1905 1 1.1 0.05 mm 1 µm 2 1 1 2004 21 2004 7 21 2005 web 2 [1, 2] 1
1: 3.3 1/8000 1/30 3 10 10 m 3 500 m/s 4 1 10 19 5 6 7 1.2 3 4 v = 2kT/m = 2RT/M k R 8.31 J/(K mole) M 18 g 1 5 a v t πa 2 vt 6 6 10 16 kg 5 10 11 7 (macroscopic) (microscopic) 2
1902 1903 8 9 1905 8 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) 1905 9 3, 4, 5 6 7 8 9 3
10 1905 2 2.1 10 4
τ 11 τ a 12 13 p τ 1) p a 2) p a 3) 1 2p 2 p 1/2 p = 1/6 1/6 2/3 1 2 3 6 τ 2: τ p a p a 2.2 1 6 1/6 1 6 11 τ 12 13 5
1/6 1 1/6 1 6 14 1 6 2, 6, 4, 5, 3, 1 1 6 15 1 6 14 2.3 1 6 i p i 1, 2,..., 6 p 1, p 2,..., p 6 15 6
1. 2. 16 1. 2. 2.3 x, y,... ˆ ˆx, ŷ,... 17 ˆx ˆx x 1, x 2,..., x n n 1, 2, 3, 4, 5, 6 16 17 X, Y,... 7
i = 1, 2,..., n ˆx x i p i p i 0 p i 1 p 1 + + p n = 1 ˆx ˆx ˆx = x 1 p 1 + x 2 p 2 + + 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 18 2.4 x t = 0 x = 0 2.1 N t = Nτ ˆx(t) ˆx(t) N ˆx(t) ˆx(t) = ê 1 + ê 2 + + ê N (4) 18 3.5 3.5 8
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 2 + + ê N = 0 (8) 0 0 (ˆx(t)) 2 (4) (ˆx(t)) 2 = (ê 1 ) 2 + (ê 2 ) 2 + + (ê N ) 2 + 2ê 1 ê 2 + 2ê 1 ê 3 + + 2ê 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 ) 2 + + (ê N ) 2 + 2ê 1 ê 2 + + 2ê 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
(ˆx(t)) 2 (ˆx(t)) 2 (12) 2Dt t t t t 3 3: 2.5 20 ρ(t, x) t x ρ(t, x) x a t a ρ(t, x) τ 20 4.2 10
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) 2.3 22 23 a 11
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 2 + + ê N = 2a 2 gn = u t (19) t = Nτ u u = 2a2 g τ > 0 (20) u g f (20) 24 f 12
3.2 2.5 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
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
27 (25) (R/N A )T f D u = R N A T (26) 28 3.3 (28) 2 (12) 2Dt (15) D D D 3.1 x = 0 u 27 28 ρ eq (x) exp[ (f x)/(kt )] 15
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) 3.2 30 η 1 10 3 kg/(m s) poise = g/(cm s) 31 2 16 (29)
(29) η R T a D N A (29) N A N A 4 4.1 0 x L L x = 0 x = L n ρ = n/l (12) 2Dt (15) D (11) ρ f 17
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
(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
33 34 (35) 20 (30) f 0 35 33 (1927) (1928) 34 (1931) 35 20
36 37 38 39 40 36 3.2 kt 37 38 39 40 Raffiniert ist der Herr Gott, aber boshaft ist er nicht. [3] [3] 21
[1], 1976 [2], 1986 [3] Abraham Pais, Subtle Is the Lord: The Science and the Life of Albert Einstein (Oxford University Press, 1982), 1987 22