The Ising is a tool used in statistical physics to study phase transitions and collective dynamics in systems with many interacting parts. Initially designed to represent ferromagnetism, it models spins on a lattice that can take one of two states, interacting with their neighbors. Its applications extend to areas like neural network activity, protein folding and quantum cluster states. The model's simplicity allows for analytical solutions in certain cases and computational approaches in others, making it broadly applicable for exploring the connection between microscopic interactions and macroscopic phenomena.
Here I'm making a small monte carlo simulation of an Ising Model with Glauber dynamics.
The energy of a configuration of spins (where ) on a lattice is given by:
Where:
The Glauber dynamics updates the spin at site based on the local energy configuration. The probability of a spin flipping to is:
Where:
For the case of a single spin flip , the energy change for the flip is:
This means that the probability of flipping a spin depends on the interaction with neighboring spins and the external field. At high temperatures (), flips occur more freely, while at low temperatures (), the system tends to settle into aligned configurations.
Note the critical temperature for an ising model should lie around
in our sim J is always 1.0, so the critical temp is 2.27.
Here we plot the lattice state for various temperatures and time steps.
The simulation calculates the lattice states using Glauber dynamics in the update_step_numba
function. At each step, a random site is picked, and the change in energy () due to flipping that spin is computed from the local interactions (the sum of the nearest neighbors’ spins and any external field). The flip is then accepted with probability . Repeating this process for many steps yields a time evolution of the spin configuration.
At low temperatures, we often see large, stable clusters where spins align and rarely flip, indicating an ordered phase. Near critical temperature (), these clusters form and dissolve in a more balanced manner, and we catch glimpses of interesting domain structures. At high temperatures, spins will flip frequently and produce a more disordered or “noisy” lattice with low magnetization.
We can plot the following graphs: magnetization, average_energy, susceptibility, and specific_heat.