Characteristics of isothermal Fokker–Planck equation for opinion-cluster involved with self-thinking

Abstract

In this paper, we investigate two types of isothermal Fokker–Planck equation to demonstrate the opinion-cluster involved with self-thinking. Firstly, the isothermal Fokker–Planck equation (IFPE type-I) is introduced from the microscopic equation of motion of opinions by extending the Hegselmann–Krause (HK) model in discrete time to the HK model in continuum time. We find that a steady solution of the opinion distribution function obtained using the IFPE type-I depends only on initial conservative variables. A steady solution of the opinion distribution function obtained using the IFPE type-I, however, deviates from that obtained using the isothermal HK model, which is solved using the direct simulation Monte Carlo (DSMC) method, when the time interval, which is used to solve the isothermal HK model, becomes large. Afterwards, we consider another type of the IFPE (IFPE type-II) from the inelastic Boltzmann equation with the cut-off opinions, which is equivalent to the microscopic model by Deffuant et al (2000 Adv. Complex Syst. 3 87). A steady solution of the opinion distribution function obtained using the IFPE type-II depends on the initial state of opinions. Such a dependency of the steady solution on the initial opinion distribution function is exclusively caused by the dependency of the propagation speed of the opinion on the value of the opinion in the IFPE type-II.