Diffusion and memory effect in a stochastic processes and the correspondence to an information propagation in a social system

Abstract

A generalized Langevin equation is suggested to describe a system with memory($u(t,t') = \frac{1}{\Gamma (\nu )}(t - t')^\nu $) as well as with positive and negative damping. The equation can be transformed into the Fokker-Planck equation by using the Kramers-Moyal expansion. The solution of Fokker-Planck equation shows that velocity obeys a Gaussian distribution. The distribution curve will flatten as the memory parameter increases, which indicates that memory can enhance the randomness of the system. There are also some other memory effects behind this distribution, which can be characterized by calculating the transport coefficients, mean square displacement and correlation between the noise and space. These discussions can be paralleled to a social system to understand the propagation of social ideology caused by memory.