Transcription

1 HP 2 Quantum Physics

2 i

3 N! Z Z Z ii

4 E N iii

5 A Legendre 106 B 107 iv

6 1 1.1 (equilibrium value) 2.8 (fluctuation) = (1.1) = 2 1 I, II 2 1

7 (asymptotic theory) (limit) lim f(x) = 1 (1.2) x x f(x) 1 x f(x) 1 ϵ ϵ X ϵ x > X ϵ x f(x) 1 < ϵ X ϵ (1.2) f(x) x 1 f(x) 1 1/10 ( ϵ) x 10 2 ( X ϵ ) f(x) 1 1/100 ( ϵ) x 10 3 ( X ϵ ) 1/1000 x 10 4 ϵ 3 4 ϵ ϵ 1/x x x 0 lim 1/x = 0 x 4 5 2

8 S B S T V V lim V S B V S T V = 0. (1.3) V S B /V S T /V 1/100 V 1m 3 1/1000 V 10m 3 V V 6 V V a n = a n, n=1 n =1 a n = a n, n=1 n =1 1 f(x)dx = f(x )dx. (1.4) a 1 + a 2 + a 3 + ( ) 2 ( ) ( ) a n = a n a n = n=1 n=1 n=1 a 2 n n=1 ( 1 2 ( 1 f(x)dx) = 0 0 ) ( 1 ) f(x)dx f(x)dx = n=1 m=1 b n,m = m=1 0 a n n=1 m=1 f(x)dx 1 0 a m = n=1 b n,m = n,m (n 1,m 1) f(y)dy = n=1 m=1 a n a m. (1.5) f(x)f(y)dxdy. (1.6) b n,m. (1.7) n 1, m 1 n, m g(x, y)dxdy = 1 0 [ 1 0 ] g(x, y)dx dy = 1 0 [ 1 0 ] g(x, y)dy dx = g(x, y)ds. (1.8) 0 x 1,0 y 1 0 x 1, 0 y 1 2 ds V

9 x Θ(x) x x f(x) (mean value) (expectation value) 10 x = x f(x) = x xθ(x), (1.9) f(x)θ(x). (1.10) x 2 = x x 2 Θ(x) x (variance) (x x ) 2 = x (x x ) 2 Θ(x) = x 2 x 2. (1.11) (standard deviation) a < b f(a) f(b) f(x) = a < b f(a) f(b) f(x) a < b f(a) < f(b) a < b f(a) > f(b) : : : > = : a a,b a a,b ( ) n : n C k k exp(x) : e x ln x : log e x b 3 a = 1, 2, 3 a=1 10 x f(x) 4

10 : x x 0 lim x x 0 f(x)/g(x) = 1 f(x) g(x) f(x) = 5x 2 4x + 3 x f(x)/(5x 2 ) 1 f(x) 5x 2 x 0 f(x)/3 1 f(x) 3 f(x) g(x) f(x) g(x) f(x) g(x) 0 : o : x x 0 lim x x 0 f(x)/g(x) = 0 f(x) = o(g(x)) f(x) = x x f(x) = o(x) f(x) = o(x 2 ) x f(x) = o(x) lim f(x) = 0 lim f(x) = x x 0 f(x)/x o(x)/x = o(1) x 0 f(x) = o(x 2 ) x 0 f(x) x 2 x 2 0 f(x) 0 f(x) x 2 x f(x) = o(x 2 ) x f(x) x 2 x 2 f(x) x 2 O : x x 0 x x 0 f(x)/g(x) f(x) = O(g(x)) 2x2 1 + x x f(x) = O(x). f(x) = O(x2 ) f(x) = O(x 3 ) f(x) = O(x) x 0 f(x) = O(x 2 ) f(x) = O(x) f(x) = O(x 2 ) f(x) = x 2 (2 + sin x) + x cos x x f(x) = O(x 2 ). f(x) = O(x 3 ) f(x) = O(x 4 ) f(x) = O(x 2 ) x 0 f(x) = O(x). f(x) = O(1) f(x) = O(x) f(x) x f(x) = O(ln x) (ln x)/ x 0 f(x)/ x 0 O : O 2x2 1 + x x f(x) = O(x) f(x) = O(x2 ) f(x) = O(x 3 ) x 0 f(x) = O(x 2 ) f(x) = O(x) f(x) = x 2 (2 + sin x) + x cos x x f(x) = O(x 2 ) f(x) = O(x 3 ) f(x) = O(x 4 ) x 0 f(x) = O(x) f(x) = O(1) Z k B T ln Z O(V ) F F = O(V ) k B T ln Z = F + o(v ) 5

11 O o(v ) O(V ) O

12 2 [1] UV N 2.1 [1] 7.1 E U E (thermal equilibrium state) (equilibrium state) (nonequilibrium state) (transition) 1 (a) V (b) E N (c) P (d) S T (equilibrium value) 1 7

13 (isolated system) (open system) (subsystem) (composite system) (internal constraint) 3 (simple system) (external field) (simple system) i = 1, 2, X i X (i) X = i X (i) (2.1) X (additive) (additive variable) V N 4.1 S 2.9 E i X X (i) V (i) X (i) = KV (i) for (2.2) K X (extensive) (extensive variable) T P µ (intensive variable) V (mean value) (fluctuation) 2 I [10] 3 8

14 1.3.2 V V, (2.3) = o(v ) (2.4) = V, (2.5) = o(1) (2.6) V o(v )/V = o(1) 0 o(v ) o(1) 4 5 N T 2 (uniform) 3.4 (a), (b) 6 4 O o V 5 6 9

15 2.1.5 (fundamental relation) S E V N S = S(E, V, N). (2.7) E E = E(S, V, N) (2.8) S = S(E, V, N) E = E(S, V, N) S A 11 Helmholtz F = F (T, V, N) (2.9) E = E(S, V, N) S = S(E, V, N) F = F (T, V, N) V Gibbs G = G(T, P, N) (2.10) E T, V, N E = E(T, V, N) E = (3/2)RNT E = E(S, V, N) S(E, V, N), E(S, V, N), F (T, V, N), G(T, P, N), E = E(S, V, N) S, V, N (natural variables) S E, V, N (fundamental relation) S E X 1,, X t S = S(E, X 1,, X t ). (2.11) E, X 1,, X t (natural variables of entropy) t + 1 f 1 1 E, X 1,, X t E, X 1,, X t ( ) (2.12) E, X 1,, X t E, X 1,, X t (thermodynamical state space) (2.11) E E = E(S, X 1,, X t ). (2.13) 10

16 S = S(E, X 1,, X t ) S = S(E, X 1,, X t ) (entropy representation) E = E(S, X 1,, X t ) (energy representation) (natural variables of energy) UV N SV N S(E, X 1,, X t ) E(S, X 1,, X t ) S F (T, X 1,, X t ) E = X 0 II-(iii) S = S(X 0, X 1,, X t ) Π 0, Π 1,, Π t Π k S(X 0, X 1, ) X k (k = 0, 1,, t). (2.14) X k (conjugate) X 1 = V Π 1 V Π V X k Π k X k Π k X 0 = E Π 0 (inverse temperature) B B S(E, X 1, ) E (k = 0, 1,, t). (2.15) (2.13) P k E(X 0, X 1, ) X k (2.16) X k (conjugate) S, X 1, E = X 0 S = X 0 X 1 = V P 1 V P V X 0 = S P 0 (temperature) T T E(S, X 1, ). (2.17) S T S, X 1, T (S, X 1, ) S V P V (pressure) P P P V = E(S, V, ). (2.18) V N P N (chemical potential) µ µ P N = E(S, V, N, ). (2.19) N B = 1/T, (2.20) Π k = P k /T (k = 1, 2, ). (2.21) 11

17 Π V = P V /T = P/T, (2.22) Π N = P N /T = µ/t. (2.23) [1] I (i) [ ] (equilibrium state) (ii) [ ] II (i) [ ] (entropy) S (ii) [ ] S E E, X 1,..., X t S = S(E, X 1,, X t ) ( ). (2.24) (fundamental relation) E, X 1,, X t (natural variables of entropy) t + 1 (iii) [ ] (2.24) E E 0 (iv) [ ] (v) [ ] i (= 1, 2, ) S (i) = S (i) (E (i), X (i) 1,, X(i) t i ) Ŝ i S (i) (E (i), X (i) 1,, X(i) t i ) (2.25) Ŝ

18 3 3.1 (microstates) m e m e 1 t = 0 3 x, y, z 3 v x, v y, v z x, y, z, v x, v y, v z 6 x, y, z, v x, v y, v z 1 3 p x, p y, p z 6 x, y, z, p x, p y, p z (canonical variables) t = 0 t = 0 x(0), y(0), z(0), p x (0), p y (0), p z (0) t t x(t), y(t), z(t), p x (t), p y (t), p z (t) N 6 k (k = 1, 2,, N) (k) x (1), y (1), z (1),, x (N), y (N), z (N), p (1) x, p (1) y, p (1) z,, p (N) x, p (N) y, p (N) z (3.1) 6N N (single-particle state) N (many-particle state) 13

19 3N q 1, q 2,, q 3N q 1 = x (1), q 2 = y (1),, q 3N = z (N) p 1, p 2,, p 3N p 2 = p (1) y q 1, p 1, q 2, p 2,, q 3N, p 3N (3.2) 3N q i, p i (i = 1, 2,, 3N) 3N ( (degrees of freedom) f q 1, q 2,, q f q p 1, p 2,, p f p q q 1, q 2,, q f, (3.3) p p 1, p 2,, p f. (3.4) f 2f q, p q, p q 1, q 2,, q f, p 1, p 2,, p f 2f 2f (phase space) 1 Γ E, X 1,, X t (t + 1) t t f f (thermodynamical state space) r(0), v(0) r(t), v(t) r(t), v(t) v(t) v(t) 2 v(t) r(0), v(0) r(t), v(t) v(t) = v(0) + v(0)t + v(0)t 2 /2 + + v (n) (0)t n /n! + o(t n ) (3.5) v(t) v r r r(0) r(t), v(t) q i q i p i q i p i 4.5 q i, p i 3.2 q, p q(t), p(t) (q, p) 1 topological space 14

20 (q, p) dq i dt dp i dt = H(q, p) p i (i = 1, 2,, f), (3.6) = H(q, p) q i (i = 1, 2,, f). (3.7) H(q, p) q, p (Hamiltonian) H(q, p) = H(q 1, q 2,, q f, p 1, p 2,, p f ) (3.8) 2f (3.6), (3.7) t = 0 q, p q(0), p(0) q(t), p(t) 2 (3.6), (3.7) d H(q, p) = 0 (3.9) dt 3.1 q, p H(q, p) E q(t), p(t) H(q, p) = E =. (3.10) H(q(0), p(0)) = E = H(q(t), p(t)) (3.11) H t = 0 E t E (3.10) (q, p) (3.10) S(E) S S Γ dim Γ dim Γ = 2f (3.12) S(E) (3.10) dim S(E) dim Γ = (3.10) (=1 ) dim S(E) = dim Γ 1 = 2f 1 (3.13) 3 S(E) 3.1 (3.9) 3.3 q, p (3.6), (3.7) q, p (linear) 2 [11] x 2 + y 2 + z 2 = = 2 15

21 f = H(q, p) H(q, p) = 1 2m p2 + mω2 2 q2. (3.14) m ω k ω 2 = k/m dq dt dp dt = H(q, p) = p p m, (3.15) = H(q, p) = mω 2 q, q (3.16) q, p p m d2 q dt 2 = mω2 q d2 q dt 2 + ω2 q = 0 (3.17) q, p A, θ q(t) = A sin(ωt + θ), (3.18) p(t) = Amω cos(ωt + θ). (3.19) q, p ω (harmonic oscillation) q p S(E) H(q, p) = 1 2m p2 + mω2 2 q2 = E (3.20) q p S(E) S(E) E S(E) S(E) E f 2 f t = 0 q, p q(0), p(0) q(t), p(t) H(q 1, q 2, p 1, p 2 ) = 1 2m p2 1 + mω2 2 q m p2 2 + mω2 2 q2 2 + mω2 2 (q 2 q 1 ) 2. (3.21) 1 Ω K Ω 2 = K/m 16

22 dq 1 dt dp 1 dt dq 2 dt dp 2 dt = H p 1 = p 1 m, (3.22) = H q 1 = mω 2 q 1 + mω 2 (q 2 q 1 ), (3.23) = H p 2 = p 2 m, (3.24) = H q 2 = mω 2 q 2 mω 2 (q 2 q 1 ). (3.25) 1 (3.23) 1 q 2 2 (3.25) q 1 1,2 1 Q 1, P 1, Q 2, P 2 1 Q 1, Q 2 (normal coordinate) Q 1 = (q 1 + q 2 )/ 2, (3.26) Q 2 = (q 1 q 2 )/ 2 (3.27) P 1, P 2 P 1 = (p 1 + p 2 )/ 2, (3.28) P 2 = (p 1 p 2 )/ 2 (3.29) 4 ( H(Q 1, Q 2, P 1, P 2 ) = H 1 (Q 1, P 1 ) + H 2 (Q 2, P 2 ), (3.30) H 1 (Q 1, P 1 ) 1 2m P mω2 2 Q2 1, (3.31) H 2 (Q 2, P 2 ) 1 2m P m(ω2 + 2Ω 2 ) Q (3.32) Q 1, P 1 H 1 Q 2, P 2 H 2 dq 1 dt dp 1 dt dq 2 dt dp 2 dt = H P 1 = H 1 P 1 = P 1 m, (3.33) = H Q 1 = H 1 Q 1 = mω 2 Q 1, (3.34) = H P 2 = H 2 P 2 = P 2 m, (3.35) = H Q 2 = H 2 Q 2 = m(ω 2 + 2Ω 2 )Q 2. (3.36) Q 1, P 1 Q 2, P 2 Q 1, P 1 Q 2, P 2 1 (3.15), (3.16) Q 1, P 1 ω Q 2, P 2 ω 2 + 2Ω H H(Q 1, Q 2, P 1, P 2 ) = E =, (3.37) H 1, H 2 H 1 (Q 1, P 1 ) = E 1 =, (3.38) H 2 (Q 2, P 2 ) = E 2 =, (3.39) 4 q 1, p 1, q 2, p 2 Q 1, P 1, Q 2, P 2 17

23 E = E 1 + E 2 (3.37) S(E) (3.38), (3.39) S 1 (E 1 ), S 2 (E 2 ) (3.37), (3.38), (3.39) 2 f q, p (integrable system) (3.13) 2f 1 dim[ ] = dim Γ f = 2f f = f (3.40) f 2f 1 q 1, q 2 (3.26), (3.27) q 1 = (Q 1 + Q 2 )/ 2, (3.41) q 2 = (Q 1 Q 2 )/ 2 (3.42) Q 1, Q 2 Q 1, Q 2 q 1, q 2 f = 2 f Q 1, P 1,, Q f, P f f H(Q 1, Q 2, P 1, P 2 ) sin cos sin cos f f (integrable system) i p 2 i K 1 i p 2 i, K 2 i p 4 i,, K f i p 2f i (3.43) 6 K 1 /N N K 1 K 2 K 1, K 2,, K f K 1, K 2,, K f 5 6 i p2f+2 i, i p2f+4 i, (3.43) p 2 i f K 1, K 2,, K f p 2 1, p2 2,, p2 f i p2f+2 i, i p2f+4 i, i p i, i p3 i, p i i p i, i p i 3, 18

24 I K 1, K 2,, K f 7 f K 1, K 2, K 3 K 1, K 2, K 3 K 4, K 5,, K f 2.2 t = 0 f K 1, K 2,, K f f = 1 1 f 1 sin cos (3.14) λ H(q, p) = 1 2m p2 + mω2 2 q2 + λ 4 q4. (3.44) S(E) q dq dt dp dt = p m, (3.45) = mω 2 q λq 3. (3.46) (λ = 0) q(t), p(t) S(E) 7 K 1 E = (1/2m) i p2 i E 19

25 S(E) S(E) S(E) S(E) 20

26 4 (2.12) (2.11) II-(ii) (microstates) 100% 21

27 1 E, X 1,, X t (microcanonical ensemble) ens(e, X 1,, X t ) W (E, X 1,, X t ) E, X 1,, X t 3 A 0 : ens(e, X 1,, X t ) ens(e, X 1,, X t ) E, X 1,, X t V V E, V, N, E/V, N/V, V 4 E, X 1,, X t (thermodynamic limit) V A 0 100% ens(e, X 1,, X t ) V 0 ens(e, X 1,, X t ) E, X 1,, X t 0 W (E, X 1,, X t ) as V (E/V, X 1 /V,, X t /V ). (4.1) ens(e, X 1,, X t ) 5 1 microstate microcanonical micro micro 2 (measure) V 5 self-consistent 22

28 n n n/n = 1/ n n n (2.12) = ( ) E, X 1,, X t ens(e, X 1,, X t ) = ( ) (4.2) 1.1 = ens(e, X 1,, X t ) (4.3) P P = P ens(e, X 1,, X t ) (4.4) 4.6 ens(e, X 1,, X t ) ens(e, X 1,, X t ) 6 : ens(e, X 1,, X t ) (microcanonical distribution) (E, X 1,, X t ) = (4.5) S(E) ens(e, X 1,, X t ) ens(e, X 1,, X t ) E, X 1,, X ens(e, X 1,, X t ) ens(e, X 1,, X t ) A A A : E, X 1,, X t A 0 A A (mixed state) 23

29 (principle of equal weights) A 0 A A 0 B A 0 A 0 8 V (a family 9 of states of systems of different sizes) S = S(E, V, N) s = s(u, v) ens(e, X 1,, X t ) A 0 = ens(e, X 1,, X t ) (4.6) ens(e, X 1,, X t ) ens(e, X 1,, X t ) ens(e, X 1,, X t ) 6.2 ens(e, X 1,, X t ) ens(e, X 1,, X t ) 8 A 9 family 24

30 equal a priori probability postulate A λ 10 λ Ψ λ Ψ λ Ψ λ 11 ens(e, X 1,, X t ) W (E, X 1,, X t ) = 1 (4.7) λ s.t. Ψ λ ens(e,x 1,,X t) Ψ λ ens(e, X 1,, X t ) λ s.t. such that Ψ λ Θ(λ) 1 (normalization) Θ(λ) = 1 (4.8) λ ens(e, X 1,, X t ) Θ(λ) 1 when Ψ λ ens(e, X 1,, X t ), Θ(λ) = W (E, X 1,, X t ) 0 otherwise (4.9) A (E, X 1,, X t ) Ψ λ A 12 Ψ λ A A λ A A (1.10) A = λ A λ Θ(λ) (4.10) = 1 W (E, X 1,, X t ) λ s.t. Ψ λ ens(e,x 1,,X t ) A λ (4.11) Ψ λ A λ λ 25

31 4.5 (q, p) (q 1, q 2,, q f, p 1, p 2,, p f ) (2πħ) f ħ h ( Js) 2π ħ h/2π Js. (4.12) dq 1 dq f dp 1 dp f ( dqdp ) (4.13) dqdp (2πħ) f (4.14) ens(e, X 1,, X t ) W (E, X 1,, X t ) = (q,p) ens(e,x 1,,X t) dqdp (2πħ) f (4.15) dq 1 dp 1 (q,p) ens(e,x 1,,X t ) 2πħ dq f dp f 2πħ (4.16) (q, p) ens(e, X 1,, X t ) 2f (4.15) (4.7) λ 1/(2πħ) f λ dqdp (2πħ) f (4.17) dqdp 1/(2πħ) f dqdp/(2πħ) f 1 (4.16) dq j dp j /2πħ) (q, p) dqdp [q 1, q 1 + dq 1 ),, [q f, q f + dq f ), [p 1, p 1 + dp 1 ),, [p f, p f + dp f ) (4.18) [q, q + dq), [p, p + dp) dqdp/(2πħ) f Θ([q, q + dq), [p, p + dp)) 14 (measure) 26

32 (4.9) ens(e, X 1,, X t ) 1/W (E, X 1,, X t ) 1 dqdp Θ([q, q + dq), [p, p + dp)) = W (E, X 1,, X t ) (2πħ) f when (q, p) ens(e, X 1,, X t ), 0 otherwise. (4.19) A (E, X 1,, X t ) [q, q+dq), [p, p+dp) (distribution function) θ(q, p) 15 Θ([q, q + dq), [p, p + dp)) = θ(q, p) dqdp (2πħ) f. (4.20) A q, p A(q, p) A A (1.10), (4.17), (4.20) A = A(q, p)θ(q, p) dqdp (2πħ) f (4.21) A(q, p) = 1 1 (normalization) θ(q, p) dqdp = 1. (4.22) (2πħ) f (4.19) (4.20) 1 when (q, p) ens(e, X 1,, X t ), θ(q, p) = W (E, X 1,, X t ) 0 otherwise. (4.23) (4.21) 1 A = A(q, p) dqdp W (E, X 1,, X t ) (2πħ) f. (4.24) (q,p) ens(e,v,n) 4.1 m K f j=1 p 2 j 2m (4.25) (4.24) K = 1 W (E, X 1,, X t ) (q,p) ens(e,v,n) 9.1 f j=1 p 2 j 2m dqdp (2πħ) f. (4.26) f = 1 2πħ πħ 1 15 (2πħ) f 16 27

33 4.2 f δq j δp j ħ/2 (j = 1, 2,, f) (4.27) κ 1 (κħ) f 1 (2πħ) f 1 (2π/κ) f dqdp/(κħ) f κ κ 4.1 κ = 2π (4.14) (dynamical variable) S T (nonmechanical variable) (genuine thermodynamic variable) 18 (4.3) A A B E (4.28) (4.34) (1), (2) A E = E (1) + E (2) E E (1), E (2) (4.7) 1, 2 V (1), V (2) W (1) (E (1) ), W (2) (E (2) ) i o(v (i) ) W (E) = W (1) (E (1) )W (2) (E E (1) ) (4.28) E (1) 2.9 B 19 E (1) Θ (1) (E (1) ) Θ (1) (E (1) ) = W (1) (E (1) )W (2) (E E (1) ) W (E) (4.29) 17 Bohr-Zommerfelt 18 II 19 B 28

34 Θ (1) (E (1) ) E E (1) E (1) W (1), W (2), W ln Θ (1) (E (1) ) = ln W (1) (E (1) ) + ln W (2) (E E (1) ) ln W (E) (4.30) E (1) 20 E (1) ln Θ(1) (E (1) ) = E (1) ln W (1) (E (1) ) + E (1) ln W (2) (E E (1) ) (4.31) E ln W (2) (E E (1) ) = (1) E ln W (2) (E (2) ) (2) E ln W (1) (E (1) ) = (1) E ln W (2) (E (2) ) (4.32) (2) E (1) V (1), V (2) 21 Θ (1) (E (1) ) E (1) E (1) (4.32) E (1) 8.2 E (1) B(= 1/T ) E (1) S(1) (E (1) ) = E (2) S(2) (E (2) ) (4.33) B (1) (E (1) ) = B (2) (E (2) ) (4.34) 2 E, X 1, X 2, E E X k (k = 1, 2, ) (4.32), (4.33) X (1) k X (1) k ln W (1) (X (1) k ) = X (2) k S (1) (X (1) k ) = X (2) k ln W (2) (X (2) k ), (4.35) S (2) (X (2) k ) (4.36) X k X k (4.32) (4.33) (4.35) (4.36) ln W S S (i) = ln W (i) + (i = 1, 2) (4.37) k B J/K (4.38) 22 (4.37) 6 (4.37) (i) S = k B ln W (4.39) V (i) E/V, N/V V V E (1) E (1) 22 SI k B = 1 29

35 o(v ) B B : E, X 1,, X t k B ln W (E, X 1,, X t ) S(E, X 1,, X t ) S(E, X 1,, X t ) = k B ln W (E, X 1,, X t ) + o(v ). (4.40) (2.11) o(v ) 4.7 B o(v ) E, V, N S TD k B ln W S SM B o(v ) S SM (E, V, N) k B ln W (E, V, N) (4.41) S TD (E, V, N) = S SM (E, V, N) + o(v ). (4.42) 4.3 S TD (E, V, N) s(e/v, N/V ) S TD (E, V, N) = V s(e/v, N/V ) (4.43) V s(e/v, N/V ) + o(v ) (4.44) E/V, N/V V o(v ) O(V ) S TD (E, V, N) (4.42) B S SM (E, V, N) S SM (E, V, N) S TD (E, V, N) S SM (E, V, N) S TD (E, V, N) for (4.45) B S SM S TD O(V ) O(V ) u, n u E/V, n N/V (4.46) S SM (V u, V, V n) s(u, n) = lim V V (4.47) (4.43) V V S TD (E, V, N) = V S SM (V u, V, V n) lim V V. (u = E/V, n = N/V ) (4.48) u, n, V (thermodynamic limit) 30

36 O(V ) o(v ) E/V, N/V V E, V, N O(V ) O(N) O(E) o(v ) o(n) o(e) E N E N u, n s(u, n) u, n s(u, n) S TD (E, V, N) = V s(u, n) (E = V u, N = V n) (4.49) E, N 5.7 S SM (E, V, N) O(V ) S TD (E, V, N) S SM (E, V, N) E, N W (E, V, N) Z(T, V, N) O(V ) W (E, V, N), Z(T, V, N) V V V c ϵ, V ϵ, 1/c < ϵ for V > V V V/γ 23 γ ϵ (0, ϵ, 2ϵ, ) E j n j ϵ (n j = 0, 1, 2, ) n j n n 1, n 2,, n V/γ (4.50) 24 toy model V V 24 n V/γ N P µ γ 31

37 V/γ 1, E/ϵ 1 (4.51) V, E γ, ϵ E, V E, V, N N E, V 4.3 V/γ E = n j ϵ (4.52) j=1 W (E, V ) = (E/ϵ + V/γ 1)! (E/ϵ)!(V/γ 1)! (4.53) O(V ) (Stirling s formula) N! 2π N N+1/2 e N as N. (4.54) o(n) ln(n!) = N ln N N + o(n). (4.55) ln W (E, V ) = ( E ϵ + V ) ( E ln γ ϵ + V ) E γ ϵ ln E ϵ V γ ln V γ + o(v ). (4.56) S(E, V ) = k B [( E ϵ + V γ ) ( E ln ϵ + V ) E γ ϵ ln E ϵ V γ ln V ]. (4.57) γ (4.48) 4.4 V B = k ( B ϵ ln 1 + ϵv ), (4.58) γe Π V = k ( B γ ln 1 + γe ). (4.59) ϵv P, V, N, T (equation of state) ( ) ( ) e ϵ/kbt 1 e γp/kbt 1 = 1. (4.60) N P, V, T V (4.37) 4.3 (4.53) 4.4 (4.48) S TD (E, V ) = V S SM (V u, V ) lim V V. (u = E/V ) (4.61) S TD (E, V ) (4.57) 32

38 A, B A  ˆρ Â, ˆρ A (4.62) Tr[ˆρÂ] Tr[( )( )] 26 entanglement entanglement ˆρ ˆρ entanglement (4.63) entanglement S S (4.62) entanglement ˆρ (4.63) S (4.64) S 27 Gibbs von Neumann Shannon Shannon Gibbs von Neumann Gibbs von Neumann W (E) E 33

39 S Gibbs von Neumann (4.64) S von Neumann von Neumann S von Neumann S Gibbs von Neumann (4.64) (4.64) S (4.64) S 34

40 5 W S = k B ln W S W 5.1 ens(e, V, N) E E l, E u (> 0) V, N [E E l, E + E u ] (5.1) ens(e, V, N) W (E, V, N) E l, E u ens(e, V, N) W (E, V, N) S(E, V, N) E l, E u E l, E u S(E, V, N) V, N E Ω(E, V, N) W (E, V, N) = Ω(E + E u, V, N) Ω(E E l, V, N). (5.2) E = O(V ) E l, E u V E l = o(v ), E u = o(v ) (5.3) (5.2) E 1 W (E, V, N) = Ω(E, V, N) ( E l + E u ) (5.4) E Ω Ω E (density of states) B Ω E [ ] Ω S(E, V, N) = k B ln E ( E l + E u ) + o(v ) (5.5) E l, E u E l, E u k B ln [ Ω E ( E l + E u) ] + o(v ) 1 ( E k B ln l + E u ) + o(v ). (5.6) E l + E u E l,, E u o(v ) V o(v ) E l, E u E l, E u S(E, V, N) 1 o(v ) o(v ) o(v ) = 0 V 1/2 V 1/3 o(v ) o(v ) 35

41 O(1) ε ε = k B T E l = E u = ε/2 S(E, V, N) = k B ln S(E, V, N) ( ) Ω E ε + o(v ) (5.7) E l o(v ) 5.2 (5.5) E l o(v ) O(V ) o(v ) S(E, V, N) (5.5) E l = o(v ) S = k B ln W + o(v ) E l = O(V ) (5.7) Ω(E, V, N) E E E 0 < E < + (5.8) V o(v ) Ω(E, V, N) O(V ) S E = 1 T > 0 (5.9) T V, N E E 0 T 0 (5.7) 2 Ω / Ω E 2 E = 1 k B T > 0. (5.10) Ω 0 E (E = 0) (5.8) E Ω E > 0, 2 Ω > 0. (5.11) E2 Ω E ε O(1) Ω E ε < Ω < Ω E E (5.12) ( ) Ω ln E ε ( ) Ω < ln Ω < ln E E (5.13) ln(e/ E l ) = o(v ) O(V ) O(V ) (5.5) (5.7) S(E, V, N) = k B ln Ω + o(v ) (5.14) Ω(E, V, N) E l = E E u = 0 W (E, V, N) E u o(v ) W (E, V, N) Ω(E, V, N) + Ω E E u ln Ω o(v ) W (E, V, N) S(E, V, N) E l, E u S(E, V, N) 36

42 S(E, V, N) E l, E u 0 E l E, 0 E u o(v ), E l + E u [ O(1) ] (5.15) (5.14) E E Ω Ω Ω E E S(E, V, N) ens(e, V, N) E l, E u W (E, V ) Ω(E, V ) (5.14) V E 5.1 Ω(E, V ) = (E/ϵ + V/γ)! (E/ϵ)!(V/γ)! (5.16) O(V ) (4.57) (5.14) 5.1 (5.16) (5.14) 5.3 SI k B S k B = 1 (5.14) S S(E, V, N) = ln Ω(E, V, N) + o(n) (5.17) S(E, V, N) = Ns(E/N, V/N) (5.18) s 4.1 N s 1 ln Ω(E, V, N) ln Ω(E, V, N) = Ns(E/N, V/N) + o(n) (5.19) 3 Ω(E, V, N) ln Ω(E, V, N) (1/N) ln Ω(E, V, N) 2 s(u, v) (u E/N, v V/N) (5.9) 5.2 s(u, v) > 0 (5.20) u 4.3 s(u, v) 2 s(u, v) 0 (5.21) u Ω (5.19), (5.20), (5.21) 37

43 16 S(U, V, N) C 1 s(u, v) C 1 2 (5.19) Ω(E, V, N) = exp [Ns(E/N, V/N) + o(n)] (5.22) s N ( Ω(E, V, N) u (= E/N) s(u, v) Ω = exp[ns(u, v)] s(u, v) = 1 u ( )u s s(( )u, v) = Ω = e Ns N = e ( )N e N = e 10 6N = e 1018 (5.23) 3 Ω(E, V, N) E % W (E, V, N) = Ω(E + E u, V, N) Ω(E E l, V, N) ( (5.2)) Ω(E, V, N) (5.24) O(V ) (5.7), (5.14) Ω(E, V, N) (5.22) s rate function B A B 5.4 E l, E u (5.11), (5.19)-(5.22) [7] 38

44 5 5 39

45 6 ( (free particles) (ideal gas) N N = 2 (a) (b) (a) (b) 6.5 ens(e, V, N) N = 2 (a) (b) W (E, V, N) Ω(E, V, N) 2 N N! 3.1 q 1, p 1, q 2, p 2,, q 3N, p 3N H = 3N j=1 (4.15) Ω(E, V, N) N! Ω(E, V, N) = 1 dqdp N! q V, H E (2πħ) 3N (6.2) 1 = N!(2πħ) 3N dq 1 dq 3N dp 1 dp 3N. (6.3) 1 q V 2m j p2 j E 1 V V N Ω(E, V, N) = N!(2πħ) 3N dp 1 dp 3N (6.4) 1 2m j p2 j E = V N (2mE) 3N/2 N!(2πħ) 3N dx 1 dx 3N. (6.5) j x2 j 1 p 2 j 2m (6.1) 40

46 x j p i / 2mE (6.6) 3N j=1 x 2 j 1 (6.7) 3N 1 3N n C n C n = π n/2 Γ(n/2 + 1) (6.8) Γ(x) 2 Γ(1) = 1, (6.9) Γ(1/2) = π, (6.10) Γ(x + 1) = xγ(x), (6.11) Γ(x + 1) 2πe x x x+1/2 as x. (6.12) x n (6.11) Γ(n + 1) = nγ(n) = n(n 1)Γ(n 1) = Γ(1) (6.9) Γ(n + 1) = n! (6.13) Γ(x + 1) n! x (6.12) (4.54) x (Stirling s formula) o(x) ln Γ(x + 1) x ln x x + o(x) as x. (6.14) (6.8) 3 C 3 = π3/2 Γ ( ) = π3/ Γ ( 1 = 2) 4π 3 2 (6.5) (6.8) Ω(E, V, N) = V N (2πmE) 3N/2 N!(2πħ) 3N Γ(3N/2 + 1) O(V ) [ ( S(E, V, N) = k B N ln V E 3/2 /N 5/2) ] + (6.15) (6.16) (6.17) m, ħ 3 3 E 0, V 0, N 0 S 0 S(E 0, V 0, N 0 ) [ ( S = N ) 3/2 ( ) ( ) ] 5/2 E V N0 S 0 + k B N ln. (6.18) N 0 N n 1 n x 2 j = 1 2 j=1 2 E 0 V 0 41

47 5.4 k B R ( 8.31J/K mol) R = N A k B (6.19) N A ( mol 1 ) N mol ( ) (N/N A )mol 4.6 (4.37) k B 6.1 (6.18) E = 3 2 k BNT, (6.20) P V = k B NT. (6.21) k B 5.2 r n C n r n ϵ (1 ϵ)r C n [(1 ϵ)r] n ϵr C n r n C n [(1 ϵ)r] n C n r n C n [(1 ϵ)r] n 1 C n [(1 ϵ)r] n = 1. (6.22) (1 ϵ) n ϵr n 1 n = 3N ϵ 1 1 (6.4) N N 5.2 S E E Ω E E E Ω 6.2 n (6.1) H = 3N j=1 p 2 j 2m k,l (k l) u(r k r l ) (6.23) u(r k r l ) k l I u

48 u u(r k r l ) r k r l 3 [1 u ] k B T (6.24) u k B ln W u = 0 u u = S(E, V, N) E, V, N W (E, V, N) ( 6.5 N! N!

49 9.2 (a) (b) p A r A p B r B (a) (b) N! ( (a) (b) N! N! N 2N, 3N, 4N, M N M (6.25) M N M!/N!(M N)! U, V, N N! M!/N!(M N)! (6.26) N M M N ln 44

50 (7.1) A (A.8) E, V, N 12 (entropy representation) E, V, N S = S(E, V, N) E, V, N E U UV N (energy representation) S, V, N E = E(S, V, N) S, V, N SV N (Legendre transform) A 11 E, V, N S, V, N S T T, V, N T V N E(S, V, N) S E/ S = T (2.17) F (T, V, N) [ E(S, V, N) ST ] (T, V, N). (7.1) F (T, V, N) (Helmholtz) T, V, N F = F (T, V, N) T, V, N T, V, N T, V, N E, V, N 45

51 T, V, N F = F (T, V, N) F (T, V, N) T (inverse Legendre transform) E(S, V, N) SV N S = S(E, V, N) E = E(S, V, N) (7.2) F = F (T, V, N) F (T, V, N) E(S, V, N) S(E, V, N) S(E, V, N) S(E, V, N) (fundamental relation) T V N (heat bath) UV N T V N T F i F (i) (T, X (i) 1, X(i) 2,, X(i) t i ) (7.3) F F = min F (T, {X (i) {X (i) 1,X(i) 2, } 1, X(i) 2, } i). (7.4) i T T V N F (T, V, N) F = P V V E(S, V, N) S V (Gibbs) G(T, P, N) [ F (T, V, N) + V P ] (T, P, N) (7.5) = [ E(S, V, N) ST + V P ] (T, P, N) (7.6) G = G(T, P, N) T P N G(T, P, N) E(S, V, N) S(E, V, N) 46

52 E, V, N S, V, N E, V, N S(E, V, N) E(S, V, N) EV N SV N 7.2 E, V, N V, N V, N λ = 1, 2, 3, λ V, N, λ E V,N,λ E V,N,λ [V, N, λ ]. (7.7) E V,N,λ E λ λ E V,N,λ = E V,N,λ λ, λ E E b + A A A 1 E E V, N E V,N,λ E λ + E t E t = E + E b (7.8) A λ Θ λ Θ λ = W b(e t E λ ) W t (E t ) (7.9) Θ λ λ λ Θ λ W b (E t E λ ) (7.10) 1 47

53 S b = k B ln W b + o(v b ) o(v b ) Θ λ exp [S b (E t E λ )/k B ]. (7.11) E t E λ S b (E t E λ ) E λ 2 S b (E t E λ ) = S b (E t ) S b(e t )E λ S b (E t )E 2 λ + (7.12) E t E b E b T b = T ( S b(e t ) S b(e b ) = 1 T b = 1 T. (7.13) E t E b E b T b S b (E t ) S b (E b ) = 1 = 0. (7.14) E b T b (7.12) + S b (E t E λ ) = S b (E t ) E λ /T (7.11) λ Θ λ e E λ/k BT = e βe λ (7.15) β 1/k B T (7.16) B (inverse temperature) Θ λ Z λ e βe λ (7.17) Θ λ λ Θ λ = 1 Θ λ = 1 Z e βe λ (7.18) (canonical distribution) T Z β T E λ V, N E λ E V,N,λ Z T, V, N Z(T, V, N) Z (partition function) Θ λ T, V, N Θ λ (T, V, N) 4.5 Z = e βh(q,p) dqdp (2πħ) f (7.19) q V V, N V q N f = 3N V, N Θ λ (canonical ensemble) Θ λ T T, V, N ens(t, V, N) T T, V, N ens(t, V, N) E, X 1,, X t X 1,, X t X 2 S b (E t E λ, V b, N b ) = V b s b (E t/v b E λ /V b, N b /V b ) s b E λ /V b V b /V 48

54 E, X X X 1,, X t E X λ E X,λ [X, λ ] (7.20) (partition function) Z(T, X) λ exp ( βe X,λ ) (β 1/k B T ) (7.21) X λ Θ λ (T, X) = 1 Z(T, X) exp ( βe X,λ) (7.22) (canonical ensemble) (canonical distribution) ens(t, X) T T, X ens(t, X) A + T, X 7.3 Θ λ e βe λ E λ (E E, E] Θ(E E, E] Ω E Θ(E E, E] = 1 Ω e βe E (7.23) Z E Ω E E E e βe Θ(E E, E] E ( E ) S = k B ln ( Ω E E) Θ(E E, E] = 1 Z e βe e S/k B = 1 Z exp [ 1 k B T = 1 Z exp [ V k B T ] {E T S(E, V, )} )}] { E V T s ( E V, (7.24) (7.25) (7.25) { } E ( E ) E/V E /V { } V (V/k B T ){ } 3 exp[ (V/k B T ){ }] Θ(E E, E] E/V = E /V 3 V V N 49

55 V V E/V E/V E /V E = E + o(1) as V. (7.26) V V E = E + o(v ) as V. (7.27) E = E (7.28) E E E E E E V T s ( E V, ) E S E {E T S(E, V, )} = 0 E 1 T E = 0 E S E = 1 T E E T E T T E, V, T (E, V, ) (7.29) T (E, V, ) = T (E, V, ) (7.30) T (E, V, ) E E = E (7.31) 7.7 T (E, V, ) E = [E ] E Z Z = λ = = = e βe λ βe Ω e E de 1 e βe e S/kB de E 1 exp E [ 1 {E T S(E, V, )} k B T ] de (7.32) exp[ { }/k B T ] { } { } E T S(E, V, ) E T S(E, V, ) 5 E {E T S(E, V, )} E=E = 0 (7.33) 4 5 S(E, V, ) E T S(E, V, ) T, V, N E E E 50

56 (E E ) 1 E T S(E, V, ) = E T S(E, V, ) 1 2! T 2 E 2 S(E, V, ) E=E (E E ) 2 + (7.34) 2 2 E 2 S(E, V, ) E=E = E 1 T (E, V, ) E=E 1 = T (E, V, ) 2 1 = T (E, V, ) 2 C V (E, V, ) E T (E, V, ) E=E (7.35) C V = [ E T (E, V, )] 1 C V O(V ) 13.1 C V O(V ) T O(1) (7.35) O(1/V ) (7.34) 2 O(1/V ) (E E ) 2 (7.32) V 2 E E O( V ) (7.36) n 1 n S(E, V, ) = En V n (E/V ) n V s(e/v, ) = 1 n V n 1 u n s(u, ) = O(1/V n 1 ) (7.37) (7.36) E n = O(1/V n 1 ) O(V n/2 ) = O(1/V n/2 1 ) (7.38) n 3 2 [ ] [ ] 1 Z = exp k B T {E 1 1 T S(E, V, )} exp E 2k B T 2 (E E ) 2 de (7.39) C V E E e ax2 dx = π a (a > 0) (7.40) [ ] 1 Z = exp k B T {E T T S(E, V, )} 2πkB C V. (7.41) E E E ln Z = 1 ( T ) {E T S(E, V, )} + ln 2πkB C V k B T E (7.42) { } F O(V ) o(v ) F = k B T ln Z + o(v ) (7.43) F T, V, N T, X) Z F 51

57 F (T, X) = k B T ln Z(T, X) + o(v ). (7.44) T V N 7.1 F Z Z W Z F Θ λ = e β(e λ F ). (7.45) o(v ) (7.32) (7.41) [ 1 exp k B T = exp ] {E T S(E, V, )} [ 1 k B T {E T S(E, V, )} de (7.46) ] [ o(v ) ] (7.47) (7.46) (7.46) 7.5 Z Z W Z W n 1, n 2,, n N n (7.48) n j n j n 7.2 λ n j j j j = 1, 2,, N 4.8 j V/γ N n E n = ε j (n j ) (7.49) j 6 52

58 j ε j ε j n j ε j (n j ) (j j n j j Z = e βen (7.50) n = e β[ε1(n1)+ε2(n2)+ +εn(nn)] (7.51) n 1 n 2 n N { } { } { } = e βε1(n1) e βε2(n2) e βεn(nn) (7.52) n 1 n 2 n N 7 z j n j e βεj(nj) (7.53) Z = j z j, ln Z = j ln z j (7.54) z j j j Z z 1 = z 2 = ( z) Z = z N, ln Z = N ln z (7.55) n j n Z M M = N! (7.55) Z = 1 M zn, ln Z = N ln z ln M (7.56) Z 4.8 j z 1 = z 2 = ( z) N = V/γ F (T, V ) = k BT V γ z = n=0 ε j (n j ) = εn j (7.57) e βεn 1 =. (7.58) 1 e βε ln Z = V γ ln 1 1 e βε = V γ ln ( 1 e βε). (7.59) ( ) ln 1 e ε/k BT = k BT V γ ( ) ln e ε/kbt 1 εv γ. (7.60) 4.8 E = E(S, V ) S F (T, V ) n j 4.5 n j 53

59 7.5.3 Z 6.1 S(E, V, N) Z F (T, V, N) j q j, p j j (j = 1, 2,, 3N) ( j = 1, 2,, N) r j p j r j H = N j=1 p 2 j 2m (7.61) M = N! (7.56) Z = 1 N! zn (7.62) ] z = exp [ β p2 d 3 rd 3 p r V 2m (2πħ) 3 (7.63) [ ] [ ] [ ] V = (2πħ) 3 exp β p2 x dp x exp β p2 y dp y exp β p2 z dp z. (7.64) 2m 2m 2m (7.40) z = V (2πħ) 3 ( ) 3 2mπ. (7.65) β F (T, V, N) = k B T (N ln z ln N!) O(N) (7.66) [ = k B T N ln V N + 3 ] 2 ln(2πk BmT ) ln(2πħ). (7.67) T 0, V 0, N 0 F F (T 0, V 0, N 0 ) F 0 [ ( F (T, V, N) = NT T F 0 k B T N ln N 0 T 0 T 0 ) 3/2 ( V V 0 ) ( ) ] N0. (7.68) N 6.1 E = E(S, V, N) S F (T, V, N) E T, V, N F (T ± 0, V, N) S(T ± 0, V, N) = T (7.69) S(T ± 0, V, N) E(T ± 0, V, N) = F (T, V, N) + S(T ± 0, V, N)T (7.70) E(T ± 0, V, N) 8 Z(T, V, N) Z(T, V, N) E = 1 E λ e βe λ (7.71) Z λ 8 T ±

60 Z (7.17) ln Z(β ± 0, V, N) E(T ± 0, V, N) = β (7.72) 9 Z β, V, N ln Z β = 1/k B T 1 E(T ± 0, V, N) S(E, V, N) B = S/ E (= 1/T ) S(E, V, N) E F F F(B, V, N) [S(E, V, N) EB](B, V, N). (7.73) (Massieu function) F(B ± 0, V, N) E(B ± 0, V, N) = B (7.72) B = k B β Z B, V, N (7.74) F(B, V, N) = k B ln Z(B, V, N) (7.75) 7.4 S(E, V, N) = k B ln W (E, V, N) ( k B k B = 1 SI k B β = B F(B, V, N) E F(B, V, N) Z(B, V, N) (2.22) Π V (= P/T ) Π V (B, V, N) = F(B, V, N) V = ln Z(B, V, N). (7.76) V T (E, V, ) E E E (7.31) E T V N T V N UV 9 Z (7.17) λ / β e βe λ 55

61 8 UV N T V N (equivalence of ensembles) E N E b N b + E, N V E, N 7.2 (7.7) E V,N,λ E X,λ E N,λ E, V, N + A A 7.2 N λ Θ N,λ Θ N,λ W b (E t E N,λ, N t N) (8.1) Θ N,λ N, λ N, λ S b = k B ln W b + o(v b ) o(v b ) Θ N,λ exp [S b (E t E N,λ, N t N)/k B ] (8.2) E t E N,λ, N t N S b (E t E N,λ, N t N) E N,λ, N 2 S b (E t E N,λ, N t N) = S b (E t, N t ) S b(e t, N t ) E N,λ S b(e t, N t ) N (8.3) E t N t S b (E t, N t ) S b(e b, N b ) = 1 = 1 E t E b T b T, (8.4) S b (E t, N t ) S b(e b, N b ) = µ b = µ N t N b T b T. (8.5) p

62 T, µ T b µ b 8 S b (E t E N,λ, N t N) S b (E t, N t ) E N,λ T + µn T (8.2) N, λ Θ N,λ e β(e N,λ µn) (8.6) (8.7) Θ N,λ Ξ N,λ e β(e N,λ µn) (8.8) Θ N,λ N,λ Θ N,λ = 1 Θ N,λ = 1 Ξ e β(e N,λ µn) (8.9) (grand canonical distribution) (8.8) Ξ e β(en,λ µn) T, µ E N,λ V E N,λ E V,N,λ Ξ T, V, µ Ξ(T, V, µ) Ξ (grand partition function) Θ N,λ T, V, µ Θ N,λ (T, V, µ) 4.5 Ξ(T, V, µ) N q V e β[h(q,p) µn)] dqdp (2πħ) f. (8.10) V q V Θ N,λ (grand canonical ensemble) Θ N,λ T, µ T, V, µ ens(t, V, µ) T µ T, V, µ ens(t, V, µ) E, V, N, X 3, X t E, N V, X 3,, X t X E, N, X E N, X λ E N,X,λ [N, X, λ ] (8.11) (grand partition function) Ξ(T, µ, X ) exp [ β (E N,X,λ µn)] (β 1/k B T ) (8.12) N,λ X N, λ Θ N,λ (T, µ, X ) = 1 Ξ(T, µ, X ) exp [ β (E N,X,λ µn)] (8.13) (grand canonical ensemble) (grand canonical distribution) ens(t, µ, X ) 57

63 T µ T, µ, X ens(t, µ, X ) E, V, N A T, µ, X Θ N,λ (E E, E] (N N, N] Θ((E E, E], (N N, N]) 7.3 Θ((E E, E], (N N, N]) E, N E, N E, N E, N E, N V E = [E ] N = [N ] E Z 7.4 Ξ 7.4 Ξ (8.8) N dn (8.14) 3 E 7.4 N Ξ = N,λ β(e µn) Ω(E, N) = e dedn E [ ] 1 1 = exp {E T S(E, V, N ) µn} dedn E k B T [ ] 1 = exp k B T {E T S(E, V, N, ) µn } [ o(v ) ] (8.15) e β(e N,λ µn) 2 (7.47) E, N E, N ln Ξ = 1 {E T S(E, V, N ) µn} + o(v ) (8.16) k B T { } E(S, V, N, ) S, N J(T, V, µ, ) [E(S, V, N, ) ST Nµ] (T, V, µ, ) (8.17) J(T, µ, X ) = k B T ln Ξ(T, µ, X ) + o(v ) (8.18) 3 f(, N, ) f(, N, ) N N N = 1 f(, N, ) = N N f(, N, ) N N 4.7 N N N = 1 o(v ) f(, N, ) = N f(, N, )dn 58

64 J T, µ, X (= T, µ, V, ) Ξ J T V µ J Z Ξ Z Ξ j N M = N! ξ j e β(εj(nj) µ) (8.19) n j Ξ Ξ = N 1 N! N ξ j. (8.20) j=1 Ξ J Θ N,λ = e β(en,λ µn J). (8.21) o(v ) Ξ Z Ξ(T, V, µ) = N Z(T, V, N)e βµn. (8.22) 8.1 J(T, V, µ) J(T, V, µ) F (T, V, N), U(S, V, N) N 8.3 UV N T V N T V µ S = S(E, V, N) = k B ln W (E, V, N) : E = E(S, V, N) F = F (T, V, N) = k B T ln Z(T, V, N) : (8.23) J = J(T, V, µ) = k B T ln Ξ(T, V, µ) : (equivalence of ensembles) 7.6 B = S/ E 59

65 (= 1/T ), Π N = S/ N (= µ/t ) S(E, V, N) F (7.73) F(B, V, N) J J(B, V, Π N ) [F(B, V, N) NΠ N ](B, V, Π N ) (8.24) S = S(E, V, N) = k B ln W (E, V, N) : F = F(B, V, N) = k B ln Z(B, V, N) : (8.25) J = J(B, V, Π N ) = k B ln Ξ(B, V, Π N ) : 8.2 F, Θ λ e βe λ Θ λ,n e β(e λ,n µn) βe = (1/k B )BE B E βµn = (1/k B )(µ/t )N = (1/k B )Π N N Π N N N X k Π k Θ exp 1 k B Π k X k (8.26) k E, V Θ λ,v e 1 k B (BE λ,v +Π V V ) = e β(e λ,v +P V ) (8.27) T -P Z Ξ Λ exp 1 k B k Π k X k (8.28) 60

66 k B T ln Λ = T S + T Π k X k (8.29) k T S S V (maximal canonical 61

67 9 9.1 N f = 3N (6.23) H(q, p) = f j=1 p 2 j 2m j + U(q) (9.1) U(q) q 1,, q f j m 1 = m 2 = m 3, m 4 = m 5 = m 6, [7] j p2 j /2m j 8.3 T j p2 j /2m j g(p) 6.1 M g(p) = 1 Z 1 g(p)e βh(q,p) dqdp M q V (2πħ) 3N 1 g(p)e β j p2 j /2mj e βu(q) dqdp M q V (2πħ) 3N = 1 e β j p2 j /2mj e βu(q) dqdp M q V (2πħ) 3N g(p)e β j p2 j /2mj dp e βu(q) dq q V = e β j p2 j /2m j dp e βu(q) dq q V g(p)e β j p2 j /2mj dp = e β. (9.2) j p2 j /2mj dp M g(p) U(q) g(p) g(p) U(q) = 0 62

68 g(p) = j p2 j /2m j j p 2 j 2m j = = = p 2 j e β j p2 j /2m j dp 1 dp f 2m j j e β j p2 j /2mj dp 1 dp f p 2 1 e βp 2 1 /2m 1 dp 1 e β f j=2 p2 j /2m j dp 2 dp f 2m 1 e βp2 1 /2m1 dp 1 e β + [1 2 ] + f j=2 p2 j /2mj dp 2 dp f p 2 1 e βp 2 1 /2m 1 p 2 2 dp 1 e βp 2 2 /2m 2 dp 2 2m 1 2m (9.3) e βp2 1 /2m 1 dp 1 e βp2 2 /2m 2 dp 2 (7.40) (7.40) a x 2 e ax2 dx = 1 π 2a a (a > 0) (9.4) j p 2 j 2m j = j f 1 2β = f 2 k BT. (9.5) 1 = 1 2 k BT. (9.6) S E U(q) = 0 E E E p 2 j E = = 3N 2m j 2 k BT. (9.7) j C V C V = ( ) E = 3 T V,N 2 k BN. (9.8) 63

69 c V c V = C V N/N A = 3 2 k BN A = 3 2 R (9.9) c V 3 T +0 c V T +0 c V 0 c V f 9.3 p (= p 1,, p f ) θ(p) θ(p)dp θ(p 1,, p f )dp 1 dp f p [p 1, p 1 + dp 1 ),, [p f, p f + dp f ) (9.10) (9.2) g(p) p p θ(p) = e β j p2 j /2mj e β = β e βp 2 j /2m j (9.11) j p2 j /2m j 2πm dp j j p j β θ j (p j ) = e βp 2 j /2m j 1 = 2πm j 2πmj k B T e p2 j /2m jk B T (9.12) v j = p j /m j θ v j (v j) θ v j (v j)dv j = θ j (p j )dp j θj v (v j ) = θ j (m j v j )m j mj = 2πk B T e m jv 2 j /2k BT (9.13) m j j v x, v y, v z 3 θ v (v) θ v (v)dv x dv y dv z v [v x, v x + dv x ), [v y, v y + dv y ), [v z, v z + dv z ) (9.14) θ v (v) = = m 2 2πk B T e mv x m /2kBT 2 2πk B T e mv y m /2kBT 2 2πk B T e mv z /2k BT ( ) 3/2 m e 1 2 m v 2 /k B T. (9.15) 2πk B T J. C. Maxwell 64

70 9.4 ) (9.15) (9.15) f[v, v + dv) N k=1 ( ) Θ v (k) [v, v + dv) (9.16) Θ ( v (k) [v, v + dv) ) k v (k) [v, v+dv) ( [v x, v x +dv x ), [v y, v y +dv y ), [v z, v z + dv z )) 1 0 f[v, v + dv) v f[v, v + dv) v, dv Θ ( v (k) [v, v + dv) ) k f[v, v + dv) f[v, v + dv) f[v, v + dv) f[v, v + dv) θ v (v)dv x dv y dv z 1 1 (9.15) 65

71 (9.1) U(q) (= U(q 1,, q f )) U int (q) U wall (q) U(q) = U int (q) + U wall (q). (9.17) q (= q 1,, q f ) p (= p 1,, p f ) (virial) V V(t) (3.6), (3.7) d dt V = j = j f q j (t)p j (t). (9.18) j=1 q j p j + j p j m j p j j q j ṗ j (9.19) q j U q j (9.20) (9.17) d dt V = 2 j p 2 j 2m j j q j U int q j j q j U wall q j (9.21) f(t) (long-time average) 1 τ f lim f(t)dt (9.22) τ τ 0 f (9.21) d dt V = lim 1 τ τ τ 0 dv dt dt (9.21) V(τ) V(0) = lim τ τ = 0 ( V ) (9.23) 2 j p 2 j 2m j = j q j U int q j + j q j U wall q j (9.24) (virial theorem) V V = (9.25)

72 (9.24) 2 j p 2 j 2m j = j q j U int q j + j q j U wall q j. (9.26) k r (k) r (1) = (q 1, q 2, q 3 ) N k=1 r (k) U wall r (k) = N k=1 r (k) f (k) wall (9.27) f (k) wall U wall r (k) (9.28) k V k ( V ) (9.27) = k ( V ) r (k) f (k) wall (9.29) V r da da da (9.29) r (k) r = r df k ( da) k ( da) f (k) wall f (k) wall = r df (9.30) (9.31) df da P df = P da (9.32) P r df (9.30) = P r da. (9.33) (9.29) da (9.29) = P V r da (9.34) 3 3 r = 3 P r da = P ( r)d 3 r = 3P d 3 r = 3P V (9.36) V (9.27) V V (9.26) = 3P V (9.37) 3 a(r) V a(r) da = ( a(r))d 3 r (9.35) V 67

73 (9.26) 3Nk B T (9.26) P V = Nk B T 1 U int q j 3 q j j (9.38) (virial theorem) U wall L U wall O(L 2 ) = o(v ) U wall q j O(L) O(L) O(L 2 ) = O(V ) 3P V df da (9.32) df i = σ ix da x + σ iy da y + σ iz da z (i = x, y, z) (9.39) 3 3 σ ii df da 2 (stress tensor) (9.32) σ ii σ ii = P δ ii (9.38) U int = 0 P V = Nk B T U int 0 P V = Nk B T U int 0 (9.38) P, T U int (q) l λ U int (λq 1,, λq f ) = λ l U int (q 1,, q f ) (9.41) q 1,, q f U int r (k) r (k ) l (9.42) k,k 4 l = l 5 d l < d (9.43) l l l < d l = l 68

74 6 (9.43) 12.5 j q j U int q j = lu int (9.44) (9.38) 7 E = j P V = Nk B T l 3 U int. (9.45) p 2 j 2m j P V = Nk B T l 3 + U int = 3 2 Nk BT + U int (9.46) ( E 3 ) 2 Nk BT. (9.47) P, V, N, T, E E (E (3/2)Nk B T ) ψ(r, t) ϕ(r, t) H int H int = H int (r, t)d 3 r (9.48) H int (r, t) (r, t) ψ, ϕ ψ ϕ ( U(q) = U int (q)+u wall (q) 2 q 1 U(q) q1, 0, qf 0 Q j q j qj 0 (9.49) 6 7 E U wall o(v ) q j U wall O(V ) 69

75 U U U 0 U = Q 1,, Q f 2 + U 0 (9.50) U(q) Q 1 = = Q f = 0 q 1 = q 0 1,, q f = q 0 f U 0 U U Q 1,, Q f 2 q 0 1,, q 0 f U wall (q) U int (q) 2 (9.26) l = 2 (9.43) (9.26) = q j Q j U U q j = Q j + U qj 0 (9.51) q j j Q j j Q j j U 1 U Q j 2 2 U 2 Q j q j (Q j = 0) 8 j U q j = 2 U. (9.52) q j (9.26) p 2 j U = = 3N 2m j 2 k BT. (9.53) j E = 3Nk B T. (9.54) 2 (9.53) (9.43) U(q) 2 U(q) 2 (9.54) V E 3Nk B T c V c V = N ( ) A (3NkB T ) = 3k B N A = 3R (9.55) N T 2 (Dulong-Petit law) c V 70

76 10 (classical statistical mechanics) 9 (quantum statistical mechanics) z 1, z 2, z 1 i 2 z i (10.1)... ψ n ψ, ψ φ 1, φ 2 1, 2 i 2 1 = 0, 2 = 1 + i.. (10.2) 3 φ φ φ (10.3) 1 71

77 1 2 ψ 2 2 φ, φ H 1 φ 1, φ 2,, φ d H H H H (dimension) dim H d dim H 3 ( ) ( i 0 ) = ( 0 i ) (10.4) ;  ˆB ; (Hamiltonian) Ĥ   = a (a ) (10.5) a   a Ĥ V (single-particle state) (many-particle state) H 1 H many 4 dim H many dim H 1 H 1 H many dim H overcomplete 4 72

78 k 1 r z (ħ/2)σ z (σ z = ±1) 1 k (r, σ z ) ν ψ ν ( H 1 ) ν = 0, 1, 2, ν (r, σ z ) ψ 0 (10.7) ψ 0 ψ 0 ψ 0 ) ψ 1 ψ 0 ψ 1 ψ 2 ψ 0, ψ 1, ψ 2, ψ 0, ψ 1, H 1 dim H 1 ψ 0, ψ 1, { ψ ν } H 1 ψ { ψ ν } ψ = ν c ν ψ ν (10.6) c ν ψ { ψ ν } ψ(x) x x x { x } ψ = ψ(x) x dx. (10.7) ψ(x) ψ ψ ψ(x) ψ(x) ψ(x) dim H 1 V dim H 1 V. (10.8) O(1) dim H 1 V ( 10.1 L x L y L z O(1) V = L x L y L z ψ ν n ν n ν (= 0, 1, 2, ) ψ ν ν (10.9) n ν ψ ν n 0, n 1, n 2, n n n = (n 0, n 1, n 2, ) (10.10) n = (9, 7, 4, 1, 0, 0, ) n 0, n 1, n 2, ψ 0, ψ 1, ψ 2, n 0, n 1, n 2, 73

79 n 0, n 1, n 2, N = ν n ν (10.11) 9, 7, 4, 1, 0, 0, N = = 21 n 0 = 9, n 1 = 7, n 0, n 1, n 2, n 0, n 1, n 2, n 0 = n 0, n 1 = n 1, n 2 = n 2, 5 n 0, n 1, n 2, n 0, n 1, n 2, 1 0, 0, 0, N N = 0 0, 0, 0, n 0, n 1, n 2, { n 0, n 1, n 2, } { n } { n } H many Ψ { n } Ψ = c n n = c n0n 1n 2 n 0, n 1, n 2, (10.12) n n 0,n 1,n 2, c n (= c n0,n 1,n 2, ) Ψ { n } 6 n 0, n 1, n 2, H many dim H many dim H many = [n 0 ] [n 1 ] [n 2 ] = [ n ν ] dim H1. (10.13) n ν 2 V dim H 1 V dim H many exp( V ) (10.14) dim H many V 7 (10.12) N ν n ν c n0 n 1 n 2 n 0, n 1, n 2, N = 1 1, 0, 0, H 1 ψ 0 H many H 1 H many H 1 H many H many N ν n ν N n 0, n 1, n 2, H many H 1 H many dim H many H 1 H many n H many 5 n 0 n 0 n 0, n 1, n 2, n 0, n 1, n 2, n H many 1, 1, 0, 0, 0, [ψ 0 (r 1 )ψ 1 (r 2 ) ψ 0 (r 2 )ψ 1 (r 1 )] / 2 Slater determinant 7 n ν V O(V ) 74

80 n ν n ν n ν = 0, 1 1 (10.15) n ν = 0, 1, 2, (10.16) ħ 1/2 n ν = 0, 1 (fermion) n ν = 0, 1, 2, (boson) ħ/2 ħ 1 (n ν = 1) (n ν = 0) 2 (n ν 2) 8 ψ ν (free particles) Ĥ 8 ψ ν 2 75

81 ψ ν ε ν ν = 0 ε 0 ε ν ψ ν n E n E n = ν ε ν n ν (10.17) W Z Ξ Ξ 8.1 N λ E N,λ n N (= ν n ν) n n 8.1 (N, λ) λ,n = N λ = (10.18) n 0 n 2 Ξ n Ξ = n n 1 e β(e n µn) (10.19) E n (10.17) 8.1 Ξ (10.17), (10.18) Ξ = n 0 = e β(ε0 µ)n0 e β(ε1 µ)n1 n 1 ( ) ( ) e β(ε0 µ)n0 e β(ε1 µ)n1 (10.20) n 0 n 1 ξ ν n ν e β(εν µ)nν = 1 + e β(ε ν µ) (fermion) 1 1 e β(ε ν µ) (boson) (10.21) Ξ = ν ξ ν (10.22) ε ν ξ ν (10.21) (10.22) Ξ ψ ν n ν n ν 9 n ν Ξ T, V, µ, n ν n ν ε ν 9 n ν

82 n 0 (10.20) n 0 = 1 n 0 e β(e n µn) Ξ n ( ) ( ) 1 = n 0 e β(ε0 µ)n0 e β(ε1 µ)n1 ν ξ ν n 0 = 1 n 0 e β(ε0 µ)n0 ξ 0 = n 0 n n 1 + e β(ε 0 e β(ε 0 µ)n 0 0 µ) n 0 =0 (1 ) e β(ε 0 µ) n 0 e β(ε 0 µ)n 0 n 0 =0 (fermion) (boson) (10.23) 2 3 (10.20) e xn 1 = 1 e x (10.24) n=0 x 10 ν 0 n ν ν n ν = 1 e β(ε ν µ) ± 1 (+: fermion, : boson). (10.25) 1 n ν = e β(εν µ) (10.26) e β(ε ν µ) 1 (free fermions) (free bosons) n ν = 0, 1 0 n ν 1 e β(ε ν µ) n ν 0 n ν 0 ε ν µ e β(ε ν µ) 1 n ν ε ν µ ε ν < µ n ν < 0 n ν 0 ε ν µ 0 for all ν µ µ ε 0 (= min ν ε ν ) (10.27) µ (10.25) E N n ν E N = n ν 10 77

83 E E = (10.17) E n = ν ε ν n ν = ν ε ν e β(ε ν µ) ± 1 (+: fermion, : boson) (10.28) N N = (10.11) N = n ν ν = ν 1 e β(ε ν µ) ± 1 (+: fermion, : boson) (10.29) N T, V, µ V ε ν µ N T, V T, V, µ T, V, N µ T, V, N e β(ε ν µ) 1 for all ν (10.30) n ν 1 for all ν (10.31) n ν n ν e β(ε ν µ) (10.32) (10.30) 10.4 N V ( ) 3/2 mkb T (10.33) N /V T 6.1 ħ n ν = f(ε ν ) (10.34) f(ε) = 1 e β(ε µ) + 1 T, µ ε (10.35) 0 f(ε) 1 (10.36) n ν 1 78

84 N /V T f(ε) T µ T +0 ε F lim µ (10.37) T +0 ε F (Fermi level) (Fermi energy) 11 βε (= ε/k B T ) /k B (10.38) ε F ε 0 βε = ε T (10.39) T F (ε F ε 0 )/k B (10.40) (degenerate temperature) T F 10 4 K n T F K T +0 lim f(ε) = T +0 lim β + 1 e β(ε ε F ) + 1 = 1 (ε < ε F ) 1/2 (ε = ε F ) 0 (ε > ε F ) (10.41) (step function) 0 n ν 1 n ν = 1 1 n ν = 1 n ν = 0 1 n ν = 0 n ν = 1 for ε ν < ε F, n ν = 0 for ε ν > ε F Ψ = 1, 1,, 1, 0, 0, 1 0 ε ν ε F (10.42) W = 1 W = O(1) S = [k B ln W O(V ) ] = 0 12 T 0 E 1 E 0 = exp[ β(e 1 E 0 )] 0 as T +0 (10.43) O(exp[ N]) lim lim T 0 N 10.2 (10.42) S N 0 (10.44) T T T F ( ) 2 T µ = ε F [O(1) ] (10.45) T F 11 µ 12 W = 2 S k B ln W O(V ) S = 0 W = O(V ) S = 0 S W 79

85 T T F µ ε F ε ε F k B T (Fermi degeneracy) T +0 S 0 T F T F µ T µ T T T F µ ε 0 µ k B T (10.30) T T F T T F T 300K T F 10 4 K n T F K 13 T 1K n ν = f (ε ν ) (10.46) f (ε) = 1 e β(ε µ) 1 (10.47) T, µ ε µ ε 0 f (ε) 0 N /V T f (ε) µ T ε 0 (10.33) µ e β(εν µ) 1 for all ν f (ε) µ ε 0 ε 0 µ k B T/2 n 0 = f (ε 0 ) 1/( e 1) f (ε) 1.5 n ν O(1) for all ν µ ε 0 µ ε 0 µ = o(1) e β(ε0 µ) 1 + β(ε 0 µ) = 1 + o(1) n 0 = 1/o(1) n 0 n 0 V N (10.48) ψ 0 (Bose-Einstein condensation) 13 80

86 T +0 n 0 N lim n 0 = N. (10.49) T +0 S = k B ln W = Ĥ = f i=1 ˆp 2 i 2m (f = 3N). (10.50) Ĥ1 Ĥ Ĥ 1 = 1 2m (ˆp2 x + ˆp 2 y + ˆp 2 z) (10.51) 4.3 Ĥ1 ( ) Ĥ 1 = ħ2 2 2m x y z 2 (10.52) ( ) ħ2 2 2m x y z 2 ψ(r) = εψ(r) (10.53) ε, ψ(r) ε, ψ(r) O(V ) L ψ(r + (L, 0, 0)) = ψ(r + (0, L, 0)) = ψ(r + (0, 0, L)) = ψ(r) for all r. (10.54) 14 k (σ = ±1) k,σ D(ε) 2 81

87 ε, ψ(r) ψ(r) ν = k ψ k (r) k = 2π L (n x, n y, n z ) (n x, n y, n z ) (10.55) ψ k (r) = 1 V e ik r (V = L 3 ) (10.56) L dx L dy L ε dz ψ k (r)ψ k (r) = δ k,k (10.57) ε k = ħ2 k 2 2m (k = k ) (10.58) ε k k ε k ε k ε 0 ε 0 = ε k=0 = N N (10.29) f(ε) N = k f(ε k ). (10.59) D(ε) D(ε)dε [ε ε + dε ] + o(v ). (10.60) D(ε) o(v ) (10.59) 1 N = D(ε) (10.55), (10.58) 0 f(ε)d(ε)dε. (10.61) D(ε) = (m)3/2 V 2 π2 ħ 3 ε (10.62) ( 5.2 Ω E D(ε) Ω E (5.22) N/V V D(ε) V N 10.3 (10.62) T +0 f µ ε F N = εf 0 D(ε)dε. (10.63) (10.62) ε N = 2 2m 3/2 3 ε V F D(ε F ) = 3 π 2 ħ 3 ε 3/2 F. (10.64) ε F (= lim T 0 µ) V, N 82

88 0 < T T F f(ε) O((T/T F ) 2 ) (10.77) g(ε) D(ε) k B = 1 N µ 0 D(ε)dε + π2 6 D (µ)t 2. (10.65) N, V (10.63) εf µ = ε F δ δ ( δ ε F ) µ D(ε)dε π2 6 D (µ)t 2. (10.66) µ ε F [ D(ε F )δ π2 6 D (ε F )T 2. (10.67) δ π2 D ( ) 2 (ε F ) 6D(ε F ) T 2 = π2 T 12 ε F. (10.68) ε F ( 1 π2 T 12 T F ) 2 ] when N/V = constant. (10.69) (10.45) δ ε F µ (10.61) 0 g(ε)f(ε)dε. (10.70) g(ε) f(ε) ε = µ f(ε) ε = µ 1 0 ε µ g(ε)f(ε)dε µ 0 0 g(ε)dε (10.71) f(ε) ε = µ [0, ) [0, µ] [µ, ) g(ε)f(ε)dε µ 0 0 g(ε)dε = = µ µ g(ε)f(ε)dε µ 0 g(ε) e β(ε µ) + 1 dε g(ε)(1 f(ε))dε µ 0 g(ε) dε (10.72) e β(ε µ) f(ε) = 1/[e β(ε µ) +1] x β(ε µ), x β(ε µ) k B = 1 T 0 g(µ + T x) βµ e x + 1 dx T 0 g(µ T x) e x dx. (10.73) + 1 T T F 2 βµ T F /T 1 1/(e x + 1) 1 for x 1 T 0 g(µ + T x) g(µ T x) e x dx. (10.74)

89 g(µ ± T x) x = 0 x = 0 (10.74) g(µ ± T x) = g(µ) ± g (µ)t x + g (µ) (T x) 2 ±. (10.75) 2 2g (µ)t 2 0 x e x + 1 dx + O(T 3 ). (10.76) π 2 /12 (10.71) 0 g(ε)f(ε)dε = µ 0 g(ε)dε + π2 6 g (µ)t 2 + O(T 3 ) (10.77) T 2 (k B T ) E E (10.28) f(ε) E = k ε k f(ε k ) (10.78) D(ε) 1 E = 0 εf(ε)d(ε)dε. (10.79) T +0 E E 0 T +0 f µ ε F E 0 = εf 0 εd(ε)dε. (10.80) D(ε) (10.62) 2(m) 3/2 V E 0 = 5 π 2 ħ 3 ε 5/2 F. (10.81) < T T F N (10.77) g(ε) = εd(ε) (10.79) N, V E E 0 + T 2 when N, V = constant. (10.82) E T c V = N ( ) A E T (10.83) N T T T 0 c V 0 0 < T T F N,V µ k B T ε µ + k B T (10.84) ε < µ k B T T (Fermi degeneracy) c V 84

90

91 E, V, N N V E A ens(e, V, N) E, V, N 1 S = ln W (E, V, N) W (E, V, N) ens(e, V, N) H H pot H kin H(q, p) = H pot (q) + H kin (p). (11.1) W (E, V, N) E H(q, p) H pot (q) H kin (p) E H(q, p) E E = [ ] + [ ] F = E T Ising H = J σ r σ r+eα hµ σ r. (11.3) r α=x,y,z r 1 86

92 J (> 0) σ r = ±1 r e x, e y, e z x, y, z h N m = 1 N µ σ r (11.4) r m = 1 N µ r σ r (11.5) H σ r, σ r+eα σ r = m /µ + δσ r (11.6) σ r+eα = m /µ + δσ r+eα (11.7) δσ r δσ r+eα H J [ ( m ) 2 ( ) ( ) ] m m + δσ r + δσ r+eα hµ σ r (11.8) µ µ µ r α=x,y,z r = 6J m σ r hµ ( ) 2 m σ r + 3J N. (11.9) µ µ r r z 6 z, 3 z/2 Z(T, N, h) [ ( Z(T, N, h) = exp β 6J m ) ( ) 2 m µ + hµ σ r 3βJ N] (11.10) µ possible states r { [ ( = exp β 6J m ) ( ) ]} 2 N m µ + hµ σ 3βJ (11.11) µ σ=±1 [ ( ) ] 2 { [ ( m = exp 3βJN 2 cosh β 6J m )]} N µ µ + hµ. (11.12) ( ) 2 { [( m F = 3JN k B T N ln 2 cosh 6J m ) ]} µ µ + hµ /k B T m h h N m [( m = µ tanh 6J m ) ] µ + hµ /k B T (11.13)

93 E, X 1,, X t H H staggered magnetization E, X 1,, X t E, X 1,, X t 1 DC ens(e, X 1,, X 1 88

94 Ĥ e βĥ/z  A  Ĥ Tr [Z 1 e βĥâ] (13.1)    Ĥ (13.2) A T = β2 β T T = β β (13.3) β A β = β  Ĥ = Ĥ Ĥ Ĥ Ĥ  Ĥ = Ĥ  Ĥ (13.4) N ( ) S C V,N T T ( ) E = T T A T = T T  Ĥ = 1 T Ĥ  Ĥ (13.5) T T Ĥ = 1 T Ĥ Ĥ (13.6) V,N V,N (13.7) = T Ĥ Ĥ = 1 T 2 ( Ĥ)2 Ĥ (13.8) 13.2 C V,N (13.7) (13.8) N C P,N C P,N T ( ) S T P,N ( ) E T P,N (13.9) 89