System size expansion for systems with an absorbing state

TL;DR

This paper introduces a generalized system size expansion that accurately captures the dynamics of systems with an absorbing state, overcoming limitations of the traditional van Kampen approach.

Contribution

A novel expansion method incorporating non-Gaussian fluctuations around absorbing states, validated against the voter model and Gillespie simulations.

Findings

The new expansion matches the exact solution in large size and long time limits.

It correctly predicts the lifetime distribution's asymptotic behavior.

Outperforms the traditional van Kampen expansion in systems with absorbing states.

Abstract

The well known van Kampen system size expansion, while of rather general applicability, is shown to fail to reproduce some qualitative features of the time evolution for systems with an absorbing state, apart from a transient initial time interval. We generalize the van Kampen ansatz by introducing a new prescription leading to non-Gaussian fluctuations around the absorbing state. The two expansion predictions are explicitly compared for the infinite range voter model with speciation as a paradigmatic model with an absorbing state. The new expansion, both for a finite size system in the large time limit and at finite time in the large size limit, converges to to the exact solution as obtained in a numerical implementation using the Gillespie algorithm. Furthermore, the predicted lifetime distribution is shown to have the correct asymptotic behavior.