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

We solve the Langevin dynamics of d-dimensional ferromagnetic spherical mod els with interactions that decay with distance as r(-(d+sigma)). The long t ime dynamics of correlations and responses are studied in detail in the dif ferent dynamical regimes and the validity of fluctuation-dissipation relati ons (or its violation) are shown. In particular, we show that the fluctuati on-dissipation ratio X(t + t(w),t(w)) is asymptotically a function only of the waiting time t(w) in the aging regime and that X --> 0 as t(w) --> infi nity. The results are valid in any finite dimension d and for 0 < sigma <2 where short-range behavior is recovered. We also solve the T = 0 Cahn-Hilli ard dynamics of this model (conserved order parameter). An analysis of the multiscaling behavior of the autocorrelation function is presented. (C) 200 1 Elsevier Science B.V. All rights reserved.