On the emergence of an 'intention field' for socially cohesive agents
We argue that when a social convergence mechanism exists and is strong enough, one should expect the emergence of a well defined 'field', i.e. a slowly evolving, local quantity around which individual attributes fluctuate in a finite range. This condensation phenomenon is well illustrated by the Deffuant-Weisbuch opinion model for which we provide a natural extension to allow for spatial heterogeneities. We show analytically and numerically that the resulting dynamics of the emergent field is a noisy diffusion equation that has a slow dynamics. This random diffusion equation reproduces the long-ranged, logarithmic decrease of the correlation of spatial voting patterns empirically found in [1, 2]. Interestingly enough, we find that when the social cohesion mechanism becomes too weak, cultural cohesion breaks down completely, in the sense that the distribution of intentions/opinions becomes infinitely broad. No emerging field exists in this case. All these analytical findings are confirmed by numerical simulations of an agent-based model.