Voronoi Polyhedron VP

Another local property of interest in the study of liquids is the Voronoi polyhedron (VP), or the Dirichlet region, defined as follows. 

Consider a specific configuration R and a particular particle /. Let us draw all the segments j = 1,. . j i) connecting the centers of particles i and j. Let Pij be the plane perpendicular to and bisecting the line lij. Each plane Pij divides the entire space into two parts. Denote by Vij the part of space that includes the point Rj. The VP of particle / for the 

A two-dimensional illustration of a construction of a VP is shown in Fig. 5.2. It is clear from the definition that the region (VP)i includes all the points in space that are nearer to than to any R - j 0- Furthermore, each VP contains the center of one and only one particle. A two-dimensional illustration is given in Fig. 5.3. 

The concept of VP can be used to generate a few local properties the one we shall be using is the volume of the VP, which we denote by 

One way of generating new properties from previous ones is by combination. For instance, the counting function for BE and the volume of the VP is 

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Note that this relation is based on the fact that the volumes of the Voronoi polyhedron (VP) of all the particles add up to build the total volume of the system. Here we have an example of an explicit dependence between V and which could have been guessed. Therefore, the partial molar volume of the 4>-species can be obtained by taking the functional derivative of V with respect to (), i.e.,... 

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