Generalized moment expansion of dynamic correlation functions in finite Ising systems

In this paper we study dynamic correlation functions of one- and two-dimensional kinetic Ising models, in particular, in situations where nonergodic behavior and critical slowing down emerge. We also investigate in how far nonexponential relaxation as described by a Williams-Watts function exp[-(t/τ${)}^{\beta }$] results in such systems. The method we apply is an expansion which simultaneously takes the high- and low-frequency behavior of observables into account (generalized moment expansion). This approximation can be applied to kinetic Ising models with arbitrary transition rate constants. Its computational effort does not increase when relaxation times diverge. However, the method involves the inversion of the transition operator and, hence, can be applied only to finite systems, the size of which depends on computational resources. We introduce a coarse graining of the state space which allows to extend the system size further and yields accurate magnetization correlation functions.