Laboratoire Matière et Systèmes Complexes, UMR CNRS 7057, Université Paris Diderot-Paris 7, 10 rue Alice Domon et Léonie Duquet, 75205 Paris Cedex 13, France.
J Chem Phys. 2013 Mar 28;138(12):12A542. doi: 10.1063/1.4792641.
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.
最近已经表明，对于硬球的玻璃化转变，模式耦合理论的预测随着维度的增加而变得越来越差，而复制理论则预测了正确的标度。然而，如果人们关注三维中动力学转变的区域，那么模式耦合的结果比复制理论的预测要可信得多。因此，似乎有必要调和这两种理论方法，以便获得一种理论，它可以在低维、模式耦合的结果和复制理论的“平均场”结果之间进行插值。尽管复制理论对动力学转变的定量结果在低维情况下并不准确，但两种不同的近似方案——小笼子展开和复制超网链（RHNC）——为转变提供了正确的定性图像，即静态序参量从零到有限值的不连续跳跃。这项工作的目的是围绕 RHNC 结果展开幂级数的系统展开，并计算该展开式中的第一个修正项。有趣的是，这个修正项涉及到液体的静态三体相关。更重要的是，我们分别证明了在转变中，展开式中的高阶项具有定量相关性，并且通常涉及液体的双体直接相关函数的模式耦合核，不能从静态计算中恢复。
