  We derive a large-scale hydrodynamic equation, including diffusive and
dissipative effects, for systems with generic static position-dependent driving
forces coupling to local conserved quantities. We show that this equation
predicts entropy increase and thermal states as the only stationary states. The
equation applies to any hydrodynamic system with any number of local,
PT-symmetric conserved quantities, in arbitrary dimension. It is fully
expressed in terms of elements of an extended Onsager matrix. In integrable
systems, this matrix admits an expansion in the density of excitations. We
evaluate exactly its 2-particle-hole contribution, which dominates at low
density, in terms of the scattering phase and dispersion of the quasiparticles,
giving a lower bound for the extended Onsager matrix and entropy production. We
conclude with a molecular dynamics simulation, demonstrating thermalisation
over diffusive time scales in the Toda interacting particle model with an
inhomogeneous energy field.
