Ising model program

This program begins with a random assignment of charges to the cells. It then computes the neighbor agreement at each cell. It then updates the entire configuration in one step, using the neighbor agreement to determine the likelihood of flipping.

Currently, the chance that a cell will "flip" is determined from a table of probabilities based on the number of agreeing neighbors:.

The default values are The default value is The default value is 0. The spin angular momentum is intrinsic to the electron and arises from a property of elementary particles called spin. The spin of an electron can be measured as being in one of two states: spin up or spin down. The second contribution to the angular momentum of an electron is called the orbital angular momentum. The orbital angular momentum is a result of the electron orbiting the nucleus of the atom. The orbital angular momentum of an electron is similar to the angular momentum encountered in classical mechanics.

As expected, the magnetic moment is proportional both to the electric charge and the angular momentum. The presence of unpaired electrons and the alignment of adjacent spins whether they are parallel or antiparallel affects the electron location and therefore their electrostatic repulsion. The interaction between electron moments is called the exchange interaction. The exchange interaction creates an energy difference between the spin up and spin down states of the dipoles.

In ferromagnetic materials, nearby spins tend to align in the same direction. Having the same spin state reduces the electrostatic repulsion and is therefore more stable. Similarly, spins tend to align with an external magnetic field:. When the magnetic dipoles in a piece of matter are aligned, pointing in the same direction, their individually tiny magnetic fields add up together to create a much larger, stronger field that we can detect. This is called magnetization. In a ferromagnetic material, the dipoles tend to align spontaneously, even in the absence of an external field.

This is called spontaneous magnetization. Here we have a graph depicting the magnetization curve for nickel nanowires at room temperature. On the horizontal axis is the external applied field measured in milli-teslas. On the vertical axis is the mass magnetization expressed in Amperes meters squared per kilogram. As you can see, the magnetization increases when the applied field increases and vice versa, until the magnetization of the material cannot be increased or decreased further saturation : The Curie temperature As the temperature of a ferromagnetic material increases, the thermal motion of the dipoles competes with their tendency to align.

When the temperature rises beyond a certain point, called the Curie temperature, the material can no longer maintain a spontaneous magnetization, so its ability to be magnetized or attracted to a magnet disappears. This is an example of a phase transition second-order, discontinuous in the second derivative of the free energy.

This graph shows the saturation magnetization i. The saturation magnetization decreases with increasing temperature until it falls to zero at the Curie temperature, where the material becomes paramagnetic. The Ising model The study of ferromagnetic phase transitions in the Ising model had a significant impact on the development of statistical physics.

The 2D square-lattice Ising model in particular is one of the simplest statistical models to show a phase transition. Each spin can interact with its neighbors. This interaction terms determine how much the spins tend to align in the same direction or point in opposite directions :.

If the interaction term is positive, as in a ferromagnetic material, spins tend to be aligned meaning that configurations in which adjacent spins are of the same sign have higher probability. If the interaction term is negative, as in an antiferromagnetic material, adjacent spins tend to have opposite signs.

The first sum is over pairs of adjacent spins where every pair is counted once. Notice that, for positive J J J , adjacent spins with the same sign result in lower energy than adjacent spins with opposite signs.

Hence aligned spins have lower energy and are therefore more stable. The second sum is over each spin site. Here, again, if the spin direction and the direction of the external magnetic field are aligned, the energy is reduced. This is called the Boltzmann distribution. Z Z Z is called the partition function. The partition function, in statistical mechanics, describes the statistical properties of a system in thermodynamic equilibrium.

Most of the aggregate thermodynamic variables of the system, such as the total energy, free energy, entropy, and pressure, can be expressed in terms of the partition function or its derivatives. Here the partition function plays the role of a normalizing constant, ensuring that the probabilities sum up to one.

In particular, the ratio of the probabilities of two states depends only on their difference in energy:. Using this model, in the limit of large numbers of spin sites, we can answer a number of statistical questions that are physically significant, such as the probability that neighboring sites have the same spin or at what temperature a phase transition occurs.

Simulating the Ising model The Ising model can be difficult to simulate if there are many states in the system. When there is a large number of sites, there is an absolutely enormous number of possible configurations. To see this, suppose there are N N N spin sites. This is the reason why the Ising model is simulated using Monte Carlo methods.

Monte Carlo methods are algorithms that rely on repeated random sampling to obtain numerical results.