Hydrodynamics without a relaxation gap: Memory effects, nonlocality, and superdiffusion

By studying a simple model of relativistic particles propagating through a background medium with an energy-dependent relaxation time that is unbounded from above, we investigate how long-term memory obstructs the emergence of local hydrodynamics in systems with a gapless nonhydrodynamic sector. In the relaxation-time approximation matching frame, we show that the full gradient expansion is generically divergent in most flows, even for Fourier modes, and that any resummation necessarily retains an infinite set of slow nonhydrodynamic degrees of freedom. The divergence of the gradient expansion therefore reflects a more fundamental breakdown of hydrodynamic locality caused by persistent nonhydrodynamic memory. We finally show that sufficiently singular relaxation spectra can invalidate ordinary diffusion itself. In these regimes, the diffusivity diverges, and the late-time dynamics become superdiffusive, governed by intrinsically nonlocal constitutive relations.