The Generalized Langevin Equation, in its original form derived by Mori using projection operators, is a powerful tool that has since then enabled the development of many coarse-graining procedures of various types. Its structure involving a convolution product mathematically results from the equilibrium assumption. Adaptations of the Mori-Zwanzig formalism to non-equilibrium processes have then been proposed in various ways, and take different forms depending on the quantitites one wishes to compute. In this talk, we will investigate the case of explicitly time-dependent microscopic dynamics, and time-dependent observables. We show in particular that the linear structure of the GLE holds true, and that a fluctuation-dissipation-like relation is also valid, but the physical interepretation of the memory kernel must be taken with care.
We then use these results to model the dynamics of phase-transitions. Motivated by recent works on crystal nucleation, we propose to describe the dynamics of phase transitions in terms of a non-stationary Generalized Langevin Equation for the order parameter. Here we do not aim at investigating the physical origin of memory effects at phase transitions in general, but rather to relate the extent of the memory kernel to quantities that are experimentally observed such as the induction time and the duration of the phase transformation process. Using a simple kinematic model and a recently developed numerical procedure, we show that the extent of the memory kernel is positively correlated with the duration of the transition and of the same order of magnitude, while the distribution of induction times does not have an effect.