Ising model statistical mechanics

Our discussion shows that the Ising model, lattice gas and binary alloy are related and present one and the same statistical mechanical problem. The solution to one provides, by means of the transcription tables, the solution to the others. Flistorically, however, they were developed independently before the analogy between the models was recognized. 

No system is exactly unifomi even a crystal lattice will have fluctuations in density, and even the Ising model must pemiit fluctuations in the configuration of spins around a given spin. Moreover, even the classical treatment allows for fluctuations the statistical mechanics of the grand canonical ensemble yields an exact relation between the isothemial compressibility K j,and the number of molecules Ain volume V ... 

In various forms, lattice-gas models permeate statistical mechanics. Consider a lattice in which each site has two states. If we interpret the states as full or empty , we have a lattice-gas model, and an obvious model for an intercalation compound. If the states are spin up and spin down , we have an Ising model for a magnetic system if the states are Atom A and Atom B , we have a model for a binary alloy. Many different approximation techniques have been derived for such models, and many lattices and interactions have been considered. 

It is not surprising that attempts have been made to derive equations of state along purely theoretical lines. This was done by Flory, Orwoll and Vrij (1964) using a lattice model, Simha and Somcynsky (1969) (hole model) and Sanchez and Lacombe (1976) (Ising fluid lattice model). These theories have a statistical-mechanical nature they all express the state parameters in a reduced dimensionless form. The reducing parameters contain the molecular characteristics of the system, but these have to be partly adapted in order to be in agreement with the experimental data. The final equations of state are accurate, but their usefulness is limited because of their mathematical complexity. 

Statistical Mechanics of the Ising Model. A tremendously important model within statistical mechanics at large, and for materials in particular, is the Ising model. This model is built around the notion that the various sites on a lattice admit of two distinct states. The disposition of a given site is represented through a spin variable a which can take the values 1, with the two values corresponding to the two states. With these kinematic preliminaries settled, it is possible to write the energy (i.e. the Hamiltonian) as a function of the occupations of these spin variables as... 

In 1925 Ising [14] suggested (but solved only for the relatively trivial case of one dimension) a lattice model for magnetism in solids that has proved to have applicability to a wide variety of other, but similar, situations. The mathematical solutions, or rather attempts at solution, have made the Ising model one of the most famous problems in classical statistical mechanics. 

The second type of test is to run an application that uses random numbers in a similar manner to your applications, but for which the exact answer is known. For statistical mechanical applications, the two-dimensional Ising model (a simple lattice spin model) is often used since the exact answer is known and it has a phase transition so one expects sensi-... 

Many important problems in computational physics and chemistry can be reduced to the computation of dominant eigenvalues of matrices of high or infinite order. We shall focus on just a few of the numerous examples of such matrices, namely, quantum mechanical Hamiltonians, Markov matrices, and transfer matrices. Quantum Hamiltonians, unlike the other two, probably can do without introduction. Markov matrices are used both in equilibrium and nonequilibrium statistical mechanics to describe dynamical phenomena. Transfer matrices were introduced by Kramers and Wannier in 1941 to study the two-dimensional Ising model [1], and ever since, important work on lattice models in classical statistical mechanics has been done with transfer matrices, producing both exact and numerical results [2]. 

We have discussed only the equilibrium properties of polymers. Of course, in many real systems, the time scales for equilibriation can be very large. It is thus of interest to study non-equilibrium properties of statistical mechanical systems on fractals. A simple prototype is the study of kinetic Ising model on fractals. Closer to our interests here, one can study, say, the reptation motion of a polymer on the fractal substrate. This seems to be a rather good first model of motion of a polymer in gels. 

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