  Large-scale electrical and thermal currents in ordinary metals are well
approximated by effective medium theory: global transport properties are
governed by the solution to homogenized coupled diffusion equations. In some
metals, including the Dirac fluid of nearly charge neutral graphene,
microscopic transport is not governed by diffusion, but by a more complicated
set of linearized hydrodynamic equations, which form a system of degenerate
elliptic equations coupled with the Stokes equation for fluid velocity. In
sufficiently inhomogeneous media, these hydrodynamic equations reduce to
homogenized diffusion equations. We re-cast the hydrodynamic transport
equations as the infimum of a functional over conserved currents, and present a
functional framework to model and compute the homogenized diffusion tensor
relating electrical and thermal currents to charge and temperature gradients.
We generalize to this system two well-known results in homogenization theory:
Tartar's proof of local convergence to the homogenized theory in periodic and
highly oscillatory media, and sub-additivity of the above functional in random
media with highly oscillatory, stationary and ergodic coefficients.
