Mapping volume scale for overlapping clusters

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Abstract
The use of the observed frequency of appearance of a cluster in a small simulation cell, immersed in a macrosystem, to specify the number of such clusters in that macrosystem is discussed, for the special case in which the cluster consists of all the molecules in the cell and the volume of the cell is one of the defining parameters of the cluster. In particular, clusters consisting of ideal gas molecules are considered. Furthermore, the clusters are allowed to “overlap” in the sense that the same molecules may be shared by more than one cluster of the same or different type. Recently, the mapping of simulation cell frequencies onto the macrosystem, for this purpose, has been discussed for the case of rare nonideal, nonoverlapping clusters of the type of interest to nucleation theory and a difference of opinion has arisen with respect to the specification of a volume scale to be used in the mapping process. Two scales have been suggested. One scale is simply the volume of the simulation cell. The other is specified by Eq. (1) of the present paper. In this paper we prove that the scale for the overlapping ideal clusters is that of Eq. (1). Since with overlapping clusters it is easy to perform computer experiments that provide a direct count of the clusters in the macrosystem, the system, although having limited physical value, is still of considerable analytical value in deciding between the two scales and, as a corollary, between the two theories underlying these scales. The indicated experiments have been performed and are reported in a following paper. They confirm the applicability of Eq. (1).