J.I. Siepmann Chem 8561 19FLUCTUATIONS• knowledge of the fluctuations of mechanical variables around their mean values isimportant because◦ it will justify why it is possible to ignore fluctuations in thermodynamics or, inother words, why in the thermodynamic limit the differences between the ensem-bles can be neglected and that we can choose the ensemble that is mathematicallymost convenient◦ some topics, such as light scattering and solutions can be analyzed in terms offluctuations◦ fluctuations are the basis for non-equilibrium statistical mechanics• the variance (standard deviation) σ = [(x − x)2]1/2is a measure of the spread of adistribution; it can be simplified by the following relation(x − x)2= x2− 2xx + x2= x2− x2• for the fluctuations of the energy in the canonical ensemble we obtainσE2=E2− E2=XjPjEj2− E2(68)• using the definition of Pjfor the canonical ensemble (see eqn. 15) we can make thefollowing substitutionXjPjEj2=1QXjEj2e−βEj= −1Q∂∂βXjEje−βEj= −1Q∂∂βEQ= −∂E∂β−E∂ ln Q∂β= kBT2∂E∂T+E2(69)• by inserting into eqn. 68 we obtain for the fluctuations in the energyσE2= kBT2∂E∂TN,V(70)J.I. Siepmann Chem 8561 20• making use of Gibbs’ ensemble postulate, we can rewrite eqn. 70 in thermodynmicterms and replace ∂E/∂T by the molar heat capacity CVσE2= kBT2CV(71)• to get a better feeling on the spread of the distribution, we use our knowledge ofthe ideal gas, namely, that E and CVare of O(NkBT ) and O(NkB), respectively;thusσEE=√kBT2CVE= O(N−1/2) (72)i.e. for a typical macroscopic system with NA, the variance is of the order 10−12E• the variance of the number of particles in the grand canonical ensemble can becalculated following an analoguous path and using the thermodynamic definitionof the isothermal compressibility κ = −(1/V )(∂V /∂p) we obtainσN2=N2kBT κV(73)• again this fluctuation is O(N−1/2), however, at the critical point the derivativeof the pressure with the volume is zero [(∂p/∂V )N,T], and hence κTc= ∞ andmacroscopic fluctuations in the density (or number of particles) can be observedJ.I. Siepmann Chem 8561 21Equivalence of thermodynamic ensembles• since the fluctuations in the number of particles in a grand canonical ensembleare very small, experimentally we will only observe numbers which differ from themean value N by a negligible amount and we can make use of the maximum-termmethodΞ(µ, V, T ) =XNQ(N, V, T ) eµN/kBTln Ξ(µ, V, T ) =pVkBT= ln Q(N∗, V, T ) +µN∗kBT(74)µN∗− pV = −kBT ln Q(N∗, V, T ) = A(N∗, V, T ) (75)• if we now take the view that µ is determined by the “independent” variablesN∗(=N), V , and T (see eqn. 75), then we find that the grand canonical ensemble“degenerates” to the canonical ensemble because of the negligible fluctuations in N• the same argument can be used to show the equivalence of the canonical and mi-crocanonical ensemblesln Q = −AkBT= ln Ω(N, V, E∗) −E∗kBT(76)S = kBln Ω(N, V, E∗) (77)• and also the entropy of the grand canonical ensemble can be expressed in terms ofthe degeneracy of the quantum states (in the microcanonical ensemble)S = kBln Ω(N∗, V, E∗) (78)J.I. Siepmann Chem 8561 22BOLTZMANN STATISTICS• to calculate the partition functions of any of the ensembles (microcanonical, canon-ical, grand canonical, and isobaric-isothermal), it is necessary to know the completeset of accessible energy states {Ej(N, V )}, but solving the Schr¨odinger equation fora macroscopic N-body system is out of reach for even the most powerful computers• however, the problem becomes attainable if the macroscopic system can be beviewed as being composed of molecules, groups of molecules, or degrees of freedom(≡ subsystems) which are effectively “independent” of each other• “independent” implies weak interactions: small enough that intermolecular forces(etc.) can be neglected, but sufficiently large to maintain thermal equilibrium byenergy exchange• if the macroscopic system is composed of independent molecules or subsystems, thenthe many-body Hamiltonian can be given as the sum of independent contributionsH = Ha+ Hb+ ··· (79)• denote the eigenvalues and eigenfunctions of Hjby ǫjand ψj, respectively, wherej = a, b, . . ., and assume that ψ = ψaψb. . .Hψ = (Ha+ Hb+ ···) ψaψb···= ψbψc···Haψa+ ψaψc···Hbψb+ ···= ψbψc···ǫaψa+ ψaψc···ǫbψb+ ···= (ǫa+ ǫb+ ···) ψ = Eψ (80)• that is the possible energy eigenvalues for the macroscopic system are the sumof the separate energies of the individual molecules (or subsystems), as we mightexpect for a system composed of independent molecules (or subsystems)J.I. Siepmann Chem 8561 23Distinguishable Particles• under th e condition that the molecules (or subsystems) are distinguishable (e.g.the Einstein crystal) the canonical partition function can be expressed in terms ofindividual moleculesQ(N, V, T ) =Xse−Es/kBT=Xi,j,...e−(ǫa,i+ǫb,j+···)/kBT=Xie−ǫa,i/kBTXje−ǫb,j/kBT···= qa(V, T )qb(V, T ) ··· (81)• since in most cases {ǫi} i s a set of molecular energy states, q(V, T ) is called amolecular partition function• in the special case that the energy states of all molecules (or subsystems) are thesame, but the molecules are nevertheless distinguishable, eqn. 81 further simplifiestoQ(N, V, T ) = [q(V, T )]N(82)• in principle (and following the same line of argument), the molecular partitionfunction could be further divided according of the various degrees of freedomqmolec= qtransqrotqvibqelecqnucl(83)J.I. Siepmann Chem 8561 24Indistinguishable Particles• in general (but not always, see Einstein crystal), molecules are indistinguishable,and we can not sum over i, j, k, . . . separately as done in eqn. 81• we have to find a way to avoid overcounting of permutations of the the indistin-guishable particles over the molecular states:◦ in cases, where all particles are in different molecular quantum states (i 6= j 6=k . . .), we find N! permutations◦ but there are many terms in eqns. 81 and 82 which are not of this type (e.q.i 6= j and j = k = l ···), i.e. all terms in which two or more indices are the samewill introduce further complications• however, under the condition that the numb er of accessible molecular quantumstates Φ(ǫ) is very large compared to the number of particles N, i.e. Φ(ǫ) ≫ N,the vast majority of