Non-Equilibrium phase transitions in large deviation theory: chemical potential and Maxwell construction

In large deviation theory, a (non-equilibrium) phase transition manifests itself in form of a 'Maxwell construction', similar as known from equilibrium thermodynamics. I will briefly summarise some results of large deviation theory and give examples how phase transitions emerge in this framework. In particular, the large deviation theory claims the existence of a 'chemical potential' for certain non-equilibrium systems. I will exemplify on a driven particle system how the chemical potential can be expressed by nearest-neighbour correlations and how the Maxwell construction allows to determine a first order phase transition in this non-equilibrium system.