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We extend a previously proposed field-theoretic self-consistent perturbationapproach for the equilibrium dynamics of the Dean-Kawasaki equation presentedin [J. Stat. Mech. 2008 P02004]. By taking terms missing in the latter analysisinto account we arrive at a set of three new equations for correlationfunctions of the system. These correlations involve the density and itslogarithm as local observables. Our new one-loop equations, which mustcarefully deal with the noninteracting Brownian gas theory, are more generalthan the historic Mode-Coupling one in that a further and well-definedapproximation leads back to the original mode-coupling equation for the densitycorrelations alone. However, without performing any further approximation step,our set of three equations does not feature any ergodic-non ergodic transition,as opposed to the historical mode- coupling approach.

Equilibrium dynamics of the Dean-Kawasaki model: MCT and beyond