  The search for the QCD critical point in heavy-ion collision experiments
requires dynamical simulations of the bulk evolution of QCD matter as well as
of fluctuations. We consider two essential ingredients of such a simulation: a
generic extension of hydrodynamics by a_parametrically_ slow mode or modes
("Hydro+") and a description of fluctuations out of equilibrium. By combining
the two ingredients we are able to describe the bulk evolution and the
fluctutations within the same framework. Critical slowing down means that
equilibration of fluctuations could be as slow as hydrodynamic evolution and
thus fluctuations could significantly deviate from equilibrium near the
critical point. We generalize hydrodynamics to partial-equilibrium conditions
where the state of the system is characterized by the off-equilibrium magnitude
of fluctuations in addition to the usual hydrodynamic variables -- conserved
densities. We find that the key element of the new formalism -- the extended
entropy taking into account the off-equilibrium fluctuations -- is remarkably
similar to the 2PI action in quantum field theory. We show how the new Hydro+
formalism reproduces two major effects of critical fluctuations on the bulk
evolution: the strong frequency dependence of the anomalously large bulk
viscosity as well as the stiffening of the equation of state with increasing
frequency or wave-number. While the agreement with known results confirms its
validity, the fact that Hydro+ achieves this within a local and deterministic
framework gives it significant advantages for dynamical simulations.
