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Prog. Theor. Phys. 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

Department of Physics, Kyushu University, Fukuoka 812

(Received March 1, 1976)

Abstract:

The macroscopic system exhibits hierarchical dynamic processes with different time scales, each level of which has a characteristic scaling. This important feature is rigorously investigated for the Brownian motion of a heavy impurity in a harmonic linear chain in thermal equilibrium which can treated analytically in the full context. The model considered is a slightly-extended one-dimensional Rubin's model with an arbitrary force constant K' between the impurity and its neighboring particles. It is shown that the characteristic scaling of the hydrodynamic process is x →Lx and t →L²t (L ≫1), x being the position of the Brownian particle, and leads to the diffusion equation. When the Brownian particle is an isotope with mass M, the kinetic process is characterized by the scaling M →L²M and t →L²t, and the asymptotic form of the probability density in µ space agrees with that of the phenomenological theory of the Brownian motion of free particles. When the Brownian particle is an impurity, the scaling K' →K'/L², M →L²M and t →L²t leads to a new non-Markov kinetic process. Thus it is shown for this model that each of the kinetic and the hydrodynamic process has a characteristic scaling, and the macroscopic scale invariance rigorously holds.

URL : http://ptp.ipap.jp/link?PTP/56/498/
DOI : 10.1143/PTP.56.498

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References:

1. H. Mori, Prog. Theor. Phys. 52 (1974), 433[PTP]; ibid. 53 (1975), 1617[PTP].
2. R. J. Rubin, J. Math. Phys. 1 (1960), 309[CrossRef]; ibid. 2 (1961), 373[CrossRef].
3. H. Mori, Prog. Theor. Phys. 33 (1965), 423[PTP].
4. S. Chandrasekhar, Rev. Mod. Phys. 15 (1943), 1[APS].
   M. C. Wang and G. E. Uhlenbeck, Rev. Mod. Phys. 17 (1945), 323[APS].
5. H. Nakazawa, Prog. Theor. Phys. Suppl. No. 36 (1966), 172 [PTP]and references cited therein.
6. H. Mori, H. Shigematsu and M. Tokuyama, Prog. Theor. Phys. 55 (1976), 627[PTP].
7. S. Takeno, S. Kashiwamura and E. Teramoto, Prog. Theor. Phys. Suppl. No. 23 (1962), 124[PTP].
8. M. J. Lighthill, Introduction to Fourier Analysis and Generalized Functions (Cambridge University Press, 1958).

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. 56 No. 4 (1976) pp. 1073-1092 :

   Statistical-Mechanical Theory of the Boltzmann Equation and Fluctuations in µ Space

   Michio Tokuyama and Hazime Mori

3. 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

4. Progress of Theoretical Physics Vol. 59 No. 5 (1978) pp. 1493-1510 :

   Kinetic Equations of Dilute Electron Plasmas in the Coherent Region

   Terumitsu Morita, Hazime Mori and Michio Tokuyama

5. Progress of Theoretical Physics Vol. 61 No. 3 (1979) pp. 850-863 :

   A New Continued-Fraction Representation of the Time-Correlation Functions of Transport Fluxes

   Takashi Karasudani, Katsuhiko Nagano, Hisao Okamoto and Hazime Mori

6. Progress of Theoretical Physics Vol. 66 No. 1 (1981) pp. 53-67 :

   Anomalous Relaxation in One-Dimensional Magnets. I

   Hisao Okamoto, Katsuhiko Nagano, Takashi Karasudani and Hazime Mori

7. Progress of Theoretical Physics Supplement No.64 (1978) pp. 50-64 :

   Scalng for the Space-Time Coarse Graining and Kinetic Equations

   Hazime Mori, Michio Tokuyama and Terumitsu Morita

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