# Approximate calculation of multidimensional first passage times

## TL;DR

This paper develops and tests approximate methods for calculating multidimensional first passage times in diffusive processes, improving on classical results by capturing non-Arrhenius behavior and aligning with exact solutions.

## Contribution

It introduces a sequence of approximations for multidimensional barrier crossing times, bridging heuristic solutions with classical results and validating them through simulations.

## Key findings

- One-dimensional approximations effectively capture non-Arrhenius dependence.
- Approximations converge to classical results in the large-barrier limit.
- Methods are validated against simulation results in two-dimensional problems.

## Abstract

The general, multidimensional barrier crossing problem for diffusive processes under the action of conservative forces is studied with the goal of developing tractable approximations. Particular attention is given to the effect of different statistical interpretations of the stochastic differential equation and to the relation between the approximations and the known, exact solutions to the one-dimensional problem. Beginning with a reasonable, but heuristic, simplifying assumption, a one-dimensional solution to the problem is developed. This is then simplified by introducing further approximations resulting in a sequence of increasingly simple expressions culminating in the classic result of Langer(Ann. Phys. 54, 258 (1969)) and others. The various approximations are tested on two dimensional problems by comparison to simulation results and it is found that the one-dimensional approximations capture most of the non-Arrhenius dependence on the energy barrier which is lost in the Langer approximation while still converging to the latter in the large-barrier limit.