1: 3.3 1/8000 1/ m m/s v = 2kT/m = 2RT/M k R 8.31 J/(K mole) M 18 g 1 5 a v t πa 2 vt kg (
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1 mm 1 µm web 2 [1, 2] 1
2 1: 3.3 1/8000 1/ m m/s v = 2kT/m = 2RT/M k R 8.31 J/(K mole) M 18 g 1 5 a v t πa 2 vt kg (macroscopic) (microscopic) 2
3 (1902/6/26), 4 (1903/1/26), 5 (1904/3/29), 6 (1905/3/17), 7 (1905/4/30), 8 (1905/5/11), 9 (1905/6/30) , 4,
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5 τ 11 τ a p τ 1) p a 2) p a 3) 1 2p 2 p 1/2 p = 1/6 1/6 2/ τ 2: τ p a p a / τ
6 1/6 1 1/ , 6, 4, 5, 3, i p i 1, 2,..., 6 p 1, p 2,..., p
7 x, y,... ˆ ˆx, ŷ, ˆx ˆx x 1, x 2,..., x n n 1, 2, 3, 4, 5, X, Y,... 7
8 i = 1, 2,..., n ˆx x i p i p i 0 p i 1 p p n = 1 ˆx ˆx ˆx = x 1 p 1 + x 2 p x n p n (1) ˆx ˆx ˆx = 1(1/6) + 2(1/6) + + 6(1/6) = 3.5 ˆx, ŷ ˆx + ŷ = ˆx + ŷ (2) ˆx, ŷ ˆx ŷ = ˆx ŷ (3) ˆx N N 1) N ˆx ˆx 2) N x i N p i x t = 0 x = N t = Nτ ˆx(t) ˆx(t) N ˆx(t) ˆx(t) = ê 1 + ê ê N (4)
9 ê i i 2.1 a p ê i = a p 0 1 2p (5) ê i = a p + ( a) p + 0 (1 2p) = 0 (6) (ê i ) 2 (ê i ) 2 = a 2 p + ( a) 2 p + 0 (1 2p) = 2pa 2 (7) ˆx(t) ˆx(t) ê i (4) (2) (6) ˆx(t) = ê 1 + ê ê N = 0 (8) 0 0 (ˆx(t)) 2 (4) (ˆx(t)) 2 = (ê 1 ) 2 + (ê 2 ) (ê N ) 2 + 2ê 1 ê 2 + 2ê 1 ê ê N 1 ê N (9) N (ê i ) 2 N(N 1)/2 2ê i ê j (i j) 19 (3) (6) 2ê i ê j = 2 ê i ê j = 0 (7) (ˆx(t)) 2 = (ê 1 ) (ê N ) 2 + 2ê 1 ê ê N 1 ê N = 2pa 2 N (10) D D = pa2 τ (11) t = Nτ (10) (ˆx(t)) 2 = 2Dt (12) 19 N = 2, 3 9
10 (ˆx(t)) 2 (ˆx(t)) 2 (12) 2Dt t t t t 3 3: ρ(t, x) t x ρ(t, x) x a t a ρ(t, x) τ
11 p a ρ(t, x) a p a ρ(t, x) 21 a p a ρ(t, x + a) p a ρ(t, x a) a ρ(t + τ, x) = a ρ(t, x) 2paρ(t, x) + pa ρ(t, x a) + pa ρ(t, x + a) (13) ρ(t, x) t x 22 Taylor ρ(t + τ, x) ρ(t, x) + τ ρ(t, x) ρ(t, x), ρ(t, x ± a) ρ(t, x) ± a + a2 2 ρ(t, x) t x 2 x 2 (14) (13) a 2 /τ a 0, τ 0 τ a t 2 ρ(t, x) = D ρ(t, x) (15) x2 D (11) (15) D N 0 x = 0 (15) ρ(t, x) = N 0 exp[ x2 4πDt 4Dt ] (16) (12) (15) a, τ, p a, τ, p (12) (15) D 23 (12) (15) D 21 a ρ(t, x) a 11
12 3 3.1 x f > 0 24 τ a ˆx(t) (4) ê i a p ag ê i = a p + ag 0 1 2p g > 0 g f g ag (1) ê i (17) ê i = a(p ag) + ( a)(p + ag) + 0 (1 2p) = 2a 2 g (18) (4) ˆx(t) ˆx(t) = ê 1 + ê ê N = 2a 2 gn = u t (19) t = Nτ u u = 2a2 g τ > 0 (20) u g f (20) 24 f 12
13 ρ(t, x) 2 ρ(t, x) = u ρ(t, x) + D ρ(t, x) (21) t x x2 (11) (20) a 2 /τ a 0, τ 0 (15) u u (16) ρ(t, x) = N 0 (x + ut)2 exp[ ] (22) 4πDt 4Dt x 0 x = 0 x f x = 0 (21) ρ eq (x) eq equilibrium= (21) 0 u d dx ρ eq(x) + D d2 dx 2 ρ eq(x) = 0 (23) α ρ eq (x) = ( ) exp( αx) α = 0, u/d 13
14 α = 0 ρ eq (x) = ρ 0 exp[ u x], x 0 (24) D x = 0 x (24) u/d f x fx x = 0 x (24) 0 0 dx fx ρ eq (x) dx ρ eq (x) 0 = f 0 dx x exp[ (u/d)x] dx exp[ (u/d)x] = f D u (25) 1.2 T (R/N A )T R 8.31 J/(K mol) 1 N A 25,26 25 [3] 26 N A R 14
15 27 (25) (R/N A )T f D u = R N A T (26) (28) 2 (12) 2Dt (15) D D D 3.1 x = 0 u ρ eq (x) exp[ (f x)/(kt )] 15
16 f u u = µ f (27) µ (27) (26) f D = R N A T µ (28) D µ (R/N A )T T 29 (28) µ = (6πηa) 1 a η 30 (28) D = RT N A 1 6πηa 1905 a 31 t D 29 (28) η kg/(m s) poise = g/(cm s) (29)
17 (29) η R T a D N A (29) N A N A x L L x = 0 x = L n ρ = n/l (12) 2Dt (15) D (11) ρ f 17
18 32 (20) u j ρ u j = ρu j σ = lim (30) f 0 f f 0 f (27) σ = ρµ (28) k = R/N A Dρ = σkt (31) 4.2 n D f j (30) σ f 0 D σ 32 18
19 (31) Dρ 2 κ = σ (32) κ κ = ( L ) 1 p(t, L, n) (33) L κ = (ρkt ) 1 (32) (31) (32) Green- σ = 1 dt J(0)J(t) (34) 2kT L 0 J(t) t σ (32) σ = h L 0 dt 1/(kT ) 0 dλ Ĵ( i hλ)ĵ(t) (35) (34) Ĵ(t) J(t) (35) (34) (35) 4.3 (28), (31), (32) 19
20 33 34 (35) 20 (30) f (1927) (1928) 34 (1931) 35 20
21 kt Raffiniert ist der Herr Gott, aber boshaft ist er nicht. [3] [3] 21
22 [1], 1976 [2], 1986 [3] Abraham Pais, Subtle Is the Lord: The Science and the Life of Albert Einstein (Oxford University Press, 1982),
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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())
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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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26 1 22 10 1 2 3 4 5 6 30.0 cm 1.59 kg 110kPa, 42.1 C, 18.0m/s 107kPa c p =1.02kJ/kgK 278J/kgK 30.0 C, 250kPa (c p = 1.02kJ/kgK, R = 287J/kgK) 18.0 C m/s 16.9 C 320kPa 270 m/s C c p = 1.02kJ/kgK, R = 292J/kgK
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