Kinetic Ising models on the square lattice with both nearest-neighbor interactions and self-interaction are studied for the cases of random sequential updating and parallel updating. The equilibrium phase diagrams and critical dynamics are studied using Monte Carlo simulations and analytic approximations. The Hamiltonians appearing in the Gibbs distribution describing the equilibrium properties differs for sequential and parallel updating but in both cases feature multispin and non-nearest-neighbor couplings. For parallel updating the system is a probabilistic cellular automaton and the equilibrium distribution satisfies detailed balance with respect to the dynamics [E. N. M. Cirillo, P. Y. Louis, W. M. Ruszel and C. Spitoni, Chaos, Solitons and Fractals, 64:36(2014)]. In the limit of weak self-interaction for parallel dynamics, odd and even sublattices are nearly decoupled and checkerboard patterns are present in the critical and low temperature regimes, leading to singular behavior in the shape of the critical line. For sequential updating the equilibrium Gibbs distribution satisfies global balance but not detailed balance and the Hamiltonian is obtained perturbatively in the limit of weak nearest-neighbor dynamical interactions. In the limit of strong self-interaction the equilibrium properties for both parallel and sequential updating are described by a nearest-neighbor Hamiltonian with twice the interaction strength of the dynamical model.
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