# Reaction rates and the noisy saddle-node bifurcation: Renormalization group for barrier crossing

## TL;DR

This paper develops a renormalization group framework to understand barrier crossing times in noisy systems, especially near the saddle-node bifurcation where the barrier vanishes, revealing universal scaling laws and distribution approximations.

## Contribution

It introduces a novel renormalization group approach to analyze the noisy saddle-node transition and derives universal scaling behavior for escape times.

## Key findings

- Derived universal scaling laws for mean escape time near the bifurcation.
- Developed an approximation for the distribution of barrier escape times.
- Linked barrier crossing phenomena with bifurcation and renormalization group theories.

## Abstract

Barrier crossing calculations in chemical reaction-rate theory typically assume that the barrier is large compared to the temperature. When the barrier vanishes, however, there is a qualitative change in behavior. Instead of crossing a barrier, particles slide down a sloping potential. We formulate a renormalization group description of this noisy saddle-node transition. We derive the universal scaling behavior and corrections to scaling for the mean escape time in overdamped systems with arbitrary barrier height. We also develop an accurate approximation to the full distribution of barrier escape times by approximating the eigenvalues of the Fokker-Plank operator as equally spaced. This lets us derive a family of distributions that captures the barrier crossing times for arbitrary barrier height. Our critical theory draws links between barrier crossing in chemistry, the renormalization group, and bifurcation theory.