Ising model (Ising model)

The Ising model (Ising model) is one of the simplest and can provide very rich physical content of the model. It can be used to describe very many physical phenomena, such as: ordered-disordered transformation in alloys, transition from liquid helium to super-flow, freezing and evaporation of liquids, properties of glass material, forest fire, urban transportation, etc. The Ising model was originally proposed to explain the phase transition of ferromagnetic material, that is, the magnet disappears when heated to a certain critical temperature, and the temperature below the critical temperature will exhibit magnetism. This is a magnetic, non-magnetic transition between two phases. is a continuous phase change (also called a two-stage phase transition). Ising model If the ferromagnetic material is composed of small magnetic needle arranged by a bunch of rules, each magnetic needle has only two directions (spin). The adjacent small magnetic needle is interacting with the energy constraint. At the same time, there will be a random change in magnetism due to the disturbance of ambient thermal noise (on or on the reverse). The size of the fluctuation is determined by the critical temperature parameters. The higher the temperature, the stronger the stochastic fluctuation disturbance. The more easy the small magnetic needle, the disordered and violent state changes. Thus, the magnetic field of the entire system disappears by offsetting the magnetism of the upper and lower two directions. If the temperature is very low, then the small magnetic needle is relatively quiet, the system is in a state of high energy constraints, a large number of small magnetic needle direction of the same, ferromagnetic system exhibits magnetic.

Scientists ' extensive interest in the model also stems from its simplest model of describing the interaction of particles (or spin). The Ising model is a very easy model that occupies a spin on each lattice point of one, two, and three dimensions.

Spin is an intrinsic property of electrons. Each spin has two quantization directions in space. That is, its pointing can be up or down. Although the model is one of the simplest physical models. Now there is only one and two-dimensional exact solution.

Consider a one-dimensional Ising model. M spins in a row, with each spin interacting with the spin of its two near neighbors. For simplicity's sake, we only consider the tendency to have a consistent interaction in the direction of the neighboring spin. The two-dimensional square Ising model is composed of n identical spin rows. Each spin not only interacts with the spin of the two nearest neighbors, but also interacts with the spin of two nearest neighbors in the spin row, and project has a two-dimensional spin array.

Three-dimensional cubic Ising model is the same as the two-dimensional spin array, each spin with its left and right, front and back, up and down six recent neighbors spin interaction. Not hard to find. As the dimension is added. Each spin of the recent adjacent spin tree is added. The interaction with the surrounding spin is also increasing.

However, the evolution of the system is not entirely determined by the total energy. Because the small magnetic needle in the noisy environment, the thermal fluctuation will cause the small magnetic needle state stochastic reversal.

We can use temperature to measure the randomness of this environmental impact. The higher the T, the greater the probability that the small magnetic needle will be reversed.

In this way, there are two forces acting on the small magnetic needle, a force derived from the small magnetic needle neighbor and the external field of its influence, such an effect tends to make the neighboring neighbors of the same state and as far as possible consistent with the outfield. That is to minimize the total amount of the system. The second force derives from the disturbance of the ambient noise. It forces the small magnetic needle to ignore the neighbor's function and take on a random state reversal.

So. Each small magnetic needle struggles between these two different forces. Not hard to imagine. If the temperature T tends to 0, then each small magnetic needle will be consistent with the outfield. So. Finally the system will be at +1 or all-1 (depending on whether the outfield h is positive or negative).

If T is particularly high, and the interaction strength J is particularly small, then the role of the neighbors can be ignored, each small magnetic needle is completely random value.

Thus, the entire Ising model has two exogenous given parameters to indicate the ambient temperature and magnetic field strength.

In the villagers ' analogy, the temperature is equivalent to the degree of freedom of the villagers to choose their opinion. The higher the temperature, the more randomly the villagers choose their views, rather than the influence of their neighbours, or the villagers ' choice depends heavily on the neighbours and the media.

Ising model (Ising model)