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We studied the critical behavior of the $J_{1}-J_{2}$ spin-{1/2} Ising modelin the square lattice by considering $J_{1}$ fixed and $J_{2}$ as randominteractions following discrete and continuous probability distributionfunctions. The configuration of $J_{2}$ in the lattice evolves in time througha competing kinetics using Monte Carlo simulations leading to a steady statewithout reaching the free-energy minimization. However, the resultingnon-equilibrium phase diagrams are, in general, qualitatively similar to thoseobtained with quenched randomness at equilibrium in past works. Accordingly,through this dynamics the essential critical behavior at finite temperaturescan be grasped for this model. The advantage is that simulations spend lesscomputational resources, since the system does not need to be replicated orequilibrated with Parallel Tempering. A special attention was given for thevalue of the amplitude of the correlation length at the critical point of thesuperantiferromagnetic-paramagnetic transition.

Nonequilibrium study of the $J_{1}-J_{2}$ Ising model with random $J_{2}$ couplings in the square lattice