Thermal transitions in a one-dimensional, finite-size Ising model

TL;DR

This paper analytically and numerically investigates the stationary and dynamical properties of a finite-size one-dimensional Ising model, revealing temperature-dependent behaviors and applications to biopolymer structural transitions.

Contribution

It provides new analytical derivations and numerical verifications of magnetization and domain wall distributions in finite Ising chains at various temperatures.

Findings

Identified temperature regimes with behaviors similar to phase transitions.

Derived probability distributions for magnetization and domain walls.

Applied results to biopolymer structural transition modeling.

Abstract

We revisit the one-dimensional ferromagnetic Ising spin-chain with a finite number of spins and periodic boundaries and derive analytically and verify numerically its various stationary and dynamical properties at different temperatures. In particular, we determine the probability distributions of magnetization, the number of domain walls, and the corresponding residence times for different chain lengths and magnetic fields. While we study finite systems at thermal equilibrium, we identify several temperatures similar to the critical temperatures for first-order phase transitions in the thermodynamic limit. We illustrate the utility of our results by their application to structural transitions in biopolymers having non-trivial intermediate equilibrium states.