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Testing Mode-Coupling Theory for a Supercooled Binary Lennard-Jones Mixture I: The van Hove Correlation Function

We report the results of a large scale computer simulation of a binary supercooled Lennard-Jones liquid. We find that at low temperatures the curves for the mean squared displacement of a tagged particle for different temperatures fall onto a master curve when they are plotted versus rescaled time $tD(T)$, where $D(T)$ is the diffusion constant. The time range for which these curves follow the master curve is identified with the $α$-relaxation regime of mode-coupling theory (MCT). This master curve is fitted well by a functional form suggested by MCT. In accordance with idealized MCT, $D(T)$ shows a power-law behavior at low temperatures. The critical temperature of this power-law is the same for both types of particle and also the critical exponents are very similar. However, contrary to a prediction of MCT, these exponents are not equal to the ones determined previously for the divergence of the relaxation times of the intermediate scattering function [Phys. Rev. Lett. {\bf 73}, 1376 (1994)]. At low temperatures the van Hove correlation function (self as well as distinct part) shows hardly any sign of relaxation in a time interval that extends over about three decades in time. This time interval can be interpreted as the $β$-relaxation regime of MCT. From the investigation of these correlation functions we conclude that hopping processes are not important on the time scale of the $β$-relaxation for this system and for the temperature range investigated. We test whether the factorization property predicted by MCT holds and find that this is indeed the case for all correlation functions investigated. The distance dependence of the critical amplitudes are in qualitative accordance with the ones predicted by MCT for some other mixtures. The non-gaussian parameter for the self part of the van