Statistical Physics in a three-state extended voter model

Abstract

[eng] This thesis aims to give a detailed study of a set of interacting particle systems with three possible particle states. The concerned systems are extensions of the conventional voter model. The proposed model, which we call The Extended Voter Model, allow particles to interact following two mechanisms: voter reactions and non-voter reactions. Voter reactions are reactions where a particle adopts a new state by imitating the state of the particle with which it has interacted. Non-voter reactions are reactions where a particle adopts an state different from the one of the particle with which it has interacted. In systems with more than three possible states there are diverse options for this new state, however, as the thesis is focused only on three-state systems, the non-voter reactions are uniquely defined. Introducing this addition to the conventional voter model gives rise to several different interacting particle systems from all the possible combinations between voter and non-voter reactions, which consensus dynamics are unknown. Therefore, the present work has first searched for methods to choose and discard the three-state systems that had emerged beyond the conventional voter model. The subsequent study of their consensus dynamics has revealed some rich, interesting behaviours different from the conventional voter model dynamics, for which analytical expressions do not exist yet. This thesis collects and classifies these systems in four classes according to their consensus times behaviour: Logarithmic, Fractional, Linear (voter model like behaviour) and Exponential class. To understand the origin of the differences, we have developed three diagrammatic descriptions to represent the systems: the Product Focused Representation, The Catalyst Focused Representation and the Boolean State Representation. For the same purpose, we have solved the deterministic equations of the systems and so, we have discovered the mechanism that are governing in each class of systems in our extended voter model. Finally, a Coarse-Grained Representation has been created as an approach to determine the consensus dynamics without carrying out a numerical study or solving the deterministic equations.