2 1 (10 5 ) 1 (10 5 ) () (1) (2) (3) (4) (1) 2 T T T T T T T T? *

1 2011 2012 1 30 1 (10 5 ) 2 2 6 2.1 (10 12 )..................... 6 2.2 (FP) (10 19 ).............. 14 2.3 2 (10 26 )...................... 26 2.4 (2. )(11 2 )..... 35 3 40 3.1 (11 9 ).......................... 40 3.2 Wiener-Khinchin (11 16 )................... 50 3.3 (11 30 )........................ 57 3.4 (12 7 ).............. 64 3.5 (12 14 )............................ 69 4 81 4.1 (12 21 )....................... 81 4.2 (1 11 )......................... 90 4.3 Thomson (1 18 )........... 100 4.4 (1 25 ).................... 105 5 ( ) (2 1 ) 112

2 1 (10 5 ) 1 (10 5 ) () 2 1900 1970 (1) (2) (3) (4) (1) 2 T T T T T T T T? *

1 () 2 () 2 1 () + ( ) (1.1) 2 3 2 1? : 1 100µm (2) 1960 : 2

4 1 (10 5 ) 1905 1908 1931 1940 1951 Calle-Walton () 1951 1955 1955 Lax 1957 1961 Zwanzig 1965 1905 1965 ( : ) (3) ( ) () ()? ( :) ( )

5 * () (4)? () : 1 (6 ) () 1 2 () 3 1 3 ()

6 2 : http://www.cmt.phys.kyushu-u.ac.jp/~a.yoshimori/hiheik11.html pdf 2 2.1 (10 12 ) () (1) (2) (3) (4) (5) X(t) : Ẋ(t) = γx(t) + R(t) (2.1.1) : Ẋ(t) = F (X(t)) + R(t) (2.1.2) γ F (X(t)) X(t) 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

2.1 (10 12 ) 7 (2-1 ) (1) www www http://k-5050-web.hp.infoseek.co.jp/1.html http://www.phys.u-ryukyu.ac.jp/wyp2005/brown.html http://www.geocities.co.jp/hollywood/5174/indexb.html 4. *: 2.1.1 (2) 1908

8 2 λv (t) V (t) R(t) 2.1.2 2 t 1 ( ): R(t) 2 ( ): λv (t) m m V (t) = λv (t) + R(t) (2.1.7) R(t) 1 R(t) δ(t t 0 ): (t 0 ) R(t) 2.1.3 ( ) ( ) p p = R t (2.1.8) R t t 0 p 0 (2.1.8) R R(t) δ(t t 0 )

2.1 (10 12 ) 9 R(t) t 0 t 0 2.1.4 t 2 R(t) 100 100 R(t) 1 R(t ) 2.1.5 R(t) 2 R(t ) R(t) R(t) 100 100 R(t) 1 R(t ) 2.1.6 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 ),... )

10 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 2 ( ) (2.1.3) (2.1.4) (2.1.3) 0 (2.1.4) (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

2.1 (10 12 ) 11 V 0 (t) Q(t) Q(t) C R 0 C I(t) V (t) = 0 V 0 (t) > 0 I(t) V (t) I(t) < 0 V 0 (t) = RI(t) (2.1.14) R 2.1.7 V (t) = 0 V 0 (t) + V (t) = RI(t) (2.1.15) I(t) = Q(t) Q V 0 (t) = Q(t) C (2.1.16) (2.1.15) R Q(t) = Q(t) C + V (t) (2.1.17) γ = 1/(RC) R(t) = V (t)/r 3 () 2.1.8 3 X(t) u(x) m mẍ(t) = λẋ(t) u(x) + R (t) (2.1.18) λẋ(t) R (t) m Ẋ(t) = 1 λ u(x) + R (t) λ (2.1.19) 4

12 2 1940 C6H5CH=CHC6H5 1 2 1 2 2 t Θ = Θ(t) Θ(t) = γ du(θ(t)) dθ(t) + R(t) (2.1.20) u(θ) Θ Θ =0 180 R(t) (5) X(t) Ẋ(t) = γx(t) + R(t) : (2.1.21) Ẋ(t) = F (X(t)) + R(t) : (2.1.22) X(t) (2.1.4) (2.1.4) : 2 (10 ) 2 a () b

2.1 (10 12 ) 13 2 2 *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 ) (2.1.23) W (t i ) ( 0 D t) X(t 1 ) t X(t) γ D γ 10 4 (20 ) 1 mẍ(t) = λẋ(t) kx(t) + R(t) (2.1.24) 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) ρ(d 1, d 2,..., t 1, t 2,... ) 1 *1 2003

14 2 2.2 (FP) (10 19 ) 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) 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) FP P (x, t) t Ẋ(t) = F (X(t)) + R(t) (2.2.2) = { 2 D F (x) + x x 2 }P (x, t) (2.2.3) 2

2.2 (FP) (10 19 ) 15 ( 12 ) : P89-98 (1) FP X = X(t) X(0) X(t) 1 2 3 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 F (X(t ))dt + t+ t t R(t )dt (2.2.5) t+ t t Ẋ(t )dt = X(t + t) X(t) (2.2.6)

16 2 1 t t+ t t F (X(t ))dt F (X(t)) t (2.2.7) 2 R(t) (2.2.7) X(t) X(t + t) X(t) W (2.2.5) t+ t t R(t 1 )dt 1 (2.2.8) X(t) = F (X(t)) t + W (2.2.9) W 1 W = 0 (2.2.10) W 2 = D t (2.2.11) (2.2.11) (2.2.8) W 2 t+ t t+ t = R(t 1 )dt 1 R(t 2 )dt 2 t t (2.2.12) = t+ t t+ t t t 1 R(t)R(t ) = Dδ(t t ) = t 1 = t+ t t+ t t t t+ t t 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)

2.2 (FP) (10 19 ) 17 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 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 +... (2.2.18) 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)

18 2 W t+ t R(t t 1 )dt 1 W R(t 1 ) t 1 t 2 *2 1 2 d 2 f dx 2 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) t 1 2 1 (3) FP 1 f(x) t f(x) 2 FP 1 (2.2.20) (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 t 1 = t 1 (2.2.22)

2.2 (FP) (10 19 ) 19 (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) 2 2 (2.2.25) 0 3 (2.2.23) (2.2.27) 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) (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 X t 2 (2.2.30) ( 9 ) 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)

20 2 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 (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

2.2 (FP) (10 19 ) 21 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) f(x) 2 P (x, t) x 2 dx (2.2.43) P (x, t) t = x {F (x)p (x, t)} + D 2 2 P (x, t) x 2 (2.2.44) FP

22 2 (4) 1 (2.1.7) 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 (2.1.17) 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) 4 1 X i i 1 W X i+1 X i = W (2.2.49)

2.2 (FP) (10 19 ) 23 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) ( 12 ) 5 (2.1.20) R(t) 1 2 3 P (θ, t) P (θ + 2π, t) = P (θ, t) FP P (θ, t) t = { γ u(θ) } P (θ, t) + D θ dθ 2 2 P (x, t) θ 2 (2.2.51) (5) FP ( (P14) ) (2.2.2) X(t) = F (X(t)) t + W f = f(x(t + t)) t + W X(t) 2 ( W ) 2 = D t 1 d 2 f dx 2 t t t 0 f 3 3 ( 1)

24 2 4 FP FP Ẋ(t) = F(X(t)) + R(t), R(t)R(t ) = D δ(t t ) (2.2.52) P (x, t) t = { x F(x) + 2 x 2 D 2 }P (x, t) (2.2.53) : 6 (15 ) F (x) P6 (2.1.3) (2.1.4) P6 1 15 15 7 (5 ) FP 6 FP 1 5 n 5n 8 (15 ) 9 (10 ) (2.2.30) W P ( W ) P ( W ) exp[ W 2 2D t ] (2.2.54) (2.2.54) (2.2.11) P14 1

2.2 (FP) (10 19 ) 25 10 (10 ) FP 4(P14) 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 (2.2.55) 12 (15 ) γ = λ/m 3 Ẋ(t) = R(t) (2.2.56) X(t) FP P (X, t) t = D 2 2 P (X, t) (2.2.57) (2.2.57) t = 0 P (X, 0) exp[ α X 2 ] t r r + r r = X r

26 2 2.3 2 (10 26 ) 2 (2nd FDT) () 2 (2nd FDT) F (x) D 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)

2.3 2 (10 26 ) 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 ) (2.1.7) λ D N A R T (1) t (2.3.10) FP F (x) D () : : ()? : m T k B v t 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) 2nd FDT

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)

2.3 2 (10 26 ) 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 (x) > 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) (2.3.11) 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) (2.1.13) 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 2 (2.3.8) (2.3.21) ( γv = D m ) 2 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 C E(q) = q2 2C S(q) = βq2 2C +, ds(q) dq = β C q (2.3.32) (2.1.17) R Q(t) = Q(t) CR + R(t), V (t) R(t) = R (2.3.33)

2.3 2 (10 26 ) 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 ) q 1 CR = Dβ 2C k BT R = D 2 V (t)v (t ) = D V δ(t t ) R(t) = V (t)/r D = D V R 2 k BT R (2.3.34) (2.3.35) = D V 2R 2 2Rk B T = D V (2.3.36) R k B T D V (2.3.36) 3 (1 ) (2.1.19) 1 (2.2.47) P eq (x) e βu(x) (2.3.37) 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 (2.3.38) (2.3.39) 1 λ = Dβ 2 (2.3.40) (4) 1 X 2 ( 12 14 15 ) {X 1, X 2,..., X n } = {X α } Ẋ α (t) = F ({X α }) + R α (t) (2.3.41) R α (t) = 0 (2.3.42) Rα (t)r β(t ) = Dαβ δ(t t ) (2.3.43)

32 2 P eq ({x α }) = e S({x α}) Ẋ α (t) = n β=1 L αβ S({X α }) X β + R α (t) (2.3.44) L αβ = D αβ 2 ( 16 ) (2.3.45) P eq (x) P eq (x) ( 13 ) S(x) 2 (FDT) 3 L, F (X)() ()D 3 2 : 13 (15 ) FP P (x, t) t = { L du(x) x dx f + D 2 } P (x, t) (2.3.46) 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 P st (x)dx = 1 (2.3.47)

2.3 2 (10 26 ) 33 14 (30 ) 2 Ẋ α = n γ αβ X β + R α (t) (2.3.48) 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) (2.3.48) Ẋ µ = λ µ X µ + R µ(t) (2.3.49) 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 (2.3.50) 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 (2.3.51) γ 15 (20 ) 1 V (t) X(t) Ẋ(t) = V (t) (2.3.52) m V (t) = λv + R(t) (2.3.53) m λv R(t) 2.2 P14 1 4 P (x, v, t) FP (2.3.52) (2.3.53) 2.2 FP D m λ T 2.3 T 16 (45 )(2.3.41) (2.3.43) 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) (2.3.54)

34 2 L µν = D µν 2 (2.3.55) T ({x µ }, {x µ}; t) T ({x µ }, {x µ}; 0) = n δ(x µ x µ) (2.3.56) µ FP S({x µ }) = n k µ 2 x2 µ (2.3.57) µ n µ L µµ k µ (2.3.55) 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))] (2.3.58) C(t) n dx µ T ({x µ }, {x µ}; t) = 1 (2.3.59) µ x µ (t) x µ (0) = x µ σ µν(t) 14 t = 0 0 n µ X µ (t)x µ (t) σ µ ν(t) = δ µν

2.4 (2. )(11 2 ) 35 2.4 (2. )(11 2 ) 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) 2.2 1 T (x, x, t, t ) FP T (x, x, t, t ) t = x {F (x)t (x, x, t, t )} + D 2 2 T (x, x, t, t ) x 2 (2.4.1) t = t T (x, x, t, t) = δ(x x ) 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)

36 2 t = 0 0 t > 0 (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) 2.4.1 t 1 t P 0 (x) 2.4.2 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 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)

2.4 (2. )(11 2 ) 37 3 ( 17 ) 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) γ = λ/m ( 19 ) v 0 (t) = v e γt (2.4.6) σ(t) = k BT m (1 e 2γt ) (2.4.7) (2.4.5) (2.4.2) t = 0 P 0 (v) t > 0 P (v) 19 2 (2.1.17) FP (2.2.46) 2 (2.3.35) 1/R = (D/2)β

38 2 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) 3 (1 ) q 0 (t) = q exp[ t CR ] (2.4.10) σ(t) = k B T C(1 exp[ 2t ]) (2.4.11) CR 2 (2.3.40) FP (2.2.48) P (x, t) t = D 2 x { βu (x) + x } P (x, t) (2.4.12) λ u(x) = 0 D T (2.3.40) λ (4) T (x, x, t) T (x, x, t) FP t = 0 T (x, x, 0) = δ(x x ) 2 1 2 2 FP D Ẋ(t) = γx(t) + R(t) T (x, x, t) = 1 exp[ (x x 0(t)) 2 ] (2.4.13) 2πσ(t) 2σ(t) x 0 (t) = x e γt (2.4.14) σ(t) = D 2γ (1 e 2γt ) (2.4.15)

2.4 (2. )(11 2 ) 39 R(t)R(t ) = Dδ(t t ) : 17 (10 ) 3 18 (30 ) 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) (2.4.16) 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 ) 19 (20 ) (2.4.5) FP 2 (2.4.5) P 0 (v) = C exp[ (v v 0) 2 ] (2.4.17) 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)

40 3 3 3.1 (11 9 ) (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)

3.1 (11 9 ) 41 (1) 3 3-5 3-2 - 3.1 3-3 2 3-4 (2) A V (t) = (1 ) t: V (t) t V (t) t 3.1.1a: 1 3.1.1b: 2 (1 ) B V (t) t V (t) t 3.1.2a: 1 3.1.2b: 2 (1 )

42 3 A B 2 A B? : 2 1 X(t) 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) (3.1.3) X(t) t + τ t + τ t t 3.1.3:

3.1 (11 9 ) 43 1 2 X(t) = 0 X(t) X(t ) X(t)X(t ) = X(t) X(t ) = 0 (3.1.4) 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) (3.1.5) X(t) = t 2 X(t)X(t ) : t t t t t + a t + a X(t)X(t ) = X(t + a)x(t + a) (3.1.6) a = t X(t t )X(0) = X(t)X(t ) (3.1.7) t t ϕ(t t ) X(t t )X(0) = X(t)X(t ) (3.1.8)

44 3 X(t) {X 1 (t), X 2 (t), } = {X µ (t)} ϕ µν (t) X µ (t)x ν (0) (3.1.9) 3 V (t) = (V x (t), V y (t), V z (t)) ϕ 11 (t) = V x (t)v x (0) (3.1.10) ϕ 12 (t) = V x (t)v y (0) (3.1.11) ϕ 31 (t) = V z (t)v x (0) (3.1.12) ϕ µν (t) () X µ (t)x ν (t ) = X µ (t t )X ν (0) = X µ (0)X ν (t t) (3.1.13) t = 0 X µ (t)x ν (0) = X µ (0)X ν ( t) (3.1.14) = X ν ( t)x µ (0) (3.1.15) µ = ν ϕ µν (t) = ϕ νµ ( t) (3.1.16) ϕ µµ (t) = ϕ µµ ( t) : ϕ µµ (t) (3.1.17) (3.1.14) t Ẋµ (t)x ν (0) t Ẋµ (t)x ν (0) = X µ (0)Ẋν( t) = X µ (t)ẋν(0) (3.1.18) (3.1.19)

3.1 (11 9 ) 45 µ = ν ϕ µµ (0) = Ẋµ (0)X µ (0) = 0 (3.1.20) 1 (3 ) x F x x X X 1 (t) X 2 (t) ϕ 11 (t) = F x (t)f x (0) (3.1.21) ϕ 12 (t) = F x (t)x(0) (3.1.22) (3.1.15) F x (t)x(0) = X( t)f x (0) (3.1.23) (3.1.17) F x (t)f x (0) = F x ( t)f x (0) (3.1.24) Ẍ = F x (3.1.19) F x (t)x(0) = Ẍ(t)X(0) = Ẋ(t) Ẋ(0) (3.1.25) Ẋ T F x (0)X(0) = Ẋ(0) Ẋ(0) = k B T/m m k B (3.1.20) F x (0)Ẋ(0) = Ẍ(0) Ẋ(0) = 0 (3.1.26) (4) 1 X(t) Ẋ(t) = γx(t) + R(t) (3.1.27)

46 3 X(0)R(t) = 0, t 0 X(0) t 0 Ẋ(t)X(0) = γ X(t)X(0) (3.1.28) ϕ(t) X(t)X(0) ϕ(t) = γϕ(t) (3.1.29) ϕ(t) = ϕ(0)e γt (3.1.30) ϕ(0) = X 2 ϕ(t) = X 2 e γt t 0 (3.1.31) X(t) 1. V (t) V (t) = γv (t) + R(t) (3.1.32) γ = λ/m((2.1.13) ) (3.1.31) V (t)v (0) = V 2 e γt (3.1.33) V 2 = k B T/m(k B : T : ) V (t)v (0) = k BT m e γt (3.1.34) A(t) = V (t) A(t)A(0) = V (t) V (0) (3.1.35) (3.1.19) = V (t)v (0) (3.1.36)

3.1 (11 9 ) 47 ϕ(t) V (t)v (0) (3.1.34) = ϕ(t) (3.1.37) = k BT m γ2 e γt (3.1.38) 2. Q(t) Q(t) = Q(t) CR + R(t) (3.1.39) C R R(t) V (t) R(t) = V (t)/r (3.1.31) Q 2 Q(t)Q(0) = Q 2 exp[ t CR ] (3.1.40) P eq (q) = A exp[ β q2 2C ] (3.1.41) A β = 1/k B T Q 2 = q 2 A exp[ β q2 2C ]dq = Ck BT (3.1.42) Q(t)Q(0) = Ck B T exp[ t CR ] (3.1.43) 2 ϕ(t) X(t)X(0) 2.4 2 1 t = 0 X(0) = x () t : X(t) x ( ) 2 x : 0 ()

= γ X(t) X(t) = X(0) exp[ γt] 48 3 Ẋ(t) = γx(t) + R(t) 1. Ẋ(t) x + R(t) x (3.1.44) x X(0) X(0) = x R(t) x = R(t) = 0 (3.1.45) Ẋ(t) x x x x x X(0) = x X(t) x = x exp[ γt] (3.1.46) 2 ϕ(t) x x X(t)X(0) = X(t) x x 0 = x 2 e γt (3.1.47) 2 2 1. t = 0 x T (x, x, t) X(t) x = xt (x, x, t)dx (3.1.48) 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 (3.1.49) (5) 1 () 2 4

3.1 (11 9 ) 49 1 ϕ µν (t) = ϕ νµ ( t): (3.1.16) 2 ϕ µµ (t) = ϕ µµ ( t): (3.1.17) 3 3.1.4 Ẋµ (t)x ν (0) 4 ϕ µµ (0) = 0 = X µ (t)ẋν(0) : (3.1.19) X(t) : (3.1.31) : (3.1.49) : 20 (15 ) 105 km 105 km 1000 km N 2 m z P eq (z) exp[ βmgz] (3.1.50) β = 1/(k B T ) g 2 21 (20 ) 2 2 22 (25 ) 23 (15 ) 2.4 2(P35) T (x, x, t, t ) = T (x, x, t t ) 24 (30 ) 3 X(t 1 )X(t 2 )X(t 3 )

50 3 3.2 Wiener-Khinchin (11 16 ) Wiener-Khinchin (WK ) (2 ) 2 2 I ω (1) (2) (3) (4) (5) 1. X(t) 2. 2 1 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)

3.2 Wiener-Khinchin (11 16 ) 51 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)

52 3 (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 0 1 ( 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

3.2 Wiener-Khinchin (11 16 ) 53 s t s t T < s < 0 T -s 0 < s < T t T + s s T 3.2.1 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 t s T τ 0 T s 1 I ω = lim T T { T T s ds e iωs X(τ + s)x(τ)dτ 0 0 0 } T +s + ds e iωs X(t)X(t s)dt T 0 (3.2.22)

54 3 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) 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) ((3.1.31) ) 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) (3.1.34) 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

3.2 Wiener-Khinchin (11 16 ) 55 1. ϕ(t) = X(t)X(0) 2. X ω 2 Wiener-Khinchin 2 X ω 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 3.2.2 1/τ ω : 25 (30 ) (3.1.31) (3.1.20) (3.1.27) R(t)R(t ) = Dδ(t t ) D = 2γ X 2 X(t)

56 3 26 (10 ) (3.1.27) X(t)X(t ) D = 2γ X 2 27 (30 ) I ω 28 (15 ) X(t) (3.2.14) I ω I ω I ω = ϕ(ω) (3.2.7) 29 (15 ) I ω I ω

3.3 (11 30 ) 57 3.3 (11 30 ) 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

58 3 (1) 3-5 3-2 - 3.1 3-3 2 3-4 : I V R I = V R : V I() (3.3.4)? x() f : I = V R + α 2V 2 + α 3 V 3 + : (3.3.5) α 2 α 3 V V/R V (2) f (f = f(t)) x ( f(t)) ( x)

3.3 (11 30 ) 59 t t 3.3.1 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) (3.3.11) () t = t 0 x = x(t 0 ) x(t) = e γ(t t 0) x(t 0 ) + t t 0 e γ(t t ) f(t )dt (3.3.12)

60 3 t 0 t 0 e γ(t t 0) = e γt+γt 0 0 1 0 2 x(t) = t e γ(t t ) f(t )dt (3.3.13) α(t) = exp[ γt] 1. Q(t) C I(t) E(t) V 0 (t) Q(t) V (t) E(t) Q(t) C Q(t)/C I(t) R Q(t) C I(t) = Q(t) + RI(t) = V (t) + E(t) (3.3.14) R Q(t) = Q(t) RC + V (t) R + E(t) R (3.3.9) X(t) = Q(t) (3.3.15) 3.3.2 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)

3.3 (11 30 ) 61 2. (2.2.47) Ẋ(t) = γ du(x(t)) + R(t) dx (3.3.20) = γk(x x 0 (t)) + R(t) (3.3.21) γ γk γ = 1/λ (3.3.9) 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

62 3 (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)

3.3 (11 30 ) 63 (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) : 30 (10 ) (3.2.27) 31 (24 ) 1 2 3 4 5 6 32 (20 )(3.3.2) x(t) 2 P57 33 (10 ) α(t) 34 (10 ) (3.3.2) x(t) x ω f(t) f ω

64 3 3.4 (12 7 ) α ω ((3.4.7) ) α ω (1) (2) (3) (4) 1. ( ) 2. W ( ) H W = f X f(t)dt = x(t) f(t)dt (3.4.1) H f = f(t) H = H 0 (X) Xf(t) H 0 (X) X x(t) X α ω α ω (α(t) ) f ω W = 1 ωα 2π ω f ω 2 dω (3.4.2) α ω α ω α α ω 0 α ω 0 = α + 1 α x dx (3.4.3) π x ω 0 = 1 α x α dx (3.4.4) π x ω 0

3.4 (12 7 ) 65 (1) (3.3.2) x(t) = ( 34) x ω = x(t)e iωt dt, f ω = t α(t t )f(t )dt (3.4.5) x ω = α ω f ω (3.4.6) f(t)e iωt dt, α ω = 0 α(t)e iωt dt (3.4.7) α ω 0 ( 1) α ω α(t) αω = α ω ( αω α ω ) α ω = α ω + iα ω α ω = α ω iα ω α ω = α ω (3.4.8) α ω = α ω (3.4.9) (2) H = H 0 (X) Xf(t) t = ± f(t) = 0 t ( ) H W = f = = = i 2π X f(t)dt (3.4.10) X f(t)dt (3.4.11) ( 35)x ω = α ω f ω x(t) f(t)dt (3.4.12) ωx ω f ωdω (3.4.13) = i ωα ω f ω 2 dω (3.4.14) 2π

66 3 α ω = α ω + iα ω = 1 2π iωα ω f ω 2 dω + 1 ωα 2π ω f ω 2 dω (3.4.15) α ω f ω = f ω 1 0 W = 1 ωα 2π ω f ω 2 dω (3.4.16) (3) α ω ω ω = ω + iω α ω = 0 α(t)e iω t ω t dt (3.4.17) t > 0 ω > 0 α ω ω > 0 C ω C C 1 R ρ ω 0 ω (ω > 0) ω 0 α ω α dω = 0 (3.4.18) ω ω 0 C

3.4 (12 7 ) 67 C R R 0 α ω α dω R 0 (3.4.19) ω ω 0 ( 35) C 1 ω0 ρ lim { α ω α α ω α α ω α dω + dω + dω} = 0 (3.4.20) ρ 0 ω ω 0 ω ω 0 =ρ ω ω 0 ω 0 +ρ ω ω 0 2 ω = ρe iθ + ω 0 α ω α dω ρ 0 iπ(α ω0 α ) (3.4.21) ω ω 0 ω ω 0 =ρ ( 35) iπ(α ω0 α ) = α ω α ω ω 0 dω (3.4.22) ω0 ρ lim{ + } (3.4.23) ρ 0 ω 0 +ρ (principal value) α ω = α ω + iα ω iπ(α ω 0 + iα ω 0 α ) = α ω + iα ω α dω (3.4.24) ω ω 0 iπα ω 0 πα ω 0 iπα = α ω + iα ω α dω (3.4.25) ω ω 0 α (3.4.25) πα ω 0 πα = α ω ω ω 0 dω (3.4.26) (3.4.25) πα ω 0 = α ω α ω ω 0 dω (3.4.27) (3.4.3) (3.4.4) (4)

68 3 (3.4.16) α ω (3.4.3) (3.4.4) : 35 (10 ) (3.4.13) (3.4.19) (3.4.21) 36 (15 ) x(t) = t α(t t )f(t )dt (3.4.28) (3.4.3) (3.4.4) 37 (10 ) (W = 0)

3.5 (12 14 ) 69 3.5 (12 14 ) 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 (3.5.1) α(t) α(t) = β Ẋ(t)X(0) (3.5.2)

70 3 f(t) = 0 X = X(t) u(x) = k(x x 0 ) 2 /2 x 0 (1) : 3-5 3-2 - 3.1 3-3 2 3-4 3.1 : ϕ(t) = X(t)X(0) (3.5.3) (X(t) ) 3.3 : x(t) = t α(t t )f(t )dt (3.5.4) 2

3.5 (12 14 ) 71 (3.5.2) (3.1.43) ϕ(t) = Q(t)Q(0) = Ck B T exp[ t CR ] (3.5.5) (3.3.19) : α E (t) α(t) α E (t) = 1 R exp[ t CR ] (3.5.6) e t/(cr) Q(t)Q(0) α E (t) ϕ(t) = Q(t)Q(0) = Ck BT CR exp[ t CR ] = k BT α E (t) (3.5.7) β = 1/(k B T ) α E (t) = β ϕ(t) (3.5.8) (2) 1 OK(62 ) 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 (3.5.9) x(t) = X(t) t f 0 f 0 x(t) = Ψ(t)f 0 + Ψ 2 (t)f 2 0 + Ψ 3 (t)f 3 0 + (3.5.10) f = 0 x(t) = 0 ()Ψ(t) α(t) *3

72 3 Ψ(t) α(t) α(t) f 0 x(t) = t (3.5.9) = t 0 α(t t )f(t )dt + Ψ 2 (t)f 2 0 + Ψ 3 (t)f 3 0 + (3.5.11) α(t t )f 0 dt + Ψ 2 (t)f 2 0 + Ψ 3 (t)f 3 0 + (3.5.12) τ = t t dτ = dt t = t τ = 0 t = 0 τ = t = t (3.5.10) 0 α(τ)dτf 0 + Ψ 2 (t)f 2 0 + Ψ 3 (t)f 3 0 + (3.5.13) Ψ(t) = t 0 α(τ)dτ (3.5.14) Ψ(t) (3.5.14) α(t) = Ψ(t) 2 ( 1) t < 0 t 0 2 2 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 ] (3.5.15) f(t) = 0 P eq (x; 0) exp[ βe 0 (x)] (3.5.16) 3 4

3.5 (12 14 ) 73 2.4 3.1 (2.4.2) P (x, t) = (2.4.3) P eq (x; f 0 ) = (3.1.49) X(t)X(0) = T (x, x, t; f 0 )P eq (x ; 0)dx (3.5.17) T (x, x, t; f 0 )P eq (x ; f 0 )dx (3.5.18) (3.5.17) 2 P 0 (X) = P eq (x; 0) x(t) = X(t) neq = xt (x, x, t; 0)dxx P eq (x ; 0)dx (3.5.19) xp (x, t)dx (3.5.20) (3.5.17) Ψ(t) T (x, x, t; f 0 ) f 0 2 3 (3.5.17) f 0 f0 2 (3.5.17) f 0 T (x, x, t; f 0 ) T (x, x, t; f 0 ) f 0 (3.5.18) P eq (x; f 0 ) f 0 3 4 (3.5.18) T (x, x, t; f 0 ) f 0 P eq (x; f 0 ) f 0 (3.5.17) P eq (x ; f 0 ) (3.5.15) (3.5.16) P eq (x; 0) = exp[ βxf 0 ]P eq (x; f 0 )C(f 0 ) (3.5.21) C(f 0 ) C(f 0 ) = 1 ( 38 ) (3.5.17) P (x, t) = T (x, x, t; f 0 ) exp[ βx f 0 ]P eq (x ; f 0 )dx (3.5.22) exp[ βx f 0 ] (3.5.18) exp[ βx f 0 ] f 0 exp[ βx f 0 ] = 1 βx f 0 + (f 0 2 ) (3.5.23) (3.5.22) 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 ) (3.5.24)

74 3 (3.5.18) = P eq (x; f 0 ) + T (x, x, t; f 0 )( βx f 0 )P eq (x ; f 0 )dx + (f 0 2 ) (3.5.25) f 0 P eq (x; f 0 ) (3.5.15) (3.5.16) P eq (x; f 0 ) = exp[βxf 0 ]P eq (x; 0) (3.5.26) f 0 = P eq (x; 0)(1 + βxf 0 + f 0 2 ) (3.5.27) (3.5.25) 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 ) (3.5.27) (3.5.28) = T (x, x, t; 0)( βx f 0 )P eq (x ; 0) + (f 0 2 ) (3.5.28) P (x, t) = P eq (x; 0) + βxf 0 P eq (x; 0) P (x, t) f 0 (3.5.29) (3.5.20) 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 ) (3.5.29) x T (x, x, t; 0)βx f 0 P eq (x ; 0)dx dx + (f 0 2 ) (3.5.30) = X + β X 2 f 0 x T (x, x, t; 0)βx f 0 P eq (x ; 0)dx dx + (f 0 2 ) (3.5.31) 3 (3.5.19) = X + β X 2 f 0 β X(t)X(0) f 0 + (f 0 2 ) (3.5.32)

3.5 (12 14 ) 75 X = 0 x(t) = β X 2 f 0 β X(t)X(0) f 0 + (f 0 2 ) (3.5.33) (3.5.10) Ψ(t) = β X 2 β X(t)X(0) (3.5.34) α(t) = Ψ(t) = β Ẋ(t)X(0) (3.5.35) (3) ( ) 4 E(x) = u(x) = k 2 (x x 0(t)) 2 = k 2 x2 kxx 0 (t) + k 2 x 0(t) 2 (3.5.36) kx 0 (t) 2 /2 x x 0 (t) E(x) E(x) = k 2 x2 kxx 0 (t) (3.5.37) f(t) = kx 0 (t) E(x) = k 2 x2 xf(t) (3.5.38) 4 x(t) = t α(t t )kx 0 (t )dt, α(t) = β Ẋ(t)X(0) (3.5.39) (3.5.9) { 0 t < 0 x 0 (t) = x 0 t 0 (3.5.40) (3.5.34) x(t) = Ψ(t)kx 0 = β X 2 kx 0 β X(t)X(0) kx 0 (3.5.41)

76 3 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 (3.5.42) X(t)X(0) = x 0 x(t) = X 2 { 1 x(t) } βkx 0 x 0 (3.5.43) 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 3.5.1 (3.5.43) P.69 1 4 (4) (3.5.9) (α(t) ) 1 (3.5.17) f 0 ( P eq (x ; 0) 2 ) T (x, x, t; f 0 ) ((3.5.17) f 0 T (x, x, t; f 0 ) )

3.5 (12 14 ) 77 f 0 (3.5.17) (3.5.18) (3.5.21) 3 4 (3.5.17) exp[ βxf 0 ] f 0 (3.5.18) P eq (x ; f 0 ) (3.5.20) 1 2 3 4 1 1 (1956 ): () 2 : 3 : 2. 3. 2. 2 2 (3.5.34) β

78 3 2 β β β 3 β 3 1 1 2 0 0 4 4 4 : 38 (15 ) (3.5.21) C(f 0 ) 1 1 C(f 0 ) 1 39 (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)) P69 1 E({x µ }) = E 0 ({x µ }) N x µ f µ (t) (3.5.44) µ α µ,ν (t) = β Ẋµ (t)x ν (0) (3.5.45) α µ,ν (t) X µ (t) = N ν t α µ,ν (t t )f ν (t )dt (3.5.46)

3.5 (12 14 ) 79 40 (10 ) α ω = ωβ 2 ϕ(ω) (3.5.47) α ω (3.3.15) (3.3.18) α E (t) α ω = 0 α E (t)e iωt dt (3.5.48) β = 1/(k B T ) ϕ(ω) (3.1.39) ϕ(ω) e iωt Q(t)Q(0) dt (3.5.49) 41 (30 ) (3.5.47) 42 (30 ) f(t) = f 0 (t)δ(r r 0 ) ρ(r) i δ(r r i) r i i 43 (20 ) P. 11 44 (30 ) 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 45 (30 ) 41 1 1 P.69 4 (31 )

80 3 46 (30 ) (P.69) α(t) (4)

81 4 4.1 (12 21 ) 1 ((4.1.1) ) (1) 4. (2) (3) (4) (5) t 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 : t ) (critical slowing down) (4.1.1)

82 4 (1) 4 4.4 4.3 Thomson 4.2 4.1 3.1 2 (2) 1 2 2 3 ( Ψ(t)) 4 () : 1 1

4.1 (12 21 ) 83 物理量 起こらない 平衡の値 時間 4.1.1 (3) x = x(t) ẋ(t) = L ds (x) dx (4.1.2) L S (x) (4.3 ) S (x) P.81 L > 0 (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) 1. 2

84 4 1 2 Q T 1 T 2 L 4.1.2 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.81 ) 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(E 2 ) S (x) = S (E 1 ) = S 1 (E 1 ) + S 2 (E E 1 ) (4.1.8) x = E 1 ds (x) = ds (E 1 ) = ds 1(E 1 ) ds 2(E 2 ) dx 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 (12 21 ) 85 (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 > 0 2 1 T 1 > T 2 Ė 1 < 0 1 2 2. 2 φ 1 φ 2 1 2 φ 1 φ 2 4.1.3 2 1 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) x: 1 q 1 S (x): 2 1 E S q T φ de = T ds + φdq (4.1.15) φdq dq ( ) S = φ q E T (4.1.16) 2 q 1 + q 2 = q 2 S (E 1, q 1 ) = S 1 (E 1, q 1 ) + S 2 (E E 1, q q 1 ) (4.1.17)

86 4 ( ) S q 1 = φ 1 E 1 T + φ 2 T = φ 2 φ 1 T (4.1.18) T 2 (4.1.14) ẋ = q 1 = T ( ) S = T ds (x) R q 1 R dx E 1 (4.1.19) L = T/R (4.1.2) L = L 22 φ 2 φ 1 q 1 = L 22 T (4.1.20) (4.1.20) L 22 > 0 φ 2 > φ 1 q 1 > 0 2 1 φ 1 > φ 2 q 1 < 0 1 2 3. 2 {x 1, x 2 } = {E 1, q 1 } (4.1.3) ẋ µ = 2 ν=1 L µν S x ν (4.1.21) S T 1 T 2 T T 2 T 1 2 L µν L µν Ė 1 = L 11 T 2 (T 2 T 1 ) + L 12 T (φ 2 φ 1 ) (4.1.22) q 1 = L 21 T 2 (T 2 T 1 ) + L 22 T (φ 2 φ 1 ) (4.1.23) 1 T 1 = T 2 (4.1.22) Ė 1 = L 12 T (φ 2 φ 1 ) (4.1.24)

4.1 (12 21 ) 87 2 2 q 1 = 0 (4.1.23) 1 2 L 21 T 2 (T 2 T 1 ) + L 22 T (φ 2 φ 1 ) = 0 (4.1.25) φ 2 φ 1 = L 21 T L 22 (T 2 T 1 ) (4.1.26) 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 4.1.4 S (x) = F (x) S (x) (4.1.2) df (x) ẋ(t) = L dx (4.1.27) F (x) = F 0 + a(t T c )x 2 + bx 4 + (4.1.28) a b F 0 x F (x)

88 4 T > T c x x 4 F (x) = F 0 +a(t T c )x 2 (4.1.27) ẋ(t) = γx (4.1.29) 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 (4.1.30) x(t) = (8L bt + C) 1/2 (x = 0) (4) (3) 2 (4.1.2) x = x(t) Yes P.81 x(t) t : S S = S(x) ds (x) dt = ds (x) ẋ (4.1.31) dx (4.1.2) = ds (x) dx L ds (x) dx ( ds = L ) 2 (x) 0 (4.1.32) dx S (x) S (x) t x S'(x) x eq x 4.1.5

4.1 (12 21 ) 89 : 47 (25 ) (4.1.2) (4.1.3) 48 (10 ) T 0 T (r, t) t = κ 2 T (r, t) (4.1.33) 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 49 (10 ) 2 x µ, µ = 1,..., n ẋ µ = ν {x µ, x ν } S x ν + ν L µν S x ν (4.1.34) ( 1996) x µ = x eq µ L µν + L νµ ( ) S 1 x µ x eq µ {x µ, x ν } = {x ν, x µ } 50 (30 ) 47 (4.1.2) (4.1.3) x(t) S(x)

90 4 4.2 (1 11 ) ( ) (1) (2) (3) (4) 1. 2. 3. {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) 3 x y X(t)(t ) Y (t) X(t)Y (0) Y (t)x(0)

4.2 (1 11 ) 91 (1) 4.2 L µν 4.4 4.3 Thomson (4.2.2) 4.2 4.1 (4.2.2) 3.1 2 : : : R, P 4.2.1 : r 1, r 2,..., p 1, p 2,... {q l, p l } 1 2 2 1 2 X(t) {q l (t), p l (t)}

92 4 X(t) = X({q l (t), p l (t)}) (4.2.3) 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)

4.2 (1 11 ) 93 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) 51 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 )})

94 4 (4.2.16) (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)

4.2 (1 11 ) 95 (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) X µ ({q l ( t)}) 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 ( 55) 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)

96 4 (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 <X μ (t)x ν (0)> t 4.2.2: X µ (t)x ν (0) 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)

4.2 (1 11 ) 97 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) (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) {ql 0, 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, {ql 0, p 0 l }), p l ( t, {ql 0, 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, {ql 0, p 0 l }), p l (t, {ql 0, 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, {ql 0, p 0 l }), p l (t, {ql 0, p 0 l })}) (4.2.45)

98 4 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) (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) : 51 (20 ) P.93 52 (10 ) x(t) t = f(x(t)) (4.2.54) x = x(t) f(x) t 1

4.2 (1 11 ) 99 53 (30 ) (4.2.5) (4.2.6) 2.4 3.1 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) 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) 54 (30 ) 53 (4.2.56) 55 (30 ) (4.2.35) ρ eq ({q l, p l }) 56 (10 ) P.90 3. 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) 57 (30 ) (4.2.1) (2.3.54) (3.1.1) (4.2.1)

100 4 : : 2 8 ( ) 4:00 http://www.cmt.phys.kyushu-u.ac.jp/~a.yoshimori/hguidan11.pdf PDF 4.3 Thomson (1 18 ) 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)

4.3 Thomson (1 18 ) 101 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) L λµ = L µλ 2 ( ) (Peltier ) (1) 4.3 L µν 4.4 4.3 Thomson (4.2.2) 4.2 4.1 (4.2.2) 3.1 2 (2) Thomson 4.1 1 ( 59 ) 2

102 4 ( ) 2 (4.1.23) 1 L 21 (T 2 T 1 ) > 0 L 21 2 L A 21 LB 21 L A 1 21 T 2 (T 2 T 1 ) A T 1 T 2 2 B L B 1 21 T 2 (T 2 T 1 ) 4.3.1 (L A 21 L B 21) 1 T 2 (T 2 T 1 ) (4.3.7) A T 1 T V 2 B V = e AB (T 2 T 1 ) (4.3.8) e AB A B V A 4.3.2 Peltier I I A B 1 I 2 4.3.3 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

4.3 Thomson (1 18 ) 103 Peltier 2 Π AB T = e AB (4.3.10) (Thomson ) (3) (4.3.5) = 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 4.3.4 (4.1.3) (4) (4.3.10)

104 4 Π AB /e AB 温度を変えて測定 4.3.5 温度 T : 58 (15 ) (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) 59 (20 ) 4.1 1 (4.1.24) Ė1 (4.1.26) 1 2 1 1 60 (10 ) P102 Peltier 2 (4.1.22) A B L A 11 LA 12 A B Q A Q B Q A Q B = 0 V 61 (20 ) ( ) Peltier 62 (30 ) S(x)

4.4 (1 25 ) 105 4.4 (1 25 ) X µ (t) X µ (t)x λ (0) X λ (t)x µ (0) Thomson (1) (2) (3) Thomson (4) 1. 4.2 2. X µ = X µ (t) (1) Ẋ µ = ν L µν S X ν + R µ (t) (4.4.1) 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)

106 4 (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) 1 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 λ (0)

4.4 (1 25 ) 107 X µ (t)x λ (0) (t 1 L µλ S ) 4 S t L µλ 5 X λ (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 L µν S({X µ }) X ν + tr µ (0) + (4.4.8) Xµ =X µ (0) 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

108 4 (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 (4.1.23) 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 (4.1.26) 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) 2 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) φ A φ B = (L A 21R A L B 21R B ) T 2 T 1 T 2 (4.4.20)

4.4 (1 25 ) 109 V = φ A φ B (4.3.8) V = e AB (T 2 T 1 ) 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 (4.1.24) 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 φ 2 φ A = R A I φ 2 φ B = R B I (4.3.9) Q A Q B = Π AB 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 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)

110 4 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) * : Thomson * Thomson P105 ( ) (4.4.2) Peltier (4.4.2) L µν

4.4 (1 25 ) 111 : 63 (30 ) {x µ } S 64 (30 ) 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 (4.4.32) = L µλ X λ Ẋ µ = ((3.1.19) ) L µλ = L λµ (4.4.13) 65 (10 ) n {x 1,..., x n } = {x µ } ẋ µ = n γ µν x ν (4.4.33) ν=1 S({x µ }) = n µ=1,ν=1 k µνx µ x ν /2 66 (25 ) Thomson {x 1, x 2 } = {E 1, q 1 } E 1 P90 3 X({q l, p l }) = X({q l, p l }) (4.4.34) 67 (25 ) Peltier L 12 L 21 R L 12 R L 21 R

112 5 ( ) (2 1 ) 5 ( ) (2 1 ) (1) (2) (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)) ()

113 2. a V (t) b 1. 1 m V (t) = t 0 M(t t )V (t )dt + R(t) (5.3) M(t) R(t) 2. 2 M(t) = R(t)R(0) m V 2 (5.4) m V (t) = λv (t) + R(t) (5.5) 1. H. Mori, Prog. Theor. Phys. 33, 423 (1965). 2. Theory of Simple Liquids, Hasen and McDonald (Academic Press) Chapter 9. 3. II-6 P67-80 4. P183-190 5. ( ) 82-3 (2004) 357 (1) {V, X, ql w, pw l } (5.1) (5.2) (5.5) R(t) {V, X, ql w, pw l } (2)

114 5 ( ) (2 1 ) 2 2 2 t < t V (t ) t M(t t )V (t )dt (5.6) 0 R(t) F ({q w l (t)}, X(t)) t F ({ql w (t)}, X(t)) = M(t t )V (t )dt + R(t) (5.7) 0 1 V (t) 2 (5.1)(5.2) ( ) (5.3) ( = ) (5.5) ()

115 (3) () A 5.1 θ 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 (5.8) P A = P A V V (5.9) P A = (A V) V V V = (A V) (V V) V (5.10) () {q l, p l } ({q l, p l } = {X, V, ql w, pw l }) A A({q l, p l }) P P (A V) AV A V A({q l, p l }) V ({q l, p l })

116 5 ( ) (2 1 ) P : V A = A({q l, p l }) P A AV V 2 V (5.11) eq = ρ eq ({q l, p l }) l dq l dp l (5.12) 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) (5.13) {q l, p l } A({q l, p l }) AV / V 2 V P A({q l, p l }) = AV V 2 V 5.2 Q 1 P P + Q = 1 A = A({q l, p l }) A = }{{} P A + QA }{{} V (5.14) (4) {q l, p l } ({q l, p l } =

117 {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) (5.15) il({q l (t), p l (t)}) l { } q l (t) q l (t) + ṗ l(t) p l (t) (5.16) dx(t) dt = il({q l (t), p l (t)})x(t) (5.17) (5.16) 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 (5.18) ṗ l (t) = H({q l(t), p l (t)}) q l (t) (5.19) il({q l (t), p l (t)}) (5.17) X(t) = e il({q l(0),p l (0)})t X(0) (5.20) ( 71)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 (5.21) il V (5.14) il = P }{{} il + QiL }{{} (5.22)

118 5 ( ) (2 1 ) 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 (5.23) ( 72 )(5.23) e til e tqil e t il P il V (t) e til F = t 0 e t il P ile (t t )QiL F dt 1 + e tqil F 2 (5.24) 2 R(t) e tqil F (5.25) (5.24) 2 R(t) 1 1 (5.25) t P (5.11) t 0 0 e t il P ile (t t )QiL F dt = t 0 e t il P ilr(t t )dt (5.26) P ilr(t t ) = [ilr(t t )]V V 2 V (5.27) e t il P ile (t t )QiL F dt = t 0 e t il [ilr(t t )]V V 2 V dt (5.28) e t il V (5.20) = t 0 [ilr(t t )]V V 2 V (t )dt (5.29) ( 72) [ilr(t)]v = R(t)[iLV ] (5.30)

119 (5.17) ilv = V (5.31) (5.1) = F m (5.32) (5.25) R = F = R m (5.33) (5.30) [ilr(t)]v = R(t)R m (5.34) t 0 t e t il P ile (t t )QiL F dt = M(t t )V (t )dt (5.35) 0 (5.3) M(t) = R(t)R m V 2 (5.36) (5) 2a t 0 M(t t )V (t )dt V (t) t 0 M(t t )dt (5.37) 2a V (t) M(t) τ = t t 2b t = V (t) M(τ)dτ (5.38) 0 V (t) M(τ)dτ = λv (t) (5.39) 0

120 5 ( ) (2 1 ) 1 0.8 sgn(1-x)*exp((x-1)*10) sin(x)*0.4+0.2 2. 0.6 1. V (t 0.4 0.2 ) M(t t ) t = t 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 t /t 5.3 : M(t) V (t) λ = 0 M(τ)dτ (5.40) (5.5) λ 2 (2ndFDT) V 2 = k B T/m R(t)R = Dδ(t) (5.40) (5.36) λ = 0 R(t)R m V 2 dt = D 2k B T (5.41) (6) R(t) 2 R(t)

121 (4) R(t) (5.25) F 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 : 68 (15 ) (4.4.10) X µ Xµ min < X µ < Xµ max X = Xµ min Xµ max P eq ({X µ }) 0 69 (30 ) 2 2 70 (15 ) 71 (30 ) (5.17) X(t) = e ilt X(0) 5.17 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)

122 5 ( ) (2 1 ) 72 (30 ) (5.23) (5.30) q l ± p l ± ρ eq ({q l, p l }) 0 73 (15 ) 1 1 ẋ(t) = v(t) (5.42) m v(t) = mω 2 x(t) (5.43) (5.3) M(t) R(t) 74 (50 ) 1