  Smoluchowski's equation is a macroscopic description of a many particle
system with coagulation and shattering interactions. We give a microscopic
model of the system from which we derive this equation rigorously. Provided the
existence of a unique and sufficiently regular solution of Smoluchowski's
equation, we prove the law of large numbers for the empirical processes. In
contrast to previous derivations we assume a moderate scaling of the particle
interaction, enabling us to estimate the critical fluctuation terms by using
martingale inequalities. This approach can be justified in the regime of high
temperatures and particle densities, which is of special interest in
astrophysical studies and where previous derivations do not apply.
