Dynamics of ferromagnetic spherical spin models with power law interactions: exact solution

We solve the Langevin dynamics of d-dimensional ferromagnetic spherical models with interactions that decay with distance as r−(d+σ). The long time dynamics of correlations and responses are studied in detail in the different dynamical regimes and the validity of fluctuation–dissipation relations (or its violation) are shown. In particular, we show that the fluctuation–dissipation ratio X(t+tw,tw) is asymptotically a function only of the waiting time tw in the aging regime and that X→0 as tw→∞. The results are valid in any finite dimension d and for 0<σ<2 where short-range behavior is recovered. We also solve the T=0 Cahn–Hilliard dynamics of this model (conserved order parameter). An analysis of the multiscaling behavior of the autocorrelation function is presented.