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Response, relaxation and fluctuation

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

Keywords

Field Effect Effective Cross Section Collision Integral Maxwell Stress Dominant Polarization 
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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
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    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
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    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
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    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
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    R. KUBO, in Stochastic Processes in Chemical Physics, ed. Shuler, (John Wiley and Sons, Inc.), 1969, p. 101.Google Scholar
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    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

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