Mean-field analysis of the glassy dynamics of an elastoplastic model of super-cooled liquids

We present a mean-field theory of a coarse-grained model of a super-cooled liquid in which relaxation occurs via local plastic rearrangements. Local relaxation can be induced by thermal fluctuations or by the long-range elastic consequences of other rearrangements. We extract the temperature dependence of both the relaxation time and the length scale of dynamical correlations. We find two dynamical regimes. First, a regime in which the characteristic time and length scales diverge as a power law at a critical temperature Tc. This regime is found by an approximation that neglects activated relaxation channels, which can be interpreted as akin to the one found by the mode-coupling transition of glasses. In reality, only a crossover takes place at Tc. The residual plastic activity leads to a second regime characterised by an Arrhenius law below Tc. In this case, we show that the length scale governing dynamical correlations diverges as a power law as , and is logarithmically related to the relaxation time.