Inhomogeneity model

We can clearly see on this figure a spread of the [Sr/Ba] ratio. This spread is larger than the expected errors, confirming the results found by previous authors (see Honda et al. 2004 [2] and reference therein). Such a large scatter found in the [Sr/Ba] ratio can be explained by inhomogeneous models of chemical evolution which predict the existence of such a large variation (see for example Ishimaru et al. 2004). 

In other cases, however, and in particular when sublattices are occupied by rather immobile components, the point defect concentrations may not be in local equilibrium during transport and reaction. For example, in ternary oxide solutions, component transport (at high temperatures) occurs almost exclusively in the cation sublattices. It is mediated by the predominant point defects, which are cation vacancies. The nearly perfect oxygen sublattice, by contrast, serves as a rigid matrix. These oxides can thus be regarded as models for closed or partially closed systems. These characteristic features make an AO-BO (or rather A, O-B, a 0) interdiffusion experiment a critical test for possible deviations from local point defect equilibrium. We therefore develop the concept and quantitative analysis using this inhomogeneous model solid solution. 

A wide range of mass transfer correlations based on Reynold and Graetz numbers have been used to characterize the performance of hollow fiber module contactors. The variation of mass transfer correlations has been attributed to the nonideality in flow distribution, deviation from simple axial flow, and fiber inhomogeneity.Modeling for concentration polarization build-up and two phase flow on the lumen side has also been developed for hollow fiber modules used in filtration.Flow distribution in modules has also been characterized using residence time distribution... 

Furthermore, j = c v was placed in the capillary and Fick s first law applied to the areas bordering the tissue, the magnitude of which d X ds v is the velocity of blood in the different capillaries, which is calculated in the inhomogeneous model according to KirchofFs laws. In the con- and countercurrent systems, the capillaries are perfused at the same velocity. The four differential quotients (dc/dn) at the areas bordering the tissue are taken in their normal directions. 

In this section we will make some algebraic manipulations on the partition function of periodic models that lead to a formula for the free energy. This is the generalization of Proposition 1.1 and of formula (1.6) to weakly inhomogeneous models. Very much like in that case, the computation of the... 

As we have already stressed, the analogy with (2.18) is evident and it is probably not surprising for the reader that from such a formula one can extract the sharp behavior of the partition function. It should however be noted that, while in the positive recurrent set-up ( > 1) the theory of Markov renewals is well developed, fewer results are available in the literature on the mass renewal function of null recurrent Markov renewals. Moreover the results, even only at the level of sharp asymptotic behavior of the partition function, are more involved. As we shall see, this complexity is not only of a technical nature, but it really reflects a substantially larger variety of phenomena that can be observed in weakly inhomogeneous models, with respect to homogeneous ones. 

The result we want to prove is that a quenched sequence of charges, even if they are centered and so on the average the charge is zero, plays successfully in favor of localization (the analogous result for weakly inhomogeneous models is proven in Proposition 3.8). This is of course due to the fact that the typical polymer trajectories target positive charges. We introduce the... 

The difficulty in proving the superior limit statement is the same as the one we have encountered in proving the convergence of the critical curve of weakly inhomogeneous models to the one of disordered models, see Figure 4.1. With the (considerable) difference that in this case one can actually show that limsup ) o c(A)/A < ihc-... 

A general consideration on this chapter is that the results presented are sensibly weaker than those in Chapter 2 and Chapter 3 for homogeneous and weakly inhomogeneous models. It is then natural to ask what should we expect to observe in the delocalized regime of disordered models This question is particularly relevant also because weakly inhomogeneous models have been studied as caricatures of disordered models. 

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