Local conservation laws

Whenever a system like a cellular automaton (or PDE) has a global conserved quantity there must always be a local conservation law which expresses the fact that every point in the system the total flux of the conserved quantity into a particular region must equal the rate of increase of the quantity inside it. (If the conserved quantity is thought of like charge, the flux is then current.) In any 1D k=2, r=1 cellular automaton, it follows from the basic structure of the rule that one can tell what the difference in values of a particular cell on two successive steps will be just by looking at the cell and its immediate neighbor on each side. But if the number of black cells is conserved, then one can compute this difference instead by defining a suitable flux, and subtracting its values on the left and right of the cell. What the flux should be depends on the rule. For rule 184, it can be taken to be 1 for each block, and to be 0 otherwise. For rule 170, it is 1 for both and . For rule 150, it is 1 for and , with all computations done modulo 2. In general, if the global conserved quantity involves blocks of size b, the flux can be computed by looking at blocks of size b+2r-1. What the values for these blocks should be can be found by solving a system of linear equations; that a solution must exist can be seen by looking at the de Bruijn network (see page 941), with nodes labelled by size b+2r-1 blocks, and connections by value differences between size b blocks at the center of the possible size b+2r blocks. (Note that the same basic kind of setup works in any number of dimensions.)