A Continuum Generalization of the Ising Model

TL;DR

This paper introduces a continuum version of the Ising model using integro-differential equations, enabling advanced analysis of phase transitions, material properties, and magnetic domain dynamics.

Contribution

It presents a novel continuum formulation of the Ising model, extending its analytical capabilities beyond the traditional discrete approach.

Findings

Allows asymptotic analysis of phase transitions

Enables study of material properties

Models dynamics of magnetic domain formation

Abstract

The Lenz-Ising model has served for almost a century as a basis for understanding ferromagnetism, and has become a paradigmatic model for phase transitions in statistical mechanics. While retaining the Ising energy arguments, we use techniques previously applied to sociophysics to propose a continuum model. Our formulation results in an integro-differential equation that has several advantages over the traditional version: it allows for asymptotic analysis of phase transitions, material properties, and the dynamics of the formation of magnetic domains.