T g T 0 T 0 fragile * ) 1 9) η T g T g /T *1. τ τ η = Gτ. G τ

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1 T g T T fragile *2 1 11) 1 9) η T g T g /T *1 τ 198 τ η = Gτ G τ T c η τ 12) strong fragile T c strong η η exp(e/k B T ) 1 2/3 E SiO 2 ( ) GeO 2 Vogel-Fulcher η exp(e/k B (T T )) 3) fragile 2, 4) 13) *1 T g 1 13 *2 Kauzmann T g T g 1

2 4, 9, 11) 14) 3 MD 15) 2 25) MD 2 (2D) 3 (3D) 2 *4 (MD) α, β( 1, 2) σ α, σ β ( ) 12 σαβ 7, 16 19) v αβ (r) = ɛ, σ αβ = 1 r 2 (σ α + σ β ), (1) r N 1 = N 2 = 2, 21) 5 σ 2 /σ 1 = 14 (2D) 12 (3D) m 2 /m 1 = 2 n 1 + n 2 = ρ = 8/σ1 d d (337 T 245 (2D) 234 T 772 (3D)) 3 *3 frustration σ 1 τ = (m 1 σ1/ɛ) 2 1/2 ɛ/k B 22, 23) 1/τ γye x (e x :x ) γ γ ( 24) 31 t 2 t j, k 1) Load( elastic) r jk (t ) = r j (t ) r k (t ) A 1 σ αβ, (2) (plastic) r j (t) j t 2 t 3 jamming j, k t (3) t jamming r jk (t + t) > A 2 σ αβ (3) (aging) A 1 = 11 A 2 = 16 (2D) A 1 = A 2 = 15 (3D) *4 *3 2

3 t N 1 F s (q, ) = 1 r j (t) e 1 t τ b = τ b (T, γ) T γ τ τ b (T, ) 4 τ α [t, t + 5τ b ] (a) (b) F s (q, t = τ α ) = e 1 (9) q = 2π/σ 1 R jk = (r j + r k )/2 t T γ τ α τ b t = 5τ b τ α = 1τb η τ α η = (2πk B T/σ1)τ 3 α (a) (b) 3) Stokes-Einstein (6) Dτ α = ξ σ1/(2π) 2 2 Dτ α T γ 6 Dτ α t 1τ α Einstein 6Dt = ( r(t)) 2 29) van Hove R jk G s (r, t) N 1 j=1 2 δ( r j(t) r) /N 1 S b (q) = exp(iq R jk ) (4) Dτ α = 1 broken bonds 6 ( r(τ α)) 2 = 1 dr 4πr 4 G s (r, τ α ) (1) 6 S b (q) 7 4πr 4 G s (r, τ α ) Ornstein-Zernike 23) t = τ α (T = 473) S b (q) = S b ()/(1 + ξ 2 q 2 ) (5) (T = 267) r > 1 Dτ α ξ = ξ(t, γ) q S b (q) S b () ξ 2 31) 8 (a) (b) 26, 27) t = 1τ α 5, (CRR) 6) 28) 32 σ D Stokes-Einstein D η Stokes-Einstein D = k B T/2πση (6) ( 3τ α ) 1 D D 1/η 1/τ α 3 1 G s (r, t) D(x, t) σ 1 1) G s (x, r, t) = [4πD(x, t)t] 3/2 exp[ r 2 /4D(x, t)t] (11) F s (q, t) = N 1 1 j e iq r j(t) (7) r j (t) = r j (t) r j () γ t dt y j (t )e x (8) 9 6Dt4πr 2 G s (r, t) r = r/ 6Dt 3

4 5% *5 (7) F s (q, t) (8) t 3τ α r 1 (r (6Dt) 1/2 ) q (11) r < 1 22, 23) t 1τ α 9 33, 34) 1 33 n 1 (r, t)n 1 (r, t ) G(R, t t ) R = r r γ(t t )ye x τ b = 1τα e x ) 23) (G(R, t) 1/τ b (T, γ) = 1/τ b (T, ) + A b γ (12) A b F (q, t) = e iq (r k() r j (t))+iq x γty j (t) (15) jk ( 57 (2D) 8 (3D)) (9) η τ α 1 q 35) η(t, γ) = η(t, )/(1 + τ η γ) (13) τ η = A b τ b (T, ) τ α (T, ) τ b (T, ) 1/τ b (T, ) < γ T eff τ b (T eff, ) = τ b (T, γ) 1/τ (τ ) τ α 1/ γ T eff η γ 24, 32) τ b (T, ) = C exp(e/k B T ) 1/ γ ɛ [ ] E/T eff = E/T k B ln 1 + A b C γ exp(e/k B T ) (16) Q nɛ γ n Q = η γ 2 η nɛ/ γ 4(c) (b) (FDT) 3 38, 39) 1 T γ ξ τ b t w FDT T τ b ξ z (14) T eff 2 z = 4 3 z = 2 18) R(t) = 1 dc(t) k B T eff dt ξ γ 1/z z R(t) C(t) z 34 1 T eff 36) τ α 36) τ γ 1/τ FDT T eff *5 g αβ (r r ) = n α (r)n β (r η γ g αβ (r) 23) ) (17) 4

5 8), : 9 (1999) ) 9) PG Debenedetti, FH Stillinger: Nature 41 (21) ) AJ Liu, SR Nagel: Nature 396 (1998) 21 MD 11) S Sastry: Nature 49 (21) ) W Götze: in Liquids, Freezing, and the Glass Transition, ed JP Hansen, D Levesque, and J Zinn-Justin, Elsevier, Amsterdam, 1991, pp ) H Tanaka: J Chem Phys 111 (1999) ) B Coluzzi, G Parisi, P Verrocchio: Phys Rev Lett 84 (2) 36; R Di Leonardo, L Angelani, G Parisi, G Ruocco: Phys Rev Lett 84 (2) ) A Onuki, A Furukawa, A Minami: in the proceedings of Statphys 22, Bangalore, 4-9 July, 24 16) MM Hurley, P Harrowell: Phys Rev E 52, (1995) 1694: DN Perera: J Phys: Condens Matter 1, (1998) ) R Yamamoto, A Onuki: J Phys Soc Jpn 66, (1997) ) R Yamamoto, A Onuki: Phys Rev E 58, (1998) ) W Kob et al: Phys Rev Lett 79, (1997) 2827; C Donati et ξ τ b al: Phys Rev Lett 8, (1998) ) : (, 1991) 21) : (, 1999) T eff 22) : 5 (1995) 2 23) A Onuki: Phase Transition Dynamics (Cambridge University 2 2nm Press, 22) 4) 24) JH Simmons, RK Mohr, CJ Montrose: J Appl Phys 53, 23) (1982) 475 JH Simmons, R Ochoa, KD Simmons, JJ Mills: J Non-Cryst Solids 15, (1988) ) : (, 23) 26) WK Kegel, A van Blaaderen: Science 287 (2) 29 abnormal 27) ER Weeks, JC Crocker, AC Levitt, A Schofield, DA butterfly Weitz: Science 287 (2) ) MD Ediger: Annu Rev Phys Chem 51 (2) 99 τ α τ R = N 2 τ α 41) 29) R Yamamoto, A Onuki: Phys Rev Lett 81, (1998) 4915 N 3) F Mezei, W Knaak, B Farago: Phys Rev Lett 58, (1987) 571; D Richter, R Frick, B Farago: Phys Rev Lett 61, (1988) γ < 1/τ R ) R Zorn: Phys Rev B 55, (1997) 6249; T Kanaya, I Tsukushi, K Kaji: Supplement to Prog Theor Phys, 126, (1997) ) R Di Leonardo, F Ianni, G Ruocco: Phys Rev E 71 (25) ) M Fuchs, M E Cates: Phys Rev Lett 89, (22) 34) K Miyazaki, DR Reichman: Phys Rev E 66 (22) 551 1) : 39 (1984) 55 4) : 33 (1998) 489 5) : 52 (23) 773 6) : 9 (1999) 117 7), : 13 (1999) 92 35) K Miyazaki, DR Reichman, R Yamamoto: Phys Rev E 7 (24) ) L Berthier, J-L Barrat: J Chem Phys 116 (22) ),, : 46 (1991) 9 37) IK Ono, CS O Hern, DJ Durian, SA Langer, A Liu, SR Nagel: Phys Rev Lett 89 (22) 9573; CS O Hern, A Liu, 3) : 48 (1993) 869; ( SR Nagel: Phys Rev Lett 93 (24) 16572, 2) 38) :, 59 (24) 9 39) : 58 (23) 58 4) EW Fischer: Physica A 21, 183 (1993) 41) R Yamamoto, A Onuki: J Chem Phys 117, 2359(22) 5

6 1: η T g T g /T (a) strong fragile 9 (b) 2: /jamming /jamming x y z /jamming 1 (c) 4: (2), (3) t = 5τ b (a) T = 254 (b) T = 337 (c) T = 337 γ = : 2 (5) ξ 6

6 E T6 E 7 Tξ 5: (4) 3 Ornstein-Zernike 1/(1 + (qξ) 2 ) 1 T = 234 267 36 352 473 772 Dτ α 1 1 1 2 1 1 1 1 1 1 2 1

4 Gaussian 3 2 T = 234 (6Dt) 1/2 4πr 2 G s (r,t) 1 5 Gaussian t=3τ α 5 t=1τ α 1 2 1 T = 473 1 2 3 4 r t=5τ α 1 2 3

7 6 E T6 E 7 Tξ 5: (4) 3 Ornstein-Zernike 1/(1 + (qξ) 2 ) 1 T = Dτ α τ α 8: t = 1τ α (a) T = 473 (b) T = 267 6: Dτ α τ α Einstein-Stokes τ α 15 t=1τ α 1 5 4πr 4 G s (r,τ α ) 4 Gaussian 3 2 T = 234 (6Dt) 1/2 4πr 2 G s (r,t) 1 5 Gaussian t=3τ α 5 t=1τ α T = r t=5τ α r(6dt) 1/2 7: 4πr 4 G s (r, τ α ) r 3τ α van Hove G s (r, t) T = 473 T = 267 Gauss x Gauss 6Dτ α 9: 5τ α < t < 7

8 τ b ( T, γ ) T = (a) L τ b ( T, γ ) ξ( T, γ ) T = (b) 772 L ξ( T, γ ) 1: τ b ξ (a) 2 (b) 3 (14) 2 z = 4 3 z = 2 1 q m =58 (a) (b) (c) (d) F ρρ (q,t)/s ρρ (q) 5 γ = t : (15) F (q, t) 2 T = 526 q = q m = 58 (a) (b) (c) (d) q={q x, q y } = q m {1, } 2 q m{1, 1} q m {, 1} 2 2 q m{ 1, 1} 2 8

1 (Berry,1975) 2-6 p (S πr 2 )p πr 2 p 2πRγ p p = 2γ R (2.5).1-1 : : : : ( ).2 α, β α, β () X S = X X α X β (.1) 1 2

1 (Berry,1975) 2-6 p (S πr 2 )p πr 2 p 2πRγ p p = 2γ R (2.5).1-1 : : : : ( ).2 α, β α, β () X S = X X α X β (.1) 1 2 2005 9/8-11 2 2.2 ( 2-5) γ ( ) γ cos θ 2πr πρhr 2 g h = 2γ cos θ ρgr (2.1) γ = ρgrh (2.2) 2 cos θ θ cos θ = 1 (2.2) γ = 1 ρgrh (2.) 2 2. p p ρgh p ( ) p p = p ρgh (2.) h p p = 2γ r 1 1 (Berry,1975) 2-6