Interacting diffusive particle systems are considered as paradigms for non-equilibrium statistical physics. Their macroscopic behaviour follows a variational principle, proposed by G. Jona-Lasinio and his collaborators, known as the Macroscopic Fluctuation Theory (MFT). Optimal fluctuations far from equilibrium are thus determined at a coarse-grained scale by two coupled non-linear hydrodynamic equations.
In this talk, we shall show that, for the exclusion process, the MFT equations are classically integrable and can be solved with the help of the inverse scattering method, originally used to study solitons in the KdV or the NLS equations. This exact solution will allow us to calculate the large deviations of the current and the optimal profiles that generates a given fluctuation, both at initial and final times.
