The “ game of everything ”

A type of reversible cellular automaton is proposed, which can be used to discuss the question of irreversibility in statistical mechanics. It lives on an infinite lattice in space-time, whose nodes are points with integer coordinates, either all even or all odd, and whose edges join all pairs of nearest-neighbour nodes. The states of the edges in the immediate future of each node are related to those in its immediate past by a 'dynamical rule' which is deter-ministic and invertible. The assumed probability measure makes the nodes at time 0 independent, and correlations develop as the time variable t increases because of the dynamical rule. Two types of entropy are defined: one, analogous to the Gibbs entropy in statistical mechanics, is shown to be independent of t, while the other, analogous to Boltzmann entropy in statistical mechanics, can change with t and is shown to increase in the sense that it is minimal at t = 0.