  The equations of fluctuating nonlinear hydrodynamics for a two component
mixture are obtained with a proper choice of slow variables which correspond to
the conservation laws in the system. Using these nonlinear equations we
construct the basic equations of the mode coupling theory (MCT) and consequent
ergodic-nonergodic (ENE) transition in a binary mixture. The model is also
analyzed in the one component limit of the mixture to study the dynamics of a
tagged particle in the sea of identical particles. According to the existing
MCT, dynamics of the single particle correlation is slaved to that of the
collective density fluctuations and hence both correlations freeze
simultaneously at the ENE transition. We show here from a non-perturbative
approach that at the ENE transition, characterized by the freezing of the long
time limit of the dynamic correlation of collective density fluctuations to a
nonzero value, the tagged particle correlation still decays to zero. Our result
implies that the point at which simulation or experimental data of self
diffusion constant extrapolate to zero would not correspond to the ENE
transition of simple MCT.
