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We investigate non-stationary heat transfer in the Kipnis-Marchioro-Presutti(KMP) lattice gas model at long times in one dimension when starting from alocalized heat distribution. At large scales this initial condition can bedescribed as a delta-function, $u(x,t=0)=W \delta(x)$. We characterize theprocess by the heat, transferred to the right of a specified point $x=X$ bytime $T$, $$ J=\int_X^\infty u(x,t=T)\,dx\,, $$ and study the full probabilitydistribution $\mathcal{P}(J,X,T)$. The particular case of $X=0$ has beenrecently solved [Bettelheim \textit{et al}. Phys. Rev. Lett. \textbf{128},130602 (2022)]. At fixed $J$, the distribution $\mathcal{P}$ as a function of$X$ and $T$ has the same long-time dynamical scaling properties as the positionof a tracer in a single-file diffusion. Here we evaluate $\mathcal{P}(J,X,T)$by exploiting the recently uncovered complete integrability of the equations ofthe macroscopic fluctuation theory (MFT) for the KMP model and using theZakharov-Shabat inverse scattering method. We also discuss asymptotics of$\mathcal{P}(J,X,T)$ which we extract from the exact solution, and also obtainby applying two different perturbation methods directly to the MFT equations.

Full Statistics of Nonstationary Heat Transfer in the Kipnis-Marchioro-Presutti Model