  Information theoretic measures have helped to sharpen our understanding of
many-body quantum states. As perhaps the most well-known example, the
entanglement entropy (or more generally, the bipartite mutual information) has
become a powerful tool for characterizing the dynamical growth of quantum
correlations. By contrast, although computable, the bipartite mutual
information (MI) is almost never explored in classical many particle systems;
this owes in part to the fact that computing the MI requires keeping track of
the evolution of the full probability distribution, a feat which is rarely done
(or thought to be needed) in classical many-body simulations. Here, we utilize
the MI to analyze the spreading of information in 1D elementary cellular
automata (CA). Broadly speaking, we find that the behavior of the MI in these
dynamical systems exhibits a few different types of scaling that roughly
correspond to known CA universality classes. Of particular note is that we
observe a set of automata for which the MI converges parametrically slowly to
its thermodynamic value. We develop a microscopic understanding of this
behavior by analyzing a two-species model of annihilating particles moving in
opposite directions. Our work suggests the possibility that information
theoretic tools such as the MI might enable a more fine-grained
characterization of classical many-body states and dynamics.
