On the Kinetics of Multi-dimensional Fragmentation

We present two classes of exact solutions to a geometric model which describes the kinetics of fragmentation of $d$-dimensional hypercuboid-shaped objects. The first class of exact solutions is described by a fragmentation rate $a({x_1},...,{x_d}) = 1$ and daughter distribution function $b({x_1},..,{x_d} | {{x_{1}^{\p}}},...,{{x_{d}^{\p}}})= {{(\a_1 + 2)x_1^{\a_1}}\over{x_1^{\p(\a_1+1)}}}...{{(\a_d+2)x_d^{\a_d}}\over {x_d^{\p(\a_d+1)}}}$. The second class of exact solutions is described by a fragmentation rate $ a({x_1},...,{x_d}) = {{{x_1}^{\a_1}}...{{x_d}^{\a_d}}/{2^d}}$ and a daughter distribution function $b({x_1},..,{x_d} | {{x_{1}^{\p}}},...,{{x_{d}^{\p}}}) = {2^d}{\d(x_1 - {{x_{1}^\p}}/2)...\d(x_d - {{x_{d}^\p}}/2)}$. Each class of exact solutions is analyzed in detail for the presence of scaling solutions and the occurrence of shattering transitions; the results of these analyses are also presented.