Role of initial conditions in the mean-field theory of spin-glass dynamics

Abstract

We discuss the dynamics of the infinite-range Sherrington-Kirkpatrick spin-glass model for which relaxation times diverge when $N$, the number of spins, tends to infinity. Calculations on a large but finite system are very difficult, so we mimic a large finite system in equilibrium by working with $N=\ensuremath{\infty}$ and imposing, by hand, a canonical distribution at an initial time. For short times, where no barrier hopping has occurred, we find that the Edwards-Anderson order parameter, ${q}_{\mathrm{EA}}$, is identical to that obtained from an analysis of the mean-field equations of Thouless, Anderson, and Palmer and, with further assumptions, gives $q(x=1)$ in Parisi's theory, in agreement with earlier work. For times longer than the longest relaxation time (of the finite system), true equilibrium is reached and our theory agrees with previous statistical-mechanics calculations using the replica trick. There is no violation of the fluctuation-dissipation theorem.