Abstract
The one-dimensional spin facilitated kinetic Ising model is studied analytically using the master equation and by simulations. The local state of the spins (corresponding to mobile and immobile cells) can change depending on the state of the neighbored spins, which reflects the high cooperativity inherent in glassy materials. The short-time behavior is analyzed using a Fock space representation for the master equation. The hierarchy of evolution equations for the averaged spin state and the time dependence of the spin autocorrelation function are calculated with different methods (mean-field theory, expansion in powers of the time, partial summation) and compared with numerical simulations. The long-time behavior can be obtained by mapping the one-dimensional spin facilitated kinetic Ising model onto a one-dimensional diffusion model containing birth and death processes. The resulting master equation is solved by van Kampen's size expansion, which leads to a Langevin equation with Gaussian noise. The predicted autocorrelation function and the global memory offer in the long-time limit a screened algebraic decay and a stretched exponential decay, respectively, consistent with numerical simulations.
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REFERENCES
W. Götze in Liquids, Freezing and the Glass Transition, Hansen et al., eds. (North Holland, Amsterdam, 1991).
W. Götze and L. Sjögren, Rep. Prog. Phys. 55:241 (1992).
J. Jäckle, Rep. Prog. Phys. 49:171 (1986).
E. Leutheusser, Phys. Rev. A 29:2765 (1984).
G. Adams and J. H. Gibbs, J. Chem. Phys. 43:139 (1965).
M. L. Williams, R. F. Landel, and J. D. Ferry, J. Am. Chem. Soc. 77:3701 (1955).
W. Götze and L. Sjögren, Zeitschrift für Physik B Cond. Matter 65:415 (1987).
W. Götze and L. Sjögren, J. Phys. C 21:3407 (1988).
T. Franosch, W. Götze, M. R. Mayr, and A. P. Singh, Phys. Rev. E 55:3183 (1997).
T. Franosch, M. Fuchs, W. Götze, M. R. Mayr, and A. P. Singh, Phys. Rev. E 55:7153 (1997).
G. H. Fredrickson and H. C. Andersen, J. Chem. Phys. 84:5822 (1985).
G. H. Fredrickson and H. C. Andersen, Phys. Rev. Lett. 53:1244 (1984).
G. H. Fredrickson, Ann. Rev. Phys. Chem. 39:149 (1988).
G. H. Fredrickson and S. A. Brawer, J. Chem. Phys. 84:3351 (1986).
M. Schulz and P. Reinecker, Phys. Rev. B 48:9369 (1993).
M. Schulz and P. Reinecker, Phys. Rev. B 52:4131 (1995).
M. Schulz, P. R. S. Sharma, and H. L. Frisch, Phys. Rev. B 52:7195 (1995).
S. Butler and P. Harrowell, J. Chem. Phys. 95:4454 (1991).
M. Schulz and S. Trimper, Int. J. Mod. Phys. B 11:2927 (1997).
J. Doi, Phys. A: Math. Gen. 9:1465 (1976).
S. Sandow and S. Trimper, Europhys. Lett. 21:799 (1993).
P. Grassberger and M. Scheunert, Fortschr. Physik 28:547 (1980).
L. Peliti, J. Physique 46:1469 (1985).
G. Schülz and S. Sandow, Phys. Rev. E 49:2726 (1994).
L. H. Gwa and H. Spohn, Phys. Rev. Lett. 68:725 (1992).
F. C. Alcarez, M. Droz, M. Henkel, and V. Rittenberg, Ann. Phys. (N.Y.) 230:250 (1994).
M. Schulz and S. Trimper, Phys. Lett. A 216:235 (1996).
N. G. van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam, 1986).
C. W. Gardiner, Handbook of Stochastic methods (Springer-Verlag, Berlin, 1983).
R. Richert, Chem. Phys. Lett. 216:223 (1993).
R. Böhmer, G. Hinze, G. Diezemann, B. Geil, and H. Sillescu, Europhys. Lett. 36:55 (1996).
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Schulz, M., Trimper, S. Analytical and Numerical Studies of the One-Dimensional Spin Facilitated Kinetic Ising Model. Journal of Statistical Physics 94, 173–201 (1999). https://doi.org/10.1023/A:1004563329723
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DOI: https://doi.org/10.1023/A:1004563329723