Cosmological boundary conditions for particles and fields are often discussed as a Cauchy problem, in which configurations and conjugate momenta are specified on an initial time slice. But this is not the only way to specify boundary conditions, and indeed in action-principle formulations we often specify configurations at two times and consider trajectories joining them. Here, we consider a classical system of particles interacting with short range two body interactions, with boundary conditions on the particles positions for an initial and a final time. For a large number of particles that are randomly arranged into a dilute gas, we find that a typical system trajectory will spontaneously collapse into a small region of space, close to the maximum density that is obtainable, before expanding out again. If generalizable, this has important implications for the cosmological arrow of time, potentially allowing a scenario in which both boundary conditions are generic and also a low-entropy state initial state of the universe naturally occurs.
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