Abstract
The aim of this paper is to study the derivation of appropriate meso- and macroscopic models for interactions as appearing in social processes. There are two main characteristics the models take into account, namely a network structure of interactions, which we treat by an appropriate mesoscopic description, and a different role of interacting agents. The latter differs from interactions treated in classical statistical mechanics in the sense that the agents do not have symmetric roles, but there is rather an active and a passive agent. We will demonstrate how a certain form of kinetic equations can be obtained to describe such interactions at a mesoscopic level and moreover obtain macroscopic models from monokinetics solutions of those. The derivation naturally leads to systems of nonlocal reaction-diffusion equations (or in a suitable limit local versions thereof), which can explain spatial phase separation phenomena found to emerge from the microscopic interactions. We will highlight the approach in three examples, namely the evolution and coarsening of dialects in human language, the construction of social norms, and the spread of an epidemic.
Highlights
The mathematical modelling of social interactions has been a topic of high recent interest
We show that for this case there is a decay of variance in the Vlasov approximation, i.e. the monokinetic solutions approximate the overall dynamics well
We have demonstrated that challenging classes of PDE systems can already arise as monokinetic equation of Vlasov approximations, further studies of the full model including a nontrivial variance in the state space are an interesting topic for future research
Summary
From a mathematical point of view, it is natural to approach the transition from microscopic interaction to macroscopic models with the methods of statistical physics and kinetic theory, yielding (systems of) partial differential equations for distributions, with wellestablished asymptotic methods to further simplify or to analyze pattern formation (cf [6, 8, 17,18,19, 22, 34, 36, 46, 50]). The final one concerns the spread of an epidemic, a classical topic of mathematical modelling, for which we use a novel approach motivated by recent findings on pandemic spread As another consequence of these exemplary cases we work out a frequent very asymmetric nature of interactions.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
