Noisy continuous-opinion dynamics

We study the Deffuant et al. model for continuous--opinion dynamics under the influence of noise. In the original version of
this model, individuals meet in random pairwise encounters after which they compromise or not depending of a confidence parameter.
Free will is introduced in the form of noisy perturbations: individuals are given the opportunity to change their opinion,
with a given probability, to a randomly selected opinion inside the whole opinion space. We derive the master equation of this
process. One of the main effects of noise is to
induce an order-disorder transition. In the disordered state the opinion distribution tends to be uniform, while for the ordered
state a set of well defined opinion clusters are formed, although with some opinion spread inside them. Using a linear stability analysis we can derive approximate conditions for the
transition between opinion clusters and the disordered state. The master equation analysis is compared with direct Monte-Carlo simulations. We find that the master equation and the Monte-Carlo simulations do not always agree due to finite-size induced fluctuations that we analyze in some detail.