  We investigate the non-equilibrium dynamics of the transverse field quantum
Ising chain evolving from an inhomogeneous initial state given by joining two
macroscopically different semi-infinite chains. We obtain integral expressions
for all two-point correlation functions of the Jordan-Wigner Majorana fermions
at any time and for any value of the transverse field. Using this result, we
compute analytically the profiles of various physical observables in the
space-time scaling limit and show that they can be obtained from a hydrodynamic
picture based on ballistically propagating quasiparticles. Going beyond the
hydrodynamic limit, we analyze the approach to the non-equilibrium steady state
and find that the leading late time corrections display a lattice effect. We
also study the fine structure of the propagating fronts which are found to be
described by the Airy kernel and its derivatives. Near the front we observe the
phenomenon of energy back-flow where the energy locally flows from the colder
to the hotter region.
