Bimodality of cluster-size distribution and condensation in a finite Lennard-Jones system

Abstract

The complete summation of Mayer's expression for the canonical-ensemble partition function for a finite number N of particles, in terms of reducible cluster integrals b_k (1≤k≤N), is performed with a recursion formula. The b_k required for this are themselves obtained by use of the same technique in the summation of Mayer's expression for them, in the volume-independent case, in terms of star (i.e., irreducible cluster) integrals ßj (1≤j≤k-1). Below the critical temperature the results, obtained with use of all the star integrals that have been evaluated for the Lennard-Jones potential, display features indicative of vapor condensation at appropriate densities, i.e., bimodality in the size distribution of Mayer's mathematical clusters and remarkable constancy (for small N) in the pressure isotherms. Thus the lower-order star integrals alone suffice for the appearance of the phenomenon of vapor condensation in the theoretical results. Indeed the first star integral β₁ alone (corresponding to a "tree approximation" for the cluster integrals b_k) is found to suffice. This suggests that the cooperative essence of the first-order transition from vapor to liquid is to be attributed primarily to a cascade in chain branching.