Logarithmic relaxation in glass-forming systems

Within the mode-coupling theory for ideal glass transitions, an analysis of the correlation functions of glass-forming systems for states near higher-order glass-transition singularities is presented. It is shown that the solutions of the equations of motion can be asymptotically expanded in polynomials of the logarithm of time t. In leading order, a $\ln \mathrm{}\left(t\right)\mathrm{}$ law is obtained, and the leading corrections are given by a fourth-order polynomial. The correlators interpolate between three scenarios. First, there are surfaces in parameter space where the dominant corrections to the $\ln \mathrm{}\left(t\right)\mathrm{}$ law vanish, so that the logarithmic decay governs the structural relaxation process. Second, the dynamics due to the higher-order singularity can describe the initial and intermediate part of the $\alpha$ process thereby reducing the range of validity of von Schweidler’s law and leading to strong $\alpha$ relaxation stretching. Third, the $\ln \mathrm{}\left(t\right)\mathrm{}$ law can replace the critical decay law of the $\beta$ process, leading to a particularly large crossover interval between the end of the transient and the beginning of the $\alpha$ process. This may lead to susceptibility spectra below the band of microscopic excitations exhibiting two peaks. Typical results of the theory are demonstrated for models dealing with one and two correlation functions.