Reduction of order, resummation and radiation reaction

TL;DR

This paper investigates the divergence of the reduction of order series in radiation reaction equations, demonstrating that Borel-Padé resummation can accurately reproduce the dynamics and highlighting limitations of the Landau-Lifshitz approximation.

Contribution

It shows the divergence of the reduction of order series and introduces Borel-Padé resummation as an effective method to recover accurate dynamics in radiation reaction models.

Findings

The reduction of order series diverges at higher orders.

Borel-Padé resummation accurately reproduces the resummed dynamics.

Landau-Lifshitz equation is optimal for large times but not for short times.

Abstract

The Landau-Lifshitz equation is the first in an infinite series of approximations to the Lorentz-Abraham-Dirac equation obtained from `reduction of order'. We show that this series is divergent, predicting wildly different dynamics at successive perturbative orders. Iterating reduction of order ad infinitum in a constant crossed field, we obtain an equation of motion which is free of the erratic behaviour of perturbation theory. We show that Borel-Pad\'e resummation of the divergent series accurately reproduces the dynamics of this equation, using as little as two perturbative coefficients. Comparing with the Lorentz-Abraham-Dirac equation, our results show that for large times the optimal order of truncation typically amounts to using the Landau-Lifshitz equation, but that this fails to capture the resummed dynamics over short times.