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()) A(p(t), q(t)) dγ() e βh (p(),q()) (5.3) 1 51
Ĥ = Ĥ + h ˆM (5.4) t = ˆρ() = e βĥ Tr e βĥ (5.5) ˆM() βĥ Tr ˆMe h Tr e βĥ t = Heisenberg ˆM(t) = e iĥt/ h iĥt/ h ˆM()e (5.6) (5.7) (5.5) (5.4) ˆM(t) h = Tr [ ] e βĥ ˆM(t) Tr e βĥ = Tr [ ( ] e βĥ 1 + βh ˆM + ) ˆM(t) Tr [ ( e βĥ 1 + βh ˆM + )] = Tr [ ] e βĥ ˆM(t) + βh Tr [ ] e βĥ ˆM ˆM(t) Tr [ ] e βĥ ˆM(t) βh Tr [ ] e βĥ ˆM Tr e βĥ Tr e βĥ Tr e βĥ Tr e βĥ = ˆM(t) + βh ˆM() ˆM(t) βh ˆM(t) ˆM() + + = ˆM + βh ( ˆM() ˆM(t) ˆM 2 ) + (5.8) ˆM(t) = ˆM (5.8) δm(t) = ˆM(t) h ˆM(t) = βh ( ˆM() ˆM ) ( ˆM(t) ˆM ) + = βh δ ˆM()δ ˆM(t) + = βh C δmδm (t) + (5.9) [ ] h(t) δm(t) = χ(t, t )h(t ) + O(h 2 ) (5.1) 52
χ(t, t ) (response function) (generalized susceptibility) χ(t, t ) = δm(t) δh(t ) (5.11) χ(t, t ) t t h(t) = h δ(t t ) δm(t) = h χ(t, t ) + O(h 2 ) (5.12) χ h(t) δm(t) χ(t, t ) : t < t χ(t, t ) = χ(t, t ) : χ(t, t ) = χ(t t ) (5.13) θ(t) Heaviside h(t) = h θ( t) (5.14) θ(x) { = for x, = 1 for x. (5.15) δm(t) = h χ(t t ) = h χ(τ)dτ = h χ(τ)dτ (5.16) t dm(t) = hχ(t) (5.17) (5.9) = β d C δmδm(t) for t >, χ(t) = for t <. (5.18) [ ] W = F dx E = W = hdm (5.19) 53
ω h(t) = Re [ h(ω)e iωt] = 1 2 ( h(ω)e iωt + h (ω)e iωt) (5.2) Ẇ = 1 T T h dm = 1 T T dh δm(t) (5.21) (ωt 1 ) (5.1) Ẇ = 1 T dh χ(t, t )h(t ) (5.22) T (5.2) Ẇ dw = 1 T = 1 T T T iω 2 iω 2 ( h(ω)e iωt h (ω)e iωt) (5.18) t dw T χ(t t )h(t ) ( h(ω)e iωt h (ω)e iωt) χ(t )h(t t ) (5.23) = 1 T iω ( h(ω)e iωt h (ω)e iωt) χ(t ) 1 ( h(ω)e iω(t t ) + h (ω)e ) iωt(t t ) 2 2 = i ω 4 h(ω) 2 χ(t ) ( e iωt e iωt ) = i ω 4 h(ω) 2 (χ( ω) χ(ω)) = ω 2 h(ω) 2 χ (ω). (5.24) χ(ω) χ(ω) χ (ω) + iχ (ω) (5.25) χ( ω) = χ (ω) Ẇ ( ) ( ) [ ] (5.18) Fourier χ(ω) = = β χ(t)e iωt dc δmδm(t) = βc δmδm (t)e iωt e iωt + βiω C δmδm (t)e iωt (5.26) 1 C δmδm ( ) = βc δmδm (t = ) 2 C δmδm (t) = C δmδm ( t) βiω C δmδm (t)e iωt = βiω 1 C δmδm (t)e iωt = i βω 2 2 C δmδm(ω) (5.27) 54
C δmδm (ω) χ (ω) = ωβ 2 C δmδm(ω) (5.28) (5.24) (fluctuation dissipation theorem) Onsager ( ): (5.28) χ (k, ω) = 1 eβ hω C(k, ω) (5.29) 2 h [Kramers-Krönig ] χ(t) Fourier χ( ω) = χ (ω) χ ( ω) = χ (ω) χ ( ω) = χ (ω) χ(t) t Fourier Fourier ω ω = ω + iω t χ(t) = χ(ω + iω ) = χ(t)e iω t e ω t (5.3) e ω t ω (ω > ) (ω ) ω C C χ(ω) ω ω dω = (5.31) 4 + ω δ R R ω +δ χ(ω ) dω + ω ω χ(ω ) dω + ω ω π π χ(ω + δe iθ ) δie iθ dθ δe iθ χ(re iθ ) Re iθ Rie iθ dθ = (5.32) R δ 1 3 ω ( ) 4 χ(ω) χ(ω ) P dω iπχ(ω ω ) = (5.33) ω 55
5.1: ω χ (ω ) + iχ (ω ) = i π P χ (ω ) + iχ (ω ) ω ω dω (5.34) Kramers-Krönig χ (ω ) = 1 π P χ (ω ) dω, ω ω (5.35) χ (ω ) = 1 π P χ (ω ) dω. ω ω (5.36) [ ] ( ) dm = 1 τ (M M eq χh) (5.37) M eq χh τ ( ) ω Fourier iωm(ω) = 1 τ M(ω) + χ h(ω) (5.38) τ χ(ω) = M(ω) h(ω) = χ 1 iωτ (5.39) χ 1 (ω) = χ χ ωτ (ω) = χ (5.4) 1 + (ωτ) 2 1 + (ωτ) 2 56
Χ Ω Χ 1..8.6.4.2 Χ Ω Χ 4 2 2 1 1 2 Ω Ω 4 2 2 4.2 ΩΤ 2 (a).4 (b) 4 5.2: : (a) (b) ( 5.2(a)) Kramers-Krönig (5.39) ω χ(ω) ω = i/τ m d2 x 2 dx = kx γ + ee (5.41) γ ω Fourier ω = k/m ω 2 x(ω) = ωx(ω) 2 + i γ m ωx(ω) + e E(ω) (5.42) m p = ex Fourier p(ω) ω 2 p(ω) = ωp(ω) 2 + iγωp(ω) + e2 E(ω) (5.43) m (γ = eγ /m) p(ω) p(ω) = 1 e 2 E(ω) (5.44) ω 2 ω 2 iγω m ( ϵ(ω) ) χ(ω) = p(ω) E(ω) = 1 e 2 ω 2 ω 2 iγω m (5.45) χ (ω 2 ω 2 )ω 2 (ω) = χ (5.46) (ω 2 ω 2 ) 2 + (γω) 2 χ γωω 2 (ω) = χ (5.47) (ω 2 ω 2 ) 2 + (γω) 2 ( 5.2(b)) χ = e2 mω 2 = e2 k (5.48) 57
χ(ω = ) ω (5.45) χ(ω) = ω ( ω 2 γ2 4 1 ) 1/2 i γ ( ω + 2 ω 2 γ2 4 e 2 ) 1/2 i γ m 2 (5.49) ω χ(ω) ±ω 5.2 Brown Brown 1827 Robert Brown Brown Gouy(1888 ) Exner(19 ) Exner 5.2.1 Einstein Brown Brown 195 Albert Einstein ( Einstein Brown 3 ) [Einstein ] Einstein Boltzmann n(x) = n e mgz/k BT (5.5) m a ρ c ρ l m = (4π/3)a 3 (ρ c ρ l ) F = mg v D j drift = nv D v D = µf µ D j diffusion = D n (5.5) j drift + j diffusion = µf n D n z = (5.51) 58
µf D = 1 n n z = F k B T (5.52) D = k B T µ (5.53) Einstein η Stokes µ = 1/6πηa D = k BT 6πηa = RT N A 1 6πηa (5.54) Einstein-Stokes (R N A Avogadro R/N A = k B Boltzmann Avogadro ) D η (5.53) [ ] 1 n(x, t) τ τ x l p τ (l) l p τ (l) = p τ ( l) n(x, t) n(x, t + τ) = n n(x, t) + n t τ = n(x, t) n(x l, t)p τ (l)dl (5.55) p τ (l)dl n lp τ (l)dl + 2 n l 2 x x 2 2 p τ(l)dl + (5.56). p τ dl = 1 n(x, t) n t = D 2 n x. (5.57) 2 D p τ (l) 2 D = 1 τ l 2 2 p τ(l)dl. (5.58) t = x = N t (5.57) n(x, ) = Nδ(x) (5.59) n(x, t) = Ne x2 /4Dt 4πDt (5.6) 59
Gauss D p τ (l) x σ x = x 2 = 1 x 2 n(x, t)dx = 2Dt (5.61) N 3 3 (5.54) σ x = t RT N A 1 3πηa (5.62) Einstein 1 6 Avogadro N A 198 Perrin ( Jean Baptiste Perrin, 187-1942) Brown Avogadro Perrin 1926 Nobel Brown 1 m = ρ(π/6)d 3 = 1.2.52 1 12 = 6.3 1 11 g ( ρ = 1.2g cm 3 ) 1 3k B T/2 6.2 1 14 erg Brown v 2 = 3k B T/m =.4cm/s 1 1 1 5 cm m/s 1 1 1 τ r mµ = 2ρa 2 /9η 2 1.2.25 1 8 /(9.135) = 5 1 8 s 5.2.2 : Langevin Brown (random force) Langevin Perrin Langevin ( Paul Langevin, 1872-1946) Langevin Einstein [1 Langevin ] 1 Brown R(t) Langevin m dv = v + R(t) (5.63) µ R(t) = (5.64) R(t)R(t ) = 2D R δ(t t ) (5.65) 6
(5.65) τ c γ = 1/mµ dv = γv + R(t) m (5.66) (5.66) v(t) = v()e γt + 1 m 1 v(t) = 1 m e γ(t t ) R(t ) (5.67) e γ(t t ) R(t ) (5.68) R(t) = v(t) = v(t)v(t ) = 1 m 2 1 t > t (5.65) 2 e γ(t+t ) e γ(t 1+t 2 ) R(t 1 )R(t 2 ) (5.69) v(t)v(t ) = 1 m 2 1 e γ(t+t ) e 2γt 1 2D R = e γ(t t) D R m 2 γ C vv ( t t ) = v(t)v(t ) = e γ t t D R m 2 γ (5.7) (5.71) C vv τ v = 1/γ D R /m 2 γ = k B T/m v(t)v(t) = k BT m D R = mγk B T = k BT µ (5.72) (5.73) D R mγ = 1/µ 2 [ ] (5.67) ( 5.3(a)) x(t) = x() + v() 1 e γt γ = x() + v() 1 e γt γ = x() + v() 1 e γt γ + 1 m + 1 m + 1 m e γ(t t ) R(t ) t e γ(t t ) R(t ) 1 ) e γ(t t R(t ) (5.74) γ 61
(x(t) x()) 2 = v() 2 (1 e γt ) 2 5.3: γ 2 + 1 m 2 1 2 1 e γ(t t 1) γ = v() 2 1 2e γt + e 2γt γ 2 = v() 2 1 2e γt + e 2γt γ 2 + 2D R m 2 γ 2 1 e γ(t t 2) R(t 1 )R(t 2 ) γ + 2D [ R t 2(1 e γt ) + (1 ] e 2γt ) m 2 γ 2 γ 2γ (5.72) (5.73) (x(t) x()) 2 = k ( BT 1 2e γt + e 2γt) + 2k BT mγ 2 mγ γt 1 [ 1 ( 1 e γ(t t 1 ) ) 2 t 2(1 e γt ) γ (5.75) + (1 ] e 2γt ) 2γ = 2k BT mγ 2 ( γt + e γt 1 ) (5.76) (x(t) x()) 2 2k BT mγ 2 1 2 γ2 t 2 = 2 kb T m t γt 1 (x(t) x()) 2 2k BT mγ (5.61) (5.77) t. (5.78) D = k BT mγ = µk BT (5.79) 62
Einstein 5.2.3 Langevin Langevin ( ) γv [Langevin ] N ϕ i (i = 1, 2,, N) dϕ i = G i({ϕ j }) + η i (t) (5.8) ϕ i G i F ({ϕ j }) F/ ϕ j G i ({ϕ l }) = j Γ ij F ({ϕ l }) ϕ j (5.81) Γ ij ϕ ϕ ( mobility) Γ ij = Γδ ij dϕ i = Γ F ({ϕ l}) ϕ i + η i (t) (5.82) [ ] t = N t G = N ϕ(t) ϕ() = η(n) t (5.83) n=1 η η(n) = (5.84) (ϕ(t) ϕ()) 2 = η(m)η(n) ( t) 2 m n = η(n) 2 ( t) 2 n = η 2 N( t) 2 (5.85) t N = t/ t ( t τ c ) η 2 = 2D η t (5.86) 63
(2 ) η(m)η(n) = 2D η δ mn t (5.87) t η(t)η(t ) = 2D η δ(t t ) (5.88) η i (m)η j (n) = η i η j δ mn = 2D η δ mn ij t (5.89) t η i (t)η j (t ) = η i η j δ(t t ) = 2D η ijδ(t t ) (5.9) D η ij ϕ Γ ij Γ ij = Γδ ij D η = k B T Γ (5.91) Brown Γ = γ/m D η = D R /m 2 D R = mγk B T 64