Hugo Jacquin, Francesco Zamponi
It has been shown recently that predictions from Mode-Coupling Theory for the glass transition of hard-spheres become increasingly bad when dimensionality increases, whereas replica theory predicts a correct scaling. Nevertheless if one focuses on the regime around the dynamical transition in three dimensions, Mode-Coupling results are far more convincing than replica theory predictions. It seems thus necessary to reconcile the two theoretic approaches in order to obtain a theory that interpolates between low-dimensional, Mode-Coupling results, and "mean-field" results from replica theory. Even though quantitative results for the dynamical transition issued from replica theory are not accurate in low dimensions, two different approximation schemes --small cage expansion and replicated Hyper-Netted-Chain (RHNC)-- provide the correct qualitative picture for the transition, namely a discontinuous jump of a static order parameter from zero to a finite value. The purpose of this work is to develop a systematic expansion around the RHNC result in powers of the static order parameter, and to calculate the first correction in this expansion. Interestingly, this correction involves the static three-body correlations of the liquid, but more importantly, we show that it provides some improvement over the quantitative results of RHNC approximation.
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http://arxiv.org/abs/1211.3468
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