This is a monte carlo simulation of the Ising and XY models in 1 and 2 dimensions.
These are toy models, but they capture some of the interesting physics that can emerge
from large numbers of interacting particles.
The phase transitions seen in these models can be seen in real physical systems,
even down to the precise numerical values of their critical exponents.
Here they are simulated on a periodic grid (if you go off one end you come back at the opposite end).
What is going on here?
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Ising model
A grid of pixels that can be in one of two states (represented as colours).
Each pixel interacts with its neighbours.
It prefers to be in the same state as its neighbours. More precisely: arrangements where neighbouring pixels
are in the same state have lower energy than arrangements where neighbouring pixels are in different states. The arrangement
with lowest possible energy is the one where all pixels are in the same state (are all the same colour). But if we raise the
temperature, thermal fluctuations can push us into arrangements with higher energy.
When the temperature is low enough for one state to dominate over the other, this is known as
"spontaneous symmetry breaking". Both states behave identically as far as the fundamental physics
is concerned (so we have a 'symmetry') but this symmetry is broken when the system spontaneously
picks one colour to prefer. This is like what happens in a magnet when you cool it below the Curie temperature,
and it picks a direction to magnetize in.
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XY model
A grid of arrows. Each arrow can point in any direction in the plane (covering a circle).
Because drawing lots of arrows looks messy, the direction of each arrow is represented with a coloured pixel.
The colour varies continuously as the direction of the arrow moves around the circle.
Each arrow interacts with its neighbours. It prefers to point in a direction that is close to the direction its
neighbours are pointing in. More precisely: arrangements where neighbouring arrows point in similar directions (are similar colours)
have lower energy than arrangements where neighbouring arrows point in different directions (are different colours). The arrangement
with lowest possible energy is the one where all arrows point in the same direction (are all the same colour).
But if we raise the temperature, thermal fluctuations can push us into arrangements of higher energy.
In two dimensions, this model can contain 'vortices', a kind of topological defect. These proliferate above the Berezinskii-Kosterlitz-Thouless
phase transition, but decay below it. This phase transition is seen in liquid helium films.
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