Analytic scaling function for island-size distributions

We obtain an explicit solution for the island-size distribution described by the rate equations for irreversible growth with the simplified capture rates of the form sigma(s) (Theta) proportional to Theta(p)(a + s - 1) for all s >= 1, where s is the size and Theta is the time-dependent coverage. The intrinsic property of this solution is its scaling form in the continuum limit. The analytic scaling function depends on the two parameters a and p and is capable of describing very dissimilar distribution shapes, both monomodal and monotonically decreasing. The obtained results suggest that the scaling features of the size distributions are closely related to the size linearity of the capture rates. A simple analytic scaling is obtained rigorously here and helps to gain a better theoretical understanding of possible origins of the scaling behavior of the island-size distributions.