  We present a theory and accompanying importance sampling method for computing
rate constants in spatially inhomogenious systems. Using the relationship
between rate constants and path space partition functions, we illustrate that
the relative change in the rate of a rare event through space is isomorphic to
the calculation of a free energy difference, albeit in a trajectory ensemble.
Like equilibrium free energies, relative rate constants can be estimated by
importance sampling. An extension to transition path sampling is proposed that
combines biased path ensembles and weighted histogram analysis to accomplish
this estimate. We show that rate constants can also be decomposed into
different contributions, including relative changes in stability, barrier
height and flux. This decomposition provides a means of interpretation and
insight into rare processes in complex environments. We verify these ideas with
a simple model of diffusion with spatially varying diffusivity and illustrate
their utility in an atomistic model of ion pair dissociation near an
electrochemical interface.
