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Ising Model Simulation

Overview

This project simulates the Ising Model using the Markov Chain Monte Carlo (MCMC) method. It provides both a command-line interface (CLI) and a graphical user interface (GUI) for running simulations.

GUI Video Demo of the Simulation

Mathematical Formulation

The Ising Model is a mathematical model used in statistical mechanics to describe ferromagnetism. It consists of discrete variables called spins, which can take values $s_i = \pm 1$ and are arranged in a lattice. The Hamiltonian (energy function) of the system is given by:

$$H = -J \sum_{\langle i,j \rangle} s_i s_j - \mu B \sum_i s_i$$

where:

  • $J$ is the interaction energy between neighboring spins.
  • $\sum_{\langle i,j \rangle}$ denotes summation over all neighboring pairs.
  • $\mu B$ represents the external magnetic field term (which can be set to zero if there is no external field).
  • $s_i, s_j$ are the spin values at lattice sites $i$ and $j$.

Metropolis-Hastings Algorithm

The system evolves using the Metropolis-Hastings algorithm, which follows these steps:

  1. Select a random spin $s_i$ in the lattice.

  2. Compute the change in energy if the spin is flipped:

    $$\Delta H = 2J s_i \sum_{j \in \text{neighbors}} s_j + 2 \mu B s_i$$

  3. Accept the flip with probability:

    $$P(\text{flip}) = \min(1, e^{-\Delta H / k_B T})$$

    where $k_B$ is the Boltzmann constant and $T$ is the temperature.

This simulation implements these steps to generate configurations and analyze phase transitions in the Ising model.

Installation

Ensure you have Python installed (>=3.8). Then, clone the repository and install the required dependencies:

pip install -r requirements.txt

Usage

CLI Simulation

Run the CLI simulation with optional parameters:

python examples/cli_simulation.py --temperature 3.0 --steps 3500
  • --temperature (default: 1.0) sets the simulation temperature.
  • --steps (default: 1500) specifies the number of Metropolis-Hastings steps.

Example CLI Output

CLI Simulation Output Example

GUI Simulation

Run the GUI-based simulation:

python examples/gui_simulation.py

The GUI allows for interactive visualization of the Ising model simulation.

GUI Preview

GUI Simulation Output Example

Project Structure

ising_model_simulation/
│── examples/        # CLI and GUI simulation scripts
│── ising_model/     # Core Ising Model implementation
│── simulator/       # Simulation logic
│── gui/             # GUI components
│── tests/           # Unit and integration tests
│── res/             # Resources (Pictures and videos) for documentation
│── README.md        # Project documentation

Contributing

Contributions are welcome! Feel free to open issues or submit pull requests.

License

This project is licensed under the MIT License.