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Some explanation |
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Ruled surfaces 2
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Ruled surfaces 9
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Classical stadium billiard 7
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Classical stadium billiard 8
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Quantum stadium billiard 2
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Toom rule interface model of snowfall
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(following Lebowitz et al): at each time step, each spin in the grid is updated according to a noisy Toom rule -- it is replaced by the majority vote of itself and its S and W neighbors, but with some small probability of error. When the error probability is small enough, even if it is not Ising-symmetric, there are two steady states. Here we have up boundary conditions along the y-axis and down boundary conditions along the x-axis, forcing there to be an interface between the two states. The paper linked above shows that this interface is governed by KPZ.
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Solidification of Lennard-Jones gas in a gravitational field
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BML traffic model. Occupation probability:
0.27,
0.29,
0.3,
0.5
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Time steps alternate between moving one set of cars up (if they are not blocked) and moving another set of cars left (if they are not blocked). Here we start with iid initial conditions with the indicated probability that a site is occupied. As you can see, there is a phase transition to a permanent traffic jam.
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Murmurations
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Dances with friends and enemies (a model of flocking due to Simon Woods)
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More murmurations
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Vicsek model of flocking
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Slime mold intelligence
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This is a crude simulation of slime mold intelligence, following the beautiful model proposed in this paper and studied in this paper. We begin with a square lattice whose links are covered with slime mold. Two food sources are placed at upper right and lower left corners of the grid. An obstacle is placed in the middle to make a kind of maze. The slime mold sets up a flow of whatever cellular fluid it is filled with, for which the food is a source or sink. The amount of current through a given link is assumed to satisfy Poiseuille (pressure-driven) flow:
$$I_{ij} = (p_i-p_j) \sigma_{ij},$$
where $p_i$ is pressure at site i, and $\sigma_{ij}$ is a conductance associated with the link ij -- think of it as the thickness of the slime mold along the link. If we interpret pressure as voltage, this is Ohm's law. The key extra ingredient that allows the slime mold to learn is that the conductivities are updated according to the rule that tubes with more current grow. That is: for each link l, $$\sigma_l(t+dt) = \sigma_l(t) + dt( -\sigma_l(t) + a |I_l(t)|) .$$ By this means, the slime mold ends up following the shortest path between the food sources. The only nonlinearity is in the absolute value.
The heatmap shown is the pressure at each site, and the conductivity on each link is shown in yellow (because that's the color of the relevant species of slime mold, Physarum Polycephalum).
I'm solving for the currents at each step by relaxation from the previous configuration.
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Sierpinski carpet automaton
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Just as Pascal's triangle mod two produces a Seirpinski gasket, here is an automaton (explained to me by Ting-Chun David Lin) that produces the Sierpinski carpet. Each square is the sum of its NW, N and W neighbors mod 3. Is there a connection to the Toom rule? What is the analog of the binomial expansion?
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Ballistic deposition
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Squares fall; if there is a square immediately below or below right or below left, they stick. I learned today from Joel Moore that the boundary of the frozen region is governed by KPZ. In particular the fluctuations of the height (averaged over space) allegedly go like $t^{1/3}$ (for $t \ll L^{z=3/2}$), as seems to be borne out numerically in this small simulation ($L=200$). Here are two references. The former cites these large-scale simulations by saying "It is not cheerful reading" because it says that the asymptotic regime is only reached for system sizes bigger than $2^{12}$.
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