Page-Curve-Like Dynamics in Integrable Systems: From Quantum Entanglement to Classical Mutual Information

Abstract

Entanglement entropy plays an important role in characterizing the buildup of correlations in quantum many-body systems, particularly in non-equilibrium settings. In integrable systems such as free fermions, its dynamical growth can exhibit Page-curve-like behavior and admits an effective large-scale description through generalized hydrodynamics. This raises the question of whether similar entropy dynamics are intrinsically quantum, or whether analogous features can arise in purely classical systems.In this work, we compare entropy and correlation dynamics in quantum free fermionic systems and the classical hard rod gas. Both systems admit a coarse-grained hydrodynamic description. In the quantum case, the Yang–Yang entropy depends only on quasiparticle mode occupation densities, whereas in the classical hard rod gas the entropy additionally involves the local particle density through an excluded-volume contribution arising from hard-core interactions between rods. While generalized hydrodynamics successfully captures entanglement entropy growth in the quantum case, classical hydrodynamic entropy does not generically reproduce non-monotonic behavior, with its evolution depending on the choice of initial conditions. To probe correlations beyond hydrodynamic entropy, we compute mutual information in classical hard rod systems, to the extent analytically tractable for this interacting system. Mutual information exhibits non-monotonic behavior—an initial rise followed by decay—with the late-time behavior depending on the system configuration, saturating to a non-zero value in finite systems and decaying to zero in the presence of an infinite reservoir. These results clarify the extent to which Page-curve-like behavior can emerge in classical systems and highlight the distinct roles played by entropy, correlations, and hydrodynamic descriptions in classical and quantum integrable systems.