8.5 [ ]<, > 2 A B Z(T,V,N) = d 3N A p N A!N B!(2π h) 3N A d 3N A q A d 3N B p B d 3N B q B e β(h A(p A,q A ;V )+H B (p B,q B ;V )) = Z A (T,V,N A )Z B (T,V,N B ) (74) F (T,V,N)=F A (T,V,N A )+F B (T,V,N B ) (75) V (partia pressure)p A,B P = P A (T,V,N A )+P B (T,V,N A )=c A P + c B P (76) P A P B A B c A,B c A = N A N A + N B c B = N B N A + N B (77) φ A,B (T ) µ A,B = k B T n k ( ) 3/2 BT ma,b k B T c A,B P 2π h 2 + φ A,B (T ) (78) µ 0 A,B µ A,B = µ 0 A,B + k B T n c A,B (79) (c A c B ) µ A = µ 0 A (T,P)+k BT n c A (80) µ B = µ 0 B(T,P) k B Tc A (8) (79) G = k B T (N A n c A + N B n c B ) (< 0) (82) 53
S = k B (N A n c A + N B n c B ) (83) [ ] B A (N A N B ) G = N B µ 0 B (T,P)+N Aψ(T,P)+N A k B T n N A en B (84) 2 A N A 3 (83) N A N B µ B = µ 0 B(T,P) c A k B T (85) µ A = ψ(t,p)+k B T n c A (86) µ 0 B (T,P) ψ(t,p) [ ]< 80, 8 > (85) µ B (T,P,c A )=µ B (T,P 2,c A2 ) (87) µ 0 B (T,P ) µ 0 B (T,P 2)=(c A c A2 )k B T (88) v B (P P 2 )=(c A c A2 )k B T (89) (osmotic pressure) P = c A v B k B T (90) 54
P = c A v B k B T = N A V k BT (9) ( : van t Hoff aw of osmotic pressure) [ 2 ]< 74 79 > 2 2 A µ () B (T,P)=µ (2) B (T,P) (92) A δt µ () B (T + δt, P) k BTc () A = µ(2) B (T + δt, P) k BTc (2) A (93) (s () B s (2) B )δt = (c () A c (2) A )k B T (94) () (2) (s (2) B s() B )T = q δt =(c () A c (2) A ) k BT 2 q (95) µ () B (T,P + δp) k B Tc () A = µ (2) B (T,P + δp) k B Tc (2) A (96) (v () B v (2) B )δp =(c () A c (2) A )k B T (97) δp =(c () A c (2) A ) k B T v () B v (2) B 2 A c (2) A δp c () k B T A v (2) B 0 v() B (98) v(2) B c () A P 0 (T ) (99) 55
P 0 ( Raout s aw) [ ]< 64, 65 > A + B C (= AB) (00) G(T,P,N A,N B,N C ) dg = G G G = 0 (0) dn C N C NA,N B N A NB,N C N B NC,N A µ A + µ B = µ C (= µ AB ) (02) ( ) ν i A i = 0 (03) i ν i µ i = 0 (04) i [ ]< 72, 73 > (78) (02) ε b c AB = ( 2π h 2) ( ) 3/2 mab 3/2 P c A c B m A m B (k B T ) 5/2 e(ε b φ AB (T )+φ A (T )+φ B (T ))/k B T (05) (aw of mass action) 56
9 S = k B n Ω /N! p p 2 p 2 p /2! p /2! 9. [ ] 2 (Boson, boson) (Fermion, fermion) Ψ(q 2,q )=±Ψ(q,q 2 ) () ( ) 2 ψ (q) ψ 2 (q) 2 Ψ(q,q 2 )= 2! (ψ (q )ψ 2 (q 2 ) ± ψ (q 2 )ψ 2 (q )) (2) Ψ B (q,,q N )= N!N!N 2! Ψ F (q,,q N )= N! N! a permutations N! a permutations ψ k (q ) ψ kn (q N ) (3) ( ) P ψ k (q ) ψ kn (q N ) = N! det (ψ i (q ki )) (4) ( ) P N i, 2,,N i i {N i } N i 0 57
9.2 [ ] ( Z(T,V,N)= exp β i a possibe configurations of N= N i N i ε i ) (5) N i 0 [ ] i i) (N i =0) E =0 ii) (N i =) E = ε i Z i (T,V,µ) = N i =0 = +e β(µ ε i) e β(e(n i ) i µn i ) (6) Z(T,V,µ)= i Z i (T,V,µ) (7) Φ(T,V,µ) = k B T i n Z i (T,V,µ) = i Φ i (T,V,µ) = k B T i n ( +e β(µ ε i) ) (8) i N i = Φ i µ βe β(µ ε i) +e β(µ ε i) = β = e β(ε i µ) + i (Fermi-Dirac statistics) 58 (9)
(Fermi distribution function) f FD (ε) = e β(ε µ) + (0) f.4.2 0.8 0.6 0.4 0.2 2 4 6 8 0 e T Figure 6: (k B T =) µ =, µ = µ =5 µ N = Φ µ = i e β(ε i µ) + () ( ) x (ε µ)/k B T f FD (ε) = = e β(ε µ) + e x + = ( tanh x ) 2 2 (2) ε = µ f FD =/2 f FD (ε) θ(µ ε) (3) ε : N i = f FD (ε) e β(ε µ) e βµ (βµ ) f FD (ε) e β(µ ε) (4) (Maxwe-Botzmann distribution) 59
[ ] i 0 Z i (T,V,µ) = = = = N i =0 e βµn i Z i (T,V,N) e βµn i e βε in i N i =0 e β(µ ε i)n i N i =0 e β(µ ε i) (5) Z(T,V,µ)= i Z i (T,V,µ) (6) Φ(T,V,µ) = k B T i n Z i (T,V,µ) = i Φ i (T,V,µ) = k B T i n ( e β(µ ε i) ) (7) i N i = Φ i µ βe β(µ ε i) = β e β(µ ε i) = e β(ε i µ) (8) i (Bose-Einstein statistics) (Bose distribution function) f BE (E) = e β(ε µ) (9) 60
f.4.2 0.8 0.6 0.4 0.2 2 3 4 5 e T Figure 7: (k B T =) µ =, µ = 0.5, µ =0, µ N = Φ µ = i e β(ε i µ) (20) ( ) III e β(ε µ) f BE (ε) e β(µ ε) (2).4.2 0.8 0.6 0.4 f 0.2 2 3 4 5 e T Figure 8: (µ =, k B T =) 6
9.3 [ ] {ε i } ε M N N M W = M C N = M! N!(M N )! {N } S = k B n W (22) ( k B M n M e N n N e (M N )n (M ) N ) e = k B M [n n n +( n )n( n )] (23) n = N /M N = M n (24) E = M ε n (25) ( ) S αn βe = S (α βε ) M n kb k B n ] n = M [n + α + βε n n = = 0 (26) e βε +α + (27) α β N = M e βε +α + 62 (28)
E = M ε e βε +α + (29) α β [ ] α β N E : S(E,V,N) ( V ) (26) T = ( S E ) V,N = ( ) ( ) S n M n E V,N V,N ( ) n = k B M (α + βε ) E V,N = k B α ( ) n M + β ( ) n M ε E E V,N V,N = k B β (30) N 2 E β /k B T µ ( ) S T = N E,V = ( ) ( ) S n M n N E,V E,V ( ) n = k B M (α + βε ) N E,V = k B α ( ) n M + β ( ) n M ε N E,V N E,V = k B α (3) α = µ/k B T [ ] {ε i } N W = M +N C N = M + N N!(M )! (32) 63
{N } S = k B n W [ k B (M + N )n (M + N ) e = k B M n M e N n N ] e M [( + n )n(+n ) n n n ] (33) N E ( S n kb ) αn βe = M [ n +n n α βε = 0 (34) ] n = e βε +α (35) α β 64