# Rate constants in spatially inhomogeneous systems

## TL;DR

This paper introduces a new theory and importance sampling method for calculating rate constants in spatially inhomogeneous systems, linking them to free energy differences in trajectory space, and demonstrating their application in complex environments.

## Contribution

The paper develops a novel approach combining biased path ensembles and weighted histogram analysis to estimate rate constants and decompose them into meaningful contributions.

## Key findings

- Validated with a diffusion model with spatially varying diffusivity.
- Applied to ion pair dissociation near an electrochemical interface.
- Provided a new interpretation framework for rare events in complex systems.

## Abstract

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.