The Simple Ising Model

Under electrochemical conditions and T, P = constant, adsorption isotherms can be derived using standard statistical considerations to calculate the Gibbs energy of the adsorbate in the interphase and the equilibrium condition for the electrochemical potentials of the adsorbed species i in the electrolyte and in the adsorbed state (eq. (8.15) in Section 8.2). A model for the statistical considerations consists of a 2D lattice of arbitrary geometry with Ns adsorption sites per unit area. In the case of a 1/1 adsorption, each adsorbed particle can occupy only one adsorption site so that the maximal number of adsorbed particles per unit area in the compact monolayer is determined by A ax = Ng. Then, this model corresponds to the simple Ising model. The number of adsorbed particles, A ads< and the number of unoccupied adsorption sites, No, per unit area are given by... 

This is a variation of the simple Ising model. Again we have M qnits which we refer to as the lattice points. Each lattice point can be either empty or occupied by a particle. The canonical PF of a system of M lattice points with N particles is... 

Consider a linear system of M sites, Na of which are occupied by molecules of type A and Nb sites are occupied by molecules of type B. If we let N = Na Nb, then we have a system the PF of which is isomorphous to the simple Ising model studied in section 4.1 i.e., each site can be in either one of two states, A or B, depending on the type of molecule occupying the site. In this section we treat a slightly more general case. Each site can be in one of three states empty, occupied by A, and occupied by B. Thus we have M sites, Na sites occupied by A and Nb occupied by B. But Na + Nb< M. 

Note that (4.7.15) cannot be factored into a product of integrals. This could have been done if G = 0 in (4.7.10). Nor can we write it as the trace of a 2 x 2 matrix as we have done in the simple Ising model. The situation is more complicated because of the hydrogen-bonding possibility included in t/(0, /). 

We consider the simple Ising model of M units, each of which can be in either of two states. The configurational PF of such a system is... 

This is not as exotic as it sounds. Even the simple Ising model, used for the qualitative description of magnetic phase transitions, exhibits this feature. 

A well-known precursor of the CE is the simple Ising model. CEs share many of the basic features of the simple Ising model regarding their setup and their statistical properties. Therefore, a brief look into this model will eventually help in understanding the CE later on. 

The most common application of the simple Ising model, however, is not the minimization of the energy, but instead of the free energy for T > 0 K,... 

This minimization also takes the configurational entropy S(thermodynamic state for temperature r, the MC simulations are used. They neither explicitly calculate S(statistical approach of the MC simulations. Even for the simple Ising model with J < 0, a phase transition is observed from the paramagnetic phase (o ) = 0 at T > T,. to the ferromagnetic phase a ) 0 at T < ff, where a ) denotes an average over all spins in the... 

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... 

CEs of Configurational Functions In 1984, Sanchez et al. [81] published the seminal paper Generalized cluster description of multicomponent systems, which became the cornerstone of the CE method. The CE provides a means to find a term-by-term expansion of a configurational function like E(many-body interactions between more than two sites of a lattice - and allows for more than two possible occupations on a site. While the latter is needed for the research of multicomponent systems, which are so significant for today s materials science, the inclusion of many-body interactions that go beyond the site-to-site interaction of the Ising model is of uttermost importance to model any many-body system in physics. 

In many physically important cases of localized adsorption, each adatom of the compact monolayer covers effectively n > 1 adsorption sites [3.87-3.89, 3.98, 3.122, 3.191, 3.214, 3.261]. Such a multisite or 1/n adsorption can be caused by a crystallographic Me-S misfit, i.e., the adatom diameter exceeds the distance between two neighboring adsorption sites, and/or by a partial charge of adatoms (A < 1 in eq. (3.2)), i.e., a partly ionic character of the Meads-S bond. The theoretical treatment of a /n adsorption differs from the description of the 1/1 adsorption by a simple Ising model. It implies the so-called hard-core lattice gas models with different approximations [3.214, 3.262-3.266]. Generally, these theoretical approaches can only be applied far away from the critical conditions for a first order phase transition. In addition, Monte Carlo simulations are a reliable tool for obtaining valuable information on both the shape of isotherms and the critical conditions of a 1/n adsorption [3.214, 3.265-3.267]. 

In fact, in studies of the Wolff algorithm for the 2D Ising model, one does not usually bother to make use of Eq. (2.5) to calculate r. If we measure time in Monte Carlo steps (i.e., simple cluster flips), we can define the corresponding dynamic exponent zsteps in terms of the correlation time rsteps of Eq. (2.5) thus ... 

The critical behavior of density fluctuations in microemulsions with a droplet structure can be treated analogously to simple fluids, because the radius is virtually constant throughout the phase separation and the droplet density may be regarded as an order parameter. Because of the nature of the droplet systems, its critical behavior is expected to belong to the 3D-Ising universality class. However, the observed critical exponents do not always coincide with the exact values of the 3D-Ising model. In particular, the well-known ternary system (WDA), consisting of an oil-rich mixture of water, n-decane, and AOT (dioctyl sulfosucdnate sodium salt) has been the subject of... 

The calculations that have been carried out [56] indicate that the approximations discussed above lead to very good thermodynamic functions overall and a remarkably accurate critical point and coexistence curve. The critical density and temperature predicted by the theory agree with the simulation results to about 0.6%. Of course, dealing with the Yukawa potential allows certain analytical simplifications in implementing this approach. However, a similar approach can be applied to other similar potentials that consist of a hard core with an attractive tail. It should also be pointed out that the idea of using the requirement of self-consistency to yield a closed theory is pertinent not only to the realm of simple fluids, but also has proved to be a powerful tool in the study of a system of spins with continuous symmetry [57,58] and of a site-diluted or random-field Ising model [59,60]. 

As a contribution to the study of these problems, stochastic models are here developed for two cases a freely-jointed chain in any number of dimensions, and a one-dimensional chain with nearest-neighbor correlations. Our work has been directly inspired by two different sources the Monte Carlo studies by Verdier23,24 of the dynamics of chains confined to simple cubic lattices, and the analytical treatment by Glauber25 of the dynamics of linear Ising models. No attempt is made in this work to introduce the effects of excluded volume or hydrodynamic interactions. 

The adsorption of diatomic or dimeric molecules on a suitable cold crystalline surface can be quite realistically considered in terms of the dimer model in which dimers are represented by rigid rods which occupy the bonds (and associated terminal sites) of a plane lattice to the exclusion of other dimers. The partition function of a planar lattice of AT sites filled with jV dimers can be calculated exactly.7 Now if a single dimer is removed from the lattice, one is left with two monomers or holes which may separate. The equilibrium correlation between the two monomers, however, is appreciable. As in the case of Ising models, the correlation functions for particular directions of monomer-monomer separation can be expressed exactly in terms of a Toeplitz determinant.8 Although the structure of the basic generating functions is more complex than Eq. (12), the corresponding determinant for one direction has been reduced to an equally simple form.9 One discovers that the correlations decay asymptotically only as 1 /r1/2. 

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-... 

Simple models such as the Ising model have the advantage of permitting analytical results to be obtained for limiting cases. Let us consider these. 

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