# WKB-type-of approximation for rare event statistics in reacting systems

## TL;DR

This paper develops a WKB-based approximation method to calculate rare event probabilities in reacting particle systems by transforming the master equation into a Schrödinger-like form and applying saddle point techniques.

## Contribution

It introduces a WKB-type approximation for the master equation in reacting systems, extending previous methods to analyze systems with analytically intractable dynamics.

## Key findings

- Derived a WKB approximation for rare event probabilities.
- Applied the method to various example systems.
- Extended previous analytical approaches to more complex systems.

## Abstract

We calculate the probabilities to find systems of reacting particles in states which largely deviate from typical behavior. The rare event statistics is obtained from the master equation which describes the dynamics of the probability distribution of the particle number. We transform the master equation by means of a generating function into a time-dependent "Schr\"odinger equation". Its solution is provided by a separation ansatz and an approximation for the stationary part which is of Wentzel-Kramers-Brillouin (WKB) type employing a small parameter. The solutions of the "classical" equations of motions and a saddle point approximation yield the proper generating function. Our approach extends a method put forward in [V. Elgart and A. Kamenev, Phys. Rev. E 70, 041106 (2004)]. We calculate the rare event statistics for systems where the dynamics cannot be entirely analyzed in an analytical manner. We consider different examples.