# Barrierless reaction kinetics : Inertial effect on different distribution functions of relevant Brownian functionals

## TL;DR

This paper explores how inertia influences barrierless electronic reaction kinetics in solution by analyzing various probability distribution functions of Brownian functionals, incorporating inertial effects in underdamped regimes for the first time.

## Contribution

It introduces a novel approach to include inertial effects in the study of Brownian functionals related to reaction kinetics, extending beyond the traditional overdamped models.

## Key findings

- Identified nontrivial scaling behavior of PDFs
- Calculated critical exponents for asymptotic limits
- Demonstrated importance of inertia in reaction dynamics

## Abstract

We investigate the effect of inertia on barrierless electronic reactions in solution by suggesting and examining different probability distribution functions (PDF) of relevant Brownian functionals associated with the lifetime and reactivity of the process. Activationless electronic reaction in solution can be modeled as a free Brownian motion with inertial term in the underdamped regime. In this context we suggest several important distribution functions that can characterize the reaction kinetics. Most of the studies on Brownian functional which has vast potential application in diverse fields, are confined in the overdamped regime. To the best of our knowledge, we are attempting first time to incorporate the much important inertial effects on the study of different PDFs related with Brownian functionals of an underdamped Brownian motion with time dependent drift and diffusion coefficients using celebrated backward Fokker-Planck and path decomposition methods. We have explored nontrivial scaling behaviour of different PDFs and calculated explicitly the critical exponents related with the asymptotic limits in time.