- This week's soup, generated by one of the demo experiments for our CAM8
mixmaster, illustrates a CA known as the HPP Lattice Gas. The most celebrated
models for diffusion are stochastic, typically based on some form of random
walk or Brownian motion. But simple ballistic automata that conserve energy
and momentum while undergoing nonlinear interactions provide efficient
parallel implementations of physical effects such as diffusion, refraction,
and turbulence. With care, the large scale behavior of these models can
even recover quantitative details of the classical partial differential
equation descriptions.
- The HPP rule, studied by J. Hardy, O. de Pazzis, and Y. Pomeau in the
mid-'70s, involves digital particles that move at unit speed in four
directions, at most one of each direction per site. Here, particles
moving North, South, East and West are colored red, blue, green and purple,
respectively. The collision rule prescribes what happens when two
or more particles land on the same cell. In all but one case the
trajectories simply pass through one another without interaction. If exactly
two particles collide head on, however, then they leave the site in the
two orthogonal directions. That's it.
- Our graphic shows the state of an HPP Gas on a (cropped) 512 by 512
array with wrap-around, after 500 steps, from a central square of cells
that were all initially occupied by particles traveling in all four
directions. Enough particles avoid interaction to maintain four coherent
clouds for quite a long time, while the nonlinear interaction gradually
diffuses the initial geometry. More sophisticated ballistic lattice gases
have been used for the past several years, at Los Alamos National Labs and
elsewhere, to simulate complex systems such as turbulent flow at high
Reynolds numbers with irregular boundary conditions. Devotees of this
paradigm argue that it competes favorably with more traditional numerical
analysis by p.d.e. approximation.
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