Equilibrium behavior in a nonequilibrium system: Ising-doped voter model on complete graph

While the Ising model belongs to the realm of equilibrium statistical mechanics, the voter model is an example of a nonequilibrium system. We examine an opinion formation model, which is a mixture of Ising and voter agents with concentrations $p$ and $1-p$, respectively. Although in our model for $p<1$ a detailed balance is violated, on a complete graph the average magnetization in the stationary state for any $p>0$ is shown to satisfy the same equation as for the pure Ising model ($p=1$). Numerical simulations confirm such a behavior, but the equivalence with the pure Ising model apparently holds only for magnetization. Susceptibility in our model diverges at the temperature at which magnetization vanishes, but its values depend on the concentration~$p$. Simulations on a random graph also show that a small concentration of Ising agents is sufficient to induce a ferromagnetic ordering.