The Cook Book

Recipe for the week of March 11 - 17

Deterministic Diffusion: the HPP Lattice Gas

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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