Chapter 9: Fundamental Physics

Section 4: Conserved Quantities and Continuum Phenomena

Conservation of vector quantities

Conservation of the total number of colored cells is analogous to conservation of a scalar quantity such as energy or particle number. One can also consider conservation of a vector quantity such as momentum which has not only a magnitude but also a direction. Direction makes little sense in 1D, but is meaningful in 2D. The 2D cellular automaton used as a model of an idealized gas on page 446 provides an example of a system that can be viewed as conserving a vector quantity. In the absence of fixed scatterers, the total fluxes of particles in the horizontal and the vertical directions are conserved. But in a sense there is too much conservation in this system, and there is no interaction between horizontal and vertical motions. This can be achieved by having more complicated underlying rules. One possibility is to use a hexagonal rather than square grid, thereby allowing six particle directions rather than four. On such a grid it is possible to randomize microscopic particle motions, but nevertheless conserve overall momenta. This is essentially the model used in my discussion of fluids on page 378.

From Stephen Wolfram: A New Kind of Science [citation]