Distribution of zeros and the equation of state. II: Phase transitions and singularities in the case of circular distribution. (Q2750090)

The phase transition of the first order of a system of infinitely large volume is analytically investigated in the case where the zeros of the grand partition function are distributed on a circle with centre at the origin in the complex \(z(=\) activity) plane. It is confirmed that the points of discontinuity of the distribution function \(\bar g(\theta)\) of zeros or its derivative of some order on the circle are analytical singularities of the density (and pressure) as a function of complex \(z\). By typical examples of circular distribution of zeros it is demonstrated that four types of phase transitions exist according to whether the transition point is an analytical singularity or not and whether the gas and liquid are described by the same analytic function or not. The equation of state and the transition point are studied from the standpoint of the theory of functions, by using the Riemann surface. The case of existence of infinitely multiple zeros or of an analytical singularity of \(\bar g(\theta)\) is discussed. The thermodynamical stability condition and the slope of the \(p-\nu\) isotherm at the condensation point are discussed, in reference to circular distributions of zeros.NEWLINENEWLINE NEWLINEPart I, see Zbl 1097.82503, Part III, see Zbl 1097.82505.