Mean-field theory of modified voter model for opinions

We present a mean-field theory for the modified three-state voter model. We obtain analytical results in asymptotic states for a general transition matrix. Numerical simulations can be well described within the proposed formulations. Distributions for both central plateau and boundary layer can be written explicitly. The central plateau is determined solely by the ratio between the internal noise and the external pressure. In the bulk, these two factors substantially compete with each other. Near the boundary regions, both the factors work together to introduce the exponential distribution of transit layer. With the scaling, simulations on a large lattice can be mapped into a small lattice. The required computer time is hence reduced significantly.