is given in Eq. (\[Eq:Carnot-fri\]). Hence knowing a refrigerator’s efficiency at cooling power $J_L$ gives us its entropy production, and we see that maximizing the former minimizes the latter.

Eqs. (\[Eq:dotS-eng\],\[Eq:dotS-fri\]) hold for systems modelled by scattering theory, because this theory satisfies the laws of thermodynamics [@Bruneau2012]$^,$[@2012w-2ndlaw]. The rate of entropy production is zero when the efficiency is that of Carnot, but becomes increasingly positive as the efficiency reduces. In this article, we calculate the maximum efficiency for given power output, and then use Eqs. (\[Eq:dotS-eng\],\[Eq:dotS-fri\]) to get the minimum rate of entropy production at that power output.

Nonlinear Scattering Theory {#Sect:ScatteringTheory}
===========================

This work uses Christen and Büttiker’s nonlinear scattering theory [@Christen-ButtikerEPL96], which treats electron-electron interactions as mean-field charging effects. Refs. \[\] added thermoelectric effects by following works on linear-response [@Engquist-Anderson1981; @Sivan-Imry1986; @Butcher1990]. Particle and heat flows are given by the transmission function, ${\cal T}_{RL}({\epsilon})$, for electrons to go from left ($L$) to right ($R$) at energy ${\epsilon}$, where ${\cal T}_{RL}({\epsilon})$ is a [*self-consistently*]{} determined function of $T_L$, $T_R$ and $V$. In short, this self-consistency condition originates from the fact that electrons injected from the leads change the charge distribution in the quantum system, which in turn changes the 