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Prog. Theor. Phys. Vol. 37 No. 3 (1967) pp. 502-537

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Moment and Continued Fraction Expansions of Time Autocorrelation Functions

Marc Dupuis

Institute of Physics, College of General Education, University of Tokyo, Tokyo

(Received October 30, 1966)

Abstract:

Two representation of the Laplace transforms of time autocorrelation functions, namely the moment expansion and Mori's continued fraction representation, are studied from the point of view of their convergence, with the aim of obtaining new general properties of time autocorrelation functions. First, the relation between them is established by using mathematical techniques in the case of a general dynamical variable, and a direct method, useful for applications, in the case of Hermitian variables. The mathematical structure associated to Mori's generalized random forces is investigated and it is shown that these random forces can be obtained by a Schmidt orthogonalization of the sequence of initial time derivatives of the dynamical variable considered. Then, the convergence criteria for both representations are examined and an illustration is given with the exactly solvable case of an isotopic impurity in a linear chain of coupled harmonic oscillators. Finally, the question of knowing whether the continued fraction expansion is convergent for any Harmitian dynamical variable and any system is discussed, with its implications for the general behaviour of time autocorrelation functions.


URL : http://ptp.ipap.jp/link?PTP/37/502/
DOI : 10.1143/PTP.37.502

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References:

  1. H. Mori, Prog. Theor. Phys. 33 (1965), 423[IPAP].
  2. H. Mori, Prog. Theor. Phys. 34 (1965), 399[IPAP].
  3. J. L. Lebowitz and O. Penrose, J. Math. Phys. 5 (1964), 841[AIP Scitation].
  4. K. Kawasaki and I. Oppenheim, Phys. Rev. 139 (1965), A 1763[APS].
  5. R. Kubo, Lectures in Theoretical Physics, edited by W. Britin and L. Dunham (Interscience Publishers, Inc., New York, 1959), Vol. I.
  6. G. Frobenius andL. Stickelberger, J. für reine u. ang. Math. 88 (1880), 146.
  7. G. Frobenius, J. für reine u. ang. Math. 90 (1880), 1.
  8. T. J. Stieltjes, Ann. Fac. Sci. Toulouse, 3 (1889), H. 1; Oeuvres complètes, II, 184.
  9. P. L. Tchebychef, Jour. de Math, IIeme Série, T. III (1858), 289; Oeuvres complètes, I, 203.
  10. N. I. Akhiezer, The Classical Moment Problem (Oliver and Boyd Ltd., Edinburgh and London, 1965).
  11. H. S. Wall, Bull. Amer. Math. Soc. 51 (1945), 97.
  12. H. S. Wall, Continued Fractions (Van Nostrand, New York, 1948).
  13. E. Borel, L\cecons sur les Séries Divergentes (Gauthier Villars, Paris, 1928), 66.
  14. T. J. Atieltjes, Ann. Fac. Sci. Toulouse 8 (1894), J., 1; ibid. 9 (1895), A, 1; or: Oeuvres complètes II, 402, or: Mémoires présentés par divers savants à l'Académie des Sciences de l'Institut National de France 33, 1.
  15. T. Carleman, Les Fonctions Quasi-Analytiques (Gauthier-Villars, Paris, 1926).
  16. P. Chr. Hemmer, Dynamic and Stochastic Types of Motion in the Linear Chain, Thesis (Trondheim, 1959).
  17. R. J. Rubin, J. Math. Phys. 1 (1960), 309[AIP Scitation]; ibid. 2 (1961), 373[AIP Scitation].
  18. S. Takeno and J. Hori, Prog. Theor. Phys. Suppl. No. 23 (1962), 177[IPAP].
  19. R. E. Turner, Physica 26 (1960), 269[CrossRef].
  20. P. Ullersma, Physica 32 (1966), 74[CrossRef].
  21. M. Toda and Y. Kogure, Prog. Theor. Phys. Suppl. No. 23 (1962), 157[IPAP].
  22. A. Sakurai, private communication.
  23. A. Blanc-Lapierre et R. Fortet, Théorie des Fonctions Aléatoires (Masson, Paris 1953), 48.
  24. J. A. Shohat and J. D. Tamarkin, The Problem of Moments, Math. Surveys, no. 1, publ. by Amer. Math. Soc. (New York, 1943), 22.
  25. N. Dunford and J .T. Schwartz, Linear Operators II (Interscience Publishers, New York, 1963), 1026.
  26. G. Valiron, Théorie des Fonctions (Masson, Paris, 1948), 163.
  27. R. C. Tolman, The Principles of Statistical Mechanics (Oxford, 1938), 491.
  28. A. I. Khinchin, Mathematical Foundations of Statistical Mechanics (Dover, New York, 1949).

Citing Article(s) :

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  2. Progress of Theoretical Physics Vol. 40 No. 6 (1968) pp. 1287-1311 :
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  3. Progress of Theoretical Physics Vol. 45 No. 5 (1971) pp. 1407-1436 :
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  4. Progress of Theoretical Physics Vol. 48 No. 4 (1972) pp. 1133-1149 :
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  5. Progress of Theoretical Physics Vol. 49 No. 3 (1973) pp. 714-730 :
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  6. Progress of Theoretical Physics Vol. 51 No. 5 (1974) pp. 1312-1331 :
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  7. Progress of Theoretical Physics Vol. 54 No. 3 (1975) pp. 918-920 :
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  8. Progress of Theoretical Physics Vol. 55 No. 1 (1976) pp. 21-32 :
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  9. Progress of Theoretical Physics Vol. 55 No. 2 (1976) pp. 411-429 :
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  10. Progress of Theoretical Physics Vol. 61 No. 3 (1979) pp. 850-863 :
    A New Continued-Fraction Representation of the Time-Correlation Functions of Transport Fluxes
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