Simplest Ising Models

We shall start in the next section with the simplest Ising model. This is a linear sequence of particles, or units, each of which can attain one of two possible states. Interactions between units extend to the nearest neighbor units only. 

After studying some of these simplest models—all are formally equivalent—we proceed to introduce the concept of molecular distribution functions. As in Chapter 3, explicit and exact expressions for the molecular distribution functions in terms of molecular parameters can be derived for such systems. In subsequent sections we generalize the Ising model in different directions Multistate units, triplet interactions, continuous systems, and so on. Finally, we apply these methods to some specific problems such as phase transition, the helix-coil transition, and one aspect of liquid water. 

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In this section, we consider the Ising model on a regular lattice where each interior site has the same number of nearest-neighbour sites. This is called the coordination number of the lattice and will be denoted by z. We assume that, in the thermodynamic limit, boundary sites can be disregarded and that, with N sites, the number of nearest-neighbour site pairs is Nz/2. The standard Hamiltonian for the the simplest Ising model is given by... 

We extend the simplest Ising model, where only nearest-neighbor interactions were assumed, to include next-nearest-neighbor interactions. 

The simplest model is the lattice-gas or Ising model. The whole space is divided into a lattice of N sites, and on each site two different states are possible a crystalline state denoted by the variable 5, = 1 and a gaseous state by Sj = -. The variable s denotes the degree of crystalline order. The cohesion of nearest-neighboring solid atoms leads to the following interaction energy... 

Several methods have been developed for the quantitative description of such systems. The partition function of the polymer is computed with the help of statistical thermodynamics which finally permits the computation of the degree of conversion 0. In the simplest case, it corresponds to the linear Ising model according to which only the nearest segments interact cooperatively149. The second possibility is to start from already known equilibrium relations and thus to compute the relevant degree of conversion 0. 

In this section we introduce the matrix method to rewrite the GPF of a linear system of m sites in a more convenient form. This is both an elegant and a powerful method for studying such systems. We start by presenting the so-called Ising model for the simplest system. We assume that each urrit can be in one of two occupational states empty or occupied. Also, we assirme only nearest-neighbor (nn) interactions. Both of these assumptions may be removed. In subsequent sectiorrs and in Chapter 8 we shall discuss four and eight states for each subunit. We shall not discuss the extension of the theory with respect to interactions beyond the nn. Such an extension is used, for example, in the theory of helix-coil transition. 

The Quasi-Chemical Approximation. The mean-field approximation ignores all correlation in the occupation of neighboring sites. This is incorrect when there is a strong interaction between adsorbates at such sites. The simplest way to include some correlation is to work with probabilities of occupations of two sites (XY) instead of one site (X). Approximations that do this are generally called pair approximations (not to be confused with pair interactions). There are more possibilities to reduce multi-site probabilities as in eqn. (8) to 2-site probabilities than to 1-site probabilities. This leads to different types of pair approximations. The best-known approximation that is used for Ising models is the Kirkwood approximation, which uses for example ... 

The approximations of the superposition-type like equation (2.3.54), are used in those problems of theoreticals physics when other-kind expansions (e.g., in powers of a small parameter) cannot be employed. First of all, we mean physics of phase transitions and critical phenomena [4, 13-15] where there are no small parameters at all. Neglect of the higher correlation forms a(ml in (2.3.54) introduces into solution errors which cannot be, in fact, estimated within the framework of the method used. That is, accuracy of the superposition-like approximations could be obtained by a comparison with either simplest explicitly solvable models (like the Ising model in the theory of phase transitions) or with results of direct computer simulations. Note, first of all, several distinctive features of the superposition approximations. 

For example, the standard synergetic approach [52-54] denies the possibility of any self-organization in a system with with two intermediate products if only the mono- and bimolecular reaction stages occur [49] it is known as the Hanusse, Tyson and Light theorem. We will question this conclusion, which in fact comes from the qualitative theory of non-linear differential equations where coefficients (reaction rates) are considered as constant values and show that these simplest reactions turn out to be complex enough to serve as a basic models for future studies of non-equilibrium processes, similar to the famous Ising model in statistical physics. Different kinds of auto-wave processes in the Lotka and Lotka-Volterra models which serve as the two simplest examples of chemical reactions will be analyzed in detail. We demonstrate the universal character of cooperative phenomena in the bimolecular reactions under study and show that it is reaction itself which produces all these effects. 

In the following sections, we shah demonstrate that the observed behavior of electro-optic activity with chromophore number density can be quantitatively explained in terms of intermolecular electrostatic interactions treated within a self-consistent framework. We shall consider such interactions at various levels to provide detailed insight into the role of both electronic and nuclear (molecular shape) interactions. Treatments at several levels of mathematical sophistication will be discussed and both analytical and numerical results will be presented. The theoretical approaches presented here also provide a bridge to the fast-developing area of ferro- and antiferroelectric liquid crystals [219-222]. Let us start with the simplest description of our system possible, namely, that of the Ising model [223,224]. This model is a simple two-state representation of the to-... 

Gi=2xi—. In the simplest ID Ising model, the random variables interact only with nearest-neighbor variables on either side. A given configuration of the lattice (a disordered sequence of As and Bs) is assumed to occur with a probability given by a Boltzmann partition ... 

Probably the simplest of all lattice models of microemulsions is the Widom-Wheeler model [10,11]. In addition to its simplicity, its appeal lies in the fact that it is isomorphic to the Ising model of magnetism, a model well studied and readily simulated. 

The simplest results for the functions/(O), g(0), and c(0) have been obtained from the charge-frustrated Ising model [37], where one finds... 

Kornyshevand Vdfan [272-274] suggested the simplest way of describing the transitions between 1x1 and 1x2 states, based on an anisotropic 2D Ising model, and derived a closed expression for the temperature-surface charge phase diagram. 

The Ising model is one of the most versatile models in physics and chemistry, and in the equivalent form of the LGM it has also been apphed to hquid-hquid interfaces. In this case it is based upon a three-dimensional, typicaUy simple cubic lattice. Each lattice site is occupied by one of a variety of particles. In the simplest case the system contains two kinds of solvent molecules, and the interactions are restricted to nearest neighbors. If we label the two types of solvent molecules and S2, the interaction is specified by a symmetric 2x2 matrix w,y, where each element specifies the interaction between two neighboring molecules of type 5, and Sj. Whether the system separates into two phases or forms a homogeneous... 

The above-mentioned results may be alternatively described in terms of the one dimensioned Ising model in order to extract the feature of the interactions. There are several theories that treat DNA-ligand interactions, but one of the simplest was conveniently chosen to express cooperative nature of the phenomena. The phosphate groups on DNA u e viewed as an array of binding sites along the poly(nucleotide) helices, each site being occupied or vacant. The partition function, z, for such system is written as. 

The lattice gas (Ising model), the simplest model that describes condensation of fluids, has played an important role in the theory of critical phenomena [1] providing crucial tests for most basic theoretical concepts. Recently, accurate numerical results for the crossover from asymptotic (Ising-like) critical behavior to classical (mean-field) behavior have been reported both for two-dimensional [29, 30] and three-dimensional [31] Ising lattices in zero field with a variety of interaction ranges. The Ginzburg number, as defined by Eq. (36), depends on the normalized interaction range R = as... 

The simplest theoretical prototype of the critical vapour-liquid transition is the lattice gas which is a reformulation of the 3D Ising model in terms of fluid variables. For the lattice gas the scaling fields are ... 

The first example we consider is arguably the simplest model displaying a quantum phase transition—the quantum Ising model in a transverse field. It can be viewed as a toy model for the magnetic quantum phase transition of LiHop4 discussed in the introductory section. For this system, we explain... 

The simplest statistical model, that of Ising, i, e, the nearest-neighbor lattice statistics, is used for a volume V, proportional to the number of total lattice sites cr. It is assumed that each of the total number of molecules n is in the neighborhood of one of these cr sites. 

The Ising model in its simplest form is described by the hamiltonian... 

The Hamiltonian of the CE is a spin Hamiltonian on a lattice and a generaHzation of the simple Ising model. In the simplest case, the only difference to Equation 11.9... 

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). ... 

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