Chains of Mean Field Models

TL;DR

This paper studies a chain of coupled Curie-Weiss spin systems, revealing oscillatory behavior in the van der Waals curve and phenomena analogous to phase transitions, with implications for error-correcting codes and graphical models.

Contribution

It introduces a novel chain model of coupled mean field systems, analyzing phase transition phenomena and oscillations in the van der Waals curve with potential applications in coding theory.

Findings

Oscillations in the van der Waals curve around the Maxwell plateau.

Oscillation period inversely proportional to chain length.

Oscillation amplitude exponentially small in interaction range.

Abstract

We consider a collection of Curie-Weiss (CW) spin systems, possibly with a random field, each of which is placed along the positions of a one-dimensional chain. The CW systems are coupled together by a Kac-type interaction in the longitudinal direction of the chain and by an infinite range interaction in the direction transverse to the chain. Our motivations for studying this model come from recent findings in the theory of error correcting codes based on spatially coupled graphs. We find that, although much simpler than the codes, the model studied here already displays similar behaviors. We are interested in the van der Waals curve in a regime where the size of each Curie-Weiss model tends to infinity, and the length of the chain and range of the Kac interaction are large but finite. Below the critical temperature, and with appropriate boundary conditions, there appears a series of equilibrium states representing kink-like interfaces between the two equilibrium states of the individual system. The van der Waals curve oscillates periodically around the Maxwell plateau. These oscillations have a period inversely proportional to the chain length and an amplitude exponentially small in the range of the interaction; in other words the spinodal points of the chain model lie exponentially close to the phase transition threshold. The amplitude of the oscillations is closely related to a Peierls-Nabarro free energy barrier for the motion of the kink along the chain. Analogies to similar phenomena and their possible algorithmic significance for graphical models of interest in coding theory and theoretical computer science are pointed out.