Ising spin dynamics

S. Alder, S. Trebst, A. K. Hartmaim, and M. Troyer (2004) Dynamics of the Wang-Landau algorithm and Complexity of rare events for the three-dimensional bimodal Ising spin glass. J. Stat. Mech. P07008... 

It should be realized that unlike the study of equilibrium thermodynamics for which a model is often mapped onto Ising system, elementary mechanism of atomic motion plays a deterministic role in the kinetic study. In an actual alloy system, diffusion of an atomic species is mainly driven by vacancy mechanism. The incorporation of the vacancy mechanism into PPM formalism, however, is not readily achieved, since the abundant freedom of microscopic path of atomic movement demands intractable number of variational parameters. The present study is, therefore, limited to a simple spin kinetics, known as Glauber dynamics [14] for which flipping events at fixed lattice points drive the phase transition. Hence, the present study for a spin system is regarded as a precursor to an alloy kinetics. The limitation of the model is critically examined and pointed out in the subsequent sections. 

Here, we adopted a spin analogy/lattice gas model, or SRS model, as shown in Fig. 1.28(a), which represents an oversimplified molecular structure yet still captures the essence of the molecule-surface interactions for describing SME profiles. Similar techniques using the Ising model to study other physical systems have been investigated [148,149,160] however, none of the literature deals with the simulation of PFPE lubricant dynamics described here. 

The impact of this is tremendous. No long-range order (LRO) can exist at finite temperature in one dimension no crystals, no magnets, no superconductors. Only special transitions are possible in two dimensions. The Ising model (n = 1 component) is an example [7]. The Kosterlitz-Thouless transition [8], without LRO, is another case for d = 2 and n = 2, discussed in Section V.C. The thermal fluctuations are very destructive in lower dimensions. Quantum fluctuations (i.e., those associated with the dynamics of a system) also tend to suppress LRO and can sometimes destroy it even at 0 K when the Mermin-Wagner theorem does not apply. Such is the case of the quantum spin- antiferromagnetic models [9] in one dimension. 

Since the action for each spin path is complex valued, a stochastic Monte Carlo evaluation of the resulting isomorphic Ising chain has to deal with the inevitable dynamical sign problem. Relying on the strategy outlined in Section II, we wish to block configurations together in an optimal way. 

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]. 

The one-dimensional Ising model also revealed the emergence of multiple time-scales in the orientational dynamics as the chain length increases. For small to intermediate-sized chains, the orientational dynamics of spins at the center acquired a decay component which is faster than in the bulk (see Figure 17.9(b)). This rapid decay component is a result of the cancellation of the polarization caging... 

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