Part II Statstcal Mechancs, Lent 25 Prof. R.R. Horgan February 14, 25 1 Statstcal Mechancs 1.1 Introducton The classcal and quantum dynamcs of smple systems wth few degrees of freedom s well understood n that we know the equatons of moton whch govern ther tme evoluton and, n most cases, we can solve these equatons exactly or, n any case, to the requred degree of accuracy. Even systems wth many degrees of freedom can be analyzed, for example, many problems n electromagnetc theory, where the feld degrees of freedom are large, can be solved usng Maxwell s equatons. However, t remans that there are physcal systems, whch are typfed by possessng a large number of degrees of freedom, for whch analyss of ths knd n napproprate or, n practce, mpossble, Whlst these systems are made up of dentfable consttuents whose nteractons are well known, dfferent concepts are needed to descrbe the evoluton of the system and the way t nteracts wth an observer. In partcular, the concepts of heat, temperature, pressure, entropy, etc. must arse from a proper analyss they are certanly not evdent n the equatons of moton, even though knowledge of the equatons of moton s ndspensable. These are deas whch come from a noton of statstcal averagng over the detaled propertes of the system. Ths s because they only make sense when thought of as propertes whch apply on the large scales typcal of the observer, rather than the mcroscopc scales of the ndvdual consttuents. Statstcal mechancs was developed to address ths conceptual problem. It enables one to answer questons lke how do you calculate the physcal propertes of a system n thermal contact wth large reservor of heat (known as a heat bath) at temperature T? Statstcal mechancs deals wth macroscopc systems wth very many degrees of freedom. For example, a gas wth a large number of partcles or a complex quantum system wth a large number of quantum states. A thermally solated or closed system s one whose boundary nether lets heat n or out. Consder such a system, S, that s not subject to any external forces. No matter how t s prepared, t s an expermental fact that after suffcent tme S reaches a steady state n that t can be specfed by a set, Σ, of tme-ndependent macroscopc or thermodynamc, varables whch usefully descrbe all the large-scale propertes of the system for all practcal purposes. The system s then sad to be n a state of thermal equlbrum specfed By macroscopc we mean that the system s large on the scale of the typcal lengths assocated wth the mcroscopc descrpton, such as the nter-atomc spacng n a sold or mean partcle separaton n a gas. That beng sad, t can be the case that systems wth as few as a hundred partcles can be successfully descrbed by statstcal mechancs. 1
1 STATISTICAL MECHANICS 2 by Σ. For example, for a gas solated n ths manner, Σ ncludes the pressure P, volume, temperature T, energy E, and entropy S. Although the state of thermal equlbrum descrbed by Σ s unque and tmendependent n ths sense, the ndvdual degrees of freedom such as partcle postons and momenta are changng wth tme. It s certanly not practcal to know the detals of how each degree of freedom behaves, and t s a basc tenet of statstcal mechancs that t s not necessary to know them n order to fully specfy the equlbrum state: they are not of any practcal mportance to calculatng the accessble expermental propertes of the equlbrum system. The task of statstcal mechancs s to use averagng technques and statstcal methods to predct the varables Σ characterzng the equlbrum state. We are generally able to descrbe the dynamcs of such systems at the mcroscopc level. For nstance, we can specfy the nteracton between neghbourng atoms n a crystal or between pars of partcles n a gas. Ths descrpton arses ultmately from our knowledge of quantum mechancs (and relatvty), but t may also be legtmate and useful to rely on classcal mechancs whch we vew as dervable from quantum mechancs by the correspondence prncple. We shall develop the deas of statstcal mechancs usng quantum mechancs n what follows. We defne (a) The mcroscopc states of S are the statonary quantum states. Ths s the Drac bra ( ) and ket ( ) notaton for states. The label stands mplctly for the complete set of quantum numbers whch specfy the state unquely. For example, H = E. The matrx element of an operator X between states and j s X j whch, n wave-functon notaton, s the same as dx φ Xφ j. Then X s the expectaton value, X, of X. (b) The macroscopc states of S are the possble states of thermodynamc equlbrum and are descrbed by the correspondng set of thermodynamc varables, Σ. These are not states n the quantum mechancal sense but nvolve a vast number of mcrostates. An mportant dea s the ergodc hypothess (whch has only been proved for a few systems ) whch states that: A system S evolves n tme through all accessble mcrostates compatble wth ts state of thermodynamc equlbrum. Ths means that tme averages for a sngle system S can be replaced by averages at a fxed tme over a sutable ensemble E of systems, each member dentcal to S n ts macroscopc propertes. An mportant feature s that the ensemble, E, can be realzed n dfferent ways whlst gvng the same results for the thermodynamc propertes of the orgnal system. The major realzatons of E are: the mcrocanoncal ensemble: each member of E has the same value of E and N. Ths s approprate to an solated system for whch the energy s fxed. There are many examples n one dmenson where the ergodc hypothess does not hold. Ths s a current research area
1 STATISTICAL MECHANICS 3 the canoncal ensemble: each member of E has N partcles but E s not fxed but the average energy over E s fxed. Ths s approprate to a system of a fxed number of partcles but whch s n thermal contact wth a heat bath wth whch t can exchange energy. The heat bath s all the other systems of the ensemble. the grand canoncal ensemble: nether E nor N s fxed for each member but the ensemble averaged values are fxed. Ths descrbes a system n thermal contact wth a heat bath wth whch partcles can also be exchanged. Members of the last two ensembles represent the orgnal system n the average. The reason why they all gve the same thermal physcs s that the fluctuatons of E and N about ther average values are very small because the number of partcles nvolved s very large. For example, consder the probablty p m of obtanng N/2 + m heads from N con tosses: ( ) p m = 2 N N 2 /N N/2 + m πn e 2m2, usng Strlng s formula (log n! = n log n n + 1 2 log 2πn) for N m 1. We see that only for m N s p m apprecable n sze. Hence the mean fluctuaton, m/n 1/ N, s very small. The densty of a volume of gas, ρ = N/, s a macroscopc quantty on whch thermodynamc varables depend. If the fluctuatons n N are O( N) then δρ ρ, whch vanshes as. Hence, we expect the results from the dfferent ensembles to be the same n the nfnte volume lmt but to dffer by correctons of O(1/ ). The large volume lmt s always assumed n what follows. In ths course we manly study the canoncal ensemble, but for fermons and bosons t wll be smpler to use the grand canoncal ensemble. 1.2 The Canoncal Ensemble We study a system S of N partcles n a volume wth N and fxed. The mcrostates of S are the complete orthogonal set. We assume that the spectrum s dscrete wth possble degeneracy. We assocate wth S a canoncal ensemble E. Ths s a very large number A of dstngushable replcas S 1, S 2, S 3,... of S. Suppose we were to measure a complete set of commutng observables for all members of E smultaneously. Then let a be the number of members found n state. Then a = A, and a E = AE, (1.2.1) where E s the fxed, tme-ndependent ensemble average energy of the members of E. In other words a /A s the (quantum mechancal) probablty for fndng the system n mcrostate. The set {a } s called a confguraton of E, and the number of ways of parttonng A nto the set {a } s W (a) = A! a!. (1.2.2)
1 STATISTICAL MECHANICS 4 The mportant prncple underlyng statstcal mechancs s that we assgn equal a pror probablty to each way of realzng each allowed confguraton. Ths means that the probablty of fndng confguraton {a } s proportonal to W (a). In the lght of the earler dscusson, the probablty dstrbuton for the {a } wll be sharply peaked and the fluctuatons about the peak value wll be very small: O(1/ A). Snce we can take A as large as we lke, the average of any varable over the ensemble wll be overwhelmngly domnated by the value t takes n the most probable confguraton, {ā }. For fxed N, and E, we shall assocate the state of thermodynamc equlbrum of S wth the most probable confguraton of E that s consstent wth the constrants (1.2.1). We fnd {ā } by maxmzng log W (a) w.r.t. {a }. Usng Strlng s formula, (log n! = n log n n + 1 2 log 2πn), we have log W A log A A a (log a 1) = A log A a log a, (1.2.3) snce a = A. We maxmze ths expresson subject to the constrants (1.2.1) by usng Lagrange multplers. Then we have ( A log A a log a α a β ) a E =,. (1.2.4) a j Thus = log a j + 1 + α + βe j =, a j = e 1 α βe j. (1.2.5) We elmnate α usng (1.2.1), and defne Z, the canoncal partton functon: A = a = e 1 α Z, wth Z = e βe = E Ω(E)e βe, (1.2.6) where Ω(E) s the degeneracy of levels wth energy E, and E runs over all dstnct values n the spectrum of the system. The fracton of members of E found n mcrostate n the macrostate of thermodynamc equlbrum s ρ = a A = 1 Z e βe. (1.2.7) Ths s the Boltzmann dstrbuton and s thought of as the (quantum mechancal) probablty of fndng n the state of thermodynamc equlbrum. The average X of a physcal observable s then X = X ρ. (1.2.8) For example, E = E ρ = 1 a E = E, (1.2.9) A
1 STATISTICAL MECHANICS 5 from (1.2.1), as we expect t should. Z s very mportant snce we shall see that t allows us to calculate the thermodynamc and large-scale propertes of a system n equlbrum from the quantum mechancal descrpton of the system. An mportant example s that ( ) log Z E =. (1.2.1) β Holdng fxed means holdng the E fxed snce these latter depend on. As emphaszed earler for the canoncal ensemble we can thnk of each ndvdual system as beng n contact wth a heat bath made up of the rest of the ensemble. What are the fluctuatons n energy of a system? Usng above results, we have E β = 1 2 Z Z β 2 + 1 ( ) Z 2 Z 2 β = E 2 + E 2 = ( E) 2. (1.2.11) For large systems typcally E N and, as E depends smoothly on β we expect E β N, = E E N N = N 1 2 1 2. (1.2.12) We have found that energy plays a vtally central rôle n determnng the probablty for fndng a system n a gven state. In prncple, the ρ can depend on other varables too. For example, the partcle number N, correspondng to the grand ensemble, or even angular momenta where relevant. In general, such quanttes must be be conserved and observable on the large scale. 1.3 Temperature Consder two systems S a and S b wth volumes a, b and partcle numbers N a, N b, respectvely. Form a jont ensemble E ab from the separate ensembles E a and E b by allowng all members of both to exchange energy whlst keepng the overall total energy fxed. By allowng S a and S b to exchange energy we are allowng for nteractons between the partcles of the two systems. These wll be suffcent to establsh equlbrum of S a wth S b but otherwse be neglgble. For nstance, they take place across a boundary common to the systems and so they contrbute energy correctons whch scale wth the boundary area and so are therefore neglgble compared wth the energy, whch scales wth volume. Now form a composte system S ab by jonng S a and S b together. Let S a have mcrostates a wth energy E a and S b have mcrostates b wth energy E b. Because the extra nteractons can be neglected n the bulk, then S ab wll have mcrostates j > ab = > a j > b wth energy Ej ab = Ea + Eb j. Now, E a and E b are separately n equlbrum characterzed by Lagrange multplers β a and β b, respectvely. Also, E ab s an equlbrum ensemble whch s characterzed by β ab, and so we must have ρ ab j (E a + E b j) = ρ a (E a )ρ b j(e b j), E a and E b j. (1.3.1) Then e β ab(e a +Eb j ) Z ab = e (βaea +β be b j ) Z a Z b, E a and E b j. (1.3.2)
1 STATISTICAL MECHANICS 6 Ths can only be satsfed f β ab = β a = β b and hence Z ab = Z a Z b. It s nterestng to note that the result (1.3.1) could be gven as the defnton of what t means for S a to be n thermal equlbrum wth S b. Ths s because we expect that n equlbrum these probabltes depend only on the energes of the systems. Then relaton (1.3.1) must hold snce energes add and probabltes multply. The only soluton to (1.3.1) s the Boltzmann dstrbuton (1.2.7). The relaton (1.3.1) can be establshed drectly by fndng the most probable dstrbuton for the ensemble E ab. Suppose a occurs a tmes, j b occurs b j tmes and j ab occurs c j tmes n E ab. Then we have a = A, b = A (1.3.3) a = j c j, b j = c j, c j = A. (1.3.4) j We only have one constrant on the energy whch s that the total energy of E ab s fxed: AE = a E a + j b j E b j, (1.3.5) AE = j c j (E a + E b j). (1.3.6) The confguraton {a, b j } arses n ways and the confguraton {c j } arses n W (a)w (b) A! A! a! j b j!, (1.3.7) W (c) A! j c j!, (1.3.8) ways. The most probable dstrbuton of {a, b j } s obtaned by maxmzng W (a)w (b) subject to (1.3.4) and (1.3.6) to get a = e 1 αa βea, ρ a = a A, b j = e 1 α b βe b j, ρ b j = b j A. Note that snce there s only one energy constrant the most probable {a } and {b j } depend on the same value of β. The temperature, T, s defned by the statement that two systems n thermal equlbrum have the same temperature. We therefore conclude that two systems n thermal equlbrum are descrbed by Boltzmann dstrbutons wth a common value of β whch can therefore be thought of as defnng the temperature The most probable dstrbuton of {c j } s obtaned by maxmzng W (c) subject to the constrants (1.3.4) and (1.3.6) to get c j = e 1 α ab β (E a +Eb j ), ρ ab j = c j A. (1.3.9)
1 STATISTICAL MECHANICS 7 So far β and β do not necessarly have the same value snce they are just Lagrange multplers mposng the energy constrant n two separate maxmzatons. However, the relatons between {c j } and {a, b j } gven n (1.3.4) can be only be satsfed f β = β. It then follows that (c.f. (1.3.1)) Z ab (β) = Z a (β)z b (β), and ρ ab j = ρ a ρ b j. (1.3.1) The temperature can be defned by β = 1 kt, (1.3.11) where k s Boltzmann s constant and T s n degrees Kelvn. Because the analyss apples to any par of systems k s a unversal constant: k = 1.38 1 23 Joules/ K. Ths defnton ensures that the average energy ncreases as T ncreases. For a dlute gas wth N partcles ths mples Boyle s law, P = NkT, as we shall see. 1.4 Thermodynamc deas A system undergoes an adabatc change f () the system s thermally solated so that no energy n the form of heat can cross the system boundary: δq =, () the change s caused by purely by external forces actng on the system, () the change takes place arbtrarly slowly. By heat Q we mean energy that s transfered n a dsordered way by random processes. In contrast, work W s energy transfered through the agency of external forces n an ordered way. Work can be done on the system by movng one of the walls (lke a pston) wth an external force, whch corresponds to changng the volume. If the change s adabatc, no energy s permtted to enter or leave the system through the walls. Because the moton s slow the appled force just balances the pressure force P A (A s the area of the wall), and the work done on the system s then δw = P AδL = P δ. If the system s n a gven mcrostate wth energy E the change s slow enough that t remans n ths state there s no nduced quantum-mechancal transton to another level. Ths s the Ehrenfest prncple. The change n energy balances the work done and so δw = δe = E δ. (1.4.1) For a system n TE the work done n an adabatc change s the ensemble average of ths result P δ = δw = ρ E δ = P = ρ E. (1.4.2) Ths s reasonable snce f P exsts then t s a thermodynamc varable and so should be expressble as an ensemble average. These deas are most easly understood n the context of an deal gas. Consder the quantum mechancs of a gas of N non-nteractng partcles n a cube of sde L. The
1 STATISTICAL MECHANICS 8 sngle partcle energy levels are En = h 2 n 2 /2mL 2 wth n Z 3. Note that for gven n, En 2/3 generally, we expect E = E( ). Then E = 2 E 3 = P = 2 3 ρ E = 2 3ɛ, (1.4.3) where ɛ = E/ s the energy densty. Ths s the formula relatng the pressure to the energy densty for a non-relatvstc system of partcles. We also deduce the energy conservaton equaton for an adabatc change (δq = ): P δ + δe =. (1.4.4) For the deal gas an adabatc change does not change the occupaton probabltes ρ of the energy levels just the energes of the levels change. Ths s the Ehrenfest prncple. For systems wth E ( ) s the ρ are stll Boltzmann equlbrum probabltes n + δ but wth a dfferent temperature s = 2/3 for the deal gas. If the change s not adabatc then heat energy can enter or leave the system. Snce E = ρ E, n the most general case we have δe = ρ δe + E δρ or δe = P δ + E δρ. (note that δρ =.) (1.4.5) To deal wth ths eventualty we ntroduce the concept of entropy. Defne the entropy, S, of a system by S = k log W (a), (1.4.6) A where, as before, W (a) s the number of ways of realzng the ensemble confguraton {a }. Snce there are A members n the ensemble ths s the entropy per system. Note that the ensemble s not necessarly an equlbrum one. However, wth ths defnton we see that for an equlbrum ensemble S eq = k A log W max, (1.4.7) and from (1.4.6) and (1.4.7) we deduce that S s maxmzed for the an equlbrum ensemble. Usng (1.2.2) we get S = k ( A log A ) a log a A = k ( A log A a log ρ log A ) a, A = S = k ρ log ρ (1.4.8) where we have used ρ = a /A, a = A. Ths s an mportant formula whch can be derved n many dfferent ways from varous reasonable requrements. However, for
1 STATISTICAL MECHANICS 9 complex systems t needs some nterpretaton but for the free gas there s no dffculty. More on entropy shortly. Let us apply the dea to a general change between nearby states of TE, e.g., an sothermal change (dt = ). Change the volume of the system arbtrarly slowly and allow heat energy to be absorbed/emtted, e.g., by contact wth a heat bath. At every stage the system wll have tme to equlbrate and so although the ρ wll be tme-dependent they wll take ther equlbrum values approprate to the condtons prevalng (.e., values of, T ). In ths case we have (S wll stand for the equlbrum value unless otherwse stated) ρ = = δs = k 1 Z(, T ) e βe ( ), δρ (log ρ + 1), = k δρ ( βe log Z + 1), = 1 E δρ T = E δρ = T δs. (1.4.9) So, for a slow change, usng (1.4.5), we get n general δe = T δs P δ, or, n exact dfferentals de = T ds P d. (1.4.1) We can thnk of ths as a comparson between two nfntesmally dfferent states of thermodynamc equlbrum. Ths equaton s an expresson of the Frst Law of Thermodynamcs whch s the statement of energy conservaton: The Frst Law of Thermodynamcs states that de = δq + δw. (1.4.11) Here de s the dfferental change n the nternal energy of S, δq s the heat energy suppled to S, and δw s the work done on S (e.g., δw = P d ). For a slow change where the system s nstantaneously always n thermal equlbrum the change s sad to be reversble. Ths s because no work done s dsspated as unrecoverable energy such as through frctonal forces or strrng of the system. In prncple, the work done on the system can be recovered by allowng t to do work on another system, for example, a slowly compressed sprng. From above, comparng (1.4.1) and (1.4.11), for a reversble change we have reversble change = δq = T ds. (1.4.12)
1 STATISTICAL MECHANICS 1 Note that nether δq or δw are exact dervatves even though de, ds and d are. For an adabatc change we have ds = δq/t =. So the defnton of an adabatc change may be taken to be ds =. From (1.4.1) we see that T = ( ) E S, P = ( ) E S. (1.4.13) For a system n equlbrum we can derve S S eq from the partton functon. We have S = k = k ρ log ρ ρ ( βe log Z) = 1 E + k log Z. (1.4.14) T We defne the free energy, F, to be F = kt log Z = F = E T S. (1.4.15) In dfferental form we then have df = de T ds SdT, or, usng the frst law n the form (1.4.1) for nearby equlbrum states ( ) F df = P d SdT, = P = T ( ) F, S = T. (1.4.16) We shall see that the dea of the free energy, F, s a very mportant one n thermodynamcs. From (1.4.16) we deduce that F s a functon of the ndependent varables and T. Smlarly, from (1.4.16), F = F (, T ), = P = P (, T ), S = S(, T ). (1.4.17) Note also that E = E(S, ) snce S, are the ndependent varables. Thus the thermodynamc state of equlbrum characterzed by P,, T, S can be determned from the partton functon, Z, through the dea of the free energy, F. The equaton whch relates P, and T, n ether of the forms shown n (1.4.16) and (1.4.17), s known as the equaton of state. Remember, all these quanttes depend mplctly on the fxed value of N and hence on the partcle densty n = N/. 1.5 Extensve and ntensve varables Suppose an equlbrum system s placed next to an exact copy and the partton between them s wthdrawn to make the system twce the sze. arables that double n ths process are called extensve and those whose value s the same as the orgnal are called ntensve. For example, N,, S, E are extensve and P, T are ntensve.
1 STATISTICAL MECHANICS 11 To elaborate. Consder two equlbrum systems at temperature T, S 1 and S 2, whch are dentcal n every way except that they have volumes 1 and 2, respectvely. Put them next to each other but do not remove the partton. Then for the whole system, S 12, we have 12 = 1 + 2, N 12 = N 1 + N 2. Snce, Z 12 = Z 1 Z 2, we see that E = E 1 + E 2, S = S 1 + S 2 etc. If these varables are extensve as clamed then these equaltes wll hold after the partton s removed. We expect ths to be the case f S 1, S 2 are homogeneous (whch means S 12 wll be, too). The partcles n each system are of the same nature and are not dstngushable and so by removng the partton we do not mx the systems n a detectable way. Hence, the entropy S 12 should reman unchanged. Then we expect that, E, S are proportonal to N. Clearly, S 1 and S 2 have the same T, P and so T, P are ntensve. Ths expectaton can fal, for example, (a) f surface energes are not neglgble compared wth volume energes the partton matters too much, (b) there are long-range forces such as gravty. The systems are no longer homogeneous. Thnk of the atmosphere dfferent parcels of ar at dfferent heghts may have the same temperature but the pressure and densty are functons of heght. 1.6 General Remarks () Let S be the sum of two systems, S = S 1 + S 2, nether of whch s necessarly n equlbrum. Although the partton functon s not useful we can stll show that the entropy defned by (1.4.8) s addtve. Let the level occupaton probabltes for S 1, S 2 be ρ, σ, respectvely. Then S = k j = k j ρ σ j log ρ σ j ρ σ j (log ρ + log σ j ) = k ρ log ρ + j σ j log σ j where we have used ρ = j σ j = 1. = S 1 + S 2, (1.6.1) () From the defnton of entropy (1.4.6) we see that the entropy s a maxmum for a system n equlbrum. Indeed, t was the entropy that we maxmzed to determne the equlbrum state. More on ths later. () Of the four varables, P,, T, S, any two can be treated as ndependent on whch the others depend. We have naturally found above, T to be ndependent, but we can defne G = F + P whch leads to dg = df + P d + dp, = dg = dp SdT. (1.6.2) G s called the Gbbs functon. Clearly, G = G(P, T ) and now and S are functons of P, T. Ths merely means that the expermenter chooses to control P, T and see how, S vary as they are changed. The transformaton G = F + P, whch changes the set of ndependent varables, s called a Legendre transformaton.
1 STATISTICAL MECHANICS 12 1.7 The example of a perfect gas A perfect gas conssts of a system of free partcles whch therefore do not nteract wth each other. The gas s n a cubcal box wth walls at temperature T, and energy wll be exchanged between the partcles and the walls when the partcle collde (nelastcally) wth the walls. The gas wll come nto equlbrum wth the box and wll have temperature T, also. 1.7.1 the densty of states Let the number of partcles be N, the box be of sde L wth volume = L 3, and the mass of each partcle be m. We use perodc boundary condtons to mpose the condton of fnte volume so that the sngle partcle wavefuncton n the box satsfes ψ(x + Le ) = ψ(x), (1.7.1.1) where the e, = 1, 2, 3 are unt vectors spannng R 3. For free non-relatvstc partcles we then have energy egenstates ψ k (x) = 1 e k x wth k = 2π L n, n Z3, (1.7.1.2) p = hk, ɛ(p) = h2 k 2 2m. (1.7.1.3) We are nterested n the lmt L and so for fxed momentum, p, we have that n, n /L fxed. Snce, n s large we can treat n as a contnuous varable. The number of states n n n + dn s d 3 n whch s therefore the number of states n k k + dk wth dk = (2π/L)dn. Thus, The number of states n k k + dk s ( ) L 3 d 3 k. (1.7.1.4) 2π Because the state label can be treated as contnuous we have ( ) L 3 d 3 d 3 k k = 2π (2π) 3 = = where g(ɛ) = (2π h) 3 g(ɛ)dɛ 4πp 2 dp = 2π 2 h 3 p 2 dp dɛ dɛ d 3 p (2π h) 3 dp 3 p2 2π2 h dɛ. (1.7.1.5) For partcles wth spn there s an extra factor of g s = 2s + 1. Then, usng (1.7.1.3), we have = m g(ɛ) = g s 3 (2mɛ) 2mɛ, 2π2 h g(ɛ) = g s 4π 2 ( ) 3 2m 2 1 ɛ h 2 2. (1.7.1.6)
1 STATISTICAL MECHANICS 13 The functon g(ɛ) s the densty of states. A general analyss n D-dmensons gves S D g(ɛ) = g s (2π h) D pd 1 dp dɛ, (1.7.1.7) where S D s the surface area of the unt sphere n D-dmensons, and = L D. Agan for non-relatvstc partcles we can use (1.7.1.3) to get D = 1 g(ɛ) = g s 2π D = 2 g(ɛ) = g s 4π ( 2m h 2 ( 2m h 2 ) 1 2 ɛ 1 2 ). (1.7.1.8) Note that for D = 2 g(ɛ) s a constant ndependent of ɛ. For relatvstc partcles of rest mass m, the analyss s dentcal and we can use the general formula (1.7.1.7) wth the relatvstc dsperson relaton ɛ(p) = p 2 c 2 + m 2 c 4. In partcular, photons have m = and g s = 2 snce there are two dstnct polarzaton states. In ths case, and for D = 3 we get g(ɛ) = ɛ 2 π 2 h 3 c 3. (1.7.1.9) Snce ɛ = hω we get the densty of states n ω ω + dω, g(ω), to be g(ω) = g(ɛ) dɛ dω = ω2 π 2 c 3. (1.7.1.1) 1.7.2 the partton functon for free spnless partcles Consder N partcles n volume = L 3. Then the sngle partcle partton functon s z = e βɛ dɛ g(ɛ)e βɛ. (1.7.2.1) From (1.7.1.6), and settng y = βɛ, ths gves z = ( ) 3 2mkT 2 4π 2 h 2 dy y 1 2 e y ( 2mπkT = In D-dmensons a smlar results holds and we fnd from (1.7.1.7) h 2 ) 3 2. (1.7.2.2) z (kt ) D/2. (1.7.2.3) Now, from before, we have Z = z N and so log Z = N log z. Then E = N β log z = N β ( D 2 log β + const.) = D NkT. (1.7.2.4) 2 Ths s an mportant result of classcal statstcal mechancs and s called the equpartton of energy whch states that The average energy per partcle degree of freedom s 1 2 kt
1 STATISTICAL MECHANICS 14 Combnng ths result wth P = 2 3E from (1.4.3) we get the equaton of state for a perfect gas. P = N kt Boyle s Law. (1.7.2.5) Note that ths result holds n D-dmensons snce the generalzaton of (1.4.3) s P = 2 D ɛ. 1.7.3 entropy and the Gbbs paradox Usng (1.4.15) and (1.4.16), whch states that we get F = kt log Z, S = ( ) F T, S = Nk log z + 3 2 Nk = Nk log + 3 2 Nk log ( 2mπkT h 2 ) + 3 Nk, (1.7.3.1) 2 whch s not extensve snce N. Ths s Gbbs paradox. The resoluton s to realze that usng Z = z N treats the partcles as dstngushable snce t corresponds to summng ndependently over the states of each partcle. In realty the partcles are ndstngushable, and we should not count states as dfferent f they can be transformed nto each other merely by shufflng the partcles. The paradox s resolved by accountng for ths fact and takng Z = z N /N!. Usng Strlng s approxmaton, we fnd ( ) S = Nk log + 3 ( ) 2mπkT N 2 Nk log h 2 + 3 Nk, (1.7.3.2) 2 whch s extensve. We should note at ths stage that dvdng by N! correctly removes the overcountng only n the case where the probablty that two partcles occupy the same level s neglgble. Ths s the stuaton where Boltzmann statstcs s vald and certanly t works for T large enough or the number densty s low enough. In other cases we must be more careful and we shall see that the consequences are very mportant. 1.8 The Increase of Entropy We saw n secton 1.4 from the defnton of entropy (1.4.6) and (1.4.7), that S s maxmzed for an equlbrum ensemble. Ths s a general prncple whch states that For a system wth fxed external condtons the entropy, S, s a maxmum n the state of thermodynamc equlbrum. By fxed external condtons we mean varables lke, E, N for an solated system or, T, N for a system n contact wth a heat bath at temperature T. For example, let S 1 be a perfect gas of volume 1 wth N 1 molecules of speces 1, and S 2 be a perfect gas of volume 2 wth N 2 molecules of speces 2. Both systems are
1 STATISTICAL MECHANICS 15 n equlbrum at temperature T. Place them next to each other wth the partton n place. The total entropy s S 1 + S 2 wth S = ( ) ( ( ) 3 kn log + kn N 2 log 2m πkt h 2 + 3 ) 2 }{{} B (T ). (1.8.1) Now remove the partton. Because the speces are dstngushable, the new system, S 12, s not ntally n equlbrum but ntally has entropy S 1 + S 2. The gases mx and after equlbrum has been attaned the partton functon of the new system, of volume = 1 + 2, s Z 12 = z 1 N 1 N 1! z 2 N 2 N 2! Note, not (N 1 + N 2 )! n the denomnator. Ths gves entropy. (1.8.2) S 12 = k [(N 1 + N 2 ) log( 1 + 2 ) N 1 log N 1 N 2 log N 2 + N 1 B 1 (T ) + N 2 B 2 (T )] [ = k N 1 log ( 1 + 2 N 1 ) + N 2 log ( ) 1 + 2 N 2 ] + N 1 B 1 (T ) + N 2 B 2 (T ). (1.8.3) So S 12 S 1 S 2 [ = k N 1 log ( ) ( )] 1 + 2 1 + 2 + N 2 log 1 2 >. (1.8.4) The entropy has ncreased because the gases have mxed. To separate the two speces requres to system to become more ordered all the boys to the left and all the grls to the rght. Ths s unlkely to happen spontaneously. From the defnton of the entropy (1.4.6) the more ordered a system s, the smaller s W (a) and the lower s the entropy. Notce that S 12 s the sum of the entropes that each gas would have f they each occuped the volume 1 + 2 separately. Ths s the law of partal entropes whch holds for deal gases. Remarks: () Entropy s a measure of nformaton. When a system s n equlbrum we know the least about t snce all avalable mcrostates are equally lkely. In contrast, the entropy s low when we know that the system s n one or very few mcrostates. Informaton theory s based on ths observaton. () The mcroscopc defnton of entropy S = k ρ log ρ, for a complex system s subtle. In partcular, reconclng the mcroscopc and thermodynamc defntons of an adabatc change (δs = ) s not smple but the perfect gas s an deal system to examne these deas. () When S 1 and S 2 separately come nto equlbrum we maxmze S 1 and S 2 separately each wth two constrants (for E 1, N 1 and E 2, N 2 ), makng four n all. When S 1 and S 2 are combned there are only two constrants (for E 1 + E 2 and N 1 + N 2 ) whch are a subset of the orgnal four. We expect that the maxmum of S 1 + S 2 subject to these two constrants wll be greater than the maxmum attaned when t s subject
2 THERMODYNAMICS 16 to the four constrants (t may be unchanged c.f. Gbbs paradox). Hence, the entropy wll generally ncrease when the systems are combned. Mxng just corresponds to removal of some constrants. (v) We have not ever dscussed how a system reaches equlbrum. In practce, t takes a certan tme to evolve to equlbrum and some parts of the system may reach equlbrum faster than others. Indeed, the tme constants characterzng the approach to equlbrum may dffer markedly. Ths can mean that although equlbrum wll be establshed t mght not occur on tme scales relevant to observaton. For example, some knds of naturally occurrng sugar molecules are found n one somerc form or handedness. In equlbrum, there would be equal of each somer but ths s not relevant to nature. The descrpton of the system at equlbrum need not nclude those forces whch are neglgble for most purposes. For example, exactly how a gas molecule nteracts wth the vessel walls s a surface effect and does not affect any thermodynamc property n the nfnte volume lmt. However, t s generally assumed that t s such small randomzng nteractons whch allow energy to be transfered between levels, allowng the most probable state to be acheved. They are vtal to the workng of the ergodc hypothess. Ths s by no means certan to hold and n one dmenson t s qute hard to establsh that t does. The pont s, that f a closed system s not n equlbrum then, gven enough tme, the most probable consequence s an evoluton to more probable macrostates and a consequent steady ncrease n the entropy. Gven ergodcty, the probablty of transton to a state of hgher entropy s enormously larger than to a state of lower entropy. Chance fluctuatons to a state of lower entropy wll never be observed. The law of ncrease of entropy states that: If at some nstant the entropy of a closed system does not have ts maxmum value, then at subsequent nstants the entropy wll not decrease. It wll ncrease or at least reman constant. 2 Thermodynamcs Consder a volume of gas wth fxed number N of partcles. From the work on statstcal mechancs we have found a number of concepts whch enable us to descrbe the thermodynamc propertes of systems: () E: the total energy of the system. Ths arose as the average ensemble energy and whch has neglgble fluctuatons. It s extensve. () S: the entropy. Ths s a new dea whch measures the number of mcrostates contrbutng to the system macrostate. S s a maxmum at equlbrum and s an extensve property of the system. It has meanng for systems not n equlbrum. () For nearby equlbrum states de = T ds P d.
2 THERMODYNAMICS 17 (v) F : the free energy. It s naturally derved from the partton functon and, lke E, s a property of the system. For nearby equlbrum states df = SdT P d. It s extensve. (v) All thermodynamc varables are functons of two sutably chosen ndependent varables: F = F (T, ), E = E(S, T ). Note, f X s extensve we cannot have X = X(P, T ). (v) Equatons of state express the relatonshp between thermodynamc varables. example, P = NkT for a perfect gas. For It s because E, S, P,, T, F etc. are propertes of the equlbrum state that we are able to wrte nfntesmal changes n ther values as exact dervatves. The changes n ther values are ndependent of the path taken from an ntal to fnal equlbrum state. Ths s not true of the heat δq absorbed or emtted n a change between nearby equlbrum states t does depend on the path: δq s not an exact dervatve. However, ds = δq/t s an exact dervatve, and 1/T s the ntegratng factor just lke n the theory of frst-order o.d.es. Energy conservaton s expressed by the fst law: δe = δq + δw for nfntesmal change E = Q + W for fnte change (2.1) where δq( Q) s the heat suppled to the system and δw ( W ) s the work done on the system. Suppose the state changes from E(S 1, 1 ) to E(S 2, 2 ) then n general we expect 2 W P d (2.2) 1 because addtonal work s done aganst frcton, generatng turbulence etc. However, de = T ds P d whch mples Q S2 S 1 T ds. (2.3) 2.1 reversble and rreversble changes These nequaltes hold as equaltes for reversble changes. Ths wll hold f the change s non-dsspatve and quasstatc. Quasstatc means a change so slow that the system s always arbtrarly close to a state of equlbrum. The path between ntal and fnal states can then be plotted n the space of relevant thermodynamc varables. Ths s necessary because otherwse thermodynamc varables such as P, T etc. are not defned on the path and we cannot express δw = P δ as must be the case for reversblty. In ths case, no energy s dsspated n a way that cannot be recovered by reversng the process. For an rreversble change the extra ncrease n entropy s due to the extra randomzng effects due to frcton, turbulence etc. A smple example s rapd pston movement as well as mxng of dstngushable gases and free expanson of gas nto a vacuum. For an solated, or closed, system Q = but S, the equalty holdng for a reversble change. In real lfe there s always some dsspaton and so even f δq = we expect δs >. These deas are compatble wth common sense notons. For reversble transtons the system can be returned to ts orgnal state by manpulatng the external condtons.
2 THERMODYNAMICS 18 E.g., a sprng that has been quasstatcally compressed. Snce the change s through a sequence of equlbrum confguratons there s no thermodynamc reason why t could not be traversed n the opposte drecton, thus reversng the change. For an rreversble change, such as mxng of dfferent gases, no such reverse transton can occur. In the case of mxng t would requre the gases to be separated wthn a closed system. It must be emphaszed that the entropes of the ndvdual parts of the closed system need not all ncrease or reman constant, but the total for the whole system must. Of course, chemcal means can be used to separate mxed gases but only at the cost of ncreasng the entropy elsewhere snce the closed system must nclude all parts of the experment. If ths were not the case then, overall, entropy would have decreased. 2.2 Applcatons 2.2.1 The Maxwell Relatons The varables we have consdered so far, F, G, E, P,, T, S, are all propertes of the equlbrum state. They each depend on an approprate par of varables chosen as ndependent and are dfferentable. These propertes are encoded n dfferental relatons between the varables. Usng de = T ds P d we have Free energy F = E T S : = df = SdT P d = F = F (T, ) Enthalpy H = E + P : = dh = T ds + dp = H = H(S, P ) (2.2.1.1) Gbbs functon G = E T S + P : = dg = SdT + dp = G = G(T, P ) Because these are all exact dfferentals we have, for example, ( 2 ) ( E 2 ) E =. (2.2.1.2) S S Snce T = ( ) E S, P = ( ) E S we have ( ) ( ) P T = S S Ths s a Maxwell relaton. Another such Maxwell relaton arses from S = ( F T ), P = ( F, (2.2.1.3). (2.2.1.4) ) T, (2.2.1.5) we fnd ( ) ( ) S P =. (2.2.1.6) T T Two more Maxwell relatons can be derved from G and H gvng four n all. However, they are not all ndependent. Ths can be seen by consderng z = z(x, y). Then δz = ( ) z δx + x y ( ) z δy = y x
2 THERMODYNAMICS 19 1 = = ( ) z x ( ) z x y y ( ) x z y ( ) z + y x and ( ) y x z. (2.2.1.7) Multply the last equaton by ( x/ z) y and use the second equaton to get ( ) z y x ( ) y x z ( ) x z y = 1. (2.2.1.8) Substtute any three of P,, T, S for x, y, z to relate the terms n Maxwell s relatons. We note that E, F, H, G are all extensve. Note that G explctly only depends on the ntensve quanttes T, P and so ts extensve character s due to N and we can wrte G = µ(p, T )N, where µ s called the chemcal potental. More on ths later. Consder E = E(, S) and change varables (, S) (, T ). Then we have ( E ) T = Usng de = T ds P d ths gves ( ) E T ( E ) S + ( E S = P + T ) ( S ( ) S T and usng (2.2.1.6) (from dfferentablty of F ) we have ( ) ( ) E P = P + T T T, ) T.. (2.2.1.9) For a perfect gas the equaton of state s P = NkT whch mples ( ) E =. (2.2.1.1) T Snce we only need two ndependent varables we have n ths case that E = E(T ) a functon of T only. Ths s an example of equpartton where E = 3 2 NkT. 2.2.2 specfc heats Because δq s not an exact dervatve there s no concept of the amount of heat stored n a system. Instead δq s the amount of energy absorbed or emtted by the system by random processes and excludng work done by mechancal or other means. If we specfy the condtons under whch the heat s suppled then we can determne δq usng δq = de + δw, (2.2.2.1) where de s an exact dervatve. It may be the case that absorpton/emsson of heat s not accompaned by a change n T. An example, s the latent heat of evaporaton of water to a gas. However, ths s a phase change whch we exclude from the current dscusson. We can wrte δq = CdT where C s the heat capacty of the system. Clearly, to gve C meanng we need to specfy the condtons under whch the change takes place. Important cases are where the change s reversble and at fxed or fxed P. Then δq = T ds and we defne C = T ( S T ), C P = T ( ) S T P, (2.2.2.2)
2 THERMODYNAMICS 2 where C s the heat capacty at constant volume and C P s heat capacty at constant pressure. Usng de = T ds P d and dh = T ds + dp. (2.2.2.3) we have also ( ) E C = T Changng varables from T, P to T, we have ( S T ) P = ( S T, C P = ) + ( S ) ( ) H T P T ( ) T P. (2.2.2.4), (2.2.2.5) and usng the Maxwell relaton (2.2.1.6) and the defntons of C P and C we have ( ) ( ) S C P = C + T T = C P = C + T For any physcal system we expect ( ) >, T P ( T T ) P ( ) P T ( P T P ). (2.2.2.6) >, (2.2.2.7) and hence that C P > C. Ths makes sense because for a gven ncrement dt the system at fxed P expands and does work P d on ts surroundngs whereas the system at fxed does no work. The extra energy expended n the former case must be suppled by δq whch s correspondngly larger than n the latter case. An mportant example s that of a perfect gas. An amount of gas wth N A (Avogadro s number) of molecules s called a mole. The deal gas law tells us that at fxed T, P a mole of any deal gas occupes the same volume. The deal gas law for n moles s P = nn A kt = nrt, (2.2.2.8) where R s the gas constant, and s the same for all deal gases. Note that at suffcently low densty all gases are essentally deal. The heat capacty per mole, c, s called the specfc heat: c = C/n. Then from (2.2.2.6) we have c p c v = R. (2.2.2.9) We defne γ = c p = c v + R = γ > 1. (2.2.2.1) c v c v If the number of degrees of freedom per molecule s denoted N F then equpartton of energy gves that the energy of 1 mole s = E = N F 2 RT c v = N F 2 R, c p = ( NF 2 + 1 ) R, γ = ( NF + 2 N F ). (2.2.2.11) For a monatomc gas N F = 3, for a datomc gas N F = 5 (3 poston, 2 orentaton), etc, and so γ = 5/3, 7/5,....
2 THERMODYNAMICS 21 2.2.3 adabatc changes δq = for an adabatc change. For n moles of an deal gas E = nc v T, P = nrt and then = RdE + RP d (extra R for convenence). (2.2.3.12) As a functon of (, T ) we have ( ) E de = T dt + ( ) E d = nc v dt, (2.2.3.13) T snce ( E/ ) T = (eqns. (2.2.1.9),(2.2.1.1)) for a perfect gas. (Note, ths wll be true for all systems where E does not depend on at fxed N.) Then Thus Thus = Rnc v dt + RP d = c v (P d + dp ) + RP d (usng P = nrt ) = c p P d + c v dp. (2.2.3.14) c p P d + c v dp =. (2.2.3.15) P γ s constant on adabatcs (2.2.3.16) Note that adabatcs, P γ constant, are steeper than sothermals, P constant. We can use these results to calculate the entropy of n moles of deal gas. We have = = T ds = de + P d = nc v dt + nrt d ds = nc v dt T + nrd S = nc v log T + nr log + S. S s the unknown constant of ntegraton. Thermodynamcs cannot determne S and does not care whether S s extensve. Ths s resolved by the more complete approach of statstcal mechancs gven earler. Ths result s the same as we found before (1.7.3.2) f all the constant parts are lumped nto S. Usng P = nrt we get S = nc v log P γ + S, (2.2.3.17) and so, as expected, S s constant on adabatcs (sentropcs). 2.2.4 van der Waals equaton In realty even a low densty gas s not deal because there are nteractons between the gas molecules. Ultmately, at temperatures low enough these nteractons cause a phase transton to occur and the gas lquefes. We can account approxmately for the nteractons by correctng the deal gas law to gve van der Waals equaton. The form of the nter-partcle potental s shown n fgure 1. There s a repulsve, hard-core, part at small separatons and an attractve part for ntermedate separatons. The effect of these two regons s qualtatvely modeled as follows:
2 THERMODYNAMICS 22 Fgure 1: () The hard-core s represented as an excluded volume snce the volume avalable s reduced. We replace b. () The pressure measured by an observer, P, s due to the partcles on the surface httng the walls. Whlst n the bulk the attractve forces roughly cancel out because the partcles are n homogeneous envronment, ths s not true at the surface snce they are pulled back nto the bulk. Consequently, P s less than the bulk pressure, P B. Now, P s gven by P = d(momentum/unt-area)/dt nmv 2 where n s the number densty. The reducton n K.E. (and hence n v 2 ) due to the unbalanced attractons near the surface s also proportonal to n. Hence the reducton n pressure takes the form for fxed N snce n = N/. P B P n 2 = P B P = a 2, (2.2.4.18) The deal gas equaton corrected for the excluded volume apples n the bulk so that P B ( b) = NkT = ( P + a ) 2 ( b) = RT. van der Waals equaton a N 2, b N, R = Nk. (2.2.4.19) 2.3 the Second Law of Thermodynamcs Kelvn-Planck: It s mpossble to construct an engne that, operatng n a cycle, wll produce no other effect than the extracton of heat from a reservor and the performance of an equvalent amount of work. Clausus: It s mpossble to construct a devce that, operatng n a cycle, wll produce no other effect than the transfer of heat from a cooler to a hotter body. These two versons can be shown to be equvalent. By a cycle we mean a seres of processes operatng on a system for whch the ntal and fnal states are the same. Clearly, the most effcent cycle s a reversble one for
2 THERMODYNAMICS 23 Fgure 2: whch all the consttuent processes are reversble. Such a cycle can be drawn as a closed curve n the 2D space of the ndependent thermodynamc varables (see fgure 3). Consder system shown n fgure 2 whch, actng n a reversble cycle, extracts heat Q 1 from the hot reservor at temperature T 1 and delvers heat Q 2 to the cold reservor at temperature T 2 and does work W = Q 1 Q 2. Because the cycle s reversble the total change n entropy s zero. Thus, S = Q 2 T 2 Q 1 T 1 =, = Q 2 = Q 1T 2 T 1. (2.3.1) The law of ncrease of entropy tells us that for realstc systems whch have frctonal losses etc. Q 2 Q 1T 2 T 1. (2.3.2) Ths clearly agrees wth the Kelvn-Planck statement of the second law whch says that Q 2 >. The effcency of the cycle s defned as η = W Q 1 = Q 1 Q 2 Q 1 T 1 T 2 T 1. (2.3.3) and we see that the reversble cycle (or engne) has the greatest effcency. The work done by the cycle s the areaof the cycle: W = P d. Ths mportant example of a reversble cycle s the Carnot cycle shown n fgure 3 n whch a sample S of perfect gas s taken around a closed curve n the P plane by a sequence of adabatc and sothermal reversble processes wth T 1 > T 2. On AB the heat bath supples heat Q 1 to S at temperature T 1. On CD the heat bath extracts heat Q 2 from S at temperature T 2. On the adabatcs, BC and DA, T γ 1 s constant so that T 1 γ 1 A = T 2 γ 1 D, T 1 γ 1 B = T 2 γ 1 C = A B = D C. (2.3.4) The drect connecton of the second law n the form of Clausus wth the law of ncrease of entropy can be seen as follows. Suppose there exsts an engne I whch s
P=RT 2 3 THE GRAND CANONICAL ENSEMBLE 24 P=RT 1 P γ = K 1 P γ = K 2 Fgure 3: Fgure 4: more effcent that a Carnot engne, C, and so volates the law of ncrease of entropy. Let I drve C backwards as a Carnot refrgerator n the manner shown n fgure 4. Note that no work s suppled or absorbed by the external surroundngs snce the output, W, of I s absorbed by C. Then η I > η C = W Q > W Q = Q < Q. (2.3.5) However, the heat suppled to the hot heat bath s then Q = Q Q > and ths has been wholly transfered from the cold heat bath wthout the external nput of work. Ths volates the Clausus statement of the second law. It also shows that the most effcent cycle s the Carnot cycle. Remember W and Q are path-dependent even for reversble paths and hence η also depends on the path used to defne the cycle. 3 The Grand Canoncal Ensemble 3.1 formalsm We follow an dentcal procedure to before. We consder a grand (canoncal) ensemble, G of A replcas of the system S n mcrostates of S wth N partcles and energes
3 THE GRAND CANONICAL ENSEMBLE 25 E. Suppose a members of G are n mcrostate so that a = A, a E = AE, a N = AN. (3.1.1) As before we assocate thermodynamc equlbrum of S wth the most probable confguraton of G subject to the gven constrants. We maxmze the same log W as before: ( δ log W α a β a E γ ) a N =. (3.1.2) δa Ths gves us a = e (1+α) e β(e µn ), (3.1.3) where we have defned the chemcal potental µ by βµ = γ. The grand partton functon s then Z = e β(e µn ). (3.1.4) The fracton of members n mcrostate s The grand ensemble average s ρ = a A Ô = = e β(e µn) Z. (3.1.5) ρ O, (3.1.6) whch gves Ê = E, ˆN = N. What we have done s exchange a fxed value of N for a value that s mantaned n the average by choosng the correspondng value for µ just as we dd for E by choosng β. Ths has bg advantages snce t s often dffcult to mpose fxed N as a constrant on a system. It s somethng that we have not needed to explctly encode htherto. By ntroducng µ the ablty to tune ˆN to the rght value usng a varable that s explctly part of the formalsm s a huge advantage, as we shall see. Then we have N = 1 ( ) log Z, β µ β, ( ) log Z E µn =. (3.1.7) β Remember, the dependence s n E E ( ). As before, we consderng adabatc changes n (e.g., constant ρ for the deal gas). Then, as before, δe = P δ and so P = ρ δe δ = 1 β µ, ( ) log Z β,µ. (3.1.8) Followng the famlar argument, more general changes δρ, n whch heat enters or leaves the system, obey δρ =, δρ N = δn, δe = P δ + δρ E. (3.1.9)
3 THE GRAND CANONICAL ENSEMBLE 26 The entropy s defned n general, as before, to be S = k A log W = k ρ log ρ. (3.1.1) For an equlbrum ensemble we then fnd δs = k = k δρ log ρ ρ 1 ρ δρ δρ ( β(e µn ) log Z + 1). (3.1.11) Ths gves the fundamental thermodynamc relaton T ds = de + P d µdn, (3.1.12) whch corresponds to the frst law de = δq+δw mech +δw chem, where δw chem = µdn s the work done n addng dn partcles to the system. Note, we have conformed to the conventon that an exact dervatve s denoted as e.g., de, whereas an nexact dervatve s denoted as δq. The chemcal potental s then gven by ( ) E µ =. (3.1.13) N S, From the defnton the entropy n equlbrum s whch can be rewrtten as S = kβ(e µn) + k log Z, (3.1.14) S = ( ) kt log Z T µ,. (3.1.15) A smlar argument to the one gven earler n secton 1.3 shows that two systems placed n thermal and dffusve contact (.e., they can exchange partcles through, say, a permeable membrane) wll reach a thermal equlbrum characterzed by a common value of β, µ. The argument s smply that when the systems are separate there are sx Lagrange multplers leadng to equlbrum characterzed by (α 1, β 1, µ 1 ), (α 2, β 2, µ 2 ), but that when they are n contact there are only three Lagrange multplers and hence common values for these two sets. The frst law states that de = T ds P d + µdn and so E can be vewed as a functon of (S,, N), and when E, S, and N are extensve (as s the case for many large systems) we get E(λS, λ, λn) = λe(s,, N) ( ) ( ) ( ) E E E = E = S + + N S,N S,N N S, = E = T S P + µn. (3.1.16) We defne the grand potental Ω for a state of thermodynamc equlbrum by the Legendre transformaton Ω = E T S µn = E µn (E µn + kt log Z) = kt log Z, = Z = e βω. (3.1.17)