Prog. Theor. Phys. Vol. 52 No. 1 (1974) pp. 54-67

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On the Yang-Lee Distribution of Zeros for a Gas Obeying van der Waals' Equation of State. I

— Fundamental Concepts and the Limit of Vanishing Attraction —

Kazuyosi Ikeda

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

(Received August 29, 1973)

Abstract:

Some fundamental concepts concerning the distribution of zeros of the grand partition function in the complex z (activity) plane are described. In order to obtain a unique distribution of zeros from the knowledge of the equation of state, we assume that the zeros are distributed on lines and the equation of state is analytically continued as far as possible in the complex plane. As an example, the distribution of zeros for a gas obeying van der Waals' equation of state, in the limit of vanishing attraction, is investigated. The Riemann surface of the pressure considered as a function of z is constructed. The line of zeros is determined from it as the "jumping line" between two sheets, which is a part of the negative real axis. The distribution function for zeros on the line is obtained. Though the problem treated in this paper has already been investigated by Hauge and Hemmer, the method of this paper including the analysis of the Riemann surface is useful in the arguments of the more complicated cases in the subsequent papers.

URL : http://ptp.ipap.jp/link?PTP/52/54/
DOI : 10.1143/PTP.52.54

References:

C. N. Yang and T. D. Lee, Phys. Rev. 87 (1952), 404; see also A. Münster, Statistical Thermodynamics, Vol. 1 (Springer-Verlag, Berlin, 1969), Chapter 4.
K. Ikeda, Modern Developments in Thermodynamics, ed. by B. Gal-Or, (Isr. Univ. Press/John Wiley & Sons, Jerusalem/New York, 1974), p. 313.
T. D. Lee and C. N. Yang, Phys. Rev. 87 (1952), 410[APS].
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E. H. Hauge and P. C. Hemmer, Physica 29 (1963), 1338[Elsevier].
P. C. Hemmer and E. H. Hauge, Phys. Rev. 133 (1964), A1010[APS].
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For example, N. M. Günter, Potential Theory and Its Applications to Basic Problems of Mathematical Physics (Frederick Ungar, New York, 1967), Chapter 2.
For example, A. Kyrala, Applied Functions of a Complex Variable (Wiley-Interscience, New York, 1972), Chapter 11; N. I. Muskhelishvili, Singular Integral Equations (P. Noordhoff N. V., Groningen, 1953), Chapters 1 and 2.
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Citing Article(s) :

Progress of Theoretical Physics Vol. 52 No. 2 (1974) pp. 415-432 :
On the Yang-Lee Distribution of Zeros for a Gas Obeying van der Waals' Equation of State. II

Kazuyosi Ikeda
Progress of Theoretical Physics Vol. 52 No. 3 (1974) pp. 840-859 :
On the Yang-Lee Distribution of Zeros for a Gas Obeying van der Waals' Equation of State. III

Kazuyosi Ikeda
Progress of Theoretical Physics Vol. 53 No. 1 (1975) pp. 66-73 :
On the Yang-Lee Distribution of Zeros for a Gas Obeying van der Waals' Equation of State. IV

Kazuyosi Ikeda
Progress of Theoretical Physics Vol. 58 No. 2 (1977) pp. 496-505 :
Distribution of Zeros and the Equation of State. I

Kazuyosi Ikeda
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. 65 No. 5 (1981) pp. 1542-1564 :
Statistical Mechanics of One-Dimensional Systems. I

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

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

Kazuyosi Ikeda