Boltzmann-like and Boltzmann-Fokker-Planck Equations as a Foundation of Behavioral Models

TL;DR

This paper develops a comprehensive mathematical framework for modeling behavioral changes in social systems using Boltzmann-like and Boltzmann-Fokker-Planck equations, incorporating social influences and individual variability.

Contribution

It introduces a general model for behavioral dynamics based on Boltzmann-like equations, including social forces and fields, and connects it to existing social science concepts.

Findings

The model encompasses many social science theories as special cases.

Introduction of social fields representing public opinion and norms.

Derivation of Boltzmann-Fokker-Planck equations for social behavior modeling.

Abstract

It is shown, that the Boltzmann-like equations allow the formulation of a very general model for behavioral changes. This model takes into account spontaneous (or externally induced) behavioral changes and behavioral changes by pair interactions. As most important social pair interactions imitative and avoidance processes are distinguished. The resulting model turns out to include as special cases many theoretical concepts of the social sciences. A Kramers-Moyal expansion of the Boltzmann-like equations leads to the Boltzmann-Fokker-Planck equations, which allows the introduction of ``social forces'' and ``social fields''. A social field reflects the influence of the public opinion, social norms and trends on behaviorial changes. It is not only given by external factors (the environment) but also by the interactions of the individuals. Variations of the individual behavior are taken into account by diffusion coefficients.