Scaled Langevin equation for complex systems: New linear scaling relation for weight factor

A set of scaled Langevin equations is proposed to study a long time tail of correlation functions for two model systems (Type I and Type II). Each system is composed of elements which are grouped into clusters according to dynamical activations for external forces. The clusters in Type I are characterized by linear scaling rules in repetitive operations, whereas the clusters in Type II are characterized by different kinds of linear scaling rules in the repetitive operations. We obtain a new linear scaling relation for a weight factor for the clusters in Type II, as a condition for the correlation function to behave a long time tail.