Equilibrium dynamics of spin-glass systems

We present a critical analysis of the Sompolinsky theory of equilibrium dynamics. By using the spherical 2+p spin-glass model we test the asymptotic static limit of the Sompolinsky solution showing that it fails to yield a thermodynamically stable solution. We then present an alternative formulation, based on the Crisanti, Horner, and Sommers [Z. Phys. B: Condens. Matter 92, 257 (1993)] dynamical solution of the spherical p-spin spin-glass model, reproducing a stable static limit that coincides, in the case of a one step replica symmetry breaking ansatz, with the solution at the dynamic free energy threshold at which the relaxing system gets stuck off equilibrium. We formally extend our analysis to any number of replica symmetry breakings R. In the limit R ->infinity, both formulations lead to the Parisi antiparabolic differential equation. This is the special case, though, where no dynamic blocking threshold occurs. The formulation does not contain the additional order parameter Delta of the Sompolinsky theory.