Limiting procedures [for cellular automata]

Several different limiting procedures all appear to yield the same continuum behavior for the cellular automata shown here. In the pictures on this page a large ensemble of different initial conditions is considered, and the density of each individual cell averaged over this ensemble is computed. In a more direct analogy to actual physical systems, one would consider instead a very large number of cells, then compute the density in a single state of the system by averaging over regions that contain many cells but are nevertheless small compared to the size of the whole system.