Ordering dynamics with two non-excluding options Bilingualism in language competition.pdf

a r X i v : p h y s i c s / 0 6 0 9 0 7 9 v 1 [ p h y s i c s .s o c - p h ] 9 S e p 2 0 0 6 Ordering dynamics with two non-excluding options: Bilingualism in language competition Xavier Castello?, V??ctor M. Egu??luz, and Maxi San Miguel IMEDEA (CSIC-UIB), Campus Universitat Illes Balears E-07122 Palma de Mallorca, Spain e-mail: xavi@imedea.uib.es Abstract: We consider a modification of the voter model in which a set of interacting elements (agents) can be in either of two equivalent states (A or B) or in a third additional mixed AB state. The model is motivated by studies of language competition dynamics, where the AB state is associated with bilingualism. We study the ordering process and associated interface and coarsening dynamics in regular lattices and small world networks. Agents in the AB state define the interfaces, changing the interfacial noise driven coarsening of the voter model to curvature driven coarsening. We argue that this change in the coarsening mechanism is generic for perturbations of the voter model dynamics. When interaction is through a small world network the AB agents restore coarsening, eliminating the metastable states of the voter model. The time to reach the absorbing state scales with system size as τ ～ lnN to be compared with the result τ ～ N for the voter model in a small world network. 1.INTRODUCTION Understanding the complex collective behavior of many particle systems in terms of a microscopic description based on the interaction rules among the particles is the well established purpose of Statistical Physics. This micro-macro paradigm [1] is also shared by Social Science studies based on agent interactions (Agent Based Models). In many cases parallel re- search in both disciplines goes far beyond superficial analogies. For example, Schelling’s model [1] of residential segregation is mathematically equivalent to the zero-temperature spin-exchange Kinetic Ising model with vacancies. Cross-fertilization between these rese