A border and a twist

One of the things that inevitably pop-up in any simulation is limit on available computational power. In particular case of simulating 2D lattice model there are 2 general ways to cope with these limitations. The first and the most straightforward thing to do is to make border of the lattice “special” in some way. For example, cells in the bulk might have 8 neighbors, cells on the border have 5 neighbors, and cells in the corners have only 3 neighbors. Usually, this means that behaviour of the system changes on the border, but when done right this does not lead to any catastrophic failure during the simulation. Here is an example of what I've got after simulating a small grid with borders:

Second way to cope with limited computational resources is to make use of periodic boundary conditions. The simplest case of periodic boundary conditions are those of asteroid game, where adjacent screens wrap on each other, and shells fired at the right edge of the screen appear at the left edge:

The overall shape of the simulation field is a torus, or more precisely, a flat torus:

But there are more twisted ways to stich the simulation fields together than just tile screens next to each other. Imagine I would take the top, twist it and glue to the bottom.

There may be even more twists to the way how ends of the screen are glued together. Here is the general blueprint of how it would look:

The overall pattern of Ising model would be different. In chaotic model these changes would be imperceptible, but when the temperature is low, the pattern would acquire certain features to it. Here how these features manifest themselves when field is annealed a number of times:

Regular patterns have a certain degree of rectangularity to them, while twisted patterns are more diagonal-like. Situations where top and bottom have opposite colors can only happen in a doubly twisted simulations.

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