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We study the dynamics of the statistics of the energy transferred across apoint along a quantum chain which is prepared in the inhomogeneous initialstate obtained by joining two identical semi-infinite parts thermalized at twodifferent temperatures. In particular, we consider the transverse field Isingand harmonic chains as prototypical models of non-interacting fermionic andbosonic excitations, respectively. Within the so-called hydrodynamic limit oflarge space-time scales we first discuss the mean values of the energy densityand current, and then, aiming at the statistics of fluctuations, we calculateexactly the scaled cumulant generating function of the transferred energy. Fromthe latter, the evolution of the associated large deviation function isobtained. A natural interpretation of our results is provided in terms of asemi-classical picture of quasi-particles moving ballistically along classicaltrajectories. Similarities and differences between the transferred energyscaled cumulant and the large deviation functions in the cases ofnon-interacting fermions and bosons are discussed.

Dynamics of large deviations in the hydrodynamic limit: Non-interacting systems