Discrete System Phase Space Modification by Vanishing Noise

We investigate two classes of discrete systems: the Hopfield model with deterministic update dynamics and elementary cellular automata. We obtain a sampling of the attractor space of the Hopfield model and the complete phase space of all minimal elementary cellular automata for small lattice sizes. Starting from the maximal entropy distribution (all configurations equiprobable), we show how the dynamics affects this distribution. We then investigate how a vanishing noise alters this phase space, connecting attractors and modifying the asymptotic probability distribution. For both the Hopfield model and Cellular Automata, this modification does not always decrease the entropy.