3 3.1 R r r + R R r Rr [ ] ˆn(r) = ˆn(r + R) (3.1) R R = r ˆn(r) = ˆn(0) r 0 R = r C nn (r, r ) = C nn (r + R, r + R) = C nn (r r, 0) (3.2) ( 2.2 ) C

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1 3 3.1 R r r + R R r Rr [ ] ˆn(r) = ˆn(r + R) (3.1) R R = r ˆn(r) = ˆn(0) r 0 R = r C nn (r, r ) = C nn (r + R, r + R) = C nn (r r, 0) (3.2) ( 2.2 ) C nn (r r ) = C nn (R(r r )) [2 ] 2 g(r, r ) ˆn(r) ˆn(r ) g(r, r ) = i j( i) = C nn ( r r ) (3.3) δ(r r i )δ(r r j ) = N(N 1) δ(r r 1 )δ(r r 2 ) 25

2 = N(N 1) d 3 r 1 d 3 r N δ(r r 1 )δ(r r 2 )e βu(r 1,,r N ) d 3 r 1 d 3 r N e βu(r 1,,r N ) (3.4) d 3 r 3 d 3 r U N e βu(r 1,r 2,r 3,,r N ) r U r r 1,r 2 fixed = 1 1 d 3 r 3 d 3 r N e βu(r 1,r 2,r 3,,r N ) = 1 β = 1 β d 3 r 3 d 3 r N e βu(r 1,r 2,r 3,,r N ) r 1 d 3 r 3 d 3 r N e βu(r 1,r 2,r 3,,r N ) ln r 1 = 1 ln β r N(N 1) 1 = 1 β d 3 r 1 d 3 r N e βu(r 1,,r N ) δ(r 1 r 1 )δ(r 2 r 2 ) d 3 r 1 d 3 r N δ(r 1 r 1 )δ(r 2 r 2 )e βu(r 1,,r N ) d 3 r 1 d 3 r N e βu(r 1,,r N ) r 1 ln g(r 1, r 2 ) (3.5) 2 r 1 2 g(r 1, r 2 ) r 1 V eff r 1 V eff (r 1, r 2 ) = dr 1 1 β r 1 ln g(r 1, r 2 ) = 1 β ln g(r 1, r 2 ) (3.6) r = r 1 r 2 2 V eff g(r) e βv eff(r) (3.7) V (r) 2 [ ] 2 U(r 1,, r N ) = 1 2 i j( i) V ( r i r j ) (3.8) Lennard-Jones [ (σ )12 ( ) ] σ 6 V (r) = 4ε (3.9) r r 26

3 V r r : Lennard-Jones ( 3.1) 1 Pauli r < σ 2 van der Waarls 2 p2 Ĥ = N 2m i V ( r i r j ) (3.10) j( i) 1 2 N(N 1) V ( r 1 r 2 ) = 1 2 N(N 1) 1 V = 1 2 N(N 1) 1 V 0 d 3 r V (r)g(r) 4πr 2 dr V (r)g(r) (3.11) Ĥ = 3 2 Nk BT Nn 4πr 2 dr V (r)g(r) (3.12) [] (hard sphere) ( 3.2(a)) 0 V (r) { = for r r0 = 2a, = 0 for r 0 (3.13) 27

4 (a) (b) 3.2: (a) () (b) 2 (P. M. Chaikin and T. C. Lubensky, Principles of condensed matter physics, (Cambridge, Cambridge, 1995) ) Q i = q i P i = p i 2mkB T (3.14) 6N {P i, Q i } V hs σ = V hs /V 3.2(b) 2 σ = hcp(hexagonal close packed) fcc(face-centered cubic) σ = σ = σ = fcc Alder f kt Η (a) -1 (b) 3.3: (a) (b)2 28

5 3.4: Alder (B. J. Alder and T. E. Wainwright, Phase transition in elastic disks, Phys. Rev. 127, 359 (1962)) Alder ( ) : 3.2 ( ) [] R {li } = l 1 a 1 + l 2 a 2 + l 3 a 3 (3.15) {l i } = {l 1, l 2, l 3 } {a i } ( 3.5) T = R {li } R {l i } (3.16) T ˆn(r + T ) = ˆn(r) (3.17) 29

6 3.5: [] T e ig T = 1 (3.18) G (reciprocal lattice) a 2 a 3 b 1 = 2π a 1 (a 2 a 3 ), b a 3 a 1 2 = 2π a 2 (a 3 a 1 ), b a 1 a 2 3 = 2π a 3 (a 1 a 2 ) (3.19) m 1 m 2 m 3 G R( T ) a i b j = 2πδ ij (3.20) G = m 1 b 1 + m 2 b 2 + m 3 b 3 (3.21) (l 1 a 1 + l 2 a 2 + l 3 a 3 ) (m 1 b 1 + m 2 b 2 + m 3 b 3 ) = 2π(l 1 m 1 + l 2 m 2 + l 3 m 3 ) (3.22) (3.21) (3.17) f(r) f(r + T ) = f(r) f(q) = d 3 rf(r)e iq r = d 3 rf(r + T )e iq (r+t ) unit cell T = T e iq T unit cell d 3 rf(r + T )e iq r (3.23) e iq T T = N cell = V V cell for q = G = 0 for q G (3.24) 30

7 f(q) = V G δ q,g f G V = (2π) 3 δ(q G)f G (3.25) G fq = 1 d 3 r f(r)e iq r (3.26) V cell unit cell f(r + T ) = f(r) f(r) = d 3 q f(q)eiq r = f(g)e ig r (3.27) (2π) 3 G ˆn(r) = G ˆn(G) e ig r (3.28) [] {R i } (2.6) k U k = U a (q) i e iq R i = V G U ag δ q,g. (3.29) U ag = U a(g)/v cell i e iq R i = Nδ q,g dσ dω V 2 U ag 2 δ q,g (3.30) G 2 Bragg k = k k = k G (3.31) k 2 = k 2 + G 2 2k G G d m1,m 2,m 3 2k G = G 2, (3.32) 2 λ sin θ = 1 (3.33) d m1,m 2,m 3 31

8 Bragg [] S(q) = 1 V I(q) = S nn(q) + 1 V ˆn(q) 2 = S nn (q) + 1 d 3 re iq r ˆn(r) V = S nn (q) + 1 V V 2 ˆn G δ q,g G = S nn (q) + V ˆn G 2 δ q,g G 2 S nn (q) + (2π) 3 ˆn G 2 δ(q G) (3.34) G ˆn G 2 Bragg (diffuse scattering) dσ dω U a(q) 2 V S(q) (3.35) 1 q S(q) q = G δ [ ] (liquid crystal) MBBA(N-(4-Methoxybenzylidene)-4-butylaniline) 3.6 l a MBBA ˆn (director) ˆn ˆn S = (ˆn ẑ) = cos2 θ 1 3 (3.36) 32

9 3.6: MBBA(N-(4- )-4- ) S 0 (nematic) n(r) = const. (smectic) n(r) n 0 + 2nq 0 cos(q 0 z) (3.37) : q 0 2π/l S(q) nq 0 2 (2π) 3 (δ(q q 0 ẑ) + δ(q + q 0 ẑ)) (3.38) 1 (quasi-long-range order) Bragg (quasi Bragg peak) 2 1 [ ] SiO 2 () T g

10 (a) (b) (c) 3.7: (a) (b) Sierpinski (T. Vicsek, Fractal Growth Phenomena, (World Scientific, Singapore, 1992) ) (c) Koch 10 DNA 3 DNA [ ] (fractal) Mandelbrot (self-similarity) S. Miyashita, Y. Saito and M. Uwaha, J. Cryst. Growth 283, 533 (2005). 34

11 (a) (c) (b) (d) 3.8: 2 (a) 0.04 mol/l, (b) 0.06 mol/l (c) n g =0.1 (d) n g = [ ] (fractal dimension) 1 M l M l d f (3.39) d f d b ( ) N(b) ( ) α N/α d f b b = αb N N = N α d f (3.40) d f d f 2 3.7(a) (Viscek ) 1/3 5 (3.40) α d f α d f = ( b b ) df = N N (3.41) 35

12 (a) (b) (c) 3.9: DLA (a) 100 (b) 1000 (c) d f d f = ln(n /N) ln(b /b) = ln 5 ln 3 = = (3.42) Sierpinski ( 3.7(b)) d f = ln 3/ln 2 = / = Koch ( 3.7(c)) d f = ln 4/ln 3 = / = [ ] (diffusion-limited aggregation: DLA) 2 DLA 3.9 () 3.8 ( 3.10) DLA d = 2 d f = 1.71 DLA [] ( ) 3.11 ( 2 2 C.-H. Lam: lam/dla/dla.html JAVA 36

13 3.10: 2 ) N R N = a i (3.43) i=1 a i a i = a 2 δ ij (3.44) R N = 0 (3.45) ( N ) 2 R 2 N N N N = a i = a i a j = a 2 = Na 2 R0 2 (3.46) i=1 i=1 j=1 i=1 R 0 : R 0 = Na n r P (r, n) dω P (r, n) = P (r a, n 1) 4π dω 3 P (r, n 1) = P (r, n 1) + a α P (r, n 1) a α a β + 4π α=1 x α 2 α=1 β=1 x α x β = P (r, n 1) P (r, n 1) 1 2 α=1 β=1 x α x β 3 a2 δ αβ + (3.47) a α = 0 a α a β = (a 2 /3)δ αβ n ( ) P (r, n) = a2 2 n 6 x y + 2 P (r, n) (3.48) 2 z 2 37

14 3.11: P (r, n) r t n = t/ t (3.48) P (r, t) t = D 2 P (r, t) (3.49) D D a2 t 6 (3.50) (3.48) P (r, 0) = δ(0) (3.51) P (r, n) = ( ) 3 3/2 ) exp ( 3r2 2πna 2 2na 2 (3.52) Gauss [ ] (3.52) (3.48) 38

15 [ ] n r W n (r) z (3.52) r W n(r) = z n (3.53) S n (r) = k B ln W n (r) = k B ln (z n P (r, n)) 3r 2 = S n (0) k B 2na 2 3r 2 S n (0) k B 2R 2 0 (3.54) F (r) = E T S = F (0) + 3 k B T r 2 (3.55) 2 R0 2 Hook ϕ R p p (1 ϕr ) N(N 1)/2 3 [ N(N 1) = exp ln (1 ϕr )] 2 3 ( exp N 2 ) ϕ 2 R 3 (3.56) S = k B ln p = k B 2 F 3 k B T R 2 + k BT N 2 ϕ 2 R0 2 2 R 3 N 2 ϕ R 3 (3.57) k B T R2 Na 2 + k BT ϕ N 2 R 3 (3.58) r R 5 ϕa 2 N 3 R N 3/5 (3.59) 39

16 ( Flory ) 1/2 3/ [ ] d 3/d + 2 [ ] p a = pa N = N/p α = p α d f = 1/p d f = 2 2 r N(r) 2 g(r) 1 dn(r) 4πr 2 dr 1 r 2 rd f 1 1 r 3 d f (3.60) 2 g(r) r α S(q) d d rg(r)e iq r dr r d 1 1 e iq r q α d r α dx x d 1 α e ix (3.61) α d = d f d f = 2 S(q) 1 q 2 (3.62) d f = 5/3 S(q) 1 q 5/3 (3.63) 40

80 4 r ˆρ i (r, t) δ(r x i (t)) (4.1) x i (t) ρ i ˆρ i t = 0 i r 0 t(> 0) j r 0 + r < δ(r 0 x i (0))δ(r 0 + r x j (t)) > (4.2) r r 0 G i j (r, t) dr 0

80 4 r ˆρ i (r, t) δ(r x i (t)) (4.1) x i (t) ρ i ˆρ i t = 0 i r 0 t(> 0) j r 0 + r < δ(r 0 x i (0))δ(r 0 + r x j (t)) > (4.2) r r 0 G i j (r, t) dr 0 79 4 4.1 4.1.1 x i (t) x j (t) O O r 0 + r r r 0 x i (0) r 0 x i (0) 4.1 L. van. Hove 1954 space-time correlation function V N 4.1 ρ 0 = N/V i t 80 4 r ˆρ i (r, t) δ(r x i (t)) (4.1) x i (t) ρ i ˆρ i t

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[ ] (Ising model) 2 i S i S i = 1 (up spin : ) = 1 (down spin : ) (4.38) s z = ±1 4 H 0 = J zn/2 i,j S i S j (4.39) i, j z 5 2 z = 4 z = 6 3 4.2 4.2.1 [ ] (Ising model) 2 i S i S i = 1 (up spin : ) = 1 (down spin : ) (4.38) s z = ±1 4 H 0 = J zn/2 S i S j (4.39) i, j z 5 2 z = 4 z = 6 3 z = 6 z = 8 zn/2 1 2 N i z nearest neighbors of i j=1

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