Is hydrodynamics ever relevant in metals?

This is a very subtle question. I learnt a lot from a nice talk that Steve Kivelson gave on the subject.

In a single component fluid [e.g. water or a gas] a hydrodynamic approach works because one has local conversation of energy and momentum. Then ALL transport properties are determined by just three quantities: the shear viscosity (eta), the second viscosity (zeta), and the thermal conductivity (kappa).

However, the electron fluid in a solid can exchange energy and momentum with the lattice and impurities. Hence, hydrodynamics is not relevant.

What about temperature ranges where electron-electron scattering dominates? Well then one has resistivity as a result of Umklapp scattering, which means that there is momentum exchange with the lattice. [I never understand all the subtle details of that]. Hence, one still does not have conditions under which hydrodynamics may apply.

So when might hydrodynamics apply? Kivelson suggests it may be relevant in 2DEGS [2-Dimensional Electron Gases] in semiconductor heterostructures close to the metal-insulator transition. Then the lattice is irrelevant and the electron-electron scattering is dominant. What causes resistivity? It can be collision of the fluid with "large" objects such as some slowly varying background impurity potential. Andreev, Kivelson, and Spivak calculated the two-dimensional resistivity as

where the averaging is over space and n0 and s0 denote the equilibrium density and entropy. An important point is that the resistivity is proportional to the viscosity. In simple kinetic theory this is proportional to the scattering time. In a Fermi liquid this will increase with decreasing temperature. Thus the resistivity will have the opposite temperature dependence to a conventional metal!

Hydrodynamics in metals opens the possibility of turbulence. Signatures of this will be nonlinear I-V characteristics, dependence on geometry, and noise.

Kivelson also highlighted relevant recent work by his string theory colleagues on breakdown of the Wiedemann-Franz law in non-Fermi liquids.

A previous post considered the problem of the viscosity of bad metals.