# Ising Model Visualization

Phase Transitions and Critical Phenomena in Statistical Mechanics - Interactive Monte Carlo Simulation

> Canonical page: https://elysiatools.com/en/visualizations/ising-model

- **Category:** Physics

## Overview

Interactive visualization of the 2D Ising Model - explore phase transitions, critical phenomena, and Monte Carlo simulations in statistical mechanics. Features real-time Metropolis-Hastings algorithm simulation on N×N spin lattice (N=20-100 adjustable), temperature control from 0.1 to 6.0 with critical temperature Tc ≈ 2.269 highlighted, external magnetic field h (-2 to +2), coupling constant J (-1 to +1 for ferromagnetic/antiferromagnetic), four initial states (random, all up, all down, striped), and simulation speed control (1-100 MC steps/frame). Real-time statistics display: temperature T, T/Tc ratio, total energy E per spin, magnetization |M|, and Monte Carlo steps count. Dual chart visualization: Energy evolution and Magnetization evolution with rolling history (200 data points). Visualization options: cluster highlighting for same-spin connected regions, grid overlay toggle, and color schemes (red/blue or classic black/white). Comprehensive educational content: Ising model theory with Hamiltonian H = -J Σ⟨ij⟩ sᵢsⱼ - h Σᵢ sᵢ, historical milestones (Lenz 1920, Ising 1925, Onsager 1944), phase transition phenomena with three regimes (TTc paramagnetic disordered phase), Metropolis-Hastings algorithm steps with acceptance probability P(accept) = min(1, exp(-ΔE/kT)), observation guide for exploring low-T ferromagnetism, critical fluctuations, high-T paramagnetism, external field effects, and antiferromagnetic phase (J<0), and interactive formula breakdown explaining energy, interaction, and field terms. Observe spontaneous emergence of order from disorder, critical opalescence-like fluctuations, and the beauty of statistical mechanics. Multi-language support (zh, en, es, fr, de, ru, pt).

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