Ushcats S. Condensation and singularities of the virial expansions in the lattice-gas model

An exact analytical expression of the phase-transition activity is obtained for an arbitrary lattice-gas model with the “hole-particle” symmetry – the activity which is the convergence radius of the virial expansions in powers of activity. A strict equality of pressure is proved for symmetric virial expansions in powers of both activity and density at the points where their isothermal bulk modulus vanishes. A new technique is developed to determine the cluster integrals of lattice gases as precise analytic functions of temperature, and the corresponding functions are obtained for the virial coefficients (irreducible cluster integrals) and reducible cluster integrals to the seventh order for a number of lattice-gas models of different geometry and dimensionality with various interaction potentials. It is established that the asymptotic behavior of high-order reducible cluster integrals for various models of matter (not only lattice gases) well corresponds to the convergence radius of activity series in accordance with the Cauchy-Hadamard theorem that, in turn, provides new broad possibilities to define the virial series in the region of very high orders. A general approximation of high-order cluster integrals is proposed for various lattice-gas models, which, at subcritical temperatures, adequately describes the behavior of such systems from gaseous to dense states, including the phase-transition region, on both qualitative and quantitative levels.