Kinetic Ising model

This last equation is valid as long as the diffusion front of the diffusing species in solution phase remains within the electrode coating, a condition that applies for times shorter than 10-20 msec (Miller and Majda, 1986,1988). Dynamics of electron hopping processes have been recently modeled by Denny and Sangaranarayan (1998) using kinetic Ising model formalism. 

Denny, R.A., and Sangaranarayan, M.V. 1998. Dynamics of electron hopping in redox polymer electrodes using kinetic Ising model. Jourrtal of Solid State Electrochemistry 2, 67-72. 

Thus oj(q = 0) vanishes as co q = 0) oc Xt % Yl" — % <2 n>> and eq. (201) hence implies the classical value Zd = 2 — ij. Although eq. (206) thus suggests a relationship between the dynamic exponent and static ones, this is not true if effects due to non-mean-field critical fluctuations are taken into account. In fact, for the kinetic Ising model (Kawasaki, 1972) extensive numerical calculations imply that z. 2.18 in d = 2 dimensions (Dammann and Reger, 1993 Stauffer, 1992 Landau et al., 1988) rather than Zc = 2 - r) = 1.75. Note also [this is already evident from eq. (206)] that not all fluctuations slow down as Tc is approached but only those associated with long wavelength order parameter variations. One can express this fact in terms of a dynamic scaling principle... 

It has been proposed recently that the faster than bulk water relaxation observed is due to frustration induced by the propagation of opposite correlations from the interior of the micellar surface towards the center of the water pool. This can be easily understood by employing a variant of the kinetic Ising model that was introduced recently in order to model this effect of nano-confinement on the orientational dynamics of water inside the reverse micelles. The model assumed that the two spins at the two ends of the onedimensional chain remained fixed in opposite directions. This mimics the orientation of water molecules fixed at diametrically opposite positions in the interior of reverse micelles. This can be made clear by Figure 17.9(a) [13]. 

R. Biswas and B. Bagchi, A kinetic Ising model study of dynamical correlations in confined fluids emergence of bofli fast and slow time scales. J. Chem. Phys., 133 (2010), 084509-1-7. 

We have discussed only the equilibrium properties of polymers. Of course, in many real systems, the time scales for equilibriation can be very large. It is thus of interest to study non-equilibrium properties of statistical mechanical systems on fractals. A simple prototype is the study of kinetic Ising model on fractals. Closer to our interests here, one can study, say, the reptation motion of a polymer on the fractal substrate. This seems to be a rather good first model of motion of a polymer in gels. 

Bandstra, J.Z., Brantley, S.L. (2008). Surface evolution of dissolving minerals investigated with a kinetic Ising model. Geochimica et Cosmochimica Acta, 72, 2587-2600. 

Stoll, E., Binder. K, Schneider, T. (1973). Monte-Carlo investigation of dynamic critical phenomena in the two-dimensional kinetic Ising model. Physical Reoiew B, 8 (7), pp. 3266. 

Here we will first compare the implication of the DRIS formalism with the predictions of the kinetic Ising model of Glauber [39] and discuss the results in relation to the theory of Shore and Zwanzig [40]. It should be noted that the model of Glauber is of fundamental importance, inasmuch as this is the first work in which the Markov character of a chain is rigorously considered and an analytical expression is provided for the time decay of correlation functions associated with a linear array of pairwise interdependent units. In the second part of this section, some empirical expressions previously proposed for describing OACFs, will be considered. 

Comparison with the Kinetic Ising Model of Glauber... 

The elements of the transition rate matrix A in the DRIS model are estimated from the multidimensional energy surface associated with the interdependent rotation of neighboring bonds u g Kramers rate theory, as described above. Accordingly, the probability of occurrence of a given isomeric state for a bond depends on the state of its first neighbors aloi the chain. Likewise, in the kinetic Ising model the transition rate Wi(Oi) of the i-th spin is assumed to be coupled to the state of its first iKighbors by the relation... 

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