The Ising model is a simple system originally created to study magnetism but is now used to examine a number of natural phenomena outside of physics.  Because this model is very simple and is mathematically well understood, it serves as an excellent building block for creating more complex models.  By examining very simple two-dimensional interactions, the Ising model has become particularly useful for studying biological behavior such as flocking in birds and synchronization of flashes in a group of fireflies.  Today, the Ising model is commonly used to study any type of aggregate behavior, be it microscopic or on macroscopic scale.

	The behavior of magnetic domains in specific substances was the first application of the Ising model.  Every atom has a magnetic field due to the flow of electrical current by electrons revolving around the nucleus (Fig. 1).  But how are magnetic domains organized if all magnetic moments are random?         Dr. Ernst Ising (Fig. 2) developed the simple two-spin state as part of his doctoral dissertation in 1924.  Most electrons are paired and spin in opposite directions (typically described as spin up or +1 and spin down or –1).
Figure 1: The magnetic field around a bar magnet		Figure 2: Dr. Ernst Ising

        As a result, the magnetic field created by each of them is typically canceled by one another.  When a large number of electrons with randomly mixed up and down spins are grouped together, all the magnetic fields add up to equal zero; this is called paramagnetism (Fig. 3). Ising’s doctoral advisor, Dr. Wilhelm Lenz, theorized that many atoms without a paired set of electrons may have magnetic fields that are aligned.  Consequently, a measurable magnetic field is created; this is called ferromagnetism.  The Ising model is nothing more than a simplified representation to illustrate this phenomenon.

Figure 3: Spin orientation of electrons in an atom

        In the Ising model, a lattice of points, each point representing an atom, is randomly assigned an orientation (either spin up + and spin down –).  Each lattice point interacts only with its closest neighbors, namely the lattice point on its right, left, above, and below (Fig. 4).

Figure 4: Lattice of the magnetic moment of each atom at that location.

        The magnetic field imposes a force on nearby charged particles.  If the charge particle is traveling in a direction opposing the magnetic field, it will have high energy; if in the same direction, it will have low energy.  A charged particle with low energy will align itself with this nearby force.  For the entire lattice, it is energetically favorable for adjacent lattice points to have the same spin.

        The model is then given periodic boundary conditions (Fig. 5). The last point on the right edge of the square is considered to be a “neighbor” of the first point on the left edge.  In other words, if you walk across the lattice, when you reach the end, the next step will take you back to the beginning.  If you continue to move in a straight line, you will traverse the same path over-and-over again in a certain period of time (depending on the size of the lattice). Same is true about the top and bottom edges. In other words, you are "gluing" the left edge to the right edge and the top to bottom, creating a torus (or donut) out of a square.

Figure 5:Periodic Boundary conditions

        To simulate the model, we use a NetLogo simulation of the Metropolis algorithm.  After creating a lattice in which each point is randomly assigned one of the spin values, we perturb the system by randomly picking a point and reversing its spin. We consider the difference in energies of the original configuration and the perturbed configuration and decide whether or not to switch to the perturbed configuration.

This procedure is repeated many times until no further major changes are observed in the system, meaning the energy of the system oscillates around a steady average or reaches equilibrium (as opposed to a system not in equilibrium, which would show entropy). The result can be considered as what would naturally exist in the given conditions, not a consequence of the initial conditions we provided.

        The pictures below (Fig 6-7) are still images from the NetLogo Ising model simulation.  The two colors represent particles of opposite spins, with –1 as light blue while +1 is dark blue. A system of alternating +1s and –1s would have the highest energy. Since they have a high degree of order, both of these systems would have low entropies. Therefore, a high-temperature system would not have any alignment of magnetic spins, and can be seen in the image on Fig 6. A system of all +1s or all –1s would have the lowest energy. So at low temperatures, magnetic spins align with their nearest neighbors and would create a state of ferromagnetism (Fig 7).

Figure 6: Screenshot of paramagnetic behavior.	Figure 7: Screenshot of ferromagnetic behavior.

        Why is it important to occasionally accept higher energy configurations if we are ultimately looking for a solution of lowest possible energy? The answer to this question is: we are trying to find create a simulation which works toward a global minimum of free energy.  If we do not accept any changes to higher energy configuration, the simulation can get trapped in a local energy minimum.  Accepting higher energy configurations by low probabilities allows the simulation to climb small mountains in an energy vs. state diagram and eventually reach the global minimum.

Although the Ising model is an extremely simple one, it is worth noting that it has numerous applications in physics and chemistry and is still used extensively in both fields.  In addition, it also lays the foundation for how the next level of complexity operates.  After you’ve experimented with several Ising model simulations, we encourage you to examine a more complicated – and certainly more powerful – version of the Ising model:  the Potts model.