Spectral formulation and WKB approximation for rare-event statistics in reaction systems

TL;DR

This paper introduces a spectral and stationary WKB approach to estimate rare-event probabilities in reaction systems, comparing it with existing methods and validating against an exactly solvable binary annihilation model.

Contribution

It presents a novel spectral formulation and stationary WKB approximation for calculating rare-event probabilities in reaction systems, extending existing semiclassical methods.

Findings

The stationary WKB approximation aligns well with exact solutions.

Comparison shows advantages over time-dependent semiclassical methods.

Validated approach on binary annihilation reaction model.

Abstract

We develop a spectral formulation and a stationary WKB approximation for calculating the probabilities of rare events (large deviations from the mean) in systems of reacting particles with infinite-range interaction, describable by a master equation. We compare the stationary WKB approximation with a recent time-dependent semiclassical approximation developed, for the same class of problems, by Elgart and Kamenev. As a benchmark we use an exactly solvable problem of the binary annihilation reaction 2A -> 0.