Prog. Theor. Phys. Vol. 52 No. 2 (1974) pp. 415-432

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

— General Considerations for All Temperatures —

Kazuyosi Ikeda

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

(Received August 29, 1973)

Abstract:

The distribution of zeros of the grand partition function in the complex z (activity) plane is investigated for a gas obeying van der Waal's equation of state, or a one-dimensional gas of hard rods with infinitesimal attractions of infinite range. The fundamental concepts and the assumptions stated in a preceding paper I are employed, and the same method of argument as used in I for treating the limit of vanishing attraction is applied. The behaviour of the pressure as a function of z near the singularities in the complex plane is studied, and the behaviour of the pressure as a real function of real z is examined by aid of the mediate variable ρ (density). With the use of these arguments the Riemann surface of the pressure considered as a function of z is constructed. From it the zero lines are determined as “jumping lines” between different sheets. The features of the zero lines below, at, and above the critical temperature are discussed. The general formulae for the distribution functions for zeros on these lines are obtained.

URL : http://ptp.ipap.jp/link?PTP/52/415/
DOI : 10.1143/PTP.52.415

References:

K. Ikeda, Prog. Theor. Phys. 52 (1974), 54, [PTP]referred to as I.
P. C. Hemmer and E. H. Hauge, Phys. Rev. 133 (1964), A1010[APS].
J. D. van der Waals, Over de Continuiteit van der Gas- en Vloeistoftoestand, Dissertation, Leiden (1873).
M. Kac, G. E. Uhlenbeck and P. C. Hemmer, J. Math. Phys. 4 (1963), 216[CrossRef].
K. Ikeda, Proceedings of the International Conference of Theoretical Physics, Kyoto and Tokyo (1953), p. 544.
K. Ikeda, Busseiron Kenkyû 57 (1952), 77.
K. Ikeda, Prog. Theor. Phys. 16 (1956), 341, [PTP](Part. II).

Citing Article(s) :

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

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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

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

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Progress of Theoretical Physics Vol. 65 No. 5 (1981) pp. 1542-1564 :
Statistical Mechanics of One-Dimensional Systems. I

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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

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