Dynamic glass transition: bridging the gap between mode-coupling theory and the replica approach

We clarify the relation between the ergodicity breaking transition predicted by mode-coupling theory and the so-called dynamic transition predicted by the static replica approach. Following Franz and Parisi [Phys. Rev. Lett. 79, 2486 (1997)], we consider a system of particles in a metastable state characterized by non-trivial correlations with a quenched configuration. We show that the assumption that in a metastable state particle currents vanish leads to an expression for the replica off-diagonal direct correlation function in terms of a replica off-diagonal static four-point correlation function. A factorization approximation for this function results in an approximate closure for the replica off-diagonal direct correlation function. The replica off-diagonal Ornstein-Zernicke equation combined with this closure coincides with the equation for the non-ergodicity parameter derived using the mode-coupling theory.