Diffusion from convection

We introduce non-trivial contributions to diffusion constants in generic many-body systems with Hamiltonian dynamics arising from quadratic fluctuations of ballistically propagating, i.e. convective, modes. Our result is obtained by expanding the current operator in terms of powers of local and quasi-local conserved quantities. We show that only the second-order terms in this expansion carry a finite contribution to diffusive spreading. Our formalism implies that whenever there are at least two coupled modes with degenerate group velocities the system behaves super-diffusively, in accordance with non-linear fluctuating hydrodynamics. Finally, we show that our expression saturates the exact diffusion constants in quantum and classical interacting integrable systems, providing a general framework to derive these expressions.