Response, relaxation and fluctuation

  • Ryogo Kubo
Conference paper
Part of the Lecture Notes in Physics book series (LNP, volume 31)


Field Effect Effective Cross Section Collision Integral Maxwell Stress Dominant Polarization 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes and references

  1. 1).
    These lectures are, in a sense, a continuation of three previous summer school lecture series (Ref. 2, 3, 4) given by the present author, and of a review report (Ref. 5) of the same author. Some of the references made previously will be omitted. A great part of this set of lectures is based on the author's lectures at the University of Tokyo and also on the chapters in Statistical Physics (Ref. 6).Google Scholar
  2. 2).
    R. KUBO, in Lectures in Theoretical Physics, vol. 1, ed. W. Brittin, (Interscience), New York, 1959, p.120.Google Scholar
  3. 3).
    R. KUBO, in Fluctuation, Relaxation and Resonance in Magnetic Systems, ed. D. ter Haar, (Oliver and Boyd) Edinburgh, 1962, p. 23.Google Scholar
  4. 4).
    R. KUBO, in Tokyo Summer Lectures in Theoretical Physics, 1965 Part I, Many-Body Theory, ed. R. Kubo, (Shokabo) Tokyo and (Benjamin) New York.Google Scholar
  5. 5).
    R. KUBO, Rep. on Progress in Physics, Vol. 29, Part I, (1966) 225.ADSGoogle Scholar
  6. 6).
    R. KUBO, in Tokei Butsurigaku (Statistical Physics) ed. R. Kubo and M. Toda, Chaper 5 (Brownian Motion), Chapter 6 (Physical Processes as Stochastic Processes), (Iwanami Pub.) Tokyo, 1973, (in Japanese).Google Scholar
  7. 7).
    Physicists may get some idea of such mathematical problems from R.E. MORTENSEN, J. Stat. Phys. 1, 271 (1969).ADSCrossRefGoogle Scholar
  8. 8).
    M.C. WANG and G.E. UHLENBECK, Rev. Mod. Phys. 17, 327 (1945). Also see N. Wax (ed.) Selected Papers on Noise and Stochastic Processes, (Dover Pub.), New York, 1954.MathSciNetADSCrossRefGoogle Scholar
  9. 9).
    L. ONSAGER and S. MACHLUP, Phys. Rev. 91, 1505 (1953).MathSciNetADSMATHCrossRefGoogle Scholar
  10. 10A).
    S. MACHLUP and L. ONSAGER, Phys. Rev. 91, 1512 (1953).MathSciNetADSMATHCrossRefGoogle Scholar
  11. 10).
    R. KUBO, J. Math. Phys. 4, 174 (1963).MathSciNetADSMATHCrossRefGoogle Scholar
  12. 11).
    R. KUBO, J. Phys. Soc. Japan, 12, 570 (1957).MathSciNetADSCrossRefGoogle Scholar
  13. 12).
    R. KUBO in Statistical Mechanics of Equilibrium and Non-Equilibrium, ed. J. Meixner, (North Holland), Amsterdam 1965, p. 80.Google Scholar
  14. 13).
    H. MORI, Prog. Theor. Phys. Kyoto 33, 423 (1965)ADSMATHCrossRefGoogle Scholar
  15. 14).
    J. KIRKWOOD, J. Chem. Phys. 14, 180 (1946)ADSCrossRefGoogle Scholar
  16. 15).
    A. KAWABATA and R. KUBO, J. Phys. Soc. Japan 21, 1765 (1966)ADSCrossRefGoogle Scholar
  17. 17A).
    R. KUBO in Cooperative Phenomena, ed. H. Haken and M. Wagner, (Springer-Verlag, 1973, p.Google Scholar
  18. 16).
    H. FROHLICH, Theory of Dielectrics, (Oxford Clarendon Press), (1949).Google Scholar
  19. 17).
    Here we have used the generalized equipartition law <J′; J′> = <J; J> = Ne2kT/m where N is the total number of electrons. This is seen from the definition of the canonical correlation (6.22) (see also eq.(9.6))Google Scholar
  20. 18).
    P. NOZIERES and D. PINES, Nuovo Cimento X 9, 470 (1958). See also articles in Many Body Problems, ed. D. Pines, (Benjamin) New York, (1962).MATHCrossRefGoogle Scholar
  21. 19).
    T. MORIYA and A. KAWABATA, J. Phys. Soc. Japan, 34, 639 (1973), 35, 669 (1973).ADSCrossRefGoogle Scholar
  22. 20).
    B.J. ALDER and T.E. WAINWRIGHT, Phys. Rev. A1, 18 (1970). Since this discovery of Alder and Wainwright, there has been a great accumulation of theoretical papers, but so far only one experimental paper has appeared claiming observation of the long tail (Y.W. KIM and J.E. MATTA, Phys. Rev. Letters, (1973)). For other references see the lectures by P. Mazur in this volume.ADSGoogle Scholar
  23. 21).
    E.H. HAUGE and A. MARTIN-LOF, J. Stat. Phys. 7, 259 (1973).MathSciNetADSCrossRefGoogle Scholar
  24. 24A).
    P. MAZUR and D. BEDEAUX, to appear in Physica.Google Scholar
  25. 22).
    There are quite a few papers on this subject. Here we only refer to: S.F. EDWARDS, Proc. Phys, Soc. 86, 977 (1965).MathSciNetADSCrossRefGoogle Scholar
  26. 26A).
    J.S. ROUSSEAU, J.C. STODDART and N.H. MARCH, J. Phys. C 5, L175 (1972).ADSCrossRefGoogle Scholar
  27. 27A).
    V.M. KENKRE and M. DRESDEN Phys. Rev. A 6, 769 (1972).ADSGoogle Scholar
  28. 27A).
    W.G. CHAMBERS, J. Phys. C 6, 2586 (1973). P.N. ARGYRES and J.L. SIGEL, Phys. Rev. Letters, 31 1397 (1973)ADSCrossRefGoogle Scholar
  29. 23).
    This can be easily seen in the following way: We write the Boltzmann-Bloch equation in the form \(\Gamma [f] = E\frac{{\partial f_O }}{{\partial p}}\) and assume f=fo(1 + ψ), linearizing the collision operator to get \(f_O \tilde \Gamma \psi = E\frac{{\partial f_O }}{{\partial p}}.\) The conductivity is expressed in the form \((p, f_O \psi ) = (p, f_O \tilde \Gamma ^{ - 1} \frac{1}{{f_O }}\frac{{\partial f_O }}{{\partial p}})\) which is a sort of average of relaxation frequencies. An approximation to the above expression would be \((p, f_O \psi ) = \frac{{(p,\frac{{\partial f_O }}{{\partial p}})^2 }}{{(p, f_O \tilde \Gamma f_O^{ - 1} \frac{{\partial f_O }}{{\partial p}})}}\) This leads to the Grueneisen formula for metallic electrons interacting with phonons in equilibrium (Kubo, 1943).Google Scholar
  30. 24).
    For example, I. MANNARI, Prog. Theor. Phys. 26, 51 (1961). This sort of approximation, for the conductivity change due to disorder, has the advantage that it expresses the resistance in terms of space-time correlations of the disorder. The spirit of the calculation is common to other kinds of relaxation processes (see Ref. 2 ).MathSciNetADSMATHCrossRefGoogle Scholar
  31. 25).
    H. MORT, Prog. Theor. Phys. 33, 424 (1965).ADSGoogle Scholar
  32. 26).
    M. DUPUIS, Prog. Theor. Phys. 37, 502 (1967).MathSciNetADSMATHCrossRefGoogle Scholar
  33. 27).
    There are a number of papers on this subject. The expression (9.16) with a constant δ3 is called the three-pole approximation. An obvious generalization of eq. (9.16) is to include natural frequencies of corresponding modes in the form \(\gamma _n [\omega ] = \frac{{\Delta _n^2 }}{{i(\omega - \Omega _n ) + \gamma _{n + 1} [\omega ]}}\) by starting from two conjugate Brownian modes (x1, x2). For more details, the reader is referred to the following articles: W. MARSHALL and S.W. LOVESEY, Theory of Thermal Neutron Scattering, (Oxford University Press), 1971; K. TOMITA and H. TOMITA, Prog. Theor. Phys. 45, 1407 (1971) H. TOMITA and H. MAKISHIMA, Prog. Theor. Phys. 48, 1133 (1972); K. TOMITA and H. MAKISHIMA, Prep. (1974); S.W. LOVESEY and R.A. MESERVE, Phys. Rev. Letters 28, 614 (1972), J. Phys. C 6, 79 (1973). It should perhaps be noted that the so-called central peak problem, which called much attention recently, seems somewhat deeper than that appears in the three-pole approximation or its simple modifications.Google Scholar
  34. 28).
    Excellent treatments of the damping theory are found in text books on quantum mechanics such as HEITLER's Quantum Theory of Radiation and MESSIAH's Quantum Mechanics. Applications to statistical mechanics have been made by many authors. Early works are: S. NAKAJIMA, Prog. Theor. Phys. 20, 948 (1958) (also R. KUBO, as quoted in this paper); R. ZWaNZIG, J. Chem. Phys. 33, 1338 (1960), also in Lectures in Theoretical Physics, Vol. III, ed. W.E. Brittin, B.W. Downs and J. Downs (Interscience Pub.), New York, 1961.Google Scholar
  35. 29).
    M. LAX, Rev. Mod. Phys. 32, 625 (1960), 38, 359 (1966), 38, 541 (1966).ADSCrossRefGoogle Scholar
  36. 30).
    H. HAKEN, Rev. Mod. Phys. to appear.Google Scholar
  37. 31).
    R. KUBO, in Stochastic Processes in Chemical Physics, ed. Shuler, (John Wiley and Sons, Inc.), 1969, p. 101.Google Scholar
  38. 32).
    R. KUBO, K. MATSUO and K. KITAHARA, J. Statist. Phys. 9, 51 (1973).ADSCrossRefGoogle Scholar
  39. 33).
    N.G. VAN KAMPEN, Can. J. Phys. 39, 551 (1961), and in Fluctuation Phenomena in Solids, ed. R.E. Burgers, (Academic Press), New York, 1965, *** DIRECT SUPPORT *** A3418025 00002ADSMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 1974

Authors and Affiliations

  • Ryogo Kubo
    • 1
  1. 1.Department of PhysicsUniversity of TokyoTokyoJapan

Personalised recommendations