Prog. Theor. Phys. Vol. 58 No. 2 (1977) pp. 496-505

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Distribution of Zeros and the Equation of State. I

— Fundamental Relations and a Theorem on a Cauchy-Type Integral —

Kazuyosi Ikeda

Department of Applied Physics, Faculty of Engineering, Osaka University, Suita 565

(Received December 10, 1976)

Abstract:

The fundamental relations between the distribution, in the complex z (=activity) plane, of the zeros of the grand partition function (for a system of infinitely large volume) and the equation of state are discussed; for linear distribution, an analysis of a Cauchy-type integral leads to a theorem which connects discontinuities of the distribution function or its derivative of some order with singularities of the thermodynamic functions (of complex z). Relations between the distribution of zeros on a circle (with centre at the origin) and the equation of state are also discussed; in particular, it is proved that if the distribution function on the circle is increasing, the density of the system as a function of (real) z is increasing. This paper serves as a preliminary argument for the investigation of the phase transitions in some examples of circular distribution in a subsequent paper.

URL : http://ptp.ipap.jp/link?PTP/58/496/
DOI : 10.1143/PTP.58.496

References:

K. Ikeda, Prog. Theor. Phys., to be published.
C. N. Yang and T. D. Lee, Phys. Rev. 87 (1952), 404[APS].
J. E. Mayer and M. G. Mayer, Statistical Mechanics (Wiley, 1940), Chap. 13.
K. Ikeda, Prog. Theor. Phys. 52 (1974), 54[PTP].
K. Ikeda, Proceedings of the International Conference of Theoretical Physics, Kyoto and Tokyo (1953), p. 544; Prog. Theor. Phys. 16 (1956), 341[PTP].
T. D. Lee and C. N. Yang, Phys. Rev. 87 (1952), 410[APS].

Citing Article(s) :

Progress of Theoretical Physics Vol. 61 No. 1 (1979) pp. 27-44 :
Distribution of Zeros and the Equation of State. II

Kazuyosi Ikeda
Progress of Theoretical Physics Vol. 68 No. 2 (1982) pp. 471-492 :
Distribution of Zeros and the Equation of State. III

Kazuyosi Ikeda
Progress of Theoretical Physics Vol. 68 No. 3 (1982) pp. 744-763 :
Distribution of Zeros and the Equation of State. IV

Kazuyosi Ikeda