Stochastic model for the glass transition of simple classical liquids

Abstract

Dynamics of atoms near the glass transition of simple classical liquids is studied on the basis of the mesoscopic stochastic-trapping diffusion model recently developed by Odagaki [J. Phys. A 20, 6455 (1987); Phys. Rev. B. 38, 9044 (1988)]. The jump rate of an atom (tracer) is assumed to have a distribution following a power-law function with exponent ρ, where ρ is a phenomenological parameter. A sharp transition is predicted at ρ=0, that is, the self-diffusion vanishes when ρ and takes a nonzero finite value when ρ>0. This transition is identified as the glass transition. With use of the coherent-medium approximation, the mean-square displacement is shown to exhibit a power-law dependence on time with exponent less than unity, and hence the incoherent scattering function for small wave vectors shows stretched exponential decay when ρ. The non-Gaussian parameter at time t=∞ is shown to be nonzero in the glassy state (ρ) and vanishes in the fluid state (ρ>0), indicating that this quantity may be used as an order parameter of the glass transition. The mean-square displacement and the non-Gaussian parameter are obtained in the intermediate time scale as well from the frequency-dependent diffusion constant. The apparent diffusion constant determined by the derivative of the mean-square displacement at an imtermediate time shows a smooth transition instead of the sharp one, which coincides with observations in molecular dynamics studies. The incoherent scattering function in the intermediate time scale agrees qualitatively with experiments and the exponent of its stretched exponential decay deviates from unity before the glass transition takes place, in agreement with observations made via computer experiments.