  We propose an efficient method to compute reaction rate constants of
thermally activated processes occurring in many-body systems at finite
temperature. The method consists in two steps: first, paths are sampled using a
transition path sampling (TPS) algorithm supplemented with a local Lyapunov
bias favoring diverging trajectories. This enhances the probability of
observing rare reactive trajectories between stable states during relatively
short simulations. Secondly, reaction constants are eventually estimated from
the unbiased fraction of reactive paths, yielded by an appropriate statistical
data analysis tool, the multistate Bennett acceptance ratio (MBAR) package.
  In order to test our algorithm, we compute reaction constants for structural
transitions in LJ38, a well studied Lennard-Jones cluster, comparing our
results to values previously reported in the literature. Additionally, we apply
our method to the calculation of reaction rates for vacancy migration in an
{\alpha}-Iron crystal, for temperatures ranging from 300 K to 850 K. Vacancy
diffusion rates associated with activation barriers at finite temperature are
then evaluated, and compared to values obtained using the standard harmonic
approximation.
