Mode-coupling theory: generalizations, high dimensions and microscopic dynamics

This work contains several applications of the mode-coupling theory (MCT) and is separated into three parts. In the first part we investigate the liquid-glass transition of hard spheres for dimensions d→∞ analytically and numerically up to d=800 in the framework of MCT. We find that the critical packing fraction ϕc(d) scales as d²2^(-d), which is larger than the Kauzmann packing fraction ϕK(d) found by a small-cage expansion by Parisi and Zamponi [J. Stat. Mech.: Theory Exp. 2006, P03017 (2006)]. The scaling of the critical packing fraction is different from the relation ϕc(d)∼d2^(-d) found earlier by Kirkpatrick and Wolynes [Phys. Rev. A 35, 3072 (1987)]. This is due to the fact that the k dependence of the critical collective and self nonergodicity parameters fc(k;d) and fcs(k;d) was assumed to be Gaussian in the previous theories. We show that in MCT this is not the case. Instead fc(k;d) and fcs(k;d), which become identical in the limit d→∞, converge to a non-Gaussian master function on the scale k∼d^(3/2). We find that the numerically determined value for the exponent parameter λ and therefore also the critical exponents a and b depend on the dimension d, even at the largest evaluated dimension d=800. In the second part we compare the results of a molecular-dynamics simulation of liquid Lennard-Jones argon far away from the glass transition [D. Levesque, L. Verlet, and J. Kurkijärvi, Phys. Rev. A 7, 1690 (1973)] with MCT. We show that the agreement between theory and computer simulation can be improved by taking binary collisions into account [L. Sjögren, Phys. Rev. A 22, 2866 (1980)]. We find that an empiric prefactor of the memory function of the original MCT equations leads to similar results. In the third part we derive the equations for a mode-coupling theory for the spherical components of the stress tensor. Unfortunately it turns out that they are too complex to be solved numerically.