Large deviations of the current in a two-dimensional diffusive system

In this notes we study the large deviations of the time-averaged current in the two-dimensional (2D) Kipnis-Marchioro-Presutti model of energy transport when subject to a boundary gradient. We use the tools of hydrodynamic fluctuation theory, supplemented with an appropriate generalization of the additivity principle. As compared to its one-dimensional counterpart, which amounts to assume that the optimal profiles responsible of a given current fluctuation are time-independent, the 2D additivity conjecture requires an extra assumption, i.e. that the optimal, divergence-free current vector field associated to a given fluctuation of the time-averaged current is in fact constant across the system. Within this context we show that the current distribution exhibits in general non-Gaussian tails. The ensuing optimal density profile can be either monotone for small current fluctuations, or non-monotone with a single maximum for large enough current deviations. Furthermore, this optimal profile remains invariant under arbitrary rotations of the current vector, providing a detailed example of the recently introduced Isometric Fluctuation Relation.