A statistical-mechanical theory of self-diffusion in glass-forming liquids was presented. A non-Markov linear Langevin equation was derived from a Newton equation by employing the Tokuyama–Mori projection operator method. The memory function was explicitly written in terms of the force-force correlation functions. The equations for the mean-square displacement, the mean-fourth displacement, and the non-Gaussian parameter were then formally derived. The present theory was applied to the glass transitions in the glass-forming liquids to consider the crossover phenomena in the dynamics of a single particle from a short-time ballistic motion to a long-time self-diffusion process via a β (caging) stage. The effects of the renormalized friction coefficient on self-diffusion were thus explored with the aid of analyses of the simulation results by the mean-field theory proposed recently by the present author. It was thus shown that the relaxation time of the renormalized memory function was given by the β-relaxation time. It was also shown that for times longer than the β-relaxation time the dynamics of a single particle was identical to that considered in the suspensions.

A Statistical-Mechanical Theory of Self-Diffusion in Glass-Forming Liquids. M.Tokuyama: Physica A, 2008, 387[21], 5003-11