Critical exponents for extended dynamical systems with simultaneous updating: the case of the Ising model

The behavior of a simultaneously updated Metropolis simulation of the two-dimensional (2D) Ising model is explored. The configurations generated by such a simulation are not constructed to be equilibrium configurations, and may therefore display a different critical behavior, assuming a phase transition is present. This is relevant to the recently posed question of how the updating scheme affects the critical exponents for chaotic extended systems that show continuous phase transitions. It is found that the simultaneously updated algorithm drives the lattice, for all non-zero temperatures, into blinking maximum energy configurations, making it of little use. A small modification, given by a reduction in the acceptance probabilities for spin flips, restores the ability of the algorithm to generate ordered and disordered states, with a well-defined phase transition between them. The critical temperature becomes a function of the acceptance probability for energy lowering spin flips. The critical exponents still fall into the 2D Ising universality class. This happens even though the simultaneously updated Metropolis algorithm does not really simulate the Ising model in equilibrium.