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Prog. Theor. Phys. Supplement No.34 (1965) pp. 91-133

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Generalized Theory of Condensing Systems. III

— Uniform Convergence of Thermodynamic Functions —

Kazuyosi Ikeda and Teruko Nakazawa

Physics Department, Faculty of Science, Kyusyu University, Hukuoka

(Received March 23, 1965)

Abstract:

The uniform convergence with respect to v (the volume per molecule) is discussed for the thermodynamic functions –especially N-1 ln ΩN, ΩN being the configurational partition function– when N (the number of molecules) tends to infinity, with a view to obtaining the physical reasonableness and the mathematical completeness of the convergence arguments in the previous papers on the theory of condensing systems. The discussion yields also a basis for the commutability of the two operations N →∞ and ∂/ ∂v in calculating the limiling pressure for an infinite system. The systems that are treated are the (0)-system, whose cluster integrals are volume-independent, and the G-system, whose cluster integrals are volume-dependent and satisfy the G-condition introduced in the previous paper. Several lemmas and theorems (concerning the cluster integrals bl(0) and bl(V)) proved in the previous papers for the (0)- and the G-system are altered into lemmas and theorems containing the statements of the uniform convergence. The following conclusions are obtained: In the region (v ≤vs) of two-phase coexistence for the (0)-system, N-1 ln ΩN converges to ∑vbl(0)zsl - ln zs uniformly for v' ≤v ≤vs, v' being any positive real number. In the gaseous state (v ≤vs) of the (0)-system, N-1 ln ΩN converges to ∑vbl(0)zl - ln z uniformly for vs ≤v ≤v2, v2 being any positive real number. Thus, for the (0)-system, N-1 ln ΩN converges uniformly in the wider sense for 0 < v < +∞. In the region (v* < v < v cond) of two-phase coexistence for the G-system, N-1 ln ΩN converges to ∑vbl(0)zl cond - ln z cond uniformly in the wider sense for v* < v < v cond. In the gaseous state (v ≥v cond) of the G-system, N-1 ln ΩN converges to ∑vbl(0)zl - ln z uniformly for v cond ≤v ≤v2, v2 being any positive real number. The present paper also contains a method of treating the gaseous state (for the (0)- and the G-system) different from the methods in the previous papers.


URL : http://ptp.ipap.jp/link?PTPS/34/91/
DOI : 10.1143/PTPS.34.91

[ Full Text PDF : FREE ACCESS (3339K) ] Citation:

References:

  1. a) K. Ikeda, Proc. Internat'l Conf. Theor. Phys. Kyoto and Tokyo (1953), p. 544.
    K. Ikeda, Prog. Theor. Phys. 16 (1956), 341[PTP];
    this paper contains the complete proofs of the lemmas and theorems given in reference 1a).
    The contents of reference 1) appeared in K. Ikeda, Busseiron Kenkyu 57 (1952), 77; ibid. 65 (1953), 145 in Japanese.
  2. a) K. Ikeda, Prog. Theor. Phys. 11 (1954), 336[PTP].
    The contents of reference 2a) appeared in K. Ikeda, Busseiron Kenkyu 52 (1952), 21 in Japanese).
    K. Ikeda, Prog. Theor. Phys. 19 (1958), 653[PTP];
    this paper contains the rigorous proofs of the discussions given briefly in reference 2a).
    The contents of reference 2) appeared in K. Ikeda, Busseiron Kenkyu 81 (1955), 66; ibid. 106 (1957), 1 in Japanese.
  3. K. Ikeda, Prog. Theor. Phys. 26 (1961), 173, [PTP]reprinted as the first paper in this issue.
  4. J. E. Mayer and M. G. Mayer, Statistical Mechanics (J. Wiley and Sons, New York, 1940), Chapters 13 and 14.
    B. Kahn and G. E. Uhlenbeck, Physica 5 (1938), 399[CrossRef].
    M. Born and K. Fuchs, Proc. Roy. Soc. A 166 (1938), 391.

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. Journal of the Physical Society of Japan 61 (1992) pp. 1520-1526 :
    Volume-Dependent Cluster Sums and Phase Transition of the Two-Dimensional Triangular Lattice Gas
    Kazuyosi Ikeda and Kunisuke Nisizima
  3. Journal of the Physical Society of Japan 61 (1992) pp. 1527-1534 :
    Volume-Dependent Irreducible Cluster Sums and Phase Transition of Lattice Gases
    Kunisuke Nisizima and Kazuyosi Ikeda
  4. Progress of Theoretical Physics Vol. 37 No. 2 (1967) pp. 245-275 :
    Generalized Theory of Condensing Systems. IV
    Kazuyosi Ikeda
  5. Progress of Theoretical Physics Vol. 37 No. 2 (1967) pp. 276-295 :
    Generalized Theory of Condensing Systems. V
    Kazuyosi Ikeda
  6. Progress of Theoretical Physics Vol. 38 No. 3 (1967) pp. 584-610 :
    On the Theory of Isothermal-Isobaric Ensemble. I
    Kazuyosi Ikeda
  7. 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
  8. Progress of Theoretical Physics Vol. 55 No. 4 (1976) pp. 1082-1092 :
    Generalized Theory of Condensing Systems. VI
    Kazuyosi Ikeda
  9. Progress of Theoretical Physics Vol. 60 No. 6 (1978) pp. 1653-1668 :
    Generalized Theory of Condensing Systems. VII
    Kazuyosi Ikeda
  10. 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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