Although basic equilibrium concepts like state variables, thermodynamic potentials, etc do not straightforwardly extend for non-equilibrium states, an analogue of Landau free energy characterising macroscopic fluctuations could be defined based on a mathematical theory of large deviations. I shall present a few old and new exact results for such a quantity in a class of non-equilibrium transport models. These are interacting classical many-body systems with relations to quantum spin chains. Our new results show that similar to a thermodynamic potential, these free-energy-analogues are robust against variations of coupling with external baths, thereby strengthening the possibility of a thermodynamic description for non-equilibrium states.
I shall explain how these exact results are derived using a tilted operator formalism and a matrix-product-representation for a specific integrable model . Then, I shall recover these results using a minimal action solution of a hydrodynamic field theory , which applies to a wider class of systems, including those which are non-integrable.Refs.  Derrida, Hirschberg, Sadhu, 182 (2021) J Stat Phys;  in preparation.