We study the non-equilibrium relaxation of the spherical spin-glass model with p-spin interactions in the N → ∞ limit. We analytically solve the asymptotics of the magnetization and the correlation and response functions for long but finite times. Even in the thermodynamic limit the system exhibits 'weak' (as well as 'true') ergodicity breaking and aging effects. We determine a functional Parisi-like order parameter P d (q) which plays a similar role for the dynamics to that played by the usual function for the statics.

We show that, in nonequilibrium systems with small heat flows, there is a time-scale dependent effective temperature which plays the same role as the thermodynamical temperature, in that it controls the direction of heat flows and acts as a criterion for thermalization. We simultaneously treat the case of stationary systems with weak stirring and of glassy systems that age after cooling and show that they exhibit very similar behavior provided that time dependences are expressed in terms of the correlations of the system. We substantiate our claims with examples taken from solvable models with nontrivial low-temperature dynamics, but argue that they have a much wider range of validity. We suggest experimental checks of these ideas. 75.40.Gb, 75.10.Nr, Typeset using REVT E X 1

Another set of experiments which basically carry the same information is those of the so-called 'Thermo-Remanent Magnetisation' (TRM) relaxation 13,14 . The system is cooled under a small magnetic field H, which is left from t = 0 (the time of the quench) to t = t w , and then suddenly switched off. The subsequent relaxation of the magnetisation M can be decomposed as 21,14c This requires to perform many independent quenches where the magnetic noise is recorded for different ages tw and then averaged over the different quenches.

We derive analytical results for the large-time relaxation of the Sherrington -Kirkpatrick model in the thermodynamic limit, starting from a random configuration.The system never achieves local equilibrium in any fixed sector of phasespace, but remains in an asymptotic out of equilibrium regime.We propose as a tool, both numerical and analytical, for the study of the out of equilibrium dynamics of spin-glass models the use of 'triangle relations' which describe the geometry of the configurations at three (long) different times.