  At high temperature, generic strongly interacting spin systems are expected
to display hydrodynamics: local transport of conserved quantities, governed by
classical partial differential equations like the diffusion equation. I argue
that the emergence of this dissipative long-wavelength dynamics from the
system's unitary microscopic dynamics is controlled by the structure of the
\textit{Liouvillean graph} of the system's Hamiltonian, that is, the graph
induced on Pauli strings by commutation with that Hamiltonian. The Liouvillean
graph decomposes naturally into subgraphs of Pauli strings of constant
diameter, and the coherent dynamics of these subgraphs determines the rate at
which operator weight spreads to long operators. This argument provides a
quantitative theory of the emergence of a dissipative effective dynamics from
unitary microscopic dynamics; it also leads to an effective model with Hilbert
space dimension linear in system size and exponential in the UV cutoff for
diffusion.
