  Adaptive Kinetic Monte Carlo combines the simplicity of Kinetic Monte Carlo
(KMC) with a Molecular Dynamics (MD) based saddle point search algorithm in
order to simulate metastable systems. Key to making Adaptive KMC effective is a
stopping criterion for the saddle point search. In this work, we examine a
recent criterion, due to S. Chill and G. Henkelman, that is based on the
fraction of total reaction rate found instead of the fraction of observed
saddles. The criterion uses the Eyring-Kramers law to estimate the reaction
rate at the MD search temperature. We also consider a related criterion that
remains valid when the Eyring-Kramers law is not. We examine the mathematical
properties of both estimators and prove their mean square errors are well
behaved, vanishing as the simulation continues to run.
