0201

2018 10 17 2019 9 19 SI J cal 1mL 1ºC 1999 cal nutrition facts label calories cal kcal 1 cal = 4.184 J heat capacity 1 K 1 J K 1 mol molar heat capacity J K mol (specific heat specific heat capacity) 1 kg mass heat capacity J K kg (volumetric heat capacity volume-specific heat capacity) J K m 1 1 1 1 1 3 C P C P = C V + R (3.2.1) R ayer s relation C V 1 / 12

1 γ γ = C P C V = C V + R C V = 1 + R C V (3.2.2) 1 V m N ideal gas free particle r v p x, y, z ( x, y, z, v x, v y, v z ) 6 topological space μ N 6N Γ Γ representative point Γ E i v i = ( v ix, v iy, v iz ) E i E 2 E i = m v m i ( vix 2 + v2 iy + v2 iz) = 2 2 N E = E i i=1 (3.3.1) (3.3.2) Γ 6N ergodic hypothesis ergodic theorem 3.3.A (3.3.2) 2 / 12

μ v C cell j = 1,, N μ j 1 1 Γ N N N μ j 0 1 1 N 1 1 n 1 N n p(n 1,, n ) = 3.3.B N! n 1! n 2! n! ( N ) N N N N (3.3.3) principle of maximum entropy p(n 1,, n ) +( N ) ln ( N ) + N ln N ln N (3.3.4) p(n 1,, n ) ln p (n 1,, n ) (n 1,, n ) n! n Stirling 3.3.C ln n! n ln n n n Stirling n n 0 1 3.3.D ln p (n 1,, n ) = ln N! ln n j! Γ gamma function Γ (x) digamma function ψ(x) Γ (x) t x 1 e t dt 0 (3.3.5) 3 / 12

0.8 0.6 ln (x + e -γ ) 0.4 y 0.2 0.0 ψ(x + 1) -0.2-0.4 0.0 0.5 1.0 x 1.5 2.0 ln(x + e γ ) ψ(x) d (3.3.6) dx ln Γ (x) Γ n! = Γ (n + 1) (3.3.4) ln p (n 1,, n ) = ln N! ln Γ ( n j + 1 ) +( N ) ln ( N ) + N ln N ln N n j (j = 1,, ) (3.3.7) ln p (n 1,, n ) n j (3.3.8) ψ(x) = d dn j ln Γ ( n j + 1 ) = ψ ( n j + 1 ) (j = 1,, ) ψ(x + 1) = γ + ( 1) k ζ(k)x k 1 k=2 (3.3.9) 4 / 12

γ Euler's constant γ = 0.57721 ζ(k) Riemann zeta function ζ(k) n=1 1 n k (3.3.10) x ψ(x + 1) ln(x + e γ ) (3.3.11) x (3.3.8), (3.3.11) ln p (n 1,, n ) n j ln ( n j + e γ ) (j = 1,, ) (3.3.12) 3.3.1 ψ(x + 1) lnx ln(x + e γ ) (3.3.7) ln p (n 1,, n ) {n j } {n j } N E n j = N E j n j = E (3.3.13) (3.3.14) method of Lagrange multiplier 3.3.E α β L (n 1,, n, α, β) ln p (n 1,, n ) α n j N β E j n j E (3.3.15) L (n 1,, n, α, β) (n 1,, n, α, β) (3.3.12) (3.3.13), (3.3.14) L (n 1,, n, α, β) n j ln ( n j + e γ ) α βe j = 0 (j = 1,, ) (3.3.16) L (n 1,, n, α, β) α = n j N = 0 (3.3.17) 5 / 12

L (n 1,, n, α, β) β = E j n j E = 0 (3.3.18) + 2 (3.3.16) ln ( n j + e γ ) +α + βe j = 0 n j e α exp ( βe j ) e γ (3.3.19) (3.3.19) (3.3.13) e α exp ( βe j ) e γ N e α N + e γ exp ( βe j) (3.3.20) (3.3.19) n j N + e γ exp ( βe j) exp ( βe j ) e γ (3.3.21) β > 0 E j n j E j β (3.3.21) (3.3.14) n j E j E j n j N + e γ exp ( βe j) E j exp ( βe j ) e γ E (3.3.22) β T β = 1 k B T k B Boltzman k B = 1.380 649 10 23 J K 1 m 2 kg s 2 K 1 (3.3.23) (3.3.21) (3.3.22) μ μ x, y, z ( x, y, z, v x, v y, v z ) 6 6 / 12

(x, y, z) 3.3.F μ x, y, z ( v x, v y, v z ) 3 j v jx v x < v jx + dv x v jy v y < v jy + dv y v jz v z < v jz + dv z i v i = ( v ix, v iy, v iz ) (3.3.1), (3.3.21), (3.3.23) p (v i) = p ( v ix, v iy, v iz ) exp ( E i k B T ) = exp m ( v2 ix + v2 iy + v2 iz) 2k B T (3.3.24) Gauss i p (v) = p ( v x, v y, v z ) exp ( E k B T ) = exp m ( v2 x + v 2 y + v 2 z ) 2k B T (3.3.25) axwell (3.3.26) axwell-boltzman Boltzman p (E ) exp ( E k B T ) 7 / 12

x, y, z L N +x 2mv x x v x 2L /v x +x F = Nm v2 x L P P = F L 2 = Nm v2 x L 3 V = L 3 (3.3.1.2) (3.3.1.1) (3.3.1.2) PV = Nm vx 2 (3.3.1.3) T x vx 2 (3.3.24) v 2 x = v 2 x exp exp m ( vx 2 + vy 2 + vz 2 ) 2k B T m ( vx 2 + vy 2 + vz 2 ) 2k B T dv x dv y dv z dv x dv y dv z = v 2 x exp ( mv2 x 2k B T ) dv x (3.3.1.3) (3.3.1.4) exp ( mv2 x 2k B T ) dv x = k B T m (3.3.1.4) PV = Nk B T (3.3.1.5) N A k B R R = N A k B n N = N A n PV = n RT (3.3.1.6) 8 / 12

ideal gas law (3.3.1.4) (3.3.1.7) vx 2 = v2 y = v2 z v2 = vx 2 + v2 y + v2 z v 2 1 2 m v2 x = 1 2 k B T 1 2 m v2 = 3 2 k B T (3.3.1.8) equipartition theorem U U = N i=1 1 2 m v2 i = 3 2 Nk B T (3.3.2.1) V du dt = 3 2 Nk B C V = N A N C V du dt = 3 2 N Ak B = 3 2 R (3.3.2.2) (3.3.2.3) T T + dt PV = Nk B T V dv = Nk B dt P (3.3.2.4) PdV = Nk B dt d (U + PV ) dt = 3 2 Nk B + Nk B = 5 2 Nk B (3.3.2.5) 9 / 12

(3.3.2.6) ayer C P = C V + R ayer C P = N A N C P d (U + PV ) dt = 5 2 N Ak B = 5 2 R 3.3.A ergodic ergodicity ergodic ergodicity έργον (ergon) όδος (hodos) 3.3.B (3.5) 3 1 5 5 3 1 1 2 1 3 1 3 1 1 2 2 1 3 5 3 5 5 3 = 2 5 3 3 5 3 3 3C 1 2 C 1 5 5 ( 5 ) ( 5 ) = 3! 5 3 3 1!1!1! 5 3 3 ( 5 ) ( 5 ) 0 1, 2, 3 0 0 C 0 = 0! 1, 2, 3 0! 0! = 1 3 4, 5 2 3! N 1! 1! 1! 0! 0! 22 3 3 n 1,, n N! n 1! n! (N )N N N N! n 1! n! ( N ) N N N 1 N = N! n 1! n! ( N ) N N N N 3.3.C Stirling Stirling n ln n! n ln n n ln 10! = 15.1044 10 ln 10 10 = 13.0259 ln 100! = 363.739 100 ln 100 100 = 360.517 ln 1000! = 5912.13 1000 ln 1000 1000 = 5907.76 10 / 12

n Stirling ψ(x + 1) lnx ψ(1) = 0.577216 ψ(2) = 0.422784 ψ(11) = 2.35175 ψ(101) = 4.61016 ψ(1001) = 6.90826 ln 0 = ln 1 = 0 ln 10 = 2.30259 ln 100 = 4.60517 ln 1000 = 6.90776 ln(0 + e γ ) = 0.577216 ln(1 + e γ ) = 0.445621 ln(10 + e γ ) = 2.35721 ln(100 + e γ ) = 4.61077 ln(1000 + e γ ) = 6.90832 ψ(10001) = 9.21039 ln 10000 = 9.21034 ln(10000 + e γ ) = 9.2104 x ψ(x + 1) ln(x + e γ ) x ψ(x + 1) lnx 3.3.D axwell μ n 0 1 Stirling ln n! n ln n n 1 3 (1966) 29 (1980) 3.3.E Lagrange g (x, y) = 0 f (x, y) (x, y) = (a, b) λ L (x, y, λ) = f (x, y) λg (x, y) x L (x, y, λ) = x f (x, y) λ x g (x, y) = 0 y L (x, y, λ) = y f (x, y) λ y g (x, y) = 0 λ L (x, y, λ) = g (x, y) = 0 (x, y, λ) = (a, b, μ) (x, y) = (a, b) Lagrange 3.3.F N Lagrange 11 / 12

N (n 1,, n ) p (n 1,, n ) (3.5) (3.9) ln p (n 1,, n ) = ln N! ln Γ ( n j + 1 ) +( N ) ln ( N ) + N ln N ln N (3.F.1) N n j = N (3.F.2) Lagrange L (n 1,, n, α ) ln p (n 1,, n ) α n j N (3.F.3) L (n 1,, n, α ) n j ln ( n j + e γ ) α = 0 (j = 1,, ) (3.F.4) (3.F.4) L (n 1,, n, α ) α = n j N = 0 (3.F.5) n j e α e γ (j = 1,, ) (3.F.6) (3.F.5) ( e α e γ ) N 0 e α N + e γ (3.F.7) (3.F.6) n j N (3.F.8) 12 / 12