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Prog. Theor. Phys. Vol. 26 No. 2 (1961) pp. 173-220

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Generalized Theory of Condensing Systems

Kazuyosi Ikeda

Physics Department, Faculty of Science, Kyushu University, Hukuoka, Kyushu

(Received March 10, 1961)

Abstract:

The theory in the previous paper II, which has dealt with the two-phase separation in condensing systems on the basis of the “cluster” concept and by means of a mathematically rigorous method, is generalized to the case of systems with volume-dependent cluster integrals.
The cluster integrals are assumed to satisfy several mathematical conditions, which are considered to represent the essential features of the real systems. One of these conditions is concerned with the effects of the volume dependence of the cluster integrals composed of a comparatively small number of molecules. Another essential condition is that the contribution, per molecule, to a cluster integral in which the number l of the constituent molecules is infinitely large and of the same order as the volume V of the system, should be definite, finite, and indepdent of the volume per molecule, V/l, in a certain range.
By means of a function l*(N) which satisfies certain conditions, the sizes of clusters are classified into “large” [l>l*(N)] and “small” [l ≦l*(N)]. The “small” cluster part of the system in consideration and that of the “(0)-system” (i.e. a system with volume-independent cluster integrals) are proved to be practically equivalent. Then, after some discussions of the “large” clusters, it is proved that, in a certain range of the specific volume, the isotherm is horizontal and a “huge” cluster (i.e. a cluster of size comparable to the volume of the system) coexists with the saturated set of “small” clusters. This corresponds to the liquid-vapour coexistence range. And it is proved that, in the gas range, the system is thermodynamically equivalent to the (0)-system and has practically no “large” clusters.
A remark is given on the analytical properties of the singularity representing the condensation. It is noted that the present theory applies to both the cases of “analytical” singularity and of “non-analytical” singularity.
A non-rigorous treatment which leads to the same results as those in the text, is given in the Appendix.


URL : http://ptp.ipap.jp/link?PTP/26/173/
DOI : 10.1143/PTP.26.173

[ Full Text PDF : FREE ACCESS (2612K) ] Citation:

References:

  1. K. Ikeda, Proc. Internat'l Conf. Theor. Phys. Kyoto and Tokyo (1953), p. 544; to be referred to as Ia.
    K. Ikeda, Prog. Theor. Phys. 16 (1956), 341 [PTP](containing the complete proofs of the lemmas and theorems given in Ia); to be referred to as I.
  2. K. Ikeda, Prog. Theor. Phys. 11 (1954), 336[PTP].
    K. Ikeda, Prog. Theor. Phys. 19 (1958), 653[PTP]; to be referred to as II.
  3. H. D. Ursell, Proc. Cambr. Phil. Soc. 23 (1927), 685.
  4. J. E. Mayer, J. Chem. Phys. 5 (1937), 67[CrossRef].
    J. E. Mayer and Ph. G. Ackermann, J. Chem. Phys. 5 (1937), 74[CrossRef].
    J. E. Mayer and S. F. Harrison, J. Chem. Phys. 6 (1938), 87[CrossRef].
    S. F. Harrison and J. E. Mayer, J. Chem. Phys. 6 (1938), 101[CrossRef].
    J. E. Mayer and M. G. Mayer, Statistical Mechanics (J. Wiley and Sons, New York, 1940), Chapters 13 and 14.
    M. Born and K. Fuchs, Proc. R. Soc. A 166 (1938), 391.
    B. Kahn and G. E. Uhlenbeck, Physica 5 (1938), 399[CrossRef].
  5. K. Ikeda and T. Nakagawa, a lecture at the meeting of the Physical Society of Japan held at Nagoya University on October 18, 1960 [cf. the abstract book of the meeting, 5, p. 1].

Citing Article(s) :

  1. Journal of the Physical Society of Japan 56 (1987) pp. 3499-3511 :
    Cluster Sums for Lattice Gases with Second Nearest Neighbour Interactions. III. Three-Dimensional Simple Cubic Lattice
    Kunisuke Nisizima and Kazuyosi Ikeda
  2. Progress of Theoretical Physics Vol. 27 No. 5 (1962) pp. 1025-1062 :
    Some Remarks on the Theory of Condensation
    Takesi Yosida and Kazuyosi Ikeda
  3. Progress of Theoretical Physics Vol. 37 No. 2 (1967) pp. 245-275 :
    Generalized Theory of Condensing Systems. IV
    Kazuyosi Ikeda
  4. Progress of Theoretical Physics Vol. 37 No. 2 (1967) pp. 276-295 :
    Generalized Theory of Condensing Systems. V
    Kazuyosi Ikeda
  5. Progress of Theoretical Physics Vol. 38 No. 3 (1967) pp. 611-625 :
    On the Theory of Isothermal-Isobaric Ensemble. II
    Kazuyosi Ikeda and Sirô Kamakura
  6. Progress of Theoretical Physics Vol. 55 No. 4 (1976) pp. 1082-1092 :
    Generalized Theory of Condensing Systems. VI
    Kazuyosi Ikeda
  7. Progress of Theoretical Physics Vol. 60 No. 6 (1978) pp. 1653-1668 :
    Generalized Theory of Condensing Systems. VII
    Kazuyosi Ikeda
  8. Progress of Theoretical Physics Vol. 71 No. 4 (1984) pp. 689-706 :
    Phase Transitions of Lattice Gases
    Kunisuke Nisizima and Kazuyosi Ikeda

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