On the law of entropy increase of some cellular automata on Zd

We consider some time-reversible cellular automata on thed-dimensional integral latticeZ ^d and study their time evolution properties. We show first that a Boltzmann-type entropy can be defined which is not less than its initial value for initial States which have no spatial correlation. For monotonic increase of the entropies for such initial States we need an additional condition which we call renewality. Under the renewality condition entropy is monotonic nondecreasing. We give some examples of cellular automata which satisfy the renewality condition