Lattice Gas Condensation and its Relation to the Divergence of Virial Expansions in the Powers of Activity

Abstract

An efficient algorithm for the calculation of high-order reducible cluster integrals on the basis of irreducible integrals (virial coefficients) has been proposed. The algorithm is applied to study the behavior of the well-known virial expansions of the pressure and density in power series of activity up to very high-order terms, as well as recently derived symmetric power expansions in the reciprocal activity, in the framework of a specific lattice gas model. Our results are consistent with those obtained in other modern studies of the partition function in terms of the density. They disclose the physical meaning of the divergence that the mentioned expansions demonstrate in the condensation region. K e y w o r d s: lattice gas, virial coefficients, reducible cluster integrals, activity, equation of state, condensation.

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

“…The vanishing of the isothermal bulk modulus at the point ρ G (z G ) means that the AVEOS divergence yields the jump of density at constant pressure and activity (chemical potential), and such behavior of the AVEOS has been confirmed for a number of statistical models of fluids [21][22][23]. Moreover, the equation of state based on the exact generating function in terms of irreducible integrals (UEOS) [17,18] (there is no activity dependence and, hence, no divergence in the UEOS) really yields the pressure constancy at any density beyond the ρ G .…”

“…This equation yields the constancy of pressure at any density beyond the point ρ G , where the isothermal bulk modulus of the virial expansion in powers of density (virial equation of state or VEOS) vanishes [19,20], that, in turn, may indicate the beginning of the condensation process at the vicinity of this point. Consequent studies of Mayer's expansion in terms of reducible cluster integrals [21][22][23] (equation of state in the parametrical form of expansions for pressure and density in powers of activity, AVEOS) have demonstrated its divergence exactly at the same point ρ G , but the actual character of this divergence [21][22][23][24] agrees with the behavior of Mayer's expansion in terms of irreducible integrals (where the activity dependence is excluded) and corresponds to the known thermodynamic signs of the first-order phase transition.…”

“…In order to establish the relationship between the condensation phenomenon and symmetrical divergence of the AVEOS, SAVEOS, the numerical studies have recently been performed for the two-dimensional Lee -Yang lattice-gas model [21]. Unfortunately, the obtained results did not demonstrate the persuasive convergence to the exact Lee -Yang solution: the equations with limited number of the known virial coefficients even yields the substantial difference of pressure at the symmetrical points of the phase-transition, ρ G (z G ) and ρ L (η L ).…”

Equation of state for all regimes of a fluid: From gas to liquid

“…All the equations mentioned above (in terms of the constant reducible as well as irreducible integrals) yield an essential discontinuity of density (the divergence to infinity) instead of the proper jump discontinuity. Although the main reason for such non-physical behavior is known in principle (it has been clearly stated in some researches [9,10,4]) the problem still remains absolutely unexplored in statistical theory.…”

“…For a wide range of lattice-gas models, this problem can simply be avoided due to the recently discovered "holeparticle" symmetry of the binodal [9,12,13] (i.e., the symmetry between ρ G and ρ L ), but, for continuous statistical models of matter, the corresponding symmetry is not so obvious, and the relation between ρ G and ρ L would have much more complex character. In this paper, the first important steps are made in a possible way to resolve the problem of the CE inadequacy at very dense regimes of model systems (i.e., regimes that correspond to the condensed states of matter).…”