Ising model defined

A binary alloy of two components A and B with nearest-neighbour interactions respectively, is also isomorphic with the Ising model. This is easily seen on associating spin up with atom A and spin down with atom B. There are no vacant sites, and the occupation numbers of the site are defined by... 

We define a fee lattice and affect at each site n, a spin or an occupation variable <7 which takes the value +1 or —1 depending on whether site n is occupied by a A or B atom. Within the generalized perturbation method , it has been shown that substitutional binary alloys AcBi-c may be described within a Ising model with effective pair interactions with concentration dependence. Thus, the energy of a configuration c = (<7i,<72,- ) among the 2 accessible configurations for one system can be written... 

This condition is similar to that used by Onsager in his own unpublished derivation of the result, Eq. (34), for the spontaneous magnetization of the Ising model. It is to be noted, however, that Devinatz retains conditions (d), (e), and (/). In our theorems, on the contrary, we retain condition (c) but replace (/) by another (slightly stronger) condition on the auxiliary coefficients p and qn defined in Eqs. (29) and (30). To state this we define the monotonically decreasing coefficients... 

In order to describe the collective-update schemes that are the focus of this chapter, it is necessary to introduce the Ising model. This model is defined on a d-dimensional lattice of linear size L (a square lattice in d = 2 and a cubic lattice in d = 3) with, on each vertex of the lattice, a one-component spin of fixed magnitude that can point up or down. This system is described by the Hamiltonian,... 

The results for the film heat capacity and the compressibility have been also found to be consistent with the values of the critical exponents 7 and a corresponding to the universality class of the two-dimensional Ising model (cf. eqn. (8)). It is noteworthy that the results presented in Fig. 2 are in a good qualitative agreement with experimentally observed dependences between the critical point temperatures for monolayer films and the dimensional incompatibility between adsorbent and adsorbate [17], defined by... 

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 ... 

Niedermayer pointed out that it is not necessary to constrain the links with which we make clusters to be only between spins which are pointing in the same direction. In general, we can define two different probabilities for putting links between sites—one for parallel spins and one for antiparallel ones. The way Niedermayer expressed it, he considered the energy contribution Ey that a pair of spins i and j makes to the Hamiltonian. In the case of the Ising model, for example... 

Assume an Ising model on a square lattice oiN = L X L spins. At site k on the lattice the possible spin orientations are = 1. Neighbor spins m and I interact with ferromagnetic energy > 0) and the reciprocal temperature is defined by K = With the SM method one starts from an empty... 

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 characteristic ratio Coo characterizes chain flexibility. It depends on the 6 and torsional potential and is determined by the chemical structure of the monomers [20]. The rotational isomeric state (RIS) model, introduced by P.J. Flory [20] is essentially an adaptation of the one-dimensional Ising model of statistical physics to chain conformations. This model restricts each torsion angle to a discrete set of states (e.g., trans, gauche , gauche"), usually defined around the minima of the torsional potential of a single bond, V((f>) (see Fig. 2d). This discretization, coupled with the locality of interactions, permits calculations of the conformational partition... 

Thus closing the circle allows us to write the PF of the system as a trace of the matrix P. This leads to an elegant solution of the one-dimensional Ising model. The trace of a matrix A is defined as the sum of all its diagonal elements. 

From now on we can proceed as in the simple two-state Ising model. We define the general matrix elements... 

The transverse-field Ising model is defined on a d-dimensional hypercubic (i.e., square, cubic, etc.) lattice. Each site is occupied by a quantum spin-. The spins interact via a ferromagnetic nearest-neighbor exchange interaction / > 0 between their z components. The transverse magnetic field couples to the x components of the spins. The Hamiltonian of the model is given by... 

In ID we can solve the nearest neighbor spin-1/2 Ising model analytically by the transfer matrix method the hamiltonian of the system is defined in Eq. (D.87). Consider a chain of N spins with periodic boundary conditions, that is, the right neighbor of the last spin is the first spin, and take for simplicity the spins to have values Si = 1. 

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). ... 

In the different cases, the response model of the electronic tongue was built employing ANNs. For this purpose, a large number of samples (mixtures of the ions considered) were generated by cumulative microadditions of standards over a defined initial volume. Per example, in the three ions case applied to river waters, series of additions employing NH4, K+, Na+ or combinations of two or three ions were alternated, generating 174 different mixtures of the three considered ions, and for each, the responses of the ISE array were recorded. The net laboratory effort implied corresponded to two work days. This initial information... 

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