Add 2D Ising model Monte Carlo simulation

modelling_simulation/ising_model.py

@@ -0,0 +1,78 @@
+"""
+Monte Carlo Simulation of the 2D Ising Model using Metropolis Algorithm.
+
+Reference:
+https://en.wikipedia.org/wiki/Ising_model
+
+This algorithm simulates spin interactions on a 2D lattice.
+It demonstrates how spins evolve toward equilibrium at a given temperature.
+
+>>> result = ising_model_simulation(lattice_size=10, temperature=2.0, sweeps=100)
+>>> isinstance(result, np.ndarray)
+True
+"""
+
+import numpy as np
+import matplotlib.pyplot as plt
+
+
+def delta_energy(matrix: np.ndarray, i: int, j: int, J: float = 1.0) -> float:
+ """Compute change in energy (ΔE) if the spin at (i, j) is flipped."""
+ n = matrix.shape[0]
+ neighbors = (
+ matrix[(i + 1) % n, j]
+ + matrix[(i - 1) % n, j]
+ + matrix[i, (j + 1) % n]
+ + matrix[i, (j - 1) % n]
+ )
+ return 2 * J * matrix[i, j] * neighbors
+
+
+def ising_model_simulation(
+ lattice_size: int = 50,
+ temperature: float = 1.5,
+ sweeps: int = 5000,
+ J: float = 1.0,
+ kB: float = 1.0,
+ visualize: bool = False,
+) -> np.ndarray:
+ """Run the 2D Ising Model Monte Carlo Simulation.
+
+ Args:
+ lattice_size: Number of spins per row/column.
+ temperature: Simulation temperature (T).
+ sweeps: Number of Monte Carlo sweeps.
+ J: Spin coupling constant.
+ kB: Boltzmann constant.
+ visualize: If True, display intermediate lattice states.
+
+ Returns:
+ Final spin lattice (numpy array) after equilibration.
+ """
+ # Initialize lattice randomly with spins {-1, +1}
+ matrix = 2 * np.random.randint(0, 2, size=(lattice_size, lattice_size)) - 1
+
+ for sweep in range(sweeps):
+ for _ in range(lattice_size * lattice_size):
+ i, j = np.random.randint(0, lattice_size, size=2)
+ dE = delta_energy(matrix, i, j, J)
+
+ # Metropolis criterion
+ if dE <= 0 or np.random.rand() < np.exp(-dE / (kB * temperature)):
+ matrix[i, j] *= -1
+
+ if visualize and sweep % 500 == 0:
+ plt.imshow(matrix, cmap="bwr", vmin=-1, vmax=1)
+ plt.title(f"Sweep {sweep}")
+ plt.pause(0.01)
+
+ if visualize:
+ plt.imshow(matrix, cmap="bwr", vmin=-1, vmax=1)
+ plt.title(f"Equilibrated Ising Model at T={temperature}")
+ plt.show()
+
+ return matrix
+
+
+if __name__ == "__main__":
+ ising_model_simulation(visualize=True)