Time’s Arrow in a Finite Universe

Abstract

In this paper, a simple model for a closed multiverse as a finite probability space is analyzed. For each moment of time on a discrete time-scale, only a finite number of states are possible and hence each possible universe can be viewed as a path in a huge but finite graph. By considering very general statistical assumptions, essentially originating from Boltzmann, we make the set of all such paths (the multiverse) into a probability space, and argue that under certain assumptions, the probability for a monotonic behavior of the entropy is enormously much larger then for a behavior with low entropy at both ends. The methods used are just very simple combinatorial ones, but the conclusion suggests that we may live in a multiverse which from a global point of view is completely time-symmetric in the sense that universes with Time’s Arrow directed forwards and backwards are equally probable. However, for an observer confined to just one universe, time will still be asymmetric.