Understanding the physics of nonequilibrium systems remains as one of the major challenges of modern theoretical physics. We believe nowadays that this problem can be cracked in part by investigating the macroscopic fluctuations of the currents characterizing nonequilibrium behavior, their statistics, associated structures and microscopic origin. This fundamental line of research has been severely hampered by the overwhelming complexity of this problem. However, during the last years two new powerful and general methods have appeared to investigate fluctuating behavior that are changing radically our understanding of nonequilibrium physics: a powerful macroscopic fluctuation theory (MFT) and a set of advanced computational techniques to measure rare events. In this work we study the statistics of current fluctuations in nonequilibrium diffusive systems, using macroscopic fluctuation theory as theoretical framework, and advanced Monte Carlo simulations of several stochastic lattice gases as a laboratory to test the emerging picture. Our quest will bring us from (1) the confirmation of an additivity conjecture in one and two dimensions, which considerably simplifies the MFT complex variational problem to compute the thermodynamics of currents, to (2) the discovery of novel isometric fluctuation relations, which opens an unexplored route toward a deeper understanding of nonequilibrium physics by bringing symmetry principles to the realm of fluctuations, and to (3) the observation of coherent structures in fluctuations, which appear via dynamic phase transitions involving a spontaneous symmetry breaking event at the fluctuating level. The clear-cut observation, measurement and characterization of these unexpected phenomena, well described by MFT, strongly support this theoretical scheme as the natural theory to understand the thermodynamics of currents in nonequilibrium diffusive media, opening new avenues of research in nonequilibrium physics.

We consider fluctuations of the time-averaged current in the one-dimensional weakly-asymmetric exclusion process on a ring. The optimal density profile which sustains a given fluctuation exhibits an instability for low enough currents, where it becomes time-dependent. This instability corresponds to a dynamical phase transition in the system fluctuation behavior: while typical current fluctuations result from the sum of weakly-correlated local events and are still associated with the flat, steady-state density profile, for currents below a critical threshold the system self-organizes into a macroscopic jammed state in the form of a coherent traveling wave, that hinders transport of particles and thus facilitates a time-averaged current fluctuation well below the average current. We analyze in detail this phenomenon using advanced Monte Carlo simulations, and work out macroscopic fluctuation theory predictions, finding very good agreement in all cases. In particular, we study not only the current large deviation function, but also the critical current threshold, the associated optimal density profiles and the traveling wave velocity, analyzing in depth finite-size effects and hence providing a detailed characterization of the dynamical transition.