Consider the geometric fragmentation process in which a rectangle splits into
four smaller fragments, as illustrated below.
(a) Verify that the Mellin transform \(c\left(s_{1}, s_{2}, t\right)\) obeys
$$
\frac{\partial c\left(s_{1}, s_{2}, t\right)}{\partial t}=\left(\frac{4}{s_{1}
s_{2}}-1\right) c\left(s_{1}+1, s_{2}+1, t\right)
$$
(b) Check that area conservation is satisfied by the above master equation and
show that the average number of fragments is equal to \(1+3 t\).
(c) Using the equation for \(c\left(s_{1}, s_{2}, t\right)\) from (a) and the
asymptotic behavior \(c\left(s_{1}, s_{2}, t\right) \sim t^{-\alpha\left(s_{1},
s_{2}\right)}\), show that
$$
\alpha\left(s_{1},
s_{2}\right)=\left[\left(s_{1}+s_{2}\right)-\sqrt{\left(s_{1}-s_{2}\right)^{2}+16}\right]
/ 2
$$
(d) Show that the area distribution admits a scaling form
$$
F(A, t) \simeq t^{-2} f(A t), \quad f(z)=6 \int_{0}^{1} d
\xi\left(\xi^{-1}-1\right) e^{-z / \xi}
$$
(e) Using the results from (c) show that the area distribution weakly diverges
in the limit of small area, \(f(z) \simeq 6 \ln (1 / z)\). Verify that in the
opposite limit the scaled area distribution exhibits an exponential decay
reminiscent of the onedimensional random scission model, but with an algebraic
correction: \(f(z) \simeq\) \(6 z^{-2} \exp (-z)\) as \(z \gg 1\)
