Diffusion, Fragmentation and Coagulation Processes: Analytical and Numerical Results

TL;DR

This paper develops analytical and numerical solutions for dynamical rate equations involving diffusion, fragmentation, and coagulation, revealing different size distribution behaviors and validating results with simulations.

Contribution

It introduces exact solutions for size distributions in fragmentation processes and maps the coagulation-including model to a Riccati equation for asymptotic analysis.

Findings

Size distribution follows a Bessel-type decay without coagulation.

Inclusion of coagulation leads to exponential decay in size distribution.

Numerical simulations confirm analytical predictions.

Abstract

We formulate dynamical rate equations for physical processes driven by a combination of diffusive growth, size fragmentation and fragment coagulation. Initially, we consider processes where coagulation is absent. In this case we solve the rate equation exactly leading to size distributions of Bessel type which fall off as $\exp(-x^{3/2})$ for large $x$-values. Moreover, we provide explicit formulas for the expansion coefficients in terms of Airy functions. Introducing the coagulation term, the full non-linear model is mapped exactly onto a Riccati equation that enables us to derive 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 $x$. These findings are checked by numerical simulations and we find perfect agreement between the theoretical predictions and numerical results.