Quasi-equilibrium in glassy dynamics: An algebraic view

We study a chain of identical glassy systems in a constrained equilibrium, where each bond of the chain is forced to remain at a preassigned distance to the previous one. We apply this description to mean-field glassy systems in the limit of a long chain where each bond is close to the previous one. We show that this construction defines a pseudo-dynamic process that in specific conditions can formally describe real relaxational dynamics for long times. In particular, in mean-field spin glass models we can recover in this way the equations of Langevin dynamics in the long time limit at the dynamical transition temperature and below. We interpret the formal identity as evidence that in these situations the configuration space is explored in a quasi-equilibrium fashion. Our general formalism, which relates dynamics to equilibrium, puts slow dynamics in a new perspective and opens the way to the computation of new dynamical quantities in glassy systems.