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Recent developments on black holes have shown that a unitarity-compatiblePage curve can be obtained from an ensemble-averaged semi-classicalapproximation. In this paper, we emphasize (1) that this peculiar manifestationof unitarity is not specific to black holes, and (2) that it can emerge from asingle realization of an underlying unitary theory. To make things explicit, weconsider a hard sphere gas leaking slowly from a small box into a bigger box.This is a quantum chaotic system in which we expect to see the Page curve inthe full unitary description, while semi-classically, eigenstates are expectedto behave as though they live in Berry's ensemble. We reproduce theunitarity-compatible Page curve of this system, semi-classically. Thecomputation has structural parallels to replica wormholes, relies crucially onensemble averaging at each epoch, and reveals the interplay between themultiple time-scales in the problem. Working with the ensemble averaged $state$rather than the entanglement entropy, we can also engineer an information"paradox". Our system provides a concrete example in which the ensembleunderlying the semi-classical Page curve is an ergodic proxy for a timeaverage, and not an explicit average over many theories. The questions weaddress here are logically independent of the existence of horizons, so weexpect that semi-classical gravity should also be viewed in a similar light.

Hints of Gravitational Ergodicity: Berry's Ensemble and the Universality of the Semi-Classical Page Curve