Magnetization in the 1D Ising Model and the Metropolis Algorithm: A Glimpse into Statistical Physics and Monte Carlo Simulations

Author: Arpan Dey

A visualization of 1D Ising spins (Alice Heiman)

Abstract

In this report, we present a comprehensive exploration of the one-dimensional (1D) Ising model using a combination of exact analytical derivations and Monte Carlo simulations. The 1D Ising model is the simplest nontrivial lattice model that is exactly solvable, yet interesting enough (shows collaborative effects at low temperatures) to be relevant in polymer physics and biophysical modeling. It is an ideal starting point for developing a proper understanding of numerical sampling techniques. Analytically, the 1D model exhibits no true thermodynamic phase transition at any finite temperature; we get a relatively high, nonzero magnetization at sufficiently low temperatures, which is a consequence of the finite size effect.

The report bridges the conceptual foundation of statistical mechanics with practical computational approaches, demonstrating how simple stochastic rules can reproduce ensemble behavior consistent with thermodynamic predictions. Through comparison of exact transfer-matrix results, mean-field approximations and Metropolis Monte Carlo data, we highlight how fluctuations and correlations – which are ignored in mean field theory – play a crucial role even in the simplest spin system. To extend beyond equilibrium averages, in the end we also visualize the space-time evolution of the spin configurations in the 1D Ising system at low, intermediate and high temperatures. These discrete 2D plots reveal the transition from long-lived ordered domains to rapidly fluctuating disordered states as temperature increases, illustrating the competition between coupling  and thermal agitation.

This report is brief and unoriginal, but unique in its pedagogical approach: instead of merely reproducing known results, it tries to use the 1D Ising model as a minimal laboratory for understanding ergodicity, equilibration and emergent order in stochastic systems. While higher-dimensional models exhibit true phase transitions, the 1D case serves as an ideal testing ground for numerical methods, offering insight into how equilibrium arises from microscopic randomness.

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