L. S. Abstract. Date: last revised on 9 Feb translated to Japanese by Kazumoto Iguchi. Original papers: Received May 13, L. Onsager and S.

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1 L. S. Abstract. Date: last revised on 9 Feb 01. translated to Japanese by Kazumoto Iguchi. Original papers: Received May 13, L. Onsager and S. Machlup, Fluctuations and Irreversibel Processes, Physical Review pp Ph. D. S. M.. 1

2 L. S RRIP II Sec.5 Gaussian random variables s ij R ij Γ k log ProbΓ = SΓ + const. 1.1 Γ Γ joint probability RRIP I II 1 L. Onsager, Reciprocal Relations in Irreversible Processes, Phys. Rev. 37, ; 38, RRIP I II

3 II, pp Casimir 4 5 RRIP RRIP 6 7. RRIP II extensive variables 1,,, n intensive variables 3 M. C. Wang and G. E. Uhlenbeck, Revs. Modern Phys. 17, J. E. Moyal, J. Roy. Statist. Soc. B11, J. L. Doob, Stochastic Processes J. Wiley and Sons, New York, H. B. Casimir, Revs. Modern Phys. 17, A. Einstein, Ann. Physik 33, , Sec.I, General Matters Relating to Boltzmann s Principle. 6 RRIP II, Sec.5 Fokker-Planck N. Hashitsume, Prog. Theoret. Phys. 8, A. N. Kolmogoroff, Foundations of the Theory of Probability, Chelsea Publishing Company, New York, 1950, Chap III, Sec.4 Borel

4 4 L. S S = S 1,, n = S. S 0 S 0 = S0,, 0. X i = S/ i, P. W. Bridgeman, Rev. Modern Phys., S. Machlup and L. Onsager, following paper, Phys. Rev. 91, X

5 5 R ij j = X i, i = 1 n;. j L ij X j = i,.3 j L R Ohm Fourier Fick R ij = R ji [R = R tr ],.4 tr transpose Ṡ = j S/ j j = j X j j,.1.5 = i,j R ij i j,..6 = i,j L ij X i X j..3.7 Φd /dt, d /dt = 1 R ij i j,.8 i,j Φ X, X = 1 L ij X i X j,.9 Φd /dt, d /dt i,j

6 6 L. S. S = S 0 1 s ij i j + higher terms..10 i,j Prob.{ } exps/k exp 1 1 s ij i j..11 k i,j X i = j s ij j..1. R ij j + s ij j = j R ij j = X i + ϵ i, 3.1 j ϵ X i 3.1 t t t

7 7 p < t < t p cumulative distribution function, c. d. f. F p 1 p t t p = Prob.{t k k, k = 1, p}. 3. c.d.f. 11 c. d. f. τ, F p 1 p = F p 1 p + τ t p + τ. 3.3 F p t,, t p 1,, p k gates 1 µ 1,,µ p =0 1 q F 1 + µ 1 1 p + µ p p p F p 1 p 1 p, 3.4 p q = p µ k. k=1 11, Harald Cramér, Mathematical Methods of Statistics, Princeton University Press, 1946

8 8 L. S. F p p.d.f. p p + 1 p.d.f. F 1 p+1 t p+1 1 p = Prob.{t p+1 = p+1 t k = k, k = 1, p}, 3.5 F p+1 1 p+1 +1 = 1 p p+1 p-fold F 1 t p+1 x1 x p df p x 1 x p. 3.6 F 1 p+1 t p+1 1 p = F 1 p+1 t p+1 p t p f p p p = f p t p p 1 t p 1 p t p t 1 p 1 t p p.d.f..11 p.d.f. t + τ 1 t 3.9 t p

9 9 3.1 ϵ i ϵ i ϵtϵt + τ = 0, τ 0. p.d.f. Sec.3 Khinchin Cramér 1 L 3... RRIP II i t i t + τ i i, t + τ, t Av = i t + τ d, 3.10 t. Callen Greene 13 1 A. Khinchin, Math. Ann. 109, H. B. Callen and R. F. Greene, Phys. Rev. 86, ; 88,

10 10 L. S. Doop covariant Aτ tr t t + τ Aτ 1 Aτ = Aτ 1 + τ, 3.11 RC. Sec R + s = ϵ, 4.1 p.d.f. t + τ 1 t { = 1 1 exp 1 π k s 1 e γτ s k } [ e γτ 1 ] 1 e γτ 4. γ = s/r 4..11, τ 1 e γτ 3.8 t, t + τ p p 1 = t, t = t + t,, t p+1 = t + τ = t + p t 14, V Sec.8. p+1 t p+1 1

11 = p 1-fold p+1 t p+1 11 p t p t 1 d d p. 4.3 Chapman Kolmogoroff 15 p 1 t 4.1 k λ k 1 = y k, 4.4 λ = 1 τ, y k = ϵt k τ/r y σy p+1 p 1-fold { exp 1 σy t p+1 1 [ p+1 λ p + + λ 1 ]} d d p ,, p p+1 t p+1 1 { exp 1 [ σy p+1 λ p + + λ 1 ]} ; 4.6 p 4.6 = 1,, t p+1 = p+1 p+1 t p+1 1 { exp 1 1 k tp+1 } R [ t + γt] dt. 4.7 min p.d.f A. Kolmogoroff, Math. Ann. 104,

12 1 L. S. t 1 = max, γ = 0, = p.d.f. = 1 = 0 t 0 [ exp 1 1 k s ]..11 L, = R[ γ] Euler Lagrange d L dt L = 0, 4.10 γ = 0, 4.11 e γt e γt t = t = t t = e γt = 0, t = 1 t R [ t + γt] dt 4 min = s R ϵ i [ 3.9] n * 4.7 L p.d.f. p.d.f. 4.7

13 13 R[ t + γt] = R + 1/Rs + d/dts, 4.13 γ = s/r S = S 0 1 s i i, [.10 ], 4.14 i Φd /dt, d /dt = 1 R i i, [.8 ], 4.15 i Ψ X, X = 1 1/R i Xi = 1 1/R i s i i, [.9 ] i i R i [ i + γ i ] = Φd /dt, d /dt + Ψ X, X d/dts, 4.17 i p.d.f.4.7 = 1, t = { exp k t f n t 1 = f i 1 t 1 i i [Φ d /dt, d /dt + Ψ X, X ddt ] } S dt min Φ Ψ X p.d.f. = 1, t = 1 f n t [ { 1 1 exp k S S 1 4 = f n t t 1 f n 1 [ Φ d /dt, d /dt + Ψ X, X ] }] dt. min 4.19 p.d.f

14 14 L. S. = 1, t =,, t p = p [ { 1 1 exp k S S p 1 tp 4 1 f p pn [ Φ d /dt, d /dt + Ψ X, X ] }] dt. min = 0 =, 1 = 1 1 t1 [ Φ d /dt, d /dt + Ψ X, 4 ] X dt min = 1 S 1 + const t t.6.7 t = Φ d /dt, d /dt + Ψ X, X = Ṡ, t p, 4.0 = 1, t =,, t p = p 1 f p pn [ exp 1 1 [ Φ d /dt, d /dt + Ψ X, 4 k ] ] X dt. 4.3 min X i II, Sec.3

15 p.d.f RRIP I Rayleigh t t = t = 1 1 [Φ d /dt, d /dt + Ψ X, 4 k X ddt ] S t = max. 4.4 Ṡ Φ d /dt, d /dt = max. 4.5 [.4.] Callen 19 R 18 Lord RayleighJ. W. Scott, Phil. Mag. 6, , H. Takasi, J. Phys. Soc. Japan 7,

16 16 L. S. Fourier

.2 ρ dv dt = ρk grad p + 3 η grad (divv) + η 2 v.3 divh = 0, rote + c H t = 0 dive = ρ, H = 0, E = ρ, roth c E t = c ρv E + H c t = 0 H c E t = c ρv T

.2 ρ dv dt = ρk grad p + 3 η grad (divv) + η 2 v.3 divh = 0, rote + c H t = 0 dive = ρ, H = 0, E = ρ, roth c E t = c ρv E + H c t = 0 H c E t = c ρv T NHK 204 2 0 203 2 24 ( ) 7 00 7 50 203 2 25 ( ) 7 00 7 50 203 2 26 ( ) 7 00 7 50 203 2 27 ( ) 7 00 7 50 I. ( ν R n 2 ) m 2 n m, R = e 2 8πε 0 hca B =.09737 0 7 m ( ν = ) λ a B = 4πε 0ħ 2 m e e 2 = 5.2977