A Toy Model Provably Featuring an Arrow of Time Without Past Hypothesis

The laws of Physics are time-reversible, making no qualitative distinction between the past and the future-yet we can only go towards the future. This apparent contradiction is known as the 'arrow of time problem'. Its current resolution states that the future is the direction of increasing entropy. But entropy can only increase towards the future if it was low in the past, and past low entropy is a very strong assumption to make, because low entropy states are rather improbable, non-generic. Recent work from the Physics literature suggests, however, that we may do away with this so-called 'past hypothesis', in the presence of reversible dynamical laws featuring expansion. We prove that this is the case, for a reversible causal graph dynamics-based toy model. It consists in graphs upon which particles circulate and interact according to local reversible rules. Some rules locally shrink or expand the graph. Almost all states expand; entropy always increases as a consequence of expansion-thereby providing a local explanation for the arrow of time without the need for a past hypothesis. This discrete setting allows us to deploy the full rigour of theoretical Computer Science proof techniques. These objects are also interesting from a Dynamical Systems point of view, as a simple generalisations of cellular automatons exhibiting non-trivial behaviours.