April 26, 2013

How Ising model works

Although there is a description of an Ising model at Wikipedia, I guess it would scare a lot of people, so here is my brief explanation of how it works.

The simplest way to understand Ising model is to imagine chessboard. In a rectangular 8-by-8 grid there are 64 black or white cells. These cells are able to change their color from time to time: there is a chance that black cell become white, and white cells become black. Probability of color flip for each cell depends on its neighborhood. The general rule of Ising model is that cells likes to be surrounded by cells of the same color. If there is single white cell, surrounded by 8 black ones, it will flip its color almost instantly. On the other hand, if a white cell has 8 white neighbors, the probability that it will turn black is much lower.

What makes Ising model so interesting is temperature. The temperature sets minimal level of noise in the system. At low temperature there is almost no chance that cell surrounded by the cells of the same color will suddenly change its color. When temperature is raised this chance also rises, and at some point cell start to change its color randomly and independently of its neighbors.

This simple model already displays profound properties, related to phase transitions. If temperature is low, cells of the same color tend to cluster together, merge into larger areas until the whole grid will have uniform color. As temperature goes up, the tendency to cluster together gives way to a noise. When temperature crosses critical value, the pattern of large areas breaks, and system transits to its chaotic state.

Evolution of the system from hot chaotic state (left) to cold ordered state (right)

The bigger the system, the sharper is transition between chaotic and ordered states. To avoid uncertainty with cells on the borders, periodic boundaries are usually used: cells from the leftmost column interact with cells from the rightmost column, and cells from the top raw interact with cells from the bottom raw as if the grid is spread on the surface of the torus.

Simulation of Ising model with temperature bouncing around critical point

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