  We derive Euler equations from a Hamiltonian microscopic dynamics. The
microscopic system is a one-dimensional disordered harmonic chain, and the
dynamics is either quantum or classical. This chain is an Anderson insulator
with a symmetry protected mode: Thermal fluctuations are frozen while the low
modes ensure the transport of elongation, momentum and mechanical energy, that
evolve according to Euler equations in an hyperbolic scaling limit. In this
paper, we strengthen considerably our previous results, where we established a
limit in mean starting from a local Gibbs state: We now control the second
moment of the fluctuations around the average, yielding a limit in probability,
and we enlarge the class of admissible initial states.
