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We introduce an heterogeneous nonlinear $q$-voter model with zealots and twotypes of susceptible voters, and study its non-equilibrium properties when thepopulation is finite and well mixed. In this two-opinion model, each individualsupports one of two parties and is either a zealot or a susceptible voter oftype $q_1$ or $q_2$. While here zealots never change their opinion, a$q_i$-susceptible voter ($i=1,2$) consults a group of $q_i$ neighbors at eachtime step, and adopts their opinion if all group members agree. We show thatthis model violates the detailed balance whenever $q_1 \neq q_2$ and hassurprisingly rich properties. Here, we focus on the characterization of themodel's non-equilibrium stationary state (NESS) in terms of its probabilitydistribution and currents in the distinct regimes of low and high density ofzealotry. We unveil the NESS properties in each of these phases by computingthe opinion distribution and the circulation of probability currents, as wellas the two-point correlation functions at unequal times (formally related to a"probability angular momentum"). Our analytical calculations obtained in therealm of a linear Gaussian approximation are compared with numerical results.

Characterization of the Nonequilibrium Steady State of a Heterogeneous Nonlinear $q$-Voter Model with Zealotry