In Earth's core, metallic electrons give a relatively large electrical
conductivity and contribute also a significant amount to the heat
conductivity. In the mantle, the dominant phases are non-metallic with a
reduced conductivity. In both regions, heat is also carried in part by
atomic motions. A poor understanding of these properties has been a
limiting factor in modeling Earth's magnetism and heat flows. It is
possible to use results from first-principles molecular dynamics to
evaluate transport coefficients, and there are good reasons to expect that
this can be done with useful accuracy.
Through Green-Kubo type formulas, the transport coefficients are expressed
in terms of time integrals of current correlation functions , j
being the electrical or thermal current. Electricity is carried primarily
by electrons, while heat is carried both by electrons and atoms. The
electronic currents are quantum operators whose expectation values are
calculable in the single-particle representation of density functional
theory. The relevant time t for electrons is fairly short, so one
typically ignores atomic motion and evaluates the time integral separately
at each time step of the MD run. A plane-wave basis, used in most of our
MD work, allows easy and accurate current matrix elements. This method has
worked for V(1-x)Alx alloys at low T and should work as well or better at
the high temperatures of the Earth's interior. Also, the computational
demands should be much simplified by the high temperature of the system,
the corresponding short mean free path of the vibrations, and rapid
thermalization.
An interesting complexity is the radiative (electromagnetic) component of
the heat current. Because of the high temperature, the electromagnetic
energy density can be large, and it is necessary to ask how this energy
diffuses. Maxwell's equations answer this in terms of the electronic
optical conductivity, which is easily computed in parallel with the dc
value discussed above, so we expect to make accurate calculations of this
component. (Allen, Gillan, Price, and Wentzcovitch)
