Ising-Model

Overview

This project implements Monte Carlo simulations of the 2D Ising Model on a 40×40 lattice, exploring phase transitions and equilibrium properties using two different Monte Carlo dynamics: Metropolis and Glauber algorithms. The simulation investigates the temperature-dependent behavior of magnetization, energy, and magnetic susceptibility.

Part 1: External Field Case (h = 1, J = 0)

Part 1A: Thermal Equilibrium Comparison (function: simulation())

This section performs Monte Carlo simulations for a non-interacting Ising system under an external magnetic field.

Setup

• Lattice size: 40 × 40 spins
• External field: h = 1
• Coupling constant: J = 0 (no spin-spin interactions)
• Temperatures: T = {0.1, 1.0, 2.0, 2.27, 3.0, 4.0}
• Monte Carlo steps: 1000
• Measurement interval: Every 50 steps

Methodology

The simulation runs both Metropolis and Glauber dynamics independently for each temperature.

1. Initialize a random spin configuration
2. Apply Monte Carlo algorithm for 1000 steps
3. Record energy (u), magnetization (m), and susceptibility (χ) every 50 steps
4. Average observables over the measurement period

Results

• Magnetization m(T): Average magnetic moment per spin
• Energy u(T): Average energy per spin
• Susceptibility χ(T): Magnetic response to field changes

Part 1B: Relaxation Curves (function: relax_curve())

This section studies how the system relaxes from a fully ordered state to thermal equilibrium.

Setup

• Initial state: all spins +1
• Temperatures: T = {0.1, 2.27, 4.0}
• Monte Carlo steps: 100
• Realizations: 100

Methodology

1. Initialize all spins up
2. Apply Monte Carlo dynamics
3. Record magnetization per step
4. Repeat 100 times
5. Average results

Part 2: Ferromagnetic Case (h = 0, J = 1)

Energy Change for Spin Flip

ΔE = 2 s_i (J Σ_nn s_j + h)

Setup

• Lattice size: 40 × 40
• J = 1
• h = 0
• Temperatures: same as above
• Steps: 1000

Results

• Clear phase transition around T ≈ 2.27
• Magnetization drops to zero above Tc
• Susceptibility peaks at Tc

Project Structure

main.py
parte1.py
parte2.py
model.py
dinamicas.py
observables.py

Summary

The project demonstrates: - Agreement between simulation and theory - Same equilibrium from different dynamics - Phase transition in ferromagnetic system