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A model for continuous-opinion dynamics is proposed and studied by takingadvantage of its similarities with a mono-dimensional granular gas. Agentsinteract as in the Deffuant model, with a parameter $\alpha$ controlling thepersuasibility of the individuals. The interaction coincides with the collisionrule of two grains moving on a line, provided opinions and velocities areidentified, with $\alpha$ being the so-called coefficient of normalrestitution. Starting from the master equation of the probability density ofall opinions, general conditions are given for the system to reach consensus.The case when the interaction frequency is proportional to the $\beta$-power ofthe relative opinions is studied in more detail. It is shown that themean-field approximation to the master equation leads to the Boltzmann kineticequation for the opinion distribution. In this case, the system alwaysapproaches consensus, which can be seen as the approach to zero of the opiniontemperature, a measure of the width of the opinion distribution. Moreover, thelong-time behaviour of the system is characterized by a scaling solution to theBoltzmann equation in which all time dependence occurs through the temperature.The case $\beta=0$ is related to the Deffuant model and is analyticallysoluble. The scaling distribution is unimodal and independent of $\alpha$. For$\beta>0$ the distribution of opinions is unimodal below a critical value of$|\alpha|$, being multimodal with two maxima above it. This means that agentsmay approach consensus while being polarized. Near the critical points and for$|\alpha|\ge 0.4$, the distribution of opinions is well approximated by the sumof two Gaussian distributions. Monte Carlo simulations are in agreement withthe theoretical results.

Approach to consensus in models of continuous-opinion dynamics: a study inspired by the physics of granular gases