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Prog. Theor. Phys. Vol. 33 No. 3 (1965) pp. 423-455

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Transport, Collective Motion, and Brownian Motion

Hazime Mori

Research Institute for Fundamental Physics, Kyoto University, Kyoto

(Received October 9, 1964)

Abstract:

A theory of many-particle systems is developed to formulate transport, collective motion, and Brownian motion from a unified, statistical-mechanical point of view. This is done by, first, rewriting the equation of motion in a generalized form of the Langevin equation in the stochastic theory of Brownian motion and then, either studying the average evolution of a non-equilibrium system or calculating the linear response function to a mechanical perturbation. (1) An expression is obtained for the damping function φ(t), the real part of whose Laplace transform gives the damping constnat of collective motion. (2) A general equation of motion for a set of dynamical variables A(t) is derived, which takes the form
(d/dt)A(t) - i\hatω ·A(t) + \intt0φ(t-s)·A(s)ds = f(t),
where \hatω is a frequency matrix determining the collective oscillation of A(t). The quantity f(t) consists of those terms which are either non-linear in A(s), t ≧s ≧0, or dependent on the other degrees-of-freedom explicitly, and its time-correlation function is connected with the damping function φ(t) by (f(t1), f(t2)*) = φ(t1 - t2)·(A, A*). (3) An expression is obtained for the linear after-effect function to thermal disturbances such as temperature gradient and strain tensor. Both the conjugate fluxes and the time dependence differ from those of the mechanical response function. The conjugate fluxes are random parts of the fluxes of the state variables, thus depending on temperature. (4) The difference in the time dependence arises from a special property of the time evolution of f(t) and ensures that the damping function and the thermal after-effect function are determined by the microscopic processes in strong contrast to the mechanical response function. The difficulty of the plateau value problem in the previous theories of Brownian motion and transport coefficients is thus removed. (5) The theory is illustrated by dealing with the motion of inhomogeneous magnetization in ferromagnets and the Brownian motion of the collective coordinates of fluids. (6) Explicit expressions are derived for the thermal after-effect functions and the transport coefficients of multi-component systems.


URL : http://ptp.ipap.jp/link?PTP/33/423/
DOI : 10.1143/PTP.33.423

[ Full Text PDF : FREE ACCESS (2026K) ] Citation:

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    A New Continued-Fraction Representation of the Time-Correlation Functions of Transport Fluxes
    Takashi Karasudani, Katsuhiko Nagano, Hisao Okamoto and Hazime Mori
  113. Progress of Theoretical Physics Vol. 62 No. 1 (1979) pp. 37-53 :
    Dynamical Structure Factor and Elementary Excitations in Normal Liquid He3
    Fukuo Yoshida and Shozo Takeno
  114. Progress of Theoretical Physics Vol. 62 No. 1 (1979) pp. 70-90 :
    Study of Langevin Type Equations by Means of a New Projection Operator Method in Nonequilibrium States
    Hiroshi Furukawa
  115. Progress of Theoretical Physics Vol. 62 No. 4 (1979) pp. 883-900 :
    Dynamical Structure Factor and Collective Modes in Classical One-Component Plasmas
    Shozo Takeno and Fukuo Yoshida
  116. Progress of Theoretical Physics Vol. 63 No. 1 (1980) pp. 62-71 :
    On the Fokker-Planck Dynamics Approach to Langevin Equations
    J. M. Sancho and M. San Miguel
  117. Progress of Theoretical Physics Vol. 63 No. 3 (1980) pp. 772-784 :
    Orbital Dynamics in 3He-A
    Haruo Takagi and Kazumasa Miyake
  118. Progress of Theoretical Physics Vol. 63 No. 6 (1980) pp. 1904-1916 :
    Reduced Equations of Motion for Generalized Fluxes and Forces in the Continued-Fraction Expansion
    Katsuhiko Nagano, Takashi Karasudani, Hisao Okamoto and Hazime Mori
  119. Progress of Theoretical Physics Vol. 64 No. 1 (1980) pp. 40-53 :
    Local-Field Correction and Dynamical Structure Factor of Electron Liquids
    Fukuo Yoshida, Shozo Takeno and Hiroshi Yasuhara
  120. Progress of Theoretical Physics Vol. 64 No. 2 (1980) pp. 448-465 :
    Dynamical Properties of the Three-Dimensional XY Model at Low Temperatures
    Tosizumi Aoki, Shigeo Homma and Huzio Nakano
  121. Progress of Theoretical Physics Vol. 64 No. 6 (1980) pp. 1893-1901 :
    Frequency Spectrum of Velocity Auto-Correlation Function of Liquid Rubidium
    G. S. Dubey, V. K. Jindal and K. N. Pathak
  122. 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
  123. Progress of Theoretical Physics Vol. 66 No. 2 (1981) pp. 437-453 :
    Anomalous Relaxation in One-Dimensional Magnets. II
    Hisao Okamoto, Takashi Karasudani, Katsuhiko Nagano and Hazime Mori
  124. Progress of Theoretical Physics Vol. 66 No. 3 (1981) pp. 759-771 :
    Dynamic Properties of One-Dimensional Harmonic Liquids. I
    Takeshi Yoshida, Kazuhisa Shobu and Hazime Mori
  125. Progress of Theoretical Physics Vol. 66 No. 3 (1981) pp. 861-878 :
    Collective Modes in the Quantum Lattice or Three-Dimensional XY Model. I
    Toshizumi Aoki, Shigeo Homma and Huzio Nakano
  126. Progress of Theoretical Physics Vol. 67 No. 5 (1982) pp. 1332-1352 :
    Real-Space Renormalization Group Approach to the Kinetic Ising Model
    Hiroshi Takano and Masuo Suzuki
  127. Progress of Theoretical Physics Vol. 67 No. 6 (1982) pp. 1724-1739 :
    Collective Modes in Quantum Lattice or Three-Dimensional XY Model. II
    Tosizumi Aoki, Shigeo Homma and Huzio Nakano
  128. Progress of Theoretical Physics Vol. 67 No. 6 (1982) pp. 1756-1775 :
    Critical Slowing Down in Stochastic Processes. I
    Masuo Suzuki, Kunihiko Kaneko and Shinji Takesue
  129. Progress of Theoretical Physics Vol. 68 No. 2 (1982) pp. 388-401 :
    Molecular Theory of Hydrodynamic Equations for Nematic Liquid Crystals
    Masahito Hosino and Huzio Nakano
  130. Progress of Theoretical Physics Vol. 68 No. 5 (1982) pp. 1460-1469 :
    Memory Function for the Autoregressed Velocity Autocorrelation Function of a Dense Liquid
    Yoshikazu Endo and Homare Endo
  131. Progress of Theoretical Physics Vol. 71 No. 5 (1984) pp. 1108-1111 :
    Langevin Equations in Quantized Field Theories
    Shijong Ryang and Takesi Saito
  132. Progress of Theoretical Physics Vol. 73 No. 6 (1985) pp. 1320-1341 :
    A. Microscopic Theory of Superfluids. III
    Shuichiro Yamasaki
  133. Progress of Theoretical Physics Vol. 74 No. 2 (1985) pp. 429-432 :
    A General Formulation of Nonequilibrium Thermo Field Dynamics
    T. Arimitsu and H. Umezawa
  134. Progress of Theoretical Physics Vol. 74 No. 6 (1985) pp. 1339-1342 :
    A Nonlinear Langevin Equation for Fluid Relaxation
    Fabio Marchesoni
  135. Progress of Theoretical Physics Vol. 77 No. 1 (1987) pp. 32-52 :
    Non-Equilibrium Thermo Field Dynamics
    Toshihico Arimitsu and Hiroomi Umezawa
  136. Progress of Theoretical Physics Vol. 77 No. 3 (1987) pp. 548-562 :
    Kinks and Breathers Associated with Collective Sugar Puckering in DNA
    Shozo Takeno and Shigeo Homma
  137. Progress of Theoretical Physics Vol. 77 No. 4 (1987) pp. 864-879 :
    Oscillatory Dynamical Correlations and Elementary Excitations in Liquid He4
    Fukuo Yoshida and Shozo Takeno
  138. Progress of Theoretical Physics Vol. 78 No. 6 (1987) pp. 1203-1208 :
    Continued Fraction Expansion of Fluctuation Spectrum and Generalized Time Correlation
    Hirokazu Fujisaka and Masayoshi Inoue
  139. Progress of Theoretical Physics Vol. 79 No. 5 (1988) pp. 1043-1060 :
    A Correction to the Hartree-Like Approximation in the Overdamped Sine-Gordon System
    Shogo Nakaya
  140. Progress of Theoretical Physics Vol. 80 No. 6 (1988) pp. 986-998 :
    The dc Transport Coefficient Calculation from Frequency Derivatives of Force-Force Time Correlation Functions. I
    N. Milinski
  141. Progress of Theoretical Physics Vol. 81 No. 5 (1989) pp. 905-909 :
    A Generalization of Gibbs' Ensemble to Nonequilibrium Statistical Mechanics
    Masakazu Ichiyanagi
  142. Progress of Theoretical Physics Vol. 82 No. 2 (1989) pp. 299-303 :
    A Simple Algebraic Technique in the Generalized Continued-Fraction Representation
    Sam Nyung Yi, Jai Yon Ryu and Sang Don Choi
  143. Progress of Theoretical Physics Vol. 82 No. 5 (1989) pp. 911-916 :
    Mori's Method and Feynman Diagram Method
    Edson Sardella and Armando F. S. Moreira
  144. Progress of Theoretical Physics Vol. 84 No. 5 (1990) pp. 810-823 :
    A Projection Operator Approach to Extended Irreversible Thermodynamics
    Masakazu Ichiyanagi
  145. Progress of Theoretical Physics Vol. 85 No. 3 (1991) pp. 493-507 :
    An Elaborate dc Electric Conductivity Formula as Result of Conjunction of Two Linear Response Formulae
    N. Milinski
  146. Progress of Theoretical Physics Vol. 87 No. 1 (1992) pp. 235-240 :
    A Derivation of a Langevin Type Map from a Many-Degrees-of-Freedom Symplectic Map and Its Invariant Measure
    Shinji Akiyama
  147. Progress of Theoretical Physics Vol. 90 No. 2 (1993) pp. 343-351 :
    Generalized Langevin Equation for Two-Dimensional Inviscid Barotropic Vorticity
    Takahiro Iwayama and Hisao Okamoto
  148. Progress of Theoretical Physics Vol. 90 No. 6 (1993) pp. 1229-1240 :
    Theory of Eddy Viscosity Coefficient for Two-Dimensional Inviscid Barotropic Fluid
    Takahiro Iwayama and Hisao Okamoto
  149. Progress of Theoretical Physics Vol. 101 No. 2 (1999) pp. 373-384 :
    Time Evolution of a Quantum System in a Medium and Mass Renormalization in the Projection Operator Method
    Tomoi Koide, Masahiro Maruyama and Fujio Takagi
  150. Progress of Theoretical Physics Vol. 104 No. 3 (2000) pp. 575-594 :
    A New Expansion of the Heisenberg Equation of Motion with Projection Operator
    Tomoi Koide and Masahiro Maruyama
  151. Progress of Theoretical Physics Vol. 104 No. 5 (2000) pp. 925-941 :
    Modeling Chaotic Evolution with Nonlinear Langevin Dynamics
    Toshiaki Tao and Hirokazu Fujisaka
  152. Progress of Theoretical Physics Vol. 107 No. 1 (2002) pp. 87-116 :
    Projection Operator Method for Collective Tunneling Transitions
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  153. Progress of Theoretical Physics Vol. 107 No. 3 (2002) pp. 525-541 :
    Derivation of Transport Equations Using the Time-Dependent Projection Operator Method
    Tomoi Koide
  154. Progress of Theoretical Physics Vol. 107 No. 5 (2002) pp. 1001-1035 :
    Projection Operator Approach to Langevin Equations in φ4 Theory
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  155. Progress of Theoretical Physics Vol. 108 No. 1 (2002) pp. 1-12 :
    Evaluation of Mean Values for a Forced Pendulum with a Projection Operator Method
    Kenji Sato and Makoto Okamura
  156. Progress of Theoretical Physics Vol. 109 No. 3 (2003) pp. 333-355 :
    Randomization and Memory Functions of Chaos and Turbulence
    Hazime Mori, Shoichi Kuroki, Hirotaka Tominaga, Ryuji Ishizaki and Nobuyuki Mori
  157. Progress of Theoretical Physics Vol. 109 No. 4 (2003) pp. 575-589 :
    Time Correlations and Power Spectra of the Duffing Equation
    Hirotaka Tominaga, Shoichi Kuroki and Hazime Mori
  158. Progress of Theoretical Physics Vol. 111 No. 5 (2004) pp. 635-660 :
    Memory Function Approach to Chaos and Turbulence and the Continued Fraction Expansion
    Hazime Mori, Shoichi Kuroki, Hirotaka Tominaga, Ryuji Ishizaki and Nobuyuki Mori
  159. Progress of Theoretical Physics Vol. 114 No. 1 (2005) pp. 1-21 :
    A New Expansion of Dynamical Correlations in Stochastic Processes and Chaotic Dynamics
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  160. Progress of Theoretical Physics Vol. 115 No. 4 (2006) pp. 701-715 :
    Determination of Chaotic Dynamical Correlations in Terms of Unstable Periodic Orbits
    Miki U. Kobayashi and Hirokazu Fujisaka
  161. Progress of Theoretical Physics Vol. 116 No. 1 (2006) pp. 47-59 :
    Evaluation of the Time Correlation Function for the Kuramoto-Sivashinsky Equation with a Projection Operator Method
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  162. Progress of Theoretical Physics Vol. 116 No. 1 (2006) pp. 61-86 :
    Theory of Non-Equilibirum States Driven by Constant Electromagnetic Fields
    Shigeki Onoda, Naoyuki Sugimoto and Naoto Nagaosa
  163. Progress of Theoretical Physics Vol. 116 No. 6 (2006) pp. 1051-1067 :
    The Memory Function and Chaos-Induced Friction in the Chaotic Hénon-Heiles System
    Ryuji Ishizaki, Hazime Mori, Hirotaka Tominaga, Shoichi Kuroki and Nobuyuki Mori
  164. Progress of Theoretical Physics Vol. 118 No. 6 (2007) pp. 1043-1052 :
    Dynamical Correlations for the Kuramoto-Sivashinsky Equation
    Miki U. Kobayashi and Hirokazu Fujisaka
  165. Progress of Theoretical Physics Vol. 119 No. 6 (2008) pp. 991-1004 :
    Effect of Soft Mode on Shear Viscosity of Quark Matter
    Takahiko Fukutome and Masaharu Iwasaki
  166. Progress of Theoretical Physics Vol. 121 No. 3 (2009) pp. 555-575 :
    Towards a Theory of Entropy Production in the Little and Big Bang
    Teiji Kunihiro, Berndt Müller, Akira Ohnishi and Andreas Schäfer
  167. Progress of Theoretical Physics Vol. 121 No. 6 (2009) pp. 1143-1156 :
    Time Correlation Calculation Method Based on Delayed Coordinates
    Kai Morino, Miki U. Kobayashi and Syuji Miyazaki
  168. Progress of Theoretical Physics Vol. 122 No. 5 (2009) pp. 1169-1188 :
    Time-Dependent Variational Approach to the Non-Abelian Pure Gauge Theory
    Yasuhiko Tsue, Tong-Gyu Lee and Hiroshi Ishii
  169. Progress of Theoretical Physics Vol. 125 No. 2 (2011) pp. 359-374 :
    Equations of Motion for the System of Interest under Time-Dependent Environment
    Fumihiko Sakata, Shiwei Yan, En-Guang Zhao, Yizhong Zhuo and Shan-Gui Zhou
  170. Progress of Theoretical Physics Vol. 127 No. 4 (2012) pp. 615-629 :
    Decay Forms of the Time Correlation Functions for Turbulence and Chaos
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  171. Progress of Theoretical Physics Supplement No.64 (1978) pp. 21-34 :
    Synergetics: Some Recent Trends and Developments
    H. Haken
  172. Progress of Theoretical Physics Supplement No.64 (1978) pp. 83-99 :
    Statistical Mechanical Theory of Non-Equilibrium Systems
    Fumiaki Shibata and Natsuki Hashitsume
  173. Progress of Theoretical Physics Supplement No.64 (1978) pp. 402-424 :
    Scaling Theory of Transient Nonlinear Fluctuations and Formation of Macroscopic Order
    Masuo Suzuki
  174. Progress of Theoretical Physics Supplement No.69 (1980) pp. 55-68 :
    Maxwell-Langevin Equations and the Kubo Formulae
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  175. Progress of Theoretical Physics Supplement No.69 (1980) pp. 439-450 :
    Molecular Dynamics Study of Velocity Autocorrelation Function in a Model of Expanded Liquid Rubidium
    Minoru Tanaka
  176. Progress of Theoretical Physics Supplement No.84 (1985) pp. 138-153 :
    Self-Consistent Theory of Anderson Localization
    Yoshiyuki Ono
  177. Progress of Theoretical Physics Supplement No.94 (1988) pp. 115-127 :
    Quantum Effect upon Friction of a Soliton in One-Dimensional φ4 System
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  178. Progress of Theoretical Physics Supplement No.99 (1989) pp. 382-398 :
    On New Nonequilibrium Effects in Stirred Fluids
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  179. Progress of Theoretical Physics Supplement No.99 (1989) pp. 399-403 :
    Realization of the Logical Operation XOR by a Synergetic Computer
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  180. Progress of Theoretical Physics Supplement No.113 (1993) pp. 1-23 :
    Soliton Diffusion
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  181. Progress of Theoretical Physics Supplement No.115 (1994) pp. 243-254 :
    The Path Probability Ansatz and Master Equation
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  182. Progress of Theoretical Physics Supplement No.126 (1997) pp. 13-20 :
    Motion of Interacting Particles in a Disordered Medium
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  183. Progress of Theoretical Physics Supplement No.126 (1997) pp. 289-299 :
    A Viewpoint, Model and Theory for Supercooled Liquids
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  184. Progress of Theoretical Physics Supplement No.126 (1997) pp. 339-342 :
    Freezing of Strongly Correlated Electrons in Bilayer Systems with Weak Disorder
    J. S. Thakur and D. Neilson
  185. Progress of Theoretical Physics Supplement No.165 (2006) pp. 18-32 :
    Effective Description of Small Non-Equilibrium Systems
    Shin-ichi Sasa
  186. Progress of Theoretical Physics Supplement No. 184 (2010) pp. 476-496 :
    Projection Operator Approaches to Complex Susceptibility
    Chikako Uchiyama

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