A pr 1 99 9 Equations of structural relaxation

In the mode coupling theory of the liquid to glass transition the long time structural relaxation follows from equations solely determined by equilibrium structural parameters. The present extension of these structural relaxation equations to arbitrarily short times on the one hand allows calculations unaffected by model assumptions about the microscopic dynamics and on the other hand supplies new starting points for analytical studies. As a first application , power–law like structural relaxation at a glass–transition singularity is explicitly proven for a special schematic MCT model. Dense liquids exhibit anomalously slow, temperature sensitive and non-exponentional relaxation processes. They are conventionally termed 'structural relaxation' because cooperative rearrangements of particles are presumably involved. The mode coupling theory (MCT) suggests that the evolution of the anomalous dynamics and its splitting off from the normal liquid dynamics is caused by bifurcation singularities in non–linear retarded equations of motion for the (normalized) intermediate scattering functions Φ q (t) = S q (t)/S q (Leutheusser 1984, Bengtzelius et al. 1984). These functions are some of the most simple ones specifying structural dynamics. The bifurcations of MCT, and the long–time dynamics in their vicinity, are solely determined by the equilibrium (static) structure factor S q , see (Götze and Sjögren 1989, Franosch et al. 1998), and thus the MCT provides a frame to single out and to define structural relaxation. In this contribution the MCT equations for the structural relaxation are extended to arbitrarily short times thus rendering them well defined even in the limit that the microscopic transient dynamics can be neglected. Whereas in (Franosch et al. 1998) this was achieved by a discrete–dynamics model, here integral equations with explicit initial conditions are formulated. Thus the dependence on the initial conditions is displayed clearly. Additionally, monotone solutions are guaranteed (Fuchs et al. 1991). Furthermore, for the obtained structural dynamics equations non–exponential relaxation can be proven for a special MCT model. §2. The equations of motion of MCT The MCT equations of motion for the autocorrelation functions of density fluctuations in simple liquids, Φ q (t), are derived from Newton's equations with the Zwanzig–Mori formalism leading to: ∂ 2 t Φ q (t) + Ω 2 q Φ q (t) + t 0 dt ′ M q (t − t ′) ∂ t ′ Φ q (t ′) = 0 , (1) Φ q (t → 0) = 1 − 1 2 (Ω q t) 2 , (2) …