Evidence for a first-order phase transition at the divergence region of activity expansions

In the example of a lattice gas with ``hole-particle'' symmetry, a convincing confirmation is obtained for the direct relationship between the condensation phenomenon and the divergent behavior of the virial expansions for pressure and density in powers of activity. For such lattice-gas models of arbitrary geometry and dimensions, a general analytical expression is derived for the phase-transition activity (the convergence radius of activity expansions) that, in particular, exactly matches the well-known phase-transition activity of the Lee-Yang model. The study proves analytically the strict equality of pressure for the low-density and high-density virial expansions in powers of density (in terms of irreducible cluster integrals or virial coefficients) exactly at the symmetrical points, where their isothermal bulk modulus vanishes, as well as for the corresponding expansions in powers of activity (in terms of reducible cluster integrals) at the same points (the points of their divergence). It is also shown that Mayer's expansion with the constant (volume-independent) cluster integrals remains correct up to the condensation beginning, and the actual density dependence must be taken into account for the high-order integrals only in more dense regimes beyond the saturation point.