Slowed relaxational dynamics beyond the fluctuation-dissipation theorem

To describe the slow dynamics of a system out of equilibrium, but close to a dynamical arrest, we generalise the ideas of previous work to the case where time-translational invariance is broken. We introduce a model of dynamics that is reasonably general, and show how all of the unknown parameters of this model may be related to the observables or to averages of the noise. One result is a generalisation of the Fluctuation-dissipation theorem of type two (FDT2), and the method is thereby freed from this constraint. Significantly, a systematic means of implementing the theory to higher order is outlined. We propose the simplest possible closure of these generalised equations, following the same type of approximations that have been long known for the equilibrium case of mode coupling theory (MCT). Naturally, equilibrium MCT equations are found as a limit of this generalised formalism.