Spin model

An even coarser description is attempted in Ginzburg-Landau-type models. These continuum models describe the system configuration in temis of one or several, continuous order parameter fields. These fields are thought to describe the spatial variation of the composition. Similar to spin models, the amphiphilic properties are incorporated into the Flamiltonian by construction. The Flamiltonians are motivated by fiindamental synnnetry and stability criteria and offer a unified view on the general features of self-assembly. The universal, generic behaviour—tlie possible morphologies and effects of fluctuations, for instance—rather than the description of a specific material is the subject of these models. 

Chapter 7 discusses a variety of topics all of which are related to the class of probabilistic CA (PCA) i.e. CA that involve some elements of probability in their state and/or time-evolution. The chapter begins with a physicist s overview of critical phenomena. Later sections include discussions of the equivalence between PCA and spin models, the critical behavior of PCA, mean-field theory, CA simulation of conventional spin models and a stochastic version of Conway s Life rule. 

There is a fundamental relationship between d-dimensional PCA and d + 1)-dimensional Ising spin models. The simplest way to make the connection is to think of the successive temporal layers of the PCA as successive hyper-planes of the next higher-dimensional spatial lattice. Because the PCA rules (at least the set of PCA rules that we will be dealing with) are (1) Markovian (i.e. the probability of a state at time t + T depends only on a set of states at time t, and (2) local, one can always define a Hamiltonian on the higher-dimensioned spatial lattice such that the thermodynamic weight of a configuration 5j,( is equal to the probability of a corresponding space-time history Si t). ... 

We thus have that the time evolution of the one-dimensional PCA system is equivalent to the equilibrium statistical mechanics of a spin model on a triangular lattice ([domany84], [geor89]). ... 

Notice that the associated spin model has the following three properties (1) it is, in general, anisotropic (i.e. a-2 / CI3), (2) its set of coupling constants hi, hij, /1123) are interdependent (this should be obvious from equation 7.63, which provides a parameterization of each of these seven constants in terms of our original four independent conditional probabilities, aj (equation 7.58)), and (3) its partition function, Z, can be calculated exactly. 

This example shows that dipolar interactions can produce unexpected effects in systems containing polynuclear clusters, so that their complete quantitative description requires a model in which the dipolar interactions between all the paramagnetic sites of the system are explicitly taken into account. Local spin models of this kind can provide a description of the relative arrangement of the interacting centers at atomic resolution and have been worked out for systems containing [2Fe-2S] and [4Fe-4S] clusters (112, 192). In the latter case, an additional complication arises due to the delocalized character of the [Fe(III), Fe(II)] mixed-valence pair, so that the magnetic moments carried by the two sites A and B of Fig. 8B must be written... 

Recently, numerous studies reported the application of homonuclear and heteronuclear selective recoupling schemes on uniformly labelled ligand interacting with membrane receptors. The polarization exchange curves were fitted with the two-spin model and showed that it is possible to determine intemuclear distances up to 4.5 A.118... 

In Kadanoff s [130, 131] two-dimensional block-spin model four neighbouring spins are assumed to have identical spins, either up or down, near the critical point. The block of four then acts like a single effective spin. The lattice constant of the effective new lattice is double the original lattice constant. The coherence length measured in units of the new lattice constant will hence be at half of its original measure. Repetition of this procedure allows further reduction in by factors of two, until finally one has an effective theory with = 1. At each step it is convenient to define renormalized block spins such that their magnitude is 1 instead of 4. The energy of such blocked spins is... 

R.M. Patel, A.K. Doufas and R.R Raradkar, Raman spectroscopy for spinline crystallinity measurements. II. Validation of fundamental fiber-spinning models, J. Appl. Polym. Sci., 109, 3398-3412 (2008). 

The interplay between the local Pb polarization and B-site ions may be described by a simple model. As discussed above the magnitude of the local off-centering of Pb " ions is always about 0.5 A, and does not depend on the environment. Thus it could be described as a pseudo-spin, and justifies the pseudo-spin model with the dipolar-dipolar interaction and the local anisotropy energy ... 

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