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Prog. Theor. Phys. Vol. 34 No. 3 (1965) pp. 399-416

[ Full Text PDF : FREE ACCESS (997K) ]

A Continued-Fraction Representation of the Time-Correlation Functions

Hazime Mori

Research Institute for Fundamental Physics, Kyoto University, Kyoto

(Received April 28, 1965)

Abstract:

A continued-fraction expansion of the Laplace transform of the time-correlation functions is obtained, which enables us to express the generalized susceptibilities and the transport coefficients in terms of the static correlation functions of a set of quantities. This expansion has a different feature from the moment and cumulant expansions, and has a convenient form to introduce the long-time approximation as well as the short-time approximation. Its application to the anomalous relaxation and transport phenomena near the second-order phase transition points is discussed.
An expansion formula is also obtained for the time evolution of dynamical quantities in order to describe the various modes of motion involved according to their characteristic time constants. These two expansions are closely related to the time-correlation function formalism of irreversible processes, and allow us to have physical intutition in calculating dissipative properties.


URL : http://ptp.ipap.jp/link?PTP/34/399/
DOI : 10.1143/PTP.34.399

[ Full Text PDF : FREE ACCESS (997K) ] Citation:

References:

  1. R. Kubo, J. Phys. Soc. Jpn. 12 (1957), 570[JPSJ].
  2. H. Mori, Prog. Theor. Phys. 33 (1965), 423[PTP].
  3. F. Johnson and A. Nethercot, Phys. Rev. 114 (1959), 705[APS].
    M. Ericson and B. Jacrot, J. Phys. Chem. Solids 13 (1960), 235[CrossRef].
    M. Fixman, J. Chem. Phys. 36 (1962), 310[CrossRef].
    A. Michels, J. Sengers and P. van der Gulik, Physica 28 (1962), 1216[CrossRef].
    R. Hill and S. Ichiki, Phys. Rev. 128 (1962), 1140[APS]; ibid. 130 (1963), 150[APS].
  4. H. Mori and K. Kawasaki, Prog. Theor. Phys. 27 (1962), 529[PTP].
  5. H. Mori and K. Kawasaki, Prog. Theor. Phys. 28 (1962), 971[PTP].
    H. Mori, Prog. Theor. Phys. 30 (1963), 578[PTP].
  6. S. Chandrasekhar, Rev. Mod. Phys. 15 (1943), 1[APS].
  7. H. Wall, Continued Fractions (D. Van Nostrand Company, Inc., Princeton, New Jersey, 1948).
  8. L. Landau and E. Lifshitz, Statistical Physics (Pergamon Press, London-Paris, 1958), Chap. XII.
    M. Wang and G. Uhlenbeck, Rev. Mod. Phys. 17 (1945), 323[APS].
  9. R. Kubo, in Lectures in Theoretical Physics, edited by W. Brittin and L. Dunham (Interscience Publishers, Inc., New York, 1959), Vol. 1.
  10. R. Kubo, J. Phys. Soc. Jpn. 17 (1962), 1100[JPSJ].
  11. K. Tomita and M. Tanaka, Prog. Theor. Phys. 29 (1963), 528[PTP].
  12. T. Matsubara, Prog. Theor. Phys. 32 (1964), 50[PTP].

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