Chapter 6 Entropy Our introduction to the concept of entropy wi be based on the canonica distribution, ρ(h) = e βh Z. (6.1) Cassicay, we can define the mean voume of phase space occupied by ρ(ē) q p = 1, (6.2) where Ē = H is the average energy. hen, we define the entropy by S = k n q p = k nρ(ē). (6.3) For the canonica distribution, n ρ(h) = const βh, so nρ(ē) = n ρ(h) = dq dp ρ nρ. (6.4) hus the entropy is S = k nρ = k dq dp ρ nρ. (6.5) Quantum mechanicay, we define the number of quantum states by and then the entropy is ρ(ē) Γ = 1, (6.6) S = k n Γ = k nρ(ē) = k nρ = krρnρ. (6.7) Note that because ρ(h) = n n p n n, (6.8) 25 ersion of February 18, 2010
26 ersion of February 18, 2010 CHAPER 6. ENROPY where the sum is over a compete set of states, the entropy can aso be written as S = k p n np n 0, (6.9) n because 1 p n 0, so the entropy is zero ony for a pure state. For the canonica distribution, this occurs ony at = 0. 1 he correspondence principe reation between the number of quantum states and the voume of phase space is q p Γ = h s, (6.10) for s degrees of freedom, where h = 2π h is Panck s constant. If we have two independent systems, so that the entropy of the composite system is ρ = ρ 1 ρ 2, r ρ 1 = rρ 2 = 1, (6.11) S = krρ 1 ρ 2 (nρ 1 + nρ 2 ) = k[r(ρ 1 nρ 1 )r ρ 2 + r (ρ 2 nρ 2 )r ρ 1 ] = k(rρ 1 nρ 1 + rρ 2 nρ 2 ) = S 1 + S 2, (6.12) that is, entropies are additive. he entropy is cosey reated to the quantity P introduced in Chapter 2, the number of ways of getting a distribution {a }: P = N! (a )!, (6.13) where the ensembe consists of N systems, and a is the number of systems in the th state, so N = a. (6.14) What we actuay maximized was n P, which is np = nn! n(a )! N nn a na, (6.15) which uses Stiring s approximation and Eq. (6.14). he probabiity of finding a system in the th state is p = a /N, so np = N nn N p (n p + nn) = N p np, (6.16) 1 his is the content of the third aw of thermodynamics, which states that if the ground state is non-degenerate, the entropy vanishes at absoute zero.
27 ersion of February 18, 2010 or k np = S; (6.17) N k np is actuay the entropy of the entire ensembe. So we see that of a distributions, with fixed N and E, the most probabe distribution, the canonica distribution, maximizes the entropy. Now, et us see what S is more expicity in the canonica distribution, for which hen the entropy is p n = e βen Z, Z = n e βen. (6.18) S = krρnρ = k n p n np n e βen = k Z [ βe n nz] n = kβu + k nz. (6.19) We have two kinds of energy appearing here: he interna energy, and the Hemhotz free energy, β nz = 1 Z he reation (6.19) says that these are reated by E n e βen, (6.20) n F = k nz, = 1 kβ. (6.21) F = U S, (6.22) which shoud be famiiar from thermodynamics. If I take the differentia of n Z, which is equivaent to d nz = dβ d dβ nz + = Udβ β Z de d de nz e βe de, (6.23) d(n Z + βu) = β(du p de ), (6.24) where by Eq. (6.19) the differentia on the eft is ds/k. Inside the parenthesis on the right, du represents the average interna energy increase, whie p de represents the average work done by the system when the energy eves are
28 ersion of February 18, 2010 CHAPER 6. ENROPY ifted from E to new energy eves E + de. We ca the atter δw, where δ is a reminder that there is no system property caed work (unike energy or entropy), and so δw depends upon the path. hen Eq. (6.24) reads ds = 1 δq, δq = du + δw, (6.25) where δq is the heat suppied to the system. hat heat either does work on the system, or changes its interna energy. 1/ is the integrating factor necessary to change the imperfect differentia δq into the perfect differentia ds. his, in fact, is the rigorous thermodynamic definition of temperature. Note that thermodynamic reations, such as df = du ds Sd = pd + ds ds Sd = pd Sd, (6.26) or = S, may aso be derived statisticay, = k nz + k 1 Z = F U = p, (6.27) ( E ) k 2 e βe = S, (6.28) because the energy eves don t change if the voume is fixed, and = k 1 ( β) E Z e βe = δw d = p. (6.29) Finay, et us rederive the thermodynamic reation, used above, du = pd + ds. (6.30) his says In fact, from we see ( ) S U ( ) S U = 1. (6.31) S = kβu + k nz, (6.32) = kβ + ku ( ) ( ) β + k U U nz = kβ = 1, (6.33)
6.1. EXAMPLES 29 ersion of February 18, 2010 where the ast two terms on the first ine cance because U nz = β nz β U = U β U, (6.34) provided the voume is hed fixed, which means that the energy eves E do not change. 6.1 Exampes 6.1.1 Idea Gas Reca that for an idea gas [Eq. (5.44)] hen the free energy is he interna energy is F = k nz = k ( ) 3N/2 2πm Z = N. (6.35) β β nz = 3N 2 (N n + 3N2 n + constant ). (6.36) 1 β = 3N 2 k, (6.37) which is the equipartition theorem. he pressure is given by the idea gas aw: p = = Nk, p = Nk. (6.38) 6.1.2 Harmonic Osciator In this case, the energy eves are ( E = + 1 ) hω, 2 = 0, 1, 2,.... (6.39) hen the partition function is Z = e βe = e β hω/2 =0 e β hω = e β hω/2 1 1 e β hω = 1 e β hω/2 e β hω/2 1 = 2 sinhβ hω/2. (6.40)
30 ersion of February 18, 2010 CHAPER 6. ENROPY hen the free energy is F = k nz = k n sinh β hω 2 he interna energy is ) β nz = 1 ( hω sinhβ hω/2 coshβ hω/2 = hω 2 2 = hω 2 e β hω/2 + e β hω/2 e β hω/2 e β hω/2 = hω 2 e β hω + 1 e β hω 1 + k n 2. (6.41) coth β hω 2 = hω 2 + hω e β hω 1, (6.42) which is the famous Panck distribution. Note the appearance in the first term of the zero-point energy. 6.1.3 wo-eve System Here the system has ony two states of energy E = 0 and E = ǫ. (6.43) his is sometimes caed a Fermi osciator. he partition function is and so the free energy is from which we find the interna energy Z = 1 + e βǫ, (6.44) F = k n ( 1 + e βǫ), (6.45) β nz = 1 1 + e βǫ( ǫ)e βǫ = which is the famous Fermi distribution. ǫ e βǫ + 1, (6.46) 6.2 Fuctuations Let us consider the spread in energies about the average, Now the interna energy is (H U) 2 = H 2 U 2. (6.47) U = d dβ nz = 1 d Z, (6.48) Z dβ
6.2. FLUCUAIONS 31 ersion of February 18, 2010 so whie so H 2 = (H U) 2 = 1 Z U 2 = ( ) 2 1 d Z dβ Z, (6.49) p E 2 = 1 e βe E 2 = 1 d 2 Z, (6.50) Z Z dβ2 d 2 ( ) 2 1 dβ 2 Z d Z dβ Z = d2 nz. (6.51) dβ2 On the other hand, the specific heat at constant voume is defined by ( ) U c = = dβ du d dβ = 1 d 2 k 2 dβ 2 nz = 1 k 2 (H U)2. (6.52) hus we define the root mean square fuctuation in the energy as we have δe = (H U) 2, (6.53) δe = kc. (6.54) If the specific heat is independent of, the fuctuation in the energy grows ineary with the temperature. For an idea gas, U = 3N 2 k, (6.55) so c = 3Nk/2, and δe = 3N 2 k, δe 2 U = 3N, (6.56) which exhibits the typica 1/ N behavior of statistica fuctuations.