around from finishing his calculaion on he blackboard here are usually a number of sudens who will objec o his resul, because he lysine side chain is never quie rigid in eiher he fullyexended conformaion or in any of he oher roamers. Insead, here are clearly many conformaional subsaes and vibraions ha conribue o enropy in each one of hese saes and because I neglec all hese subsaes my caroonish represenaion of lysine will lead o compleely meaningless number for he enropy change. Afer all here mus be housands of subsaes for each one of he roamers. How can we know exacly how many subsaes here are and how do you define wha a subsae is anyway? These are all very good poins (And i is always good o use your common sense and experience wih oher suff o double check wha you learn in his class or in any oher class.) Bu before you all hrow up your hands and conclude ha doing all he mah ha go us o his poin was a complee wase of ime, I would encourage you o use his very mah o see, if he presence of his large number of subsaes or he choice when we call somehing a separae subsae acually impacs our resuls. So for he sake of he argumen, les assume ha he exended conformaion acually consiss of 10,000 conformaional subsaes and ha he same is rue for each one of our roamers.
4 W ordered 1"10,000 so S 0 ordered Rln(1"10,000) and W disordered 81"10,000 S 0 disordered Rln(81"10,000) #S 0 ordered $disordered Rln(81"10,000) % Rln(1"10,000) wih ln(a " b) ln a + lnb #S 0 ordered $disordered R(ln81+ ln10,000) % R(ln1+ ln10,000) #S 0 ordered $disordered R " ln81% R " ln1 #S 0 ordered $disordered R " 4.4 As you see, he fac ha each of our roamers may acually consis of many, many subsaes does no effec he enropy from going o he ordered sae o he disordered sae, nor does he way we define wha makes a subsae a subsae. We could say ha each subsae acually consiss of anoher1000 subsubsaes and we would ge he very same resul. Of course, he calculaion above assumes, ha each one of he roamers has he same number of subsaes and his may no be rue. However, we also do no have any reason o believe ha any one of he roamers would have a much larger number of subsaes han all he oher ones. Feel free o do he mah o see how our resuls change if you have only half as many subsaes in he exended form han in all he oher forms. You will see ha even such a drasic difference in he number of subsaes only resuls in a 10% change in he enropy. So in he absence of any informaion o he conrary, assuming ha all of hem have he same number of subsaes is a very good assumpion and, even if he number of subsaes differs subsanially, he error resuling from violaing ha assumpion is no very daramaic. Anoher poin One poin I have o concede hough is ha no all of he 81 saes will be occupied equally. For example, we did no accoun for he fac ha here will be some combinaions of roamers ha will no be occupied because of seric collisions. So if we really waned o know he enropy, we could perform a molecular dynamics simulaion in a compuer of he free lysine sidechain.
5 We could hen plo ou he probabiliy, wih which each of he roamers ges adoped. And use he formula S 0 "R# p i o calculae he enropy of he disordered form ha way. If resrains are presen a sysem will end owards he maximal enropy configuraion ha is consisen wih hese resrains. Bu wha is going on here, didn we jus say ha he enropy of a sysem is maximal if he disribuion if compleely fla? Should a sysem no always end o maximal enropy (principle of maximum mulipliciy a.k.a. he principle of maximum enropy)? Well yes, a sysem is always going o go owards maximal enropy, bu ofen here are resrains imposed on a sysem. For example he number of differen saes ha are accessible is limied. If his is he only resrain, he sysem will simply populae each accessible sae wih equal probabiliy and hus achieve he maximal enropy possible. In general, if resrains are presen, he sysem adops he maximal enropy configuraion ha is compaible wih hose resrains. To undersand his principle, les look a dice. Because we only have 6 sides he maximal enropy sae of a sysem of dice will be he case where each side faces up wih a probabiliy of 1/6. Resrains on he average properies of a sysem Les see wha happens if we inroduce a resrain in our dice example.
6 If we apply our principle of maximal enropy, we will ge a disribuion ha has equal numbers of and 6. p 1 p 2 p 3 p 4 p 5 p 6 1/6 bu his maximal enropy configuraion has a defined average score. "#$ % p i &# i i So he only way o change our average score is o change he values of he differen p i which as we know will decrease our enropy, because S "k # i p i and S is minimal if all p i are exacly equal (p 1 p 2 p 3 p ). So our populaion, if lef alone, will end owards adoping ha even disribuion wih is associaed average <ε>. However in many cases in real life here are environmenal influences ha dicae he average propery of a sysem. The example ha springs o mind righ away is he average kineic energy of a subsance, which is direcly conrolled by he emperaure. If we hea a sysem, he sysem will change is average kineic energy and his will change is enropy. We are now asking wha is he probabiliy disribuion ha gives he maximal enropy while saisfying he imposed resrains. Les say we wan o have an average value of 5 insead of 3.5. How could we ge a disribuion wih ha value? 1) we could make all dice show 5. Tha would obviously be a low enropy. As a maer of fac he enropy would be zero. 2) We could make half of he dice show 4 and half 6. Then our enropy would be k ln0.5 k ln0.5 k ) Or we could ge 1/3 of all dice o show 4, 5 and 6. Then our enropy would be k ln k ln0.333 k So we see ha we can mainain an average score of 5 and sill ge quie a boos in enropy by spreading ou our probabiliy disribuion over a few values for he dice, bu wha is he absoluely highes enropy we can achieve while mainaining ha average of 5. Imposing resrains on sysems leads o exponenial disribuion funcions. The answer is an exponenial disribuion funcion. Ken Dill nicely derives his very cenral resul in his Book (see page 86). I am no going o show his here, so eiher ake my word for i or look i up.
7 Specifically for a sysem wih saes he exponenial funcion ha will give us he maximal enropy will look like his p i % e"#$ pi e "#$ pi Noice how he probabiliy of all p i depends only on ha propery of sae i ha conribues o he average propery of he populaion and on he parameer bea. So if we wan o find he maximal enropy configuraion of a sysem wih a cerain average propery, all we need o do is o urn up or down bea. And use "#$ % p i & # i o calculae our average propery. i On he nex page I show you wha our disribuions for he dice example look like for differen beas and wha he enropy of hese differen saes are. Noe ha bea 0 gives you he fla disribuion, and ha his disribuion has he highes enropy as well as he average value 3.5 (shown in quoaion marks). Also noe ha he exponenial disribuion wih an average value of 5 has an enropy of 1.367, significanly beer han he enropy of he 1/3 4 s, 1/3 5 s, 1/3 6 s disribuion.
8 The equaion ha describes he exponenial disribuion giving us he maximal enropy under resrained condiions p i % e"#$ pi e "#$ pi is called he MaxwellBolzmann disribuion. The form in which we will see i mos of he ime is
9 p i e" # G i e " G i where he propery of sae i is he free energy of his sae and he facor bea is 1/. This las equaion describes a very fundamenal propery of sysems wih large numbers of paricles. In one form or anoher he MaxwellBolzmann disribuion shows up in all sors of differen places in he naural sciences. For example, if we wan o know he relaive probabiliy of wo differen saes and we know he sae s energies we can use his formula o calculae he former from he laer. Conversely, if we know he relaive probabiliy of wo differen saes a a given emperaure we can ge heir relaive energies. p a p b # e " G a # e " G i e " G b e " G i G a e" e " G b e " $G Summary I hope I have convinced you ha enropy is nohing myserious or ha i is paricularly difficul o undersand, when we approach i from he molecular level. Enropy is proporional o he mulipliciy W of microscopic saes ha conribue o a macroscopic sae. This is o say, a sysem ends owards a sae of maximal enropy, simply because here are more microscopic ways o be in a macroscopic sae wih high mulipliciy/enropy. The Bolzmann equaion allows us o calculae he enropy from he mulipliciy. S N A k lnw We can conver he Bolzmann equaion ino a differen form ha relaes probabiliy disribuions o enropy: S "N A k# p i wih # p i 1 A differen way of looking a his is o look a he disribuion of probabiliies, wih which a molecule adops a number of differen saes. Given a fixed number of microscopic
10 saes, he sae of maximal enropy is he sae ha has he flaes disribuion i.e. he sae in which each microscopic sae has exacly he same probabiliy. So if we place no resrains on our sysem oher han he number of saes ha are accessible we will always end owards a sae wih a compleely fla disribuion of probabiliies. Finally we realized ha he maximum enropy sae of a sysem will deermine many of he macroscopic properies of a sysem. If we now force our sysem o change hose average properies, we will auomaically force he sysem ino a sae of lower enropy. The quesion now is, Wha disribuion across he microscopic saes reains he maximal enropy, while achieving ha average propery. The answer is An exponenial disribuion. Specifically i is he MaxwellBolzmann disribuion, which has he following equaion. p i e" # G i e " G i This equaion relaes he free energy of a specific sae G i o he probabiliy of populaing sae i via he emperaure of our sysem and he energies of all oher saes accessible o our sysem.
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