Diffusion, fragmentation and merging processes in ice crystals, alpha helices and other systems

We investigate systems of nature driven by combinations of diusive growth, size fragmentation and fragment coagulation. In particular we derive and solve analytically rate equations for the size distribution of fragments and demonstrate the applicability of our models in very dierent systems of nature, ranging from the distribution of ice crystal sizes from the Greenland ice sheet to the length distribution of -helices in proteins. Initially, we consider processes where coagulation is absent. In this case the diusion-fragmentation equation can be solved exactly in terms of Bessel functions. Introducing the coagulation term, the full non-linear model can be mapped exactly onto a Riccati equation that has various asymptotic solutions for the distribution function. In particular, we find a standard exponential decay, exp(x), for large x, and observe a crossover from the Bessel function for intermediate values of x.