Logarithmic decay laws in glassy systems

The equations for the beta relaxation dynamics as obtained within mode-coupling theory for the glass transition are solved asymptotically for parameters near Whitney cusp singularities. The solution is given by a two-parameter scaling law, where the time t enters as ln t and where the scaling times depend exponentially with a Vogel-Fulcher like form on the control parameters. The master function is given in terms of Weierstrass' elliptic function. It describes crossovers from critical relaxation Phi (t) varies as 1/ln²t to a constant f₀, to a power-law decay 1/t ^a, to Phi (t) varies as -lnt, or to Phi (t) varies as ln²t depending on the sector in parameter space. The relaxation data for the Cu-Mn spin-glass alloy can be described by the theory for a time interval of eight decades.