The ''Game of Everything''

O. Penrose

1996, v.2, Issue 1, 167-182


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 deterministic 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 $t=0$.

Keywords: entropy,irreversibility,direction of time,reversible cellular automata,probability measures on infinite graphs,Gibbs states


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