An Analytical Approach to the Fredrickson–andersen Model With Vacancies

Abstract

The dynamics of the n-spin facilitated kinetic Ising model (Fredrickson-Andersen model) with mobile vacancies as a model for the glassy materials are studied analytically by means of the Fock-space representation of the master equation. The system is mapped onto a three state model characterizing mobile, immobile and vacant cells. The characteristic cooperativity for glass forming systems are introduced by restrictions influencing the local dynamics and subsequently the local mobility of different lattice cells. In a moderate temperature regime the relaxation time versus the inverse temperature T −1 reveals two processes. Whereas the slow process can be identified with the conventional α-process of the supercooled liquid, the fast one originated by the additional empty sites is suggested to be the β JG -process due to Johari-Goldstein. The results are accordant with numerical simulations and suggest that the modified n-spin facilitated kinetic Ising model is able to describe qualitatively the behaviour of a supercooled liquid near the glass transition temperature Tg. Int. J. Mod. Phys. B 1999.13:1379-1396. Downloaded from www.worldscientific.com by MONASH UNIVERSITY on 02/04/15. For personal use only.

Diffusion with restrictions

“…In this manner the inherent cooperativity manifested in a cooperative rearrangement of certain regions in order to change mobile states within the subsequent time steps into immobile ones and vice versa. The inclusion of mobile vacancies in the facilitated model 15,16 yields to the occurence of nearly nonactivated relaxation additional to the main α-process.…”

Kinetic Ising model in two and three dimensions with constraints

“…Also, extensions to the model were very often considered [19,26,30,32,33]. However, in most of those studies the original ideas of Fredrickson and Andersen [14] have been simplified to a special case, where the interactions between spins ij J are neglected.…”

An extended network model with a packet diffusion process

“…An inherent cooperativity is included assuming that the local dynamics and subsequently the local mobilities are restricted by the occupation of neighboring cells. Depending on a temperature-like parameter h −1 (interconnectivity) the diffusive process of the packages (information) can be almost stopped, thus we get a well separation of the time regimes and a quasi-localization for the intermediate range at low temperatures.interest in this context concern the crystal growth, transport (traffic) models, diffusion processes and supercooled liquids [1][2][3][4][5][6].Here, we will apply the kinetics of the Fredrickson-Andersen model (FAM) recently discussed in the framework of the glass transition and related phenomena [5][6][7][8][9][10][11][12]. But we show that this model may be used on other fields like stock trading, citation networks, company relations or internet communications as well [13][14][15][16].…”

“…interest in this context concern the crystal growth, transport (traffic) models, diffusion processes and supercooled liquids [1][2][3][4][5][6].…”

“…Here, we will apply the kinetics of the Fredrickson-Andersen model (FAM) recently discussed in the framework of the glass transition and related phenomena [5][6][7][8][9][10][11][12]. But we show that this model may be used on other fields like stock trading, citation networks, company relations or internet communications as well [13][14][15][16].…”