A systematic derivation of exact generalized Brownian motion theory

We present here a simple unified derivation of the exact Fokker-Planck equation obtained earlier by Zwanzig and the exact Langevin and transport equations derived by Mori. The derivation, based on the use of a Hilbert space formulation of the dynamics, leads to substantial generalizations of these results in a straightforward manner. We obtain nonlinear Langevin equations for classical systems and discuss the extension of the theory to driven transport and to quantum dynamics based either on the use of density matrices or Γ-space densities as suggested by Wigner. Remaining limitations of the theory are pointed out.

[1]  F. Haake.  On a non-Markoffian master equation , 1969 .

[2]  H. L. Frisch,et al.  QUANTUM-MECHANICAL, MICROSCOPIC BROWNIAN MOTION , 1966 .

[3]  H. Haken.  Exact generalized Fokker-Planck equation for arbitrary dissipative and nondissipative quantum-systems , 1969 .

[4]  Robert Zwanzig,et al.  Memory Effects in Irreversible Thermodynamics , 1961 .

[5]  K. Kawasaki.  Correlation-Function Approach to the Transport Coefficients near the Critical Point. I , 1966 .

[6]  H. Mori.  Statistical-Mechanical Theory of Kinetic Equations Kinetic Equations for Dense Gases and Liquids , 1973 .

[7]  R. Kapral,et al.  Nonlinear Transport Phenomena: Analysis in Terms of Density Expansions , 1973 .

[8]  R. Zwanzig,et al.  Strategies for fluctuation renormalization in nonlinear transport theory , 1974 .

[9]  Melville S. Green,et al.  Markoff Random Processes and the Statistical Mechanics of Time-Dependent Phenomena , 1952 .

[10]  James D. Gunton,et al.  Theory of Nonlinear Transport Processes: Nonlinear Shear Viscosity and Normal Stress Effects , 1973 .

[11]  G. Sewell.  Quantum-statistical theory of irreversible processes: Dynamics of gross variables☆ , 1965 .

[12]  A. Muriel,et al.  Projection techniques in non-equilibrium statistical mechanics: II. The introduction of outside fields☆ , 1969 .

[13]  Kyozi Kawasaki,et al.  Simple derivations of generalized linear and nonlinear Langevin equations , 1973 .

[14]  C. R. Willis,et al.  Time-dependent projection-operator approach to master equations for coupled systems , 1974 .

[15]  R. Kubo.  Statistical-Mechanical Theory of Irreversible Processes : I. General Theory and Simple Applications to Magnetic and Conduction Problems , 1957 .

[16]  S. Chandrasekhar.  Stochastic problems in Physics and Astronomy , 1943 .

[17]  E. Wigner.  On the quantum correction for thermodynamic equilibrium , 1932 .

[18]  J. E. Moyal.  Quantum mechanics as a statistical theory , 1949, Mathematical Proceedings of the Cambridge Philosophical Society.

[19]  J. Neumann.  Zur Operatorenmethode In Der Klassischen Mechanik , 1932 .

[20]  H. Mori.  Transport, Collective Motion, and Brownian Motion , 1965 .

[21]  Melvin Lax,et al.  Fluctuations from the Nonequilibrium Steady State , 1960 .

[22]  B. O. Koopman,et al.  Hamiltonian Systems and Transformation in Hilbert Space. , 1931, Proceedings of the National Academy of Sciences of the United States of America.

[23]  J. Lebowitz,et al.  Dynamical study of brownian motion , 1963 .

[24]  R. Zwanzig.  Ensemble Method in the Theory of Irreversibility , 1960 .

[25]  I. Oppenheim,et al.  Bilinear Hydrodynamics and the Stokes-Einstein Law , 1973 .

[26]  H. Mori,et al.  On Nonlinear Dynamics of Fluctuations , 1973 .

[27]  G. Uhlenbeck,et al.  On the Theory of the Brownian Motion II , 1945 .

[28]  Paul C. Martin,et al.  Kinetic Theory of a Weakly Coupled Fluid , 1970 .