A diagrammatic theory of time correlation functions of facilitated kinetic Ising models

Abstract

We present a diagrammatic formulation of the kinetic theory of time correlation functions for facilitated kinetic Ising models with directed constraints. Such models have been studied because their relaxation properties are similar to those of glass forming liquids and because some of them undergo ergodic–nonergodic transitions. Using a Hilbert space formulation of the dynamics of Markov processes that satisfy a detailed balance condition, we derive a diagrammatic series for the spin autocorrelation function, its memory function, and its irreducible memory function. Using graphical methods we derive various approximations for the irreducible memory function, including approximations equivalent to various versions of the simplified and extended mode coupling theory of Götze and co-workers. An extended mode coupling theory for the “East model” is compared with simulation data. The theory is accurate at short and intermediate times for all up-spin concentrations and for all times at high up-spin concentrations. The major failing of the extended mode coupling theory for this model is the prediction of long time decay that is too rapid and too exponential, as compared with the simulation results.