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1 /JST kitahata@physics.s.chiba-u.ac.jp

2 : Belousov-Zhabotinsky (BZ )

3 A BZ 44 B 46 B B

4 1 1.1??????? (Newton) 3

5 19 20 (Einstein) c c (Planck constant) h h GPS Global Positioning System: [1] 4

6 1.3 : A B B A 0 0 m d2 x = F, (1.1) dt2 m x F k (1.1) m d2 x = kx, (1.2) dt2 x 1 m d2 x = kx, (1.3) dt2 x(t) =A cos ( ) ( ) k k m t + B sin m t, (1.4) 5

7 (Lagrange) (Hamilton) ( ) 4 β F ( ) r 2 ( m 1 m 2 ) ( ) F = G m 1m 2 r 2, (1.5) G G = [m 3 s 2 kg 1 ], (1.6) x 1 x ( )F 12 F 12 = Gm 1 m 2 x 2 x 1 x 2 x 1 3 = Gm 1m 2 1 x 2 x 1 2 x 2 x 1 x 2 x 1, (1.7) x 2 x x 2 x ( )F F 21 = Gm 1 m 2 x 1 x 2 x 1 x 2 3 = Gm 1m 2 1 x 1 x 2 2 x 1 x 2 x 1 x 2, (1.8) F 12 = F 21, (1.9) ( ) (Coulomb) (Lorentz) ( ) 6

8 (a) t = T (b) t = T t = 0 x = x2 t = 2T x = x2 x = x1 x = x1 1.1: (a)t =0 x = x 1 t = T x = x 2 (b)t = T (a) t =2T x = x q 1 q F 12 F 12 = 1 x 2 x 1 q 1 q 2 4πɛ 0 x 2 x 1 3 = 1 1 x 2 x 1 q 1 q 2 4πɛ 0 x 2 x 1 2 x 2 x 1, (1.10) ɛ 0 = [F m 1 ], (1.11) 2 1 F 21 F 12 = F 21, (1.12) 1 x 1 v 1 X i 1 F 1 F 1 = f (X i x 1 ), (1.13) i 1 (1.1) 1.1 t =0 x = x 1 t = T x = x 2 ( ) t = T x = x 2 t = T F 1 X i x 1 X i x(t) t =0 x = x 1 t = T x = x 2 x(t) =x(2t t), (1.14) x(t) t = T x = x 2 T t =2T x(0) 1 7

9 q 1 1 E B 1 F 1 ( F 1 = q 1 E + dx ) 1 dt B, (1.15) E B Maxwell E = ρ ɛ 0, (1.16) B =0, (1.17) E + B =0, (1.18) t E B μ 0 ɛ 0 t = μ 0j, (1.19) ɛ 0 μ 0 ρ j j = ρv, (1.20) [2]

10 V U N 3 P T ( ) ρ cm 1 l K 3 (Avogadro) n n

11 + 2.1: ( ) (detailed balance) ( ) ( 2.1 ) A B ( ) B C A C A B 2 T A = T B T B = T C T A = T C, (2.1) U ( 2.2) du =d Q +d W, (2.2) 2 (Nernst) 0 10

12 du 2.2: du d Q d W (Clausius) (Thomson) (Kelvin) 4 (Ostwald ) d Q du U T V ( ) U Q W ( ) 4 100% (Clausius) energy en τρoπη [3] 11

13 P A B 0 D C V 2.3: PV A B C D A (A B C D) (B C D A) A B A B? (Carnot) P V 2.3 n mol A B T H C D T C U A B = Q A B + W A B = Q A B nrt H (ln V B ln V A )=0, (2.3) U B C = W A B = nr (T H T C ), (2.4) U C D = Q C D + W C D = Q C D nrt C (ln V D ln V C )=0, (2.5) U D A = W D A = nr (T H T C ), (2.6) W W = (W A B + W B C + W C D + W D A ), (2.7) 12

14 Q Q = Q A B, (2.8) η η = W Q = nrt H (ln V B ln V A )+nrt C (ln V D ln V C ). (2.9) nrt H (ln V B ln V A ) V γ 1 T = const. (2.10) ( γ 6 ) V A = V D, (2.13) V B V C η = T H T C, (2.14) T H η (Joule) (2.3) (2.5) Q A B = Q C D, (2.15) T H T C (2.15) d Q =0, (2.16) T ds = d Q T, (2.17) S S S S S ( ) 1 6 γ = C P, (2.11) C V C V C P C P = C V + R, (2.12) C V = 3 2 R C V = 5 2 R 13

15 1 2 U 1, P 1, V 1, S 1, T 1 U 2, P 2, V 2, S 2, T 2 2.4: U 1, P 1, V 1, S 1, T 1 U 2, P 2, V 2, S 2, T 2 2.5: ds 1 = du 1 T ds 2 = du 2 ( ) T ds =ds 1 +ds du 1 = du 2 ds =0 ds = du 1 T 1 + du 2 T 2 = ( 1 T 1 1 T 2 du 1 ) du 1 =0. (2.18) T 1 = T 2, (2.19) T 1 >T 2 ds >0 du 1 < ds 1 = du 1 + P 1 dv 1, T (2.20) ds 2 = du 2 + P 2 dv 2, T (2.21) 14

16 V,N 2V,N 2.6: du 1 = du 2 dv 1 = dv 2 ds =ds 1 +ds 2 ( 1 ds = 1 ) ( P1 du 1 + P ) 2 dv 1. (2.22) T 1 T 2 T 1 T 2 ds =0 du 1 dv 1 T 1 = T 2 P 1 = P 2 2 V N V T 2V ( 2.6 ) d W =0 d Q =0 U U = N N A C V T, (2.23) 2V 2V U N V 2V

17 V,N 2V,N T T 2.7: T ( ) ds = du T + P dv. (2.24) T PV = Nk B T du =0 ds = Nk B dv, (2.25) V ΔS = 2V V Nk B V dv = Nk B[ln V ] 2V V = Nk B ln 2, (2.26) 2.2? ( ) N 16

18 1 x y z 6 6N 6N N ( )?? V N ( ) x y z x S x x v x Δt v x Δt = v xs x Δt, (2.27) 2V/S x 2V x 1 mv x mv x 1 2mv x 7 Δt SNmv 2 x Δt/V S x Nm v 2 x /V Nm v 2 x /V P P = Nm v 2 x, (2.28) V v 2 = v 2 x + v 2 y + v 2 z, (2.29) x y z v 2 = 1 2 vx = 1 2 vy = 1 2 vz, (2.30) P = Nm v 2, (2.31) 3V 7 17

19 n mol PV = nrt, (2.32) R nn A = N(N A ) 3RT v 2 = m 2N A 2, (2.33) E k B = R/N A E = 3 2 k BT, (2.34) U U = 3 2 nrt = 3 2 Nk BT, (2.35) [4, 5] N 3N 3N 1 ( ) 3N 3N 1 3N 1 ( ) (ergodic) Ξ ξ +dξ V (ξ)dξ, (2.36) 18

20 N,V,E N,2V,E 2.8: Ξ( Ξ ) Ξ = ξv (ξ)dξ V (ξ)dξ, (2.37) (Ξ Ξ ) 2 (ξ ξ ) 2 V (ξ)dξ =, (2.38) V (ξ)dξ Ξ W (ξ) ξw (ξ)dξ Ξ =, (2.39) W (ξ)dξ (Ξ Ξ ) 2 (ξ ξ ) 2 W (ξ)dξ =, (2.40) W (ξ)dξ (microcanonical distribution) 2.2.3? (Boltzmann) W S = k B ln W, (2.41) ( ) k B 2.8 W (N,V,E) = V N (2πmE) 3 2 N h 3N N!Γ ( 3 (2.42) 2N +1), 19

21 8 S = k B ln W V ΔS = k B ln (2V ) N k B ln V N = k B N ln 2, (2.46) ΔS ( ) W 1 W 2 2 W 2 =2 N W 1 ΔS = k B ln W 2 k B ln W 1 = k B ln ( 2 N ) W 1 kb ln W 1 = k B 2 N = Nk B ln 2. (2.47) (T )? S S S tot S tot E tot S S E W tot (E) S W (E) W (E) S hb W tot (E) =W (E) W (E tot E), (2.48) S hb (E tot E) =k B ln W (E tot E), (2.49) 8 3 m N E E +de W de = 1 N!h 3N dr 1 dr N dp 1 dp N, (2.43) E p p N 2 E+dE 2m N! r k V N p k 2m(E +de) 3N 2mE 3N ( 2mE 3N de ) r n W de = V N N!h 3N [ π 3N 2 (2m(E +de)) 3N 2 Γ ( 3N 2 +1) S = k B ln W = k B N = k B N π 3N 2 (2mE) 3N 2 Γ ( 3N 2 +1) ] = V N (2πmE) 3N ( 2 3N N!h 3N 3N Γ +1) 2 2 [ln V + 32 ln(2πme) 3lnh ln N ln ( 32 N ) [ ( ) V ln N N (N ) ] π n 2 r n Γ ( n 2 +1) de E. (2.44) + 3 ( ) 4πmE 2 ln 3h ]. (2.45) N 2 20

22 (2.48) ( ) W tot (E) =W (E) W Shb (E tot E) (E tot E) =W (E) exp, (2.50) E tot E (Taylor) ( Shb (E tot ) W tot (E) =W (E) exp 1 S hb k B k B E E +ø( E 2) ), (2.51) ( Shb (E tot ) W tot (E) =W (E) exp ( E exp E k BT E W (E) exp k B E S hb = T, (2.52) ) ( exp E k B k B T +ø( E 2) ), (2.53) ) T ( E ), (2.54) k B T (canonical distribution) T Ξ ξ E(ξ) Ξ ) Ξ(ξ) exp ( E(ξ) k BT dξ Ξ = ), (2.55) exp ( E(ξ) k BT dξ Ξ E ) E(ξ) exp ( E(ξ) dξ U = E = exp ( E(ξ) k BT k BT ) dξ = ( 1 k BT ( ) ln ( exp E(ξ) ) ) dξ, (2.56) k B T Z = ( exp E(ξ) ) dξ, k B T (2.57) β = 1 k B T, (2.58) U = ln Z, (2.59) β Z (partition function) F F = k B T ln Z, (2.60) 21

23 i. (relaxation) 1 ( ) (Gibbs)

24 i-a. T T 0 k dt dt (T 0 T ). (3.1) T (t) =T 0 +[T (0) T 0 ] e kt, (3.2) (Fluctuation-dissipation theorem) [4, 6, 7] i-b. ( ) ( ) ( ) ii. (Nonequilibrium open systems) 2 ( ) ( ) ( ) (Prigogine) (dissipative systems) [8] ii-a. [8, 9]

25 ( )T H ( )T C 3.1: ii-b. 2 ii-a. - (Belousov-Zhabotinsky) (BZ ) (Schrödinger) What is life? [10] 24

26 A+B C+C. (4.1) k + k d[a] dt d[b] dt d[c] dt = k + [A][B] + k [C] 2, (4.2) = k + [A][B] + k [C] 2, (4.3) = k + [A][B] k [C] 2, (4.4) [A] A A B A B [A] [B] [C] t [A] u = [B], (4.5) [C] du = F (u), (4.6) dt (dynamical systems) du dt = F (u,t), (4.7) 25

27 F (u,t) t u u m d2 x = kx, (4.8) dt (m k ) v = dx dt, (4.9) ( d dt ) ( x = v ) v k m x, (4.10) 4.1.3? N N 1 N N ( (limit cycle; ) ) N 3 ( (torus)) (chaos) 1 (bistable)

28 1 1 dx dt = F (x), (4.11) dx dt = F (x) =0, (4.12) x = x 0 x = x 0 +Δx d dt (x 0 +Δx) =F (x 0 +Δx), (4.13) ( ) = dδx dt, (4.14) ( ) = F (x 0 )+ df dx Δx + o(δx 2 ). (4.15) x=x0 x = x 0 F (x 0 )=0 dδx = df dt dx Δx, (4.16) x=x0 Δx Δx 0 Δx =Δx 0 exp(at), (4.17) a = df dx. x=x0 (4.18) a = df dx 0, x=x0 (4.19) a = df dx > 0, x=x0 (4.20) F (x) ( ) 27

29 F(x) O x 4.1: x dx x x dt x x x - (Stuart- Landau) dx dt = αx ωy (x 2 + y 2 )x, (4.21) dy dt = αy + ωx (x 2 + y 2 )y, (4.22) ( α >0 ) 1 dr dt = r dx x dt + r dy y dt 1 x = r cos θ, (4.27) y = r sin θ, (4.28) r = x 2 + y 2, (4.29) ( y ) θ = arctan, (4.30) x dx dt dy dt = αx ωy (x 2 + y 2 )(x βy), (4.23) = αy + ωx (x 2 + y 2 )(y + βx), (4.24) dr dt dθ dt = αr r 3, (4.25) = ω βr 2, (4.26) 28

30 = cos θ [ αr cos θ ωr sin θ r 2 r cos θ ] + sin θ [ αr sin θ + ωr cos θ r 2 r sin θ ] = αr r 3, (4.31) dθ dt = θ dx x = sin θ = ω, dt + θ dy y dt r [ αr cos θ ωr sin θ r 2 r cos θ ] + cos θ r [ αr sin θ + ωr cos θ r 2 r sin θ ] (4.32) α>0 - α (bifurcation) ( ) [11, 12] 2 [13, 14, 15] (Brown) 3 (random walk) 4 2 (entrainment) (synchronization)

31 1 i x i ( x i +1 1 ), 2 x i+1 = ( x i 1 1 ) (4.33), 2 x i+1 = x i + ξ i, (4.34) ( 1 1 ), 2 ξ i = ( 1 1 ) (4.35), 2 x 0 =0 n 1 x n = ξ i, (4.36) i=0 < x i > x i ξ i = ( 1) = 0, (4.37) 2 ξ i ξ j = δ ij, (4.38) 5 x n = n 1 n 1 ξ i = ξ i =0, (4.40) i=0 i=0 ( n 1 ) 2 x 2 n = ξ i = n 1 n 1 ξ i i=0 i=0 j=0 i=0 j=0 n 1 n 1 ξ j = ξ i ξ j = n, (4.41) n n P (n, t) i n P (n, i +1)= 1 [P (n +1,i)+P (n 1,i)]. (4.42) 2 P (n, i +1) P (n, i) = 1 [P (n +1,i)+P (n 1,i) 2P (n, i)], (4.43) 2 i+1 t+δt i t n x n+1 x+δx n 1 x Δx P (x, t +Δt) P (x, t) = 1 [P (x +Δx, t)+p (x Δx, t) 2P (x, t)]. (4.44) 2 5 ξ i ξ j i j δ ij (Kronecker delta) δ ij = { 1 (if i = j), 0 (if i j), (4.39) 30

32 P (x, t +Δt) P (x, t) Δt = 2(Δx)2 Δt P (x +Δx, t)+p (x Δx, t) 2P (x, t) (Δx) 2 (4.45) (Δx)2 Δt (= C) Δx 0 Δt 0 P t = C ( ) P = C 2 P 2 x x 2 x 2, (4.46) u N u = NP u u t = D 2 u x 2, (4.47) (diffusion equation) D (diffusion coefficient; diffusion constant) [ 2 / ] (D = C/2) 1 u t = D 2 u, (4.48) 2 =Δ= 2 x y (Laplacian) ( z2 3 ) 6 u u = J, (4.51) t J 7 D J = D u, (4.52) u t = (D u) =D 2 u, (4.53) 6 ( (Langevin) ) m d2 x dt 2 dx = γ + ξ(t), (4.49) dt ( m γ ξ(t) ) ( (Fokker-Planck) ) [ ] P t = D (1) i (x,t)+ 2 D (2) ij x i x i x (x,t) P, (4.50) j ( D (1) D (2) - (Kramers-Moyal coefficients)) [16] 7 (Fick) (Fourier) 31

33 D 1 (Green function) u(x, t =0)=u 0 (x) ( ) x ± u 0 du dx 0 u(x, t =0)=δ(x) (4.54) G(x, t) 8 u(x, t =0)=u 0 (x) u(x, t) = dx G(x x,t)u 0 (x ), (4.58) 9 G(x, t) (Fourier Transform) G(x, t) = D 2 G(x, t) t x 2, (4.61) G(x, t =0)=δ(x), (4.62) G(x, t) G(k, t) = dxg(x, t) exp(ikx), (4.63) G(k,t =0)= dxg(x, t = 0) exp(ikx) = dxδ(x) exp(ikx) =1, (4.64) 8 δ(x) (Dirac) f(x) = dx f(x )δ(x x ), (4.55) 9 t =0 u(x, t =0)= δ(x) =δ( x). (4.56) δ(x) = 1 dke ikx, (4.57) 2π dx δ(x x )u 0 (x )=u 0 (x), (4.59) u t = t = dx G(x x,t)u 0 (x ) dx G(x x,t) u 0 (x ) t = dx D 2 G(x x,t) x 2 u 0 (x ) = D 2 x 2 dx G(x x,t)u 0 (x ) = D 2 u x 2. (4.60) G(x, t) u(x, t =0)=u 0 (x) 32

34 G(k,t) t = dx exp(ikx)g(x, t) t = dx exp(ikx)d 2 G(x, t) 2 x = D dx exp(ikx) 1 dk ( k 2 ) exp( ik x) 2π G(k,t) = D dx dk k 2 exp(i(k k )x) 1 2π G(k,t) = D dk k 2 G(k,t)δ(k k) = Dk 2 G(k,t) (4.65) δ(k) = 1 2π dx exp( ikx), (4.66) t G(k,t) = exp( Dk 2 t), (4.67) G(x, t) G(x, t) = 1 dk exp( ikx) exp( Dk 2 t) 2π = 1 [ ( dk exp Dt k + ix ) ] 2 x2 2π 2Dt 4Dt ) 1 = exp ( x2 4πDt 4Dt (4.68) (4.69) (Gauss) + dx exp( ax 2 )= π a, (4.70) 10 ( ) (Gauss) t u 0 x =0 10 [ dx exp( ax 2 )] 2 = = dx exp( ax 2 ) dx dy exp( ay 2 ) dy exp[ a(x 2 + y 2 )] 2π = rdr dθ exp( ar 2 ) 0 0 = 2π r exp( ar 2 ) = 0 [ ] 1 2π 2a exp( ar2 ) 0 = π a, (4.71) 33

35 x = + dxxg(x, t) = + ) 1 dxx exp ( x2 =0. (4.74) 4πDt 4Dt ( x x ) x 2 + = dxx 2 G(x, t) =2Dt. (4.75) + dxx 2 exp( ax 2 )= 1 2 π a 3, (4.76) ? du dt = f(u), (4.79) 2 u i (i =1, 2) du 1 dt du 2 dt = f(u 1 )+K (u 2 u 1 ), (4.80) = f(u 2 )+K (u 1 u 2 ), (4.81) K b x = x b + + dx exp( ax 2 )= dx exp [ a ( x b ) 2 ] = π a, (4.72) b z = I(b) [ ( ) ] dx exp( ax 2 )= dx exp a x 2 π b = a, (4.73) π dx exp( ax 2 )= a, (4.77) a + x 2 dx exp( ax 2 )= 1 π 2 a 3, (4.78) 34

36 u i u i+1 u i+2 x i x i+1 x i+2 x 4.2: du i dt = f(u i)+k (u i+1 u i )+K (u i 1 u i ), (4.82) Δx u i+1 u i 1 u i u i+1 = u i + u x Δx u x=xi 2 x 2 Δx 2 + o ( Δx 3), (4.83) x=xi u i 1 = u i u x Δx u x=xi 2 x 2 Δx 2 + o ( Δx 3). (4.84) x=xi u i x x i du i dt = f(u i)+kδx 2 2 u x 2. (4.85) x=ui u i u KΔx 2 D u t = f(u)+d 2 u x 2. (4.86) (reaction-diffusion equation) u t = f(u)+d 2 u, (4.87) [17, 18](Turing pattern) i) ii) u t = u3 + u 4v + D u 2 u v t = u 3v + a + D v 2 v (4.88) 35

37 (a) (b) 4.3: (4.88) (a)a =0 (b)a =0.05 D u =1 D v =20 a =0 u t = v =0, (4.89) t u = v =0, (4.90) u = 0+Δu, (4.91) v = 0+Δv, (4.92) ( d dt Δu Δv ) ( = )( Δu Δv ) ( A Δu Δv ), (4.93) A λ = 1, (4.94) u = v =0 ( ) 1 u = 0+ Δu(k)e ikx dk, (4.95) v = 0+ Δv(k)e ikx dk, (4.96) Δu(k) t ( e ikx dk = 4 3 Δu(k)e dk) ikx + Δv(k)e ikx dk + D u 2 36 x 2 Δu(k)e ikx dk Δu(k)e ikx dk, (4.97)

38 ( Δu Δv Δu(k)e ikx dk) 3 2 x 2 Δu(k)e ikx dk = k 2 Δu(k)e ikx dk, (4.98) [ ] Δu(k) Δu(k)+4Δv(k)+D u k 2 Δu(k) e ikx dk =0, (4.99) t x 0 Δu(k) = ( 1 D u k 2) Δu(k) 4Δv(k). (4.100) t Δv(k) =Δu(k) ( 3 D v k 2) Δv(k). (4.101) t ( ) ( )( ) ( ) Δu(k) 1 Du k 2 4 Δu(k) Δu(k) = A. (4.102) t Δv(k) 1 3 D v k 2 Δv(k) Δv(k) k A 2 λ 2 + ( 2+(D u + D v )k 2) λ + ( 1 D u k 2)( 3 D v k 2) +4=0, (4.103) 1 2 ( 1 Du k 2)( 3 D v k 2) +4> 0, (4.104) ( k 2 D ) 2 v 3D u + 4D ud v (D v 3D u ) 2 2D u D v 4(D u D v ) 2 > 0, (4.105) k 2 D v 3D u D v 3D u 4D u D v (D v 3D u ) 2 (D v D u )(D v 9D u ) > 0. (4.106) D v 3D u > 0 D v D u D v 9D u < 0, (4.107) D v < 9D u D v > 9D u k D v 9D u k 2 = D v 3D u 2D u D v, (4.108) 1 k = 3, (4.109) 2D u 2D v

39 5 5.1 Belousov-Zhabotinsky (BZ ) - (Belousov-Zhabotinsky) BZ BZ 5.1 BZ ( /TCA ) ( (target pattern) ) [19] - [20, 21, 22] BZ 1972 (Field) (Körös) (Noyes) BZ 10 [23] 3 FKN BZ (Oregonator) FKN BZ 2 HBrO 2 ( ) 5 A+Y k1 X+P, (5.1) X+Y k2 2P, (5.2) A+X k3 2X + 2Z, (5.3) 2X k4 A+P, (5.4) Z+B k5 hy. (5.5) A BrO 3 B CH 2(COOH) 2 +BrCH(COOH) 2 P HOBr X HBrO 2 Y Br Z h 1 Br (h ) 38

40 5.1: BZ A B X Y Z 3 dx dt = k 1AY k 2 XY + k 3 AX 2k 4 X 2, (5.6) dy dt = k 1AY k 2 XY + hk 5 BZ, (5.7) dz dt =2k 3AX k 5 BZ. (5.8) X Y Z 3 X Y Z k 1 k 5 (5.1) (5.5) 3 [24] (Tyson) 3 Y 2 X Z Y Y Y X Z 2 ( ) [25] 2 ( X U Z V ) ɛ du dt U q = U(1 U) fv f (U, V ), (5.9) U + q dv = U V g (U, V ). (5.10) dt U HBrO 2 V ( [Fe(phen) 3 ] 3+ ( (ferriin) ) (ferroin) ) 3 f q ɛ (5.9) (5.10) 1 ( ) 1 ( ) 39

41 (a) (b) 5.2: ( ) BZ (a) (target pattern) (b) (spiral pattern)? 2 3 BZ BZ BZ BZ 1 BZ ( ) 5.2 (target pattern) (spiral pattern) 1 continuous stirred tank reactor (CSTR) 40

42 (a) (b) 5.3: 2 (a) (b) 2 Oregonator 2 ( 5.3 ) 5.2 (Navier-Stokes) xy Γ z = f(x, y) at Γ, (5.11) 2 41

43 x y z = z(x, y), (5.12) 2 ( ) F F =2γ 1+ z 2 dxdy, (5.13) Γ γ 1 z F = [1+ 12 ] z 2 dxdy, (5.14) F Γ δf δz =0, (5.15) 2 z =0, (5.16) z(x, y) =f(x, y) at Γ, (5.17) f(x, y) = const. k, (5.18) z(x, y) =k, (5.19) 3? 0 2 z = 2 z x z =0, (5.20) y2 1 2 (κ 1 + κ 2 )= 1 2 ( ) = R 1 R 2 ( 2 ) z x z y 2, (5.21) 0 42

44 6 43

45 A Belousov-Zhabotinsky(BZ) Belousov-Zhabotinsky A.1 I II III II III ( II) ( III) 1 (CeSO 4 ) Ru(bpy) 3 Cl 2 ) (Br 2 ) 2 (FeSO 4 )1 mol (C 12 H 8 N 2 )3 mol 2 A.1: BZ I II III NaBrO mol/l 0.3 mol/l 0.3 mol/l H 2 SO mol/l 0.6 mol/l 0.45 mol/l CH 2 (COOH) mol/l 0.1 mol/l 0.1 mol/l NaBr 0.03 mol/l 0.03 mol/l 0.03 mol/l Fe(phen) 3 SO mol/l mol/l mol/l 1 BZ 2 1 mol/l 3 mol/l 1 mol/l 0.3 mol/l 0.02 mol/l 44

46 BZ 45

47 B B.1 [26, 27] B.1 l F l 2l 3 γ F =2γl. (B.1) [N/m] Δx ΔW (B.1) ΔW = F Δx, ΔW =2γlΔx = γδs, (B.2) (B.3) ΔS Δx (a) (b) F F F B.1: (a) F l γ (b) F Δx 3 2 2l F F +ΔF F F ΔF 6 46

48 [J/m 2 ] [N/m] [J/m 2 ] l 3 l 1 l ρgl 3 = γl, (B.4) l = γ ρg, (B.5) ρ g l l c (capillary length) B.2? B.2(a) (van der Waals) [28] ( B.2(a) ) B.2(b) a a ɛ n u = 4 3 πa3 nɛ, (B.6) 47

49 (a) (b) 0 a z B.2: (a) ( ) ( ) (b) z =0 z a V U = nv u = πa3 n 2 ɛv, (B.7) z(z a) Ṽ (z) = a z ( ) π a 2 z 2 2 dz = π 3 (a z)2 (2a + z), (B.8) Δu(z) πnɛ Δu(z) =ɛnṽ (z) = 3 (a z)2 (2a + z), (B.9) S z = z z +Δz nsδz a Ũ = Sɛn 2 Ṽ (z)dz = π 4 Sn2 ɛa 4, (B.10) Ũ 0 E = U + Ũ = k VV + k S S, (B.11) k S γ 7 γ = π 4 n2 ɛa 4. (B.12) n [1/ ] ɛ [ ] a [ ] γ [ / ] 7 k V (chemical potential) 48

50 [1] R. P. Feynman, R. B. Leighton, M. L.Sands, The Feynman lectures on physics, Vol I (Addison-Wesley, Reading, 1965). :, I (, 1967). [2] (, 1973). [3], (, 1972). [4], (, 1972). [5], (, 2002). [6], (, 1997). [7], (, 2000). [8] G. Nicolis and I. Prigogine, Self-organization in nonequilibrium systems: From dissipative structures to order through fluctuations (Wiley, New York, 1977). : (, 1980). [9] I. Prigogine and D. Kondepudi, Modern thermodynamics: from heat engines to dissipative structures (Wiley, New York, 1998). :,, (, 2001). [10] E. Schrödinger, What is life? : the physical aspect of the living cell (Cambridge University Press, Cambridge, 1945). :,, (, 1951). [11] S. H. Strogatz, Nonlinear dynamics and chaos (Perseus Books, 2000). [12] S. Wiggins, Introduction to applied nonlinear dynamical systems and chaos (Springer- Verlag, New York, 1990). :, (Springer, 2000). [13] (, 2005). [14] Y. Kuramoto Chemical oscillations, waves and turbulence (Springer-Verlag, 1984). [15] A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization (Cambridge University Press, Cambridge, 2001). [16] H. Risken, The Fokker-Planck equation (Springer-Verlag, 1989). [17] A. M. Turing, Phil. Trans. Roy. Soc. London B, 327, 37 (1952). 49

51 [18] (, 2006). [19] A. N. Zaikin and A. M. Zhabotinsky, Nature, 225, 535 (1970). [20] R. Kapral and K. Showalter, Chemical Waves and Patterns (Kluwer Academic, Dordrecht, 1995). [21], (, 1992). [22],,, III (, 1997). [23] R. J. Field, E. Körös, and R. M. Noyes, J. Am. Chem. Soc., 94, 8649 (1972). [24] R. J. Field and R. M. Noyes, J. Chem. Phys., 60, 1877 (1972). [25] J. J. Tyson and P. C. Fife, J. Chem. Phys., 73, 2224 (1980). [26] P.-G. de Gennes, F. Brochard-Wyart, D. Quéré, Capillarity and wetting phenomena: drops, bubbles, pearls, waves (Springer, New York, 2004). :, :,,, (, 2003). [27], (, 1980). [28] J. N. Israelachvili, Intermolecular and surface forces, (Academic Press, London, 1985). :,, (, 1991) : (, 1996). 50

52 ( ) ( )

I ( ) 1 de Broglie 1 (de Broglie) p λ k h Planck ( Js) p = h λ = k (1) h 2π : Dirac k B Boltzmann ( J/K) T U = 3 2 k BT

I ( ) 1 de Broglie 1 (de Broglie) p λ k h Planck ( Js) p = h λ = k (1) h 2π : Dirac k B Boltzmann ( J/K) T U = 3 2 k BT I (008 4 0 de Broglie (de Broglie p λ k h Planck ( 6.63 0 34 Js p = h λ = k ( h π : Dirac k B Boltzmann (.38 0 3 J/K T U = 3 k BT ( = λ m k B T h m = 0.067m 0 m 0 = 9. 0 3 kg GaAs( a T = 300 K 3 fg 07345