This report was set up to answer the question “why do real metals oppose the flow of current?” We’ve expanded on the simple “because electrons can’t be accelerated indefinitely” answer, to talk about the mechanism of energy dissipation. An argument by \citet{Drude_1900} shows that the conductivity of a system of weakly interacting constituents is proportional to the characteristic decay time of such mechanism.
In the framework of Landau Fermi liquids, it isn’t bare electrons that conduct electricity, but dressed electrons, or Landau quasiparticles. When excited above the Fermi Energy by an external field, these quasiparticles decay back to the Fermi sea with a decay time proportional to \(T^{-2}\) at temperature \(T\). Also, due to Umklapp scattering, scattering that sends a crystal momentum outside the first Brillouin zone, the whole Fermi sea looses momentum that is then transferred to the crystal.
Two other energy dissipation mechanisms are scattering by the impurities in the crystal lattice and by phonons. The former leads to a temperature independent resistivity, while the latter leads to a decay time scaling as \(T^{-1}\).
Finally, we looked at what happens when we add an external magnetic field. In this case, the current in directions orthogonal to the magnetic field will stop being parallel to the electric field, resulting in non-zero off-diagonal components to the conductivity tensor. As long as a material has a constant decay time throughout, conductivity will not depend on magnetic field, resulting in the familiar Hall effect. However, as is the case for Landau quasiparticles, when a material has charge carriers with varying decay times, the conductivity will depend on the magnetic field, resulting in the effect known as magnetoresistance.