Restricted Wiedemann-Franz law in 1D conductors

We show that under external electric fields or thermal gradients, carrier distributions in one-dimensional (1D) conductors with linear $E$-$k$ dispersion have different temperatures for forward and backward (branch) carrier populations, as a consequence of self-consistent carrier-heat transport. We derive the moment equations of the Boltzmann transport equation, in the presence of elastic scattering, for which: (a) The Wiedemann-Franz law is restricted to each branch with its specific temperature; (b) thermoelectric power vanishes due to electron-hole symmetry. The model depicts different regimes such as ballistic and diffusive and shows excellent agreement with diffusive carrier transport in 1D conductors.