  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.
