Full statistics of nonstationary heat transfer in the Kipnis-Marchioro-Presutti model

We investigate non-stationary heat transfer in the Kipnis-Marchioro-Presutti (KMP) lattice gas model at long times in one dimension when starting from a localized heat distribution. At large scales this initial condition can be described as a delta-function, u(x, t = 0) = Wδ(x). We characterize the process by the heat transferred to the right of a specified point x = X by time T, J = ∫ X ∞ u ( x , t = T ) d x , and study the full probability distribution P ( J , X , T ) . The particular case of X = 0 has been recently solved by Bettelheim et al (2022 Phys. Rev. Lett. 128 130602). At fixed J, the distribution P as a function of X and T has the same long-time dynamical scaling properties as the position of a tracer in a single-file diffusion. Here we evaluate P ( J , X , T ) by exploiting the recently uncovered complete integrability of the equations of the macroscopic fluctuation theory (MFT) for the KMP model and using the Zakharov-Shabat inverse scattering method. We also discuss asymptotics of P ( J , X , T ) which we extract from the exact solution and also obtain by applying two different perturbation methods directly to the MFT equations.