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Hydrodynamic projections, the projection onto conserved charges representingballistic propagation of fluid waves, give exact transport results in many-bodysystems, such as the exact Drude weights. Focussing one one-dimensionalsystems, I show that this principle can be extended beyond the Euler scale, inparticular to the diffusive and superdiffusive scales. By hydrodynamicreduction, Hilbert spaces of observables are constructed that generalise thestandard space of conserved densities and describe the finer scales ofhydrodynamics. The Green-Kubo formula for the Onsager matrix has a naturalexpression within the diffusive space. This space is associated withquadratically extensive charges, and projections onto any such charge givegeneric lower bounds for diffusion. In particular, bilinear expressions inlinearly extensive charges lead to explicit diffusion lower bounds calculablefrom the thermodynamics, and applicable for instance to genericmomentum-conserving one-dimensional systems. Bilinear charges are interpretedas covariant derivatives on the manifold of maximal entropy states, andrepresent the contribution to diffusion from scattering of ballistic waves. Ananalysis of fractionally extensive charges, combined with clustering propertiesfrom the superdiffusion phenomenology, gives lower bounds for superdiffusionexponents. These bounds reproduce the predictions of nonlinear fluctuatinghydrodynamics, including the Kardar-Parisi-Zhang exponent 2/3 for sound-likemodes, the Levy-distribution exponent 3/5 for heat-like modes, and the fullFibonacci sequence.

Diffusion and superdiffusion from hydrodynamic projections