  We present an in-depth study of the non-equilibrium statistics of the
irreversible work produced during sudden quenches in proximity to the
structural linear-zigzag transition of ion Coulomb crystals in 1+1 dimensions.
By employing both an analytical approach based on a harmonic expansion and
numerical simulations, we show the divergence of the average irreversible work
in proximity to the transition. We show that the non-analytic behaviour of the
work fluctuations can be characterized in terms of the critical exponents of
the quantum Ising chain. Due to the technological advancements in trapped ion
experiments, our results can be readily verified.
