Dynamical replica theory for disordered spin systems.

We present a method to solve the dynamics of disordered spin systems on finite time scales. It involves a closed driven diffusion equation for the joint spin-field distribution, with time-dependent coefficients described by a dynamical replica theory which, in the case of detailed balance, incorporates equilibrium replica theory as a stationary state. The theory is exact in various limits. We apply our theory to both the symmetric and the nonsymmetric Sherrington-Kirkpatrick spin glass, and show that it describes the ~numerical! experiments very well. Recently it has become clear 1 that even mean-field models exhibit the ageing phenomena, familiar from experiments on real spin glass, 2 that were hitherto assumed to be typical for short-range systems. This has led to a renewed interest in dynamical studies of mean-field spin-glass models and to valuable new insights into spin-glass dynamics away from equilibrium, see, e.g., Ref. 3. In this paper we present an approach to analyzing the dynamics of spin-glass models on finite time scales, leading to a dynamical replica theory, which, in the case of detailed balance, incorporates equilibrium replica theory as a stationary state ~including replica symmetry breaking, if it occurs!. The formalism is built on a closure procedure with which we obtain a closed diffusion equation for the joint spin-field distribution. It builds on and extends earlier studies. 4‐6 Our theory is proven to be exact for short times and in equilibrium. For intermediate times we can prove that it is exact if in the thermodynamic limit the joint spin-field distribution indeed obeys a closed dynamic equation. Here we discuss only the underlying physical ideas and the results of applying our theory to both the symmetric 7 and the nonsymmetric 8 Sherrington-Kirkpatrick spin glass. Full mathematical details will be published elsewhere. 9 We believe the agreement between theory and ~simulation! experiment to be quite convincing. The generalized ~asymmetric! version of the Sherrington