On a kinetic opinion formation model for pre-election polling.

Motivated by recent successes in model-based pre-election polling, we propose a kinetic model for opinion formation which includes voter demographics and socio-economic factors like age, sex, ethnicity, education level, income and other measurable factors like behaviour in previous elections or referenda as a key driver in the opinion formation dynamics. The model is based on Toscani's kinetic opinion formation model (Toscani G. 2006 Kinetic models of opinion formation. Commun. Math. Sci. 4, 481-496.) and the leader-follower model of Düring et al. (Düring B. et al. 2009 Boltzmann and Fokker-Planck equations modelling opinion formation in the presence of strong leaders. Proc. R. Soc. A 465, 3687-3708.), and leads to a system of coupled Boltzmann-type equations and associated, approximate Fokker-Planck-type systems. Numerical examples using data from general elections in the UK show the effect different demographics have on the opinion formation process and the outcome of elections. This article is part of the theme issue 'Kinetic exchange models of societies and economies'.

An Elo-type rating model for players and teams of variable strength.

The Elo rating system, which was originally proposed by Arpad Elo for chess, has become one of the most important rating systems in sports, economics and gaming. Its original formulation is based on two-player zero-sum games, but it has been adapted for team sports and other settings. In 2015, Junca and Jabin proposed a kinetic version of the Elo model, and showed that under certain assumptions the ratings do converge towards the players' strength. In this paper, we generalize their model to account for variable performance of individual players or teams. We discuss the underlying modelling assumptions, derive the respective formal mean-field model and illustrate the dynamics with computational results. This article is part of the theme issue 'Kinetic exchange models of societies and economies'.

An anisotropic interaction model for simulating fingerprints.

Evidence suggests that both the interaction of so-called Merkel cells and the epidermal stress distribution play an important role in the formation of fingerprint patterns during pregnancy. To model the formation of fingerprint patterns in a biologically meaningful way these patterns have to become stationary. For the creation of synthetic fingerprints it is also very desirable that rescaling the model parameters leads to rescaled distances between the stationary fingerprint ridges. Based on these observations, as well as the model introduced by Kücken and Champod we propose a new model for the formation of fingerprint patterns during pregnancy. In this anisotropic interaction model the interaction forces not only depend on the distance vector between the cells and the model parameters, but additionally on an underlying tensor field, representing a stress field. This dependence on the tensor field leads to complex, anisotropic patterns. We study the resulting stationary patterns both analytically and numerically. In particular, we show that fingerprint patterns can be modeled as stationary solutions by choosing the underlying tensor field appropriately.

Kinetic equations modelling wealth redistribution: a comparison of approaches.

Kinetic equations modelling the redistribution of wealth in simple market economies is one of the major topics in the field of econophysics. We present a unifying approach to the qualitative study for a large variety of such models, which is based on a moment analysis in the related homogeneous Boltzmann equation, and on the use of suitable metrics for probability measures. In consequence, we are able to classify the most important feature of the steady wealth distribution, namely the fatness of the Pareto tail, and the dynamical stability of the latter in terms of the model parameters. Our results apply, e.g., to the market model with risky investments [S. Cordier, L. Pareschi, and G. Toscani, J. Stat. Phys. 120, 253 (2005)], and to the model with quenched saving propensities [A. Chatterjee, B. K. Chakrabarti, and S. S. Manna, Physica A 335, 155 (2004)]. Also, we present results from numerical experiments that confirm the theoretical predictions.