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 the mode-coupling theory and the so-called dynamic transition predicted by the static replica approach. Following Franz and Parisi (Phys. Rev. Lett., 79 (1997) 2486), 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 non-trivial part of the 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. Copyright (C) EPLA, 2010