The lognormal-like distributions in the linear growth equations

The distributions of the global interfacial widths, correlation functions, and the local interfacial widths of the growth process described by the one-dimensional Edward-Wilkinson equation are shown to be denumerable convolutions of exponential distributions. The same conclusions can also be extended to the distributions of the global interfacial widths for another linear growth equation, describing some super-rough growth processes, in both one- and two-dimensional cases. Most of these distributions display the lognormal-like behavior. We propose that the mechanism provided by the accumulation of exponential random variables may contribute to a lot of the lognormal-like behavior observed in the social and natural sciences.