  In this work we elaborate on two recently discovered invariance principles,
according to which transport coefficients are, to a large extent, independent
of the microscopic definition of the densities and currents of the conserved
quantities being transported (energy, momentum, mass, charge). The first such
principle, gauge invariance, allows one to define a quantum adiabatic energy
current from density-functional theory, from which the heat conductivity can be
uniquely defined and computed using equilibrium ab initio molecular dynamics.
When combined with a novel topological definition of atomic oxidation states,
gauge invariance also sheds new light onto the mechanisms of charge transport
in ionic conductors. The second principle, convective invariance, allows one to
extend the analysis to multi-component systems. These invariance principles can
be combined with new spectral analysis methods for the current time series to
be fed into the Green-Kubo formula to obtain accurate estimates of transport
coefficients from relatively short molecular dynamics simulations.
