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

Abstract

Abstract 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.