Extreme current fluctuations in a nonstationary stochastic heat flow

We employ the Hamiltonian formalism of macroscopic fluctuation theory to study large deviations of integrated current in the Kipnis-Marchioro- Presutti (KMP) model of stochastic heat flow when starting from a step-like initial condition. The KMP model belongs to the hyperbolic universality class where diffusion remains relevant no matter how large the fluctuating current is. The extreme current statistics for the KMP model turns out to be sub-Gaussian, as distinguished from the super-Gaussian statistics found for the symmetric simple exclusion process and other models of the elliptic class. The most probable time history of the system, which dominates the extreme current statistics of the KMP model, involves two large-amplitude solitary pulses: of the energy density field and of the conjugate 'momentum' field. The coupled pulses propagate with a constant speed, but their amplitudes slowly grow with time, as the energy density pulse collects most of the available energy on its way.