Exact Solution and Multifractal Analysis of a Multivariable Fragmentation Model

Abstract

Most theoretical kinetic approaches proposed so far to describe fragmentation processes rely on the assumption that the fragments can be characterized by a unique relevant variable, for example their size or mass h. We investigate the consequences of introducing additional variables by considering a fragmentation model in which the fragments are described by two internal variables, h, w, and a breakup rate proportional to hw~. The exact solution. already partly presented in reference ill, is derived and discussed in details with an emphasis on the fragment distribution function. It is shown that~ because of the random multip)icative nature of the process, no reduction to an effective one-variable problem is usually possible. The fragment mass distribution has no simple scaling properties; it rather exhibits multiscaling, a feature that is interpreted in the framework of a multifractal formalism.