(9 30 ) (10 7 ) (FP) (10 14 ) (10 21 ) (2
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1 (9 30 ) (10 7 ) (FP) (10 14 ) (10 21 ) (2. )(10 28 ) (11 4 ) Wiener-Khinchin (11 12 ) (11 18 ) (11 25 ) (12 2 ) (12 9 ) (12 16 ) (1 13 ) ( ) (1 20 ) 110
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3 3 1 () 2 () 2 1 () + ( ) (1.1) 2 2 1? : 1 100µm (2) 1960 : 2
4 4 1 (9 30 ) Calle-Walton () Lax Zwanzig ( : )
5 5 (3) () ( ) ()? ( :) ( ) * () (4)? () : 1 (6 ) () 1 2 () 3 1 3
6 6 1 (9 30 ) ()
7 7 : pdf (10 7 ) () (1) (2) (3) (4) (5) : Ẋ(t) = γx(t) + R(t) (2.1.1) : Ẋ(t) = F (X(t)) + R(t) (2.1.2) R(t) = 0 (2.1.3) R(t 1 )R(t 2 ) = Dδ(t 1 t 2 ) (2.1.4) (D > 0) t = 0 X(0) : X(0)R(t) = 0 t 0 (2.1.5) : g(x(0))r(t) = 0 t 0 (2.1.6) g(x) X
8 8 2 (2-1 ) (1) www www *: (2) 1908
9 2.1 (10 7 ) 9 λv (t) V (t) R(t) m 2 t 1 ( ): R(t) 2 ( ): λv (t) m V (t) = λv (t) + R(t) (2.1.7) R(t) 1 R(t) δ(t t 0 ): (t 0 ) R(t) ( ) ( ) p p = R t (2.1.8) R t t 0 p 0 (2.1.8) R R(t) δ(t t 0 ) R(t) t 0 t 0 t 2 R(t)
10 10 2 R(t) 1 R(t ) R(t) 2 R(t ) R(t) R(t) R(t) 1 R(t ) R(t) 2 R (t ) R(t ) R(t) R(t 1 )R(t 2 ) f(x 1, x 2,... ) f(r(t 1 ), R(t 2 ),... ) ( 5 ) i R(t) R i (t) 1 R(t) = lim n n R(t)R(t 1 ) = lim n n f(r(t), R(t 1 ),... ) = lim n n n R i (t) (2.1.9) i=1 n R i (t)r i (t ) (2.1.10) i=1 n f(r i (t), R i (t ),... ) (2.1.11) n i=1 R(t) ( ) (2.1.3) (2.1.4) (2.1.3) 0 (2.1.4)
11 2.1 (10 7 ) 11 (3) V X(t) (2.1.5) t > 0 R(0) X(t) R(0)X(t) = 0 X(0) R(t) R(t)X(0) = R(t) X(0) = 0 (2.1.12) (4) 1 (2.1.7) V (t) = γv (t) + R(t) m, γ = λ m (2.1.13) X(t) = V (t) (2.1.1) 2 () V 0 Q(t) C I(t) Q(t) V (t) C R V 0 Q V 0 = Q(t) C (2.1.14) V 0 V (t) I V 0 + V (t) = RI(t) (2.1.15) R I(t) = Q(t) R Q(t) = Q(t) C + V (t) (2.1.16) γ = 1/(RC) R(t) = V (t)/r 3
12 12 2 レーザー 力が働く 3 X(t) u(x) m mẍ(t) = λẋ(t) u(x) + R (t) (2.1.17) コロイド粒子 λẋ(t) R (t) m Ẋ(t) = 1 λ u(x) + R (t) λ (2.1.18) (5) X(t) Ẋ(t) = γx(t) + R(t) : (2.1.19) Ẋ(t) = F (X(t)) + R(t) : (2.1.20) X(t) (2.1.4) (2.1.4) : 2 (10 ) 2 a () b
13 2.1 (10 7 ) *1 2 3 (15 ) (2.1.1) (2.1.3)-(2.1.5) X(t) t t i (i = 1,..., n) (2.1.1) X(t i+1 ) X(t i ) = γx(t i ) t + W (t i ) (2.1.21) W (t i ) ( 0 D t) X(t 1 ) t X(t) γ D γ 4 (20 ) 1 mẍ(t) = λẋ(t) kx(t) + R(t) (2.1.22) X(t) m R(t) (2.1.3) (2.1.4) (2.1.5) t = 0 Ẋ(0) = 0 X(0) = x 0 Ẋ(t) {Ẋ(t)}2 2 Ẋ(t) 5 (10 ) R(t) δ(t t 0 ) R(t) R(t) = i d iδ(t t i ) R(t) {d 1, d 2,... } {t 1, t 2,... } {d 1, d 2,... } {t 1, t 2,... } ρ(d 1, d 2,..., t 1, t 2,... ) (2.1.3) (2.1.4) *1 2003
14 (FP) (10 14 ) FP FP P (x, t) t X x x + dx FP X(t) f(x(t)) t t f(x(t)) X(t) 2 FP 4 FP (1) FP (2) X(t) (3) FP (4) (5) 1 R(t) = 0, R(t)R(t ) = Dδ(t t ) 2 X(t) R(t ) t < t 3 R(t) 4 P (x, t) x ± P (x, t) 0, P (x, t) x 0 (2.2.1) Ẋ(t) = F (X(t)) + R(t) (2.2.2) FP P (x, t) t = { 2 D F (x) + x x 2 }P (x, t) (2.2.3) 2 (2-2 )
15 2.2 (FP) (10 14 ) 15 : P89-98 (1) FP X = X(t) X(0) X(t) X P (x, t) P (x, t): t X x x + dx = P (x, t)dx t = 0 P (x, t) t = 0 P (x, t) P (x, t) FP f(x) f(x) f(x) = f(x)p (x, t)dx (2.2.4) (2) X(t) Ẋ(t) = F (X(t)) + R(t)((2.2.2) ) t t + t ( ) t+ t t Ẋ(t )dt = t+ t t t+ t t 1 t t+ t t F (X(t ))dt + t+ t t R(t )dt (2.2.5) Ẋ(t )dt = X(t + t) X(t) (2.2.6) F (X(t ))dt F (X(t)) t (2.2.7)
16 R(t) (2.2.7) X(t) X(t + t) X(t) (2.2.5) W 1 W t+ t t R(t 1 )dt 1 (2.2.8) X(t) = F (X(t)) t + W (2.2.9) W = 0 (2.2.10) W 2 = D t (2.2.11) (2.2.11) W 2 t+ t t+ t = R(t 1 )dt 1 R(t 2 )dt 2 = t t+ t t+ t t t 1 R(t)R(t ) = Dδ(t t ) = = t+ t t+ t t t+ t t t t (2.2.12) R(t 1 )R(t 2 ) dt 1 dt 2 (2.2.13) Dδ(t 1 t 2 )dt 1 dt 2 (2.2.14) Ddt 2 = D t (2.2.15) f(x) f(x(t + t)) X(t) t () f(x(t + t)) = f(x(t)) + df dt t + 1 d 2 f 2 dt 2 t2 + (2.2.16) = f(x(t)) + df dx dx t + t 2 (2.2.17) dt
17 2.2 (FP) (10 14 ) 17 t f(x) 1 2 X(t) X(t) f(x(t+ t)) = f(x(t))+ df dx X(t)+ 1 d 2 f X=X(t) 2 dx 2 X(t) (2.2.18) X=X(t) (2.2.9) f(x(t + t)) = f(x(t)) + df dx {F (X(t)) t + W } X=X(t) + 1 d 2 f 2 dx 2 {F (X(t)) t + W } 2 + X (2.2.19) X=X(t) df f(x(t + t)) = f(x(t)) + dx {F (X(t)) t + W } X=X(t) + 1 d 2 f 2 dx 2 {F (X(t)) t + W } 2 + X (2.2.20) X=X(t) t (2.2.20) d 2 f/dx 2 3 W 2 1 d 2 f 2 dx 2 W 2 X=X(t) (2.2.21) W t+ t t 2 *2 1 d 2 f 2 dx 2 R(t 1 )dt 1 W R(t 1 ) t 1 t W 2 = X=X(t) 1 2 d 2 f W 2 dx 2 X=X(t) (2.2.22) (2.2.11) = 1 2 d 2 f dx 2 X=X(t) D t (2.2.23) *2 t 1 = t 1 (2.2.22)
18 18 2 t (3) FP 1 f(x) t f(x) 2 FP 1 (2.2.20) 2 df dx {F (X(t)) t + W } X=X(t) df = df dx F (X(t)) t + X=X(t) dx W X=X(t) (2.2.24) (2.2.24) 2 (2.2.22) 2 df df dx W = W = 0 (2.2.25) X=X(t) dx X=X(t) (2.2.10) W = 0 df df dx {F (X(t)) t + W } = X=X(t) dx F (X(t)) t X=X(t) (2.2.26) (2.2.20) 3 1 d 2 f 2 dx 2 {F (X(t)) t + W } 2 X=X(t) 1 d 2 f = 2 dx 2 F (X(t)) 2 t 2 + X=X(t) 1 2 d 2 f dx 2 2F (X(t)) t W X=X(t) 1 d 2 f + 2 dx 2 ( W ) 2 (2.2.27) X=X(t)
19 2.2 (FP) (10 14 ) (2.2.25) 0 3 (2.2.23) (2.2.27) 1 2 d 2 f dx 2 {F (X(t)) t + W } 2 X=X(t) 1 d 2 f = 2 dx 2 F (X(t)) 2 t 2 + X=X(t) (2.2.20) (2.2.26) (2.2.28) 1 2 d 2 f dx 2 X=X(t) D t (2.2.28) f(x(t + t)) df = f(x(t)) + 1 d 2 f dx F (X(t)) t + X=X(t) 2 dx 2 F (X(t)) 2 t 2 X=X(t) 1 d 2 f + 2 dx 2 D t + X (2.2.29) X=X(t) 3 R(t) W ( 9 ) X t 2 (2.2.30) d dt f(x(t + t)) f(x(t)) f(x(t)) lim t 0 t df = dx F (X(t)) + D d 2 f 2 dx 2 (2.2.31) (2.2.32) f = f(x) X = X(t) (2.2.32) f(x) 2 FP f(x(t)) = f(x)p (x, t)dx (2.2.33) d dt f(x(t)) = P (x, t) f(x) dx (2.2.34) t
20 20 2 (2.2.32) 1 df dx F (X(t)) df = F (x)p (x, t)dx (2.2.35) dx df dx F (X(t)) = [f(x)f (x)p (x, t)] f(x) {F (x)p (x, t)}dx (2.2.36) x 4 df dx F (X(t)) = f(x) {F (x)p (x, t)}dx (2.2.37) x 2 D d 2 f 2 dx 2 = D d 2 f P (x, t)dx (2.2.38) 2 dx2 = D 2 [ ] df P (x, t) D dx 2 df P (x, t)dx (2.2.39) dx x 4 1 = D 2 = D 2 df P (x, t)dx (2.2.40) dx x [ ] P (x, t) f(x) + D x 2 f(x) 2 P (x, t) x 2 dx (2.2.41) 4 = D 2 f(x) 2 P (x, t) x 2 dx (2.2.42) f(x) P (x, t) dx = f(x) t x {F (x)p (x, t)}dx + D 2 f(x) 2 P (x, t) x 2 dx (2.2.43)
21 2.2 (FP) (10 14 ) 21 f(x) P (x, t) t = x {F (x)p (x, t)} + D 2 2 P (x, t) x 2 (2.2.44) FP (4) 1 () m V (t) = λv (t) + R(t) X = V γ = λ/m F (V ) = γv R(t)R(t ) = Dδ(t t ) FP P (v, t) t = D 2 P (v, t) {γvp (v, t)} + v 2 v 2 (2.2.45) D = D/m 2 2 R Q(t) = Q(t)/C +V (t) X = Q F (Q) = Q/CR V (t)v (t ) = D V δ(t t ) FP P (q, t) t = q { q CR P (q, t) } + D 2 2 P (q, t) q 2 (2.2.46) D = D V /R 2 3 (1 ) 1 X(t) 1 Ẋ(t) = u (X(t)) λ + R(t) (2.2.47) u (X) u(x) X R(t) 1 λ R(t)R(t ) = Dδ(t t ) P (x, t) = { } u(x) P (x, t) + D t x λ 2 2 P (x, t) x 2 (2.2.48)
22 X i i 1 W X i+1 X i = W (2.2.49) t = i t X(t) = X i X(t + t) = X i+1 X(t + t) X(t) = W W i (2.2.9) F (X) = 0 t 0 P (x, t) P (x, t) t = D 2 2 P (x, t) x 2 (2.2.50) D W 2 = D t t (2.2.50) t = 0 P (x, 0) = δ(x) P (x, t) ( 13 ) T P (x, T ) C6H5CH=CHC6H t Θ = Θ(t) Θ(t) = γ du(θ(t)) dθ(t) + R(t) (2.2.51) u(θ) Θ =0 180 R(t) R(t) P (θ, t) P (θ + 2π, t) = P (θ, t) FP ( 12) P (θ, t) t = { γ u(θ) } P (θ, t) + D θ dθ 2 2 P (x, t) θ 2 (2.2.52)
23 2.2 (FP) (10 14 ) 23 (5) ( ) X(t) = F (X(t)) t + W f = f(x(t)) t + W X(t) 2 ( W ) 2 = D t 1 d 2 f dt 2 t t t 0 f 3 3 ( 1) 4 FP FP Ẋ(t) = F(X(t)) + R(t), R(t)R(t ) = D δ(t t ) (2.2.53) P (x, t) t = { x F(x) + 2 x 2 D 2 }P (x, t) (2.2.54) : 6 (30 ) 2
24 24 2 P7 (2.1.3) (2.1.4) P7 7 (10 ) FP 2 6 FP (15 ) 9 (10 ) (2.2.30) W P ( W ) P ( W ) exp[ W 2 2D t ] (2.2.55) (2.2.55) (2.2.11) 1 10 (10 ) P14 FP 4 0 L (2.2.2) F (X) = 0 x = 0, L P (x, t)/ x = 0 x = 0, L P (x, t) = 0 FP (2.2.3) FP 11 (10 ) 4 P (x, t) = P (x + L, t) f(x) = x FP (2.2.3) FP F (x) F (x) = F (x + L) f(x) = L 0 f(x)p (x, t)dx (2.2.56) 12 (10 ) (2.2.52) u(θ + 2π) = u(θ) f(θ) f(θ + 2π) = f(θ) 11
25 2.2 (FP) (10 14 ) (15 ) γ = λ/m 3 Ẋ(t) = R(t) (2.2.57) X(t) FP P (X, t) t = D 2 2 P (X, t) (2.2.58) (2.2.58) t = 0 P (X, 0) exp[ α X 2 ] t r r + r r = X r
26 (10 21 ) 2 (2nd FDT) D FP () 2 (2nd FDT) 3 2nd FDT (1) (2) 2 (2nd FDT) (3) (4) X X = X(t) Ẋ(t) = F (X(t)) + R(t) (2.3.1) R(t) = 0 (2.3.2) R(t)R(t ) = Dδ(t t ) (2.3.3) FP FP P eq (x) P eq (x) { 2 D F (x) + x x 2 2 }P eq(x) = 0 (2.3.4) J eq (x) = { F (x) + D 2 } P eq (x) (2.3.5) x x ± J eq (x) = 0 (2.3.6)
27 2.3 2 (10 21 ) 27 P eq (x) = e S(x) (2.3.7) F (x) = D 2 F (x) = LdS(x)/dx ds(x) dx (2.3.8) L = D 2 (2.3.9) (2.3 ) λ D N A R T (1) t (2.3.10) FP F (x) D () : : ()? : m T k B t v m P eq (v) = 2πk B T exp[ m 2k B T v2 ] (2.3.11) F (x) D 2nd FDT: F (x) D P eq (x)
28 28 2 2nd Fluctuation Dissipation Theorem ( 2 ) 2 F (x) P eq (x) D D P eq (x) F (x) (2) 2 P (x, t) P (x, t) t J(x, t) = x (2.3.12) J(x, t) x (2.3.12) x x + dx J(x, t) J(x + dx, t) J(x) FP P (x, t) t = { 2 D F (x) + x x 2 }P (x, t) (2.3.13) 2 { J(x, t) = F (x) + D 2 } P (x, t) (2.3.14) x (2.3.6) P eq (x) (2.3.4) (2.3.5) J eq (x) (2.3.15) J eq(x) x = 0 (2.3.15) J eq (x) = C : x (2.3.16) x ± J eq (x) = 0 C = 0 J eq (x) = 0 (2.3.17) (2.3.14) J eq (x) = { F (x) + D 2 } P eq (x) = F (x)p eq (x) D x 2 P eq (x) x (2.3.18)
29 2.3 2 (10 21 ) 29 P eq (x) = e S(x) S(x) ln P eq (x) (2.3.18) 2 D 2 P eq (x) x = D 2 d dx es(x) = D 2 ds(x) dx es(x) = D 2 ds(x) dx P eq(x) (2.3.19) J eq (x) = F (x)p eq (x) D 2 { ds(x) dx P eq(x) = F (x) D 2 } ds(x) P eq (x) = 0 (2.3.20) dx P eq > 0 F (x) = D 2 ds(x) dx (2.3.21) F (x) S(x) F (x) = LdS(x)/dx Ẋ = LdS(X)/dx + R(t) L = D 2 (2.3.22) 2 (FDT) (3) 1 (1 ) P eq (v) S(v) = m m 2k B T v2 + ln 2πk B T (2.3.23) ds(v) dv = m k B T v (2.3.24) V (t) = γv (t) + R (t) (2.3.25) γ = λ m, R (t) = R(t) m, R (t)r (t ) = D δ(t t ), D = D m 2 (2.3.26)
30 (2.3.8) (2.3.21) γv = D 2 ( m ) k B T v (2.3.27) γ = D m 2k B T (2.3.28) γ D (2.3.26) λ m = D 2mk B T (2.3.29) λk B T = D 2 (2.3.30) D λ k B T λ k B T D (2.3.30) N A = R/k B k B (2.3.30) k B = D/(2λT ) N A = 2λRT D (2.3.31) 2 P eq (q) e βe(q) ( ) β = 1/(k B T ) E(q) q E(q) = q2 2C S(q) = βq2 2C +, ds(q) dq = β C q (2.3.32) Q(t) = Q(t) CR + R(t) (2.3.33)
31 2.3 2 (10 21 ) 31 F (q) = q/(cr) 2 (2.3.8) (2.3.32) q CR = D 2 { βc q } R(t)R(t ) = Dδ(t t ) (2.3.34) 1 CR = Dβ 2C k BT R = D 2 (2.3.35) R(t) = V (t)/r V (t)v (t ) = D V δ(t t ) D = D V R 2 k BT R = D V 2R 2 2Rk B T = D V (2.3.36) R k B T D V (2.3.36) (4) 1 X 2 ( ) {X 1, X 2,..., X n } = {X α } Ẋ α (t) = F ({X α }) + R α (t) (2.3.37) R α (t) = 0 (2.3.38) Rα (t)r β(t ) = Dαβ δ(t t ) (2.3.39) P eq ({x α }) = e S({x α}) Ẋ α (t) = n β=1 L αβ S({X α }) X β + R α (t) (2.3.40) L αβ = D αβ 2 (2.3.41) ( 17 ) P eq (x) P eq (x) ( 14 )
32 32 2 S(x) 2 (FDT) 3 L, F (X)() ()D 3 2 : 14 (15 ) FP P (x, t) t = { L du(x) x dx f + D 2 } P (x, t) (2.3.42) x P (x, t) U(x) P (x, t) = P (x + L, t) U(x) = U(x + L) f = 0 f x (2.3.14) F (x) = LdU(x)/dx + f J(x) 0 P st (x) L 0 15 (30 ) 2 P st (x)dx = 1 (2.3.43) Ẋ α = n γ αβ X β + R α (t) (2.3.44) β R α (t) = 0 R α (t)r β (t ) = D αβ δ(t t ) X α (0)R β (t) = 0(t 0) (2.3.44) Ẋ µ = λ µ X µ + R µ(t) (2.3.45)
33 2.3 2 (10 21 ) 33 t = 0 X µ = 0 X µ(t)x ν(t) R µ (t)r ν(t ) = D µνδ(t t ) t X µ(t)x ν(t) = X µ X ν eq X µx ν (λ eq µ + λ ν ) = D µν (2.3.46) t = 0 X µ = 0 X α (t)x β (t) {γ αγ X γ X β eq + γ βγ X γ X α eq } = D αβ (2.3.47) γ 16 (20 ) 1 V (t) X(t) Ẋ(t) = V (t) (2.3.48) m V (t) = λv + R(t) (2.3.49) m λv R(t) 2.2 P P (x, v, t) FP (2.3.48) (2.3.49) 2.2 FP D m λ T 2.3 T 17 (45 )(2.3.37) (2.3.39) F µ ({x µ }) = n ν L µν S({x µ })/ x ν P eq ({x µ })T ({x µ }, {x µ}; t) = P eq ({x µ})t ({x µ}, {x µ }; t) (2.3.50) L µν = D µν 2 (2.3.51) T ({x µ }, {x µ}; t) T ({x µ }, {x µ}; 0) = n δ(x µ x µ) (2.3.52) FP n k µ S({x µ }) = 2 x2 µ (2.3.53) µ µ
34 34 2 n µ L µµ k µ (2.3.51) T ({x µ }, {x µ}; t) = C(t) exp[ n µν 1 2 σ µν(t)(x µ x µ (t))(x ν x ν (t))] (2.3.54) C(t) n dx µ T ({x µ }, {x µ}; t) = 1 (2.3.55) µ x µ (t) x µ (0) = x µ σ µν(t) 15 t = 0 0 n µ X µ (t)x µ (t) σ µ ν(t) = δ µν
35 2.4 (2. )(10 28 ) (2. )(10 28 ) 2 t X = x t X x x + dx X = X(t) FP FP 2 2 FP (1) (2) 2 (3) 2 (4) X = X(t) T (x, x, t, t ) FP T (x, x, t, t ) t = x {F (x)t (x, x, t, t )} + D 2 t = t T (x, x, t, t) = δ(x x ) 2 T (x, x, t, t ) x 2 (2.4.1) 2 T (x, x, t, t ) = T (x, x, t t ): ( ) 3 P (x, t) T (x, x, t) t = 0 P 0 (x) P (x, t) = FP T (x, x, t)p 0 (x )dx (2.4.2) t = 0 0 t > 0
36 36 2 (1) : X = X(t) T (x, x, t, t ): t X = x t X x x + dx = T (x, x, t, t )dx t t x x t X = X(t) X x' t' t 時刻 t 1 t P 0 (x) X t' t 時刻 t P (x, t) P 0 (x) FP T (x, x, t, t ) P 0 (x) = δ(x x ) FP (2) 2 (2.4.2) P (x, t) 2
37 2.4 (2. )(10 28 ) 37 1 t = 0 X(0) = x t : T (x, x, t) ( ) 2 t = 0 : P 0 (x ) () P eq P eq (x) = T (x, x, t)p eq (x )dx (2.4.3) 3 ( 20 ) t = 0 x 1 P (x, 0) = δ(x x ) t T (x, x, t) 1 t = 0 ( 2 ) P 0 (x) t T (x, x, t) P 0 (x) x T (x, x, t) P 0 (x ) P (x, t) (3) 2 1 (2.1.7) m (2.1.13) FP (2.2.45) 2 (2.3.28) P (v, t) t = D 2 { βmv + } P (v, t) (2.4.4) v v D = D/m 2 β = 1/(k B T ) T k B (2.4.4) T (v, v, t) = 1 exp[ (v v 0(t)) 2 ] (2.4.5) 2πσ(t) 2σ(t) v 0 (t) = v e γt (2.4.6) σ(t) = k BT m (1 e 2γt ) (2.4.7)
38 38 2 γ = λ/m ( 22 ) 2 (2.1.16) FP (2.2.46) 2 (2.3.35) 1/R = (D/2)β P (q, t) t = D 2 { βq q C + } P (q, t) (2.4.8) q T (q, q, t) = 1 exp[ (q q 0(t)) 2 ] (2.4.9) 2πσ(t) 2σ(t) q 0 (t) = q exp[ t CR ] (2.4.10) σ(t) = k B T C(1 exp[ 2t ]) (2.4.11) CR 3 (1 ) (2.1.18) 1 FP (2.2.48) P eq (x) e βu(x) (2.4.12) ds(x) dx du(x) = β dx R(t)R(t ) = Dδ(t t ) 2 1 du(x) λ dx = D ( β du(x) ) 2 dx (2.4.13) (2.4.14) 1 λ = Dβ 2 (2.4.15)
39 2.4 (2. )(10 28 ) 39 FP P (x, t) = D { βu (x) + } P (x, t) (2.4.16) t 2 x x λ u(x) = 0 D T (2.4.15) λ (4) T (x, x, t) T (x, x, t) FP t = 0 T (x, x, 0) = δ(x x ) FP D Ẋ(t) = γx(t) + R(t) T (x, x, t) = 1 exp[ (x x 0(t)) 2 ] 2πσ(t) 2σ(t) (2.4.17) x 0 (t) = x e γt (2.4.18) σ(t) = D 2γ (1 e 2γt ) (2.4.19) R(t)R(t ) = Dδ(t t ) : 18 (15 ) (2.4.1) 19 (15 ) P (x, t) 2 (P.36-P.37 ) X = X(t) Ẋ(t) = F (X(t)) (2.4.20) (2.4.20) X = f(t, X 0 ) T (x, x, t, t ) X 0 20 (10 ) 3
40 (25 ) S(t) 2.2 x P (x, t) t = x {F (x)p (x, t)} + D 2 2 P (x, t) x 2 + S(t) (2.4.21) t = 0 P (x, 0) = 0 P (x, t) T (x, x, t, t ) T (x, x, t, t ) (2.4.1) t = t T (x, x, t, t ) = δ(x x ) 22 (20 ) (2.4.5) FP 2 (2.4.5) P 0 (v) = C exp[ (v v 0) 2 ] (2.4.22) 2σ 0 (2.4.2) (x v ) P (v, t) (2.4.4) v C v 0 σ 0 P eq (v) (2.4.3) 23 (15 ) 1 P (x, t) t = D 2 2 P (x, t) x 2 (2.4.23) x P (x, t) t P 0 (x) = C exp[ x2 x 2 ] (2.4.24) 0 (2.4.2) P (x, t) P (x, t) (2.4.23) C x 0 24 (20 ) t < t FP FP P (x, t) P (x, t) P (x, t) t P (x, t + t) P (x, t) lim t 0 t (2.4.25)
41 2.4 (2. )(10 28 ) 41 t < t < 0
42 (11 4 ) (Time Correlation Function: TCF) TCF TCF 2 (1 2) (TCF) TCF (3) (1) 3 (2) (3) (4) (5) X µ = X µ (t)(µ = 1,..., n) ( ) T (x, x, t) (3) 1 ϕ µν (t) X µ (t)x ν (0) ϕ µν (t) = ϕ νµ ( t) µ = ν 2 Ẋµ (t)x ν (0) = X µ (t)ẋν(0) µ = ν ϕ µµ (0) = 0 3 n = 1 X(t)X(0) = P eq (x) xt (x, x, t)dxx P eq (x )dx (3.1.1) V (t) A(t) = V (t) A(t)A(0)
43 3.1 (11 4 ) 43 (1) (2) A V (t) = (1 ) V (t) t V (t) t 1 2 (1 ) B V (t) t V (t) t 1 2 (1 ) A B 2
44 44 3 A B? : 2 1 X(t) ((2.1.10) ) i X i (t) X(t)X(t 1 ) lim N N N R(t)R(t ) 2 () X(t)X(t 1 ) lim T T T 0 N X i (t)x i (t ) (3.1.2) i X(t + τ)x(t + τ)dτ (3.1.3) 1 1 X(t) X(t) t + τ t + τ t t 1 2 X(t) = 0 X(t) X(t ) X(t)X(t ) = X(t) X(t ) = 0 (3.1.4)
45 3.1 (11 4 ) 45 X(t)X(t ) = 0 t < t X(t)X(t ) = 0 t X t X t t X t X(t)X(t ) 0 (3) : ( 1 ) X(t) * : a t t + a 1 X(t) t t + a X(t) = X(t + a) X(t) = t 2 X(t)X(t ) : t t t t t + a t + a X(t + a)x(t + a) = X(t)X(t ) (3.1.5) a = t X(t t )X(0) = X(t)X(t ) (3.1.6) t t ϕ(t t ) X(t t )X(0) = X(t)X(t ) (3.1.7) X(t) {X 1 (t), X 2 (t), } = {X µ (t)} ϕ µν (t) X µ (t)x ν (0) (3.1.8)
46 V (t) = (V x (t), V y (t), V z (t)) ϕ 11 (t) = V x (t)v x (0) (3.1.9) ϕ 12 (t) = V x (t)v y (0) (3.1.10) ϕ 31 (t) = V z (t)v x (0) (3.1.11) () X µ (t)x ν (t ) = X µ (t t )X ν (0) = X µ (0)X ν (t t) (3.1.12) t = 0 X µ (t)x ν (0) = X µ (0)X ν ( t) (3.1.13) = X ν ( t)x µ (0) (3.1.14) µ = ν ϕ µν (t) = ϕ νµ ( t) (3.1.15) ϕ µµ (t) = ϕ µµ ( t) : ϕ µµ (t) (3.1.16) (3.1.13) t Ẋµ (t)x ν (0) t Ẋµ (t)x ν (0) = X µ (0)Ẋν( t) = X µ (t)ẋν(0) (3.1.17) (3.1.18) µ = ν ϕ µµ (0) = Ẋµ (0)X µ (0) = 0 (3.1.19) 1 (3 ) x F x x X X 1 (t) X 2 (t) ϕ 11 (t) = F x (t)f x (0) (3.1.20) ϕ 12 (t) = F x (t)x(0) (3.1.21)
47 3.1 (11 4 ) 47 (3.1.14) F x (t)x(0) = X( t)f x (0) (3.1.22) (3.1.16) F x (t)f x (0) = F x ( t)f x (0) (3.1.23) Ẍ = F x (3.1.18) F x (t)x(0) = Ẍ(t)X(0) = Ẋ(t) Ẋ(0) (3.1.24) Ẋ T F x (0)X(0) = Ẋ(0) Ẋ(0) = k B T/m m k B (3.1.19) F x (0)Ẋ(0) = Ẍ(0) Ẋ(0) = 0 (3.1.25) (4) 1 X(t) Ẋ(t) = γx(t) + R(t) (3.1.26) X(0)R(t) = 0, t 0 X(0) t 0 Ẋ(t)X(0) = γ X(t)X(0) (3.1.27) ϕ(t) X(t)X(0) ϕ(t) = γϕ(t) (3.1.28) ϕ(t) = ϕ(0)e γt (3.1.29)
48 48 3 ϕ(0) = X 2 ϕ(t) = X 2 e γt t 0 (3.1.30) X(t) 1. V (t) V (t) = γv (t) + R(t) (3.1.31) γ = λ/m((2.1.13) ) (3.1.30) V (t)v (0) = V 2 e γt (3.1.32) V 2 = k B T/m V (t)v (0) = k BT m e γt (3.1.33) A(t) = V (t) A(t)A(0) = V (t) V (0) (3.1.34) (3.1.18) = V (t)v (0) (3.1.35) ϕ(t) V (t)v (0) = ϕ(t) (3.1.36) (3.1.33) = k BT m γ2 e γt (3.1.37) 2.
49 3.1 (11 4 ) 49 Q(t) Q(t) = Q(t) CR + R(t) (3.1.38) C R R(t) V (t) R(t) = V (t)/r (3.1.30) Q 2 Q(t)Q(0) = Q 2 exp[ t CR ] (3.1.39) P eq (q) = A exp[ β q2 2C ] (3.1.40) A β = 1/k B T Q 2 = q 2 A exp[ β q2 2C ]dq = Ck BT (3.1.41) Q(t)Q(0) = Ck B T exp[ t CR ] (3.1.42) 2 ϕ(t) X(t)X(0) t = 0 X(0) = x () t : X(t) x ( ) 2 x : 0 () Ẋ(t) = γx(t) + R(t) 1. Ẋ(t) = γ X(t) x x + R(t) x (3.1.43) X(0) X(0) = x R(t) x = R(t) = 0 (3.1.44) Ẋ(t) = γ X(t) x x X(t) x = X(0) x exp[ γt] x X(0) = x X(t) x = x exp[ γt] (3.1.45)
50 ϕ(t) x x X(t)X(0) = X(t) x x 0 = x 2 e γt (3.1.46) 2 1. t = 0 x T (x, x, t) X(t) x = xt (x, x, t)dx (3.1.47) T (x, x, t) X(0) = x 2. x P eq (x) X(t)X(0) = X(t) x x 0 = xt (x, x, t)dxx P eq (x )dx (3.1.48) (5) 1 () ϕ µν (t) = ϕ νµ ( t): (3.1.15) 2 ϕ µµ (t) = ϕ µµ ( t): (3.1.16) ϕ μμ (t) 3 t Ẋµ (t)x ν (0) = X µ (t)ẋν(0) : (3.1.18)
51 3.1 (11 4 ) 51 4 ϕ µµ (0) = 0 X(t) : (3.1.30) : (3.1.48) : 25 (15 ) 105 km 105 km 1000 km N 2 m z P eq (z) exp[ βmgz] (3.1.49) β = 1/(k B T ) g 2 D FP 26 (10 ) (2.2.51) FP (2.2.52) 2 D γ 27 (15 ) 2 28 (20 ) 29 (15 ) 2.4 2(P35) T (x, x, t, t ) = T (x, x, t t ) 30 (20 ) 3 X(t 1 )X(t 2 )X(t 3 ) 31 (10 ) (1 ) X u(x) = kx 2 /2 X(t)X(0) X = X(t) Ẋ(t) = u (X) λ + R(t) (3.1.50)
52 52 3 u (X) u(x) X X(0) 2 T 32 (10 ) (2.1.13) 1 V (t) V (0) m γ k B T t
53 3.2 Wiener-Khinchin (11 12 ) Wiener-Khinchin (11 12 ) Wiener-Khinchin (WK ) (2 ) 2 2 I ω (1) (2) (3) (4) (5) 1. X(t) ϕ(t) X(t)X(0) lim T T T 0 X(t + τ)x(τ)dτ (3.2.1) 1. : X ω X(t)e iωt dt (3.2.2) X ω X ω = 2πδ(ω + ω ) ϕ(ω) (3.2.3) ϕ(ω) e iωt ϕ(t)dt (3.2.4) 2. : 1 I ω lim T T X ω(t ) X ω (T ) (3.2.5)
54 54 3 X ω (T ) T 0 X(t)e iωt dt (3.2.6) X ω (T ) X ω (T ) I ω = ϕ(ω) (3.2.7) I ω (1) X(t) X ω = X(t)e iωt dt (3.2.8) X(t) X ω Wiener-Khinchin X ω 2 1. : 2. X ω 2. (2) (3.2.2) X ω X ω = dt e iωt dt e iω t X(t)X(t ) (3.2.9) (1) X(t)X(t ) = ϕ(t t ) X ω X ω = dt e iωt dt e iω t ϕ(t t ) (3.2.10)
55 3.2 Wiener-Khinchin (11 12 ) 55 (3.2.4) X ω X ω = 2πδ(ω + ω ) e iωs ϕ(s)ds (3.2.11) X ω X ω = 2πδ(ω + ω ) ϕ(ω) (3.2.12) (3.2.12) X ω X ω ω = ω (3) X ω (T ) = T 0 X(t)e iωt dt (3.2.13) () 1 I ω lim T T X ω(t ) X ω (T ) (3.2.14) X ω (T ) X ω (T ) 1 (3.2.14) (3.2.13) 1 I ω = lim T T = lim T 1 T T 0 T 0 T dt e iωt dt e iωt X(t)X(t ) (3.2.15) 0 T dt dt e iω(t t ) X(t)X(t ) (3.2.16) 0 2 (t, t ) (s, t) s = t t ( ) ( ) ( ) ( ) s t t s t t t t = ( 1) = 1 (3.2.17) t t t iω(t t ) = iωs t = t s 1 I ω = lim dsdt e iωs X(t)X(t s) (3.2.18) T T t 0<t <T,0<t<T s t
56 56 3 s t s t T < s < 0 T -s 0 < s < T t T + s s T t s s t 1 t s 0 < s < T s < t < T (3.2.19) T < s < 0 0 < t < T + s (3.2.20) t s s 2 (3.2.18) { 1 T T 0 } T +s I ω = lim ds dt e iωs X(t)X(t s) + ds dt e iωs X(t)X(t s) T T 0 s T 0 (3.2.21) 1 t τ = t s 1 I ω = lim T T { T T s ds e iωs X(τ + s)x(τ)dτ } T +s + ds e iωs X(t)X(t s)dt T 0 (3.2.22) I ω = 0 ds e iωs lim T 1 T T s 0 + X(τ + s)x(τ)dτ 0 ds e iωs lim T 1 T T +s 0 X(t)X(t s)dt (3.2.23)
57 3.2 Wiener-Khinchin (11 12 ) 57 ϕ(t) ( 2)(3.2.1) (3.2.23) 1 2 s T ϕ(s) ϕ( s) I ω = 0 ds e iωs ϕ(s) + 0 ϕ( s) = ϕ(s)(1) ds e iωs ϕ( s) = ds e iωs ϕ(s) = ϕ(ω) (3.2.24) (4) ϕ(t) X(t)X(0) = X 2 e γt (3.2.25) ((3.1.30) ) t 0 ϕ(t) = ϕ( t) t ϕ(t) = X 2 e γ t (3.2.26) I ω = ϕ(ω) = 2 X 2 γ ω 2 + γ 2 : (3.2.27) (3.1.33) t < 0 V (t)v (0) = k BT m e γ t (3.2.28) I ω = 2k BT γ m 1 ω 2 + γ 2 (3.2.29) (5) X(t) 2 1. ϕ(t) = X(t)X(0) 2. X ω 2 Wiener-Khinchin 2 X ω
58 58 3 X ω X(t)X(0) I ω ϕ(t) I ω ϕ(t) (3.2.1) I ω (3.2.5) I ω X(t) ϕ(t) = X 2 e t /τ (3.2.30) τ X(t) t ϕ(t) τ I ω ω ωi ω ω = 1/τ ϕ(t) ωi ω ちょうど 1/τ でピーク 時定数 τ が分かりにくい t 1/τ ω : 33 (25 ) (3.1.19) (3.1.30) (3.1.19) (3.1.26) R(t)R(t ) = Dδ(t t ) D = 2γ X 2 X(t) 34 (30 ) I ω 35 (15 ) FP 2.2 Wiener-Khinchin 2
59 3.2 Wiener-Khinchin (11 12 ) (10 ) (3.1.42) I ω
60 (11 18 ) α(t)((3.3.2) ) (1) (2) (3) (4) 1. a f(t) b f(t) = 0 c ( ) 2. X(t) Ẋ(t) = γx(t) + R(t) + f(t) (3.3.1) (R(t) ) f(t) x(t) x(t) = X(t) 1. x(t) = t α(t t )f(t )dt (3.3.2) 2. α(t) = e γt (3.3.3) 1 X = X(t) u(x) = k(x x 0 (t)) 2 /2 x(t) = X(t) x 0 (t) = A cos ωt
61 3.3 (11 18 ) 61 (1) WK I = V R : V I() (3.3.4)? x() f : I = V R + α 2V 2 + α 3 V 3 + : (3.3.5) V/R V (2) f (f = f(t)) x ( f(t)) ( x) t t
62 62 3 f(t') 累積 t t' t t x(t, t ) x(t) = t x(t, t )dt (3.3.6) 1a f(t ) () x(t, t ) = α(t, t )f(t ) (3.3.7) f(t ) x(t, t ) t t t (1b) α(t, t ) = α(t t ) (3.3.8) (3.3.2) t 1c (3) f(t) R(t) Ẋ(t) = γx(t) + R(t) + f(t) (3.3.9) x(t) = X(t) (2) R(t) = 0 f(t) = f(t) Ẋ(t) = γ X(t) + f(t) (3.3.10) Ẋ(t) = d X(t) /dt = ẋ(t) ẋ(t) = γx(t) + f(t) (3.3.11)
63 3.3 (11 18 ) 63 (3.3.11) () t = t 0 x = x(t 0 ) x(t) = e γ(t t0) x(t 0 ) + t t 0 e γ(t t ) f(t )dt (3.3.12) t 0 t 0 e γ(t t 0) = e γt+γt x(t) = t e γ(t t ) f(t )dt (3.3.13) α(t) = exp[ γt] 1. V 0 Q(t) C I(t) E(t) Q(t) V (t) E(t) Q(t)/C Q(t) C I = Q(t) + RI = V (t) + E(t) (3.3.14) Q(t) = Q(t) RC + V (t) R + E(t) R (3.3.15) (3.3.9) X(t) = Q(t) R γ = 1 RC, E(t) f(t) = R (3.3.16) (3.3.13) q(t) = Q(t) = t e (t t ) CR E(t ) R dt (3.3.17) q(t) = t α E (t t )E(t )dt (3.3.18) E(t) α E (t) = 1 R exp[ t CR ] (3.3.19)
64 (3.3.9) Ẋ(t) = γ du(x(t)) + R(t) dx (3.3.20) = γk(x x 0 (t)) + R(t) (3.3.21) γ γk f γkx 0 (t) x(t) = X(t) = t α(t t )γkx 0 (t )dt (3.3.22) α(t) = e γkt (3.3.23) x 0 (t) = A cos ωt = RAe iωt (R ) (3.3.23) (3.3.22) t x(t) = R e γk(t t ) γkae iωt dt (3.3.24) t = Re γkt γka exp[γkt + iωt ]dt (3.3.25) = Re γkt [ γka exp[γkt + iωt ] γk + iω ] t (3.3.26) t e γkt 0 = RγkA eiωt γk + iω (3.3.27) γk ω = γka γ 2 k 2 cos ωt + γka + ω2 γ 2 k 2 sin ωt + ω2 (3.3.28) sin ωt ω x(t) 0
65 3.3 (11 18 ) 65 (4) : ()x f (3.3.2) α(t) (3.3.3) (3.3.19) : α(t) t x(t) = }{{} α(t t ) }{{} f(t ) dt (3.3.29) }{{} α(t) x(t) 2 α(t) f(t) : (3.3.11) (3.3.12) f(t) = 0 ẋ(t) = γx(t) x(t) = Ce γt (3.3.30) C (3.3.11) C = C(t) x(t) = C(t) exp[ γt] (3.3.11) ẋ(t) = Ċ(t) exp[ γt] γc(t) exp[ γt] Ċ(t)e γt = f(t) (3.3.31) Ċ(t) = f(t)e γt (3.3.32) t C(t) = f(t )e γt dt (3.3.33) x(t) = e γt t f(t )e γt dt (3.3.34)
66 66 3 (3.3.34) x(t) = Ae γt + t e γ(t t ) f(t )dt (3.3.35) A t = t 0 x(t) t = t 0 x = x(t 0 ) (3.3.12) : 37 (24 ) (10 )(3.3.2) x(t) 2 P60 39 (10 ) (3.3.2) x ω = α ω f ω x ω f ω x(t) f(t) α ω = 0 α(t)e iωt dt (3.3.36) 40 (15 ) t ± f(t) 0 W = x(t) f(t)dt (3.3.37) α ω f ω f ω f(t) α ω (3.3.36) α ω (3.3.2) W 41 (20 )
67 3.4 (11 25 ) (11 25 ) α(t) 3 (2 4) 3 (1) (2) (3) (4) 3.3 f(t) x(t) 1. X = X(t) X neq x(t) = X(t) neq 2. X 0 3. E(x) exp[ βe(x)] β = 1/(k B T ) 4. E(x) = E 0 (x) xf(t) E 0 (x) 0 x(t) = t α(t t )f(t )dt (3.4.1) α(t) α(t) = β Ẋ(t)X(0) (3.4.2)
68 68 3 f(t) = 0 X = X(t) u(x) = k(x x 0 ) 2 /2 x 0 (1) : WK : ϕ(t) = X(t)X(0) (3.4.3) (X(t) ) 3.3 : x(t) = t α(t t )f(t )dt (3.4.4) 2 (3.4.2) (3.1.42) (3.3.19) : α E (t) α(t) Q(t)Q(0) = Ck B T exp[ t CR ] (3.4.5) α E (t) = 1 R exp[ t CR ] (3.4.6)
69 3.4 (11 25 ) 69 e t/(cr) Q(t)Q(0) α E (t) ϕ(t) = Q(t)Q(0) = Ck BT CR β = 1/(k B T ) exp[ t CR ] = k BT α E (t) (3.4.7) α E (t) = β ϕ(t) (3.4.8) (2) 1 OK(65 ) 2 3 * 3 1 α(t) f(t) f(t) f(t) = 0 f(t) f(t) = f 0 t f(t) = { 0 t < 0 f 0 t 0 (3.4.9) x(t) = X(t) t f 0 f 0 x(t) = Ψ(t)f 0 + Ψ 2 (t)f Ψ 3 (t)f (3.4.10) f = 0 x(t) = 0 ()Ψ(t) α(t) Ψ(t) α(t) α(t) f 0 x(t) = t α(t t )f(t )dt + Ψ 2 (t)f Ψ 3 (t)f (3.4.11) *3
70 70 3 (3.4.9) = t 0 α(t t )f 0 dt + Ψ 2 (t)f Ψ 3 (t)f (3.4.12) τ = t t dτ = dt t = t τ = 0 t = 0 τ = t = t (3.4.10) 0 α(τ)dτf 0 + Ψ 2 (t)f Ψ 3 (t)f (3.4.13) Ψ(t) = t 0 α(τ)dτ (3.4.14) Ψ(t) (3.4.14) α(t) = Ψ(t) 2 ( 1) t < 0 t f(t) = f 0 T (x, x, t; f 0 ) f(t) = 0 T (x, x, t; 0) f 0 f(t) = f 0 P eq (x; f 0 ) exp[ βe(x)] = exp[ βe 0 (x) + βxf 0 ] (3.4.15) f(t) = 0 P eq (x; 0) exp[ βe 0 (x)] (3.4.16) (2.4.2) P (x, t) = (2.4.3) P eq (x; f 0 ) = (3.1.48) X(t)X(0) = T (x, x, t; f 0 )P eq (x ; 0)dx (3.4.17) T (x, x, t; f 0 )P eq (x ; f 0 )dx (3.4.18) xt (x, x, t; 0)dxx P eq (x ; 0)dx (3.4.19)
71 3.4 (11 25 ) 71 (3.4.17) 2 P 0 (X) = P eq (x; 0) x(t) = X(t) neq = xp (x, t)dx (3.4.20) (3.4.17) Ψ(t) T (x, x, t; f 0 ) f (3.4.17) f 0 f0 2 (3.4.17) f 0 T (x, x, t; f 0 ) T (x, x, t; f 0 ) f 0 (3.4.18) P eq (x; f 0 ) f (3.4.18) T (x, x, t; f 0 ) f 0 P eq (x; f 0 ) f 0 (3.4.17) P eq (x ; f 0 ) (3.4.15) (3.4.16) P eq (x; 0) = exp[ βxf 0 ]P eq (x; f 0 )C(f 0 ) (3.4.21) C(f 0 ) C(f 0 ) = 1 ( 43 ) (3.4.17) P (x, t) = T (x, x, t; f 0 ) exp[ βx f 0 ]P eq (x ; f 0 )dx (3.4.22) exp[ βx f 0 ] (3.4.18) exp[ βx f 0 ] f 0 exp[ βx f 0 ] = 1 βx f 0 + (f 0 2 ) (3.4.23) (3.4.22) P (x, t) = T (x, x, t; f 0 )P eq (x ; f 0 )dx + T (x, x, t; f 0 )( βx f 0 )P eq (x ; f 0 )dx + (f 0 2 ) (3.4.24) (3.4.18) = P eq (x; f 0 ) + T (x, x, t; f 0 )( βx f 0 )P eq (x ; f 0 )dx + (f 0 2 ) (3.4.25)
72 72 3 f 0 P eq (x; f 0 ) (3.4.15) (3.4.16) P eq (x; f 0 ) = exp[βxf 0 ]P eq (x; 0) (3.4.26) f 0 = P eq (x; 0)(1 + βxf 0 + f 0 2 ) (3.4.27) (3.4.25) 2 T (x, x, t; f 0 ) f 0 1 βx f 0 f 0 1 P eq (x ; f 0 ) f 0 1 T (x, x, t; f 0 )( βx f 0 )P eq (x ; f 0 ) (3.4.27) (3.4.28) = T (x, x, t; 0)( βx f 0 )P eq (x ; 0) + (f 0 2 ) (3.4.28) P (x, t) = P eq (x; 0) + βxf 0 P eq (x; 0) P (x, t) f 0 (3.4.29) (3.4.20) x(t) = xp eq (x; 0)dx + T (x, x, t; 0)βx f 0 P eq (x ; 0)dx xβxf 0 P eq (x; 0)dx + (f 0 2 ) (3.4.29) x T (x, x, t; 0)βx f 0 P eq (x ; 0)dx dx + (f 0 2 ) (3.4.30) = X + β X 2 f 0 x T (x, x, t; 0)βx f 0 P eq (x ; 0)dx dx + (f 0 2 ) (3.4.31) 3 (3.4.19) = X + β X 2 f 0 β X(t)X(0) f 0 + (f 0 2 ) (3.4.32) X = 0 x(t) = β X 2 f 0 β X(t)X(0) f 0 + (f 0 2 ) (3.4.33)
73 3.4 (11 25 ) 73 (3.4.10) Ψ(t) = β X 2 β X(t)X(0) (3.4.34) α(t) = Ψ(t) = β Ẋ(t)X(0) (3.4.35) (3) ( ) 4 E(x) = u(x) = k 2 (x x 0(t)) 2 = k 2 x2 kxx 0 (t) + x 0 (t) 2 (3.4.36) x 0 (t) 2 x E(x) E(x) = k 2 x2 kxx 0 (t) (3.4.37) f(t) = kx 0 (t) E(x) = k 2 x2 xf(t) (3.4.38) 4 x(t) = t α(t t )kx 0 (t )dt, α(t) = β Ẋ(t)X(0) (3.4.39) (3.4.9) { 0 t < 0 x 0 (t) = x 0 t 0 (3.4.40) (3.4.34) x(t) = Ψ(t)kx 0 = β X 2 kx 0 β X(t)X(0) kx 0 (3.4.41) P eq (x) e βkx2 /2 X 2 = x2 P eq (x)dx = k B T/k x(t) = x 0 β X(t)X(0) kx 0 (3.4.42)
74 74 3 X(t)X(0) = x 0 x(t) = X 2 { 1 x(t) } βkx 0 x 0 (3.4.43) 2 (βk) 1 = X 2 x(t) X(t)X(0) Ψ(t) x(t) x 0 レーザーを動かす前の位置 レーザーを動かし た後の安定な位置 緩和関数 Ψ(t) t (3.4.43) P
75 3.4 (11 25 ) 75 (4) (3.4.9) (α(t) ) 1 (3.4.17) f 0 ( P eq (x ; 0) 2 ) T (x, x, t; f 0 ) ((3.4.17) f 0 T (x, x, t; f 0 ) ) f 0 (3.4.17) (3.4.18) (3.4.21) 3 4 (3.4.17) exp[ βxf 0 ] f 0 (3.4.18) P eq (x ; f 0 ) (3.4.20) (1956 ): () 2. :
76 : (3.4.34) β 2 β β β 3 β : 42 (10 ) q(t) = γ{q(t) q 0 } + R(t) (3.4.44)
77 3.4 (11 25 ) 77 q(t) q(t) 2 1 () q 0 = 0 () q 0 = q ex q 0 0 q ex 43 (15 ) (3.4.21) C(f 0 ) 1 1 C(f 0 ) 1 44 (30 ) N ({X µ (t)} = X 1 (t), X 2 (t),..., X N (t)) ({f µ (t)} = f 1 (t), f 2 (t),..., f N (t)) P67 1 N E({x µ }) = E 0 ({x µ }) x µ f µ (t) (3.4.45) µ α µ,ν (t) = β Ẋµ (t)x ν (0) (3.4.46) α µ,ν (t) X µ (t) = N ν t α µ,ν (t t )f ν (t )dt (3.4.47) 45 (30 ) 1 46 (10 ) 3.3 2(P.60) α ω = ωβ 2 ϕ(ω) (3.4.48) α ω (3.3.36) α ω β = 1/(k B T ) ϕ(ω) (3.2.4) 47 (30 ) f(t) = f 0 (t)δ(r r 0 ) ρ(r) i δ(r r i) r i i
78 (10 ) P (25 ) a X µ = X µ (t), µ = 1,..., n (FP) b X µ 0 c d f µ (t), µ = 1,..., n E({x µ }) = E 0 ({x µ }) N µ x µf µ (t) X 1 X 2 50 (30 ) (37 ) 51 (30 ) (P.67) α(t) (4)
79 (12 2 ) 1 ((4.1.1) ) (1) 4. (2) (3) (4) (5) x = x(t) ẋ(t) = L ds (x) dx (4.1.1) S (x) 1 x L > 0 x(t) t T c T > T c T = T c (t α ) (critical slowing down) (4.1.1)
80 80 4 (1) (2) ( Ψ(t)) 4 () : 1 1 (3) x = x(t) ẋ(t) = L ds (x) dx (4.1.2) L S (x) (4.3 ) S (x) P.79 L > 0
81 4.1 (12 2 ) 81 (4.1.2) {x µ (t)} = {x 1 (t), x 2 (t),..., x n (t)} x µ = x µ (t) ẋ µ (t) = n ν L µν S ({x µ (t)}) x ν (4.1.3) Q T 1 T 2 L L κ 2 1 Q Q = κs T L (4.1.4) S T 2 (T 2 T 1 ) 1 E 1 E 1 = Q (4.1.5) (4.1.4) E 1 = κs T L (4.1.6) (4.1.6) (4.1.2) x: 1 E 1 S (x): 2 ( P.79 ) S 1 S 2 E 1 E 2 S 1 = S 1 (E 1 ), S 2 = S 2 (E 2 ), S = S 1 (E 1 ) + S 2 (E 2 ) (4.1.7) 2 E 1 + E 2 = E S (x) = S (E 1 ) = S 1 (E 1 ) + S 2 (E E 1 ) (4.1.8)
82 82 4 x = E 1 ds (x) dx = ds (E 1 ) = ds 1(E 1 ) ds 2(E 2 ) de 1 de 1 de 2 (4.1.9) E2 =E E 1 = 1 T 1 1 T 2 1 T 2 (T 2 T 1 ) (4.1.10) T 1 T 2 T 1 T 2 T T = T 2 T 1 (4.1.6) ẋ = E 1 = κs L T 2 ds (x) dx (4.1.11) (4.1.11) L = κst 2 /L (4.1.6) (4.1.2) L = L 11 (4.1.12) L 11 > 0 Ė 1 = L 11 T 2 (T 2 T 1 ) (4.1.12) T 2 > T 1 Ė 1 > T 1 > T 2 Ė 1 < φ 1 φ I R I = 1 R (φ 2 φ 1 ) (4.1.13) 1 q 1 q 1 = I q 1 = 1 R (φ 2 φ 1 ) (4.1.14) (4.1.14) (4.1.2)
83 4.1 (12 2 ) 83 x: 1 q 1 S (x): 2 1 de = T ds + φdq (4.1.15) φdq dq ( ) S = φ q E T (4.1.16) 2 q 1 + q 2 = q S (E 1, q 1 ) = S 1 (E 1, q 1 ) + S 2 (E E 1, q q 1 ) (4.1.17) ( ) S q 1 = φ 1 E 1 T + φ 2 T = φ 2 φ 1 T (4.1.18) (4.1.14) ẋ = q 1 = T ( ) S R q 1 E 1 = T R ds (x) dx (4.1.19) L = T/R (4.1.2) L = L {x 1, x 2 } = {E 1, q 1 } (4.1.3) ẋ µ = 2 ν=1 L µν S x ν (4.1.20) S T 1 T 2 T T 2 T 1 2 Ė 1 = L 11 T 2 (T 2 T 1 ) + L 12 T (φ 2 φ 1 ) (4.1.21) q 1 = L 21 T 2 (T 2 T 1 ) + L 22 T (φ 2 φ 1 ) (4.1.22)
84 T 1 = T 2 (4.1.21) Ė 1 = L 12 T (φ 2 φ 1 ) (4.1.23) 2 2 q 1 = 0 (4.1.22) 1 2 L 21 T 2 (T 2 T 1 ) + L 22 T (φ 2 φ 1 ) = 0 (4.1.24) φ 2 φ 1 = L 21 T L 22 (T 2 T 1 ) (4.1.25) T 1 T 2 : S (x) F x x F = F (x) T>Tc F(x) T<Tc F(x) 平衡値 平衡値 x=0 x x=0 x S (x) = F (x) S (x) (4.1.2) df (x) ẋ(t) = L dx (4.1.26)
85 4.1 (12 2 ) 85 F (x) = F 0 + a(t T c )x 2 + bx 4 + (4.1.27) F (x) T > T c x x 4 F (x) = F 0 +a(t T c )x 2 (4.1.26) ẋ(t) = γx (4.1.28) γ = 2L a(t T c ) x(t) = x(0)e γt (x = 0) T = T c x 2 0 x x 4 x 6 ẋ(t) = 4L bx 3 (4.1.29) x(t) = (8L bt + C) 1/2 (x = 0) (4) (3) 2 (4.1.2) x = x(t) Yes P.79 x(t) t : S S = S(x) ds (x) dt = ds (x) ẋ (4.1.30) dx (4.1.2) = ds (x) dx L ds (x) dx ( ds = L ) 2 (x) 0 (4.1.31) dx S (x) S (x) t x
86 86 4 S'(x) x eq x : 52 (25 ) (3) 1. 2 n (4.1.3) 1 S n 53 (30 ) 1 (4.1.2) x = x(t) S (x) (4.1.2) 2 54 (30 ) (4.1.2) (4.1.3) 55 (10 ) T 0 T (r, t) t = κ 2 T (r, t) (4.1.32) T (r, t) t 3 r κ S(t) = T (r, t) ln[t (r, t)/(et 0 )]dr t T (r, t) T 0 56 (30 ) (4.1.2) (4.1.3) x(t) S(x)
87 4.2 (12 9 ) (12 9 ) ( ) (1) (2) (3) (4) {X µ } = {X 1, X 2,...} l q l X µ = X µ ({q l }) X µ (t)x ν (0) = X µ ( t)x ν (0) (4.2.1) X µ ( t)x ν (0) = X ν (t)x µ (0) X µ (t)x ν (0) = X ν (t)x µ (0) (4.2.2) 2 Θ = Θ(t)(t ) N = N(t) Θ(t)N(0) N(t)Θ(0)
88 88 4 (1) 4.2 L µν (4.2.2) (4.2.2) : : : R, P : r 1, r 2,..., p 1, p 2,... {q l, p l } X(t) {q l (t), p l (t)} X(t) = X({q l (t), p l (t)}) (4.2.3)
89 4.2 (12 9 ) 89 q l (t), p l (t) q l (0), p l (0) X(t) q l (0), p l (0) t X(t) = X({q l (t), p l (t)}) = f(t, {q l (0), p l (0)}) (4.2.4) q l (0), p l (0) q l (t), p l (t) X(t) q l (0), p l (0) ρ({q l, p l }) X(t) = dγf(t, {q l, p l })ρ({q l, p l }) (4.2.5) q l (0) = q l p l (0) = p l dγ = l dq ldp l ρ eq ({q l, p l }) X(t)X(0) = dγf(t, {q l, p l })X({q l, p l })ρ eq ({q l, p l }) (4.2.6) X 2 X µ (t)x ν (0) = X(t) (t=0) X dγf µ (t, {q l, p l })X ν ({q l, p l })ρ eq ({q l, p l }) (4.2.7) (2) n {X 1, X 2,..., X n } = {X µ } Ẋ µ (t) = F ({X µ (t)}), µ = 1,..., n (4.2.8)
90 90 4 t t = h(t), X µ X µ = g µ ({X µ }) t t = j(t ) X µ(t ) X µ(t ) g µ ({X µ (j(t ))}), µ = 1,..., n (4.2.9) X µ (t) X µ(t ) *4 (4.2.8) Ẋ µ(t ) = F ({X µ(t )}), µ = 1,..., n (4.2.10) X µ (t) (4.2.8) X µ(t ) (4.2.8) 57 V ({q l (t)}) m q l (t) = p l(t) m (4.2.11) ṗ l (t) = V ({q l(t)}), l = 1,..., n (4.2.12) q l (t) () ( ) t t = t, (4.2.13) q l q l = q l, p l p l = p l (4.2.14) q l(t ) q l ( t ), p l(t ) p l ( t ) (4.2.15) q l (t) = q l(t ), p l (t) = p l(t ) (4.2.16) t t = t q l (t) = q l(t ), ṗ l (t) = ṗ l(t ) (4.2.17) *4 X µ = g µ({x µ }) X µ = k({x µ }) X µ(t) = k({x µ (t )})
91 4.2 (12 9 ) 91 (4.2.15) (4.2.17) (4.2.11) (4.2.12) t = t q l(t ) = p l (t ) m ṗ l(t ) = V ({q l (t )}) q l (t ) (4.2.18) (4.2.19) (4.2.11) (4.2.12) q l (t), p l (t) q l ( t), p l ( t) : (1 ) q(t) = p(t) (4.2.20) m ṗ(t) = 0 (4.2.21) q(t) = vt + c (4.2.22) p(t) = mv (4.2.23) v c q (t ) = q( t ) = vt + c (4.2.24) p (t ) = p( t ) = mv (4.2.25) q( t), p( t) (4.2.20) (4.2.21) q l (t), p l (t) q 0 l, p0 l q l (0) = q 0 l, p l(0) = p 0 l q l( t), p l ( t) q l (0), p l (0) q 0 l, p0 l q l (t) q l (t) = q l (t, {ql 0, p0 l }) q l( t) q l ( t, {ql 0, p0 l }) ql 0, p0 l q l(t, {ql 0, p0 l }) q l ( t, {ql 0, p0 l }) = q l(t, {ql 0, p0 l }) (4.2.26)
92 92 4 (3) X µ = X µ (t), µ = 1,..., n q l (X µ (t) = X µ ({q l (t)})) (4.2.4) q l, p l X µ (t) = X µ ({q l (t)}) = f µ (t, {q l, p l }) (4.2.27) (4.2.27) X µ ( t) = X µ ({q l ( t)}) = f µ ( t, {q l, p l }) (4.2.28) (4.2.26) q l ( t) = q l ( t, {q l, p l }) = q l (t, {q l, p l }) (4.2.29) X µ ( t) = X µ ({q l ( t)}) = X µ ({q l (t, {q l, p l })}) = f µ (t, {q l, p l }) (4.2.30) f µ ( t, {q l, p l }) = f µ (t, {q l, p l }) (4.2.31) (4.2.1) X µ ( t) = f µ ( t, {q l, p l }) (4.2.7) X µ ( t)x ν (0) = dγf µ ( t, {q l, p l })X ν ({q l })ρ eq ({q l, p l }) (4.2.32) (4.2.31) = dγf µ (t, {q l, p l })X ν ({q l })ρ eq ({q l, p l }) (4.2.33) p l p l = p l X µ ( t)x ν (0) = dq l dp lf µ (t, {q l, p l})x ν ({q l })ρ eq ({q l, p l}) (4.2.34) l ρ eq ({q l, p l }) = ρ eq ({q l, p l }) (4.2.35) p l p l ( 61) X µ ( t)x ν (0) = dq l dp lf µ (t, {q l, p l})x ν ({q l })ρ eq ({q l, p l}) l = X µ (t)x ν (0) (4.2.36)
93 4.2 (12 9 ) 93 (4.2.1) 1 1 {q l, p l } = {q, p} (4.2.32) f µ ( t, {q, p})x ν (q)ρ eq ({q, p}) f µ ( t, {q, p}) = f µ (t, {q, p}) ρ eq ({q, p}) = ρ eq ({q, p}) f µ ( t, {q, p})x ν (q)ρ eq ({q, p}) = f µ (t, {q, p})x ν (q)ρ eq ({q, p}) (4.2.37) p f µ (t, {q, p})x ν (q)ρ eq ({q, p}) (4.2.1) (4) + : : q l (t) q l ( t) (4.2.2) 2 1. (4.2.1) µ = ν 2. X µ ( t)x ν (0) = X ν (t)x µ (0) 2 : 3 {X µ } = {X 1, X 2,...} q l, p l X µ = X µ ({q l, p l }) X µ ({q l, p l }) = ɛ µ X µ ({q l, p l }) ; ɛ µ = ±1 (4.2.38) X µ (t)x ν (0) = ɛ µ ɛ ν X µ ( t)x ν (0) (4.2.39) X µ ( t)x ν (0) = X ν (t)x µ (0) X µ (t)x ν (0) = ɛ µ ɛ ν X ν (t)x µ (0) (4.2.40)
94 94 4 (1 ) {q l, p l } = {R, r 1, r 2,..., P, p 1,...}: R P r i p i i X 1 ({q l, p l }) = q 1 = R ɛ 1 = 1 X 2 ({q l, p l }) = p 1 /M = P/M ɛ 2 = 1 X 3 ({q l, p l }) = i p2 i /(2m) ɛ 3 = 1 (4.2.26) p l (t) p l (t) = p l (t, {ql 0, p0 l }) (4.2.26) p l ( t, {ql 0, p 0 l }) = p l (t, {ql 0, p 0 l }) (4.2.41) (4.2.4) t t X µ ( t) = X µ ({q l ( t), p l ( t)}) = f µ ( t, {q l, p l }) (4.2.42) {q 0 l, p0 l } {q l, p l } q l ( t) p l ( t) q l ( t, {q l, p l }) p l ( t, {q l, p l }) X µ ({q l ( t), p l ( t)}) = X µ ({q l ( t, {q 0 l, p 0 l }), p l ( t, {q 0 l, p 0 l })}) (4.2.43) (4.2.26) (4.2.41) X µ ({q l ( t), p l ( t)}) = X µ ({q l (t, {q 0 l, p 0 l }), p l (t, {q 0 l, p 0 l })}) (4.2.44) (4.2.44) (4.2.38) X µ ({q l ( t), p l ( t)}) = ɛ µ X µ ({q l (t, {q 0 l, p 0 l }), p l (t, {q 0 l, p 0 l })}) (4.2.45) f µ (t, {q l, p l }) (4.2.4) = ɛ µ f µ (t, {q 0 l, p 0 l }) (4.2.46) f µ ( t, {q 0 l, p 0 l }) = ɛ µ f µ (t, {q 0 l, p 0 l }) (4.2.47)
95 4.2 (12 9 ) 95 (4.2.7) X µ ( t)x ν = dγf µ ( t, {q l, p l })X ν ({q l, p l })ρ eq ({q l, p l }) (4.2.48) = dγɛ µ f µ (t, {q l, p l })X ν ({q l, p l })ρ eq ({q l, p l }) (4.2.49) p l = p l = dq l dp lɛ µ f µ (t, {q l, p l})x ν ({q l, p l})ρ eq ({q l, p l}) (4.2.50) l (4.2.38) = dq l dp lɛ µ f µ (t, {q l, p l})ɛ ν X ν ({q l, p l})ρ eq ({q l, p l}) (4.2.51) l (4.2.35) = dq l dp lɛ µ f µ (t, {q l, p l})ɛ ν X ν ({q l, p l})ρ eq ({q l, p l}) (4.2.52) l = ɛ µ ɛ ν X µ (t)x ν (4.2.53) : 57 (10 ) P (20 ) KdV u t + 3 u x 3 + 6u u x = 0 (4.2.54) 2 u = 2 sech 2 (x 4t) 59 (30 ) (4.2.5) (4.2.6) T ({q l, p l }, {ql 0, p0 l }, t) X(t) X(t)X(0) T ({q l, p l }, {ql 0, p0 l }, t) f(t, {ql 0, p 0 l }) = U({ql 0, p 0 l }, {q l, p l }, t)x({q l, p l })dγ (4.2.55)
96 96 4 U({ql 0, p0 l }, {q l, p l }, t) U({ql 0, p 0 l }, {q l, p l }, t) = T ({q l, p l }, {ql 0, p 0 l }, t) (4.2.56) (4.2.5) (4.2.6) 60 (30 ) 59 (4.2.56) 61 (30 ) (4.2.35) ρ eq ({q l, p l }) 62 (10 ) P X µ l p l X µ = X µ ({q l, p l }) X µ ({q l, p l }) = X µ ({q l, p l }) (4.2.1) 63 (30 ) 4.1 ((4.1.2) (4.1.3) ) 64 (30 ) (4.2.1) P.33 (2.3.50) P.42 (3.1.1) (4.2.1)
97 4.3 1(12 16 ) (12 16 ) Thomson Peltier Thomson ( ) (1) 4.3 (2) Thomson (3) (4) : X µ = X µ (t) Ẋ µ = ν L µν S X ν + R µ (t) (4.3.1) R µ (t) R µ (t) = 0 (4.3.2) R µ (t 1 )R ν (t 2 ) = D µν δ(t 1 t 2 ) (4.3.3) g({x µ (0)})R µ (t) = 0 t 0 (4.3.4) g({x µ (0)}) {X µ (0)} = X 1 (0), X 2 (0), (4.1.3) Ẋ µ = ν L µν S X ν (4.3.5) (4.3.1) (4.3.5) L µν = L µν S = S (4.3.6)
98 98 4 L λµ = L µλ 2 ( ) (Peltier ) (1) 4.3 L µν (4.2.2) (2) Thomson ( 66 ) 2 ( ) 2 (4.1.22) 1 L 21 (T 2 T 1 ) > 0 L 21 2 L A 21 LB 21 A T 1 T 2 B L A 1 21 T 2 (T 2 T 1 ) L B 1 21 T 2 (T 2 T 1 ) (L A 21 L B 21) 1 T 2 (T 2 T 1 ) (4.3.7)
99 4.3 1(12 16 ) 99 A T 1 T V 2 B V = e AB (T 2 T 1 ) (4.3.8) e AB A B V A Peltier I I A B 1 I 2 Q Q A 1 Q A B Q B I 1 Q A Q B = Π AB I (4.3.9) Π AB A B Thomson Peltier 2 Π AB T = e AB (4.3.10) (Thomson ) (3) (4.3.5) =
100 100 4 X(t) 緩和過程の式で表せる ゆらぎ : ランジュバン方程式で表せる 平衡の値 時刻 t (4.1.3) : 65 (10 ) (4.2.11) (4.2.12) a q l = q l + a p l = p l t = t (4.2.11) (4.2.12) () q l (t) p l (t) (4.2.26) 66 (20 ) (4.1.23) Ė1 (4.1.25) (10 ) P (4.1.22) A I A = L A 21(T 2 T 1 )/T 2 + L A 22(φ 2 φ 1 )/T B I B = L B 21(T 2 T 1 )/T 2 + L B 22(φ 2 φ 1 )/T I A + I B = 0 I A T 2 T 1 T L A 21 LB 21 A B R A R B 68 (20 ) ( ) Peltier
101 4.3 1(12 16 ) (25 ) S(x)
102 102 4 : : 1 27 ( ) 4:00 PDF (1 13 ) X µ (t) X µ (t)x λ (0) X λ (t)x µ (0) Thomson (1) (2) (3) Thomson (4) X µ = X µ (t) (1) Ẋ µ = ν L µν S X ν + R µ (t) (4.4.1)
103 4.4 2 (1 13 ) 103 R µ (t) (4.3.2) (4.3.3) (4.3.4) 3. : X µ = X µ (t) Ẋ µ = ν L µν (4.4.1) (4.4.2) S X ν (4.4.2) L µν = L µν (4.4.3) (4.4.2) L λµ = L µλ (4.4.4) Thomson (1) (1) X µ (t)x ν (0) = X ν (t)x µ (0) (4.4.5) (3): (4.4.3) L µν = L µν } X µ (t)x ν (0) = X ν (t)x µ (0)
104 X µ (t) t = 0 X µ (t) = X µ (0) + tẋµ(0) + (4.4.6) 2 Ẋµ(0) (4.4.1) (X µ (t) t 1 L µν S ) 3 X µ (t) X ν (0) X µ (t)x ν (0) (t 1 L µν S ) 4 S t L µν 5 X ν (t)x µ (0) 6 (4.4.5) L λµ = L µλ 7 (4.4.3) L λµ = L µλ (2) X µ (t) t = 0 t X µ (t) = X µ (0) + tẋµ(0) + (4.4.7) (4.4.1) = X µ (0) + t ν S({X µ }) L µν X ν + tr µ (0) + (4.4.8) Xµ =X µ (0)
105 4.4 2 (1 13 ) 105 X λ (0) X λ = X λ (0) X λ R µ (0) = 0 X λ X µ (t) = X λ X µ + t ν S({X µ }) L µν X λ + (4.4.9) X ν S({X µ }) = ln P eq ({X µ }) S({X µ }) X λ = δ λν (4.4.10) X ν X µ ± P eq ({X µ }) 0 (4.4.10) (4.4.9) X λ X µ (t) = X λ X µ tl µλ + (4.4.11) X µ X λ (t) = X µ X λ tl λµ + (4.4.12) (4.4.5) X λ X µ (t) = X µ X λ (t) t L λµ = L µλ (4.4.13) L µν = L µν (4.4.14) L λµ = L µλ (4.4.15) (3) Thomson (4.1.22) q 1 = L 21 T 2 (T 2 T 1 ) + L 22 T (φ 2 φ 1 ) (4.4.16) 2 R L 22 /T = 1/R q 1 = 0 (4.1.25) φ 1 φ 2 = L 21 T L 22 (T 2 T 1 ) = L 21 R T 2 T 1 T 2 (4.4.17)
106 AB L 21 R AB L A 21 RA L B 21 RB φ A A T 1 T 2, φ V 2 φ B B φ A φ 2 = L A 21R A T 2 T 1 T 2 (4.4.18) φ B φ 2 = L B 21R B T 2 T 1 T 2 (4.4.19) (4.4.18) (4.4.19) V = φ A φ B (4.3.8) V = e AB (T 2 T 1 ) φ A φ B = (L A 21R A L B 21R B ) T 2 T 1 T 2 (4.4.20) V = (L A 21R A L B 21R B ) T 2 T 1 T 2 (4.4.21) e AB = LA 21R A L B 21R B T 2 (4.4.22) Π AB T 1 = T 2 Q A φ A I A φ 2 φ B Q B B 1 I 2 (4.1.23) Ė 1 = L 12 T (φ 2 φ 1 ) (4.4.23) Ė1 Q A = LA 12 T (φ 2 φ A ) (4.4.24) Q B = LB 12 T (φ 2 φ B ) (4.4.25) Q B L 21 AB L A 12 LB 12
107 4.4 2 (1 13 ) 107 φ 2 φ A = R A I φ 2 φ B = R B I Q A = LA 12 T RA I (4.4.26) Q B = LB 12 T RB I (4.4.27) Q A Q B = LA 12R A L B 12R B I (4.4.28) T (4.3.9) Q A Q B = Π AB I Π AB = LA 12R A L B 12R B T (4.4.29) {X 1, X 2 } = {E 1, q 1 } L A 12 = L A 21, L B 12 = L B 21 (4.4.30) Thomson Π AB T = e AB (4.4.31) (4) L λµ = L µλ 1 X µ (t) = f µ (t, {q l (0), p l (0)}) (4.4.5) 2 (4.4.1) * 3 ( ) (4.4.1) R µ (t) (4.4.2) *
108 108 4 : Thomson * Thomson P102 ( ) (4.4.2) Peltier (4.4.2) L µν : 70 (10 ) (4.4.10) X µ Xµ min < X µ < Xµ max X = Xµ min Xµ max P eq ({X µ }) 0 L λµ L µλ X µ = Xµ min, Xµ max P eq ({X µ }) 71 (10 ) 1 P87 3 P93 X µ (t)x ν (0) = X ν (t)x µ (0) 72 (30 ) 2 r(1 ) {x 1, x 2 } = {r, ṙ} ṙ r S k ( m S = βe + = β 2 ṙ2 + k ) 2 r2 + (4.4.32) m k β = (k B T ) S r = L 21 r + L S 22 ṙ (4.4.33) 71 L (35 ) {x µ } S
109 4.4 2 (1 13 ) (35 ) X λ = X λ (0) X µ = X µ (0) X λ Ẋ µ Ẋµ (4.4.1) X λ Ẋ µ = ν (4.4.10) X λ Ẋ µ X µ Ẋ λ S({X µ }) L µν X λ X ν (4.4.34) = L µλ X λ Ẋ µ = ((3.1.18) ) L µλ = L λµ (4.4.13) 75 (20 ) Thomson {x 1, x 2 } = {E 1, q 1 } E 1 P87 3 X({q l, p l }) = X({q l, p l }) (4.4.35) 76 (20 ) Peltier L 12 L 21 R L 12 R L 21 R 77 (25 ) (4.4.21) A B B 76
110 110 5 ( ) (1 20 ) : (10) α ω = ωβ 2 ϕ(ω) (4.36) α ω (3.3.15) (3.3.18) α E (t) α ω = 0 α E (t)e iωt dt (4.37) β = 1/(k B T ) ϕ(ω) (3.1.38) ϕ(ω) e iωt Q(t)Q(0) dt (4.38) 5 ( ) (1 20 ) (1) (2)
111 111 (3) (4) (5) (6) 1. (1 ) X V {q w l, pw l }( ) m V (t) = F ({q w l (t)}, X(t)) (5.1) q w l (t) = pw l (t) m w ṗ w l (t) = V ({qw l (t)}, X(t)) ql w(t) (5.2) m m w F ({ql w (t)}, X) V ({ql w (t)}, X(t)) () 2. a V (t) b 1. 1 t m V (t) = M(t t )V (t )dt + R(t) (5.3) 0 M(t) 2. 2 m V (t) = λv (t) + R(t) (5.4) 1. H. Mori, Prog. Theor. Phys. 33, 423 (1965). 2. Theory of Simple Liquids, Hasen and McDonald (Academic Press) Chapter II-6 P P ( ) 82-3 (2004) 357
112 112 5 ( ) (1 20 ) (1) {V, X, ql w, pw l } (5.1) (5.2) (5.4) R(t) {V, X, ql w, pw l } (2) t < t V (t ) t R(t) F ({q w l F ({q w l (t)}, X(t)) = 0 M(t t )V (t )dt (5.5) t 0 (t)}, X(t)) M(t t )V (t )dt + R(t) (5.6) 1 V (t) 2
113 113 (5.1)(5.2) ( ) (5.3) ( = ) (5.4) () (3) () A P A V A V A V A V P A A V θ A cos θ V (A V) P A = A cos θ = A A V = (A V) V (5.7) P A = P A V V (5.8) P A = (A V) V V V = (A V) (V V) V (5.9) () {q l, p l } ({q l, p l } = {X, V, ql w, pw l })
114 114 5 ( ) (1 20 ) A A({q l, p l }) P P (A V) AV A V A({q l, p l }) V ({q l, p l }) P : V A = A({q l, p l }) P A AV V 2 V (5.10) eq = ρ eq ({q l, p l }) l dq l dp l (5.11) ρ eq ({q l, p l }) 1. P 2 = P 2. A = A({q l, p l }) B = B({q l, p l }) P (A + B) = P A + P B () 3. P = P () O O (OA)B = A(O B) (5.12) {q l, p l } A({q l, p l }) AV / V 2 V P A({q l, p l }) = AV V 2 V
115 115 Q 1 P P + Q = 1 A = A({q l, p l }) A = }{{} P A + QA }{{} V (5.13) (4) {q l, p l } ({q l, p l } = {X, V, q w l, pw l }) {q l, p l } X(t) = X({q l (t), p l (t)}) (5.1) (5.2) dx({q l (t), p l (t)}) dt = l { q l (t) X({q l(t), p l (t)}) + ṗ l (t) X({q } l(t), p l (t)}) q l (t) p l (t) (5.14) il({q l (t), p l (t)}) l { } q l (t) q l (t) + ṗ l(t) p l (t) (5.15) dx(t) dt = il({q l (t), p l (t)})x(t) (5.16) (5.15) q l (t), ṗ l (t) (5.1) (5.2) H({q l, p l }) q l (t) = H({q l(t), p l (t)}) q l (5.17) ṗ l (t) = H({q l(t), p l (t)}) q l (t) (5.18) il({q l (t), p l (t)}) (5.16) X(t) = e il({q l(0),p l (0)})t X(0) (5.19) ( 81)X(0) X il({q l (0), p l (0)}) il X F (t) = F ({q w l F ({q l (t)}) (t)}, X(t)) = F (t) = e ilt F (5.20)
116 116 5 ( ) (1 20 ) il V (5.13) il = P }{{} il + QiL }{{} (5.21) e til P il QiL e ilt = e QiLt e P ilt e til = t 0 e t il P ile (t t )QiL dt + e tqil (5.22) ( 83 )(5.22) e til e tqil e t il P il V (t) e til F = 2 t 0 e t il P ile (t t )QiL F dt 1 + e tqil F 2 (5.23) R(t) e tqil F (5.24) (5.23) 2 R(t) 1 1 (5.24) t P (5.10) t 0 0 e t il P ile (t t )QiL F dt = t 0 e t il P ilr(t t )dt (5.25) P ilr(t t ) = [ilr(t t )]V V 2 V (5.26) e t il P ile (t t )QiL F dt = t 0 e t il [ilr(t t )]V V 2 V dt (5.27)
117 117 e t il V (5.19) = t 0 [ilr(t t )]V V 2 V (t )dt (5.28) ( 84) [ilr(t)]v = R(t)[iLV ] (5.29) ilv (5.16) dv/dt F/m (5.24) R = F ilv = R/m (5.29) [ilr(t)]v = R(t)R m (5.30) t 0 t e t il P ile (t t )QiL F dt = M(t t )V (t )dt (5.31) 0 (5.3) M(t) = R(t)R m V 2 (5.32) (5) 2a t t M(t t )V (t )dt V (t) M(t t )dt (5.33) 0 0 2a V (t) M(t) τ = t t 2b t = V (t) M(τ)dτ (5.34) 0 V (t) M(τ)dτ = λv (t) (5.35) 0 λ = 0 M(τ)dτ (5.36)
118 118 5 ( ) (1 20 ) sgn(1-x)*exp((x-1)*10) sin(x)* V (t ) M(t t ) t = t t /t 1 : M(t) V (t) (5.4) λ 2 (2ndFDT) V 2 = k B T/m R(t)R = Dδ(t) (5.36) (5.32) λ = 0 R(t)R m V 2 dt = D 2k B T (5.37) (6) R(t) 2 R(t) (4) R(t) (5.24) F
119 119 V X({q l, p l }) ( 1973) X(t) ( Gunton1973) r i i X({q l, p l }) = i δ(r r i)( ) : M(t) M(t) Gunton : 78 (15 ) 4.4 X µ (t) 5 79 (30 ) (10 ) 81 (20 ) (5.16) X(t) = e ilt X(0) 5.16 il({q l (t), p l (t)}) {q l (t), p l (t)} X(t) = e ilt X(0) {q l (0), p l (0)} X(t) = e ilt X(0) 82 (30 ) 81
120 120 5 ( ) (1 20 ) til 83 (25 ) (5.22) X(t) = e Ẋ(t) = ilx(t) = P ilx(t) + QiLX(t) Ẋ(t) = QiLX(t) + f(t) 84 (20 ) il (5.17) (5.18) (il) = il (5.12) q l ± p l ± ρ eq ({q l, p l }) 0 (5.29) 85 (50 ) 1
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1 1. (1) a + b = i +3i + k () a b =5i 5j +3k (3) a b =1 (4) a b = 7i j +1k. a = 14 l =/ 14, m=1/ 14, n=3/ 14 3. 4. 5. df (t) d [a(t)e(t)] =ti +9t j +4k, = d a(t) d[a(t)e(t)] e(t)+ da(t) d f (t) =i +18tj
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1 variation 1.1 imension unit L m M kg T s Q C QT 1 A = C s 1 MKSA F = ma N N = kg m s 1.1 J E = 1 mv W = F x J = kg m s 1 = N m 1.
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201711grade1ouyou.pdf
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d (K + U) = v [ma F(r)] = (2.4.4) t = t r(t ) = r t 1 r(t 1 ) = r 1 U(r 1 ) U(r ) = t1 t du t1 = t F(r(t)) dr(t) r1 = F dr (2.4.5) r F 2 F ( F) r A r
![d (K + U) = v [ma F(r)] = (2.4.4) t = t r(t ) = r t 1 r(t 1 ) = r 1 U(r 1 ) U(r ) = t1 t du t1 = t F(r(t)) dr(t) r1 = F dr (2.4.5) r F 2 F ( F) r A r](/thumbs/85/92098187.jpg "d (K + U) = v [ma F(r)] = (2.4.4) t = t r(t ) = r t 1 r(t 1 ) = r 1 U(r 1 ) U(r ) = t1 t du t1 = t F(r(t)) dr(t) r1 = F dr (2.4.5) r F 2 F ( F) r A r")
2.4 ( ) U(r) ( ) ( ) U F(r) = x, U y, U = U(r) (2.4.1) z 2 1 K = mv 2 /2 dk = d ( ) 1 2 mv2 = mv dv = v (ma) (2.4.2) ( ) U(r(t)) r(t) r(t) + dr(t) du du = U(r(t) + dr(t)) U(r(t)) = U x = U(r(t)) dr(t)
meiji_resume_1.PDF
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
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
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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
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5 5.1 [ ] ) d f(t) + a d f(t) + bf(t) : f(t) 1 dt dt ) u(x, t) c u(x, t) : u(x, t) t x : ( ) ) 1 : y + ay, : y + ay + by : ( ) 1 ) : y + ay, : yy + ay 3 ( ): ( ) ) : y + ay, : y + ay b [],,, [ ] au xx
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I 1 1 1.1 1. 3 m = 3 1 7 µm. cm = 1 4 km 3. 1 m = 1 1 5 cm 4. 5 cm 3 = 5 1 15 km 3 5. 1 = 36 6. 1 = 8.64 1 4 7. 1 = 3.15 1 7 1 =3 1 7 1 3 π 1. 1. 1 m + 1 cm = 1.1 m. 1 hr + 64 sec = 1 4 sec 3. 3. 1 5 kg
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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
( ) ( 40 )+( 60 ) Schrödinger 3. (a) (b) (c) yoshioka/education-09.html pdf 1
2009 1 ( ) ( 40 )+( 60 ) 1 1. 2. Schrödinger 3. (a) (b) (c) http://goofy.phys.nara-wu.ac.jp/ yoshioka/education-09.html pdf 1 1. ( photon) ν λ = c ν (c = 3.0 108 /m : ) ɛ = hν (1) p = hν/c = h/λ (2) h
5 H Boltzmann Einstein Brown 5.1 Onsager [ ] Tr Tr Tr = dγ (5.1) A(p, q) Â 0 = Tr Âe βĥ0 Tr e βĥ0 = dγ e βh 0(p,q) A(p, q) dγ e βh 0(p,q) (5.2) e βĥ0
![5 H Boltzmann Einstein Brown 5.1 Onsager [ ] Tr Tr Tr = dγ (5.1) A(p, q) Â 0 = Tr Âe βĥ0 Tr e βĥ0 = dγ e βh 0(p,q) A(p, q) dγ e βh 0(p,q) (5.2) e βĥ0](/thumbs/99/141438380.jpg "5 H Boltzmann Einstein Brown 5.1 Onsager [ ] Tr Tr Tr = dγ (5.1) A(p, q) Â 0 = Tr Âe βĥ0 Tr e βĥ0 = dγ e βh 0(p,q) A(p, q) dγ e βh 0(p,q) (5.2) e βĥ0")
5 H Boltzmann Einstein Brown 5.1 Onsager [ ] Tr Tr Tr = dγ (5.1) A(p, q) Â = Tr Âe βĥ Tr e βĥ = dγ e βh (p,q) A(p, q) dγ e βh (p,q) (5.2) e βĥ A(p, q) p q Â(t) = Tr Â(t)e βĥ Tr e βĥ = dγ() e βĥ(p(),q())
Note.tex 2008/09/19( )
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1. (8) (1) (x + y) + (x + y) = 0 () (x + y ) 5xy = 0 (3) (x y + 3y 3 ) (x 3 + xy ) = 0 (4) x tan y x y + x = 0 (5) x = y + x + y (6) = x + y 1 x y 3 (
1 1.1 (1) (1 + x) + (1 + y) = 0 () x + y = 0 (3) xy = x (4) x(y + 3) + y(y + 3) = 0 (5) (a + y ) = x ax a (6) x y 1 + y x 1 = 0 (7) cos x + sin x cos y = 0 (8) = tan y tan x (9) = (y 1) tan x (10) (1 +
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ii p ϕ x, t = C ϕ xe i ħ E t +C ϕ xe i ħ E t ψ x,t ψ x,t p79 やは時間変化しないことに注意 振動 粒子はだいたい このあたりにいる 粒子はだいたい このあたりにいる p35 D.3 Aψ Cϕdx = aψ ψ C Aϕ dx
i B5 7.8. p89 4. ψ x, tψx, t = ψ R x, t iψ I x, t ψ R x, t + iψ I x, t = ψ R x, t + ψ I x, t p 5.8 π π π F e ix + F e ix + F 3 e 3ix F e ix + F e ix + F 3 e 3ix dx πψ x πψx p39 7. AX = X A [ a b c d x
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128 3 II S 1, S 2 Φ 1, Φ 2 Φ 1 = { B( r) n( r)}ds S 1 Φ 2 = { B( r) n( r)}ds (3.3) S 2 S S 1 +S 2 { B( r) n( r)}ds = 0 (3.4) S 1, S 2 { B( r) n( r)}ds
127 3 II 3.1 3.1.1 Φ(t) ϕ em = dφ dt (3.1) B( r) Φ = { B( r) n( r)}ds (3.2) S S n( r) Φ 128 3 II S 1, S 2 Φ 1, Φ 2 Φ 1 = { B( r) n( r)}ds S 1 Φ 2 = { B( r) n( r)}ds (3.3) S 2 S S 1 +S 2 { B( r) n( r)}ds
4. ϵ(ν, T ) = c 4 u(ν, T ) ϵ(ν, T ) T ν π4 Planck dx = 0 e x 1 15 U(T ) x 3 U(T ) = σt 4 Stefan-Boltzmann σ 2π5 k 4 15c 2 h 3 = W m 2 K 4 5.
A 1. Boltzmann Planck u(ν, T )dν = 8πh ν 3 c 3 kt 1 dν h 6.63 10 34 J s Planck k 1.38 10 23 J K 1 Boltzmann u(ν, T ) T ν e hν c = 3 10 8 m s 1 2. Planck λ = c/ν Rayleigh-Jeans u(ν, T )dν = 8πν2 kt dν c
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
No.2 1 2 2 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 δx j (5) δs 2 = δx i δx i + 2 u i δx i δx j = δs 2 + 2s ij δx i δx j
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(1.2) T D = 0 T = D = 30 kn 1.2 (1.4) 2F W = 0 F = W/2 = 300 kn/2 = 150 kn 1.3 (1.9) R = W 1 + W 2 = = 1100 N. (1.9) W 2 b W 1 a = 0
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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Gauss Gauss ɛ 0 E ds = Q (1) xy σ (x, y, z) (2) a ρ(x, y, z) = x 2 + y 2 (r, θ, φ) (1) xy A Gauss ɛ 0 E ds = ɛ 0 EA Q = ρa ɛ 0 EA = ρea E = (ρ/ɛ 0 )e
7 -a 7 -a February 4, 2007 1. 2. 3. 4. 1. 2. 3. 1 Gauss Gauss ɛ 0 E ds = Q (1) xy σ (x, y, z) (2) a ρ(x, y, z) = x 2 + y 2 (r, θ, φ) (1) xy A Gauss ɛ 0 E ds = ɛ 0 EA Q = ρa ɛ 0 EA = ρea E = (ρ/ɛ 0 )e z
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IA 31 4 11 1 1 4 1.1 Planck.............................. 4 1. Bohr.................................... 5 1.3..................................... 6 8.1................................... 8....................................
IA hara@math.kyushu-u.ac.jp Last updated: January,......................................................................................................................................................................................
18 I ( ) (1) I-1,I-2,I-3 (2) (3) I-1 ( ) (100 ) θ ϕ θ ϕ m m l l θ ϕ θ ϕ 2 g (1) (2) 0 (3) θ ϕ (4) (3) θ(t) = A 1 cos(ω 1 t + α 1 ) + A 2 cos(ω 2 t + α
18 I ( ) (1) I-1,I-2,I-3 (2) (3) I-1 ( ) (100 ) θ ϕ θ ϕ m m l l θ ϕ θ ϕ 2 g (1) (2) 0 (3) θ ϕ (4) (3) θ(t) = A 1 cos(ω 1 t + α 1 ) + A 2 cos(ω 2 t + α 2 ), ϕ(t) = B 1 cos(ω 1 t + α 1 ) + B 2 cos(ω 2 t
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V(x) m e V 0 cos x π x π V(x) = x < π, x > π V 0 (i) x = 0 (V(x) V 0 (1 x 2 /2)) n n d 2 f dξ 2ξ d f 2 dξ + 2n f = 0 H n (ξ) (ii) H
199 1 1 199 1 1. Vx) m e V cos x π x π Vx) = x < π, x > π V i) x = Vx) V 1 x /)) n n d f dξ ξ d f dξ + n f = H n ξ) ii) H n ξ) = 1) n expξ ) dn dξ n exp ξ )) H n ξ)h m ξ) exp ξ )dξ = π n n!δ n,m x = Vx)
II ( ) (7/31) II ( [ (3.4)] Navier Stokes [ (6/29)] Navier Stokes 3 [ (6/19)] Re
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25 7 18 1 1 1.1 v.s............................. 1 1.1.1.................................. 1 1.1.2................................. 1 1.1.3.................................. 3 1.2................... 3
m dv = mg + kv2 dt m dv dt = mg k v v m dv dt = mg + kv2 α = mg k v = α 1 e rt 1 + e rt m dv dt = mg + kv2 dv mg + kv 2 = dt m dv α 2 + v 2 = k m dt d
m v = mg + kv m v = mg k v v m v = mg + kv α = mg k v = α e rt + e rt m v = mg + kv v mg + kv = m v α + v = k m v (v α (v + α = k m ˆ ( v α ˆ αk v = m v + α ln v α v + α = αk m t + C v α v + α = e αk m
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[ ] IC. f(x) = e x () f(x) f (x) () lim f(x) lim f(x) x + x (3) lim f(x) lim f(x) x + x (4) y = f(x) ( ) ( s46). < a < () a () lim a log xdx a log xdx ( ) n (3) lim log k log n n n k=.3 z = log(x + y ),
chap1.dvi
1 1 007 1 e iθ = cos θ + isin θ 1) θ = π e iπ + 1 = 0 1 ) 3 11 f 0 r 1 1 ) k f k = 1 + r) k f 0 f k k = 01) f k+1 = 1 + r)f k ) f k+1 f k = rf k 3) 1 ) ) ) 1+r/)f 0 1 1 + r/) f 0 = 1 + r + r /4)f 0 1 f
2 1 x 2 x 2 = RT 3πηaN A t (1.2) R/N A N A N A = N A m n(z) = n exp ( ) m gz k B T (1.3) z n z = m = m ρgv k B = erg K 1 R =
1 1 1.1 1827 *1 195 *2 x 2 t x 2 = 2Dt D RT D = RT N A 1 6πaη (1.1) D N A a η 198 *3 ( a =.212µ) *1 Robert Brown (1773-1858. *2 Albert Einstein (1879-1955 *3 Jean Baptiste Perrin (187-1942 2 1 x 2 x 2
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微分積分 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. このサンプルページの内容は, 初版 1 刷発行時のものです.
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