Reduction from non-Markovian to Markovian dynamics: The case of aging in the noisy-voter model

TL;DR

This paper demonstrates how certain non-Markovian aging effects in the noisy-voter model can be effectively approximated by a Markovian process, simplifying analysis while preserving key dynamics.

Contribution

It introduces a method to reduce non-Markovian aging dynamics to an effective Markovian model by assuming rapid age distribution equilibration.

Findings

Effective Markovian process replicates non-linear noisy-voter phenomenology

Age distribution reaches a quasi-steady state quickly

Global system state remains out of equilibrium

Abstract

We study memory dependent binary-state dynamics, focusing on the noisy-voter model. This is a non-Markovian process if we consider the set of binary states of the population as the description variables, or Markovian if we incorporate "age", related to the time one has spent holding the same state, as a part of the description. We show that, in some cases, the model can be reduced to an effective Markovian process, where the age distribution of the population rapidly equilibrates to a quasi-steady state, while the global state of the system is out of equilibrium. This effective Markovian process shares the same phenomenology of the non-linear noisy-voter model and we establish a clear parallelism between these two extensions of the noisy-voter model.