ON THE EXISTENCE OF A SCALING RELATION IN THE EVOLUTION OF CELLULAR-SYSTEMS

A mean field approximation is used to analyse the evolution of the dis tribution of sizes in systems formed by individual ''cells'', each of which grows or shrinks, in such a way that the total number of cells d ecreases (e.g. polycrystals, soap froths, precipitate particles in a m atrix). The rate of change of the size of a cell is defined by a growt h function that depends on the size (x) of the cell and on moments of the size distribution, such as the average size (xBAR). Evolutionary e quations for the distribution of sizes and of reduced sizes (i.e. x/xB AR) are established. The stationary (or steady state) solutions of the equations are obtained for various particular forms of the growth fun ction. A steady state of the reduced size distribution is equivalent t o a scaling behaviour. It is found that there are an infinity of stead y state solutions which form a (continuous) one-parameter family of fu nctions, but they are not, in general, reached from an arbitrary initi al state. These properties are at variance from those that can be deri ved from models based on von Neumann-Mullins equation.