marks the Curzon-Ahlborn efficiency. ](figure2.pdf){width="\columnwidth"}

Contents of this article
------------------------

This article provides detailed derivations of the results in Ref. \[\]. The first part of this article is an extended introduction. Section \[Sect:literature\] is a short review of the relevant literature. Section \[Sect:Unique\] discusses how we define temperature, heat and entropy. Section \[Sect:entropy-prod\] recalls the connection between efficiency and entropy production in any thermodynamic machine. Section \[Sect:ScatteringTheory\] reviews the nonlinear scattering theory, which section \[Sect:over-estimates\] uses to make very simple over-estimates of a quantum system’s maximum power output.

The second part of this article considers how to optimize a system which is free of relaxation and has no phonons or photons. Section \[Sect:guess-heat\] gives a hand-waving explanation of the optimal heat engine, while Section \[Sect:eng\] gives the full derivation. Section \[Sect:guess-fri\] gives a hand-waving explanation of the optimal refrigerator, while Section \[Sect:fri\] gives the full derivation. Section \[Sect:chain\] proposes a system which could in principle come arbitrarily close to the optimal properties given in sections \[Sect:eng\] and \[Sect:fri\]. Section \[Sect:in-parallel\] considers many quantum thermoelectrics in parallel.

The third part of this article considers certain effects neglected in the above idealized system. Section \[Sect:ph\] adds the parasitic effect of phonon or 