The equal-size binary breakage problem: Evolution toward a steady shape or periodic behavior?

Abstract

During the last few years, the self-similar particle size distribution for a particle population undergoing breakage in equal size fragments has been derived using approximating, numerical, and analytical means. But very recently it was shown [N.V. Mantzaris, J. Phys. A Math. Gen. 38 (2005) 5111] through transient simulation of the breakage process that the particle size distribution in case of breakage in two equal fragments, never attains a steady shape, i.e., a self-similar form. The new results give rise to questions about the real meaning and utility of the previously derived self-similar distributions for these systems. The scope of the present work is to answer these questions and it is attempted using only analytical (exact) means for the solution of the transient breakage problem. In doing so, the very interesting and rich underlying structure and properties of the solutions of the equal size breakage problem (seemingly, very simple) are revealed. It appears that the utility of the known self-similar distributions for this particular problem has to be redefined but yet not entirely abandoned.