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.

Numerical analysis of the relation between interactions and structure in a molecular fluid

Coarse graining is a popular technique used in physics to speed up the computer simulation of molecular fluids. An essential part of this technique is a method that solves the inverse problem of determining the interaction potential or its parameters from the given structural data. Due to discrepancies between model and reality, the potential is not unique, such that stability of such method and its convergence to a meaningful solution are issues.rnrnIn this work, we investigate empirically whether coarse graining can be improved by applying the theory of inverse problems from applied mathematics. In particular, we use the singular value analysis to reveal the weak interaction parameters, that have a negligible influence on the structure of the fluid and which cause non-uniqueness of the solution. Further, we apply a regularizing Levenberg-Marquardt method, which is stable against the mentioned discrepancies. Then, we compare it to the existing physical methods - the Iterative Boltzmann Inversion and the Inverse Monte Carlo method, which are fast and well adapted to the problem, but sometimes have convergence problems.rnrnFrom analysis of the Iterative Boltzmann Inversion, we elaborate a meaningful approximation of the structure and use it to derive a modification of the Levenberg-Marquardt method. We engage the latter for reconstruction of the interaction parameters from experimental data for liquid argon and nitrogen. We show that the modified method is stable, convergent and fast. Further, the singular value analysis of the structure and its approximation allows to determine the crucial interaction parameters, that is, to simplify the modeling of interactions. Therefore, our results build a rigorous bridge between the inverse problem from physics and the powerful solution tools from mathematics. rn