  Heat production and dissipation induced by current flow in nanostructures is
of primary importance to understand the stability of these systems. These
effects have contributions from both electron-phonon and electron-electron
interactions. Here, we consider the effect of the local electron and ionic
heating on the conductance of nanoscale systems. Specifically we show that the
non-linear dependence of the conductance on the external bias may be used to
infer information about the local heating of both electrons and ions. We
compare our results with available experimental data on transport in
$\mathrm{D}_2$ and $\mathrm{H}_2$ molecules. The comparison between experiment
and theory is reasonably good close to the lowest phonon mode of the molecule,
especially for the $\mathrm{D}_2$ molecule. At higher biases we cannot rule out
the presence of other effects like, e.g., current-induced forces that make the
scenario more complex.
