Mesoscopic description of the annealed Ising model and Multiplicative noise

TL;DR

This paper introduces a mesoscopic Langevin equation with multiplicative noise that models the annealed Ising model's phase transitions, revealing different underlying mechanisms for disorder-to-order transitions despite similar phenomenology.

Contribution

A novel Langevin equation derived from the annealed Ising model that captures its phase transition behavior and clarifies different mechanisms behind noise-induced transitions.

Findings

The Langevin equation reproduces the AIM's phase transition phenomenology.

It reveals that disorder-to-order transitions have different controlling mechanisms.

The model demonstrates two distinct phase transitions related to temperature and noise amplitude.

Abstract

A new type of Langevin equation exhibiting a non trivial phase transition associated with the presence of multiplicative noise is introduced. The equation is derived as a mesoscopic representation of the microscopic annealed Ising model (AIM) proposed by Thorpe and Beeman, and reproduces perfectly its basic phenomenology. The AIM exhibits a non-trivial behavior as the temperature is increased, in particular it presents a disorder-to-order phase transition at low temperatures, and a order-to-disorder transition at higher temperatures. This behavior resembles that of some Langevin equations with multiplicative noise, which exhibit also two analogous phase transitions as the noise-amplitude is increased. By comparing the standard models for noise-induced transitions with our new Langevin equation we elucidate that the mechanisms controlling the disorder-to-order transitions in both of them are essentially different, even though for both of them the presence of multiplicative noise is a key ingredient.