One-dimensional models are a theoretical playground for the investigation of strongly interacting many-body dynamics that have in recent years become experimentally realizable. While mean-field methods and other standard approaches are not applicable, the models can often be investigated with techniques of integrability and generalized hydrodynamics. Fluctuations around hydrodynamic values display remarkable robustness and have recently been observed to be anomalous, indicating additional processes besides normal diffusion.

We propose a theory of diffusive-scale dynamics in integrable and closely related models. The theory is similar to nonlinear fluctuating hydrodynamics but with important structural differences, including linear degeneracy that ensures its stability. Diffusive-scale dynamics of integrable models stems from fluctuations of ballistically propagating modes, a mechanism termed convective diffusion, which can coexist with normal diffusion. A family of cellular automata with ballistically propagating charged particles and stochastic scattering provides an exactly solvable toy model where the hydrodynamic predictions can be matched to microscopic results, revealing how anomalous fluctuations are generated by convective diffusion.

We explain the structural similarity between the hydrodynamics of the cellular automata and the quantum XXZ spin chain in the easy-axis regime where a similar phenomenology of anomalous fluctuations has been observed in equilibrium spin current fluctuations. We also discuss the Heisenberg point where the spin structure factor matches that of the Kardar-Parisi-Zhang universality class while spin fluctuations are anomalous but distinct from those of the KPZ class.