Systematic expansion in the order parameter for replica theory of the dynamical glass transition

Jacquin, Hugo; Zamponi, Francesco
March 2013
Journal of Chemical Physics;Mar2013, Vol. 138 Issue 12, p12A542
Academic Journal
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. More importantly, we separately demonstrate that higher order terms in the expansion are quantitatively relevant at the transition, and that the usual mode-coupling kernel, involving two-body direct correlation functions of the liquid, cannot be recovered from static computations.


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