Facilitated spin models, mode coupling theory, and ergodic–nonergodic transitions

Abstract

Facilitated kinetic Ising spin models are models for cooperative relaxation in liquids. Some of them have ergodic–nonergodic transitions of the type predicted by the mode coupling theory of supercooled liquids. We discuss two mode coupling theories, that of Kawasaki and one developed by us, and compare their predictions with the properties of several facilitated kinetic spin models, including the hierarchically constrained kinetic Ising model in one dimension (the East model), the North-east model, and the class of (a,a−1)-Cayley tree models. We present new simulation data for the East model. For models with low dimensionality and low coordination number, there is little or no relationship between the transitions predicted by the mode coupling theories and the actual behavior of the spin systems, with the mode coupling theories generally predicting transitions for models that don’t have them and attributing qualitatively incorrect properties to those transitions that do occur. The mode coupling theories describe the relaxation of the East model well for short times but fail at long times in the vicinity of the incorrectly predicted transition and for the states that are incorrectly predicted to be nonergodic. Simulation evidence is presented for scaling behavior of the relaxation for low temperatures and long times in the East model, but no extant mode coupling theory predicts this behavior correctly. An analogy between liquids and facilitated spin models is proposed, whereby the slightly supercooled liquid regime is analogous to the spin system states near the spurious mode coupling transition, and the low temperature supercooled liquid near its glass transition is analogous to the spin model states just above the actual ergodic–nonergodic transition (in the case of models that have such a transition) or in the low temperature scaling regime (in the case of models with this type of low temperature behavior). According to this analogy, the actual transition or the low temperature scaling behavior of the spin models is analogous to the behavior at or near the thermodynamic transition that is sometimes proposed as the basis for the glass transition in liquids.