Generation of charge imbalance in a superconductor by a temperature gradient

The charge-imbalance voltage in a superconductor carrying a current in the presence of a temperature gradient is calculated from the Boltzmann equation in the clean limit. We demonstrate why the Green's-function approach in the dirty limit, first performed by Schmid and Schön, generates the same Boltzmann-like equation for the distribution function. In addition, the charge-imbalance voltage in the absence of an impressed current is calculated. It is shown to depend on ${\nabla }^{2}T+\frac{{\left(\nabla T\right)}^2}{T}$, and not solely on ${\left(\nabla T\right)}^2$. The calculations are limited to the temperature regime near $T_c$, such that $\Delta \ll T$.