Equilibrium Pure States and Nonequilibrium Chaos

TL;DR

This paper investigates the long-term behavior of nonequilibrium spin glass systems, showing that they rarely settle into a single pure state and that the configuration space is mostly boundary, challenging traditional equilibrium assumptions.

Contribution

It demonstrates that after a deep quench, systems either do not settle into a single pure state or depend on initial conditions, with proofs for 2D ferromagnets and implications for equilibrium concepts.

Findings

Deeply quenched 2D ferromagnets do not settle into a single pure state.

Most initial configurations lie on the boundary between pure states.

Time averages may align with Boltzmann averages despite multiple pure states.

Abstract

We consider nonequilibrium systems such as the Edwards-Anderson Ising spin glass at a temperature where, in equilibrium, there are presumed to be (two or many) broken symmetry pure states. Following a deep quench, we argue that as time goes to infinity, although the system is usually in some pure state locally, either it never settles permanently on a fixed lengthscale into a single pure state, or it does but then the pure state depends on both the initial spin configuration and the realization of the stochastic dynamics. But this latter case can occur only if there exists an uncountable number of pure states (for each coupling realization) with almost every pair having zero overlap. In both cases, almost no initial spin configuration is in the basin of attraction of a single pure state; that is, the configuration space (resulting from a deep quench) is all boundary (except for a set of measure zero). We prove that the former case holds for deeply quenched two-dimensional ferromagnets. Our results raise the possibility that even if more than one pure state exists for an infinite system, time averages don't necessarily disagree with Boltzmann averages.