Large deviations of density in the non-equilibrium steady state of boundary-driven diffusive systems

A diffusive system coupled to unequal boundary reservoirs reaches a non-equilibrium steady state. While the full-counting-statistics of current fluctuations in these states are well understood for generic systems, results for steady-state density fluctuations remain limited to only a few integrable models. By obtaining an exact solution of the Macroscopic Fluctuation Theory, we characterize steady-state density fluctuations through large deviations for a wide range of boundary-driven diffusive systems. This allows us to identify two distinct classes of systems, one with only short-range correlations and another displaying long-range correlations. We also quantitatively describe the irreversible dynamical paths leading to these rare fluctuations in such systems. For very generic systems in arbitrary dimensions, we use a perturbation around the equilibrium state to solve for large deviations and the corresponding fluctuation paths. We find that non-locality in the large deviations emerges only at quadratic order in the perturbation, revealing non-trivial features of long-range correlations in non-equilibrium steady states.