HOME    |    BROWSE    |    SEARCH    |    SUBSCRIPTION   |    SUBMISSION    |    REGISTRATION

Quick Search:
Author: Title/Abstract: Vol./No: Page:
PTP Online Top > Volume 52 (1974) > Number 2
| Next Article >

Prog. Theor. Phys. Vol. 52 No. 2 (1974) pp. 433-452

[ Full Text PDF : FREE ACCESS (1527K) ]

Dynamics of Macrovariables for Nonuniform Systems

— Scaling Theory —

Hazime Mori

Department of Physics, Kyushu University, Fukuoka

(Received February 20, 1974)

Abstract:

A scale transformation of the nonequilibrium macroscopic system to larger similar systems is introduced to find kinetic equations for the evolution and fluctuation of the macrovariables. In the scale transformation, we postulate that the probability distribution for the fluctuation of the macroscopic degrees of freedom and the quantities determined by the microscopic degrees of freedom par unit volume are invariant. The characteristic length of the macroscopic state l, the macroscopic state variables yk and their fluctuation variables zk are transformed by lL = Ll, (L ≫1), ykL = L-αyk and zkL = L-βzk, respectively. The probability distribution then takes the form P({ zklβ }, { ql }, Ω/ld, t/lθ), where q, Ω, d and t denote the wave vectors, volume, dimensionality and time, respectively. If α< β, then the master equation is reduced to a linearly generalized Fokker-Planck equation with time-dependent coefficients and the probability distribution is normal around the mean evolution. If αβ, then the nonlinear drift terms are important and a renormalization of kinetic coefficients must be done to determine the mean evolution. For the isotropic Heisenberg ferromagnets near the Curie point, α= β= (d - 2 + η)/2 and θ= (d + 2 - η)/2, where η is the correlation critical exponent. For the isotropic homogeneous turbulence, α= 1, β= -1/3 and θ= 2/3, where Kolmogorov's spectrum is assumed. For example, this indicates that the turbulent viscosity has the form q-4/3V(ωq-2/3), ω being the frequency.


URL : http://ptp.ipap.jp/link?PTP/52/433/
DOI : 10.1143/PTP.52.433

[ Full Text PDF : FREE ACCESS (1527K) ] Citation:

References:

  1. H.Mori, H. Fujisaka and H. Shigematsu, Prog. Theor. Phys. 51 (1974), 109[PTP]. This is referred to as I.
  2. N. G. van Kampen, Can. J. Phys. 39 (1961), 551; in Fluctuation Phenomena in Solids, ed. R. E. Burgess (Academic Press, New York, 1965).
  3. R. Kubo, K. Kitahara and K. Matsuo, J. Stat. Phys. 9 (1973), 51[CrossRef].
  4. L. P. Kadanoff, W. Gotze, D. Hamblen, R. Hecht, E.A.S. Lewis, V. V. Palciauskas, M. Rayl. J. Swift, D. Aspnes and J. Kane, Rev. Mod. Phys. 39 (1967), 395[APS].
  5. L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Pergamon Press, London-Paris, 1960), Chap. III.
  6. M. Nelkin, Phys. Rev. Lett. 30 (1973), 1029[APS]; Phys. Rev. A 9 (1974), 3880[APS].
    R. H. kraichnan, Phys. Rev. Lett. 31 (1973), 744[APS].
  7. K. Kawasaki, Ann. of Phys. 61 (1970), 1[CrossRef].
  8. Y. Kuramoto, Prog. Theor. Phys. 51 (1974), 1712[PTP].
  9. H. Mori, Prog. Theor. Phys. 49 (1973), 1516[PTP].
  10. H. Mori and H. Fujisaka, Prog. Theor. Phys. 49 (1973), 764[PTP].
  11. P. Résibois and C. Piette, Phys. Rev. Lett. 24 (1970), 514[APS].
    P. Résibois and G. Dewel, Ann. of Phys. 69 (1972), 299[CrossRef].
  12. B. I. Halperin and P. C. Hohenberg, Phys. Rev. Lett. 19 (1967), 700[APS].
    M. Nelkin, Phys. Rev. 177 (1969), 952[APS].
  13. L. Van Hove, Phys. Rev. 95 (1954), 1374[APS].
  14. M. Suzuki, Prog. Theor. Phys. 51 (1974), 1257[PTP].
  15. B. I. Halperin, P. C. Hohenberg and S. Ma, Phys. Rev. Lett. 29 (1972), 1548[APS].
    M. Suzuki and G. Igarashi, Prog. Theor. Phys. 49 (1973), 1070[PTP].
  16. R. A. Ferrell, N. Menyhard, H. Schmidt, F. Schwabl and P. Szé pfalusy, Phys. Rev. Lett. 18 (1967), 891[APS]; Ann. of Phys. 47 (1968), 565[CrossRef].

Citing Article(s) :

  1. Journal of the Physical Society of Japan 61 (1992) pp. 4367-4380 :
    A Unified Approach to Nonequilibrium Phenomena in a Crystalline Lattice. III. Typical Examples of the Linear Theory
    Toshio Fukuta and Masaru Sugiyama
  2. Progress of Theoretical Physics Vol. 53 No. 6 (1975) pp. 1617-1640 :
    Stochastic Processes of Macroscopic Variables
    Hazime Mori
  3. Progress of Theoretical Physics Vol. 53 No. 6 (1975) pp. 1657-1674 :
    Statistical Mechanics of Non-Equilibrium Systems. I
    Masuo Suzuki
  4. Progress of Theoretical Physics Vol. 54 No. 1 (1975) pp. 60-71 :
    Asymptotic Evaluation of Fluctuation of Macrovariables near Instability Points
    Yoshiki Kuramoto and Toshio Tsuzuki
  5. Progress of Theoretical Physics Vol. 54 No. 2 (1975) pp. 370-383 :
    On Extensive Properties of Probability Distribution Functions in Non-Equilibrium States
    Hiroshi Furukawa
  6. Progress of Theoretical Physics Vol. 54 No. 4 (1975) pp. 1050-1066 :
    Fluctuations and Phase Transitions Far from Equilibrium
    Hiroyuki Mashiyama, Akira Itō and Takao Ohta
  7. Progress of Theoretical Physics Vol. 55 No. 2 (1976) pp. 383-399 :
    Statistical Mechanics of Non-Equilibrium Systems. II
    Masuo Suzuki
  8. Progress of Theoretical Physics Vol. 56 No. 2 (1976) pp. 477-493 :
    Scaling Theory of Non-Equilibrium Systems near the Instability Point. II
    Masuo Suzuki
  9. Progress of Theoretical Physics Vol. 56 No. 2 (1976) pp. 498-514 :
    Kinetic and Hydrodynamic Scalings in an Exactly-Soluble Model for the Brownian Motion
    Terumitsu Morita and Hazime Mori
  10. Progress of Theoretical Physics Vol. 56 No. 3 (1976) pp. 754-772 :
    Reduction of Variables and Processes in Macrodynamics
    Hirokazu Fujisaka and Hazime Mori
  11. Progress of Theoretical Physics Vol. 57 No. 3 (1977) pp. 734-745 :
    Theoretical Study of a Chemical Turbulence
    Hirokazu Fujisaka and Tomoji Yamada
  12. Progress of Theoretical Physics Vol. 57 No. 3 (1977) pp. 770-784 :
    Critical Dimensionality for Normal Fluctuations of Macrovariables in Nonequilibrium States
    Hazime Mori and Kenneth J. McNeil

[PTP HOME] [BROWSE] [SEARCH] [SUBSCRIPTION] [SUBMISSION] [REGISTRATION]

Copyright © Progress of Theoretical Physics 2013 All rights reserved.