September 25, ( ) pv = nrt (T = t( )) T: ( : (K)) : : ( ) e.g. ( ) ( ): 1
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1 September 25, p = nr = t : : K : : e.g : 1
2 intensive variable e.g. extensive variable e.g Equation of State e.g. p = nr X = A Quantity of Heat Q > 3 > 2 Q Q = C = C C 1 : 1 heat of capacity C 2 : 2 C Q = C 2
3 2.1.2 : Calorie 1cal=1g =1cal/gK Lateut heat e.g. e.g. L=80cal/g e.g. L=540cal/g calorimeter thermometer calorique: work: W = W = ps h N m = 1J 1N = 1kg m/s 2 1J = 1kg m 2 /s 2 MKS 3
4 ln = log e natural logarithm e 3 = ln 20 = W = 2 = nr pd = nr ln d 2 e = lim n = n n { y = ln x x = e y d ln x dx = 1 x cal=4.186J ˆ Rumford ˆ Mayor ˆ Jouls : 2000kcal 1 = = Q = 2000kcal = cal = J 4
5 Q W 1 : P 1, 1, +Q,+W 2 : P 2, 2, 2 Q + W = U = Q + W = U 2 U 1 U U Q W p- 1 2 A B = Q A Q B A B W A = pd A W B = pd B pd A B pd = W A + W B Q A + W A = Q B + W B W A + W B = Q A Q B A: + 5
6 B: 1 1mol 15 2 = 2 Q = P 1 2 pd 2 = 2, p = nr B 2 1 Q = p 1 R 1 d 2 = p 1 R 1 ln = R 1 1 ln 2 = J = J = 2.3 ˆ dt, dp, d, du ˆ d W, d Q d W = pd du = d Q + d W = d Q pd d Q = du + pd 6
7 2.3.1 du d d, d du U, du = U U d + d U U U U f x, y f x = lim f x + x, y fx, y x 0 x f y = lim f x, y + y fx, y y 0 y f x, y x, y a, b, c f x, y = ax 3 + bxy 2 + cy 3 f x = 3ax2 + by 2 7
8 f y f x = 2 x 2 f y 2 = y 2 f x y = x = 2bxy + 3cy2 f = 6ax x f = 2bx + 6cy y f = 2by y 2 f y x = y f = 2by x f x, y x, y 2 f x y = 2 f y x f x y = f x x, y df dx = f x 2 z=fx,y f x x 0, y 0, f x y 0, y 0 x 0, y 0, z 0 = f x 0, y 0 x y U U : : 8
9 adiabetic: quasi-static: : : dq = du + pd = du = C v d p = nr p = nr d d C v d + nr d = C v d = nr d C v d = nr 1 d 9
10 C v ln 2 1 = nr ln 2 ln 2 1 = nr ln C v = ln 2 2 nr Cv 2 1 = = 2 2 nr Cv nr 1 Cv nr C v 1 1 = 2 2 nr C v nr C v = C r C v C v = C r C v 1 = γ γ = Cr C v 1 γ 1 1 = 2 γ 1 2 γ 1 = cond p = nr p p γ 1 = const 10
11 p γ = const p = nr = const isotherm p γ = const adiabat ˆ ˆ U = C v 2 1 < n 1 2 W W = pd = c d p γ = c = p γ 1 γ 1 = p 2 γ 2 p = c γ x α dx = 1 α + 1 xα+1 x, α γ 11
12 1 [ W = c ] 1 γ 2 1 γ c [ ] = 1 γ 2 γ 1 1 γ 1 [ ] = p2 γ 2 γ 2 p 1 γ 1 γ 1 1 γ 1 = 1 γ [p 2 2 p 1 ] 1 = 1 γ [nr 2 nr 1 ] = nr 1 γ 2 1 C p C v = nr γ = C p C v nr 1 γ = C p C v 1 C p C v = C p C v C v C p C v = C v W = C v 2 1 = U 2 U 1 = U = h 10km p h p = p h + h p h = gρ h 12
13 ρ : g : h p h + h = p h + dp h dh dp h = gρ h dh dp dh = gρ d Q = du + pd = dh dp d Q = 0 H = U + p dh dp = 0 dh = C p d dh = dp U = C v + U 0 H = U + p = U + nr = C v + nr + U 0 = C p + U 0 13
14 dh = C p d = dp = dp dh dh = gρ dh d = gρ C p dh = gm C p dh M = ρ : c p = C p M : d = g C p dh c p = C p M C p C v = cr γ = C p C v = 1.41 C p = C p 1 γ = nr C p = nr 1 1 γ = γ γ 1 nr c p = C p M = γ nr γ 1 M M n = m: m = 28.8J/K mol 14
15 d = γ 1 gm γ R dh = m/s kg/mol dh kg m 2 /s 2 /k mol = dh K/m 100m 1K 0.6K : γ 2: : 1-2,3,9, Kinetic theory = = x v = v x, v y, v z v = v x, v y, v z p x = 2mv x p = v 2 x = v 2 y = v 2 z = 1 3 v2 p y = p z = 0 N 2 v x p x = N m vx 2 N = m1 v 2 : 3 15
16 p = 1 3 Nm v 2 = 2 3 N 1 2 m v 2 = 2 3 U : p = nr nr = 2 3 U U = 3 2 nr U = C v + U 0 C v = 3 2 nr C p = C v + nr = 5 2 nr γ = C p C v = = 5 3 = U = N 1 2 m v 2 = N 1 2 m vx 2 + v 2 y + v 2 3 z = 2 nr 1 2 m v 2 = 3 nr 2 N = 3 2 k B nr N = R N A = k B = J/K : N A = N n : k B O 2, N U = 1 2 k B = 5 2 k B 16
17 U = N 5 2 k B = N 5 nr 2 N = 5 2 nr C v = 5 2 nr C p = 5 2 nr + nr = 7 2 nr γ = = : : 3.1 a P. Clausius b = 1. a b b a b a Q 17
18 a 2. b a heat engine : heat resewair R 2 Q 2 R 1 Q 1 < Q 2 Q 2 Q 1 W Sadi Carnot A B 2 1 C D 1 2 Q 2 Q 1 W = Q 2 Q 1 efficiency η = W Q 2 = Q 2 Q 1 Q 2 = 1 Q 1 Q 2 < 1 Heat Pump
19 3.2.2 c,c c Q 2 Q 1 = W Q 2 Q 1 = W η = W Q 2 W = ηq 2 η = W Q 2 W = ηq 2 Q 2 = Q 2 W = η Q 2 = η Q 2 W W = η Q 2 ηq 2 = η η Q 2 W W = Q 2 Q 1 Q 2 Q 1 = Q 1 Q 1 η > η Q 1 Q 1 W W b η η c η η η = η = η = W Q 2 = Q 2 Q 1 Q 2 = 1 Q 1 Q 2 19
20 1 η c = Q 1 Q 2 = n p = nr Q 1 Q Q 2 = 2 pd = nr 2 2 = nr 2 ln 1 2 d Q 1 = pd = nr 1 2 = nr 1 ln = nr 1 ln d γ 1 2 = 1 γ
21 1 γ 1 4 = 2 γ γ 1 = 3 4 γ 1 2 = 3 4 Q 1 Q 2 = nr 1 ln nr 2 ln = η = 1 Q 1 Q 2 η c = Q 1 Q Q 1 1 Q S = Q 2 2 = Q entropy Q 2 2 < Q
22 S 2 = Q 2 2 R 1 S 1 = Q 1 1 R 2 S 1 S 2 = Q 1 1 Q 2 2 > Q S = Q Q=0 S = p S 13 = Q 13 S 32 = 0 S 12 = S 13 + S 32 = S 13 22
23 S 12 = S 15 + S 54 + S 46 + S 62 S 54 = S 62 = S 53 + S 64 = 0 S 46 = S 64 = S 53 S 23 = S 15 + S 46 = S 15 + S 53 = S 13 = S 12 S 12 = S 13 S = Q 2 p 1 p n 2 23
24 Q = 2 pd 2 1 = nr = nr ln 2 d S = Q = nr ln 2 nr ln 2 J/K n S 12 = S 13 + S 32 1 = nr ln γ 1 1 γ 1 1 = 2 γ γ 1 =
25 γ 1 ln 1 3 = ln 2 1 S 12 = nr γ 1 ln 2 1 nr = C p C v γ = C r C v nr γ 1 = C p C v C p 1 = C v C v S 12 = C v ln ds = dq c.f. d W = pd d Q = ds du = d Q + d W = ds pd
26 ds = du + pd U S, = : S U, U U, p = S 1 S =, U U p S = U ds = du+pd du = C v d p = nr 1 nr ds = C v d + d S 2 S 1 = C v d + nr 1 2 = C v ln + nr ln d
27 2 S, = C v ln + nr ln + S isotherms 2 isocores 2 adiabats S, ds = S d + S d S ds = du + pd du = U d + U d ds = 1 [ U 1 d + U + P ] d S = 1 U = C v
28 U = C v S = 1 U + P = 2 S = 1 2 U = 1 U + 2 U 2 p U p U = p p p = 0 p S = = p p p + p S = p Maxwell 28
29 3.4.4 p = nr p = nr p = nr S = ds = C v nr d + d S, = C v ln + nr ln + S U = nr p = p p = van der Wals p = nr b a 2 S, U, F = a: b: S p = = nr b 29
30 C v ds = C v d + nr b d b S, = C v ln + nr ln + S b U nr nr = b b a 2 = a 2 U U du = d + d = C v d + a d 2 U, = C v a + U g g 0 S 1 S 1 = Q cal/g 100g 4.2J/cal = 273K = 123J/K
31 ds = = du + pd dh dp = dh = C pd C p = H p C p = 1cal/g K 4.2J/cal = 420J/K 1 C p 2 1 S 2 = d = C p ln = 420J/K ln 273 = 22.5J/K S = S 1 + S 2 = = 145J/K > S = Q Q = Q >
32 S = nr ln 2 > p 1 > p 2 = p 1 = nr = p 2 2 S = nr ln 2 > 0 p 1 p 2 = 2 > mixing entropy 2 = A + B S = S A + S B S A = Cv A ln + n A R ln 0 A 0 + S 0 S B = C B v ln + n B R ln 0 B 0 + S 0 32
33 S = S A + S B S A = Cv A ln + n A R ln + S S B = C B v ln + n B R ln + S S = S S = n A + n B R ln n A R ln 0 = n A R ln + n B R ln > 0 A B A 0 n B R ln B : p S du = ds pd du = 0 ds pd = 0 ds = pd ds = 2 1 S 2 S 1 = 2 S 2 S 1 1 S 2 S 1 = Q 2 Q 1 33
34 W = Q 2 Q 1 pd = ds p S : p S p, S, 2 pd = S p S = t p = S = p d S 3.6 : : W U,, N N U S U,, N = k B l n W N W R = N 0 N 34
35 N 0 : U N p 2 = p 2 2m N = U U N 2m W p W p U 2m N p 0 8N W = W R W p N = N 0 U 2m N p 0 3N p 0 ; S = k B ln W N = k B ln N 0 U 2m N p 0 3N = Nk B ln 3N U N 0 2 k N B ln 2m p 2 0 d S = d + pd 1 S = U = 3 2 Nk 1 B U U = 3 2 Nk B = 3 2 nr p S = U U 35 = Nk B 1
36 p = Nk B = nr 4 : : 4.1 Q S W U = Q W W = Q U S U = S 2 S 1 U 2 U 1 W U 1 S 1 U 2 S 2 F = U S4.1.1 W F 1 F W = 2 pd = nr ln 36 2
37 U = 0 F 2 F 1 = F = U S = nr ln = W 2 W = F 1 F 2 W = 0 F 1 F 2 F 2 F 1 F F n 1, n 2 = + 2 = F,, = F 1, + F 2, = F F, = 0 F, = = F1 F1 F2 + 2 F2 2 2 = 0 37
38 2 = F, = U S df = du ds Sd = pd Sd F = p p 1 = p 2 p 1 = p F = U S S df = Sd pd du = ds pd F S = F p = 38
39 4.2 Gibbs W = p 2 F 1 F 2 F 2 + p 2 F 1 + p G = F + p = U S + p 42.1 G 2 G 1 G G, P G 4.3 U S, du = ds pd = p = U 4.3.1a S U 4.3.1b S = 2 U S = 2 U S p = S 39
40 S, U p = 4.3.1c S S : F = U S df = du ds Sd = Sd pd, F df = Sd pd F S = 4.3.2a F p = 4.3.2b S = 2 F = 2 F p = S = p 4.3.2c : G = F + p dg = df + pd + dp = Sd 40
41 dg = Sd + dp4.3.3, p G S = = S p G 4.3.3a p G 4.3.3b p = 4.3.3c p : H = U + p dh = du + pd + dp = ds dh = ds + dp4.3.4 S, p H = = H 4.3.4a S p H 4.3.4b p S = 4.3.4c p S S p du = ds pd df = Sd pd dg = Sd + dp 41
42 dh = ds + dp F = U S G = F + p = U S + p H = U + p = G + S S p : 1 f x f x x = x 0 f x 0 = v 0 y = v 0 x x 0 + f x 0 y ψ 0 ψ 0 = v 0 x 0 + f x 0 f x v = f x = df dx ψ v = f x vx f x ψ v f x envelope f x ψ v 42
43 4.3.2 f x = 1 2 x2 x + 1 v = df dx = x 1 x = v + 1 ψ v = f x vx ψ v = 1 2 x2 x + 1 vx = 1 2 v + 12 v v v + 1 = 1 2 v2 + v v v2 v = 1 2 v2 v ψ v f x x = dψ dv = v + 1 f x = ψ v + vx = 1 2 v2 v vx = 1 2 x 12 x x 1 x , S, p, 4 U, F, G, H 2 43
44 5 closed system open system 5.1,, 2 N,, 2 N = N 1 + N 2 N 1, N 2 F = F 1 1,, N 1 + F 2, 2, N 2 F N 1,,N F1 = N 1 = 0, + F2 N2 N 2, 2 N 1 N F1 N 1, = F2 N 2, 2 F,, N η = F N, µ = η 1 = η G = g, p = G N,p N 44
45 µ 1 G = µn F = G p = µn p U = F + S = S p + µn d = ds pd + µdn dp = sd + vdp5.1.14, p, µ µ, p µ, p S = v = µ a p µ b p dµ = S N d N dp = sd vdp s = S N : 1 v = N : phase: 45
46 5.2.1 A = B p A = p B µ A = µ B, p µ A, p = µ B, p 5.2.1, p 2, p, p +, p + p µ A +, p + p = µ B +, p + p, p, p µa µa µ A +, p + p = µ A, p + + p + p p µa µa dµ A = d + dp p p µ A +, p + p = µ A, p + s A + v A p + µ B +, p + p = µ B, p + s B + v B p + 2 µ A, p = µ B, p s A + v A p = s B + v B p 46
47 v A v B p = s A s B p = s A s B v A v B dp d = s A s B v A v B = s A s B v A v B l AB = v A v B l AB : A B 1 v A v B : 1 N dp d = l AB N v A v B N L AB = A B dp d = L AB A B L AB, A B p g N N = 1 N 18 A 47
48 L AB = l AB N = 540cal/g = J/g A = v A N = 1 ρ A ρ A = 0.60kg/m 2 = g/m 2 B = v B N = 1 ρ B ρ B = g/m 2 ρ B ρ A A B p = L AB A B L AB A L ABρ A p = L AB ρ A = J/g J/m K = J/m 2 K = Pa/K p = Pa = = 5.6K 48
49 A B L AB A A B A = nr p dp d = L AB A B L AB A = L AB nr p 2 dp p = L AB 1 nr d 2 p ln = L AB 1 p 0 nr + 10 p = p 0 e L AB 1 nr p = p 0e L AB nr = 373K p = 10 5 Pa p van der Waals p = nr b a 2 49
50 { v = N nr = k B N p = k B v b 0 a 0 v { a = a 0 N 2 b = b 0 N = p v ˆ ˆ v A, C, B ˆ A, B v ˆ C v A B A B p A = p B F = G p = µn pvn = µ pv N 50
51 f = F N = µ pv5.3.2 dµ = sd + vdp df = sd pdv5.3.3 p = f v p = k B v b 0 a 0 v 2 f = f, v v f = = v 0 pdv v v 0 kb a 0 pdv v b 0 v 2 v b0 1 f, v = k B ln a 0 + h v 0 b 0 v f v, double tangent v A, B ˆ A, B ˆ A, B p = f v : ˆ A, B y y f µ f = f v + ψ v = pv + ψ 51
52 ψ = f + pv = µ µ A = µ B µ A = f v A + p 0 v A µ B = f v B + p 0 v B µ A µ B = 0 f v B f v A + p 0 v B v A = 0 vb vb p v dv + p 0 dv = 0 v A v A vb v A p 0 p v dv = 0 p = p 0 2 f v A, B 5.4 phase rule k, mixture F,, N 1, N 2,, N k N i : i df = Sd pd + µ 1 dn 1 + µ 2 dn µ k dn k dg = Sd dp + µ 1 dn 1 + µ 2 dn µ k dn k G, p, N 1, N 2,, N k = µ 1 N 1 + µ 2 N µ k N k N = N 1 + N N k 52
53 c i = N i : concentration N G = Ng, p, c 1, c 2,, c k k i=1 c i = 1 c 1,, c k g = µ 1 c 1 + µ 2 c 2 + µ k c k f N j i : j i c j i = N j i : j i N j N j = N j 1 + N j N j k, p i = 1, 2,, k k f 1, p 2 µ 1 i = µ 2 i = µ 3 i = = µ f i f k c 1 1,, c f 1 c 1 k.,, cf k c i 1 + c i c i k = 1 k ν = 2 + f k f k f 1 = 2 f + k
54 f: k: k = f = 1 ν = = 2, p k = 1, f = 2 ν = = 1, p k = 1, f = 3 ν = = 0, p k = 2, f = 1 ν = = 3, p, G k = 2, f = 2 ν = = 2, p < 100 C 54
55 6. 0?k = 0, f = 1 ν = = 1 p = c 4 : 3 5,8,13,15 7/31 BOX 55
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5 1.2, 2, d a V a = M (1.2.1), M, a,,,,, Ω, V a V, V a = V + Ω r. (1.2.2), r i 1, i 2, i 3, i 1, i 2, i 3, A 2, A = 3 A n i n = n=1 da = 3 = n=1 3 n=1
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現代物理化学 1-1(4)16.ppt
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1 1 1.1 1.) T D = T = D = kn 1. 1.4) F W = F = W/ = kn/ = 15 kn 1. 1.9) R = W 1 + W = 6 + 5 = 11 N. 1.9) W b W 1 a = a = W /W 1 )b = 5/6) = 5 cm 1.4 AB AC P 1, P x, y x, y y x 1.4.) P sin 6 + P 1 sin 45
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I II II 1 dw = pd = 0 1 U = Q (4.10) 1K (heat capacity) (mole heat capacity) ( dq / d ) = ( du d C = / ) (4.11) du = C d U = C d (4.1) 1 1 du = dq + dw dw = pd dq = du + pd (4.13) p dq = d( U + p ) p (4.14)
50 2 I SI MKSA r q r q F F = 1 qq 4πε 0 r r 2 r r r r (2.2 ε 0 = 1 c 2 µ 0 c = m/s q 2.1 r q' F r = 0 µ 0 = 4π 10 7 N/A 2 k = 1/(4πε 0 qq
49 2 I II 2.1 3 e e = 1.602 10 19 A s (2.1 50 2 I SI MKSA 2.1.1 r q r q F F = 1 qq 4πε 0 r r 2 r r r r (2.2 ε 0 = 1 c 2 µ 0 c = 3 10 8 m/s q 2.1 r q' F r = 0 µ 0 = 4π 10 7 N/A 2 k = 1/(4πε 0 qq F = k r
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II 14 14-7-8 8/4 II (http://www.damp.tottori-u.ac.jp/~ooshida/edu/fluid/) [ (3.4)] Navier Stokes [ 6/ ] Navier Stokes 3 [ ] Reynolds [ (4.6), (45.8)] [ p.186] Navier Stokes I 1 balance law t (ρv i )+ j
II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2
![II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2](/thumbs/101/149159646.jpg "II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2")
II No.1 [n/] [1]H n x) H n x) = 1) r n! r!n r)! x)n r r= []H n x) n,, H n x) = 1) n H n x) [3] H n x) = 1) n dn x e dx n e x [4] H n+1 x) = xh n x) nh n 1 x) ) d dx x H n x) = H n+1 x) d dx H nx) = nh
r 1 m A r/m i) t ii) m i) t B(t; m) ( B(t; m) = A 1 + r ) mt m ii) B(t; m) ( B(t; m) = A 1 + r ) mt m { ( = A 1 + r ) m } rt r m n = m r m n B
1 1.1 1 r 1 m A r/m i) t ii) m i) t Bt; m) Bt; m) = A 1 + r ) mt m ii) Bt; m) Bt; m) = A 1 + r ) mt m { = A 1 + r ) m } rt r m n = m r m n Bt; m) Aert e lim 1 + 1 n 1.1) n!1 n) e a 1, a 2, a 3,... {a n
9 1. (Ti:Al 2 O 3 ) (DCM) (Cr:Al 2 O 3 ) (Cr:BeAl 2 O 4 ) Ĥ0 ψ n (r) ω n Schrödinger Ĥ 0 ψ n (r) = ω n ψ n (r), (1) ω i ψ (r, t) = [Ĥ0 + Ĥint (
9 1. (Ti:Al 2 O 3 ) (DCM) (Cr:Al 2 O 3 ) (Cr:BeAl 2 O 4 ) 2. 2.1 Ĥ ψ n (r) ω n Schrödinger Ĥ ψ n (r) = ω n ψ n (r), (1) ω i ψ (r, t) = [Ĥ + Ĥint (t)] ψ (r, t), (2) Ĥ int (t) = eˆxe cos ωt ˆdE cos ωt, (3)
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[ ] 7 0.1 2 2 + y = t sin t IC ( 9) ( s090101) 0.2 y = d2 y 2, y = x 3 y + y 2 = 0 (2) y + 2y 3y = e 2x 0.3 1 ( y ) = f x C u = y x ( 15) ( s150102) [ ] y/x du x = Cexp f(u) u (2) x y = xey/x ( 16) ( s160101)
H21環境地球化学6_雲と雨_ ppt
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9 2 1 f(x, y) = xy sin x cos y x y cos y y x sin x d (x, y) = y cos y (x sin x) = y cos y(sin x + x cos x) x dx d (x, y) = x sin x (y cos y) = x sin x
2009 9 6 16 7 1 7.1 1 1 1 9 2 1 f(x, y) = xy sin x cos y x y cos y y x sin x d (x, y) = y cos y (x sin x) = y cos y(sin x + x cos x) x dx d (x, y) = x sin x (y cos y) = x sin x(cos y y sin y) y dy 1 sin
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1 17 () BAC9ABC6ACB3 1 tan 6 = 3, cos 6 = AB=1 BC=2, AC= 3 2 A BC D 2 BDBD=BA 1 2 ABD BADBDA ABC6 BAD = (18 6 ) / 2 = 6 θ = 18 BAD = 12 () AD AD=BADCAD9 ABD ACD A 1 1 1 1 dsinαsinα = d 3 sin β 3 sin β
I-2 (100 ) (1) y(x) y dy dx y d2 y dx 2 (a) y + 2y 3y = 9e 2x (b) x 2 y 6y = 5x 4 (2) Bernoulli B n (n = 0, 1, 2,...) x e x 1 = n=0 B 0 B 1 B 2 (3) co
16 I ( ) (1) I-1 I-2 I-3 (2) I-1 ( ) (100 ) 2l x x = 0 y t y(x, t) y(±l, t) = 0 m T g y(x, t) l y(x, t) c = 2 y(x, t) c 2 2 y(x, t) = g (A) t 2 x 2 T/m (1) y 0 (x) y 0 (x) = g c 2 (l2 x 2 ) (B) (2) (1)
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9. 05 L x P(x) P(0) P(x) u(x) u(x) (0 < = x < = L) P(x) E(x) A(x) P(L) f ( d EA du ) = 0 (9.) dx dx u(0) = 0 (9.2) E(L)A(L) du (L) = f (9.3) dx (9.) P
9 (Finite Element Method; FEM) 9. 9. P(0) P(x) u(x) (a) P(L) f P(0) P(x) (b) 9. P(L) 9. 05 L x P(x) P(0) P(x) u(x) u(x) (0 < = x < = L) P(x) E(x) A(x) P(L) f ( d EA du ) = 0 (9.) dx dx u(0) = 0 (9.2) E(L)A(L)
20 6 4 1 4 1.1 1.................................... 4 1.1.1.................................... 4 1.1.2 1................................ 5 1.2................................... 7 1.2.1....................................
master.dvi
4 Maxwell- Boltzmann N 1 4.1 T R R 5 R (Heat Reservor) S E R 20 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 )
1 No.1 5 C 1 I III F 1 F 2 F 1 F 2 2 Φ 2 (t) = Φ 1 (t) Φ 1 (t t). = Φ 1(t) t = ( 1.5e 0.5t 2.4e 4t 2e 10t ) τ < 0 t > τ Φ 2 (t) < 0 lim t Φ 2 (t) = 0
1 No.1 5 C 1 I III F 1 F 2 F 1 F 2 2 Φ 2 (t) = Φ 1 (t) Φ 1 (t t). = Φ 1(t) t = ( 1.5e 0.5t 2.4e 4t 2e 10t ) τ < 0 t > τ Φ 2 (t) < 0 lim t Φ 2 (t) = 0 0 < t < τ I II 0 No.2 2 C x y x y > 0 x 0 x > b a dx
6 2 T γ T B (6.4) (6.1) [( d nm + 3 ] 2 nt B )a 3 + nt B da 3 = 0 (6.9) na 3 = T B V 3/2 = T B V γ 1 = const. or T B a 2 = const. (6.10) H 2 = 8π kc2
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Introduction to Numerical Analysis of Differential Equations Naoya Enomoto (Kyoto.univ.Dept.Science(math))
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β β β (q 1,q,..., q n ; p 1, p,..., p n ) H(q 1,q,..., q n ; p 1, p,..., p n ) Hψ = εψ ε k = k +1/ ε k = k(k 1) (x, y, z; p x, p y, p z ) (r; p r ), (θ; p θ ), (ϕ; p ϕ ) ε k = 1/ k p i dq i E total = E
No δs δs = r + δr r = δr (3) δs δs = r r = δr + u(r + δr, t) u(r, t) (4) δr = (δx, δy, δz) u i (r + δr, t) u i (r, t) = u i x j δx j (5) δs 2
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