Here we describe the extension to non-equilibrium which Lieb and Yngvason provide to their equilibrium approach presented in Chapters 2 to 9. This involves an extension of the space of equilibrium states to include non-equilibrium states with their mathematical relationship with equilibrium states being defined and explored. However, the physical nature of such states is left open. We rectify this omission with a brief presentation of classical irreversible thermodynamics (CIT). The interesting feature of the juxtaposition of these two very different pictures of non-equilibrium thermodynamics is that it shows that CIT could allow for the possibility of a unique non-equilibrium entropy function otherwise lacking in the Lieb–Yngvason formulation.