A NOTE ON THE ABOAV-WEAIRE LAW

In this paper we present an alternative derivation of the Aboav-Weaire law. By first making the assumption that the mean of the number of si des surrounding a cell is a function of the time, leads to M(n), the m ean number of sides surrounding a cell of sides n, as a linear functio n of the second moment mu, and is independent of n. This indicates tha t the local mean increases with time. When written in the form of the general Aboav-Weaire relation we find in their notation that a = 1 and b = 6/7. The analysis also leads to the relation that the deviation f rom the ensemble average, 6, is proportional to mu, and b is the coeff icient of proportionality. The initial assumption is removed and the a ssumption that M(n) is now a function of time and the number of sides is made. This assumption leads to M(n) as a linear function of the rat io mu/(n + 1). This applies in the limit of small mu and the assumptio n that M(n) can be expanded as a Maclaurin series. When written in the form of the general Aboav-Weaire law a = 1 and b = 0. This implies th at the deviation from the mean 6 is then proportional to mu/(n + 1). A similar analysis is applied to three dimensions and when the faces of the cells are considered we find that the average number of faces of the cells surrounding a cell of face f is a linear function of mu/(f 1), where mu is the second moment of the faces with mean 14. This ana lysis of the mean number of sides surrounding a cell is extended to fi nding the mean area surrounding a cell of sides n and area A, M(n)A. I t is found that the mean area of the cells M(n)A is given by M(n)A = A (a) + (A(a) - A)/n, where A(a) is the local cell area average which in the first approximation can be treated as the ensemble average area. Similarly the mean volume of cells surrounding a cell of faces f and v olume v is given by M(f)v = v(a) + (v(a) - v)/f, where v(a) is the ave rage local volume. In this case cells of small areas (volumes) are sur rounded by cells of areas (volumes) greater than the average area (vol ume). We can reverse the argument to show that the large area (volume) cells are surrounded by small area (volume) cells. The results of thi s analysis indicates that the cells are not randomly distributed.