Cluster Size Distribution in a System of Randomly Spaced Particles

Abstract

The distribution function of particles over clusters is proposed for a system of identical intersecting spheres, the centers of which are uniformly distributed in space. Consideration is based on the concept of the rank number of clusters, where the rank is assigned to clusters according to the cluster sizes. The distribution function does not depend on boundary conditions and is valid for infinite medium. The form of the distribution is determined by only one parameter, equal to the ratio of the sphere radius (‘interaction radius’) to the average distance between the centers of the spheres. This parameter plays also a role of the order parameter. It is revealed under what conditions the distribution behaves like well known log-normal distribution. Applications of the proposed distribution to some realistic physical situations, which are close to the conditions of the gas condensation to liquid, are considered.

Appendix

In what follows we shall consider the algorithm for calculating the cluster structure and illustrate it for the system shown in Fig. 1.

Let the matrix $R_{ij}$ define the distances between the particles ($i<j$ ). We construct a square matrix Q with zero diagonal elements and zeros for all elements below the main diagonal. The remaining elements (above the diagonal) consist of zeros and ones, so that

$$\begin{aligned} Q_{ij}= \left\{ \begin{array}{rcl} 0, &{} \text{ if } &{} R_{ij}>R; \\ 1, &{} \text{ if } &{} R_{ij} \le R; \end{array}\right. \end{aligned}$$

(A)

where R is the “interaction radius”; $(i,j)=1,$...$,N_0$; $N_0$ is the number of particles in the system. The number of ones in such a matrix is equal to the number of binary bonds in the system. The matrix Q corresponding to the system in Fig. 1 is shown in Fig. 17. Instead of ones, the elements of the matrix are shown by bold points. The matrix in this case contains 15 points, corresponding to 15 bonds between particles.

A system without bonds (i.e., without clusters) corresponds to a zero matrix. Further, the indices of the rows and columns i, j of the matrix for some k-th bond will be denoted as $(a_k,b_k)$, where $a_k$ and $b_k$ are integers, numbering the rows and columns of the matrix Q.

Fig. 17

The matrix Q for the system of 21 particles shown in Fig. 1. Each point corresponds to a binary bond between particles, i.e. to the element of type ($a_k, b_k$) in Eq. (B)

The set of m links (bonds)

$$\begin{aligned} (a_1,b_1),(a_2,b_2),...,(a_m,b_m), \end{aligned}$$

(B)

will belong to one cluster, if one of the numbers $a_i$ or $b_i$ (or both) of any pair occurs at least once in the remaining pairs in (B). For example, a cluster of six particles in Fig. 1 corresponds to a set of five bonds

$$\begin{aligned} (2,4),\, (2,5), \, (4,11),\, (4,13),\, (11,15). \end{aligned}$$

(C)

We see that at least one of the numbers in any pair is contained at least once in one of the other pairs. From the construction (C) we conclude that this cluster consists of 6 particles with numbers: 2, 4, 5, 11, 13 and 15. The rows and columns of the matrix Q, which correspond to the configuration (C), form a grid that defines the particles that belong to the same cluster.

Collections of pairs of type (B), and hence the distribution of particles in clusters, can be found by analyzing the rows of the matrix Q, starting with the first row. All non-zero elements of a row or column of the matrix Q belong to the same cluster. The bonds with the same row or column numbers belong to the same cluster as well (for example, when we consider another row with the same number as the column number in the original matrix element, see below). If all elements of the k-th row and k-th column are equal to zero, then the k-th particle does not form bonds and is a cluster of one particle.

Analysis of the cluster structure of the matrix Q begins with the 1st row, which defines all the bonds of the first particle with the other particles (for a given system configuration, the particles are numbered in an arbitrary order). All these particles are included in the first cluster. Further, we do not mean the ranked cluster number. Clusters will be numbered in the order of their formation. The ranked number will be determined after identifying all the clusters in the system followed by lining up in order of decreasing the number of particles in them.

If a bond (i.e., a unit matrix element) appears in the 1st row and k-th column, then the particles with numbers 1 and k belong to the same cluster. Similarly, the remaining bonds in the first line (1, j), $j>k$, are revealed. After that, we go to the line (matrix row) with the number k and reveal all the bonds in this line. The particles forming these bonds are also included into the cluster 1. Then, it is analyzed whether there are links in the remaining rows with numbers j that have already been encountered as column numbers in the 1st row. In subsequent calculations, all rows and columns belonging to the same cluster are no longer considered.

After all the particles belonging to cluster 1 formed by the first particle (i.e., the 1st matrix row) are identified, then we go to the second matrix row. If the number 2 has already entered the sequence of links (B) included in the first cluster, then we go to the 3rd matrix row, otherwise we analyze the matrix row 2, as described above. And so on, we iterate through the rows of the matrix Q.

As an example, consider how the sequence of bonds (C) is formed. This sequence is formed by the second row of the matrix Q in Fig. 17. There are the unit elements (2, 4) and (2, 5) on this line. Therefore, we go first to the 4th row. It forms bonds (4, 11) and (4, 13). Lines 5 and 13 do not form bonds, but line 11 contains the link (11, 15). This gives the sequence (C). Further, the rows and columns with numbers included in the sequence (C), fall out of the analysis, and we go to the matrix line 3, etc.

As a result, with the help of the matrix Q in Fig. 17, we arrive at the following sequence of calculation of the cluster structure of the system shown in Fig. 1.

• Cluster 1: (1,6), 2 particles with numbers 1 and 6;

• Cluster 2: is defined by the sequence (C);

• Cluster 3: one particle with number 3;

• Cluster 4: (7,10), 2 particles with numbers 7 and 10;

• Cluster 5: (8,12), (8,17), (8,19), (12,17), (19, 21), 5 particles with numbers 8, 12, 17, 19, 21;

• Cluster 6: one particle with number 9;

• Cluster 7: (14,20), (16,18), (16,20), 4 particles with numbers 14, 16, 18, 20.

All matrix lines after the 14th fall out of the analysis, since all particles are already distributed in clusters.