2,200 WEB * Ξ ( ) η ( ) DC 1.5 i

mailto: tomita@phys.h.kyoto-u.ac.jp 2002 2003 1.5 2003 2004 2005

2,200 WEB * Ξ ( ) η ( ) DC 1.5 i

1 1 1.1............................. 1 1.2..................................... 2 1.3................................. 2 1.4.................................. 5 1.5................................. 6 1.6................................ 8 1.7 -..................... 12 1.8 H *.......................... 13 2 18 2.1............................ 18 2.2.............................. 19 2.3................................. 20 2.4................................... 22 2.5.................................. 23 2.6 *..................................... 24 3 27 3.1 -..................... 27 3.2.............................. 29 3.3.............................. 31 3.4................................ 34 4 37 4.1................................. 37 4.2................................. 39 4.3........................... 41 4.4............................. 43 5 47 5.1 2................................. 47 5.2 2..................................... 50 5.3..................................... 52 5.4 *................................ 53 5.5 *............................. 57 ii

iii 6 61 6.1............................ 61 6.2 -............................. 61 6.3............................. 62 6.4................................ 64 6.5.............................. 67 7 69 7.1..................................... 69 7.2.................................. 71 7.3.............................. 72 7.4 - *........................ 73 7.5 *.......................... 77 8 82

1 1.8 H 1.1 1828 1829 20 One Point 1

2 1 1.2 V N V/N 1 0 3 v =(v x,v y,v z ) i v i i N i N i /N i N i = N (1.1) i dv x dv y dv z N i V = f(v x,v y,v z )dv x dv y dv z (1.2) f(v x,v y,v z )dv x dv y dv z = n (= N/V ) (1.3) f(v x,v y,v z ) v = v = v x2 + v y2 + v z2 1.3 V N

1.3. 3 1.1 x- da dt v i da v ix dt v i 1 v i N i /V ν ( ) νdadt = v ix >0 N i V v ix dt da (1.4) x- v x > 0 x 2 ν = 1 2 i N i V v ix = N 2V i N i N v ix = n 2 v x (1.5) n = N/V B = i N i N B(v i) (1.6) 2 v x = v y = v z = 1 2 v (1.7) ν = n 4 v (1.8) 1 v ix dt 2 v z = v 4π π 0 cos θ 2π sin θ dθ = v π/2 0 cos θ sin θ dθ = v 1 0 sin θ dsinθ = v 2

4 1 v i p i 2p ix P da dt (P da) dt = = N V v ix >0 i 2p ix N i V v ix dt da N i N p ixv ix dt da = n p x v x dt da (1.9) 2 p x v x = p y v y = p z v z = 1 3 p v (1.10) P = n 3 p v (1.11) m p = mv ɛ = mv 2 /2 ɛ ( ) 3 P = 1 V p v =2ɛ (1.12) u = U/V = Nɛ/V = nɛ P = 2 3 u (1.13) N N A RT = N V k = R/N A R N A T = nkt (1.14) u = 3 2 nkt U = 3 2 NkT (1.15) (1.13) H = U + PV = 5 3 U (1.16) 3 ɛ = cp P = u/3

1.4. 5 C V = 3 2 R, C P = 5 2 R, γ= C P /C V =5/3 (1.17) γ ( ) 1 2 mv x 2 = 1 2 mv y 2 = 1 2 mv z 2 = 1 3 1 2 mv2 = 1 kt (1.18) 2 kt/2 2 C V =5R/2 1 (300K) v 2 480m/s 1.4 a M Z M 2a 1.3 ν 4π(2a) 2 (1.8) M 4 2 v Z =4 2πa 2 nv (4πa 2 ) (1.20) 4 v v 1 V =(v +v 1 )/2 u = v 1 v mv 2 +mv 1 2 = MV 2 + µu 2 M =2m, µ = m/2 1.6 exp( mv 2 /2kT) exp( mv 1 2 /2kT) d 3 vd 3 v 1 =exp( MV 2 /2kT) exp( µu 2 /2kT) d 3 V d 3 u (1.19) u 1.3 u = 8kT/πµ = 16kT/πm = 2 v 2 (v 1 v) 2 = v 12 2v 1 v + v 2 =2v 2

6 1 Z Z 1 λ ( ) v λ = vz 1 = 1 4 2πa 2 n (1.21) V =22.4l N A =6.0 10 23 n = N A /V 2.7 10 25 m 3, n 1/3 3.3 10 9 m a 0.5 10 10 m 2.0 10 3 m/s λ 8.3 10 7 m, Z 2.4 10 9 s 1, 10 10 10 33 2 T P A ν n = P/kT v 1.5 A

1.5. 7 1.2 w v ix (> 0) (v ix +2w) w v ix 1 2 m(v ix +2w) 2 1 2 mv ix 2 2mwv ix =2wp ix (1.22) v ix V/A dt ( Z 1 ) v ix dt 2V/A 5 dt v i dɛ i =2wp ix v ixdt 2V/A = wadt V p ix v ix (1.23) wadt dv d W = i ( ) ( ) N i n N i dɛ i = i V p ixv ix dv = 3 p v dv (1.24) (1.11) d W = P dv U = i N i ɛ i du = i ɛ i dn i + i N i dɛ i (1.25) du =d W +d Q d Q = i ɛ i dn i (1.26) 5

8 1 1.3 i ɛ i {N i } 1.6 3 N 1 N 1 2 N 2 G = N! N 1! N 2!... (1.27) {N i } G

1.6. 9 log G (Stirling) 6 log G =logn! i log N i!=n(log N 1) i N i (log N i 1) (1.28) 2 N = i N i (1.29) E = i N i ɛ i (1.30) δ log G = i [log N i ] δn i =0 {δn i } 2 δn = i δn i =0, δe = i ɛ i δn i =0 2 α β δ log G + αδn βδe = i [ log N i + α βɛ i ] δn i =0 0 log G log N i = α βɛ i N i = A exp( βɛ i ) (A =e α ) (1.31) ɛ i = mv 2 /2 N i (1.2) [ ] f(v x,v y,v z )dv x dv y dv z = a exp βmv2 dv x dv y dv z (1.32) 2 A a a β (1.3) ɛ = a n mv 2 2 exp [ 6 N ] βmv2 dv x dv y dv z = 3 2 2 kt log N! N(log N 1) N 10 log G N

10 1 7 1.4 f(v x,v y,v z )dv x dv y dv z = n ( m ) 3/2 [ mv 2 ] exp dvx dv y dv z (1.33) 2πkT 2kT ( 1.8 H v v +dv F (v)dv dv x dv y dv z =4πv 2 dv ( m F (v)dv =4πn 2πkT ) 3/2 v 2 exp [ mv2 ] dv (1.34) 2kT v v = 1 n 0 vf(v)dv = 8kT πm (1.35) v 2 v 2 = 3kT/m 3π v 2 = 8 v 1.085 v 7 n =0 1 x 2n 1 3 5 (2n 1) π e ax2 dx = 2 n+1 a n a 0 0 x 2n+1 e ax2 dx = n! 2a n+1

1.6. 11 (v v) 2 = v 2 (v) 2 0.422 v (1.36) n F (x 1,x 2,..., x n ) r (x 1 0,x 2 0,..., x n 0 ) G k (x 1,x 2,..., x n )=0 (k =1, 2,.., r) (1.37) δg k = n i=1 G k x i δx i =0 (k =1, 2,..., r) (1.38) δf = n i=1 F x i δx i = 0 (1.39) n {δx i } r 1 (1.38) r {δx j }(j = n r +1,..., n) (n r) {δx i } (i =1, 2,..., n r) δf = n r i=1 [ F + x i n j=n r+1 F x j ] x j δx i = 0 (1.40) x i (n r) {δx i } (i =1, 2,..., n r) (1.40) =0 r {λ k } (k =1, 2,..., r) (n r)+r = n δf r k=1 λ k δg k = n i=1 [ F x i r k=1 ] G k λ k δx i = 0 (1.41) x i n {δx i } (i =1, 2,.., n) F x i r k=1 λ k G k x i =0 (i =1, 2,..., n) (1.42) n {x i } {λ k } x i 0 (λ 1,λ 2,..., λ r ) (1.43) r (1.37) r {λ k } (1.43) F n {x i }

12 1 G = ax + by + cz 1=0 F = x 2 + y 2 + z 2 δf λδg =(2x λa)δx +(2y λb)δy +(2z λc)δz =0 x = λa/2, y = λb/2, z = λc/2, G =0 G = λ(a 2 + b 2 + c 2 )/2 1=0 λ =2/(a 2 + b 2 + c 2 ) x 0 = a/(a 2 + b 2 + c 2 ), y 0 = b/(a 2 + b 2 + c 2 ), z 0 = c/(a 2 + b 2 + c 2 ) F F min =1/(a 2 + b 2 + c 2 ) 1.7 - φ(r) ( ) ɛ i = 1 2 mv2 + φ(r) (1.44) f(r, v) dxdydzdv x dv y dv z ( ) m 3/2 [ = a exp 1 ( 1 2πkT kt 2 mv2 + φ(r))] dxdydzdv x dv y dv z (1.45) - (1.45) T r v n(r) =n 0 exp [ φ(r) ] kt (1.46) a φ =0 n 0 n(z) =n(0) exp [ mg kt z]

1.8. H * 13 n P = nkt M R mg/kt = Mg/RT ω ( ) φ(r) = mω 2 r 2 /2 3 z z +dz RT/Mg m 8500m 1.8 H * v r f(r, v,t) f(r, v,t) t r, v df dt = f t + f dx x dt + f dy y dt + f dz z dt + f dv x v x dt + f dv y v y dt + f dv z v z dt = f t + ṙ f + v v f (1.47) v =( / x, / y, / z) v =( / v x, / v y, / v z ) ṙ = v K v = K/m 8 ( t + v + K ) ( ) f m v f = (1.48) t 8 2.2 µ f/ t coll.

14 1 1.5 M S v v 1 v v 1 v v v + v 1 = v + v 1, v2 + v 2 1 = v 2 + v 1 2 (1.49) v v 1 3 6 v, v 1 a ω ( ) (v, v 1, ω) S[(v, v 1, ω) (v, v 1 )] S v M ω v 1 (1.4) M ω dω M 2a da =(2a) 2 dω dt v 1 v (1.4) x- ω v M S v 1 = f(v 1 ) (v 1 v) ω dt (2a) 2 dω (v 1 v) ω < 0 r f(r, v 1,t) f(v 1 ) r S v f(v) { 4a 2 (v C(S) = 1 v) ω (v 1 v) ω < 0 0 (v 1 v) ω > 0

1.8. H * 15 1.6 S S S ω 9 v S [(v, v 1, ω) (v, v 1)] ( ) f = dv 1 dω C(S)f(v)f(v 1 )+ dv 1 dω C(S )f(v )f(v 1 t ) coll. S S C(S) = C(S ) (1.50) M v 1 v v 1 v ω (v 1 v) ω = (v 1 v ) ω S S[( v, v 1, ω) ( v, v 1)] ω C(S) = C(S) x, y, z S S (1.50) (1.48) ( ) f = dv 1 dω C(S) [ f(v)f(v 1 ) f(v )f(v 1 t ) ] (1.51) coll. v f 9 σ I Iσ C(S) dω 4πa 2 v 1 v σ =4πa 2

16 1 - (1.33) (1.49) f(v)f(v 1 )=f(v )f(v 1) (1.52) H H H 10 H(t) = dv f(v,t)logf(v,t) (1.53) dh dt 0 (1.52) dh/dt =0 (1.52) H H H H f 0 = f(v,t), f 1 = f(v 1,t), f 0 = f(v,t), f 1 = f(v 1,t) 10 e η

1.8. H * 17 dv df 0 dt =0 dh dt = dv df 0 dt log f 0 0 (1.51) ( ) dh = C(S) [f 0 f 1 f 0 f 1 ] logf 0 dt coll. S S 0 1 C(S) [f 0 f 1 f 0 f 1 ] logf 0 = C(S) [f 1 f 0 f 1 f 0 ] logf 1 = S C(S )[f 0 f 1 f 0 f 1 ] logf 0 = S S C(S )[f 1 f 0 f 1 f 0 ] logf 1 S (1.50) S S 4 4 log f 0 +logf 1 =logf 0 f 1 ( ) dh = 1 C(S) [f 0 f 1 f 0 f 1 ] [log f 0 f 1 log f 0 f 1 ], dt coll. 4 S (x y)(log x log y) 0 x = y dh/dt 0 (1.49) (v 1 v) 2 = 2(v 1 2 + v 2 ) (v 1 + v) 2 = 2(v 12 + v 2 ) (v 1 + v ) 2 =(v 1 v ) 2 (1.54) ω c v 1 v 1 = cω = (v v) (1.55) (v 1 v ) (v 1 v) =2cω 2cω (1.54) (v 1 v ) cω =(v 1 v)+cω, 2c (v 1 v ) ω =2c (v 1 v) ω (v 1 v ) ω = (v 1 v) ω (1.55) ω

2 N 6N 2.1 N 3N 3N 6N 3N 1 {q 1,q 2,q 3,,q 3N } {p 1,p 2,p 3,,p 3N } 6N (q 1,q 2,q 3,,q 3N,p 1,p 2,p 3,,p 3N ) (ˆq, ˆp) 6N Γ ( ) 6 µ ( ) 2 Γ Φ(q 1,q 2,q 3,,q 3N ) ( ) H(ˆq, ˆp) = 3N i=1 p i 2 2m + Φ(q 1,q 2,q 3,,q 3N ) (2.1) (ˆq, ˆp) H(ˆq, ˆp) H(ˆq, ˆp) = constant (2.2) 6N Γ ṗ i = Φ q i, p i = m q i (i =1, 2,, 3N) 1 2 q i p i (2.3) 2 Γ µ Gas molecule 18

2.2. 19 2.1 (2.1) q i = H p i, ṗ i = H q i (2.3) Γ 6N ˆv =(ˆ q,ˆṗ) (2.3) 2.2 Γ 1 {ˆq(t), ˆp(t)} Q(ˆq, ˆp) Q τ = 1 τ τ 0 Q(ˆq(t), ˆp(t)) dt (2.4) τ ( ) τ τ = 1 τ Q = lim Q(ˆq(t), ˆp(t)) dt (2.5) τ τ 0

20 2 τ Γ dˆqdˆp =dq 1 dq 2 dq 3 dq 3N dp 1 dp 2 dp 3 dp 3N ρ(ˆq, ˆp)dˆqdˆp (2.5) Q = Q(ˆq, ˆp)ρ(ˆq, ˆp) dˆqdˆp (2.6) Q = Q (2.7) N 2.3 (2.6)

2.3. 21 ρ(ˆq, ˆp) Γ Γ 6N ρ(ˆq, ˆp) ( ) 3 3 { ρ t = ˆ ρˆv 3N (ρ qi ) = + (ρṗ } i) (2.8) q i p i ˆ 3 div i=1 a = a x x + a y y + a z z = i=x,y,z 6N (2.3) 6N ˆv =(ˆ q, ˆṗ) ˆ ˆv = 3N i=1 { qi + ṗ } i = q i p i 3N i=1 a i x i { H } H =0 (2.9) q i p i p i q i 4 (2.8) 5 ρ t = 3N i=1 ( ρ q i + ρ ) ṗ i q i p i = 3N i=1 ( ρ H ρ ) H q i p i p i q i (2.10) Γ ρ/ t Γ Γ ˆq(t), ˆp(t) t dρ dt = ρ 3N ( t + ρ q i + ρ ) ṗ i q i p i (2.10) i=1 dρ dt = 0 (2.11) 3 4 5 Q(ˆq, ˆp)

22 2 2.2 Γ Γ Γ ρ t =0 ρ(ˆq, ˆp) H(ˆq, ˆp) (ˆq, ˆp) ρ = ρ 0 (H(ˆq, ˆp) ) ρ = dρ 0 H ρ, = dρ 0 H q i dh q i p i dh p i (2.10) ρ/ t =0 2.4

2.5. 23 H(ˆq, ˆp) (ˆq, ˆp) ρ 0 (H(ˆq, ˆp)) E 0 2.3 Γ E E + E constant E H(ˆq, ˆp) E + E ρ 0 (H(ˆq, ˆp)) = (2.12) 0 6 2.5 6 1/ ˆ H(ˆq, ˆp) ˆ 6N { ˆ H 3N ( ) 2 H 3N ( ) } 1/2 2 H = + q i p i i=1 i=1

24 2 2.3 1/2 6 1/6 10 1010 10 10 10 10 10 10 10 (2.7) 2.6 * 2.3 Γ T T (t 1,t 2 )=T(t 1,t)T(t, t 2 ) 2

2.6. * 25 2.4 S={(x, y); 0 <x 1, 0 <y 1} S (2x, y/2) 0 <x 1 2 F (x, y) = (2.13) 1 (2x 1, (y +1)/2) <x 1 2 2.3 2.4 Γ Γ Γ 2.2 7 1 7 2.2

26 2 ρ(x, y, z, t) v(x, y, z, t) j = ρv ρ j dx dy dz x- dydz x ρdxdydz t ρ(x, y, z, t)dxdydz = [j x(x, y, z, t) j x (x +dx, y, z, t)]dydz + [j y (x, y, z, t) j y (x, y +dy,z,t)]dzdx + [j z (x, y, z, t) j z (x, y, z +dz,t)]dxdy = j x x dx dydz j y y dy dzdx j z dz dxdy z (2.14) ρ t = j x x j y y j z z ( / x, / y, / z) = ( x, y, ) z (2.15) (2.16) a b = a x b x + a y b y + a z b z (2.17) ρ t = j = ρv (2.18) div ρ t = div(ρv) (2.19) ρv =( ρ) v + ρ( v) (2.20) grad ρ =gradρ (2.21)

3 - Γ 1 2 3 Γ Γ 3.1 - Γ µ 1.2 3 6 µ i =1, 2, 3, i i g i 1 i N i {N i } {N i } N N i N i N!/N 1! N 2! N 3! N i g i g N i i G({N i })= N! N 1! N 2! N 3! gn 1 1 g N 2 2 g N 3 3 (3.1) E W i 1 ɛ i 27

28 3 N i = N, N i ɛ i = E (3.2) i i {N i } (3.1) W = G({Ni }) (3.3) {N i } E W N (3.3) G 1.6 1.6 1.6 (1 p.9 ) (3.1) 1 log G N(log N 1) + i N i (log g i log N i + 1) (3.4) (3.2) {N i } α β δ log G + αδ( N i ) βδ( N i ɛ i )=0 i i [log g i log N i + α βɛ i ] δn i =0 i log g i log N i + α βɛ i =0 N i = g i A e βɛ i A =e α (3.5) A β (3.5) (3.2) β N (1) N (2) 2 (3.2) (1) (2) (1) Ni = N (1) (2), Ni = N (2) (1), Ni ɛ i + (2) Nj ɛ j = E j i i i 1 N i /N i p i 4 log G N i log(n i /N g i ) N 1 log G p i log(p i /g i )= log(p i /g i ) i i

3.2. 29 1 β 2 β 2 α µ dxdydzdp x dp y dp z i g i φ(r) ɛ i = p2 2m + φ(r) (3.5) 1.7 - β =1/kT A 3.2 - µ n G(n) W G(n) = N! n!(n n)!, N W = G(n) =2 N (3.6) g 1 = g 2 (= g) g N (3.6) G max n = N/2 2 n=0 log G max N(log N 1) 2 N 2 (log N 2 1) = N log 2 = log W G max W log M! M(log M 1) + 1 log 2πM 2 log G max log W =1 ( log N N ) 2 N n =(N 1)/2 n =(N +1)/2 N

30 3 3.1 N N log G max log W 2 1 n n P (n) = G(n) W = 1 N! 2 N n!(n n)! f(s) = N n=0 P (n)s n = 1 (1 + s)n 2N N n = np (n) =f (1) = N 2 n=0 N n(n 1) = n(n 1)P (n) =f (1) = n=0 n 2 = n(n 1) + n = N(N +1) 4 N(N 1) 4 (n n) 2 = n 2 n 2 = N 4, (n n) 2 = N 2

3.3. 31 x = n/n x = 1 (x 2, (n n) 2 x) 2 = = 1 N 2 N N 3.1 N 1/2 x p(x) x 1/2 log p(x) p(x)dx = 2N π exp [ 2N ( x 1 ) ] 2 dx 2 ( ) µ 2 N i N 3 / log G max log W =1 (3.8) N N A =6 10 23 1 52 1 1 1 52 W = 52! G = (26!) 2 G/W = (26!) 2 /52! = 2.0 10 15 5.0 10 14 1600 1 (3.7) 3.3 log G max (3.5) A (3.2) / A = N g i e βɛ i G max / N (max) i = Ng i e βɛ i g j e βɛ j (3.9) j 3 G max /W 1 2 3 100 10 N i

32 3 (max) (3.1) log G max = N(log N 1) + i N i [ log N + βɛ i +log( j g j e βɛ j )+1] = N log( j g j e βɛ j )+βe 1 Nd [log( j g j e βɛ j )] = N j / (ɛ j dβ + βdɛ j )g j e βɛ j g j e βɛ j j = ( j N j ɛ j )dβ β j N j dɛ j = Edβ β j N j dɛ j 4 d(log G max ) = Edβ β j N j dɛ j +d(βe) = βde β j N j dɛ j 2 j N j dɛ j 1.5 d W de = T ds +d W d(log G max )=β(de d W )=βtds = k 1 ds S = k log G max (3.10) N (3.8) (3.10) S = k log W W (3.11) W (3.11) S 4 g j U E

3.3. 33 1 1 1 1 (3.10) (3.11) G G 5 G S G max W 9 11 1.5 ( ) 2 W W S W 2 W W (1) W (2) W = W (1) W (2) S S (1) S (2) S = S (1) + S (2) 5 log G (3.4) 1 8 H (1.53) dv g i f(v i )=N i /g i p.28 Γ ρ(ˆq, ˆp, t) H H(t) = ρ(ˆq, ˆp, t)logρ(ˆq, ˆp, t) dˆqdˆp ρ(ˆq, ˆp, t) (2.10) dh/dt =0 H

34 3 S = k log W k 3.4 S W (3.11) W E V (3.11) S E V S(E,V ) E U ( ) S E V = 1 ( ) S T, = P V E T T P H = E + PV F = E TS G = F + PV V N E W µ 1 h 3 h µ dxdydzdp x dp y dp z 1 g i = dxdydzdp xdp y dp z h 3 1 q p = h µ ( ) H(ˆq, ˆp) E E + E N W (E) = 1 N! E H(ˆq, ˆp) E+ E dˆq dˆp h 3N 1/N! H(ˆq, ˆp) = 3N i=1 p i 2 2m

3.4. 35 1 V p i = 2ms i W (E) = ( ) 3N 2m V N h N! E s 2 1 +s 2 2 + +s 2 3N E+ E ds 1 ds 2 ds 3N = ( ) 3N 2m V N h N! 2π 3N/2 ( E) 3N 1 Γ(3N/2) E 2 E (3.12) 3 3N E E E + E Γ(x) x 0 Γ(x +1)=x! x log Γ(x +1) x(log x 1) (3.12) N log W = N (log V N + 3 2 log E ) N + constant π m h constant E E S(E,V ) = k log W = Nk (log V N + 3 2 log E ) N + constant (3.13) ( ) 1 S T = E V = 3 ( ) Nk 2 E, P S T = = Nk V E V E = 3 2 NkT, PV = NkT Nk = R (3.10) (3.11) k 2 (3.12) W E E 3N (2/3N) π 3N/2 ( E) 3N / Γ(3N/2) Ω(E) ( ) S = k log Ω

36 3 h 3N 1/N! h 3N (3.13) ( ) 1 V/N E/N N V/N N W 1/N! N N! N N! 1 2 generic phase specific phase Γ N! - N! 3 N i ( i N i = N) V N S 1/N! S N! N 1!N 2!... log N! i log N i! i N i log(n/n i ) x i = N i /N S = Nk i x i log x i

4 4.1 I T II I II I II I Γ i i E i i N G i I II I II E 0 W I+II W I+II (E 0 )= i G i W II (E 0 E i ) (4.1) W II II I II (4.1) I i P i = G i W II (E 0 E i ) (4.2) W I+II (E 0 ) II S II (E) (3.11) W II (E 0 E i )=exp[s II (E 0 E i )/k] (4.3) I II E 0 S II (E 0 E i ) E i S II (E 0 E i ) = S II (E 0 ) E i S II 37 E + E2 i 2 2 S II E 2 +

38 4 ( ) S E V = 1 ( 2 ) T, S E 2 V = S II (E 0 ) E i T = [ ( ) 1 E T V 1+ E ] i + 2TCV II = 1 ( ) T T 2 E V = 1 T 2 C V (4.4) II CV II (4.4) [] 2 (4.3) I (4.2) P i = 1 ( Z G i exp E ) i (4.5) kt Z exp( E i /kt ) (4.5) - - 1 ɛ i (4.5) E i N 1 I II G i = g i = dxdydzdp xdp y dp z h 3 E i = ɛ i P i = N i /N (4.5) - 1 N N 1 Γ µ - N i (3.5) (4.5) G p.28 (3.4) log G = N i P i log(p i /G i )= N log(p i /G i ) (N )

4.2. 39 P i (4.5) log G (3.10) 1 1 S = N 1 k log G max = k log(p i /G i ), P i = Z 1 G i exp( E i /kt ) (4.6) 1 4.2 (4.5) Z i P i =1 Z = i G i e βe i ( β =1/kT ) (4.7) E = i P i E i = 1 Z G i E i e βe i = 1 i Z Z β = log Z β (4.8) Z Z 2 U 4.3 Z U = E (4.8) log Z log(p i /G i )= log Z E i /kt (4.6) S = E T + k log Z E = U S =(U F )/T F = kt log Z (4.9) 1 P i /G i i G i (4.6) W 1/W S = k log W 2

40 4 Γ (4.5) 3.3 d(log G max ) (4.7) (4.7) d(log Z) = Z 1 i G i (E i dβ + βde i )e βe i = Edβ + β i P i de i (4.10) 2 d W ( β) F = U TS ( ) F d T ( ) U = d T S ( ) 1 = Ud + 1 (du T ds) T T ( ) 1 = Ud + 1 T T d W U = E β =1/kT (4.9) 3.3 (4.10) 2 V ( ) log Z = βp (4.11) V 19 ) T T V Z(T,V ) (4.7) F (T,V ) U ( ) F T V ( ) F U = F + TS = F T T V = S, ( ) F V T ( ) = T 2 F T T V = P = ( ) F T 1 T V = log Z β (4.8) U F -

4.3. 41 T V N U P 3.4 Γ G i (4.7) Z = 1 N! exp[ βh(ˆq, ˆp)] dˆqdˆp h 3N (4.12) N! 3.4 H = ˆq V N 3N i=1 p 2 i /2m Z = V N ( ) 3N e p2 /2mkT V N ( ) 3N/2 2πmkT dp = h 3N N! N! h 2 Γ 1 p.9 F (T,V )= kt log Z = NkT (log V N + 3 ) 2 log T + constant ( ) F S = T V = F T + 3 ( ) F 2 Nk, P = V T U = log Z β = 3 NkT = F + TS 2 = NkT V = nkt 4.3 E E +de 3 G(E)dE G(E)dE = G i E E i E+dE E E +de P (E)dE P (E)dE = Z 1 G(E) e βe de (4.13) Z = G(E) e βe de (4.14) 3 3.4 W (E) W (E) = G(E) E 3.4 2 Ω(E) G(E) =dω/de G(E) (4.14) G(E)

42 4 4.1 E Z (4.8) E U T C V β =1/kT C V = ( ) E T V [ ] 1 = 1 kt 2 ( 1 β Z ) Z = 1 1 2 ( Z 1 β kt 2 Z β 2 Z ) 2 Z β 1 2 Z Z β 2 = 1 Z G(E)E 2 e βe de = P (E)E 2 de = E 2 2 (4.8) (E) 2 C V = 1 kt 2 [E2 (E) 2 ]= 1 kt 2 (E E)2 (4.15) E E 4 (4.15) (E E) 2 E kt 1 C V E N (4.15) E = (E E) 2 N E E 1 N N E E (4.13) E e βe G(E) 4 T V

4.4. 43 4.1 E E + E 0 4.4 N (4.1) (4.4) W S N (4.4) S II (E 0 E i,n 0 N i ) S II (E 0,N 0 ) E i T + µn i T (4.16) µ 1 5 ( ) S = µ N T E,V (4.13) E E +de N P (E,N)dE = Ξ 1 G N (E) e β(e µn) de (4.17) Ξ ( ) α = βµ 6 Ξ(V,β,α) = G N (E)e β(e µn) de = N=0 e αn Z N (T,V ) (4.18) N=0 5 1 µ =( G/ N) T,P dg = SdT + V dp + µdn G = U TS + PV (U, V, N) ds = 1 T du + P T dv µ T dn 6 β βµ

44 4 N G Z G N Z N (4.17) (4.18) N Q(N) Q(N) = P (E,N)dE = Ξ 1 e αn Z N (T,V ) (4.19) E N ( E = Q(N) 1 ) Z N = 1 ( ) ( ) e αn ZN log Ξ = Z N β Ξ β β N=0 N=0 N = Q(N)N = 1 Ne αn Z N = 1 N=0 Ξ N=0 Ξ V ( ) Ξ = α V, β ( ) log Ξ (4.11) N ( ) kt log ZN P N = V T = kt ( ) ZN Z N V T α V, β V, α (4.20) (4.21) (4.19) Q(N) P = kt Ξ N=0 e αn ( ZN V ) T = kt Ξ ( ) Ξ = V α,β ( ) kt log Ξ V α, β (4.22) kt log Ξ F = kt log Z J J = kt log Ξ = PV (4.23) dj = SdT P dv Ndµ (4.24) (4.23) dj = P dv V dp (4.24) N G = Nµ ( G/ N) T,P = µ - SdT + V dp Ndµ = 0 (4.25) κ T ( ) 4.3 (4.21) ( ) N α V,β = 2 log Ξ α 2 = 1 Ξ 2 ( ) 2 Ξ 1 α 2 Ξ Ξ α = N 2 (N) 2 = (N N) 2 (4.26)

4.4. 45 α = βµ ( ) ( ) N N = kt α V,β µ V,T ( ) N µ V,T = N 2 κ T V, κ T = 1 V ( ) V P N,T (4.27) (N N) 2 N = nkt κ T (4.28) n = N/V nkt P N 1 N = (N N) 2 N N 1 N N 1 N N (4.23) (4.6) (4.17) S = k log [P (E,N)dE/G N (E)dE] = (kt log Ξ µn + E)/T E = U Nµ = G kt log Ξ = G U + TS = PV (4.22) log Ξ V V V 2 V 1 V 2 N 1 N N 1 N 1 =0, 1, 2,..., N 3.4 p.36 2 (4.18) G N (E) G N (E)dE = N N 1 =0 de 1 G (1) N 1 (E 1 ) G (2) N N 1 (E E 1 )d(e E 1 ) Z N (T,V 1 + V 2 ) = = = de G N (E)e βe N N 1 =0 N N 1 =0 de 1 G (1) N 1 (E 1 )e βe 1 de G (2) N N 1 (E E 1 )e β(e E 1) Z N1 (T,V 1 )Z N N1 (T,V 2 )

46 4 (4.18) Ξ(V 1 + V 2,β,α) = N=0 N 1 =0 e αn N N 1 =0 Z N1 (T,V 1 )Z N N1 (T,V 2 ) = e αn 1 Z N1 (T,V 1 )e α(n N 1) Z N N1 (T,V 2 ) N=N 1 = Ξ(V 1,β,α) Ξ(V 2,β,α) log Ξ V log Ξ(V 1 + V 2,β,α)=logΞ(V 1,β,α)+logΞ(V 2,β,α) (4.24) J = β 1 log Ξ (4.20) (4.22) [ log Ξ dj = β 2 1 ( ) ] log Ξ dβ 1 ( ) log Ξ dα 1 ( ) log Ξ dv β β α,v β α β,v β V α,β = 1 β (PV + E)dβ N dβ (µdβ + βdµ) P dv ( β β = dt, dα = µdβ + βdµ) T E + PV µn = dt Ndµ P dv = SdT Ndµ P dv T (4.27) V ( ) N = V µ V,T ( µ ) N V V,T = V ( ) n µ V,T - (4.25) µ (T,P) n = N/V (T,P) (T,µ) n V N ( N µ ) V,T = V ( n µ ) V,T = V ( ) n µ N,T = N V ( ) V µ N,T T µ P - (4.25) ( P/ µ) T = N/V ( ) ( ) ( ) V V P = = Nκ T µ N,T P N,T µ N,T (4.27) 1 T -P Y (T,P,N)= 0 e PV/kT Z N (T,V )dv, G(T,P,N)= kt log Y (T,P,N) ktκ T = (V V ) 2 /V (4.29) 2 T,P T,µ N V V V N = n(t,p)v N N = n(t,p) V N/N = V/V κ T 2 (4.28) (4.29)

5 5.1 2 2 2 5.1 θ ( ) ϕ ( ) 1 H 1 = p x 2 + p y 2 + p z 2 2M + p θ 2 + p ϕ 2 / sin 2 θ 2I (5.1) 1 M 2 1 I θ ϕ p θ = I θ, p ϕ = I ϕ sin 2 θ 1 Z 1 = V h 5 dθdϕ Z N = 1 N! Z 1 N dp x dp y dp z dp θ dp ϕ e H 1/kT (5.2) N! 3 H 1 5 2 1 E N = 5 kt (5.3) 2 (5.1) p 2 ϕ sin2 θ p ϕ p.9 αs 2 / /2 ds e 2 e αs2 αs2 /2 ds = 1 (5.4) 2 1 47

48 5 5.1 2 G 2 H = f i=1 s i 1 2 α is i 2 1 2 α i s i2 = 1 2 kt E N = f 2 kt (5.5) 2 f =5 (5.3) s i 2 1 kt f C V = 1 2 N Afk = 1 2 fr, C P = C V + R = 1 f +2 (f +2)R, γ= 2 f (5.6) γ ( ) γ = C P /C V 5.2 (5.1) 2

5.1. 2 49 5.2 ɛ>kt q i p i H(ˆq, ˆp) lim H = lim H = (5.7) q i p i q i H q i = kt, p i H p i = kt (5.8) q i X i X i = H/ q i f q i X i = fkt (= 2 ) (5.9) i=1 q i X i Z H q i = 1 H q i e βh dˆqdˆp = 1 e βh q i dˆqdˆp q i Z q i βz q i = 1 [ e βh (qi e βh ] ) dˆqdˆp dˆqdˆp = 1 βz q i β (5.10) 1 Z 2 (5.7) q i 0 p i 1 cx 2n (c>0) kt/2n 2 (5.8) q i H/ q j p i H/ p j δ ij kt

50 5 5.2 2 N { ɛ0 ɛ = 0 2 2 Z = Z 1 N, Z 1 = ɛ e βɛ =1+e βɛ 0 E E = β log Z = Nɛ 0e βɛ0 1+e βɛ 0 = Nɛ 0 1+e βɛ 0 = Nɛ 0 1+e ɛ 0/kT (5.11) ( Nɛ 0 1 ɛ ) 0 2 2kT + kt/ɛ 0 1 E Nɛ 0 e ɛ0/kt (1 e ɛ0/kt + ) kt/ɛ 0 1 (5.12) C = de dt ( de = kβ2 dβ = Nkβ2 2 ɛ 0 ɛ0 (1 + e βɛ 0 ) 2 eβɛ 0 /kt = Nk 1+e ɛ 0/kT ) 2 e ɛ 0/kT (5.13) 5.3 ( ) ɛ0 2 C Nk e ɛ 0 /kt kt C 0 kt/ɛ 0 =0.391 E = nɛ 0 N n ɛ 0 W (E) = N! n!(n n)!, W (E) =2 N E 2 N! N

5.2. 2 51 5.3 2 S = k log W = k[n(log N 1) n(log n 1) (N n)(log(n n) 1)] [ n = Nk N log n ( N + 1 n ) ( log 1 n )] N N (5.14) 1 T = S E = 1 S ɛ 0 n = k [ log n +log(n n)] = k log N n ɛ 0 ɛ 0 n N n n =e ɛ 0/kT E = nɛ 0 = (5.11) Nɛ 0 1+e ɛ 0/kT (5.12) T Nɛ 0 /2 2 ɛ =0 ɛ = ɛ 0 ɛ =0 1 1+e ɛ 0/kT < e ɛ0/kt 1+e ɛ 0/kT T<0 T = W (E) E E = Nɛ 0 /2 n = N/2 T 1 = S/ E

52 5 5.3 n m H H = m H = mh cos θ (5.15) θ ( ) H m m n Z = Z 1 Z 1 = 1 π e βmh cos θ 2π sin θdθ = 1 1 e βmhz dz = sinh(βmh) 4π 0 2 1 βmh e x ± e x cosh x = 2 sinh x (5.16) E = log Z β = n [mh coth(βmh) β 1 ] (5.17) coth x =coshx/ sinh x M M = n m cos θ = E H = nm [coth(βmh) 1 βmh ] (5.18) coth x ( )/ ( ) coth x = 1+ x2 2! + x 1+ x2 3! + H kt/m M χh, χ = ( ) M = C H H=0 T = 1 x ( 1+ 1 ) 3 x2 + (C = nm 2 /3k) (5.19) χ ( ) T 4 (4.15) M = n m cos θ = 1 Z Z h ( h = βh)

5.4. * 53 5.4 χ = M H = β h 1 Z Z h = β 1 2 ( Z 1 Z h 2 Z ) 2 Z h = 1 kt [M 2 (M) 2 ]= 1 kt (M M)2 (5.20) (5.15) 5.4 * m ±1 2 σ ( ) 3 H 1 H 1 = σh (5.21) 1 m σ = ±1 e βh 1 =e ±βh m = σ = eβh e βh = tanh(βh) (5.22) e βh +e βh tanh x =sinhx/ cosh x ( ) m χ 0 (T )= = C ( C =1/k) (5.23) H H=0 T 3 i τ i =1 τ i =0 τ i σ i =2τ i 1 (i, i ) K (ii ) τ i τ i τ i = τ i =1 K 0

54 5 2 H = Hσ i J σ i σ i (5.24) i (ii ) i (i i ) J >0 J <0 1 1 N N N H = Hσ i J σ i σ i+1, σ N+1 = σ 1 (5.25) i=1 i=1 Z = = {σ i =±1} {σ i =±1} exp [ ] N β (Hσ i + Jσ i σ i+1 ) i=1 [ β N exp i=1 ( H σ )] i + σ i+1 + Jσ i σ i+1 2 (5.26) s A ss =TrA (A N ) ss = s 2 (5.26) 2 2 ( {+1, +1} {+1, 1} A = { 1, +1} { 1, 1} s 3 A ss2 A s2 s 3 A sn 1 s N A sn s s N ) ( exp[β(h + J)] exp[ βj] = exp[ βj] exp[β( H + J)] ) (5.27) Z =TrA N (5.28) A 2 λ 1 λ 2 λ 1 >λ 2 Z =TrA N = λ N 1 + λn 2 = λn 1 [1 + (λ 2/λ 1 ) N ]

5.4. * 55 N 1 log Z = N log λ 1 (5.29) 2 2 A λ 1 =e βj cosh βh +e βj 1+e 4βJ sinh 2 βh (5.30) 1 σ βh = h σ = 1 N log Z h = log λ 1 h = 1 λ 1 λ 1 h χ = σ H = e2j/kt kt (5.31) T 0 T 1 2 T C 2 T C lim σ 0 H 0 2 3 1 i 1 σ i H i = (H + J σ i ) σ i (5.32) i i i 2 4 3 6 σ i σ i m = σ z (5.32) ( ) H = H + zjm

56 5 5.5 (5.22) H σ i m m ( ) H + zjm m =tanh kt m =tanh(βh) (5.33) (5.34) H m (5.23) (5.34) m χ 0 (T )(H + zjm) (5.35) m χ(t )= m H = χ 0 1 zjχ 0 = C T T C (T C = zj/k) (5.36) - T C T T C +0 T C lim σ 0 H 0 (5.34) H m T 4 H(m, T ) = T log 1+m kt C 2T C 1 m m (5.37) 5.5(a) T<T C H =0 m 0 m S (T ) (5.37) H =0 / T =2m S log 1+m S T C 1 m S 4 y =tanhx x =tanh 1 y =log (1 + y)/(1 y)

5.5. * 57 5.5(b) 5.5(a) T<T C χ =( m/ H) T < 0 5.5(a) (b) T<T C χ(t )= ( ) m H H=±0 C 2(T C T ) T T C 0 H =0 T T C 2 T C H H =0 m(t,h) ±m S - 1 5.5(c) T - T<T C 1 z =2 (5.32) σ i σ i m 5.5 * n n PV NkT =1+B 2(T )n + B 3 (T )n 2 + (5.38) 2 B 2 (T ) φ(r) B 2 (T )= 0 [1 e φ(r)/kt ]2πr 2 dr (5.39)

58 5.. 1 φ(r)... φ(r) = 1 r 12 1 r 6 r 0 =1.1224. 0 r 0. 2. r.... 5.6-5.6 0 3 (5.39) B 2 (T ) 1 3 a φ(r) = (r <2a), φ(r) =0(r 2a) B 2 (T ) B 2 =16πa 3 /3=1 4 6.1 Ψ ( ) Φ ( ) ξ ( ) i r i N Ψ(r 1, r 2, )=Φ(r 1, r 2, )+ ξ(r i ) i=1 (5.9) f =3N N N r i i Φ + r i i ξ =3NkT (5.40) i=1 i=1 i =( / x i, / y i, / z i ) 2 i ξ i i ξ i f i 2 1 N N r i i ξ = r i f i = r P nds i=1 i=1 = P r nds = P ( r) dv =3PV (5.41) r = x/ x + y/ y + z/ z =3 (5.41)

5.5. * 59 5.7 2 G(r) e φ(r)/kt (5.40) PV = NkT 1 N r i i Φ (5.42) 3 i=1 2 φ( r i r j ) Φ = 1 2 N N φ( r i r j ) i j i N N i Φ = i φ( r i r j )= j φ( r i r j ) j i j i N N N N N r i i Φ = r i i φ( r i r j ) = r i j φ( r i r j ) i i j i i j i = 1 N N i r j ) i φ( r i r j )= 2 i j i(r 1 N N dφ(r ij ) r ij (5.43) 2 i j i dr ij N r i i Φ = 1 N N dφ(r ij ) N(N 1) r ij = i 2 i j i dr ij 2 r dφ(r) dr (5.44) r ij = r i r j N 1 (5.42) PV = NkT N 2 6 r dφ(r) dr (5.45) 1 r 2

60 5 G(r) 2 r dφ dr = 1 V 0 r dφ dr G(r)4πr2 dr dφ(r)/dr 1 V 1 (5.42) PV NkT =1 2πn 3kT 0 r 3 dφ(r) G(r)dr (5.46) dr 2 G(r) X (5.46) 2 G(r) n (5.46) (5.38) n G(r) n 0 2 2 1 φ(r) G(r) e φ(r)/kt G( ) =1 (5.46) r 3 dφ(r) 0 dr e φ(r)/kt dr = kt r 3 d 0 dr [1 e φ(r)/kt ]dr = ktr 3 [1 e φ(r)/kt ] 3kT [1 e φ(r)/kt ]r 2 dr (5.47) 1 φ( ) =0 φ(0) = 0 2 B 2 (T ) (5.39) 0 0

6 6.1 H 1 = p2 2m + mω2 q 2 (6.1) 2 m ω ν = ω/2π 1 1 [ ( p 2 Z 1 = exp β 2m + mω2 q 2 )] dqdp 2 h = 2π βhω = 1 βhν E 1 = log Z 1 β = 1 β = kt (6.2) q 2 1 2 mω2 q 2 = 1 2m p2 = 1 kt (6.3) 2 ν E 1 =2 (kt/2) = kt 6.2-5.6 6.1 3 3N 3N (6.2) E =3NkT 61

62 6 6.1 1 N = N A C V =3N A k =3R (6.4) C P C V C P 3R - 3 1 lim C P (T ) = lim C V (T ) = 0 (6.6) T 0 T 0 C P (T ) T 3 (6.7) T 3 6.3 E 1 hν ɛ n = ( n + 1 ) hν, (n =0, 1, 2, ) (6.8) 2 1 3 TS(T ) d ds(t ) 0 = lim S(T ) = lim = lim TS(T ) = lim [S(T )+T ] T 0 T 0 T T 0 dt T 0 dt = lim S(T ) + lim C P, lim C P =0 (6.5) T 0 T 0 T 0

6.3. 63 6.2 q 2 2E 1 /mω 2 + p2 2mE 1 =1 A = π (2E 1 /mω 2 )(2mE 1 )=E 1 /ν A = ( n + 1 ) h, (n =0, 1, 2, ) 2 h 3 1 h 1 (4.7) g i =1 Z 1 = e βɛn = e β(n+1/2)hν = e βhν/2 (6.9) 1 e βhν n=0 n=0 ν kt hν ( 6.5(b) kt/hν ) E 1 = log Z 1 β = hν 2 + hν e βhν 1 (6.10) 1 (6.8) ( ) 1/2 ( 1) ν 3N (6.10) ( ) 2 de 1 hν dt = k e hν/kt (6.11) kt (e hν/kt 1) 2

64 6 6.3 hν/k Θ D 3N A C V =3R ( ) 2 hν e hν/kt (6.12) kt (e hν/kt 1) 2 hν kt C V 3R - (6.4) hν kt ( ) 2 hν C V 3R e hν/kt kt (6.6) 6.2 hν kt kt 1 (6.12) - 3R 1 3N A 0 [3R C V (T )]dt = hν 2 6.4 C V T 0 0 (6.12) T 3

6.4. 65 g(ν)dν = { (9N/νD 3 ) ν 2 dν (ν ν D ) 0 (ν>ν D ) (6.13) ν D νd 0 g(ν)dν =3N (6.14) N = N A 1 (6.11) (6.13) C V = 9N Ak = 3R νd ν 3 D 0 ( ) T 3 ΘD /T ΘD (hν/kt ) 2 e hν/kt ν 2 dν (e hν/kt 1) 2 0 e x 3x 4 dx (6.15) (e x 1) 2 Θ D = hν D /k T Θ D 3x 4 e x (e x 1) 2 =3x2 (1 + 5 12 x2 + ) 3x 2 - T Θ D 2 0 e x 3x 4 4π4 dx = (e x 1) 2 5 (6.16) C V T 3 2 x z 1 dx e x 1 = 0 n=1 0 C V 12π4 R 5 1 e x 1 = x z 1 e nx dx = ( T ΘD e x 1 e x = e nx n=1 n=1 1 n z ) 3 (6.17) 0 x z 1 e x dx = Γ(z)ζ(z) Γ(z) p.35 z 1 Γ(z) =(z 1)! ζ(z) ζ(z) = n=1 1 n z (z>1) ζ(2) = π 2 /6 ζ(4) = π 4 /90 (6.16) 3x 4 e x dx ( ) 1 (e x 1) 2 = 3x 4 x e x dx =12 3 dx 4π4 1 e x =12Γ(4)ζ(4) = 1 5 0 0 0

66 6 6.4 Θ D =0 21 (6.13) L 3 (l, m, n) ( ) ( ) ( ) lπx mπy nπz u(x, y, z, t) =sin(2πνt)cos cos cos L L L c 6.1 1 2 u c 2 t 2 = 2 u x 2 + 2 u y 2 + 2 u z 2 (2πν) 2 =(πc/l) 2 (l 2 + m 2 + n 2 ) ν = c l 2L 2 + m 2 + n 2, (l, m, n =1, 2, 3, ) ν (l, m, n) (l, m, n) 3 1 (l, m, n > 0) 1 ν ν +dν g(ν)dν (l, m, n) l 2 + m 2 + n 2 =2Lν/c 2L(ν +dν)/c 1 ) g(ν)dν = 1 4π 8 3 ( ) 2L 3 [(ν +dν) 3 ν 3 ]= 4πV c c 3 ν2 dν, (V = L 3 ) c/ν ν c 1 c 2 2 ( 1 g(ν)dν =4πV c 3 + 2 ) ν 2 dν (6.18) 1 c 3 2 (6.13) (6.18) (6.14)

6.5. 67 6.5 T 2 (6.18) g(ν)dν = 8πV c 3 ν2 dν (6.19) c 2 c ν 1 (6.10) 6.5 ν ν +dν E(ν)dν = 8π hν 3 dν (6.20) c 3 e hν/kt 1 (6.10) V =1 λ = c/ν I(λ)dλ = 8πhc λ 5 dλ e hc/λkt 1 (6.21) T λ 6.5(b) λ max T 1 kt (6.21) I cl (λ)dλ = 8πkT λ 4 - u (6.20) 65 u = 8π c 3 0 dλ hν 3 dν e hν/kt 1 = 8πh ( ) 4 kt x 3 dx c 3 h 0 e x 1 = αt 4 (6.22) α = 8π5 k 4 15c 3 h 3 =7.566 10 16 J/m 3 K 4

68 6 6.5 (a) 1 (b) P u P = u/3-3 (1.8) c n u Q = cu 4 = σt 4 - σ = cα/4 σ = 2π5 k 4 15c 2 h 3 =5.6705 10 8 W/m 2 K 4 P = u/3 25 (6.22) T T 3 2 λ max λ max T =2.898 10 3 mk 3 du = T ds P dv ( ) U = T V T ( ) S P = T V T ( ) P P = T du T V 3 dt u 3 U = u(t )V ( U/ V ) T = u du/u =4dT/T u T 4

7 7.1 3 1 1 {ɛ i } (i =1, 2, 3, ) {n i } n i 2 (a) - BE n i =0, 1, 2, 3,, (b) - FD 2 n i =0, 1 (a) (b) 4 He 3 He 1 1 2 2 1 2 N! 69

70 7 N N! - MB N = i n i E = i n i ɛ i 22 {n i } 1 (4.18) G N (E) =1 {n i } n i =0, 1, 2, Ξ(T,V ) = {n i } e β(e µn) = = i n 1 =0 n 2 =0 n i =0 e β(ɛ i µ)n i = i e β (ɛ i µ)n i 1 1 e β(ɛ i µ) log Ξ(T,V )= i log(1 e βɛ i+α ), (α = βµ) (7.1) βµ = α (4.20) (4.21) E = log Ξ β = i ɛ i e βɛ i+α 1 e βɛ i+α = i ɛ i e β(ɛ i µ) 1 (7.2) N = log Ξ α = i e βɛ i+α 1 e βɛ i+α = i 1 e β(ɛ i µ) 1 ɛ i 2 n i = 1 e β(ɛ i µ) 1 (7.3) (7.4) - (7.3) N µ N T V (7.4) µ =0 ɛ i = hν hν g(ν) (6.20) ν hν µ =0 2 βµ i n i = i α in i log Ξ(T,V )= i log[1 exp( βɛ i+α i )] n i = log Ξ/ α i =1/[exp(βɛ i α i ) 1] α i = βµ

7.2. 71 24 n i =0, 1 Ξ(T,V ) = 1 1 e β(e µn) = e β (ɛ i µ)n i {n i } n 1 =0 n 2 =0 = 1 e β(ɛ i µ)n i = (1 + e β(ɛi µ) ) i n i =0 i log Ξ(T,V )= i log(1 + e βɛ i+α ), (α = βµ) (7.5) E = log Ξ β = i ɛ i e βɛ i+α 1+e βɛ i+α = i ɛ i e β(ɛ i µ) +1 (7.6) N = log Ξ α = i e βɛ i+α 1+e βɛ i+α = i ɛ i n i = 1 e β(ɛ i µ) +1 1 e β(ɛ i µ) +1 (7.7) (7.8) ±1 7.2 V ɛ = p 2 /2m ɛ ɛ +dɛ 1 µ V 4πp 2 dp =2πV (2m) 3/2 ɛdɛ p = 2mɛ h 3 g(ɛ)dɛ = 2πV h 3 (2m)3/2 ɛdɛ (7.9)

72 7 1 2 2 7.5 (7.3) (7.7) 2π ɛdɛ h 3 (2m)3/2 0 e β(ɛ µ) 1 = N V (7.10) + µ(t,v,n) 7.3 (7.4) (7.8) e β(ɛ µ) 1 n i e βµ e βɛ i ( 1 ) (7.11) - (3.5) n i 1 1 2 - - - (7.10) ɛ = x 2 (p.10 ) N V 2π ( ) 3/2 2πmkT h 3 (2m)3/2 e βµ ɛe βɛ dɛ = e βµ (7.12) 0 h 2 e βµ 1 ( ) 3/2 N 2πmkT V (7.13) h 2 h/ 2πmkT 3 (7.13) ( V N ) 1/3 h 2πmkT (7.14) 3 p 2 /2m =3kT/2 p T = 3mkT x T = h/ p T = h/ 3mkT (V/N) 1/3 3.3 10 9 m p.6 O 2 h/ 2πmkT 1.8 10 11 m

7.4. - * 73 2 BE FD MB 3 N V (a) BE N V 1 W BE = N+V 1 C N = (N + V 1)! N!(V 1)! (b) FD 1 2 V N N V W FD = V C N = V! N!(V N)! (c) MB N! V W MB = V N (N + V 1)! = V (V +1)(V +2) (V + N 1) >V N (V 1)! (be) V! (V N)! = V (V 1)(V 2) (V N +1)<VN (fd) N! W FD <W MB <W BE MB 1 N/V N/V 1 2 (be),(fd) V N MB MB MB N! 7.4 - * µ (7.10) 2π(2m) 3/2 h 3 0 ( ) 3/2 ɛ dɛ 2πmkT e β(ɛ µ) 1 = 2 x dx π h 2 0 e x βµ 1 = N V (7.15)

74 7 n = N/V T ɛ = p 2 /2m 0 0 µ <0 µ =0 x µ =0 4 ( 2πmkT h 2 ) 3/2 ζ(3/2) < N V (7.16) (7.15) n = N/V (7.12) µ kt (log n 3 ) 2 log T + constant < 0 (7.17) 7.1(a) µ ( ) 3/2 2πmkTC ζ(3/2) = n (7.18) h 2 T C (n) µ =0 µ>0 T<T C µ =0 n T C T <T C (n) V N 0 = N V 2π(2m)3/2 ɛ dɛ h 3 0 e βɛ 1 ( ) 3/2 2πmkT = N V ζ(3/2) N 0 p =0 ɛ = p 2 /2m =0 - (7.18) h 2 N 0 (T ) N =1 ( T T C ) 3/2 (T<T C ) (7.19) 7.1(b) 4 ζ(z) p.65 ζ(3/2) = 2.612375... ζ(5/2) = 1.341487... Γ(1/2) = π Γ(3/2) = π/2 Γ(5/2) = 3 π/4 n=1 s n = φ(z,s) (z>1, s 1 z>0, s < 1) nz φ(z,1) = ζ(z) βµ 0 1 x z 1 dx Γ(z) 0 e x βµ 1 = φ(z,eα ), (α = βµ) dφ(z,e α )/dα = φ(z 1, e α )

7.4. - * 75 7.1 - (a) (b) T C (7.18) E = V 2π(2m)3/2 ɛ ɛ dɛ h 3 0 e β(ɛ µ) 1 ( ) 3/2 2πmkT 3kT 4 = V h 2 2 3 x 3/2 dx π 0 e x βµ 1 ( ) 3/2 2πmkT 3kT = V φ(5/2, e α ), (α = βµ) (7.20) h 2 2 E = 3 ( ) T 5/2 φ(5/2, e α ) NkT C 2 T C ζ(3/2) (7.21) φ(z, e α ) T >T C α(= βµ) (7.15) T C ( TC T ) 3/2 = 1 ζ(3/2) 2 π 0 xdx e x βµ 1 = φ(3/2, eα ) ζ(3/2) (7.22) T α T/T C C V Nk = 15 φ(5/2, e α ) 4 φ(3/2, e α ) 9 φ(3/2, e α ) 4 φ(1/2, e α ) (7.23) T T C ( φ(z, s) =s 1+ s ) 2 + s2 z 3 + < s s (s 1) z 1 s φ(z, e βµ ) e βµ

76 7 E 3 2 NkT, C V 3 2 Nk T T C µ =0 φ(z, 1) = ζ(z) E = 3 ( ) ζ(5/2) T 5/2 ( ) T 5/2 =0.77027 (7.24) NkT C 2 ζ(3/2) TC TC T T C C V Nk = 15 ζ(5/2) 4 ζ(3/2) ( T TC ) 3/2 ( ) T 3/2 =1.9257 (7.25) TC T =0 ɛ =0 W =1 0 3 lim C V =0 T 0 7.1(b) T C 5 C V /Nk T C 1.9257... 3/2-3 23 PV =2E/3 (7.20) T n n V P ( ) 3/2 2πmkT n = n C (T )=ζ(3/2) (V C = N/n C ) P (= 2E/3V ) (7.20) α =0 h 2 P = ζ(5/2) ζ(3/2) n C(T )kt =0.51351 n C (T )kt - 2 p =0 5 φ(1/2, 1) =

7.5. * 77 ɛ =0 p =0 =0 = - 1 4.2K ρ =1.25 10 2 kg/m 3 m =6.68 10 27 kg n = ρ/m T C h =6.63 10 34 Js k =1.38 10 23 JK 1 T C 2.8K T λ =2.17K λ - T λ λ 2 7.5 * (7.8) 1 f(ɛ) = (7.26) e β(ɛ µ) +1 (7.9) g(ɛ) 6 g(ɛ)f(ɛ)dɛ = N (7.27) (7.26) T 0 β 1 ɛ<µ f(ɛ) = 1/2 ɛ = µ 0 ɛ>µ (7.28) T =0 µf g(ɛ) dɛ = N (7.29) µ F T =0 1 E 0 = µf ɛg(ɛ)dɛ (7.30) 6 ɛ ɛ 0 g(ɛ)

78 7 7.2 W =1 3 lim S =0 T 0 lim C V =0 T 0 ɛ 0 µ F ɛ 0 T F =(µ F ɛ 0 )/k (7.31) T T F f(ɛ) 7.2 ɛ µ 4kT T f (ɛ) f (ɛ) = βeβ(ɛ µ) (e β(ɛ µ) +1) = β 2 (e β(ɛ µ) + 1)(e β(ɛ µ) +1) (7.32) ɛ µ ɛ = µ 4kT ϕ(ɛ) ( ) ϕ(ɛ)f(ɛ)dɛ = µ ϕ(ɛ)dɛ + π2 ϕ (µ) 6 (kt) 2 + 7π4 ϕ (µ) (kt) 4 + (7.33) 360 (7.29) (7.27) N = = µ g(ɛ)f(ɛ)dɛ g(ɛ)dɛ + π2 g (µ) (kt) 2 + (7.34) 6

7.5. * 79 N µ µ F g(ɛ)dɛ = π2 g (µ) 6 E 0 = E = = µf µ = E 0 + (kt) 2 + (7.35) µ = µ F π2 g (µ F ) 6g(µ F ) (kt)2 + (7.36) ɛg(ɛ)dɛ (7.37) ɛg(ɛ)f(ɛ)dɛ ɛg(ɛ)dɛ + π2 (g(µ)+µg (µ)) (kt) 2 + 6 µ ɛg(ɛ)dɛ + π2 (g(µ)+µg (µ)) µ F 6 (kt) 2 + (7.38) E = E 0 + π2 g(µ F ) (kt) 2 + (7.39) 6 C V π2 g(µ F ) k 2 T (7.40) 3 1 2 (7.9) 2 g(ɛ) = 4πV (2m)3/2 3N ɛ = ɛ, (ɛ 0) (7.41) h 3 2µ 3/2 F µ F (7.29) (7.30) E 0 = ( ) µ F = h2 3N 2/3 (7.42) 2m 8πV 3N µf ɛ 3/2 dɛ = 3Nµ F 2µ 3/2 0 5 F E = 3Nµ F 5 1+ 5π2 (kt) 2 + (7.43) 12µ 2 F

80 7 7.3 C V π2 Nk 2 T = π2 Nk T (7.44) 2µ F 2 T F T F = µ F /k C V =3Nk/2 - C V =3Nk +3Nk/2=9Nk/2 T T F T F K Θ D K T 3 T 2 1 1 T F M =63.5 10 3 kg/mol N A =6.02 10 23 mol 1 ρ =8.93 10 3 kg/m 3 n h =6.63 10 34 Js k =1.38 10 23 JK 1 m =9.11 10 31 kg T F = 82000K Θ D = 340K (6.17) 3K (7.33) Φ(ɛ) = ϕ(ɛ)f(ɛ)dɛ = ɛ ϕ(ɛ )dɛ, Φ( ) =0 Φ (ɛ)f(ɛ)dɛ = Φ(ɛ)f (ɛ)dɛ

7.5. * 81 Φ(ɛ) ɛ = µ Φ(ɛ) =Φ(µ)+Φ (µ)(ɛ µ)+ Φ (µ) (ɛ µ) 2 + 2! (7.32) f (ɛ) f (ɛ)dɛ = f( ) f( ) =1 (ɛ µ) 2n 1 f (ɛ)dɛ =0 1 (ɛ µ) 2n f 1 (ɛ)dɛ = 2 (2n)! (2n)! µ 2n = 2 (2n)! µ = 2 (kt) 2n 1 ϕ(ɛ)f(ɛ)dɛ = Φ(µ)+ π2 Φ (µ) 6 (ɛ µ) 2n f (ɛ)dɛ (ɛ µ) 2n 1 f(ɛ)dɛ Γ(2n) ( = 2 1 1 2 2n 1 0 x 2n 1 dx e x +1 ) ζ(2n) (kt) 2n (7.45) (kt) 2 + 7π4 Φ (4) (µ) (kt) 4 + 360 (7.45) 1 x z 1 ( dx Γ(z) 0 e x +1 = φ(z, 1) = 1 1 ) 2 z 1 ζ(z) p.65 ζ(2) = π 2 /6, ζ(4) = π 4 /90,

8 * ** 1 A p.6 2 T P m A mv 2 i /2 2 (T 1,P 1 ) (T 2,P 2 ) A m T 1 T 2 3 2V T P m A T p.6 1 4 1 1 5 m 1 m 2 2 1 2 w 12 T 1 2 a 1 a 2 n 1 n 2 p.5 6 T n m R M( m) v u ω x u v x > u 2m(v x + u) x y x 2 ds = y 2 ds = (x 2 + y 2 )ds = πr 4 /4 82

83 7 ω T m 1 m 2 2 8 P u P =2u/3 PV 5/3 = 9 x =0 x = L x- v m x-p x = L v u dl 10 3 µ 6 λ TV 2/3 = 11 m l E ω E/ω E 12 n N 1 13 x τ 1 1/2 a t x x 2 t N = t/τ n N n x =[n (N n)]a =(2n N)a 14 a, b 2 N 2 ɛ 0 x X = ( F/ x) T ɛ 0 =0 15 ɛ ɛ = cp P = u/3 C V C P ɛ = c (mc) 2 + p 2 ɛ = cp

84 8 ɛ = p 2 /2m * 16 Φ(r 1, r 2,...) Q N (V,T) = 1 N! e Φ/kT dr 1 dr 2... 2 (1) V b 1 2 (2) N/V 2Na/V (a >0) V/b N V Nb kt Na/V 17 2 ±Q N 2N 2 L V = L 2 m 2 Q 1,Q 2 2 r 12 U 12 = aq 1 Q 2 ln(1/r 12 ) a (1) T (2) (1) P =0 T C T C (3) T T C T>T C - [0,L] [0, 1] L 18 19 Z N (T,V ) ( log Z N / V ) T = P/kT F = kt log Z N,P = ( F/ V ) T x = L ξ(x i L) ξ(x i L)/ x i = ξ(x i L)/ L ξ(x L) =0(x<L), ξ(x L) = + (x L)

85 20 m A m B 2 A B N A N B A B AB ɛ 0 A B AB 3 N A +N AB = N B + N AB = log Z N A,N B,N AB 21 1 3 φ(q) =cq 2 /2 bq 3 (b> 0) q a( kt/c) ba 3 ca 2 /2 q =0 22-23 P =2u/3 PV = kt log Ξ ɛ =(2/3)dɛ 3/2 /dɛ 24 22 µ =0 25 15 P = u/3 23 log Ξ 26 15 g(ɛ)dɛ =2 V 4πp 2 dp/h 3 = V (8π/c 3 h 3 )ɛ 2 dɛ 27 σ = ±1/2 η B H ±η B H ɛ = p 2 /2m ± η B H η B µ B T =0 ±η B p =0 p ± p 2 ±/2m η B H p ± 28 20 A B AB T =0 m A = m B (= m) N A = N B (= N) ɛ 0 (i) (ii) (iii) 3

53 18 18 18 18 68 H 16 H 16 5, 47 4 48 20 18-16 43 71 10 38 38 9 35 Γ 18 53-44 36 36, 73-40 15 25 56 52, 54-56 54 55 77 12 43 54 69 36 25 15 52 19 15 10 72, 78 78-68 - 68 6 16 41 39 5 13, 15 3 14, 16 5 50 9, 29 31 2, 27 2 2 25, 27 3 76 52 18 43 8 5 33 33 31 77 86

87 65, 74, 81 78 T 80 T -P 46 T 3 62 65-62 54 8 44 23 21 22 20 47 13 5 30 2 60 66 37 3 62, 76 24 18 19 10 54 15 5 49 57 49 57 77 69 77-69 71 77 69 70 71 51 1 67 16 55 6 16 39 38 5 5 55 39 74-74 - 69 70 69 30 24 69 38 4 32 13 8-70 - 12, 29 23 23 µ 18 31 42 11 λ 77 83 22 21 24 44

88 43 6 34 69 63-67 51-58 21