A study of self-organizing processes of nonlinear stochastic variables

A general theory is given for the time evolution of nonlinear stochastic variables a(t) = {ai(t)} whose statistical distribution is changing due to the self-organization of “macroscopic” order. The dynamics of a(t) is conveniently expressed by self-consistent equations for the ensemble average x(t) = 〈a(t)〉, the supersystem, and for the deviations ξ(t) = a(t)−x(t), the subsystem; the systems are connected to each other by feedback loops in their dynamics. The time dependence of the variance and the correlation function ofξ(t) are studied in terms of relaxation toward local equilibrium underx(t) and dynamical coupling withx(t). A special example shows that the stochastic motions of subsystems are pulled together by the motion of the supersystem through feedback loops, and that this pull-together phenomenon occurs when symmetry-breaking instability exists in nonlinear systems.