Long-Time Predictability in Disordered Spin Systems Following a Deep Quench

TL;DR

This paper investigates how the influence of initial conditions on the final state of disordered spin systems diminishes with increasing dimension or system size, combining numerical simulations and analytical models.

Contribution

It provides new insights into the decay of initial state influence in disordered spin systems and supports conjectures with both numerical and theoretical analyses.

Findings

Initial state influence decays with increasing dimension in short-range systems.

In infinite-range models, initial conditions fully determine the final state.

Frustration in mean-field models significantly affects predictability.

Abstract

We study the problem of predictability, or "nature vs. nurture", in several disordered Ising spin systems evolving at zero temperature from a random initial state: how much does the final state depend on the information contained in the initial state, and how much depends on the detailed history of the system? Our numerical studies of the "dynamical order parameter" in Edwards-Anderson Ising spin glasses and random ferromagnets indicate that the influence of the initial state decays as dimension increases. Similarly, this same order parameter for the Sherrington-Kirkpatrick infinite-range spin glass indicates that this information decays as the number of spins increases. Based on these results, we conjecture that the influence of the initial state on the final state decays to zero in finite-dimensional random-bond spin systems as dimension goes to infinity, regardless of the presence of frustration. We also study the rate at which spins "freeze out" to a final state as a function of dimensionality and number of spins; here the results indicate that the number of "active" spins at long times increases with dimension (for short-range systems) or number of spins (for infinite-range systems). We provide theoretical arguments to support these conjectures, and also study analytically several mean-field models: the random energy model, the uniform Curie-Weiss ferromagnet, and the disordered Curie-Weiss ferromagnet. We find that for these models, the information contained in the initial state does not decay in the thermodynamic limit-- in fact, it fully determines the final state. Unlike in short-range models, the presence of frustration in mean-field models dramatically alters the dynamical behavior with respect to the issue of predictability.