  Upcoming experimental programs will look for signatures of a possible
critical point in the QCD phase diagram in fluctuation observables. To
understand and predict these signatures, one must account for the fact that the
dynamics of any critical fluctuations must be out-of-equilibrium: because of
critical slowing down, the fluctuations cannot stay in equilibrium as the
droplet of QGP produced in a collision expands and cools. Furthermore, their
out-of-equilibrium dynamics must also influence the hydrodynamic evolution of
the cooling droplet. The recently developed Hydro+ formalism allows for a
consistent description of both the hydrodynamics and the out-of-equilibrium
fluctuations, including the feedback between them. We shall explicitly
demonstrate how this works, setting up a Hydro+ simulation in a simplified
setting: a rapidity-independent fireball undergoing radial flow with an
equation of state in which we imagine a critical point close to the $\mu_B=0$
axis of the phase diagram. Within this setup, we show that we can
quantitatively capture non-equilibrium phenomena, including critical
fluctuations over a range of scales and memory effects. Furthermore, we
illustrate the interplay between the dynamics of the fluctuations and the
hydrodynamic flow of the fireball: as the fluid cools and flows, the dynamical
fluctuations lag relative to how they would evolve if they stayed in
equilibrium; there is then a backreaction on the flow itself due to the
out-of-equilibrium fluctuations; and, in addition, the radial flow transports
fluctuations outwards by advection. Within our model, we find that the
backreaction from the out-of-equilibrium fluctuations does not yield
dramatically large effects in the hydrodynamic variables. Further work will be
needed in order to check this quantitative conclusion in other settings but, if
it persists, this will considerably simplify future modelling.
