Evolution of order parameters in disordered spin systems — a closure procedure

Abstract

In this paper we have discussed a theory to describe the dynamics of disordered spin systems in terms of a closed set of deterministic flow equations for a finite number of macroscopic order parameters. The theory is based on the systematic removal of microscopic memory effects. We have used the theory to analyse two archetypical disordered spin systems: the Hopfield [1] neural network model near saturation and the Sherrington-Kirkpatrick [2] spin-glass model. In addition we have studied an exactly solvable toy model, in order to obtain a quantitative understanding of the potential and restrictions of our theory. Full details of the derivations involved in the three case studies can be found in [6, 7, 14]. Our equations are by construction exact in three limits: (i) removal of the disorder, (ii) for t = 0 (upon choosing appropriate initial conditions), and (iii) for t = ∞. Replica theory enters as a tool in calculating the local field distribution, and involves dynamical generalisations of familiar objects from equilibrium replica theory, like the overlap distribution P(q) and of AT [9] and zero-entropy lines. At fixed-points of our flow equations we recover the full equilibrium replica theory, including replica symmetry breaking if it occurs, and the corresponding phase diagrams. For intermediate times our equations capture the main features of the flow in the order parameter plane (for homogeneous initial conditions). The theory fails, however, in that for large times the relaxation towards equilibrium is slower than predicted by the theory. Our results show that for homogeneous initial conditions the impact of microscopic memory effects on the evolution of the macroscopic order parameters appears to be mainly an overall slowing down.