Order Parameter Flow in the SK Spin-Glass I: Replica Symmetry

TL;DR

This paper develops a deterministic flow theory for the dynamics of the SK spin-glass model under Glauber dynamics, using replica symmetry to describe the evolution of order parameters and local field distributions.

Contribution

It introduces a novel macroscopic flow framework based on replica symmetry, extending static results to dynamic behavior in spin glasses.

Findings

The theory reproduces equilibrium results from statistical mechanics.

Numerical simulations support the accuracy of the flow equations in the replica-symmetric regime.

The approach provides dynamical generalizations of known static phase boundaries.

Abstract

We present a theory to describe the dynamics of the Sherrington- Kirkpatrick spin-glass with (sequential) Glauber dynamics in terms of deterministic flow equations for macroscopic parameters. Two transparent assumptions allow us to close the macroscopic laws. Replica theory enters as a tool in the calculation of the time- dependent local field distribution. The theory produces in a natural way dynamical generalisations of the AT- and zero-entropy lines and of Parisi's order parameter function $P(q)$. In equilibrium we recover the standard results from equilibrium statistical mechanics. In this paper we make the replica-symmetric ansatz, as a first step towards calculating the order parameter flow. Numerical simulations support our assumptions and suggest that our equations describe the shape of the local field distribution and the macroscopic dynamics reasonably well in the region where replica symmetry is stable.