Simple one-particle diffusional model to mimic some properties of the glass transition

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Abstract
A new simplified model is proposed to mimic some properties of the glass transition. The physical system undergoing glass transition is modeled as Brownian particle diffusing in one- or two-dimensional space with obstacles. In one dimension obstacles are points which cannot be crossed by Brownian particles, in two dimensions obstacles are randomly distributed sections of straight lines which are impenetrable for the diffusing particle. The obstacles have a finite lifetime τ. After time τ the obstacle disappears and reappears in some new random position. In another modification of the model the obstacle barrier can be opened for short time and then closed again. Both cases are studied for one-dimensional diffusion, while in two dimensions only the first modification of the model is considered. The main feature of the model is that the mean lifetime of obstacles τ is connected with the diffusion coefficient of the Brownian particle through the coupling equation Dτ=K, with K being the coupling constant. This idea is borrowed from the theory of reptations in polymer liquids. Both analytical calculations and results of computer simulations are presented. The model is shown to reflect some of the features of glass transition. It was found that a slight change in the model, i.e., opening and closing of a barrier in the same position versus transfer of a barrier to a new position, leads to a drastic change in the diffusion kinetics of the system.

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