F=lnM!-∑{ nin n-n1)-a(∑ny-M)-阝(∑nE1-E Notice that this function F is exactly equal to the InQ2 function we wish to maximize whenever the ( n, variables obey the two constraints. So, the maxima of f and of InQ2 are identical if the(n) have values that obey the constraints. The two Lagrange multipliers a and p are introduced to allow the values of in, that maximize f to ultimately obey the two constraints. That is, we will find values of the n,) variables that make F maximum; these values will depend on a and p and will not necessarily obey the constraints However, we will then choose a and p to assure that the two constraints are obeyed This is how the Lagrange multiplier method works Taking the derivative of F with respect to each independent n variable and setting this derivative equal to zero gives In -βEk=0 This equation can be solved to give nx-exp(-a)exp(-BEx) Substituting this result into the first constraint equation gives M= exp( a)2j exp(-BE), which allows us to solve for exp(-a)in terms of M. Doing so, and substituting the result into the expression for nk p(阝EQ where PAGE
PAGE 11 F = ln M! - SJ {nJ ln nJ – nJ ) - a(SJnJ – M) -b(SJ nJ eJ –E). Notice that this function F is exactly equal to the lnW function we wish to maximize whenever the {nJ} variables obey the two constraints. So, the maxima of F and of lnW are identical if the {nJ} have values that obey the constraints. The two Lagrange multipliers a and b are introduced to allow the values of {nJ} that maximize F to ultimately obey the two constraints. That is, we will find values of the {nJ} variables that make F maximum; these values will depend on a and b and will not necessarily obey the constraints. However, we will then choose a and b to assure that the two constraints are obeyed. This is how the Lagrange multiplier method works. Taking the derivative of F with respect to each independent nK variable and setting this derivative equal to zero gives: - ln nK - a - b eK = 0. This equation can be solved to give nK = exp(- a) exp(- b eK). Substituting this result into the first constraint equation gives M = exp(- a) SJ exp(- b eJ ), which allows us to solve for exp(- a) in terms of M. Doing so, and substituting the result into the expression for nK gives: nK = M exp(- b eK)/Q where
Q=∑1exp(-β Notice that the n are, as we assumed earlier, large numbers if M is large because nk is proportional to M. Notice also that we now see the appearance of the partition function Q and of exponential dependence on the energy of the state that gives the boltzmann population of that state It is possible to relate the p Lagrange multiplier to the total energy e of the M containers by using E=M∑Eexp(-阝Q M(alnQ/aβNy This shows that the average energy of a container, computed as the total energy e divided by the number M of such containers can be computed as a derivative of the logarithm of the partition function Q. As we show in the following Section, all thermodynamic properties of the N molecules in the container of volume V can be obtained as derivatives of the logarithm of this Q function. This is why the partition function plays such a central role in statistical mechanics To examine the range of energies over which each of the M single-container system ranges with appreciable probability, let us consider not just the degeneracy Q2(n*) of that set of variables ( n*=n*,n*2,. that makes Q2 maximum, but also the PAGE 12
PAGE 12 Q = SJ exp(- b eJ ). Notice that the nK are, as we assumed earlier, large numbers if M is large because nK is proportional to M. Notice also that we now see the appearance of the partition function Q and of exponential dependence on the energy of the state that gives the Boltzmann population of that state. It is possible to relate the b Lagrange multiplier to the total energy E of the M containers by using E = M SJ eJ exp(- b eK)/Q = - M (¶lnQ/¶b)N,V. This shows that the average energy of a container, computed as the total energy E divided by the number M of such containers can be computed as a derivative of the logarithm of the partition function Q. As we show in the following Section, all thermodynamic properties of the N molecules in the container of volume V can be obtained as derivatives of the logarithm of this Q function. This is why the partition function plays such a central role in statistical mechanics. To examine the range of energies over which each of the M single-container system ranges with appreciable probability, let us consider not just the degeneracy W(n*) of that set of variables {n*} = {n*1 , n*2 , …} that makes W maximum, but also the
degeneracy Q2(n) for values of (n, n,,... differing by small amounts (8n,8n,,.) from the optimal values(n*). Expanding In Q2 as a Taylor series in the paramters(n,,n2,.i and evaluating the expansion in the neighborhood of the values (n*), we find lng=ln9({n*1,n*2,…})+∑(lnOn)6n1+1/2∑x( aIng/an, an)6n6n<+ We know that all of the first derivative terms(alnQ2/an,) vanish because InQ2 has been made maximum at( n*. The first derivative of InQ2 as given above is aInQ/an,=-In(n) so the second derivatives needed to complete the Taylor series through second order are (aInS/0n,OnK)=-OJK n We can thus express Q2(n) in the neighborhood of (n* as follows n9n)=ln9(n*)-1/2∑(n)/n3* or, equivalently, 2(n)=g2(n*)exp-12∑(6n)/n*] This result clearly shows that the degeneracy, and hence by the equal a priori probability hypothesis, the probability of the M-container system occupying a state having ( n, n2,j falls off exponentially as the variables n, move away from their"optimal"values (n*) PAGE 13
PAGE 13 degeneracy W(n) for values of {n1 , n2 , …} differing by small amounts {dn1 , dn2 , …} from the optimal values {n*}. Expanding ln W as a Taylor series in the paramters {n1 , n2 , …} and evaluating the expansion in the neighborhood of the values {n*}, we find: ln W = ln W({n*1 , n*2 , …}) + SJ (¶lnW/¶nJ ) dnJ + 1/2 SJ,K (¶ 2 lnW/¶nJ¶nK) dnJ dnK + … We know that all of the first derivative terms (¶lnW/¶nJ ) vanish because lnW has been made maximum at {n*}. The first derivative of lnW as given above is ¶lnW/¶nJ = -ln(nJ ), so the second derivatives needed to complete the Taylor series through second order are: (¶ 2 lnW/¶nJ¶nK) = - dJ,K nj -1 . We can thus express W(n) in the neighborhood of {n*} as follows: ln W(n) = ln W(n*) – 1/2 SJ (dnJ ) 2 /nJ*, or, equivalently, W(n) = W(n*) exp[-1/2SJ (dnJ ) 2 /nJ*] This result clearly shows that the degeneracy, and hence by the equal a priori probability hypothesis, the probability of the M-container system occupying a state having {n1 , n2 , ..} falls off exponentially as the variables nJ move away from their “optimal” values {n*}
As we noted earlier, the n are proportional to M (i. e, n, *=Mexp-Bs)Q=f, M), so when considering deviations Sn, away from the optimal n*, we should consider deviations that are also proportional to M: Sn,=MSf. In this way, we are treating deviations of specified percentage or fractional amount which we denote f. Thus, the ratio(Sn /n,* that appears in the above exponential has an M-dependence that allows Q(n)to be written as 2(n)=g(n*)exp[M2∑(δf)*] where f* and &f, are the fraction and fractional deviation of containers in state J: f* n * /M and 8f,=8n /M. The purpose of writing $2(n) in this manner is to explicitly show that, in the so-called thermodynamic limit, when M approaches infinity, only the most probable distribution of energy (n*) need to be considered because only (8f = 0) is important as M approaches infinity Let's consider this very narrow distribution issue a bit further by examining fluctuations in the energy of a single container around its average energy Eave -E/M.We already know that the nunmber of containers in a given state K can be written as nk= M exp(-BEK)/Q. Alternatively, we can say that the probability of a container occupying the state J P1=exp(-βEkQ Using this probability, we can compute the average energy eave PAGE 14
PAGE 14 As we noted earlier, the nJ* are proportional to M (i.e., nJ* = M exp(-beJ )/Q = fJ M), so when considering deviations dnJ away from the optimal nJ*, we should consider deviations that are also proportional to M: dnJ = M dfJ . In this way, we are treating deviations of specified percentage or fractional amount which we denote fJ . Thus, the ratio (dnJ ) 2 /nJ* that appears in the above exponential has an M-dependence that allows W(n) to be written as: W(n) = W(n*) exp[-M/2SJ (dfJ ) 2 /fJ*], where fJ* and dfJ are the fraction and fractional deviation of containers in state J: fJ* = nJ*/M and dfJ = dnJ /M. The purpose of writing W(n) in this manner is to explicitly show that, in the so-called thermodynamic limit, when M approaches infinity, only the most probable distribution of energy {n*} need to be considered because only {dfJ=0} is important as M approaches infinity. Let’s consider this very narrow distribution issue a bit further by examining fluctuations in the energy of a single container around its average energy Eave = E/M. We already know that the nunmber of containers in a given state K can be written as nK = M exp(- b eK)/Q. Alternatively, we can say that the probability of a container occupying the state J is: PJ = exp(- b eK)/Q. Using this probability, we can compute the average energy Eave as:
Eae=∑1P1E1=SE1exp(-Bck)Q=-(lnQOβ)v To compute the fluctuation in energy, we first note that the fluctuation is defined as the average of the square of the deviation in energy from the average (E-Eave) ave=2,(Ey-Eave)P)=2P)(E/-2Ey Eave +Eave )=2]P,Ej-eave) The following identity is now useful for further re-expressing the fluctuations (aInQ/aB hv=a(-2G, exp(-BEy)/Q)aB ∑E2exp(-B)Q-{Eexp(-B)/Q}{ΣEexp(-BE1)Q} Recognizing the first factor immediately above as 2,E P], and the second factor as eave and noting that 2,P=1, allows the fluctuation formula to be rewritten as (E-Eave ave.=(OIn Q/OB NN =-(a(Eave)aB)N Because the parameter B can be shown to be related to the Kelvin temperature t as p 1/(kT), the above expression can be re-written as (E-Eave)'ave=-(a(Eave )OB)N v)=kT(a(Eave)aTN PAGE 15
PAGE 15 Eave = SJ PJ eJ = SJ eJ exp(- b eK)/Q = - (¶lnQ/¶b)N,V. To compute the fluctuation in energy, we first note that the fluctuation is defined as the average of the square of the deviation in energy from the average: (E-Eave)) 2 ave. = SJ (eJ –Eave) 2 PJ = SJ PJ (eJ 2 - 2eJ Eave +Eave 2 ) = SJ PJ (eJ 2 – Eave 2 ). The following identity is now useful for further re-expressing the fluctuations: (¶ 2 lnQ/¶b2 )N,V = ¶(-SJeJ exp(-beJ )/Q)/¶b = SJ eJ 2 exp(-beJ )/Q - {SJ eJexp(-beJ )/Q}{{SL eLexp(-beL )/Q} Recognizing the first factor immediately above as SJ eJ 2 PJ , and the second factor as Eave 2 , and noting that SJ PJ = 1, allows the fluctuation formula to be rewritten as: (E-Eave)) 2 ave. = (¶ 2 lnQ/¶b2 )N,V = - (¶(Eave)/¶b)N,V). Because the parameter b can be shown to be related to the Kelvin temperature T as b = 1/(kT), the above expression can be re-written as: (E-Eave)) 2 ave = - (¶(Eave)/¶b)N,V) = kT2 (¶(Eave)/¶T)N,V