Ergodicity and hydrodynamics: From quantum to classical spin systems.

We show that in classical spin systems, the precise nature of the late-time hydrodynamic tails of the autocorrelation functions of a generic observable is determined by (i) the dynamical critical exponent and (ii) the equilibrium thermodynamic properties of the corresponding observable. We provide numerical results for one- and two-dimensional systems and present theoretical considerations that only rely on the notion of ergodicity. Our result extends to the classical framework the relaxation-overlap inequality, first introduced in [Phys. Rev. X 15, 011059 (2025)2160-330810.1103/PhysRevX.15.011059] for quantum many-body systems satisfying the eigenstate thermalization hypothesis.