Ising problem

The calculation of the partition function for the ensemble of chain molecules from Eq. (2.9) is closely related to solving the two-dimensional Ising problem with an external field. Since thae are no exact solutions for this problem yet, further approximating a umptions must be made to allow simple estimates. 

The kinetics of cooperative processes in macromolecular structures, synthetic or biological, was developed further with his student R. H. Lacombe [Simha and Lacombe, 1971]. The authors also examined cooperative equilibria in copolymer systems of specified sequence structures. This implied solutions of the classical Ising problem for linear lattices. It had already been treated by the methods of statistical mechanics for homogeneous chains and, most recently, for copolymers. Lacombe and Simha showed how these problems could be dealt with advantageously by the method of detailed balancing of opposing rates [Lacombe and Simha, 1973,1974]. The results were examined for a spectrum of linear structures, chain lengths, and sequential distributions, such as he had computed, for example, with Jack Zimmermann for polypeptides [Zimmerman et al., 1968]. 

Despite the high selectivity of sodium glass ISEs, problems arise in clinical use, and the contaminated membrane surfaces need special treatment at regular intervals (7,51). Polymeric membrane ISEs with ionophores (20) and (21) are less affected and permit sodium measurements in undiluted serum and urine (51). 

The term equilibrium copolymerization was introduced by Alfrey and Tobolsky in 1959, who stated that, mathematically, the equilibrium in copolymerization is identical with the Ising problem in ferromagnetism, which leads to the same solution. Consequently, the authors formulated the most important... 

The Ising model is isomorphic with the lattice gas and with the nearest-neighbour model for a binary alloy, enabling the solution for one to be transcribed into solutions for the others. The tlnee problems are thus essentially one and the same problem, which emphasizes the importance of the Ising model in developing our understanding not only of ferromagnets but other systems as well. 

The relationship between tlie lattice gas and the Ising model is also transparent in the alternative fomuilation of the problem, in temis of the number of down spins [i] and pairs of nearest-neighbour down spins [ii]. For a given degree of site occupation [i]. 

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. 

In 1925 Ising [14] suggested (but solved only for the relatively trivial ease of one dunension) a lattiee model for magnetism m solids that has proved to have applieability to a wide variety of otiier, but similar, situations. The mathematieal solutions, or rather attempts at solution, have made the Ising model one of tlie most famous problems in elassieal statistieal meehanies. 

The standard analytic treatment of the Ising model is due to Landau (1937). Here we follow the presentation by Landau and Lifschitz [H], which casts the problem in temis of the order-disorder solid, but this is substantially the same as the magnetic problem if the vectors are replaced by scalars (as the Ising model assumes). The themiodynamic... 

That analyticity was the source of the problem should have been obvious from the work of Onsager (1944) [16] who obtained an exact solution for the two-dimensional Ising model in zero field and found that the heat capacity goes to infinity at the transition, a logarithmic singularity tiiat yields a = 0, but not the a = 0 of the analytic theory, which corresponds to a finite discontinuity. (Wliile diverging at the critical point, the heat capacity is synnnetrical without an actual discontinuity, so perhaps should be called third-order.)... 

Suppose now that the sites are not independent, but that addition of a second (and subsequent) ligand next to a previously bound one (characterized by an equilibrium constant K ) is easier than the addition of the first ligand. In the case of a linear receptor B, the problem is fonnally equivalent to the one-dimensional Ising model of ferromagnetism, and neglecting end effects, one has [M] ... 

This problem must be broken up into two parts, first considering the walls with their refractory-backed tubes. To imaginary planes A of area 1.83 by 3.05 m (6 by 10 ft) and located parallel to and inside the rows of radiant tubes, the tubes emit radiation gTiAj i2, which equals gTiA2 3 2i- To find S 2i. ise Fig. 5-17, curve 5, from which F21 = 0.81. Then from Eq. (10-200)... 

In this work, the most important problems, connected with development of ISEs for double-charged anions, have been analyzed and possible ways for their overcoming have been discussed. The main difficulties in creating such ISEs are caused by ... 

The //2-optimal eontrol problem is to find a eontoller c(t) sueh that the 2-norm of the ISE (written e(t) 2) is minimized for just one speeifie input v(t). 

These apparent restrictions in size and length of simulation time of the fully quantum-mechanical methods or molecular-dynamics methods with continuous degrees of freedom in real space are the basic reason why the direct simulation of lattice models of the Ising type or of solid-on-solid type is still the most popular technique to simulate crystal growth processes. Consequently, a substantial part of this article will deal with scientific problems on those time and length scales which are simultaneously accessible by the experimental STM methods on one hand and by Monte Carlo lattice simulations on the other hand. Even these methods, however, are too microscopic to incorporate the boundary conditions from the laboratory set-up into the models in a reahstic way. Therefore one uses phenomenological models of the phase-field or sharp-interface type, and finally even finite-element methods, to treat the diffusion transport and hydrodynamic convections which control a reahstic crystal growth process from the melt on an industrial scale. 

The logarithmic response of ISEs can cause major accuracy problems. Very small uncertainties in the measured cell potential can cause large errors. (Recall that an... 

Miniaturized catheter-type ISE sensors, such as the implantable probe shown in Figure 5-20 represent the preferred approach for routine clinical in-vivo monitoring of blood electrolytes. For these intravascular measurements the reference electrode is placed outside die artery (in die external arm of die catheter), tints obviating biocompatability and drift problems associated with its direct contact with the blood. 

When using the selectivity constant or coefficient (k) mentioned by ISE suppliers, one must be sure that if the ion under test and the interfering ion have different valence the exponent in the activity term according to Nikolski has been taken into account it has become common practice to mention the interferent concentration that results in a 10% error in the apparent ion concentration these data facilitate the proper choice of an ISE for a specific analytical problem. Often maximum levels for no interference are indicated. 

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