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

TL;DR

This paper demonstrates that the unitarity-compatible Page curve can emerge from a single quantum chaotic system through ensemble averaging, linking semi-classical approximations to ergodic behavior and Berry's ensemble.

Contribution

It shows that the Page curve's unitarity features are not exclusive to black holes and can arise from a single system via ensemble averaging, with structural parallels to replica wormholes.

Findings

Reproduces the Page curve semi-classically in a quantum chaotic system.

Highlights the role of ensemble averaging as an ergodic proxy for time averages.

Provides a concrete example where semi-classical gravity relates to ergodic behavior.

Abstract

Recent developments on black holes have shown that a unitarity-compatible Page curve can be obtained from an ensemble-averaged semi-classical approximation. In this paper, we emphasize (1) that this peculiar manifestation of unitarity is not specific to black holes, and (2) that it can emerge from a single realization of an underlying unitary theory. To make things explicit, we consider 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 in the full unitary description, while semi-classically, eigenstates are expected to behave as though they live in Berry's ensemble. We reproduce the unitarity-compatible Page curve of this system, semi-classically. The computation has structural parallels to replica wormholes, relies crucially on ensemble averaging at each epoch, and reveals the interplay between the multiple 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 ensemble underlying the semi-classical Page curve is an ergodic proxy for a time average, and not an explicit average over many theories. The questions we address here are logically independent of the existence of horizons, so we expect that semi-classical gravity should also be viewed in a similar light.