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We investigate the non-equilibrium large deviations function of the particledensities in two steady-state driven systems exchanging particles at avanishing rate. We first derive through a systematic multi-scale analysis thecoarse-grained master equation satisfied by the distribution of the numbers ofparticles in each system. Assuming that this distribution takes for largesystems a large deviations form, we obtain the equation (similar to aHamilton-Jacobi equation) satisfied by the large deviations function of thedensities. Depending on the systems considered, this equation may satisfy ornot the macroscopic detailed balance property, i.e., a time-reversibilityproperty at large deviations level. In the absence of macroscopic detailedbalance, the large deviations function can be determined as an expansion closeto a solution satisfying macroscopic detailed balance. In this case, the largedeviations function is generically non-additive, i.e., it cannot be split astwo separate contributions from each system. In addition, the large deviationsfunction can be interpreted as a non-equilibrium free energy, as it satisfies ageneralization of the second law of thermodynamics, in the spirit of theHatano-Sasa relation. Some of the results are illustrated on an exactlysolvable driven lattice gas model.

Non-additive large deviations function for the particle densities of driven systems in contact