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We study a zero-dimensional version of the fluctuating nonlinearhydrodynamics (FNH) of supercooled liquids originally investigated by Das andMazenko (DM) [Phys. Rev. A {\bf 34}, 2265 (1986)]. The time-dependentdensity-like and momentum-like variables are introduced with no spatial degreesof freedom in this toy model. The structure of nonlinearities takes the similarform to the original FNH, which allows one to study in a simpler setting theissues raised recently regarding the field theoretical approaches to glassforming liquids. We study the effects of density nonlinearities on the timeevolution of correlation and response functions by developing field theoreticformulations in two different ways: first by following the originalprescription of DM and then by constructing a dynamical action which possessesa linear time reversal symmetry as proposed recently. We show explicitly that,at the one-loop order of the perturbation theory, the DM-type field theory doesnot support a sharp ergodic-nonergodic transition, while the other admits one.The simple nature of the toy model in the DM formulation allows us to developnumerical solutions to a complete set of coupled dynamical equations for thecorrelation and response functions at the one-loop order.

Density nonlinearities in field theories for a toy model of fluctuating nonlinear hydrodynamics of supercooled liquids