Memory effects in the relaxation of Ising models

– It is analytically shown that the one-dimensional Ising model with Glauber dynamics exhibits short time memory effects when submitted to an abrupt change in the temperature. These effects are qualitatively similar to those experimentally observed in the compaction of vibrated granular materials. Moreover, a critical time separating regimes of “normal” and “anomalous” responses to the perturbation is found. The dynamics of granular systems exhibits a rich phenomenology that has been the subject of intensive study in the last years [1]. In particular, when submitted to vertical vibration starting from a loose packing configuration, granular media compact behaving in a way that is reminiscent of conventional structural glasses [2,3]. The glassy nature of granular compaction clearly shows up when the response of the system to a sudden change in the vibration intensity is measured [4–6]. The results show the presence of relevant short-termmemory effects that are in contrast with the long time behaviour observed at constant intensity. A decrease (increase) in the vibration acceleration leads to an increase (decrease) of the compaction rate at short times. For large times, the previous history is forgotten and the relaxation rate becomes the same as for a constant intensity process. Here we study the existence of a similar “anomalous” response in the context of the one-dimensional Ising model with Glauber dynamics [7]. This is a model with short range interactions that exhibits many of the qualitative features of conventional structural glasses, like slow non-exponential relaxation, a kinetic phenomenon resembling a laboratory glass transition, aging, and hysteresis effects [8–19]. On the other hand, it is simple enough to allow exact analytical calculations in situations where more realistic models are untractable. In particular, we will be able to calculate the short-time response of the system to a sudden small change in the temperature, which plays in the model a role analogous to the vibration intensity in granular compaction. As long as the change is not made at the early stages of the relaxation process, the system shows an “anomalous” response. On the other hand, if the perturbation is introduced very soon, the response is “normal”, in the sense that the modification of the relaxation rate has the same sign as the temperature change. Let us mention that these two qualitatively different time regimes have been also identified within a general theoretical framework developed to analyze memory effects in granular compaction [6], although it has not been experimentally verified yet to the best of our knowledge.