5 5.1 E 1, E 2 N 1, N 2 E tot N tot E tot = E 1 + E 2, N tot = N 1 + N 2 S 1 (E 1, N 1 ), S 2 (E 2, N 2 ) E 1, E 2 S tot = S 1 + S 2 2 S 1 E 1 = S 2 E

5 5.1 E 1, E 2 N 1, N 2 E tot N tot E tot = E 1 + E 2, N tot = N 1 + N 2 S 1 (E 1, N 1 ), S 2 (E 2, N 2 ) E 1, E 2 S tot = S 1 + S 2 2 S 1 E 1 = S 2 E 2, S 1 N 1 = S 2 N 2 2 (chemical potential) µ S N = µ T (5.1) 5.2 T µ 44

(Grand canonical ensemble) S T µ S S S E, E R E tot N, N R N tot E tot = E + E R, N tot = N + N R S R (E R, N R ) E R N R S E N exp(s R /k) p(e, N) p(e, N) exp [S R (E tot E, N tot N)/k] [ 1 = exp k S R(E tot, N tot ) E S R N ] S R + k E R k N R exp [ (E µn)/kt ] (5.2) (Grand canonical distribution) 5.3 E N A(E, N) A = i,n A(E i(n), N)e β(e i(n) µn) i,n e β(e i(n) µn) (5.3) Z G = e β[ei(n) µn] = Z N e Nβµ (5.4) i,n i,n Z N N (Grand partition function) 45

E µn = ln Z G β N = 1 ln Z G β µ E = ln Z G β + µ β ln Z G µ (5.5) Helmholtz (5.4) N Z G = N e βµn e βf (T,N) = N β[f (T,N) µn] e F (T, N) N N e β[f (T,N) µn] e β[f (T,N ) µn ] N 1 β ln Z G = F (T, N) µn (5.6) N F (T, N) N = µ (5.7) 1 1 F (T, N + 1) F (T, N) F (T, N) N (5.7) (5.7) N = µ 5.4 46

E e β[e T S(E,N)] = e βf (T,N), F (T, N) = 0 (5.8) N (5.7) µ = 0 5.5 5.5.1 µ N Z N ( [ 2πm V e Nβφ ( ) ] 3/2 N 2πm Z N = V N e Nβφ N!h 3N β ) 3N/2 = 1 N! φ p 2 /2m + φ (5.4) Z G Z G = λ N Z N = N=0 N=0 [ 1 λv e βφ N! h 3 h 3 ( ) ] 3/2 N 2πm [λv e βφ ( ) ] 3/2 2πm = exp β h 3 β λ = e βµ N β (5.9) N = 1 β ln Z G µ = V ( ) 3/2 2πm h 3 e β(µ φ) β N/V ( ) βh 2 3/2 e βφ (5.10) e βµ = N V 2πm ln Z G β ln Z G β = µλv ( ) 3/2 2πm h 3 + λv β h 3 = µn 3 N 2 β ( 2πm β ) 1/2 ( 3 2 ) 2πm β 2 47

(5.10) µ E ( µ E = β µ β ) ln Z G = µn (µn 3N/2β) = N ( ) 3 2 kt + φ (5.10) A, B φ A, φ B N/V e βφ n A n B = e β(φ B φ A ) V/N r 0 T 0 4π 3 r3 0 = V N, h 2 2mr 2 0 = 1 2 kt 0 r 0 19 e βµ = 3 4π [ µ = t kt 0 ( ) 3/2 ( 1 λt 2π log 3 4π ( 1 2π ) 2 = 3 ( 1 r 0 4π 2π ) ] 3/2 3 2 log(t) ) 3/2 ( ) 3/2 T0 (5.11) T, (t = T/T 0 ) T 0 /T 1 e βµ 1 10.0 0.0 µ/kt 0 10.0 20.0 0.0 1.0 2.0 t 19: 48

Z G p r 0 1 z(p, r) 1 + e β(p2 /2m µ) (5.12) e βµ 1 Z G = [ 1 Π r Π p z(p, r) = exp h 3 dr = [ V e βµ ] exp h 3 dpe βp2 /2m = exp ] dp log z(p, r) [ V e βµ h 3 ( ) ] 3/2 2πm β (5.13) log z (5.13) 1/h 3 log z(p, r) = log[1 + e β(p2 /2m µ) ] e β(p2 /2m µ) ( ) z(p, r) = e log[z(p,r)], Π i exp(a i ) = exp A i (5.13) (5.9) (5.10) { [ F = N ( ) ] } 3/2 V 2πm ln β N βh 2 + 1 = N β = Nµ N/β i [ ln e βµ + 1 ] N/β = pv F + pv = Nµ F + pv Nµ Gibbs 5.5.2 T µ ω N Z N ( ) N 1 Z N = β hω 49

Z G = λ N Z N = N=0 N=0 ( ) N λ = β hω 1 1 λ β hω ( ) λ exp β hω λ = e βµ (5.5) N = 1 β ln Z G µ = 1 λ β 2 hω µ = λ β hω β ln Z G β = λ β 2 hω + 1 λ β hω β = N β + µn ( µ E = β µ ) ln Z G = µn (µn N/β) = N/β β 5.5.3 Langmuir (Adsorption) ( ) N 0 1 ε (ε > 0) 2 T µ 20: Z ad G = [ 1 + λe βε] N 0, (λ = e βµ ) 50

1 1 + e β( ε µ) (5.5) N ad N ad = 1 β ln Z ad G µ = N 0λe βε 1 + λe βε (5.14) N V λ = N V ( βh 2 2πm ) 3/2 = p ( ) βh 2 3/2 (5.15) kt 2πm (5.15) (5.14) θ = N ad /N 0 θ = p p + p 0 (T ), p 0(T ) = kt e βε ( ) 3/2 2πm βh 2 Langmuir θ 1/p 0 (T ) p 0 (T ) θ 1 (5.14) N ad N 0 N ad S(N ad ) = [ ] N 0! k log (N 0 N ad )!N ad! = k[n 0 log N 0 (N 0 N ad ) log(n 0 N ad ) N ad log N ad ] N ad S(N ad ) N ad = k[log(n 0 N ad ) log N ad ] (5.16) (5.1) N ad S N = S E T E N + S N = 1 ( ε µ) (5.17) E T (5.16) (5.17) (5.14) 5.6 Gibbs 2 21 51

E, V E r, V r E t, V t 2 E t = E + E r, V t = V + V r E r, V r Ω r = exp [S r (E r, V r )/k] E V p(e, V ) p(e, V ) exp [S r (E t E, V t V )/k] [ exp E S r k E V ] S r = exp [ β(e + pv )] k V p S p E, V R E R, V R 21: Z = dv de exp [ β(e + pv T S(E, V ))] Z exp [ βg(t, p)] G(T, p) Helmholtz G(T, p) = E + pv T S(E, V ) = F (T, V ) + pv (5.18) S(E, V ) E = 1 T, S(E, V ) V = p T 52

Helmholtz G(T, p) Gibbs T, p Gibbs Helmholtz Gibbs ( ) G F V p = V + p p + V = V Helmholtz ( ) G F V N = V + p N + F N = µ (5.19) Gibbs Nµ Gibbs Gibbs Extensive ξ G(T, p, ξn) = ξg(t, p, N) ξ ξ = 1 G N G(T, p, N) N = G(T, p, N) N G G(T, p, N) = g(t, p)n N g(t, p) N (5.19) Gibbs N G(T, p, N) = Nµ(T, p) α Gibbs G(T, p, N) = α N α µ α (5.18) Z G 1 β ln Z G = F µn = G pv + µn = pv 5.7 x, y, z, F (x, y, z, ) df (x, y, z, ) = Xdx + Y dy + Zdz + (5.20) 53

X, Y, Z, x, y, z, X(x, y, z, ) = F x, F Y (x, y, z, ) = y, (5.21) F X, Y, Z, G(X, Y, Z, ) G(X, Y, Z, ) = F (x, y, z, ) Xx Y y Zz (5.22) (5.21) x, y, z, F G G dg(x, Y, Z, ) = xdx ydy zdz (5.23) (5.22) X ( G F X = x + x X = x ) x X + ( F ) y ( F ) z y Y X + z Z X + 1 2 (5.21) G(X, Y, Z, ) F (x, y, z, ) F (x, y, z, ) = G(X, Y, X, ) + Xx + Y y + Zz + (5.24) X, Y, x(x, Y, Z, ) = G X, G y(x, Y, Z, ) = Y, (5.25) G(T, p, N) = F (T, V, N) pv, F (T, V, N) = G(T, p, N) + pv, F V = p G p = V 54

6 19 20 Planck T 2 Planck Einstein [ 19 ] Maxwell Einstein λ ν (c ) ε(p) = cp, ν = c, ( c = λν) (6.1) λ ε(p) = hν = hω, p = 2π (6.2) λ 2 1. T 2. 6.1 (6.1) H = i cp i (6.3) T V N Z = V N N!h 3N Π i d 3 p i exp[ βh] = V N [ ] N 8π N!h 3N (βc) 3 55

Helmholtz F (T, V, N) F = N [ ln V ] β N + 1 + ln 8π (βch) 3 2 2 N/2 N V N [ ] N 8π Z = (N/2)!(N/2)!h 3N (βc) 3, F = N [ ln 2V ] β N + 1 + ln 8π (βch) 3 E p (6.4) E = ln Z β p = F V = N β = 3N β 1 V = E 3V (6.5) p E/V 1/3 (6.1) p 2 p 2 2/3 N (6.4) F N F N = 1 [ ln 2V ] β N + 1 + ln 8π (βch) 3 + N 1 β N = 0 N V = 2 8π ( ) 3 kt (βch) 3 = 16π (6.6) ch (6.5) E V = 48πk4 (ch) 3 T 4 4 Stefan-Boltzmann T 4 (6.6) 1 l/2 V/N = 4π(l/2) 3 /3 l l(t ) = ( 3 ) 1/3 ch 8π 2 kt (6.7) 56

(p x, p y, p z ) δp x δp y δp z f(p)δp x δp y δp z f(p) Maxwell-Boltzmann f(p) f(p) exp[ cp/kt ], p = (p 2 x + p 2 y + p 2 z) 1/2 p p+dp ρ(p)dp ρ(p) ρ(p) = 8πV h 3 p2 exp[ cp/kt ] p (6.6) T p ω ρ(ω) = V ω2 π 2 c 3 exp[ hω/kt ] ω hω u(ω) u(ω) = hωρ(ω) = V hω3 π 2 exp[ hω/kt ] (6.8) c3 Wien (1896) Wien Stefan-Boltzmann Wien 57

6.2 Maxwell λ ω = 2πc/λ T kt V L V = L 3 λ λ = 2L/n, (n = 1, 2, 3, ) k λ 2π/λ (k x, k y, k z ) ( k µ ) k x = n xπ L, k y = n yπ L, k z = n zπ L n x, n y, n z (π/l) 3 1 2 k k + δk 4πk 2 dk/8 k µ 8 ρ(k)dk ρ(k) ρ(k)dk = 1 (π/l) 3 24πk2 8 dk = V π 2 k2 dk (6.9) V (6.9) V (2π) 3 dk = V 2π 2 k 2 dk = 1 h 3 dr dp (6.10) 2 p = hk h kt (6.9) u(ω) u(ω) = V hω2 π 2 kt (6.11) c3 Rayleigh-Jeans 58

6.3 Planck Reyleigh-Jeans (6.11) ( ) Wien 2 2 (6.8) (6.11) u(ω) = V ω2 π 2 kt f( hω/kt ) (6.12) c3 x = hω/kt f(x) (6.7) l(t ) x = hω kt = ch λkt = ( 8π 2 3 ) 1/3 l(t ) λ x 5 x x 1 l(t ) λ x 1 l(t ) λ 5: (6.12) f(x) x 1 x 1 xe x 2 Planck f(x) 2 22 π 2 u(ω)(kt/ hc) 3 /V x = hω/kt u(ω) = V ω2 π 2 c 3 hω e hω/kt 1 (6.13) Planck Planck (6.13) u(ω) = V ω2 π 2 c 3 n=1 (n hω)e n hω/kt ω hω ω ε hω z = 1 + e β hω + e 2β hω + e 3β hω + = 59 1 1 e β hω (6.14)

2.0 1.5 1.0 0.5 0.0 0.0 2.0 4.0 6.0 8.0 10.0 x 22: : Wien Rayleigh-Jeans (6.13) ε = 1 z k (k hω)e kβ hω = β ln z = β ln(1 e β hω hω ) = e β hω 1 (6.14) 6.4 1 p, p = h/λ, λ 2 Wien Rayleigh-Jeanes 60

λ l(t ) λ l(t ) 2 (6.14) 2 0 1 µ e βµ e βµ = 1 2 2 p 2 (5.12) 0 (6.14) 0 1 2 2 61