1. ( ) L L L Navier-Stokes η L/η η r L( ) r [1] r u r ( ) r Sq u (r) u q r r ζ(q) (1) ζ(q) u r (1) ( ) Kolmogorov, Obukov [2, 1] ɛ r r u r r 1 3

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1 Kolmogorov Toward Large Deviation Statistical Mechanics of Strongly Correlated Fluctuations - Another Legacy of A. N. Kolmogorov - Hirokazu FUJISAKA Abstract Recently, spatially or temporally strongly correlated fluctuations are observed in many different contexts such as price fluctuations in economic dynamics as well as, e.g., turbulence, intermittency in coupled chaotic systems. They are ubiquitous in nonlinear, nonequilibrium systems. Statistical mechanics so far developed for statistically independent or weakly correlated fluctuations faces the problem, how to construct the emergency mechanism as well as to characterize the fundamental statistics of strongly correlated fluctuations. They often exhibit self-similarity characteristics. The aim of the present paper is to suggest the possibility of constructing statistical mechanics for strongly correlated fluctuations by proposing a unified approach to several kinds of examples from the phenomenological viewpoint based on the large devition statistics in the probability theory. Keywords Turbulence, On-off intermittency, Price fluctuations, Strongly correlated fluctuations, Self-similarity, Multifractals 1

2 1. ( ) L L L Navier-Stokes η L/η η r L( ) r [1] r u r ( ) r Sq u (r) u q r r ζ(q) (1) ζ(q) u r (1) ( ) Kolmogorov, Obukov [2, 1] ɛ r r u r r 1 3 ɛ 1 3 r Sq(r) ɛ ɛ q r r τ(q) (2) 2

3 ζ(q) = q 3 + τ ( q 3 ) (3) [1, 3, 4] 1941 Kolmogorov (7) ɛ r r (ɛ r ɛ L ) τ(q) = min [ qz] (9) τ(q) = 0 ζ(q) = q 3 z E(k) ɛ 2 3 L k 5 3, (L 1 Kolmogorov(1962) k η 1, Kolmogorov ) (K62) = 1 2µ (z + µ 2 )2 K41 u r [2] (3) [5] µ ɛ r r 1 r 2 r 3 Taylor r P (ɛ j, r j ɛ k, r k ) r k [5] SL ɛ r ɛ k r j She-Leveque ɛ j ) η r 3 [3] r 2 r 1 L τ(q) = 2 ( ) q ] 2 [1 3 q + 2 (10) 3 P (ɛ 3, r 3 ɛ 1, r 1 ) = P (ɛ 3, r 3 ɛ 2, r 2 )P (ɛ 2, r 2 ɛ 1, r 1 )dɛ 2 (4) [4] K62 [2] P (7) P (ɛ j, r j ɛ k, r k ) ɛ 1 j ( ) S(z(ɛj,r j ɛ k,r k )) rk r j z(ɛ j, r j ɛ k, r k ) = P r (ɛ) ɛ 1 ( L r (5) ln ɛj ɛ k ln r k rj (6) Kolmogorov(1962) r k = L, r j = z r r r = L ɛ L ɛ r ( ) S(zr(ɛ)) )[5], (7) z r (ɛ) = ln ɛ ɛ L ln L r (8) [5] u r r 1 3 ɛ 1 3 r τ(q) Legendre 2 3 = 2 + z ( 2 3 ln 3 ln z ) 2 2e ln 3 (11) 2 (4) ɛ r r z r (12) z r (Extended Self- Similarity, ESS) 3

4 r/η =19 A t = 1 t t 0 r s ds (13) t T τ t T t T τ τ t T A q t t φ(q) (14) z 1: n t n = e n T, (15) (Generalized Extended Self-Similarity, GESS)[6] (n = 0, 1, 2, 3,, N(= ln τ 1 )) [5, 7] 3. A tn+1 A tn = e zn (16) z n z n n ( ) z j N A tn = A T e n z n, (17) z n 1 n 1 z j (18) n n z n [9] Q n (z) e n (19) A T A [9] T A t P r t t (a) ( ) T S(zt (a)) T P t (a) a 1, (20) t j=0 z t (a) = ln a/a T ln T/t (21) 4

5 4. 2 Ising ξ T T c ν ξ = ψ 2 G(r) ψ(r + r 0 )ψ(r 0 ) r (d 2+η), (24) z d η Fisher Ising r 2: m τ = 1, T = r (x 0 ) = 1 ψ(x 0 + r)dr, (25) V r r <r [10] φ(q) φ(q) = min z [ qz] (22) (V r r d ) q µ q (r) m q r r φ(q), (26) φ(q) 2 m r r t+1 = r t exp (λ r t + f t ), (23) (λ > 0, f t : ) µ 2 (r) r (d 2+η) z φ(2) = (d 2 + η) (27) (0 r t ) 2 Onsager φ(2) = η = 1 4 ( ) z m r r z r (28) [10] z r z r z r Q r (z) r S(z r(m)), (29) z r (m) = ln m m 0 ln r a (30) 5

6 z r r=64 m 0 a m 0 = 1 m r ( P r (m) m 1 r ) S(zr(m)), (31) a ( r a, a ) φ(q) = min z [ qz] z r : Ising φ(2)/2 z r z r 3 2 Ising ( ) Monte Carlo [14] P (s) s t r t (s) = ln P (s) (32) P (s t) 2 t r t (s) t 2 t( 1 ) [11, 12] r t r t volatility clustering ( ) (New York ) t s (1 ) [13] T ( 1 ) z r=32 r=128 6

7 t/ t 1 y t (s) r t (s k t) = r t (s) (33) k=0 20 t= n t = t n T e n, (n = 0, 1, 2,, N, N ln(t/t s )) y n = 14 y tn r t (s) 17 0 t=20 t= y n+1 y n = e z n (34) s (10 4 samples) z n z n n 4: y t t z(t) (35) z(t) z(t n ) = n 1 n 1 j=0 z j z j z t (y) = ln y y T ln T t (36) (T/t) 40 y t P t (y) y 1 ( T t ) S(zt(y)) (37) 5 (NYSE, New York ) S (37) t s t T 5: t t t = z 7

8 6. Kolmogorov - - ( ) [15] Tsallis [16] (a),, (b) z (c) Andrei Nikolaevich Kolmogorov ( ) (K41 ) 1962 Kolmogorov Frisch Kolmogorov ([8] ) (The Legacy of A. N. Kolmogorov) 8

9 [1] Kolmogorov [17] 30 ( ) (Kolmogorov ) Appendix Kolmogorov (K62 ) 30 (ζ(q) ) ( Navier-Stokes ) 1970 K62 (η r Kolmogorov L) Reynolds Re L/η = Re 3 4 Reynolds Kolmogorov 7. : ESS GESS 1993 Benzi (1) ζ(q) Sq u (r) Sp u (r)(p ) Sq u (r) [ Sp u (r) ] α(q p), (38) α(q p), r Langevin 9

10 α(q p) = ζ(q) ζ(p) (39) [ ] q [ Sq u (r) u q r ζ(q) L f(r) 1(r)] L g, (46) (Extended GESS r = η L Self-Similarity, ESS) α(q p) ζ(3) = 1 p = 1 ESS, ζ(q) GESS ESS (39) [ [ Sq u (r) u q r ζ(q) L 1(r)] L g Sq(r) ɛ ɛ q r τ(q), (ul = (Lɛ L ) 1 L 1(r)] 3 ) (40) L g, (47) (2) g 1 (r) η r L 1 r = η L 2 (47) 1 ESS η < r 3 < r 2 < L r 1 < L (4) ESS ESS ˆr = r g 1 (r), (48) S ɛ q(r) [ S ɛ p(r) ] β(q p), (41) β(q p) = τ(q) τ(p), (42) ( ˆr P r (ɛ) ɛ 1 L Benzi (1996) ESS zˆr (ɛ) ln ɛ (Generalized Extended Self-Similarity, GESS) (47) τ(q) (9) G q,p (r) Su q (r) ( S u p (r) ). (43) q/p Reynolds G q,p (r) [G q,p (r)]γ(q,p q,p ), (44) γ(q, p q, p ) r q, p, q, p (44) Sq u (r) (dˆr/dr > 0) (4) ) S(zˆr (ɛ)), (49) [1] U. Frisch, Turbulence: The Legacy of A. q γ(q, p q, p pζ(p) ζ(q) ) = (45) N. Kolmogorov, (Cambridge Univ. Press, q p ζ(p ) ζ(q ) Cambridge, 1995). ɛ L ln Ḽ r, (50) (r = η, L) 10

11 [2] A. N. Kolmogorov, J. Fluid Mech. 13, 82 (1962), A. M. Obukov, J. Fluid Mech. 13, 77 (1962). [3] Z.-S. She and E. Leveque, Phys. Rev. Lett. 72, 336 (1994). [4] T. Watanabe and H. Fujisaka, J. of Phys. Soc. Jpn 69, 1672 (2000) (1989). 4 ( 1998) 54 6, 423(1999) No.436, 29 (1999). C. Beck and F. Schlögl, Thermodynamics of Chaotic Systems, An Introduction, (Cambridge University Press, 1993) ( 1997) [5] H. Fujisaka, Y. Nakayama, T. Watanabe and S. Grossmann, Scaling hypothesis leading to generalized extended selfsimilarity in turbulence, Phys. Rev. E 65, (2002). [6] ESS(Extended Self-Similarity): R. Benzi et al., Phys. Rev. E 48, R29 (1993). R. Benzi et al., Physica D 80, (1995). GESS(Generalized Extended Self- Similarity): R. Benzi et al., Phys. Rev. E 53, R3025 (1996). R. Benzi et al., Physica D 96, 162 (1996). [7] H. Fujisaka and S. Grossmann, Phys. Rev. E 63, (2001). [8] [11] W. Stroock, An Introduction to the Theory of Large Deviations, (Springer, Berlin,,, 56 27pZC ), P. S. Ellis, Entropy, Large Deviations, and [12] Statistical Mechanics, (Springer, Berlin, N. Ito and M. Suzuki, Prog. Theor. Phys. 1985), 77, 1391 (1987). A. D. Wentzell, Limit Theorems on Large [13] : Deviations for Markov Stochastic Processes, (Kluwer Academic, Dortrecht and ( ) ( 2000). London, 1990). P. Gopikrishnan et al., Physica A 287, 362 (2000). : 11, 322 (2001). H. Fujisaka and M. Inoue, Prog. Theor. Phys. 77, 1334 (1987); Phys. Rev. A 39, [9] : 51 11, 813 (1996). 9 1, 28 (1999); No.462, 47 (2001).. : J. Becker et al., Phys. Rev. E 59, 1622 (1999). T. John, R. Stannarius and U. Behn, Phys. Rev. Lett. 83, 749 (1999). [10] : H. Fujisaka, H. Suetani and T. Watanabe, Prog. Theor. Phys. Suppl., No. 139, 70(2000). 11

12 [14] Y. Fujiwara and H. Fujisaka, Physica A, 294, Issue 3-4, 439 (2001). H. E. Stanley et al., Physica A 302, 126 (2001). [15] D. C. Lin and R. L. Hughson, Phys. Rev. Lett. 86, 1650 (2001). Y. Ashkenazy et al., Phys. Rev. Lett., 86, 199 (2001). T. H. Mäkikallio et al., J. of the American College of Cardiology 37, 1395 (2001). [16] Tsallis : Nonextensive Statistical Mechanics and Its Applications, eds. S. Abe and Y. Okamoto, (Springer-Verlag, Berlin, 2001). [17] H. Mori and H. Fujisaka, Prog. Theor. Phys. 49, 764 (1973). 12

,, Andrej Gendiar (Density Matrix Renormalization Group, DMRG) 1 10 S.R. White [1, 2] 2 DMRG ( ) [3, 2] DMRG Baxter [4, 5] 2 Ising 2 1 Ising 1 1 Ising

,, Andrej Gendiar (Density Matrix Renormalization Group, DMRG) 1 10 S.R. White [1, 2] 2 DMRG ( ) [3, 2] DMRG Baxter [4, 5] 2 Ising 2 1 Ising 1 1 Ising Model 1 Ising 1 Ising Model N Ising (σ i = ±1) (Free