Ising Model)

The Ising model is the simplest and rich physical content model that can be used to describe many physical phenomena, such: ordered-disordered transition in the alloy, transition from liquid helium to ultra-flow state, freezing and evaporation of liquid, properties of glass, forest fire, urban traffic, etc. The Ising model was initially proposed to explain the phase transition of the ferrite, that is, the magnetic properties will disappear when the magnet is heated to a certain critical temperature, and the magnetic properties will be displayed when the temperature drops below the critical temperature. This kind of transformation between magnetic and non-magnetic phases is a continuous phase change (also called a secondary phase change ). The Ising model assumes that the ferrite is composed of a pile of small magnetic needles which are arranged in a regular arrangement. Each magnetic needle has only two directions (spin ). The adjacent magnetic needles interact with each other through energy constraints, and random magnetic changes (from top to bottom or vice versa) may occur due to the interference of ambient thermal noise ). The fluctuation is determined by the key temperature parameters. The higher the temperature, the stronger the disturbance of random fluctuations. the more likely the smaller the magnetic needle is to change in a disordered and violent State, so that the magnetic fluctuations in the upper and lower directions can offset each other, the whole system disappears magnetic. If the temperature is very low, the small magnetic needle is relatively quiet, and the system is in a state of high energy constraints. A large number of small magnetic needles are in the same direction, and the magnetic system shows magnetic properties.

Scientists are also interested in this model because it is the simplest model of particles (or spin) that describe the interaction. The Ising model is a very simple model that occupies a spin on each grid point in one, two, and three dimensions. Spin is an internal property of electrons. Each spin has two quantitative directions in space, that is, it points up or down. Although this model is the simplest physical model, there is only one and two dimensional exact solutions.

Considering the one-dimensional Ising model, M spin is arranged in a row, and each spin interacts with the spin of the two nearest neighbors. For simplicity, we only consider the mutual effect that tends to make the directions of the neighboring spin consistent. The two-dimensional square Ising model is composed of N identical spin bins. Each spin not only interacts with the spin of the two nearest neighbors, but also interacts with the spin of the two nearest neighbors in the adjacent spin bins, engineering a Two-Dimensional Spin array. The three-dimensional cubic Ising model has an identical Two-Dimensional Spin array. Each spin interacts with the spin of the Left, right, back, and up and down six nearest neighbors. It is not difficult to find that as the dimension increases, the number of neighboring spin trees in each spin increases, and the interaction with the surrounding spin also increases.

However, the evolution of the system is not entirely determined by the total energy. Because the needle is in a noisy environment, the thermal fluctuations will trigger random reversal of the state of the needle. We can use temperature to measure the randomness of such environmental impact. The higher the value of T, the higher the probability of a mirror.

In this way, there are two kinds of force acting on the magnetic needle, one is from the influence of the neighbor of the magnetic needle and the field on it. This influence tends to make the adjacent neighbors stay consistent with each other and try to be consistent with the field, that is, try to minimize the total energy of the system. Another force is derived from environmental noise disturbance, which forces the magnetic needle to randomly reverse the State regardless of its neighbors. As a result, each small magnet is struggling between these two different forces. It is hard to imagine that if the temperature T tends to be 0, then each magnetic needle will be consistent with the field, the final system will be in the state of + 1 or all-1 (depending on whether the field H is positive or negative ). If t is particularly high, and J is very small, the role of the neighbors can be ignored, each small magnetic needle is completely random value.

In this way, the entire Ising model has two external parameters to indicate the temperature and magnetic intensity of the environment. In the villagers' analogy, the temperature is equal to the degree of freedom for the villagers' opinion selection. The higher the temperature, the more random the villagers' views are, regardless of their neighbors; otherwise, the villagers' choice is heavily dependent on neighbors and media propaganda.

Ising Model)