master.dvi

Size: px

Start display at page:

Download "master.dvi"

ありあのしろ
5 years ago
Views:

1 4 Maxwell- Boltzmann N T R R 5 R (Heat Reservor) S E R 20

2 E 4.2 E E R E t = E + E R E R Ω R (E R ) S R (E R ) Ω R (E R ) = exp[s R (E R )/k] E, E E, E E t E E t E exps R (E t E) exp S R (E t E ) E E p(e)/p(e ) = exp[s R (E t E)/k S R (E t E )/k] E p(e) exp S R (E t E) p(e) exp S R (E t E)/k (4.1) 2 p(e) S R (E t E) E S R (E t E) = S R (E t ) E S R E + E=ER = S R (E t ) E T + (4.2) T S R (E R ) E R 21 = 1 T

3 E R E ( E/E R 1 ) (4.2) (4.1) E p(e) exp( βe), (β = 1/kT) (4.3) exp( βe) 1/kT β Liouville p(e i ) = 1 Z exp( βe i) (4.4) H(q, p) p(q, p) exp[ βh(q, p)] 1 (4.4) Z Z = exp( βe i ) (4.5) i (4.4) A = i p(e i )A(E i ) = i A(E i)e βei i e βei (4.6) i E = E ie βei i e βei 4.3 (4.4) Z (Partition Function) 22

4 (Sum over States) (4.4) H(q, p) Z Π i dq i dp i e βh(q,p) (4.7) H(q, p) e βh(q,p) E = 1 Z Z β = lnz β (4.7) Π Π n i=1a i = a 1 a 2 a n Σ ρ(e) ρ(e) = i δ(e E i ) (4.8) E 0 E 1 E1 E 0 deρ(e) E E + δe ρ(e)δe E Ω(E) E Ω(E) = de ρ(e ), ρ(e) = dω(e) de Ω(E) E 3N/2 N ρ(e) Ω(E) ρ(e) = E3N/2 E = 3N 2 E3N/2 1 E 3N/2 S(E) ρ(e) ρ(e) Ω(E) = exp S(E)/k 23

5 : x = a x = dxδ(x a)f(x) = f(a) (4.8) f(x) = 1 E 0 E 1 E i 24

6 4.3.2 Helmholtz Helmholtz Z (4.5) Z = e βei = dee βe δ(e E i ) = dee βe ρ(e) i i = dee S(E)/k βe = dee βf(e) (4.9) F(E) = F(T) + f 2 2 (E E ) 2 + F(T) E F(T) = E TS(E ), S(E) E E=E E (4.9) Z = exp[ βf(t)] dee β[f2(e E ) 2 /2+ ] = 1 kt = exp[ βf(t) (1/2) ln(βf 2 /2π) + ] exp[ βf(t)] (4.10) (4.9) F(E) F(E) = E 3 NkT lne = NkT[f(ε) lnnkt] 2 f(ε) = ε 3 2 lnε f(ε) ε 6 F(T) E kt F(T) = 3 2 NkT 3 NkT ln(3nkt/2) (4.11) 2 (4.10) (3.8) T (4.10) F(T) F(T) F(T) = 1 β lnz, Z = e βf(t) (4.12) 25

7 f(ε) ε 6 f(ε) (4.10) F T = E T T S E E S = S (4.13) T S V 0 N V 0 V 1 V 1 /V 0 N (V 1 /V 0 ) N V N S(E, V ) A, B A, B E A, E B V A, V B E tot V tot 26

8 V 1 V 0 7 V 0 N E tot = E A + E B, V tot = V A + V B V E A, V A A E B, V B B 8 2 A, B Ω A, Ω B Ω A (E A, V A ) = exp[s A (E A, V A )/k], Ω B (E B, V B ) = exp[s B (E B, V B )/k] Ω A Ω B S A (E A, V A ) + S B (E B, V B ) 27

9 S A (E A, V A ) = S B(E B, V B ) E A E B (4.14) EB=E tot E A S A (E A, V A ) = S B(E B, V B ) V A V B (4.15) VB=V tot V A (4.14) 2 (4.15) B 9 A δv A B W p E A = W = pδv A, E A V A = p δv A p E A, V A A 9 A E A V A ds A (E A, V A ) = S A(E A, V A ) E A ( pδv A ) + S A(E A, V A ) V A δv A = 0 S A E A 1/T A S A (E A, V A ) V A = p T 2 F F V S = (1 T E ) E V T S V (4.16) = p (4.17) 28

10 4.5 (4.13) (4.17) F(T, V ) F(T, V ) T F(T, V ) = S(T, V ), = p V df(t, V ) = S(T, V )dt pdv (4.18) S(T, V ) ds = 1 T de + p T dv, TdS = de + pdv S E F(T, V ) (4.10) F = E TS S F Helmholtz S = (E F)/T = 1 TZ = k i e βei Z E i e βei + 1 kt lnz T i ln(e βei ) + k i e βei Z lnz = k i (e βei /Z)ln(e βei /Z) = k i p i lnp i (4.19) (4.10) (4.5) (4.12) p i = e βei /Z 29

11 4.6 (E i = ε, ε + ) T Z F Z = e βε + e β(ε+ ) = e βε (1 + e β ) F = kt log Z = kt log[e βε (1 + e β )] = ε kt ln(1 + e β ) (4.18) S = F(T) = k ln(1 + e β ) + kt e β T 1 + e β kt [ 2 = k ln(1 + e β ) + ] 1 kt 1 + e β (4.20) /kt T 0 = /k (4.20) e β 1, T T 0 S = k kt e β + 0, (T/T 0 0) e β = 1 kt +, T T 0 [ ] /kt S = k ln(2 /kt + ) /kt + [ = k ln 2 2kT + ] 2kT + O(( /kt)2 ) 10 3 ln 2 kt (4.19) 30

12 1.0 S(T) kt/ q p q p h/2 q p q p = h 1 N 3 h 3N 1 = 1 h 3N Π i dr i Π j dp j dqdp (Action) h 31

13 d 3 r 1 d 3 r 2 = V 2 L x 11 L x 2 L x 2 L L x 1 0 L x 1 11 N N! T N H = 1 2m p2 i i 32

14 Z [ 1 Z = h 3N Π N N! i=1d 3 r i d 3 p i exp β ] p 2 i/2m i = V N ( ) 3N/2 2πm h 3N (4.21) N! β V N (3.11) F S C F = 1 [ ( ) ] 3 2πm β lnz = N β 2 ln + ln(v/h 3 ) lnn + 1 β S = F [ ( ) ( ) ] 3 2πm V T = Nk 2 ln + ln β Nh Nk/2 [ (2πm ) ] 3/2 V e 5/2 = Nk ln β Nh 3 (4.22) C = T S T = 3Nk/2 Sackur-Tetrode N! 2 (Extensive Variable) (Intensive Variable) (4.22) N! ln(v/n) lnv 2 N ln 2 λ p p = h/λ h kt/2 λ T h 2 2mλ 2 T = 1 2 kt [ S = Nk ln (2π) 3/2 e 5/2 v ] λ 3 = 32 ] [2πe Nk ln 5/3 TT0 h 2, T 2mv = 1 2/3 2 kt 0 v = V/N v 1/3 S/Nk

15 S(t)/Nk T/T 0 12 Sackur-Tetrode ω i N H(q, p) = ( ) 1 2m p2 i + mω2 i 2 q2 i i Z = 1 [ 1 h N Π i dq i dp i exp[ βh(q, p)] = Π i dq i dp i exp β ] h 2m p2 i βmω2 i qi 2 2 [ ( ) 1/2 ( ) ] 1/2 ( ) 1 2πm 2π 1 = Π i h β βmωi 2 = Π i β hω i (4.23) N! (4.23) F = 1 β lnz = kt i ln (β hω i ) S = F T = k i C = T S T = Nk ln (β hω i ) + Nk N! Extensive 34

16 4.6.4 µ i E, µ i B µ i E 13 z 1 ( ) h = µ 0 E cosθ, (µ 0 = µ i ) θ z φ (θ, φ) 1 z rot = 1 dφsin θdθ exp[βµ 0 E cosθ] = 2π dve βµ0ev, 1 = 4π βµ 0 E sinh (βµ 0E) (v = cosθ) Z 1 N zrot N F rot = N β lnz rot = NkT ln 4π sinh(βµ 0E) βµ 0 E (4.24) 35

17 S = F T = Nk ln 4π sinh(βµ 0E) + NkT βµ 0 E = Nk [ µ 0E [ ln 4π sinh(βµ 0E) βµ 0 E coth(βµ 0 E) + 1 βµ 0 E s(t) = ln[4πt sinh(1/t)] 1 t coth(1/t) + 1 kt 2 coth(βµ 0E) + 1 T ] = Nks(t) ] t = kt/µ 0 E s(t) 14 { ln(4π) + 1 5/6t s(t) 2, t 1 ln(4πt) 1/t, t s(t) t 14 z- Ω 1 µ z i = µ 0 βµ0e cos θ dω cosθe z rot (4.24) µ z i = N β i lnz r E = F E = Nµ 0L(y), (y = βµ 0 E) 36

18 L(y) Langevin L(y) = coth y 1 y Langevin y { y/3 y 1 L(y) 1 1 y kt µ 0 E 1 N i µ z i { µ0 T µ 0 E/k µ 2 0 E/3kT T µ 0E/k Debye µ z i /µ 0 t = kt/µ 0 E t 15 : = µ 0 E z rot = e βεi = e βµ0e + e βµ0e = 2 cosh(βµ 0 E) i=1,2 F = N β ln[2 cosh(βµ 0E)] S = F T = Nk ln[2 cosh(βµ 0E)] Nk µ 0E kt tanh(βµ 0E) = Nks(t) s(t) = ln[2 cosh(1/t)] 1 t tanh(1/t) 37

19 t = kt/µ 0 E s(t) H t kt/µ 0 H kt/µ 0 (H/2) = 2t s(2t) H T A A B C B A s C D t 16 38

20 4.6.5 He, Ne, Ar ( 3) 3 2 ( ) Lagrangian m 1, m 2 1, 2 R G r 1, r 2 17 m 1 r 1 + m 2 r 2 = 0, r 1 r 2 = r (4.25) r r 1 = m 2 M r, r 2 = m 1 M r, (M = m 1 + m 2 ) m 1 m 2 r 2 r 1 G R 17 (4.25) T 39

21 T = m 1 2 (Ṙ + r 1) 2 + m 2 2 (Ṙ + r 2) 2 = M 2 Ṙ2 + m 1 2 r m 2 2 r 2 2 = M 2 Ṙ2 + µ 2ṙ2 1, µ = (4.26) m 1 m 2 µ (reduced mass) (r, θ, φ) r = (r cosφsin θ, r sinφsin θ, r cosθ) ẋ = ṙ cosφsin θ r sinφ φ sin θ + r cosφcos θ θ ẏ = ṙ sin φsin θ + r cosφ φ sin θ + r sin φcosθ θ ż = ṙ cosθ r sinθ θ µ 2ṙ2 = µ [ ṙ 2 + r 2 ( 2 θ 2 + sin 2 θ φ ] 2 ) (4.26) r V (r) L = T V (r) = M 2 Ṙ2 + µ 2 ṙ I( θ 2 + sin 2 θ φ 2 ) V (r), (I = µr 2 ) I r, θ, φ p r, p θ, p φ P = L Ṙ = MṘ, p r = L ṙ = µṙ, p θ = L = I θ, p θ φ = L φ = I sin2 θ φ h = P Ṙ + p rṙ + p θ θ + pφ φ L = P 2 2M + 1 2µ p2 r + V (r) + 1 2I ( p 2 θ + 1 ) sin 2 θ p2 φ M Z t Z 1 z N I r r r 0 Z = Z t z N, z = z rot z vib 40

22 z rot, z vib 1 z rot = 1 h 2 dθdp θ dφdp φ exp [ βh] = 1 [ h 2 dθdp θ dφdp φ exp β 2I (p2 θ + 1 ] sin 2 θ p2 φ) = 2πI βh 2 dφdθ sin θ = 8πI βh 2 β E rot E rot = N lnz rot β = N ln(8πi/βh2 ) β = NkT C rot = de rot dt = Nk 2 V (r) r = r 0 V (r) = V (r 0 ) V (r 0 )δr 2 +, δr = r r 0 r h v = V (r 0 ) + 1 2µ p2 r + 1 V 2 µω2 v δr2 +, ω v = (r 0 ) µ ω v E v = NkT, C v = de v dt = Nk 2 4 7Nk/2 5Nk/2 41

23 (k) 3/ Gibbs 18 V A V B 18 V V A, V B A, B 2 N A, N B p m A, m B pv A = N A kt, pv B = N B kt p(v A + V B ) = (N A + N B )kt N A /V A = N B /V B = (N A + N B )/(V A + V B ) V N Sackur-Tetrode { [ (2πm ) ] } 3/2 e 5/2 S = kn ln β h 3 + ln V N (4.27) 42

24 2 Z = V ( A NA 2πmA N A!h 3NA β ) 3NA/2 V B N B N B!h 3NB ( 2πmB β ) 3NB/2 (4.28) V A V B A, B 2 N A, N B (4.27) Z = V NA N A!h 3NA ( 2πmA β ) 3NA/2 V NB N B!h 3NB ( 2πmB β ) 3NB/2 (4.29) (4.28) (4.29) V S = N A ln(v/n A ) + N B ln(v/n B ) N A ln(v A /N A ) N B ln(v B /N B ) = N A ln(v/v A ) + N B ln(v/v B ) (4.30) = N A ln(n/n A ) + N B ln(n/n B ) N A = N B N ln 2 Gibbs (4.29) Z = V NA+NB (N A + N B )!h 3(NA+NB) ( ) 3NA/2 ( 2πmA 2πmB β β ) 3NB/2 (4.31) (4.29) (4.31) N A!N B! (N A + N B )! (4.31) Gibbs 43

4.6 (E i = ε, ε + ) T Z F Z = e βε + e β(ε+ ) = e βε (1 + e β ) F = kt log Z = kt log[e βε (1 + e β )] = ε kt ln(1 + e β ) (4.18) F (T ) S = T = k = k

4.6 (E i = ε, ε + ) T Z F Z = e ε + e (ε+ ) = e ε ( + e ) F = kt log Z = kt loge ε ( + e ) = ε kt ln( + e ) (4.8) F (T ) S = T = k = k ln( + e ) + kt e + e kt 2 + e ln( + e ) + kt (4.20) /kt T 0 = /k (4.20)