The distribution functions of the penetrable-sphere models of liquid-vapour equilibrium

Abstract

The distribution functions of penetrable-sphere models are expressed in terms of those of the related binary mixtures of species a and b. The transcription takes a simple form for approximations, such as the mean-field (MF) or Percus-Yevick (PY), in which the direct correlation functions c_aa (r) and c_bb (r) are zero. For the MF the equation of state and the distribution functions are classical at the critical point, and it is shown that these results become exact at infinite dimensionality for the two most widely studied versions of the penetrable-sphere model. The transcription of the PY approximation leads to an inconsistency in the correlation functions at the critical point, but the equation of state is still classical. The hyper-netted chain (HNC) approximation leads to a pressure equation of state (for any mixture) which is symmetrical in the two components a and b, but to a compressibility equation which is not. For one of the models studied here, however, both lead to a thermodynamically non-classical critical point in three dimensions, and to behaviour that is, at least, much closer to classical in four and five dimensions.