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The problem of the equivalence of the spherical and mean spherical models,which has been thoroughly studied and understood in equilibrium, is consideredanew from the dynamical point of view during the time evolution following aquench from above to below the critical temperature. It is found that thereexists a crossover time $t^* \sim V^{2/d}$ such that for $t < t^*$ the twomodels are equivalent, while for $t > t^*$ macroscopic discrepancies arise. Therelation between the off equilibrium response function and the structure of theequilibrium state, which usually holds for phase ordering systems, is found tohold for the spherical model but not for the mean spherical one. The lattermodel offers an explicit example of a system which is not stochasticallystable.Comment: 11 pages, 1 figure, references corrected, to appear in Phys.Rev.

Relaxation and overlap-probability function in the spherical and mean-spherical models