Distribution of Zeros and the Equation of State. III: Cluster Series, the Ideal Fermi-Dirac Gas and Other Problems

Abstract

The radius of convergence of the cluster series (expressing the equation of state) is discussed in connection with the distribution of zeros of the grand partition function on the complex z ( = activity) plane, by giving various examples of circular distribution. Anomalous phase transitions and phase transitions of third order are considered by showing some examples of circular distribution of zeros. For the ideal Fermi-Dirac gas, the distribution function of zeros, lying on the part of the negative real axis from - .iI3 to - co [where .iI = h(27CmkTt1!2l, is calculated, and the function-theoretical structure of the equation of state is investigated. The distribution of zeros for this gas is compared with that for Tonks' gas (having purely repulsive interparticle forces). The two-dimensional and one-dimensional Fermi-Dirac gases are dealt with from the point of view of the distribution of zeros. In the present series of papers we investigate the relations between the distribution of zeros of the grand partition function on the complex z( = activity ) plane and the thermodynamic properties including the equation of state, on the basis of the function-theoretical concepts, In paper 1,1) we discussed the fundamental relations concerning this problem; and, in particular, for the case of linear distribution, we established a theorem which connects discontinuities of the distribution function or of its derivative of some order with singularities of the thermodynamic functions. We also examined the relations between the circular distribution (with centre at the origin) and the equation of state. In paper 11,2) on the basis of the arguments in I, we discussed some examples of circular distribution (with centre at the origin) which lead to phase transitions of first order; by these examples we showed the existence of four essential types of analytical behaviour of the functions describing phase transition. In particular, we rigorously investigated the function-theoretical structure of the equation of state by using the Riemann surface. We also treated the problems of the thermodynamic stability, the continuity or discontinuity of the slope of the P-v isotherm at the condensation point, and others, in connection with the distribution of zeros. In the present paper, we first study the cluster series and their radius of convergence, on the basis of various examples of circular distribution, and we