A simple analytical description of the non-stationary dynamics in Ising spin systems

The analytical description of the dynamics in models with discrete variables (e.g. Isingspins) is a notoriously difficult problem, that can be tackled only undersome approximation.Recently a novel variational approach to solve the stationary dynamical regime has beenintroduced by Pelizzola [Eur. Phys. J. B, 86 (2013) 120], where simpleclosed equations arederived under mean-field approximations based on the cluster variational method. Here wepropose to use the same approximation based on the cluster variational method also for thenon-stationary regime, which has not been considered up to now within this framework. Wecheck the validity of this approximation in describing the non-stationary dynamical regime ofseveral Ising models defined on Erdos-R ́enyi random graphs: westudy ferromagnetic modelswith symmetric and partially asymmetric couplings, models with randomfields and also spinglass models. A comparison with the actual Glauber dynamics, solvednumerically, showsthat one of the two studied approximations (the so-called ‘diamond’approximation) providesvery accurate results in all the systems studied. Only for the spin glass models we find somesmall discrepancies in the very low temperature phase, probably due to the existence of alarge number of metastable states. Given the simplicity of the equations to be solved, webelieve the diamond approximation should be considered as the ‘minimalstandard’ in thedescription of the non-stationary regime of Ising-like models: any new method pretending toprovide a better approximate description to the dynamics of Ising-like models should performat least as good as the diamond approximation.