  We propose a stochastic model of a fragmentation process, developed by taking
into account fragment lifetime as a function of their size based on the Gibrat
process. If lifetime is determined by a power function of fragment size,
numerical results indicate that size distributions at different times can be
collapsed into a single time-invariant curve by scaling size by average
fragment size (i.e., the distribution obeys the dynamical scaling law). If
lifetime is determined by a logarithmic function of fragment size, the
distribution does not obey the scaling law. The necessary and sufficient
condition that the scaling law is obeyed is obtained by a scaling analysis of
the master equation.
