April 3, 2016

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.

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