Optimal paths and dynamical symmetry breaking in the current fluctuations of driven diffusive media

Large deviation theory provides a framework to understand macroscopic fluctuations and collective phenomena in many-body nonequilibrium systems in terms of microscopic dynamics. In these lecture notes we discuss the large deviation statistics of the current, a central observable out of equilibrium, using mostly macroscopic fluctuation theory (MFT) but also microscopic spectral methods. Special emphasis is put on describing the optimal path leading to a rare fluctuation, as well as on different dynamical symmetry breaking phenomena that appear at the fluctuating level. We start with an overview of trajectory statistics in driven diffusive systems as described by MFT. We discuss the additivity principle, a simplifying conjecture to compute the current distribution in one-dimensional nonequilibrium systems, and extend this idea to higher dimensions, where the nonlocal structure of the optimal current vector field becomes crucial. Next we explore dynamical phase transitions (DPTs) in current fluctuations, which manifest as symmetry-breaking events in trajectory statistics. These include particle-hole symmetry-breaking DPTs in open channels, for which we work out a Landau-like theory as well as the joint statistics of the current and the order parameter. Time-translation symmetry-breaking DPTs in periodic systems are also discussed, where coherent traveling condensates emerge to facilitate current deviations. We also discuss the microscopic spectral mechanism leading to these DPTs, which is linked to an emerging degeneracy of the leading eigenspace. Using this spectral perspective, we find the signatures of the recently discovered time-crystal phases of matter in traveling-wave DPTs, and use Doob's transform to propose a packing-field mechanism to create programmable time-crystals in driven systems. Finally, we address open challenges and future directions in this rapidly evolving field.