  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.
