Area, perimeter, density and entropy of objects generated by deposition of particles

Abstract

Each time a particle is added in order to study growth phenomena with a discrete
model, the mass of the object increases and its perimeter changes. This change of perimeter
may have different values and signs, depending on the place where the particle is added and
on the computational model used. We study these changes of perimeter, for the deposition
of particles on a two dimensional euclidean space, starting with a line of seeds and using
two types of lattices, one composed of square cells and another one in which the cells are
equilateral triangles. Functions relating perimeter, density and entropy with the area are
studied. Different types of random walks are considered in a particular fashion through
the selection of five different possible directions. The usual procedure to study the way the
perimeter of an object varies with its area is by means of the well known power-law function.
An alternative method is herein proposed to define a similar approach, but it is based upon
the probabilities of the changes of perimeter. No attempt has been made in this study to
compare numerical results with those of real phenomena.