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We study the stochastic relaxation dynamics of the Ising p-spin model on arandom graph, a well-known model with glassy dynamics at low temperatures. Weintroduce and discuss a new closure scheme for the master equation governingthe continuous-time relaxation of the system, that translates into a set ofdifferential equations for the evolution of local probabilities. The solutionto these dynamical mean-field equations describes very well theout-of-equilibrium dynamics at high temperatures, notwithstanding the keyobservation that the off-equilibrium probability measure contains higher-orderinteraction terms, not present in the equilibrium measure. In thelow-temperature regime, the solution to the dynamical mean-field equationsshows the correct two-step relaxation (a typical feature of the glassydynamics), but with a relaxation timescale too short. We propose a solution tothis problem by identifying the range of energies where entropic barriers playa key role and defining a renormalized microscopic timescale for the dynamicalmean-field solution. The final result perfectly matches the complexout-of-equilibrium dynamics computed through extensive Monte Carlo simulations.

Improved mean-field dynamical equations are able to detect the two-steps relaxation in glassy dynamics at low temperatures