Sharp Page transitions in generic Hamiltonian dynamics

We consider the entanglement dynamics of a subsystem initialized in a pure state at high energy density (corresponding to negative temperature) and coupled to a cold bath. The subsystem's R\'enyi entropies $S_\alpha$ first rise as the subsystem gets entangled with the bath and then fall as the subsystem cools. We find that the peak of the min-entropy, $\lim_{\alpha \to \infty} S_\alpha$, sharpens to a cusp in the thermodynamic limit, at a well-defined time we call the Page time. We construct a hydrodynamic ansatz for the evolution of the entanglement Hamiltonian, which accounts for the sharp Page transition as well as the intricate dynamics of the entanglement spectrum before the Page time. Our results hold both when the bath has the same Hamiltonian as the system and when the bath is taken to be Markovian. Our ansatz suggests conditions under which the Page transition should remain sharp even for R\'enyi entropies of finite index $\alpha$.