Non-equilibrium diffusion characteristics of a particle system and the correspondence to a social system.

A Langevin equation is suggested to describe a system driven by correlated Gaussian white noise as well as with positive and negative damping demarcated by a critical velocity. The equation can be transformed into the Fokker-Planck equation by the Kramers-Moyal expansion. The solution exhibits some non-equilibrium phenomena. In the beginning the distribution curve of velocity/energy takes on a random oscillation, and then a near-equilibrium distribution described by the Boltzmann distribution is gradually established. However, a spike appears on the distribution curve and breaks this stable distribution. The spike moves in the direction of velocity/energy decreasing and is nonlinearly enlarged so as to sustain. The final distribution is a sharp peak formed by a monotonically ascending segment and a monotonically descending one. The calculating results of the statistical quantities demonstrate that the process is a sub-diffusion, and the spike originates from the correlation between noise and space. Based on a basic hypothesis that the generalized displacement is regarded as the information carried by an opinion particle, and the velocity is taken as the sensitivity of an agent to an event, we map the observations in this system to a social system to understand the propagation of public opinion or message.