  We study the dynamics of the statistics of the energy transferred across a
point along a quantum chain which is prepared in the inhomogeneous initial
state obtained by joining two identical semi-infinite parts thermalized at two
different temperatures. In particular, we consider the transverse field Ising
and harmonic chains as prototypical models of non-interacting fermionic and
bosonic excitations, respectively. Within the so-called hydrodynamic limit of
large space-time scales we first discuss the mean values of the energy density
and current, and then, aiming at the statistics of fluctuations, we calculate
exactly the scaled cumulant generating function of the transferred energy. From
the latter, the evolution of the associated large deviation function is
obtained. A natural interpretation of our results is provided in terms of a
semi-classical picture of quasi-particles moving ballistically along classical
trajectories. Similarities and differences between the transferred energy
scaled cumulant and the large deviation functions in the cases of
non-interacting fermions and bosons are discussed.
