H.Haken Synergetics 2nd (1978)

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1 ) Ising Landau Synergetics Fokker-Planck F-P Landau F-P Gizburg-Landau G-L G-L Bénard/ Hopfield

2 H.Haken Synergetics 2nd (1978)

3 (1) Ising m T T C 1: m h Hamiltonian H = <i,j> J ij S i S j h i S i, (1) J ij = J H = J <i,j> S i S j h i S i, (2) Ising J > 0 J < 0 S i = {1, 1} < i, j >

4 i-th m m =< S i >, S i = m + δs i H = J <i,j> (m + δs i )(m + δs j ) h i = Jm 2 N B + Jm 2 zn (Jmz + h) i S i S i, (3) N N B z 1 Jmz Self-consistency S m =< S i > = 1 S N S i e βh S 1 S N e βh = tanh[β(jmz + h)] (4) 1 y 0-1 y=tanh( J z m) y=m m 2: βjz > 1

5 Landau m f(m) = f 0 (β) + a(β)m 2 + b(β)m 4 + (5) Landau m order parameter m f(m) df(m) dm = 0. (6) 2 f - f 0 a > 0 0 a < 0 m : m m a < 0 m 0 f(m) = f 0 + am 2 + bm 4 hm Landau 1 log Z(m) = f(m), (7) Nβ Z(m) = e βh(m), (8) S 1 S N Z(m) = e βjm2 (N B zn) [2 cosh β(jmz + h)] N, (9) f(m) = 1 N Jm2 (N B zn) 1 log[2 cosh β(jmz + h)], (10) β

6 m Landau self-consistency 1 N N i=1 S i = m Ising Onsarger m Z(m) = Tr[δ(m 1 N i S i)e βh ] Z Z = dmz(m) (11) m( r) =< m( r) > +δm( r) δm( r) Landau Ginzburg-Landau G-L Green < δm( r)δm( r ) > = G( r, r ), (12) δm( r) = β d r G( r, r )δh( r ). (13) Green G-L F F = d r f(m( r)), (14) f(m( r)) = f 0 (β) + A(β)m 2 ( r) + B(β)m 4 ( r) + D 2 m( r) 2 h( r)m( r).(15) δf δm = 0 2Am( r) + 4Bm 3 ( r) D 2 m( r) = h( r), (16) h( r) = h + δh( r), m( r) = m + δm( r) (2A + 12B m 2 D 2 )δm( r) = δh( r). (17)

7 m Landau Green G( r r ) = 1 e r r /ξ 4πβD r r. (18) D 2A ξ = = D = 4A D 2a(T T c ) D 4a(T c T ) if T > T c if T < T c < δm( r)δm( r ) > 1. (19) r r d 2 Lagevin δf m( r, t) = Γ + ζ( r, t), (20) t δm( r, t) < ζ( r, t)ζ( r, t ) > = 2T Γδ( r r )δ(t t ), (21) t m( r, t) = Γ(2Am + 4Bm3 D 2 m) + ζ( r, t). (22) time dependent Ginzburg Landau TDGL Γ2A (T T c ) critical slowing down

8 Synergetics F-P Landau q = (q 1, q N ) Langevin d dt q = K( q), (23) d dt q = K( q) + F (t), (24) < F i (t)f j (t ) > = Q ij δ(t t ), (25) q P ( q, t) F-P t P = q( KP ) Q ij = Q = const. P eq i,j Q ij 2 q i q j P (26) KP eq Q qp eq = 0. (27) K = q V 2V ( q) P eq ( q) exp{ Q } (28) V ( q) V, as q q V

9 Landau d df m = Γ dt dm (29) m = 0, m 1 Langevain f (m) 0 m 1 m 4: d m dt = df Γ + F (t), dm (30) < F (t)f (t ) > = Qδ(t t ) (31) F-P P eq (m) exp{ 2Γf(m) Q } (32) m = m 1 m 1 Landau f(m) m P eq (m) Z(m) = e βnf(m) (33) P eq (m) = e βnf(m) Z(m)dm. (34) 2Γ βn Q

10 F-P G-L m G-L G-L functional f f({m( r)}) = f 0 + d r[ α 2 m2 ( r) + β 4 m4 ( r) + γ 2 m( r) 2 ], (35) P eq ({m( r)}) = e βf({m( r)}) D{m}e βf (36) Γδf m( r, t) = t δm( r) = αm βm 3 + γ 2 m (37) t m = αm βm3 + γ 2 m + F (38) F-P t P = δ d r{ δm( r) (αm + βm3 γ 2 m) + Q 2 δ 2 }P (39) δm( r) 2 P eq ({m( r)}) exp{ 2Γ f({m( r)})} (40) Q G-L order parameter ξ TDGL t ξ = αξ β ξ 2 ξ + γ 2 ξ + F, (41) P eq ({ξ}) exp{ 2Γ d r[α ξ 2 + β Q 2 ξ 4 + γ ξ 2 ]} (42) t U µ = G µ (, U; σ) + D µ 2 U µ + F µ (t),

11 G µ U, σ σ < σ c U 0 σ > σ c U 0 U µ = U 0 µ + q µ ( r, t) ( K( ))q = g(, q) + F (t), (43) t ( K( ))q = 0, (44) t marginal stable mode σ marginal ξ ξ ξ slaving principle H.Haken σ

12 Ginzburg-Landau I.Prigogine Bénard/ z y d x 5: Boussinesq u = 0, (45) t u + u u = ν 2 u 1 π + αgẑθ, ρ 0 (46) t θ + u θ = κ 2 θ + βw, (47) u = (u, v, w), (48) θ = T (T 0 βz). (49)

13 u = θ = π = 0 π t 2 w = ν( 2 ) 2 w + αg( 2 x + 2 )θ. (50) 2 y2 k = (k k, k y ) (w, θ) = (w m, k (t), θ m, k (t)) sin mπz d exp(i k r), (51) (w m, k (t), θ m, k (t)) e λt. (52) k c marginal R = g d 3 T > 0 < 0 R C ~ k c ~ k = k d 6: m = 1 x w(x, y, z) = (Z + Z) sin πz d, (53) Z = W e ikcx, (54) W super-critical,r > R c R R c = µr c, λ( k c ) = µλ 1 W (t) W saturation d dt W = µλ 1W g W 2 W, (55)

14 g ky kc kx 7: k W µ 8: (x, y) super-critical k c k = k c + δ k. (µ 1) δ k k c W λ( k) = µλ 1 D(k k c ) 2, D > 0, (56) = µλ 1 D(k x k c + k2 y 2k c ) 2. (57)

15 t W = µλ 1W g W 2 W + D( x i 2 2k c y 2 )2 W. (58) Newell-Whitehead N-W Ψ t W = δψ, (59) δ W Ψ = d r( µλ 1 W 2 + g 2 W 4 + D W x i 2 W 2k c y 2 2 ). (60) Ψ k c λ( k) = µλ 1 D 1 (k x k cx ) 2 D 2 (k y k cy ) 2 (61) Ginzburg-Pitaevskii F-P (H.Haken) Bousinesq H.Haken

16 µ Hoph x ν x ν = α ν Z, (62) Z = W e iω 0t W λ (63) t W = µλ 1W. (64) t W = µλ 1W g W 2 W. g : complex, (65) λ(k) = iω 0 + µλ 1 Dk 2, (66) D G-L t W = µλ 1W g W 2 W + D 2 W, (67) λ 1, g, D G-L

17 Hopfield Hopfield Hopfield ξ(t ) 1 1 θ 1 ξ 1 (t +1) ξ i (t ) w ij j θ j ξ i (t +1) ξ N (t ) N ξ N (t +1) θ N 9: Hopfield

18 N ξ i (t + 1) = sgn( w ij ξ j (t) θ i ), (68) sgn(x) = { j=1 1 if x > 0 1 if x < 0 ξ = {1, 1} θ i I i = I w ij = 1 N P ξ µ i ξµ j. (69) µ P N N µ=1 ξ 1 µ ξ 2 µ ξ N µ 10: H(t) = 1 w ij ξ i (t)ξ j (t) + θ i ξ i (t) (70) 2 i,j i d H (ξ µ i, ξ i) = 1 N (ξ µ i ξ i ) 2, (71) 4 i=1 = N N ξ µ i ξ i. (72) i=1

19 0 d H N E E = 1 2N P [N 2d H ({ξ µ i }, {ξ i})] 2. (73) µ=1 d H ({ξ µ i }, {ξ i}) 0 N ξ i = ξ µ i ξ i = ξ µ i E E = 1 2N P [N 2( N 2 1 ξ µ i 2 ξ i)] 2, µ=1 H = 1 P ξ µ i 2N ξµ j ξ iξ j, µ=1 i,j = 1 w ij ξ i ξ j. (74) 2 i,j i H(t) = E(t) + i θ i ξ i (t). (75) ξ I (t + 1) = ξ I (t) H = H(t + 1) H(t) = 0 ξ I (t + 1) = ξ I (t) H = E + θ I (ξ I (t + 1) ξ I (t)), (76) = 2ξ I (t + 1) j I w Ij ξ j (t) + θ I (ξ I (t + 1) ξ I (t)) = 2ξ I (t + 1) j w Ij ξ j (t) + 2ξ I (t + 1)w II ξ I (t) + θ I (ξ I (t + 1) ξ I (t)) = 2sgn( j w Ij ξ j (t) θ I )( j w Ij ξ j (t) θ I ) 2w II 2θ I ξ I (t + 1) + θ I (ξ I (t + 1) ξ I (t)) = 2 j w Ij ξ j (t) θ I 2w II < 0. (77)

20 H ξ i ξ i µ 11: H H({ξ i }) ξ i = ±ξ µ i GA

21 Hopfield TDGL

1 s 1 H(s 1 ) N s 1, s,, s N H({s 1,, s N }) = N H(s k ) k=1 Z N =Tr {s1,,s N }e βh({s 1,,s N }) =Tr s1 Tr s Tr sn e β P k H(s k) N = Tr sk e βh(s k)

1 s 1 H(s 1 ) N s 1, s,, s N H({s 1,, s N }) = N H(s k ) k=1 Z N =Tr {s1,,s N }e βh({s 1,,s N }) =Tr s1 Tr s Tr sn e β P k H(s k) N = Tr sk e βh(s k) 19 1 14 007 3 1 1 Ising 4.1................................. 4................................... 5 3 9 3.1........................ 9 3................... 9 3.3........................ 11 4 14 4.1 Legendre..............................