Ising’s model

Kvasnikov, I. A. The application of Ising s model in the statistical theory of high elasticity. Vysokomolekul. Soedin. 3, 1617 (1961). 

One possible structure would consist of irregular forms among which spheres and cylinders with alternatively hydrophobic and hydrophilic outer surfaces are formed according to Figure 8. This suggestion has an immediate attraction its features are similar to critical phenomena [cf., two-dimensional Ising s model (40)]. The resemblance between micellar associations and the fast fluctuation aggregates before phase separations... 

Let us consider the cooperated behavior of the dipoles (i.e., O H O bonds) in the domain on the film using Ising s model with regard to friction— that is, phonons of the film. In Ising s model the ordering of dipoles is characterized by the mean value of the z component of the pseudospin (Sz). 

Thus let the electric field that is applied to the film be oriented along the z axis. The polarization current appears as a result of rotation of the dipoles in the direction of the field that is, the dipoles periodically change the orientation and are now straight and in parallel direction and antiparallel direction in relation to the z axis. These positions of the dipoles in Ising s model are linked with two projections of the zth component of the pseudospin (Sz) 1/2 and —1/2, respectively. Since the water molecules are situated on the surface of the film, we should include in the consideration the interaction with the film, which in turns introduces the interaction with the film phonons. 

R. Peierls, On Ising s model of ferromagnetism. Proc. Cambridge Philos. Soc. 32,477 (1936)... 

Kawamoto T, Abe S. Thermal hysteresis loop of the spin-state in nanoparticles of transition metal complexes Monte Carlo simulations on an Ising-like model. Chemical Communications. 2005 No. 31, 3933-3935. DOI 10.1039/b506643c. 

The label a is a generalized index to indicate the energy of a particular excitation associated with a given configuration. We note that the present discussion emphasizes the local atomic-level relaxations and how they can be handled within the context of the Ising-like model introduced earlier. However, there are additional effects due to long-range strain fields that require further care (they are usually handled in reciprocal space) which are described in Ozoli s et al. (1998). 

Sanchez-Lacombe [1976,1978] The Sanchez and Lacombe [1976,1978] equation of state (S-L) is based on the Ising fluid model. The authors followed the Guggenheim [ 1966] approach, placing A -mers and No holes in an A -lattice. Hard-core volumes of the s-mer, as well as its flexibility, were assumed independent of Tand P. Furthermore, only the nearest neighbors of nonbonded mers contributed to the lattice energy ... 

Figure 1.2.1. Ising s lattice for modelling of different. systems. The system parameter can take one of two possible values at every site the particle spin is directed up or down in a magnetic (o), there is a particle or there is no particle in a liquid-vapour. system (A), a particle of component I or 2 is present in a binary sy.stem (c), site-to-sile walk of connected particles in the mode) of a macromolecule (d)...

The discussion of a positive feedback as the deciding point of excitation models leads us directly to a class of models in which a cooperative mechanism from statistical physics is applied to nerve excitation. This cooperative mechanism is the Ising model it is frequently used to describe phase transitions in solids and liquids and its application to nerve excitation is very suggestive. The first one who worked out an excitation theory on the basis of the Ising model was ADAM (1968 1970). Meanwhile, a number of variations of this idea has been proposed by BLUMEN-THAL, CHANGEUX, LEFEVER (1970), HILL, CHEN (1971), BASS, MOORE (1973), KARREMANN (1973), GOTOH (1975). Since the basic idea is the same in all these theories, let us restrict ourselves to a brief outline of Adam s model. 

Our discussion of solids and alloys is mainly confined to the Ising model and to systems that are isomorphic to it. This model considers a periodic lattice of N sites of any given symmetry in which a spin variable. S j = 1 is associated with each site and interactions between sites are confined only to those between nearest neighbours. The total potential energy of interaction... 

The Ising model has been solved exactly in one and two dimensions Onsager s solution of the model in two dimensions is only at zero field. Infomiation about the Ising model in tliree dunensions comes from high- and low-temperature expansions pioneered by Domb and Sykes [104] and others. We will discuss tire solution to the 1D Ising model in the presence of a magnetic field and the results of the solution to the 2D Ising model at zero field. 

Onsager s solution to the 2D Ising model in zero field (H= 0) is one of the most celebrated results in theoretical chemistry [105] it is the first example of critical exponents. Also, the solution for the Ising model can be mapped onto the lattice gas, binary alloy and a host of other systems that have Hamiltonians that are isomorphic to the Ising model Hamiltonian. 

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... 

Suppose we wanted to model an Ising model in the most straightforward possible manner using a CA approach. For argument s sake, let s consider modeling a basic Ising model without any external fields. Using binary site variables <7 = (1 — 5) to make any possible connection to conventional CA transparent, our task is to come up with a CA rule ,... 

In both of these cases, represents the energy of two noninteracting Ising spin systems, one system having spins s over one sublattice and the other spins s over the other sublattice. Indeed, the Q2R CA actually provides an efficient microcanonical algorithm for performing parallel simulation of the Ising model (see discussion in section 7.1.5). 

---