L. S. Abstract. Date: last revised on 9 Feb 01. translated to Japanese by Kazumoto Iguchi. Original papers: Received May 13, 1953. L. Onsager and S. Machlup, Fluctuations and Irreversibel Processes, Physical Review 91 1953 pp. 1505 151. Ph. D. S. M.. 1
L. S. 1. 1 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, 405 1931; 38, 65 1931. RRIP I II
3 3 4-7 II, pp.65 74 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, 33 1945. J. E. Moyal, J. Roy. Statist. Soc. B11, 150 1849 J. L. Doob, Stochastic Processes J. Wiley and Sons, New York, 1953 4 H. B. Casimir, Revs. Modern Phys. 17, 343 1945. 5 A. Einstein, Ann. Physik 33, 175 1910, Sec.I, General Matters Relating to Boltzmann s Principle. 6 RRIP II, Sec.5 Fokker-Planck N. Hashitsume, Prog. Theoret. Phys. 8, 461 195 7 A. N. Kolmogoroff, Foundations of the Theory of Probability, Chelsea Publishing Company, New York, 1950, Chap III, Sec.4 Borel
4 L. S. 8 3 9 S = S 1,, n = S. S 0 S 0 = S0,, 0. X i = S/ i,.1 10 8 P. W. Bridgeman, Rev. Modern Phys., 56 1950. 9 S. Machlup and L. Onsager, following paper, Phys. Rev. 91, 151 1953. 10.5 X
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 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 = 0..13 j 3. 3.1... R ij j = X i + ϵ i, 3.1 j ϵ X i 3.1 t t t
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 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. 3.7 3.6 3.7 1 f p p p = f p t p p 1 t p 1 p t p t 1 p 1 t p 1 1. 3.8 p.d.f..11 p.d.f. t + τ 1 t 3.9 t p
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, 604 1933. 13 H. B. Callen and R. F. Greene, Phys. Rev. 86, 70 195; 88, 1387 195.
10 L. S. Doop 14.11.13 covariant Aτ tr t t + τ Aτ 1 Aτ = Aτ 1 + τ, 3.11 RC. Sec. 4.1.. 4. 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
= 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 4.3 16 p+1 p 1-fold { exp 1 σy t p+1 1 [ p+1 λ p + + λ 1 ]} d d p. 4.5 4.,, 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. 1 15 A. Kolmogoroff, Math. Ann. 104, 415 1931. 16 4.7
1 L. S. t 1 = max, 4.8 4.7 + γ = 0, 4.9.13 = 1 4.7 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 = 1. 4.1 4... s R ϵ i [ 3.9] n * 4.7 L p.d.f. p.d.f. 4.7
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 ]. 4.16 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 1 4 1 k t f n t 1 = f i 1 t 1 i i [Φ d /dt, d /dt + Ψ X, X ddt ] } S dt. 4.18 min Φ Ψ X p.d.f. = 1, t = 1 f n t [ { 1 1 exp k S 1 + 1 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. 4.18 3.8
14 L. S. = 1, t =,, t p = p [ { 1 1 exp k S 1 + 1 S p 1 tp 4 1 f p pn [ Φ d /dt, d /dt + Ψ X, X ] }] dt. min 4.0 4.0 = 0 =, 1 = 1 1 t1 [ Φ d /dt, d /dt + Ψ X, 4 ] X dt min = 1 S 1 + const. 4.1 4.10.13.13 t t.6.7 t = Φ d /dt, d /dt + Ψ X, X = Ṡ, 4. 4.1 17 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 4.3 17 II, Sec.3
15 4.3.. 4.3 p.d.f. 4.3 4.4.. RRIP I Rayleigh 18 4.18 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.] 4.5.. Callen 19 R 18 Lord RayleighJ. W. Scott, Phil. Mag. 6, 776 1913. 19 13, H. Takasi, J. Phys. Soc. Japan 7, 439 195.
16 L. S. Fourier