1 (Berry,1975) 2-6 p (S πr 2 )p πr 2 p 2πRγ p p = 2γ R (2.5).1-1 : : : : ( ).2 α, β α, β () X S = X X α X β (.1) 1 2

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1 2005 9/ ( 2-5) γ ( ) γ cos θ 2πr πρhr 2 g h = 2γ cos θ ρgr (2.1) γ = ρgrh (2.2) 2 cos θ θ cos θ = 1 (2.2) γ = 1 ρgrh (2.) 2 2. p p ρgh p ( ) p p = p ρgh (2.) h p p = 2γ r 1

2 1 (Berry,1975) 2-6 p (S πr 2 )p πr 2 p 2πRγ p p = 2γ R (2.5).1-1 : : : : ( ).2 α, β α, β () X S = X X α X β (.1) 1 2

3 X X, N 1, N 2 A Ni S = N i Ni α N β i = Γ i A (i = 1, 2) (.2) Γ i N < N α + N β (.2) Γ i = 1 A N i V n β i + V α (n β i nα i )] (.) n i (.) i = 1 i = 2 V α ] Γ 2 Γ 1 n β 2 nα 2 n β 1 nα 1 = 1 A (N 2 V n β 2 ) (N 1 V n β 1 )nβ 2 nα 2 n β 1 nα 1 (.4) Γ 2,1 (V, N i, n α,β i ) 2-2 A N S (.2) ΓA Γ α β Γ 1 = N1 α α N 1( ) p18 α (n α 1, nα 2 ) β

4 2 F = 2γl F x = 2γlx γ F df = du T ds SdT = pdv γda df = pdv γda - df = γda = 2γlx -4 α β V α = 1 R (R α ) ]ω (.5) V β = 1 (Rβ ) R ]ω (.6) A = ωr 2 (.7) R α R β ηdω dw = ηdω = p α dv α + p β dv β γda (.8) (.5),(.6),(.7) ω γ = 1 R(pα p β ) + 1 R 2 ( η 1 pα (R α ) + 1 pβ (R β ) ) (.9, 10) η R γ R (.9) R = 1 (pα p β ) 2R (η 2 1 pα (R α ) + 1 ) pβ (R β ) (.9) p α p β = 2γ R + (.11) (R, p α = p β ) γ = 0 γ (R, p α = p β ) (.11) 0=0 4

5 ω R α,β dw = p α dv α + p β dv β γda AdR (.1) ] γ AdR.4 ] γ = 0 p α p β = 2γ s R s (.14) γ s (.9) (.9) (.14) γ ( R 2 γ = γ s s R 2 + 2R ) R s γ R = R s γ s (.15) ] γ = 0 ɛ = δ R s R = R s + δ (.15) γ = γ s (1 + ɛ 2 + O(ɛ )) δ γ γ s.5-4 (.1) du = T ds p α dv α p β dv β + γda + AdR (.16) df = SdT p α dv α p β dv β + γda + AdR (.17) ds 0 ds dq T du = dq du T ds 5

6 df SdT 2 F T, V, N 1, N 2 µ 1 =, µ 2 = (.18) N 1 T,V,N 2 N 2 T,V,N 1 1,2 1 (.18) F δf = β N β α N α = µβ µ α = 0 µ α = µ β df = µ 1 dn 1 + µ 2 dn 2 df = SdT p α dv α p β dv β + γda + AdR + µ 1 dn 1 + µ 2 dn 2 (.19) ν f(x 1, x 2,, x ν ) f(λx 1, λx 2,, λx ν ) = λ n f(x 1, x 2,, x ν ) n x 1 f x 1 + x 2 f x x ν f x ν = nf 1 λ F λf F = V + N 1 + N 2 V T,N 1,N 2 N 1 T,V,N 2 N 2 (.20) p, µ 1, µ 2 T,V,N 1 F = pv + N 1 µ 1 + N 2 µ 2 (.22) -4 R α, R β, T ω, N 1, N 2 λ λ F = ω + N 1 + N 2 ω R α,r β,t,n 1,N 2 N 1 R α,r β,ω,t,n 2 N 2 R α,r β,ω,t,n 1 6

7 (.21) F = ηω + µ 1 N 1 + µ 2 N 2 (.2) ηω p2 ηω = γa p α V α p β V β F = p α V α p β V β + γa + µ 1 N 1 + µ 2 N 2 (.24) γ R p α = p β.6 (.22) F α = p α V α + µ 1 N1 α + µ 2 N2 α (.25) F β = p β V β + µ 1 N β 1 + µ 2N β 2 (.26) (.24) F S = γa + µ 1 N1 S + µ 2 N2 S (.27) µ 1 N1 S + µ 2 N2 S = 0 (.28) F S = γa (.29) (cf..) (.29) γ (.20) F α, F β (.19) F S df S = S S dt + γda + µ 1 dn S 1 + µ 2 dn S 2 + (.2) df S Adγ + S S dt + N S 1 dµ 1 + N S 2 dµ 2 = + AdR (.2) AdR (.) A dγ + σdt + Γ 1 dµ 1 + Γ 2 dµ 2 = dr (.4) 7

8 σ = SS A (.4) (.4) Γ 1 = 0(Γ 2,1 = Γ 2 ) dγ = Γ 2,1 dµ 2 (.6) P V = nrt S = Nc p ln T Nk ln p + const 4 S = (N 1 c 1p + N 2 c 2p ) ln T N 1 k ln p 1 N 2 k ln p 2 + const p i = NikT V µ i = N i = T S N i + U N i = kt ln p i + φ i (T ) (i = 1, 2) (.7) φ(t ) 5 (.7) (.6) dγ = kt Γ 2,1 d ln p 2 (i = 1, 2) (.8) 2 Γ 2,1 2 6 (.6) Γ 2,1 > 0 γ.7 (cf. (.28),(.29)) 0 (.) Adγ + S S dt + N S dµ = AdR (.9) R = R 0 dγ 0 = dr 0 R=R (.8) 8

9 ] γ 0 γ = (.40) 0 R=R 0 ] γ = 0 (.40) ] (.40) γ γ 0 R 0 γ, γ S R, (.15) dγ = 1 ( R 2 S R 2 + 2R ) dγ S + 2 ( γ RS S R 2 R ) RS 2 d γ S R2 S R dr (γ S, = ) = 2 γ S (R0 RS ) R=R 0 R0 (.41) R 0 (γ S ) γ 0 = 1 ( R 2 S 0 R R 0 S γ S ) γs γ S 1 R2 S R0 ( RS R 2 0 R ) 0 RS 2 (.40) 2 ( R 2 S R R ) ( 0 RS dγ S = 2γ S R0 2 R ) 0 RS 2 d γ S γ S = 2(R 0 RS ) S RS + 2R 0 (.41) (.40) γ 0 R 0 = 2 γ S (R0 RS ) 0 R0 2R = 2γ ( 0 R 0 RS) S RS + 2R 0 (.42) (.4) (.15) (.40),(.41),(.42),(.4) R 0 = + δ ] ln γ = ln γ 0 = ln γ S = ln R R=R 0 ln R 0 ln 2 δ 1 + δ + 1 ( δ 1 + δ + 1 ) ] 2 δ ( ) ] 2 (.44) δ 7 7 (.42) (.4) γ 0 (.44) 9

10 ln γ S RS = γ 2 δ ( δ R) 2 ] R 1 + δ 2 R δ R 1 + δ R + ( 1 δ 2 ]dr (.45) R) γ δ R )] RS ln ( 1 + 2γ R ( γ S = γ 1 2δ ) + ( γ 0 (R 0 ) = γ 1 2δ ) + R 0 (.46) (.47) δ δ δ R h = 2γ ρ β gr p α = p β 2γρα ρ β R (.48) (.49) 1 v α, v β p α = p β 2γvβ p α RkT p β p 2γvβ kt R p α = p 1 + 2γvβ kt R p e 2γvβ kt R << 1 p 1 2γvβ kt R (.50) (1) 8 ln pα p = 2γvβ kt R ln pβ p = 2γvα kt R (.51) (.52) 8 (.51) (.52) 10

11 -4 SdT V dp + Ndµ = 0 (.55) α v α dp α = dµ (.56) 2 dµ = kt dpβ p β v α dp α = kt dpβ p β (.11) p α 2γ R + v α (p α p ) = kt pβ p γ ] = kt pβ ln vα p (pβ p ) (.57) ln pβ p = vα kt { 2γ ]} γ R + R = (.40) ln pβ p = 2vα γ S = vα 2γ0 + γ ] 0 kt kt R 0 0 (α:,β: ) ln pα p = 2vβ γ S kt = vβ kt 2γ0 + γ ] 0 R 0 0 (.58) (.59) (.60) 9 (.59) (.60).9 (.24) U = T S p α V α p β V β + γa + µn U 0 = T 0 S + p 0 V + µ 0 N W = U U 0 = (T T 0 )S V α (p α p β ) V (p β p 0 ) + N(µ µ 0 ) + γa (.61) 9 (.59) (.60) 11

12 ψ = (T T 0 )S V (p β p 0 ) + N(µ µ 0 ) (.61) W = γa V α (p α p β ) R W = 4πK (.6) (.10) (.11) W = 4πR2 γ { ]} ln γ 1 ln R (.64) (R = ) W = Aγ S (.65) (.40) W = 4πR2 0γ 0 1 ln γ ] 0 ln R 0 (.66) x ψ s 1 R 1 R = dψ ds dz dx = tan ψ ds 2 = dx 2 + dz 2, dz ds = sin ψ 1 R = dψ dz cos ψ = d dz ds dz (4.) 12

13 10 ψ = 0 p 0 z p α p β = (ρ α ρ β )gz (4.4) (4.2) (4.),(4.4) a 2 d cos ψ dz = 2z (4.5) a 2 = 2γ (ρ α ρ β )g (4.5) ψ = 0 z 0 z = z a2 sin 2 ψ 2 (4.6) (4.7) (4.5),(4.7) dz dx = tan ψ z dx = ak (1 2 cos 2 φ) dφ (4.8) 2 1 k2 sin 2 φ k 2 = 2a2 z0 2 +, φ = π ψ 2a2 2 ψ = 0 z 0 = 0 k = 1 h ψ ψ = π 2 θ (4.7) θ (4.8) (4.9) h = a 1 sin θ (4.10) :5:8:

14 (1) θ > π 2 θ < π 2 (2) () (cf..6) (4) (5.11) 4 14

1. z dr er r sinθ dϕ eϕ r dθ eθ dr θ dr dθ r x 0 ϕ r sinθ dϕ r sinθ dϕ y dr dr er r dθ eθ r sinθ dϕ eϕ 2. (r, θ, φ) 2 dr 1 h r dr 1 e r h θ dθ 1 e θ h

1. z dr er r sinθ dϕ eϕ r dθ eθ dr θ dr dθ r x 0 ϕ r sinθ dϕ r sinθ dϕ y dr dr er r dθ eθ r sinθ dϕ eϕ 2. (r, θ, φ) 2 dr 1 h r dr 1 e r h θ dθ 1 e θ h IB IIA 1 1 r, θ, φ 1 (r, θ, φ)., r, θ, φ 0 r