  We study cumulants of the chiral order parameter as function of beam energy
as a possible signal for the presence of a critical end point and first-order
phase transition in the QCD phase diagram. We model the expansion of a
heavy-ion collision by a fluid dynamic expansion coupled to the explicit
propagation of the chiral order parameter sigma via a Langevin equation. We
evolve the medium until a parametrized freeze-out condition is met where we
calculate event-by-event fluctuations and cumulants of sigma which are expected
to follow the trend of net-proton number cumulants. We emphasize the role of a
nonequilibrium first-order phase transition: The presence of an unstable phase
causes the well-known bending of the trajectories in the space of temperature
and baryochemical potential. For these cases at lower beam energies, the system
crosses the freeze-out line more than once, allowing us to calculate a range of
cumulants for each initial condition which are overall enhanced for the second
hit of the freeze-out line. We thus find not only the critical end point but
also the phase transition of the underlying model clearly reflected in the
cumulants. The impact of volume fluctuations is demonstrated to play a
measurable role for fluid dynamical evolutions that last significantly long.
